ganitam

 

 

 

 

 

Published by Willmann-Bell, Inc.

P.O. Box 35025, Richmond, Virginia 23235

Copyright O 1991 and 1998 by Willmann-Bell, Inc. First English Edition 1991

Second English Edition 1998

All rights reserved. Except for brief passages quoted in a review, no part of this book may be reproduced by any mechanical, photographic, or electronic process, nor may it be stored in any information retrieval system, transmitted, or otherwise copied for public or private use, without the written permission of the publisher. Requests for permission or further information should be addressed to Permis- sions Department, Willmann-Bell, Inc. P.O. Box 35025, Richmond, VA 23235.

Printed in the United States of America

Library of Congress Cataloging-in-Publication Data.

Meeus, Jean.

Astronomical algorithms / Jean Meeus. -- 2nd ed.

p. cm.

Includes bibliographical references and index. ISBN 0-943396-61-1

1. Astronomy - Data processing. 2.Astronomy - Problems, exercises, etc. 3. Algorithms. I. Title.

 

QB51.3.E43M42 1998 520’.285--dc21

98 99 00 01 02 03 04 05 06 9 8 7 6 5 4 3 2 1

 

98-55091

CIP

 

Foreword {to the first edition}

People who write their own computer programs often wonder why the machine gives inaccurate planet positions, an unreal eclipse track, or a faulty Moon phase. Sometimes they insist, bewildered, “and I used double precision, too.” Even commercial software is sometimes afflicted with gremlins, which comes as quite a shock to anyone caught up in the mystique and presumed infallibility of computers. Good techniques can help us avoid erroneous results from a flawed program or a simplistic procedure — and that’s what this book is all about.

In the field of celestial calculations, Jean Meeus has enjoyed wide acclaim and respect since long before microcomputers and pocket calculators appeared on the market. When he brought out his Astronomical Formulae for Calculators in 1979, it was practically the only book of its genre. It quickly became the “source among sources”, even for other writers in the field. Many of them have warmly acknowledged their debt (or should have), citing the unparalleled clarity of his instructions and the rigor of his methods.

And now this Belgian astronomer has outdone himself yet again! Virtually every previous handbook on celestial calculations (including his own earlier work) was forced to rely on formulae for the Sun, Moon, and planets that were developed in the last century — or at least before 1920. The past 10 years, however, have seen a stunning revolution in how the world’s major observatories produce their almanacs. The Jet Propulsion Laboratory in California and the

U.S. Naval Observatory in Washington, D.C. , have perfected powerful new machine methods for modeling the motions and interactions of bodies within the solar system. At the same time in Paris, the Bureau des Longitudes has been a

beehive of activity aimed at describing these motions analytically, in the form of explicit equations.

Yet until now the fruits of this exciting work have remained mostly out of reach of ordinary people. The details have existed mainly on reels of magnetic tape in a form comprehensible only to the largest brains, human or electronic. But Astronomical Algorithms changes all that. With his special knack for computations of all sorts, the author has made the essentials of these modern techniques available to us all.

 

We also stand at a confusing crossroads for astronomy. In just the last few years the International Astronomical Union has introduced subfle changes in the reference frame used for the coordinates of celestial objects, both within and far beyond our solar system. So sweeping are these revisions that a highly respected work for professional astronomers, the Explanatory Supplement to the Astronomical Ephemeris, published in 1961, is now seriously out of date. While the technical journals have seen a flurry of scientific papers on these issues, the book you’re holding now is the first to offer succinct and practical methods for coping with the changeover. It will be many years before astronomical data bases and catalogues are fully converted to the new system, and anyone who needs a detailed understanding of what’s going on will appreciate this book’s many comments about the FK4 and FK5 reference frames, “equinox error”- and the distinction between “J” and “B” when placed before an epoch like 2000.0.

Scarcely any formula is presented without a fully worked numerical example

— so crucial to the debugging process. The emphasis throughout is on testing, on the proper arrangement of formulae, and on not pushing them beyond the time span over which they are valid. Chapter 2 contains much wisdom of this sort, growing out of the author’s long experience with various computers and their languages. He alerts us to other pitfalls throughout the text. Anyone who tries to chart the path of a comet, for instance, soon encounters Kepler’s equation. It has so vexed astronomers over the years that literally hundreds of solutions have been proposed; the striking graphs in Chapter 30 give a good idea why.

Whenever I read about interpolation techniques, as in Chapter 3, I’m reminded poignanfly of Comet Kohoutek. News of its discovery caused a great stir in the spring of 1973, and then it let observers down with a lackluster performance. But this comet also taught me an lmportant mathematical lesson. After preparing a chart of its motion from a list of ephemeris points, I noticed that it was going to pass very near the Sun and tried several interpolation schemes in hopes of finding out what the exact time and minimum distance would be. Much to my surprise, they all failed to give an answer matching what was perfecfly obvious from my chart! Readers of this book can save themselves a similar frustration by paying close attention to the remarks on page 111.

When he’s not busy writing or conducting seminars on computing techniques, Meeus likes to seize hold of an astronomical problem with great zeal, especially if he senses it is a calculation that has never been done before. Once I asked him about the dates in the past and future when the Moon reaches its most extreme near and far distances from the Earth. Within weeks he had created a table much like that given in Table 50.C of this book. He later confided that this calculation had taken 470 hours on his HP-85 computer, consuming 12 kilowatt-hours of electricity.

On another occasion I heard about a program that was much too large for the mainframe computer he was using at the time. So he devised a scheme to avoid

 

FOREWORD iii

storing the vast number of coefficients in the computer’s limited memory; his Fortran program simply read and rewound the same magnetic tape 915 times in the course of generatîng the hour-by-hour lunar ephemeris he sought. No problem, except that the computer-room operators began to take notice, getting mildly perturbed!

Astronomical calculations have a variety of uses, some scarcely foreseen by the person making them. As long ago as 1962, for example, Meeus published an article in the British Astronomical Association Jotirnnf about a rare and remarkable forthcoming event. If any observers happened to be on Mars on 1984 May 11, he explained, they should be able to see the silhouette of Earth pass direcfly across the face of the Sun. Among his readers was the science-fiction writer Arthur C. Clarke, who later incorporated the calculations in a short störy, Transit of Earth. The piece tells of an astronaut, stranded on the red planet, who barely mariages to witness this event before his oxygen supply runs out.

Many of the topics in this book are targeted at serions observers of the sky. Thus, Chapter 53 can help in predicting the illuminaûon at a specific spot on the Moon, for any date and time. Observers often want to know the exact moments when sunlight will just glance across a particular crater, sinuous rille, or gently sloping lunar dome, because oblique lighûng is ideal for telescopic scrutiny, making subde reliefs stand out better than in most of NASA’s closeup spacecraft photographs. This chapter can also help us find when the Moon will undergo extreme librations, turning craters near the limb our way.

Chapter 44 holds a special treat for students of Jupiter. First there is a simple method for locating the four famous satellites, quite adequate for identifying them in your own telescope or on historical drawings back to the time of Galileo. Then cornes a second set of formulae of the utmost accuracy. Here the computer hobbyist can have a field day, creating observing schedules not only for ordinary satellite eclipses and transits but also for the mutual events between one satellite and another. Astronomy journals have been lax in forecasting thèse dramatic events, so that many of them have gone unobserved except by accident. For handling the Jovian moons, the routines presented in this book rival or exceed in accuracy those used by the great national almanac offices.

Other unusual topics are offered, like the method in Chapter 52 for computing the dates when the Moon’s declination becomes extreme. This is no frivolous calculation, for the very issue came up in recent findings about a century-old murder trial involving the Illinois lawyer and soon-to-be U.S. President Abraham Lincoln. Historians had long tried io reconcile conflicting testimony about the Moon and its role in allowing a witness io see the details of the murder. Some suggested that Lincoln, as lawyer for the defense, may have tampered with an almanac. Not until 1990 was this curious situation explained, and Lincoln’s integrity upheld, when Donald W. Olson and Russell Doescher noticed something quite unusual about the Moon on the night in question: 1857 August 29. As any user of this book can confirm, the Moon had a far southerly

 

declination that night, nearly the most extreme value possible in its 18.6-year cycle, and this circumstance made the time of moonset appear quite at odds with its phase. Here is a beautiful instance of astronomers stepping in, bringing their special knowledge and calculations to bear on a longstanding puzzle for historians.

We now live in a thrilling time for practitioners of the number-crunching art. The four-function pocket calculators that were so costly 20 years ago are now incorporated as a gimmick on certain wristwatches. The memory capacity of the lK RAM board in the pioneering MITS Altair microcomputer is exceeded 500-fold by a single chip in some of today’s laptop and notebook computers. Who knows what other marvels lie just ahead? By presenting these astronomical algorithms in standard mathematical notation, rather than in the form of program listings, the author has made them accessible to users of a wide variety of machines and computer languages — including those not yet invented.

Roger W. Sinnott

Sky & Telescope magazine

 

Introduction

When, in 1978, I wrote the first (Belgian) edition of my Astronomical Fomiulae for Calculators, the industry of microcomputers was just starting its worldwide expansion. Because these “personal computers” were not yet within reach of everybody, the aforesaid book was written mainly for the users of pocket calculating machines and therefore calculation methods requiring a large amount of computer memory, or many steps in a program, were avoided as far as possible, or kept to a minimum.

The present work is a greatly revised version of the former one. It is, in fact, a completely new book. The subjects have been expanded and the content has been improved. Changes were needed to take into account new resolutions of the International Astronomical Union, particularly the adoption of the new standard epoch J2000.0, while moreover I profited by the new planetary and lunar theories constructed at the Bureau des Longitudes, Paris.

As Gerard Bodifée wrote in the Preface of my previous work:

Anyone who endeavours to make astronomical calculations has to be very familiar with the essential astronomical conceptions and rules and he must have sufficient knowledge of elementary mathematical techniques. As a matter of fact he must have a perfect command of his calculating machine, knowing all possibilities it offers the competent user. However, all these necessities don’t suffice. Creating useful, successful and beautiful programs requires much practice. Experience is the mother of all science. This general truth is certainly valid for the art of programming. Only by experience and practice can one learn the innumerable tricks and dodges that are so useful and often essential in a good program.

Astronomical Algorithms intends to be a guide for the (professional or amateur) astronomer who wants to do calculations. An algorithm (from the Arabic mathematician Al-KhOrezmi) is a set of rules for getting something done ; for us it is a mathematical procedure, a sequence of reasonings and operations which provides the solution to a given problem.

This book is not a general textbook on astronomy. The reader will find no theoretical derivations. Some definitions are kept to a minimum. Nor is this a textbook on mathematics or a manual for microcomputers. The reader is assumed to be able to use his machine properly.

1

 

Except in a few rare cases, no programs are given in this book. The reasons are clear. A program is useful only for one computer language. Even if we consider BASIC only, there are so many versions of this language that a given program cannot be used as such by everybody without making the necessary changes. Every calculator thus must learn to create his own programs. There is the added circumstance that the precise contents of a program usually depend on the specific goals of the computation, that are impossible to anticipate by anybody else.

The few programs we give are in standard BASIC. They can easily be converted into FORTRAN or any other programming language.

Of course, in the formulae we still use the classical mathematical symbols and notations, not the symbolism used in program languages. For example, we write instead of SQR(A), or a(1 — e) instead of A • (1 — E), or cos z instead of CDS(X)“2 or cos(X) •    2.

The writing of a program to solve some astronomical problem will require a study of more than one chapter of this book. For instance, in order to create a program for the calculation of the altitude of the Sun for a given time on a given date at a given place, one must first convert the date and time to Julian Day (Chapter 7), then calculate the Sun’s longitude for that instant (Chapter 25), its right ascension and declination (Chapter 13), the sidereal time (Chapter 12) and finally the required altitude of the Sun (Chapter 13).

This book is restricted to the “classical", mathematical astronomy, although a few astronomy oriented mathematical techniques are dealt with, such as interpolation, fitting curves, and sorting data. But astrophysics is not considered at all. Moreover, it is clear that not all topics of mathematical astronomy could have been covered in this book. So nothing is said about orbit determination, occultations of stars by the Moon, meteor astronomy, or eclipsing binaries. For solar eclipses, the interested reader will find Besselian elements and many useful formulae in Elements of Solar Eclipses 1951 to 22(O by the undersigned (1989). Elements and formulae about transits of Mercury and Venus across the Sun’s disk are provided in my Transits (1989). These two books are published by Willmann-Bell, Inc.

The author wishes to express his gratitude to Dr. S. De Meis (Milan, Italy), to

A. Dill (Germany), and to E. Goffin and C. Steyaert (Belgium), for their valuable advice and assistance.

Jean Meeus

Note to the second edition

In this second edition several misprints and errors have been corrected. The principal change in the new edition is the addition of some material, such as expressions for the times of the stations of the planets (Chapter 36), a list of constants (Appendix I), expressions for the heliocentric coordinates of the giant planets from 1998 to 2025 (Appendix IV), and new chapters about the Jewish and Moslem Calendars, and the satellites of Saturn.

 

Contents

Pa ge

Some Symbols and Abbreviations ....... .. . . 5

1.  Hints and Tips 7

2.   About Accuracy . .. ... .. . . ... 15

3. Interpolation 23

4. Curve Fitting 35

5. Iteration 47

6. Sorting Numbers 55

7. Julian Day . .. .. . 59

8. Date of Easter 67

9. Jewish and Moslem Calendars . .. . . . .. 71

10. Dynamical Time and Universal Time 77

11. The Earth’s Globe 81

12. Sidereal Time at Greenwich 87

13. Transformation of Coordinates ..... 91

14. The Parallactic Angle, and three other Topics ... 97

15. Rising, Transit, and Setting 101

16. Atmospheric Refraction 105

17. Angular Separation 109

18. Planetary Conjunctions 117

19. Bodies in Straight Line . . ... ... . . 121

20. Smallest Circle containing three Celestial Bodies 127

21.   Precession .. . . . . . 131

22. Nutation and the Obliquity of the Ecliptic 143

23. Apparent Place of a Star 149

24. Reduction of Ecliptical Elements from one Equinox to another one . 159

25. Solar Coordinates . . .. .. . . .. 163

26. Rectangular Coordinates of the Sun . . . . .. 171

27. Equinoxes and Solstices 177

28.   Equation of Time . .. .. .. .. .. 183

29. Ephemeris for Physical Observations of the Sun 189

30. Equation of Kepler 193

3

 

31. Elements of the Planetary Orbits ...

32. Positions of the Planets

33.  Elliptic Motion ...............

34. Parabolic Motion

35. Near-parabolic Motion ...

36. The Calculation of some Planetary Phenomena

37. Pluto .

38. Planets in Perihelion and in Aphelion

39. Passages through the nodes

40. Correction for Parallax

41. Illuminated Fraction of the Disk and Magnitude of a Planet

42. Ephemeris for Physical Observations of Mars

43. Ephemeris for Physical Observations of Jupiter .

44. Positions of the Satellites of Jupiter

45. The Ring of Saturn

46. Positions of the Satellites of Saturn

47. Position of the Moon

48. Illuminated Fraction of the Moon’s Disk

49. Phases of the Moon

50. Perigee and Apogee of the Moon

51. Passages of the Moon through the Nodes ............

52. Maximum Declinations of the Moon

53. Ephemeris for Physical Observations of the Moon .

54. Eclipses

55. Semidiameters of the Sun, Moon, and Planets .

56. Stellar Magnitudes .

57. Binary Stars .

58. Calculation of a Planar Sundial . .

 

209

217

223

241

245

249

263

269

275

279

283

287

293

301

317

323

337

345

349

355

363

367

371

379

389

393

397

401

 

 

Appendix I Appendix II Appendix IH Appendix IV

Index

 

Constants

Some Astronomical Terms Planets: Periodic Terms

Coefficients for the Heliocentric Coordinates of

Jupiter to Neptune, 1998 —2025

 

407

409

413

455

473

 

Some Symbols and Abbreviations

a Semimajor axis (of an orbit)

e Eccentricity (of an orbit)

h Altitude above the horizon

i Orbital inclination Mean daily motion

q Perihelion distance, in AU

r Radius vector, or distance of a body to the Sun, in AU v True anomaly

A Azimuth

Hour angle

M Mean anomaly

Distance from Earth to Sun, in AU

T Time in Julian centuries (36525 days) from 12000.0

Right ascension

Declination

Obliquity of the ecliptie (q is used for the mean obliquity)

Sidereal time (8 is the sidereal time at Greenwich)

Longitude of perihelion

Time in Julian millennia (365250 days) from J2000.0 Geographical latitude

Geocentric latitude

 

 

 

 

Ae

 

AU

INT

io IDE TD UT

 

Distance to the Earth, in AU

is used to indicate a correction or a difference, for instance As

Difference TD — UT Nutation in obliquity Nutation in longitude

Astronomical Unit Integer part of a number Julian Day

Julian Ephemeris Day Dynamical Time Universal Time

5

 

Following an old, general astronomical practice, small superior symbols are placed immediately above the decimal point, not after the last decimal. For instance, 28‘.5793 means 28.5793 degrees. See, for instance, the Circulars of the International Astronomical Union, or the great astronomical almanacs.

Moreover, note carefully the difference between hours with decimals, and hours—minutes—seconds. For example, 1!30 is rtoi 1 hour and 30 minutes, but 1.30 hours, that is 1 hour and 30 hundreths of an hour, or 1 hour and 18 minutes.

Do not use the symbols ' and “ for minutes and seconds of time: they are used for minutes and seconds of a degree (or arcminutes and arcseconds, respectively). Minutes and seconds of time have the symbols m and s. For example,

the angle 23°26’44”, but the instant l5 h22‘ 07'.

Indeed, we have

1’ = one minute of arc -— 1/60th of a degree

1" = one minute of time   -- 1/60th of an hour

Do not use the symbol + for “approximately”. That symbol means “plus or minus” (or "plus and minus”). For instance, the square root of 25 is + 5, which means +5 or —5. Writingr   = +3 is incorrect, becauser   is equal to neither +3 nor —3. The correct symbol to be used here is = . For example, 1002 - 1000.

In general, we shall use the “scientific” form for calendar dates, which reads from the largest to the smallest unit of time, for example 1993 November 6. It contrasts with the common “American” form (November 6, 1993) and with the “European” form (6 November 1993). Anyway, it is recommended to spell out the month, because one person’s “11/6/93” is another’s “6/11/93”.

It is recommended to write the year number out in full, not trimmed to the last two digits. For example, the solar eclipse of February 1998, not February 98 nor February ’98.

 

Chapter 1

Mints and Tips

To explain how to calculate or to program on a computer is out of the scope of this book. The reader should, instead, study carefully his instructions manual. However, even writing good programs cannot be learned in the lapse of time of one day. It is an art which can be acquired only progressively. Only by practice can one learn to write better and shorter programs. In this first Chapter, we will give some practical hints and tips, which may be of general interest.

Trigonometric auctions of large angles

Large angles frequently appear in astronomical calculations. In Example 25.a we find that on 1992 October 13.0 the mean longitude of the Sun is —2318.19280 degrees. Even larger angles are found for rapidly moving objects such as the Moon and the bright satellites of Jupiter, or the rotations of the planets (see, for instance, the angle lY in step 9 of Example 42.a).

It may be necessary to reduce the angles to the interval 0—360 degrees, because some pocket calculators or some program languages give incorrect values for the trigonometric functions of large angles. Try, for instance, to calculate the sine of 36000030 degrees. The result must be 0.5 exactly.

Angle modes

The majority of calculating machines do not calculate directly the trigonometric functions of an angle which is given in degrees, minutes and seconds. Before performing the trigonometric functions, the angle should be converted to degrees and decimals. Thus, to calculate the cosine of 23°26’49”, first convert this angle to 23.446944 44 degrees, and then use the COS function.

There is the added complication that most programming languages can calculate only in radians, not in degrees. It is an infernal nuisance having to convert degrees to radians all the time, but in most computer languages this has to be done before calculating a trigonometric function of an angle given in degrees. To convert an angle from degrees to radians, multiply it by w/180 = 0.017 453 292 519 942...

7

 

Right ozcenâions

Right ascensions are generally expressed in hours, minutes, and seconds of time. To calculate the trigonometric function of a right ascension, it is necessary to convert that value to degrees (and then in radians, if needed). Remember that one hour corresponds to 15 degrees.

Example 1.a — Calculate tan n, where n  = 9h14" 55'8. We first convert n to hours and decimals:

9‘14*55'.8 = 9 + 14/60 + 55.8/3600 = 9.248 833 333 hours.

Then, multiplying by 15, we obtain n = 138.°73250.

Multiplying this value by  w/180 = 0.017 453 2925... gives n in radians. We then find tan n = —0.877 517.

The correct quadrant

When the sine, the cosine or the tangent of an angle is known, the angle itself can be obtained by using the “inverse” function arcsine (ASN or ASIN), arccosine (ACS or ACOS), or arctangent (ATN or ATAN). Note that, unfortunately, the functions arcsine and arccosine are absent in many programming languages.

The inverse trigonometric functions (arcsine, arccosine, arctangent) are not single valued. For instance, if sin n = 0.5, then n = 30°, 150°, 390°, etc. For this reason, the programming languages return inverse trigonometric functions correctly over only half the range of 0 to 360 degrees: arcsine and arctangent give an angle lying between —90 and +90 degrees (that is, between — w/2 and + r/2 radians), while arccosine gives a value between 0 and + 180 degrees (between 0 and

r radians).

For example, try cos 147°. The answer is —0.8387, which reverts to 147° when you take the inverse function. But now try cos 213°. The answer is again

—0.8387 which, when you take its arccosine, gives 147°.

Hence, whenever the inverse function of SIN, COS, or TAN is taken, an ambiguity arises which has to be cleared up by one or other means when ii is necessary. Each problem must be examined separately.

For instance, formulae (13.4) and (25.7) give the sine of the declination of a celestial body. The function arcsine then will always give this declination in the correct quadrant, because all declinations lie between —90 and +90 degrees. So, no special test should be performed here.

This is also the case for the angular separation whose cosine is given by formula (17.1). Indeed, any angular separation is in the range of 0‘ to + 180°, which matches the range of the inverse cosine function.

 

1. HINTS AND TIPS 9

But consider the conversion from right ascension (o) and declination (6) to celestial longitude (h) and latitude (9) by means of the following formulae

cos jS sin k = sin 6 sin c + cos 6 cos c sin a cos jS cos X = cos 6 cos a

Call A and B the second members. Then, dividing the first equation by the second one, we obtain tan k = A f B. Applying the function arctangent to the quotient A lB will yield the angle k between —90° and +90°, with an ambiguity of + 180°. This ambiguity can be removed with the following test: if B < 0, add 180‘ to the result. However, some computer languages contain the useful “second” arctangent function, ATN2 or ATAN2, which uses the two arguments A and B separately and returns the angle in the proper quadrant. For instance, suppose that A —- —0.5712, B -— —0.9139 ; then ATN(A/B) will give the angle 32°, while ATN2(A, B) will yield the correct value — 148°, or +212°.

The input of negative angles

Angles expressed in degrees, minutes, and seconds can be input as three different numbers (in BASIC: INPUT D, M, S). For instance, the angle 21°44'07“ can be entered as the three numbers 21, 44, and 7. Then, in the program the angle H in degrees is calculated by means of the instruction H = D + M/60 + S/3600.

In such a case, care must be taken for negative angles. If the angle is, for example, — 13°47’22”, then this means — 13° and —47’ and —22”. In this case, the three numbers are D = — 13, M = —47, and S = —22. All three numbers have the same sign!

Mislead by the notation — 13°47’22“, one can have the tendency to input — 13,

+47, and +22 instead, and in that case the angle entered would actually be

— 12°12'38“. It is possible to write the program in such a way that similar errors are corrected automatically:

200 INPUT D, M, S

210 IF D < 0 THEN M = —ABS(M) : S = —ABS(S)

220 H = D + M/ 60 + S / 3600

In line 210, the minutes and seconds are made negative when the degrees are negative. The two ABS functions make sure that no error is made when M and S are actually entered as negative numbers.

This procedure does not work, however, when the angle is between 0° and

— 1°. If the angle is, for instance, equal to —0°32'41”, then we have D = —0, which a computer automatically converts to 0, which is not negative, so the machine will conclude that the angle is +0°32'41“ instead. One solution (in BASIC) is to enter the degrees as a “string” instead of a numeric variable, hence by means of INPUT D$ instead of INPUT D. Then one can use the VAL function and test on the first character of the string D$.

 

Powers o/ time

Some quantities are calculated by means of a formula containing powers of the time ‹T, r 2, T , ...). It is important to note that such polynomial expressions are valid only for values of T that are not too large. For instance, the formula

e —- 0.046381 22 — 0.000027 293 T + 0.000000 0789 F 2 (1.1)

gives the eccentricity e of the orbit of Uranus; T is the time measured in Julian centuries (36525 days) from the beginning of the year 2000. It is evident that this formula is valid for only a limited number of centuries before and after A.D. 2000, for instance for T lying between —30 and + 30. For | T I much larger than 30, the

above expression is no longer valid. For T —- —3307. 9 the formula would give e

= 1, and an incompetent person, thinking that “the computer cannot make errors", would deduce that in the year —328790 the orbit of Uranus was parabolic and hence that this planet originates from outside our solar system — bringing us in the realm of pseudoscience.

In fact, the eccentricity e of a planet’s orbit varies rather irregularly in the course of time, though it cannot exceed a well-defined upper limit. But for a time interval of a few millennia the eccentricity can be accurately represented by a polynomial of the second degree such as (1.1).

One should further carefully note the difference between periodic terms (terms in sine and/or cosine), which remain small throughout the centuries, and secular terms (terms in T, T , T , . ..) which increase more and more rapidly with time. A term in r 2, which is very small when r is small, becomes increasingly important for larger values of | T   . Thus, for large values of | T I it is meaningless to take

into account small periodic terms if terms in T , etc. , are neglected in the

calculation.

Avoiding powers

Suppose that one wants to calculate the value of the polynomial

y —— A + Bx + Cx2 + Dz3 + Ex

with A, B, C, D, and E constants, and z a variable. Now, one may write the program to calculate this polynomial directly term after term and adding all terms, so that for each given x the machine obtains the value of the polynomial. However, instead of calculating all the powers of z, it appears to be wiser to write the polynomial as follows:

y —— A + x {B + x {C + x (D + zZ)))

In this expression all power functions have disappeared and only additions and multiplications are to be performed. This way of expressing a polynomial is called

 

Homer’s method, z:n approach especially well suited for automatic calculation

because powers are avoided.

Also, it may be wise to calculate the square of a number A by means of A • A instead of using the power function. We calculated the squares of the first 200 positive integers on the HP-85 microcomputer. Using the procedure

FOR I = 1 TO 200 K = I“2

NEXT I

The complete calculation took 10.75 seconds. But when the second line was replaced by K = I a I, then the calculation time was only 0.96 second!

To shorten • Program

To make a program as short as possible is not always an art for art’s sake, but sometimes a necessity as long as the memory capacities of the calculating machine have their limits.

There exist many tricks to make a program skorter, even for simple calculations. Suppose that one wants to calculate the sum 5 of many terms:

S —— 0.0003233 sin (2.6782 + 15.54204 T)

+ 0.0000984 sin (2.6351 + 79.62980 r)

+ 0.0000721 sin (1.5905 + 77.55226 T)

+ 0.0000198 sin (3.2588 + 21.32993 T)

First, because the coefficients of all sines are small numbers, one can avoid typing in all those decimals by taking as unit the last decimal (10*7 in this case). So, instead of 0.0003233, etc. , we use 3233, etc. Then, nder the sum of the terms has been calculated, we divide the result by 10'.

Secondly, it would be unwise to write all those terms explicitly in the program. Instead, we could make use of a so-called loop. Each of the above terms is of the form A sin (B + CT), so we put all values A, B, C as DATA In the program. Suppose there are 50 terms. Then the program will look like this:

100 S = 0

110 RESTORE 170

120 FOR I = 1 TO 50

130 READ A, B, C

140 S = S + A * SIN(B + C T)

150 NEXT J

160 S = 5/10000000

170 DATA 3233, 2.6782, 15.54204, 984, etc. . . .

 

Safer lesls

Include a safety test in case an “impossible* situation might occur, for example in order to stop the calculation when, after a specified number of iterations, the required accuracy has not been reached.

Or consider the case of the occultation of a star by the Moon. In a program for local circumstances, the times of disappearance and of reappearance of the star are calculated. It may happen, however, that the star is not occulted as seen from the given place; in such a case, the times of ingress and egress do not exist, and trying to calculate them would correspond to calculating the square root of a negative number. To avoid this problem, the program should be written in such a way that first of all the value of the star’s least distance to the center of the lunar disk (as seen from the given place) is calculated; if, and only if, this distance is smaller than the radius of the Moon’s disk, can the times of ingress and egress be calculated.

Debugging

After a program has been written, it must be checked for errors, which are called bugs. The process of locating the bugs and COfTecting them is known as debugging. Several types of errors can occur when programming in any language:

a. syntax errors violate the rules of the language, such as spelling, a forgotten parenthesis, or other conventions specific to each language. For instance, in BASIC,

A = SIM(B) should be      A = SIN(B)

P = SQR(ABS(A + B)      should be      P = SQR(ABS(A + B))

b. semantic errors, such as a forgotten line. For instance, GOTO 800 when no line

labelled 800 exists in the program.

c. run-time errors, which occur during the execution of a program. For example: A = SQR(B). The variable B is calculated during execution of the program, but its value happens to be negative;

ON X GOTO 1000, 2000, 3000, but X is larger than 3.

d. other programmer’s errors. The following ones happen frequently:

• Typing the letter O (“oh”) instead of the digit zero (0 or B), or vice versa, or typing the digit 1 instead of the letter I.

• The name of a variable is used twice in the program (with different meanings).

• A variable has not been defined, and therefore the program assumes its value is zero.

• Error in copying down a numerical constant (such as 127.3 instead of 127.03), or 15 instead of .15), typing an instead of a +, etc.

 

Incorrect units are used. For instance, an angle is expressed in degrees instead of radians, or a right ascension expressed in hours has not been converted to degrees or radians.

The angle is in the wrong quadrant. See *The correct quadrant" on page 8.

The natural logarithm of a number has been used instead of its logarithm to the base 10 — see Chapter 56.

Rounding errors. For example, the cosine of an angle d has been calculated, from which one wants to deduce that angle. This does not work well when the angle is very small. Indeed, if d is very small, its cosine is almost equal to 1 and varies quite slowly as a function of d. In that case, the value of d is ill- defined and cannot be calculated accurately.

For instance, cos 15“ = 0.999 999 997 but cos 0” is I exactly. If one expects that the angle d can be very small, then its value should be calculated by means of another method. See, for instance, Chapter 17.

Single precision is used instead of double precision. In QuickBA5IC, even if the

varGiable has been declared  to be of  the double-precision type,  the statement

G = . 1 gives a result of lower accuracy, namely 0.100 000 014 901 16. One should write G = . 1# here.

An iteration procedure which does not guarantee convergence in some cases. See Chapters S (Iteration) and 30 (Equation of Kepler).

An incorrect method of calculation has been used. For example, to interchange two numbers X and Y, an extra variable A is needed (*) :

Incorrect procedure Correct procedure

Y = X A = Y

X = Y Y = X

X = A

In QuickBASIC, GWBASIC, and some other BASIC versions, there exists the SWAP function: SWAP(X, Y) interchanges the numbers X and Y.

(*) This is not quite exact. Theoretically, it is possible to interchange two numbers without using a third, auxiliary variable, as follows :

X = X + Y Y = X — Y X = X — Y

But, of course, this is rather a curiosity than a useful method, because the execution of these operations requires extra computer time, and because rounding errors can Occuf.

 

Checking the results

Of course, a program should not only be “grammatically” correct: it must give correct results. Test your program using a known solution. If, for instance, you wrote a program for the calculation of planetary positions or for the times of lunar phases, compare your results with the values given in an astronomical almanac.

Test your program for some “special” cases. For instance, are the results still correct for a negative value of the declination? Or for a declination lying between 0° and — 1°? Or if the observer’s latitude is exactly zero? Or for negative values of the time T*

 

Chapter 2

About Accuracy

The following topics will be considered in this Chapter: the accuracy needed for a particular problem, the accuracy with which a given programming language works, and finally the accuracy of the published results.

The accuracy needed for a given problem

The accuracy needed in a calculation depends on its aim. For example, if one wants to calculate the position of a planet with the goal of obtaining the times of rising and setting for a given place, an accuracy of 0.001 or even 0.01 degree will be sufficient. The reason is evident: the apparent diurnal motion of the celestial sphere corresponds to a rotation over one degree during a time interval of four minutes, and so an error of 0.01 degree in the object’s position will result in an error of only 0.04 minute (approximately) in its time of rising or setting. Taking hundreds of periodic terms into account in order to obtain the planet’s position to an accuracy of 0“.01 would just be a waste of effort and of computer time for this problem.

But if the position of the planet is needed to calculate the occultation of a star by that planet, then an accuracy of better than 1“ will be necessary by reason of the small size of the planet’s disk.

A program written for one aim may not be suitable for another application. Suppose that, for the calculation of the position of a star, a program uses the low- accuracy method for the precession (see Chapter 21). While the results will be good enough for the observer who wants to find celestial objects with a telescope on a parallactic mounting, that program will be completely worthless when accurate results are required, for instance in occultation work, or for the calculation of close conjunctions.

If a given accuracy is required, one has to use an algorithm that really provides this precision. John Mosley [1] mentions a commercially available program which calculates planetary positions; but because perturbations are not taken into account, the positions of Saturn, Uranus, and Neptune can be up to 1 degree off, even though displayed to the nearest arcsecond!

15

 

To obtain a better accuracy it is often necessary to use another method of calculation, not just to keep more decimals in the result of an approximate calculation. For example, if one has to know the position of Mars with an accuracy of 0.1 degree, it suffices to use an unperturbed elliptical orbit (Keplerian motion). But if the position of Mars is to be known with a precision of 10” or better, perturbations due to the other planets have to be calculated and the program will be a much longer one.

The programmer, who knows his formulae and the desired accuracy in a given problem, must himself consider which terms, if any, may be omitted in order to keep the program handsome and as short as possible. For instance, the mean geometric longitude of the Sun, referred to the mean equinox of the date, is given by

L -- 280°27’59”.245 + 129 602 771”.380 T + 1“.0915 'r 2

where T is the time in Julian centuries of 36525 ephemeris days from the epoch 2000 January 1.5 TD. In this expression, the last term (secular acceleration of the Sun) is smaller than 1“ if | T | < 0.95, that is, between the years 1905 and 2095. If an accuracy of 1“ is sufficient, the term in T may thus be dropped for any instant in that period. But for the year + 100 we have T —— — 19, so that the last term becomes 394”, which is larger than 0.1 degree.

The computer’s  accuracy

This is a much more complex problem. The program language should work with a sufficient number of significant digits. Note that this is not the same as the number of decimals! For instance, the number 0.0000183 has seven decimals, but only three significant digits. The significant digits of a number are those digits which are left over when the leading and trailing zeros are suppressed.

On a machine rounding   operations   to 6 significant figures,   the result of 1 000 000 + 2 will just be 1 000 000.

There can be dangerous situations, for instance when the difference is made of two nearly-equal numbers. Suppose that the following subtraction is performed:

6.92736 — 6.92735 = 0.00001.

Each number is given to six figures, but subtracting them gives a number with just one significant figure! Moreover, the two given numbers perhaps have already been rounded. If such is the case, then the situation can even be worse. Suppose that the two numbers are actually 6.927 3649 and 6.927 3451. Then the correct result of the subtraction is 0.000 0198, which is almost twice the previous result!

Six or eight significant digits, as was the general rule for the early microcomputers, or is nowadays often the case in "single precision”, are generally not sufficient for mathematical astronomy.

 

For many applications, it is necessary that the machine calculates with a larger number of significant digits than it is required in the final result. Let us consider, for example, the following formula giving the mean longitude L’ of the Moon for any given instant, in degrees (Chapter 47) :

L’ —— 218.316 4477 + 481 267.881 234 21 T - 0.0015786 T + 0.000 0019 T

where 'r is the time measured in Julian centuries of 36525 days elapsed since the standard epoch 2000 January 1.5 TD (JDE 2451545.0). Suppose now that we wish to obtain the Moon’s mean longitude to an accuracy of 0.001 degree. Because longitudes are restricted to the interval 0—360 degrees, one might think that a language calculating with only six significant digits internally will be just sufficient for our purpose (3 digits before, and 3 digits after the decimal point). This is not the case in the present problem, however, because L ' can reach large values before it is reduced to less than 360 degrees.

For instance, let us calculate L’ for T —- 0.4 which corresponds to 2040 January 1 at 12‘ TD. We find L' -— 192 725.°469, which reduced to 125.°469, the correct answer. But if the machine works with only six significant digits, it will not find L ' = 192 725.°469, but rather 192 725° (six digits!), which will reduce to 125°, so in this case the final result is only to the nearest degree, and the error is 0.469 degree or 28’ ; and this happens for only 40 years after the starting epoch. Under such circumstances it is just impossible to calculate eclipses or occultations.

To find out with which internal accuracy a programming language works, the following short program (in BASIC) can be used.

10 X = 1

20 J = 0

30 X = X 2

40 IF X + 1 < > X THEN 60

50 GOTO 80 60 J = J + 1

70 GOTO 30

80 PRINT I, J 0.30103

90 END

Here, is the number of significant bits in the mantissa of a floating number, while 0.30103 7 is the number of significant digits in a decimal number. The constant 0.30103 is log 2. For instance, the HP-85 computer gives I = 39, whence 11.7 digits. With the HP-UX Technical Basic 5.0, working on the HP- Integral microcomputer, we find J -- 52, whence 15.6 internal digits. The QuickBASlC 4.5 gives = 63, whence 19.0 digits.

However, this accuracy refers only to simple arithmetics, not to the trigonometric functions. Although the GWBASIC has 7 = 55, that is 16,6 internal digits, it gives the sines with only 7 correct decimals; the last nine figures are all wrong!

 

One rapid way to check the accuracy of trigonometric functions is PRINT

4   ATN(1). If the computer works in radians, this must give the famous number z = 3.14 15 92 65 35 89 79... Or one may calculate the sine of an angle whose value is accurately known, for instance SIN(0.61 rad) — 0.572 867 460 100 48...

Rounding is inevitable in a computer. Consider for instance the value 1/3 = 0.33333333... Because the machine cannot handle an infinite number of decimals, such a number must necessarily be truncated somewhere.

Rounding errors can accumulate from one calculation to the next. In most cases this is of no important because the errors almost cancel each other, but in some arithmetical applications the accumulated error can increase beyond any limit. Although this topic is outside of the scope of this book, we shall mention two cases.

Consider the following program.

10 X = 1/3

20 FOR J = 1 TO 30

30 X = (9 • X + 1) X — 1

40 PRINT I, X

50 NEXT J

60 END

The operation on line 30 actually replaces X by itself. Yet on most computers the results diverge. The above-mentioned HP-UX Technical Basic yields

0.333 333 333 333 308 after 4 steps

0.333 326 162 117 054 after 14 steps

0.215 899 338 763 055 after 19 steps

286.423... after 24 steps

and a value of the order of 1021’ after 30 steps!

The difference in accuracy between microcomputers or even hand-held calculators can be demonstrated by a simple test [2]: repeatedly squaring the number 1.0000001. After 27 times, the result to ten significant figures must be 674 530.4707. The results for some machines or programming languages are as follows:

674 494.06 on the HP-67 calculator

674 514.87 on the HP-85 and on the HP-48s calculator 674 520.61 on the TI-58 calculator

674 530.4755 on the HP-Integral (HP-UX Technical Basic) 674 530.4755 in QuickBASIC 4.5

But that is still not the end of the story. There are two basically different ways for the internal representation of numerical information into a computer. Some machines, such as the older HP-85, use the BCD (Binary Coded Decimal) scheme for representing numbers internally, but in most other cases the binary representation is used.

 

BCD is a scheme where the actual value of each digit of a number is stored individually. This allows numbers to be represented exactly, to the specified digits of precision of the given machine or programming language. Binary, on the other hand, represents all numbers as some combination of powers of 2. In binary, fractions are also represented as being powers of 2, so it is impossible to represent numbers which are not exact combinations of negative powers of 2 in a binary system. For instance, 1/10 is not rationally expressed as combinations of negative powers of 2, because 1/10 = 1/16 + 1/32 + 1/ 128....

Binary arithmetic functions are usually faster in their execution than BCD counterparts, but the inconvenience is that some numbers, even with a small number of decimals, are not represented exactly.

As a consequence, the result of an arithmetic operation may be incorrect, even when numbers with only a few decimals are involved. Suppose that X — 4.34. Then the correct result of the operation H = INT(100  (X — INT (X))) is 34. However, many computer languages give H = 33 here. The reason is that in this case the value of X is represented internally as 4.3399999998, or something like that.

Another surprising example is

2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 — 3

On many computers, the result is not zero! On the HP-Integral, using the HP-UX

Technical Basic 5.0, the result is 8.88 X 10 16. But on the same machine 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 2 — 3

does give zero, so the order in which the operations are performed can be of importance here!

Surprisingly, 2 + (5 • 0.2) — 3 gives exactly zero on the HP-Integral, and so does the following:

A = 0.2 + 0.2 + 0.2 + 0.2 + 0.2 B = 2 + A

C = B — 3

PRINT C

Consider the following program:

10 FOR I = 0 TO 100 STEP 0.1

20 U = I

30 NEXT I

40 PRINT U

50 END

Here, i and U take the successive values from 0 to 100 with steps of 0, 1, and the last value of U must be exactly 100. The HP-85 does give 100 indeed, but QuickBASIC 4.5 gives 99.999 999 999 9986, which can have a disastrous con-

 

sequence in some applications. The error is due to the fact that the step value of 0.1 is translated into binary as 0.0999999.... The difference with 0.1 is very small, but because there are 1000 steps, the final error is 1000 times as large as that small difference. In this case, one remedy may consist in taking an integer value for the step:

10 FOR I = 0 TO 1000 20 I = I/ 10

30 U = I

40 NEXT J

50 PRINT U

60 END

We may find other surprises with A = 3 • (1/3), PRINT INT(A), whose result is correctly 1 in some programming languages, but zero in others. Or try, for instance, A = 0.1, PRINT INT(1000 • A).

Another interesting test is

INPUT A B = A / 10

C = 10 • B

PRINT A — C

The result must be zero. But for some numbers A the answer can be different.

One easy way to find out if a computer language works in BCD or not, consists of looking at the largest possible integer value, that is, a number defined as an INTEGER. If this is a “nice, round” number, this indicates that the machine works in BCD. For example, on the HP-85 that largest integer is 99 999 (or 105 — 1). But if the largest possible integer is a “strange” number (in fact, a power of 2 minus one), this means that the computer does not work in BCD. On the old TRS-80, that largest integer is 32767 (or 21 — 1), while for QuickBASIC 4.5 it is 2 147 483 647

(Cry. 231 — 1).

Rounding by inexact arithmetics can yield other surprising results. In most programming languages, the result of SQR(25) — 5 is trol zero! This can be a problem when testing on the result. Is 25 a perfect square? One might think the answer is no, since the computer tells us that SQR(25) — INT(SQR(25)) is not zero!

Important! If you are comparing INTEGER numbers, no special precautions are necessary. However, if you are comparing so-called REAL values, especially those which are the results of calculations and functions, it is possible to run into problems. The equality test may fail due to rounding or other errors caused by the inherent limitations of machines. A repeating decimal or irrational number cannot be represented exactly in any finite machine.

 

Rounding the final result

Results should be rounded correctly and meaningfully, where it is necessary. Rounding should be made to the nearest value. For instance, 15.88 is to be rounded to 15.9, or to 16, not to 15. However, calendar dates and years are exceptions. For example, March 15.88 denotes an instant belonging to March 15: it means 0.88 day after March 15, 0'. So, if we read that an event occurs on March 15.88, it takes place on March 15, not on March 16. Similarly, 1987. 69 denotes an instant belonging to the year 1987, not 1988 ; it is 0.69 year after the start of A.D. 1987.

Only meaningful digits should be retained. For example, Müller’s formula for calculating the visual magnitude of Jupiter is

m = —8.93 + 5 log rA

where r is Jupiter’s distance to the Sun, A its distance to the Earth (both in astronomical units), and the logarithm is to the base 10. Now, on 1992 May 14, at 0' TD, we have

r = 5.417 149

A = 5. 125 382

whence m = — 1.712514 898. But giving all thèse decimals, under the pretext that they were given like this by the computer, would be ridiculous and would give the reader a false impression of high accuracy. Since the constant —8.93 in Müller’s formula is given to 0.01 magnitude, no higher accuracy can be expected in the result. And, in any case, the meteorological phenomena in the atmospkeie of Jupiter are such that the magnitude of that giant planet cannot be predicted with an accuracy better than 0.01 or even 0. 1.

As another example, John Mosley [3] mentions a commercially available program giving rising and setting times où heavenly bodies to the nearest 0.1 second, which is impossibly precise.

Some “feeling" and sufficient astronomical knowledge are necessary here. For instance, it would be completely irrelevant to give the illuminated fraction of the Moon’s disk accurate to 0.000000 001.

The rounding should be performed afier the whole calculation has been made, not before the start or before the input of the data into the computer.

Example: Calculate 1.4 + 1.4 to the nearest integer. If we first round the given numbers, we obtain 1 + 1 — 2. In fact, 1.4 + 1.4 = 2.8, which rounds to 3.

Here is another example. At its opposition date, 1996 July 18, the declination of Neptune was 8 = —20°24'. What was the planet’s altitude hq at the transit through the southern meridian at Sonneberg Observatory, Germany, to the nearest degree? The Observatory’s latitude is ‹,o = + 50°23'. The formula to be used is

ñp = 90° — p + 6

The answer is hp -- 90° — 50°23' — 20°24’ = 19°13‘, whence 19°. Rounding

 

e and 6 to the nearest degree before the calculation would yield the incorrect result 90° — 50° — 20° = 20°.

A similar error occurs when distances, already rounded to the nearest mile, are converted to kilometers. In this case the value of 17 km, for instance, will never be reached, because

10 miles will give 16.09 km, which is rounded to 16 km, 11 miles will give 17.70 km, which is rounded to 18 km.

Right ascensions and declinations. — Since 24 hours correspond to 360 degrees, one hour corresponds to 15°, one minute of time corresponds to 15 minutes of arc, and one second of time to 15 seconds of arc: during a time interval of one second the Eanh rotates over an arc of 15“.

For this reason, if the declination of a celestial body is given, for instance, to 1“, then its right ascension should be given to the nearest tenth of a second of time, since otherwise the declination would be given with a much greater accuracy than the right ascension. The following table gives the approximate correspondence between the accuracies in right ascension (n) and in declination (6). For example, if 6 is given with an accuracy of 1', then o must be given to the nearest 0.1 minute of time. As examples, the position of Nova Cygni 1975 with different accuracies is given.

Example (Nova Cygni 1975)

1‘ 0°.1 o = 21h10* h = + 47.° 9

0‘.1 1' 21'09'?9 +47°57'

1’ 0'.1 21'09‘53‘ +47°56'.7

0 S. 1 1 “ 21 09"52’.8 +47° 56’41“

0'.01 0”. 1 21'09"52\ 83 +47° 56’41“. 2

As a final remark, let us mention that trailing zeros can be important. For instance, 18.0 is not the same as 18. The former value means that the actual number lies between 17.95 and 18.05, while the second value has been rounded to the nearest integer and can actually be equal to any number between 17.5 and 18.5. For this reason, trailing zeros must be given in the result to indicate the accuracy: a star of magnitude 7 is not the same as a star of magnitude 7.00.

RE FE REN CES

1. John Mosley, Slcy and Telescope, Vol. 78, p. 300 (September 1989).

2. F. Gruenberger, “Computer Recreation”, Scientific American, Vol. 250, p. 10 (April 1984).

3. John Mosley, Sky and Telescope, Vol. 81, p. 201 (February 1991).

 

Chapter 3 interpolation

The astronomical almanacs or other publications contain numerical tables giving some quantities y for equidistani values of an argument x. For example, y is the right ascension of the Sun, and the values z are the different days of the year at 0h.

Interpolation is the process of finding values for instants, quantities, etc., intermediate to those given in a table.

Of course, the “table” should not necessarily be taken from a book, but may have been calculated in a computer program. Suppose that the position of the Sun is to be calculated for many (> 3) instants of the same day. Then one may calculate the Sun’s position for 0', 12', and 24' of that day, and then use these values to perform the interpolation for every given instant. This will require less computer time than calculating the position of the Sun directly for every instant,

In this Chapter we will consider two cases: interpolation from three or from five tabular values. In both cases we will also show how an extremum or a zero of the function can be found. The case of only two tabular values will not be considered here, for in that case the interpolation can but be linear, and this will give no difficulty at all.

Three tabular values

Three tabular values y , y , y   of the function y are given, corresponding to the values zt , z2 , z, of the argument x. Let us form the table of differences

(3.1)

where a —- yt — y and b —— y3 — y  art  called the 9TsJ d@erences, The second

difference c is equal to b - a, that is

2 3

 

Generally, the differences of the successive orders are gradually smaller in absolute value. Interpolation from three tabular values is permitted when the second differences are almost constant in that part of the table, that is, when the third differences are almost zero. Some good sense and experience are needed here. For example, the Moon’s position can be interpolated accurately from three positions given at hourly interval, but not when the interval is one day.

Let us consider, for instance, the distance of Mars to the Earth from 5 to 9 November 1992, at 0' TD. The values are given in astronomical units, and the differences are in units of the sixth decimal:

 

—6904

—6883

—6860

—6835

 

+21

+23 +2

+25 +2

 

Since the third differences are almost zero, we may interpolate from only three

tabular values.

The central value x2 must be chosen in such a way that it is that value of x that is closest to the value of x for which we want to perform the interpolation. For example, if from the table above we must deduce the value of the function for November 7 at 22h14*, then y2 is the value for November 8.00. In that case we should consider the tabular values for November 7, 8, and 9, namely the table

November 7 y = 0.884226

 

8 y —- 0.877366

9 yz —- 0.870531

and the differences are

a —— —0.006 860

 

(3.2)

 

b ——  —0.006 835

 

c = +0.000025

 

Let n be the interpolating factor. That is, if the value y of the function is required for the value z of the argument, we have n = x — x2 in units of the tabular interval. The value n is positive if x > z , that is for a value “later" than x2, or from x, towards the bottom of the table. If z precedes x2 , thi n n < 0.

If y2 has been correctly chosen, then n will be between —0.5 and +0.5, although the following formulae will also give correct results for all values of n between —1 and + 1.

The interpolation formula is

y -- y2 + ' ta + b + nc) (3.3)

 

3. INTERPOLAT10N 25

<*- h e 3.a — From the table (3.2), calculate the distance of Mars to the Earth on

1992 November 8, at 4'21‘ TD.

We have 4'21" = 4.35 hours and, since the tabular interval is 1 day or 24 hours, n = 4.35/24 = +0.18125.

Formula (3.3) then gives y -- 0.876 125, the required value.

If the tabulated function reaches an extremum (that is, a maximum or a minimum value), this extremum can be found as follows. Let us again form the difference table (3.1) for the appropriate part of the ephemeris. The extreme value of the function is

(3.4)

 

and the corresponding value of the argument z is given by

nq —— _   a + b 

2 c

 

(3.5)

 

in units of the tabular interval, and again measured from the central value x2.

Example 3.b — Calculate the time of passage of Mars through the perihelion in May 1992, and the value of its radius vector at that instant.

The following values for the distance Sun—Mars have been calculated at intervals of four days:

1992 May 12.0 TD 1.3814294 AU

16.0 1.381 2213

20.0 1.3812453

 

The differences are

 

a = —0.000 2081

b -- +0.000 0240

 

c = +0.000 2321

 

from which we deduce

yq -- 1.381 2030   and   rig = +0.39 660

Hence, the least distance from Mars to the Sun is 1.3812030 astronomical units. The corresponding time is found by multiplying 4 days (the tabular interval) by + 0.39 660. This gives 1.58640 days, or 1 day and 14 hours later than the central time, that is 1992 May 17, at 14' TD.

[Of course, if rig were negative, the extremum would take place earfier than the central time.]

 

The value of the argument z for which the function y becomes zero can be found by again forming the difference table (3.1) for the appropriate part of the ephemeris. The interpolating factor corresponding to a zero of the function is then given by

 

  — 2yt

a + b + cn0

 

(3.6)

 

This equation can be solved by first putting • — 0 in the second member. Now the formula gives an approximate value for rig. This value is then used to calculate the right hand side again, which gives a still better value for •      This process,

called iteration (Latin: iterare —— to repeat), can be continued until the value found

for nd no longer varies, to the precision of the computer.

Example 3.c — Given the following values for the declination of Mercury, 1973 February 26.0 TD —0° 28' 13”.4

27.0 +0   06 46.3

28.0 +0 38 23.2

calculate when the planet’s declination was zero.

Firstly, we convert the tabulated values into seconds of a degree and then form

the differences:

 

y, =  — 1693.4 p

 

= +2099.7

 

Hz ' + 406.3 b -- + 1896.9 y, = +2303 2

 

c =  —202. 8

 

Formula (3.6) then becomes

_ —812.6

 

0 +3996.6 — 202.8 nq

Putting n0 = 0 in the second member, we find nt = —0.20332. Repeating the calculation, we find successively —0.20125 and —0.20127. Hence, n0 = —0.20127. The tabular interval being one day, Mercury crossed the celestial equator on

1973 February 27.0 — 0.20127 = February 26.79873

= February 26, at 19‘10” TD.

For the calculation of the value of the interpolating factor n0 for which the function is zero, formula (3.6) is excellent when, as in Example 3.c, the function is “almost a straight line” in the interval considered. If, however, the curvature of the function is important, use of the formula may require a large number of iterations; moreover, it can lead to divergence even when starting from an almost

 

3.   INTERPOLATION 27

correct value for up In this case, a better method for calculating rt is as follows: the correction to the assumed value of no is

 

   *y  + x (o + b + c ) 

a + b + 2c

 

(3.7)

 

The calculation should be repeated, using the new value of rit, until rig no longer varies.

Example 3.d — Consider the following values of a function:

*i —1 li

 

' ‘z

 

0    ' /z “/

 

These three points actually define the parabola y = 3 + 2r — 3z 2, which has a strong curvature between z = — 1 and z = + 1 (see the Figure at left).

Starting with ii0 = 0, formula (3.6) gives successively

—0.461 538...

—0.886 363. . .

—0.643 902. . .

—0.763 027. ..

—0.699 450. ..

and so on. The correct value of the sñi/2 decimal is obtained after not less than 24 iterations. But if we use formula (3.7), again starting with n0 = zero, we find successively

— 1.5

—0.886 363 636 364

—0.732 001 693 959

—0.720 818 540 935

—0.720 759 221 726

—0.720 759 220 056

—0.720 759 220 056

so the l2th decimal is correctly obtained with only six iterations in this case.

 

Pive tabular values

When the third differences may not be neglected, more than three tabular values must be used. Taking five consecutive tabular values, y l to y, , we form, as before, the table of differences

A

y B E H

y F K

•f C G I

y D

where A -— y2 — y , H = F — E, etc. If it is the interpolating factor, measured from the central value y, in units of the tabular interval, positively towards y4 , the interpolating formula is

 

 

y —— 3 + 2

 

(B + C)

 

it2

+ 2 F +

 

 n{n’ — l)  

12 K

 

which may also be written

y —— yz + n B + C H + J 

2 12 +

 

  + J + n’

2 24 12

 

(3.8)

 

 

Example 3.e — Consider the following values of the equatorial horizontal parallax of the Moon:

1992 February 27.0 TD 54'36“.125

27.5 54 24.606

28.0 54 15.486

28.5 54 08.694

29.0 54 04.133

The differences in arcseconds are

 

A —- — 11.519

 

£ = +2.399

 

B —— —9.120 If = —0.071

N = +2.328 K —— —0.026

C ——  — 6.792 I = —0.097

G = +2.231

D -- —4.561

 

We see that the third differences (d and J) may not be neglected, unless an accuracy of 0‘.1 is sufficient.

Let us now calculate the Moon’s parallax on February 28 at 3'20'" TD. The tabular interval being 12 hours, we have

 

_ 3‘20” _ 3.333 333

" 2 " 12

 

= +0.277 7778

 

Formula (3.8) then gives

y -— 54’15".486 - 2".117 = 54'13".369

The interpolating factor nq corresponding to an extremum of the function can be obtained by solving the equation

6B + 6C — If — I + 3 rig° (H -F I) + 2gn 3K

 

rig ——

 

K — 12 F (3.9)

 

As before, this may be performed by iteration, firstly putting rig = 0 in the second member. Once rig is found, the corresponding value of the function can be calculated by means of formula (3.8).

The interpolating factor n0 corresponding to a zero of the function may be found

—24y + n 2 {K - 12 F) - 2 3 (H + I) — K

2 (6B + 6C — H — I) (3.10)

where, again, n0 can be found by iteration, starting by putting = 0 in the second

member.

The remark made on pages 26—27 about formula (3.6) holds here too. If the curvature of the function in the considered interval is important, a better method for calculating rt0 is as follows. Calculate

 

M —— K

24

 

N ——         + 

12

 

P ——  F +  

2 2

 

Then the correction to the assumed value of n0 is

(3.11)

and, again, the calculation should be repeated with the new value of nd until ii0 no longer varies.

 

Exe rc i se. — From the following values of the heliocentric latitude of Mercury, find the instant when the latitude was zero, by using formula (3.10).

 

1988 January 25.0 TD

26.0

27.0

28.0

29.0

 

— 1°11’ 21“.23

—0 28 12.31

+0  16 07.02

+ 1 01 00.13

+ 1 45 46.33

 

A n sw er:  Mercury reached the ascending node of its orbit for = —0.361413, that is on 1988 January 26.638 587, or January 26 at 15h20" TD.

Using only the three central values and formula (3.6), one would find =

—0.362 166, a difference of 0.000 753 day, or 1.1 minute, with respect to the previous result.

1. Interpolation cannot be performed on complex (*) quantities directly. These quantities should be converted, in advance, into a single, suitable unit. For instance, angles expressed in degrees, minutes, and seconds should be converted either to degrees and decimals, or to arcseconds, before they can be used for interpolation.

2. IWerpolating times and riglir ascensions. — We draw attention to the fact that times and right ascensions jump to zero when the value of 24 hours is reached. This should be taken into account when interpolation is performed on tabular values. Suppose, for example, that we wish to calculate the right ascension of Mercury for the instant 1992 April 6.2743 TD, using the three following values:

 

1992 April 5.0 TD

6.0

7.0

 

n  =  23‘5l‘56'.04

23 56 28.49

0 01 00.71

 

Not only is it necessary to convert these values to hours and decimals, but the last value should be written as 24h01*00'.71, otherwise the machine will consider that, from April 6.0 to 7.0, the value of n decreases from 23'56‘.... to 0'01°....

We find a similar situation in some other cases. For instance, here is the longitude of the central meridian of the Sun for a few dates:

(*) By definition, a complex number is a number composed of different units, having among them a ratio different from a power of 10. Examples of “complex” quant- ities are 10h29'°55'; 23° 26'44“ ; £, shillings, pence ; yd, ft, inch ; a + b i.

 

1992 June 14.0 UT

15.0

16.0

17.0

 

37.’96

24.72

11.48

358.25

 

It is evident that the variation is approximately — 13.24 degrees per day. Hence, one should not interpolate directly between 11.48 and 358.25. Either the first value should be written as 371°.48, or the second value should be considered as being equal to —1.75 degrees.

3. As much as possible, avoid making an interpolation for | n | > 0.5. In any case, the interpolating factor n should be restricted between the limits — 1 and + 1. This same rule applies to the calculation of an extremum (np) or a zero ( ) of the function. Choose the central value of y in such a way that this is the tabular value which is closest to the extremum or to the zero. Of course, the exact value of nq or  is not known in advance, but an approximate value can be calculated first,

after which the choice of the central value G3 or ) of the function can be changed accordingly.

If the chosen value is too far from the zero or from the extremum, the formulae given in this Chapter for calculating these points will give incorrect or even absurd results. Let us give an example. We know that sin z reaches a maximum for x = 90°. But consider the following sines, with ten decimals:

sin 29° 0.484 809 6202

sin 30‘ 0.500 000 0000

sin 31° 0.515 038 0749

sin 32° 0.529 919 2642

sin 33° 0.544 639 0350

Using the three central values, formula (3.4) gives yp = 1.22827 instead of 1 exactly, and (3.5) yields nz —— +95.35, indicating that the maximum occurs for 31°

+ 95°.35 = 126°.35, instead of 90°.

Using allJve values, formula (3.9) gives rig = +57.30, whence the maximum taking place at 88.°30, from which the value of 0.99348 is found for that maximum. Although these results are much better than those obtained with only three points, they are still unsatisfactory!

/n/ezpo/afion to halves

If the values yt , y2 , y, , y of the function are given for four equally-spaced abscissae x , z2 , x3 , and z4 , then the value of the function for the point exactly half-way between z 2 and z, is easily calculated by means of the following formula, which is valid when the fourth differences of the tabulated values are negligible:

 

16 (3.12)

Example 3.f — Given the following values for the apparent right ascension of the Moon, calculate the right ascension for 11‘00"' TD.

1994 March 25 8‘ TD n = l0‘18“48'732

10 10 23 22.835

12 10 27 57.247

14 10 32 31.983

Converting the minutes and seconds, after 10‘, into seconds, we change the four given data into

y   —— 1128.732 seconds

y   —— 1402.835

yz -- 1677.247

y4 =   1951.983

Formula (3.12) then gives y = 1540.001 seconds = 25*40‘.001, so that the required right ascension is n = 10'25‘40.001.

interpolation with unequally-spaced abscissae : Lagrange’s interpolation  fomiula

When the abscissae (the values of the independent z coordinate) of the given points are not equally spaced, the interpolation formula of Lagrange may be used. (Of course, this formula may also be used when the points are evenly spaced).

This simple formula, developed by the French mathematician J.L. Lagrange (1736—1813), determines a polynomial of degree it — 1 matching n given points exactly. If the given points are z; , y, (i = 1 to n), the formula is, for a given x,

  (3.13)

where

The H means that the product of the fractions should be calculated for all values

j —— 1 to n, except for y = i. That is,

 

3. INTERPOLATION 33

Important : The values x, of the given points must all be different!

The following program in BASIC can be used.

10 DIM X(50), Y (50)

20 PRINT "NUMBER OF GIVEN POINTS = ";

30 INPUT N

40 IF N < 2 OR N > 50 THEN 20

50 PRINT

60 FOR I = 1 TO N

70 PRINT "X, Y FOR POINT No."; I

80 INPUT X(I), Y(I)

90 IF I = 1 THEN 130

100 FOR J = 1 TO I — 1

110 IF X(I) = X(J) THEN PRINT "THIS VALUE OF X HAS ALREADY BEEN USED !" : GOTO 70

120 NEXT I

130 NEXT I

140 PRINT : PRINT "POINT X FOR INTERPOLATION = ";

150 INPUT Z

160 V = 0

170 FOR I = 1 TO N

180 C = 1

190 FOR J = 1 TO N

200 IF J = I THEN 220

210 c = c •(z — x('))›‹x(I) — x‹'))

220 NEXT J

230 v - v + c • v(i)

240 NEXT I

250 PRINT ‘ PRINT "INTERPOLATED VALUE = "; V

260 PRINT : PRINT "STOP (0) OR INTERPOLATION AGAIN (1) ";

270 INPUT A

280 IF A = 0 THEN END

290 IF A = 1 THEN 140

300 GOTO 260

The program first asks how many known values you are going to enter from a table and allows you to input these one at a time. Then it asks you repeatedly for intermediate values of interest, returning the interpolated value for each.

 

A remarkable feature of Lagrange interpolation is that the values entered initially do not have to be in order, or evenly spaced. Accuracy is usually better with uniform spacing, however.

As an exercise, try the program on the following six given points.

x -- angle in degrees y -— sine

29.43 0.491 359 8528

30.97 0.514 589 1926

27.69 0.464 687 5083

28.11 0.471 165 8342

31.58 0.523 688 5653

33.05 0.545 370 7057

Asking for the sine of 30°, you should obtain 0.5 exactly. It is remarkable that, even for the remote values x = 0° and x —— 90°, the Lagrange interpolation formula performed with these six data points yields the still rather good values +0.0000482 and + 1.00007, respectively, the correct values being 0 and + 1 exactly.

The expression (3.13) is a polynomial of degree n — 1, and it is the unique

polynomial of that degree which takes the values yl , y2 , . . , yn for x = x , x2 ,

, on. But Lagrange’s formula has the disadvantage that in itself it gives no indication of the number of points required to secure a desired degree of accuracy. However, when we wish to express the interpolating polynomial explicitly as a function of the variable z rather than making an actual interpolation, the use of Lagrange’s formula is advantageous.

Example 3.g  — Construct the (unique) 3rd-order polynomial passing through the following values:

  z .’ 1 3 4 6

y .- —6 6 9 15

By substituting the given values of x and y into (3. 13), we obtain

 

y = (—6) (z — 3) (z — 4) (z — 6)  

(1 — 3) (1 — 4) (1 — 6)

 

+ (6)

 

(z-1)(i-4)(i-6)

(3- 1)(3-4)(3-6)

 

+ (9)

 

 (z — 1) (z — 3) (z — 6) + (15)  (z — 1) ( — 3) (z — 4) 

 

(4 — 1) (4 — 3) (4 — 6)

which upon simplification reduces to

1 (z 3 — 13z 2 + 69z — 87)

 

(6 — 1) (6 — 3) (6 — 4)

 

 

 

Chapter 4

Cuwe Fitting

In many cases, the result of a large number of observations is a series of points in a graph, each point being defined by an z-value and an y-value. It may be necessary to draw, through the points, the “best” fitting curve.

Several curves can be fitted through a series of points: a straight line, an exponential, a polynomial, a logarithmic curve, and so on.

To avoid indivi- dual judgment, it is necessary to agree on a derinition of a “best fitting” curve. Consider Figure 1 in which the N data points are given by (>i›   <i)›   ( z›   Y   y

, (X N , x N).  The

values of X are sup-

posed to be rigor- ously exact, while the     r-values     are

measured quantities, hence subject to an

error.

For a given Figure I

value of X, say X I , there will be a dif-

ference between the value Y and the corresponding value as determined from the curve C. As indicated in the figure, we denote this difference by D , which is sometimes referred to as deviation, error, or residual and may be positive, negative,

or zero.  Similarly, corresponding  to the value- s >z •    - -• Xp  we obtain  the deviations

D ’2 ’ ’  ' , fi N•

A measure of the “goodness of fit” of the curve C to the given data is provided by the quantity   D   2 + D 2 +   . .      +      N2    *f this is small the fit is good; if it is large the fit is bad. We therefore make the following definition : of all curves

35

 

approximating a given set of data points, the curve having the property that ID,

is a minimum, is called a best fitting curve. The E means "sum of".

A curve having this property is said to nt the data in the least square sense and is called a least square curve.

As has been said above, all values of the independent variable X are supposed to be exact. Of course, it is possible to define another least square curve by considering perpendicular distances from each of the data points to the curve instead of vertical distances; however, this is not used too often.

In this Chapter we will consider principally the case where the best fitting curve is a straight line, a problem called linear regression.

The name "regression" may seem strange, because in the calculation of the best curve nothing "regresses" ! Alt [1] writes:

Die Benennung Regression wurde von Galton (1822-1911) eingeführt, der die Körperlängen von Eltern und Kindern verglich und dabei beobachtete, daß zwar im allgemeinen große Väter große Söhne haben, daß diese Beziehung jedoch nicht immer stimmt, da die Körpergröße der Söhne inn Mittel etwas kleiner ist, als die dcr Väter, umgekehrt aber kleine Eltern inn Mittel etwas grösere Kinder haben. Diesen ‘Rückschlag’ in Richtung auf die Durchschnittgrõße der Bevölkerung bezeichnete er als Regression.

A better term is curve fitting, and in the case of a straight line it is a linear curve fitting.

Linear cuwe fitting lfinear regression}

We wish to calculate the coefficients of the linear equation

y -- ax + b (4.1)

using the least-squares method. The slope o and the y-intercept b can be calculated by means of the formulae

(4.2)

b ——

where N is the number of points. Note that both fracńons have the same denominator. The sign Z indicates the summation. Thus, Dx is the sum of all the z-values, by the sum of all y-values, Ez 2 the sum of the squares of all x-values,

 

Zxy the sum of the products xy of all the couples of values, etc. Note that Sx y is not the same as Ex x Ey (the sum of the products is not the same as the product of the sums), and that (Zx)2 is not the same as F•z 2 (the square of the sum is not the same as the sum of the squares)!

An interesting astronomical application is to find the relation between the intrinsic brightness of a comet and its distance to the Sun. The apparent magnitude m of a comet can generally be represented by a formula of the form

m = g + 5 log A + x log r

Here, A and r are the distances in astronomical units of the comet to the Earth and to the Sun, respectively. The logarithms are to the base 10. The absolute magnitude g and the coefficient   must be deduced from the observations. This can be performed when the magnitude m has been measured during a sufficiently long period. More precisely, the range of r should be sufficiently large. For each value of m, the values of A and r must be deduced from an ephemeris or calculated from orbital elements.

In this case, the unknowns are g and x. The formula above can be written m — 5 log A = c log r + g

which is of the form (4.1), when we write y —— m - 5 log A, and x = log r. The quantity y may be called the “heliocentric" magnitude, because the effect of the variable distance to the Earth has been removed.

Example 4.a — Table 4.A contains visual magnitude estimates m of the periodic comet Wild 2 (1978b), made by John Bortle. The corresponding values of r and A have been calculated from orbital elements [2].

The quantities z and y are used to calculate the sums Ez, Zy, Ex2, and F•zy. We find

N —— 19 Ex = 4.2805

Ey = 192.0400 Xz 2 =

Lzy = 43.7943

whence, by formulae (4.2),

a = 13.67 b —— 7.03

Consequently, the “best" straight line fitting the observations is

y -- 13.67 z  + 7.03

or m — 5 log A = 13.67 log r + 7.03 Hence, for the periodic comet Wild 2 in 1978, we have

m = 7.03 + 5 log A + 13.67 log r

 

TABLE 4.A

1978,

UT

 

r

  * - log r y -

tn - 5 log A

Febr. 4.01 11.4 1.987 1.249 0.2982 10.92

5.00 11.5 1.981 1.252 0.2969 11.01

9.02 11.5 1.958 1.266 0.2918 10.99

10.02 11.3 1.952 1.270 0.2905 10.78

25.03 11.5 1.865 1.335 0.2707 10.87

March 7.07 11.5 1.809 1.382 0.2574 10.80

14.03 11.5 1,772 1.415 0.2485 10.75

30.05 11.0 1.693 1.487 0.2287 10.14

April 3.05 11.1 1.674 1.504 0.2238 10.21

10.06 10.9 1.643 1.532 0.2156 9.97

26.07 10.7 1.582 1.592 0.1992 9.69

May 1.08 10.6 1.566 1.610 0.1948 9.57

3.07 10.7 1.560 1.617 0.1931 9.66

8.07 10.7 1.545 1.634 0.1889 9.63

26.09 10.8 1.507 1.696 0.1781 9.65

28.09 10.6 1.504 1.703 0.1772 9.44

29.09 10.6 1.503 1.707 0.1770 9.44

June 2.10 10.5 1.498 1.721 0.1755 9.32

6.09 10.4 1.495 1.736 0.1746 9.20

Coefficient of Correlation

A correlation coefficient is a statistical measure of the degree to which two variables are related to each other. In the case of a linear equation, the coefficient of correlation is

r -— N Y•x y -  Dz Ey (4.3)

This coefficient is always between + 1 and — 1. A value of + 1 or —1 would indicate that the two variables are totally correlated; it would denote a perfect linear relationship, all the points representing paired values of z and y falling exactly on the straight line representing this relationship. If r = + I, an increase of z corresponds to an increase of y (Figure 2). If r —— — 1, there is again a perfect linear relationship, but y decreases when x increases (see Figure 3).

 

F-igure 2

Perfect linear relationship ; positive correlation

F-igure 3

Perfect linear relationship ; negative correlation

Figure 4

No correlation

Figure 5

Some correlation

When r is zero, there is no relationship between x and y (Figure 4). In practice, however, when there is no relationship, one may find that r is not exactly zero, due to fortuitous coincidences that generally occur except for an infinite number of points.

When | r | is between 0 and 1, there is a trend between x and y, although there is no strict relationship (Fig. 5). Here, again, / there is actually a strict relationship

 

between the two variables, the calculation may give a value of r that is not exactly equal to + 1 or to --1,  by reason of inaccuracies  inherent to all measures.

Note that r is a dimensionless quantity: it does not depend on the units employed. The sign of r only tells us whether y is increasing or decreasing when x increases. The important fact is not the sign, but the magnitude of r, because it is this magnitude which indicates how well the linear approximation is.

It must be emphasized that the computed value of r in an y case measures the degree of relationship relative to the assumed type of function, namely the linear equation. Thus, if the value of r appears to be nearly zero, it means that there is almost no linear correlation between the variables. However, it does not necessarily mean that there is no correlation at

all, since there may actually be a high non-linear correlation between the variables. As an example, consider the seven points

 

Formula (4.3) yields r -- zero, al- though the points lie exactly on the parabola y = 2 -- 2z -- x 2 (Fig. 6).

We must be careful not to improperly deduce causation from correlation. A high correlation coef- ficient (that is, near + 1 or --1) does not necessarily indicate a direct, physical dependence of the variables. Thus, if we consider a sufficiently large number of administrative

territories, one can find a high correlation

 

Figure 6

between the number of beds in

 

x

the

 

psychiatric hospitals and the number of television receivers of each territory. A high

mathematical correlation, indeed, but a physical nonsense.

Example 4.b -- Table 4.B gives, for each of the twenty-two sunspot maxima which have occurred from 1761 to 1989, the time interval z, in months,

since the previous sunspot minimum, and the height y of the maximum (highest smoothed monthly mean). We find

Ez = 1120; T•y --- 2578.9; Ez 2 = 60608 ; Ey 2 = 340225.91;

Lx y -- 122337.1 ; N -- 22 ; and then, by formulae (4.2) and (4.1),

y = 244.18 -- 2.49a (4.4)

 

 

200 -

150  -

100 -

50 -

 

Rise (mon ths)

10 20 30 40 50 60 70 80

Fipure 7

T A B L E 4. B

 

Epoch of maximum Epoch of

imum

1761 June 73 90.4 1884 Jan. 61 78.1

1769 Oct. 38 125.3 1893 Aug. 42 89.5

1778 May 35 161.8 1905 Oct. 4S 63.9

1787 Nov. 42 143.4 1917 Aug. 50 112.1

1804 Dec. 78 52.5 1928 June 62 82.0

1816 March 68 50.8 1937 May 44 119.8

1829 June 74 71.5 1947 July 3S 161.2

1837 Feb. 42 152.8 1957 Nov. 43 208.4

1847 Nov. 52 131.3 1969 Feb. 54 111.6

1860 July 54 98.5 1979 Nov. 44 167.1

1870 July 39 144.8 1989 Oct. 37 162.1

 

Equation (4.4) represents the best straight line fitting the given 22 points. These

points and the line are shown in Figure 7.

From formula (4.3) we find r -— —0.767. This shows that there exists an evident trend to connection, and the negative sign of r indicates that the correlation between x and y is negative: the longer the duration of the rise from a minimum to the next maximum of the sunspot activity, the lower this maximum generally is.

Note that here, as in all statistic studies, the sample must be sufficiently large in order to give a meaningful result. A correlation coefficient close to + 1 or to — 1 has no physical meaning if it is based on too small a number of cases. With too few cases the correlation coefficient can accidentally be quite large.

TA BLE 4. C

year x y year x y year x y year x y

1901 2.7 700 1925 44.3 1075 1949 134.7 521 1973 38.0 690

1902 5.0 762 1926 63.9 896 1950 83.9 951 1974 34.5 1039

1903 24.4 854 1927 69.0 837 1951 69.4 878 1975 15.5 734

1904 42.0 663 1928 77.8 882 1952 31.5 926 1976 12.6 541

1905 63.5 912 1929 64.9 688 1953 13.9 557 1977 27.5 855

1906 53.8 821 1930 35.7 953 1954 4.4 741 1978 92.5 767

1907 62.0 622 1931 21.2 858 1955 38.0 616 1979 155.4 839

1908 48.5 678 1932 11.1 858 1956 141.7 795 1980 154.6 913

1909 43.9 842 1933 5.7 738 1957 190.2 801 1981 140.5 1016

1910 18.6 990 1934 8.7 707 1958 184.8 834 1982 115.9 800

1911 5.7 741 1935 36.1 916 1959 159.0 560 1983 66.6 689

1912 3.6 941 1936 79.7 763 1960 112.3 962 1984 45.9 931

1913 1.4 801 1937 114.4 900 1961 53.9 903 1985 17.9 758

1914 9.6 877 1938 109.6 711 1962 37.5 862 1986 13.4 946

1915 47.4 910 1939 88.8 928 1963 27.9 713 1987 29.2 908

1916 57.1 1054 1940 67.8 837 1964 10.2 785 1988 1£D.2 1005

1917 103.9 851 1941 47.5 744 1965 15.1 1073 1989 157.6 639

1918 80.6 848 1942 30.6 841 1966 47.0 1054 1990 142.6 759

1919 63.6 980 1943 16.3 738 1967 93.8 707 1991 145.7 794

1920 37.6 760 1944 9.6 766 1968 105.9 776 1992 94.3 916

1921 26.1 417 1945 33.2 745 1969 105.5 776 1993 54.6 857

1922 14.2 938 1946 92.6 861 1970 104.5 727 1994 29.9 894

1923 5.8 917 1947 151.6 640 1971 66.6 691 1995 17.5 763

1924 16.7 849 1948 136.3 792 1972 68.9 710 1996 8.6 745

 

As an exercise, show that there is no correlation between the rainfall at the Uccle Observatory, Belgium, and the sunspot activity, using the data of Table 4.C, where

z = yearly mean of the definitive Zürich sunspot numbers,

y —— total annual rainfall at Uccle, in millimeters.

Answer: The correlation coefficient is r = —0.054, which shows that there is no significant correlation between x and y.

Quadratic cuwe fitting

Suppose that we wish to draw, through a set of N given points (x, y), the best quadratic function

y —— az 2 + bx + c

This is a parabola with vertical axis. Let

R —— ZX  3

S = I ¡z 4

 

V = I¡Wz),

D -- NQS + 2 PQIt — Q’ — P 2S — NR’

Then we have

a N V + PRT + PQU - @ 2'r — P’V — NRU

b NSU -1- PQV + QRT - Q’U — PST - NRV

D

 

(4.5)

(4.6)

 

QST + QRU + PRV — 0 2 Y — PS U - 2t 2 P

C D

 

General curre fitting {multiple linear regression j

The principle of the best fitting straight line can be extended to other functions

and with more than two unknown linear coefficients.

Let us consider the case of a linear combination of three functions. Suppose that we know that

y -— ay (•) + b f {x) + c f {x)

where ft, f , and f 2 art    three known functions of z, but that the coefficients o, b, and c are not known. Suppose, moreover, that the value of y is known for at least three values of x. Then the coefficients a, b, c can be found as follows.

Calculate the sums

M —— D/ 2

R —— Z/ 2

S -- E f  f

T —— Z f

 

Then

 

D -— MR T + 2 POS -  MS’ — R Q 2 — r P 2

 

 

 

b ——

 

U {S Q — PT)  +

 

V(MT - @ 2) + W(P Q — MS)

D

 

(4.7)

 

_ U (PS — RQ) + V{PQ - MS) + W{MR —P 2)

C D

Example 4. c — We know that y is of the form

y = â sin x + b sin L + c sin 3z and that y takes the following values:

 

x (degrees) y

3 0.0433

20 0.2532

34 0.3386

50 0.3560

75 0.4983

88 0.7577

111 1.4585

129 1.8628

143 1.8264

160 1.2431 0°

183 —0.2043

200 —1.2431

218 —1.8422

230 —1.8726

248 —1.4889

269 —0.8372

290 —0.4377

303 —0.3640

320 —0.3508

344 —0.2126

Find the values of the coefficients o, I›, c.

We leave it as an exercise to the reader. The fiinction is y = 1.2 sin z — 0.77 sin 2z + 0.39 sin 3z

and is illustrated in the Figure above. The reader will not find 1.2, —0.77, and

+0.39 exactly, because in the table the values of y are given with only Your decimals.

Let us consider the special case y = cx° + bx + c. Here we have

 

resulting in T -- N (the number of given points) and @ = it. The formulae (4.7) then reduce to (4.5) and (4.6), with other notations.

As another special case, consider y —— n/(x) with only one unknown coefficient. The latter is easily found from

(4.8)

 

 

Example 4.d — y -- n (x > -- 0)

Find a for the best fitting curve through the following points:

x : 0 1 2 3 4 5

y : 0 1.2 1.4 1.7 2.1 2.2

Here, f{x) = , so EQ° is simply the sum of the x-values. Formula (4.8) gives

a —— 15.2437 

15

so the required function is

y = 1.016 Mz

RE FE REN CES

1. Helmut Alt, Angewandte Mathematik, Finanz-Mathematik, Statistik, Informatik für UPN-Rechner, p. 125 (Vieweg, Braunschweig, 1979).

2. International Astronomical Union Circular No. 3177 (1978 February 24).

 

Chapter 5 Iteration

Iteration (from the Latin iterare -- to repeat) is a method consisting of repeating a calculation several times, until the value of an unknown quantity is obtained. Generally, after each repetition of the calculation, one obtains a result that is closer to the exact solution. We have already seen the use of iteration in Chapter 3, for solving equations (3.6), (3.7), (3.9), (3.10), and (3.11).

Iteration is used, for instance, when there is no method for calculating the unknown quantity directly in an easy way. Examples are:

— solving the equation of the fifth degree z + 17a — 8 = 0 ;

— the calculation of the times of beginning and end of a solar eclipse, or of an occultation of a star by the Moon, for a given place at the Earth’s surface;

— the equation of Kepler E —— M + e min E (see Chapter 30), where E is the

unknown quantity.

To perform an iteration, one must start with an appTOXimate value for the unknown quantity, and use must be made of a formula, or of a set of formulae, in order to obtain a better value for the unknown. This process is then repeated (iteration) until the required accuracy is reached.

A classical example is the calculation of the square root of a number. Of course, this method has nowadays lost its interest (except in special cases), because all pocket calculators and all program languages already possess the function for SQR. The calculation proceeds as follows.

Let N be the number whose square root is requested. Starl with an approximate value ii for this root; if none is known, the value 1 can be used. Divide N by n, and take the arithmetic mean of the quotient and n. The result is a better value for the square root. In other words, a better value is given by {n + Nln)l2. Then the calculation must be repeated.

47

 

 

Example 5.a — Calculate to eight decimals.

We know that 12 x 12 = 144, so that 12 is an approximate value of the square root of 159. We divide 159 by 12, and find the quotient 13.25. The arithmetic mean of 12 and 13.25 is 12.625, which is a better value for the required square root.

We now divide 159 by 12.625; the quotient is 12.59406. The mean of 12.625 (the previous result) and 12.59406 is 12.60953, which is a still better value for the square root.

In that way, we find successively

12 = starting value 12.625 000 00 12.609 529 70 12.609 520 21 12.609 520 21

As you see, 12.609 52021 yields 12.609 520 21 again, so this is the required square root of 159.

Example 5.b — Calculate the (only) real root of the equation

x   + 17a - 8 = 0 (5.1)

Because there is no method or formula for the direct calculation of the roots of an equation of the fifth degree, we will have recourse to the iteration procedure. In equation (5.1) we put x’ in the second member and solve for x ; this gives

 

g y 5

 

(5.2)

 

The unknown quantity is now present in the right-hand member too, but that does not matter, as we shall see. We start by letting z = 0 in the right-hand member. Formula (5.2) then yields

z = 8/17 = 0.470 588 235

which is already a better value than x = 0. We now put the value z = 0.470 588 235 in the right-hand member, and now the formula gives z = 0.469 230 684. After four more iterations, we obtain the definitive value, namely z = 0.469 249 878.

The iteration process is not always without problems, however, as it is shown in the following example.

 

 

Example 5.c — Consider the equation z 5 + 3z — 8 = 0.

As in the preceding example, we put z in the right-hand member, and we obtain

8 —* 5

 

3

If we start, here again, with z = 0, we obtain successively

0.0000 (starting value)

2.6667

—42.2826

45 049 099

—6.18 x 10’ 7

etc .. . .

and so the method does not work in this case! The successive results diverge; in absolute value they grow bigger and bigger. They go “in the wrong direction”.

Why did the method work in Example 5.b, but not in Example 5. c* When z lies between 0 and 1, then x5 too is between 0 and 1. Moreover, z is then smaller than x. This is the reason why in Example 5.b the results of the successive lterations converge to a well-defined value, the root of the equation. This root lies between 0 and 1.

But, as we shall see, the root of the equation in Example 5.c is larger than 1. When z b    1, then x’     > x > 1, and a small increase of z gives rise to a much larger increase of z'. For z = 2, we have already z' = 32.

Consequently, the iteration procedure, performed in the same way as in Example 5.b, cannot converge to the required result: the successive values diverge. However, it is possible to get the answer, on the condition that we write the iteration formula in another form.

Example S.d — Let us again consider the equation z 5 + 3s - 8 = 0, but now we take into account the fact that the root is larger than 1, and hence

that z’ z. For this reason, we do nnr put z 5 in the right-hand member here.

Instead, we keep z 5 in the first member, so the equation becomes

z’   = 8 — 3z or  

Starting again with z = 0, we obtain the required root aber 14 iterations, namely, x = 1.321 785 627.

 

5O ASTRONOMICAL ALGORITHMS

In example 5.b, we searched for the root of the equation

5 + 17z — 8 = 0

However, we can write this equation as

z (x 4 + 17)  = 8, whence x —- 8

z’ + 17

We now can use this latter formula instead of (5.2). As an exercise, solve this equation by iteration; you should obtain the same result as in Example 5.b.

If we wish to work similarly for the equation of Example 5.c, we obtain the iteration formula

8

 

z + 3

If we again start by putting the value x =0 in the right-hand member, we obtain z = 8/3 = 2.666.... But then comes the surprise: after a few iterations, the successive results jump unceasingly from 2.666 223 459 to 0.149 436 927, and back. As you see, the iteration method does not succeed in all cases; much depends on the form of the iteration formula.

As another example, consider the equation sin p = 3 cos p. Putting p = 0° in the right member yields sin p = 3, an impossibility. Putting, instead, e = 90° in the second member gives sin p = 0, whence p = 0‘, which brings us back to the first case.

But if we write the equation as cos ‹,o = (sin e )/3 then, starting with p = 0°, we reach the solution p = 71°.565051 after a few iterations.

Or consider the equation sin p = cos 2p. Evidently, the solution is p = 30°, because sin 30‘   = cos 60°. If we start by putting p = 29° in the second member of that equation, the results of the successive iterations diverge. If, however, we write the equation the other way, namely, cos *e = sirl ‹,r, then the successive results converge!

As a further illustration of the iteration procedure, let us consider Newton’s method for searching the solution of an equation with one unknown by successive approximations.

Let /(z) be a function of x, and we want io find for what value of x that functiOn 1s zero. Let /’(x) be the derivative func- tion of /(z). If x, is an assumed vaiue for the root X, then calculate the value y of the

 

function /(z), and the value I i Of the derivative f'{x), for that value of z. The value yi is the slope of the tangent to the curve at the point z , y — see the Figure on the preceding page. Then, a better value for the unknown quantity is given by

 

The calculation is then repeated using this new value of z, until the final value X is reached.

In this procedure, the choice of a good starting value for z can be a problem. For example, for the equation

x’ — 3s — 8 = 0

the derivative function is 5x 4 — 3 and, if we start with x = 0, we obtain oscillating values, as shown in the box at right.

The reason is that the function reaches a maximum value for x = —0.88, so that the tangents on both sides of that point have slopes in opposite directions.

But if we start with x —- 1, then the correct value (to 9 decimal places) is reached after 11 iterations, as shown in the second box.

 

—2.666 666 667

—2.126 929 222

— 1.672 392 941

— 1.227 532 073

—0.376 965 299

—2.749 036 974

—2.194 266 642

— 1.751 201 846

— 1.293 218 529

—0.588 844 800

—3. 216 865 068

—2.572 967 056

—2.049 930 312

— 1.603 831 481

— 1.145 086 796

 

rest on “smaller than”

+6.000 000 000

+ 4.803 458 391

+ 3.850 111 311

When an iteration procedure is used, one should — as + 3.095 824 107

has been mentioned above — repeat the calculation until +2.510 476 381

the result no longer varies. In other words, as long as the + 2.080 081 724

last result differs from the previous one, a new iteration must be performed. But here we are faced with a small problem, due to the fact that the computer does not calculate “exacfly”.

Consider the following equation of the third degree

s’ + 3s — lY = 0

which appears in the calculation of the motion in a parabolic orbit (see Chapter 34). iY is a given constant, while s is the unknown quantity. This equation can very easily be solved by iteration. Start from an.y value; a good choice is s = 0. Then a better value for s is

  2s’   + W 

3 (s2 +   1)

 

52 ASTRONOMICAL ALGORITHMS

After some iterations the correct value of s is obtained. Take, for instance, the case lY = 0.9. The calculation performed on the HP-85 microcomputer gives the following successive results:

0.000 000 000 000

0.300 000 000 000

0.291 743 119 266

0.291724 443 641

0.291 724 443 546

0.291 724 443 548

0.291 724 443 548

and hence the exact value (with twelve significant digits) is 0.291724 443 548. But if we repeat the calculation on the same machine for W = 1.5, we have a surprise: the machine does not stop and finds successively:

0.000 000 000 000

0.500 000 000 000

0.466 666 666 667

0.466 220 600 162

0.466 220 523 909

0.466 220 523 911

0.466 220 523 910

0.466 220 523 908

0.466 220 523 911

0.466 220 523 910

0.466 220 523 908

and forever again ...911, ... 910, ... 908. However, we tried this calculation (for W = 1.5) with two other programming languages, and the iteration procedure did converge; but then it did not converge for other values of the constant W.

A remedy for this trouble consists of testing on “smaller than" instead of on “equal to*. In other words, let the iteration process stop when the difference between the new value of s and the previous one is, in nbrnfme in/ur, less than a given quantity, for instance 10*'°.

The binary search

There is a procedure which is absolutely foolproof, because it can neither stall nor diverge, and always converges in a fixed amount of time to the most exact value of the root the programming language is capable. The method does not try to find successively better values of the root. Instead, it just uses a binary search to locate the correct value of the root.

 

Let us explain the procedure by reconsidering the equation of Example 5.b, namely x’ + 17a — 8 = 0.

For x = 0 and z = 1, the first member of this equation takes the values —8 and + 10, respectively. So we know that the root lies between 0 and 1 (*).

Let us now try z = 0.5, which is the arithmetical mean of 0 and 1. For z = 0.5, the function takes the value +0.53125, which has the opposite sign of the function’s value for x = 0. So we now know that the root is between 0 and 0.5.

We now try z = 0.25, which is the arithmetical mean of 0 and 0.5. And so on.

After each step, the interval in which the root necessarily must be, is halved. After 32 steps the value of the root is known with nine exact decimals. (In Example 5.b, the same accuracy was obtained after only six steps. But, as we already pointed out, the binary search is a method which is absolutely safe, and ii can be used when the “ordinary" iteration procedure is likely to fail).

With the binary search, one knows in advance the accuracy after n steps: it is the initial interval divided by 2".

For the example given above, the program in BASIC can be written as follows. Line 60 is not actually needed; it has been included to show the successively better values of x.

10 DEF FNA(X) = X (X“4 + 17) — 8

20 X1 = 0 Y1 = FNA(X1)

30 X2 = 1 : Y2 = FNA(X2)

40 FOR I = 1 TO 33 50 X = (X1 + X2)/2

60 PRINT J, X

70 Y = FNA(X)

80 IF Y = 0 THEN PRINT J, X : END

90 IF Y • YI > 0 THEN 120 loo X2 = X : Y2 = Y

110 GOTO 130

120 X1 = X : Yl = Y

130 NEXT J

140 END

(*) This is true only if the function is continuous in the interval considered, From the fact that tan 86° > 0 and tan 93° < 0, we may nor conclude that tan z becomes zero for a value of x between 86° and 93°.

 

 

 

Chapter 6

Sorting Numbers

Computers are more than calculating machines. They can store and handle data. One example of handling is to rearrange or sort data. Sorting is a function with almost universal application for all users of computers. In astronomy, examples are: sorting stars by right ascension, or by declination; sorting times chronologically; sorting minor planets by increasing semimajor axis, or sorting their names alphabetically. Different algorithms are available to perform sorting. In this Chapter we shall give three methods, provide the BASIC programs, and compare the calculittlOfl times.

One of the simplest sorting algorithms is given in Table 6.A under the name “SIMPLE SORT”. We start from N numbers X(1), X(2), . . , X(N). The values of these elements are arbitrary, and the same value may occur more than once.

After the execution of the routine the numbers X(i) are sorted in increasing order. If one wants them in decreasing order, one should, on line 120, replace > = by < = ; or, alternatively, one may replace X(?) by —X(?).

At each step, two elements are permuted. Successively, the smallest element is placed in front (for f = 1), then the second, and so on, up to N — 1. Note that on line 100 the index i should go till N - 1, not till N.

This method is also called “straight insertion”. The time needed to sort N numbers depends, of course, on the type of the computer and on the program language, but in any case the sorting time wih approximately be propouional to N2. This means that the method is unsuitable for large N.

The method we called “BETTER* is somewhat faster, but again the sorting time is approximately proportional to N 2. ItS prinGlple is simple: fifld the smallest element, and place it in front by permuting two elements.

When the set of ‹rata to be sorted is large, a much better method is “QUICKSORT”, which was invented by C.A.R. Hoare. The program itself is longer, but the computer time is considerably shorter. Moreover, when N is sufficiently large, the computer time is approximately proportional to N, not to N . (In fact, it is nearly proportional to N log N).

55

 

TA BLE 6. A

Three sorting programs in BA SIC

SIMPLE SORT QU IC KSORT

i00 FOR I = 1 TO N—1 100 DIM L(30), R(30)

110 FOR J = I+1 TO N 110 S = 1 L(1) = 1 : R(1) = N

120 IF X(J) > = X(I) THEN 160 120 L = L(S) : R = R(S)

130 A = X(I) 130 S = S — 1

140 X(I) = X(J) 140 I = L J = R

150 X(J) = A 150 V = X(INT((L + R)/2))

160 NEXT J 160 IF X(I) > = V THEN 190

170 NEXT 1 170 I = I + 1

180 GOTO 160

190 IF V > = X(J) THEN 220

200 I = 1 - 1

210 GOTO 190

220 lF I > I THEN 250

230 W = X(I) : X(I) = X(I)

BETTER

240 X(J) = W

I = I + 1 I = J — 1

100 FOR I = 1 TO N—I 250 IF I < = I THEN 160

110 M = X(I) 26D IF J — L < R - I THEN 320

120 K = 1 270 IN L > = I THEN 300

130 FOR I = I+1 TO N 280 S = S + 1

140 IF X(I) < M THEN 290 L(S) = L : R(S) = J

M = X(J) : K = J 300 L = I

150 NEXT J 310 GOTO 360

160 A = X(I) : X(I) = M : X(K) = A 320 IF I > = R  THEN 350

170 NEXT 1 330

340 S = S + 1

L(S) = I : R(S) = R

350 R = I

360 IF L < R THEN 140

370 IF S < > 0 THEN 120

The QUICKSORT sorting technique needs two small auxiliary one-dimensional arrays: L(M) and R(M). M is at least the smallest integer larger than log2 N. A value of M = 30 is certainly sufficient for all practical purposes.

In Table 6.B we mention the calculation times for some values of N on the HP- 85 microcomputer for the three programs mentioned in Table 6. A. As we akeady said, the times will be different on other computers, btlt lfl any case we find that these times increase rapidly for larger values of N, except for the QUICKSORT algorithm.

 

6.   SORTING   NUMBERS 57

T A B LE 6. B

Calculation times  (in seconds)  a I the three sor/‘/oy algorithms on the I-IP-B5 microcomputer

N SIMPLE SORT BETTER QUICKSORT

10

20

40

60

80

loo

150

200

300

too

1000

1500

2000 0.73

3.92

15.4

38.0

63.8

104.3

254

453

1002 0.51

2.11

7.81

17.0

29.1 44.6 98.6 174 387 0.70

1.84

4.43

8.63

11.3

14.6

24.1

32.9

56.7

97.7 218 342 472

To gain some idea of the calculation speeds for larger values of N, we did appeal to a faster computer; the programs were written in FORTRAN and were compiled. The results are given in Table 6.C. The superiority of QUICKSORT is conspicuous here. For N —- 300, the calculation time is still 15 4 of that with BETTER (Table 6.B); but for 15000 numbers it is only one third of 1 per cent!

T A B LE 6. C

Calculation times (in secondsl o f the (hree sorting algorithms on a “big“ computer

N SIMPLE SORT BETTER QUICKSOR T

1 000 13 10 < 1

2 000 51 40

 

3 000 114 90

 

4 000 206 159

 

5 000 321 249 2

10 000 1272 994 S

15 000 2236 7

20 000 TO

25 000 2

30 000 IJ

 

In some cases there is even no need to write a program. For instance, the old TRS-80 Model I contained a built-in function which sorted 1000 numbers in 9 seconds, and 8000 numbers in 83 seconds. It appears that the sorting time is approximately proportional to N here, not to N 2, so probably the QUICKSORT method was used.

To conclude, we can recommend the “straight insertion” (SIMPLE SORT) if the set of data to be sorted is not too large, for instance for V < 200. For larger sets it is well worth while to use QUICKSORT.

Besides numerical data, often strings (names) are to be sorted, such as X$(1) = “Ceres”, X$(2) = “Pallas”, etc. Each character has its own value. The complete list with all signs constitutes the so-called ASCII table, a part of which is given in Table 6.D. [ASCH = “American Standard Code for Information Interchange”.]

TA B LE 6. D

Visible ASCII Characters

After each character its decimal code is piven

space 32 8 56 P 80 h 104

! 33 9 57 Q 81 i 105

" 34 58 R 82 j 106

# 35 , 59 S 83 k 107

$ 36 < 60 T 84 1 108

P• 37 = 61 U 85 m 109

& 38 > 62 V 86 n 110

39 ? 63 W 87 o 111

( 40 @ 64 X 88 p 112

) 41 A 65 Y 89 q 113

* 42 B 66 Z 90 r 114

+ 43 C 67 { 91 s 115

44 D 68 \ 92 t 116

- 45 E 69 ] 93 u 117

  F 70 “ 94 v 118

/ 47 G 71 95 w 119

0 48 H 72 96 x 120

1 49 1 73 a 97 y 121

2 50 J 74 b 98 z 122

3 51 K 75 c 99 ( 123

4 52 L 76 d 100 1 124

5 53 M 77 e 101 } 125

6 54

7 55 N 0 78

79 f g 102

103 126

 

Chapter 7

Julian Day

In this Chapter we give a method for converting a date, given in the Julian or in the Gregorian calendar, into the corresponding Julian Day number (JD), or vice versa.

General remarks

The Julian Day number or, more simply, the Julian Day (*) (JD) is a continuous count of days and fractions thereof from the beginning of ihe year —4712. By tradition, the Julian Day begins at Greenwich mean noon, that is, at 12' Universal Time. If the JD corresponds to an instant measured in the uniform scale of Dynamical Time, the expression Julian Ephemeris Day (JDE) (**) is often used. For example,

1977 April 26.4 UT   =   JD 2443 259.9

1977 April 26.4 TD   =   JDE 2443 259.9

In the methods described below, the Gregorian calendar reform is taken ifltO account. Thus, the day following 1582 October 4 (Julian calendar) is 1582 October 15 (Gregorian calendar).

 

(*)

(**)

 

In many books we read “Julian Date” instead of “Julian Day”. A date consists of a year number, a month, and a day of the month, in any calendar. For me, a Julian date is a date in the Julian calendar, just as a Gregorian dare refers to the Gregorian calendar. The JD has nothing to do with the Julian calendar.

Not JED as it is sometimes written. The “E” is a sort of index appended to ‘JD": JDE = (Julian Day)Ept,p gg,. The name Ephemer“u: comes from ‘Ephemeris Time", the old name for the uniform Dynamical Time. The abbreviation JDE has been used in the Minor Planet Circulars until 1991 inclusively, when it was

changed to JDT. Here the “T” means Terrestrial Dynamical Time (see Chapter 10). But what if we want to refer to the Barycentric Dynamical Time, or in cases

where the very small difference between TDT and TDB does nor matters For this reason, I prefer to continue to use the abbreviation JDE.

59

 

The Gregorian calendar was not at once officially adopted by all countries. This should be kept in mind when making historical research. In Great Britain, for instance, the change was made as late as in 1752, and in Turkey not before 1927.

The Julian calendar was established in the Roman Empire by Julius Caesar in the year —45 and reached its final form about the year +8. Nevertheless, we shall follow the astronomers’ practice consisting of extrapolating the Julian calendar indefinitely to the past. In this system we can speak, for instance, of the solar eclipse of August 28 of the year — 1203, although at that remote time the Roman Empire was not yet founded and the month of August was still to be conceived!

There is a disagreement between astronomers and historians about how to count the years preceding the year 1. In this book, the “B.C.” years are counted astronomically. Thus, the year before the year + I is the year zero, and the year preceding the latter is the year — 1. The year which the historians call 585 B. C. is actually the year —584. (Do nor use the mention “B.C.” when using negative years! “ —584 B.C.”, for instance, is incorrect.)

The astronomical counting of the negative years is the only one suitable for arithmetical purposes. For example, in the historical practice of counting, the rule of divisibility by 4 revealing the Julian leap years no longer exists; these years are, indeed, 1, 5, 9, 13, .. . B.C. In the astronomical sequence, however, these leap years are called 0, —4, —8, — 12 . .. , and the rule of divisibility by 4 subsists.

We will indicate by INT(z) the greatest integer less than or equai to x. For example:

INT(7/4) = 1 INT(5.02) = 5

INT(8/4) = 2 INT(5.9999) = 5

There may be a problem with negative numbers. In most programming languages, INT(x) has the definition given above. In that case we have, for instance, INT(—7.83) = —8, because —7 is indeed larger than —7.83.

But in other languages, such as FORTRAN 77, INT is the integer part of the wri’tten number, that is, the part of the number that precedes the decimal point. In that case, INT(—7.83) is —7. This is called truncation, and some programming languages have both functions: INT(z) having the first of the above-mentioned meanings, and TRUNC(x) or FIX(z).

Hence, take care when using the INT function for negative numbers. (For positive numbers, both meanings yield the same result). In the formulae given in this Chapter, however, the argument of the INT function is always positive.

Calculation of the JD

The following method is valid for positive as well as for negative years, but not for negative JD.

Let Y be the year, M the month number (1 for January, 2 for February, etc. , to 12 for December), and D the day of the month (with decimals, if any) of the given calendar date.

 

• If M > 2, leave Y and M unchanged.

If M —— I or 2, replace Y by Y - 1, and ñf by M + 12.

In other words, if the date is in January or February, it is considered to be in the 13th or 14th month of the preceding year.

• In the Gregorian calendar, calculate

 

A —— INT

In the Julian calendar, take B —— 0.

• The required Julian Day is then

 

B —— 2 — A + INT   A

 

 

JD = INT (365.25 (r + 4716)) + INT (30. 6001 (3f + l))

+ D + B — 1524.5

 

(7.1)

 

The number 30.6 (instead of 30.6001) will give the correct result, but 30.6001 is used so that the proper integer will always be obtained. [In fact, instead of 30.6001, one may use 30.601, or even 30.61.) For instance, 5 times 30.6 gives 153 exactly. However, most computer languages would not represent 30.6 exactly — see in Chapter 2 what we said about BCD — and hence might give a result of 152.9999998 instead, whose integer part is 152. The calculated ID would then be incorrect.

In formula (7.1), the constant 4716 has been added to the argument of the first INT function, in order to avoid trouble for negative years.

Example 7. a — Calculate the JD corresponding to 1957 October 4.81, the time of launch of Sputnik 1.

Here we have Y -- 1957, M —— 10, D — 4.81.

Because If > 2, we leave r and M unchanged.

The date is in the Gregorian calendar, so we calculate

A -— INT(1957/ 100) = INT(19.57) = 19

B —— 2 — 19 + INT(19/4) = 2 — 19 + 4 = — 13

JD = INT(365.25 x 6673) + INT(30.6001 x 11) + 4.81 — 13 — 1524.5

JD = 2436 116.31

Example 7.b — Calculate the JD corresponding to January 27 at 12' of the year 333.

Because 3f = 1, we have F = 333 — 1 = 332 and M = 1 12 = 13. Because the date is in the Julian calendar, we have B —— 0.

JD = INT(365.25 x 5048) + INT(30.6001 x 14) + 27.5 + 0 — 1524.5

JD = 1842 713.0

 

The following list gives the JD corresponding to some calendar dates. These data may be useful for testing a program.

2000 Jan. 1.5 2451545.0 1600 Dec. 31.0 2305 812.5

1999 Jan. 1.0 2451179.5 837 Apr. 10.3 2026 871.8

1987 Jan. 27.0 2446 822.5 —123 Dec. 31.0 1676496.5

1987 June 19.5 2446966.0 —122 Jan. 1.0 1676497.5

1988 Jan. 27.0 2447 187.5 —1000 July 12.5 1356001.0

1988 June 19.5 2447 332.0 —1000 Feb. 29.0 1355 866.5

1900 Jan. 1.0 2415020.5 —1001 Aug. 17.9 1355 671.4

1600 Jan. 1.0 2305447.5 —4712 Jan. 1.5 0.0

If one is interested only in dates between 1900 March 1 and 2100 February 28, then in formula (7.1) we have B -- —13.

In some applications it is needed to know the lulian Day JD corresponding to January 0.0 of a given year. This is the same as December 31.0 of the preceding year. For a year in the Gregorian calendar, this can be calculated as follows.

Y —— year — 1

JD0 = INT(365.25 r) — A + INT   A + 1721 424.5

For the years 1901 to 2099 inclusively, this reduces io

JD = 1721 409.5 + INT(365. 25 x  (year — 1))

 

The Modified Julian Day (MID) sometimes appears in modem work, for instance when mentioning orbital elements of artificial satellites. Contrary to the JD, the Modified Julian Day begins at Greenwich mean midniglu. It is equal to

MID = JD — 2400 000.5

and therefore MID = 0.0 corresponds to 1858 November 17 at 0‘ UT.

Calculation of the Calendar Date from the JD

The following method is valid for positive as well as for negative years, but not for negative Julian Day numbers.

Add 0.5 to the JD, and let Z be the integer part, and I the fractional (decimal) part of the result.

If Z < 2299 161, take A —— Z.

If Z is equal to or larger than 2291 161, calculate

= INT — 1867 216.25 

36524.25

A — Z + 1 + n — INT

Then calculate

B - A + 1524

C —- INT   B 2’’ 365.2

D — INT(365.25 C)

E = INT     B - D 

\ 30.6001

The day of the month (with decimals, if any) is then

B - D — INT(30.6001 £) + F

The month number m is fi — 1 if E < 14

ñ   — 13 if E —— 14 or 15

The year is C — 4716 if m > 2

C — 4715 if m = 1 or 2

Contrary to what has been said about formula (7.1), in the formula for 6 the number 30.6001 may not be replaced by 30.6, even if the computer calculates exactly. Otherwise, one would obtain February 0 instead of January 31, or April 0 instead of March 31.

 

 

2fxample 7. c - Calculate the calendar date corresponding to JD 2436 116.31.

2436 116.31 + 0.5 = 2436 116.81

Z = 2436 116 and F -—  0.81

Because Z > 2299 161, we have

=   INT   2436 116J 65216.25 _

 

Then we find

 

6528

A = 2436 116 + 1 + 15 - INT 15

4

 

= 2436 129

 

B -— 2437 653

day of month = 4.81

 

C -- 6673 D - 2437 313 E —— 11

 

month m = E - 1 = 10 (because E < 14) year = C - 4716 = 1957 (because m > 2)

Hence, the required date is 1957 October 4.81.

Exercise : Calculate the calendar dates corresponding to

JD = 1842 713.0 and  JD = 1507 900.13.

Answers:   333 January 27.5   and —584 May 28.63.

Time interval  in dogs

The number of days between two calendar dates can be found by calculating the difference between their corresponding Julian Days.

Example 7.d — The periodic comet Halley passed through the perihelion of its orbit on 1910 April 20 and on 1986 February 9. What is the time interval between these two passages?

1910 April 20.0 corresponds to JD 2418 781.5

1986 Febr. 9.0   corresponds to JD 2446 470.5

The difference is 27 689 days.

Exercise : Find the date exactly 10 000 days after 1991 July 11.

Answer: 2018 November 26.

 

 

The day of the week corresponding to a given date can be obtained as follows. Compute the ID for that date at 0' UT, add 1.5, and diviöe the result by 7. The remainder of this division will indicate the weekday, as follows: if the remainder is 0, it is a Sunday, 1 a Monday, 2 a Tuesday, 3 a Wednesday, 4 a Thursday, 5 a Friday, and 6 a Saturday.

The week was not modified in any way by the Gregorian reform of the lulian calendar. Thus, in 1582, Thursday October 4 was followed by Friday October 15.

Example 7. e — Find the weekday of 1954 June 30. 1954 June 30.0 corresponds to JD 2434 923.5

2434 923.5 + 1.5 = 2434 925

The remainder of the division of 2434 925 by 7 is 3. Hence i1 was a Wednesday.

Day of the Year

The number N of a day in the year can be computed by means of the following formula [1].

N —— INT 275 )   — K x INT (M 9 + D - 30 9 12

where M is the month number, D the day of the month, and

K —— 1 for a leap (bissextile) year,

K —— 2 for a common  year.

N takes integer values, from 1 on January 1, to 365 (or 366 in leap years) on December 31.

Example 7. f — 1978 November 14.

Common year,   3f = 11,   D -- 14, K —— 2. One finds N 318.

Example 7.g — 1988 April 22.

Leap year, M —— 4, D —— 22, K —- 1.

One finds N 113.

 

Let us now consider the reverse problem: the day number N in the year is known, and the corresponding date is required, namely the month number 3f and the day D of that month. The following algorithm was found by A. Pouplier, of the Société Astronomique de Liége, Belgium [2].

As above, take

K -- 1 in the case of a leap year,

K —— 2 in the case of a common year.

M = 9 (K + N)  + 0.98

275

If N < 32, then  3f = 1

 

D -— N — INT

 

 275 M  +   R x IM M + 9   + 30

 

9 12

REFEREN CES

1. Nautical Almanac Office, U.S. Naval Observatory, Washington, D. C. , Almanac for Compulers for the Year 1978, page B2.

2. A. Pouplier, letter to Jean Meeus, 1987 April 10.

 

Chapter 8

Date of Master

In this Chapter we give a method for calculating the date of the Christian Easter Sunday of a given year. For the Jewish Pesach, see next Chapter.

Gregorian Easter

The following method has been given by Spencer Jones in his book General Astronomy (pages 73—74 of the edition of 1922). It has been published again in the Journal of the British Astronomical Association, Vol. 88, page 91 (December 1977) where it is said that it was devised in 1876 and appeared in Butcher’s Ecclesiastical Calendar.

Unlike the formula given by Gauss, this method has no exception and is valid for all years in the Gregorian calendar, hence from the year 1583 on. The procedure for finding the date of Easter is as follows:

Divide by Quotient Remainder

the year x the year x b

b + 8

b — f + 1

19c + b — d - g + 15 c

32 + 2e + 2i - h -   k

a + 11/i + 22I

h + 1 — 7m + 114 19

100

4

25

3

30

4

7

451

31 —

b d f g

—

i

m

 

a c e

—

—

h

 

 

 

67

 

Then    n = number of the month (3 = March, 4 = April),

p + 1 = day of that month upon which Easter Sunday falls.

If the programming language has no “modulo” function or no “remainder” function, the calculation of the remainder of a division must be programmed carefully. Suppose that the remainder of the division of 34 by 30 should be found. On the HP-48s calculator, for instance, we find

34/30 = 1.133 333 333 33

the fractional part of which is 0.133 333 333 33. When multiplied by 30, this gives 3.999 999 9999. This result differs from 4, the correct value, and may give a wrong date for Easter at the end of the calculation.

Try your program on the following years:

 

1991 —• March 31

1992 —• April 19

1993 —• April 11

 

1954 —• April 18

2000 —• April 23

1818 —• March 22

 

The extreme dates of Easter are March 22 (as in 1818 and this) and April 25 (as in 1886, 1943, 2038).

The rule for finding the date of Easter Sunday is well known: Easter is the first Sunday afier the Full Moon that happens on or next after the March equinox. Actually, the rules for finding the Easter date were fixed long ago by the Christian clergy. For the purposes of these rules, the Full Moon is reckoned according to an ecclesiastical computation and is not the real, astronomical Full Moon. Likewise, the equinox is always assumed to fall on March 21; actually, it can occur a day or two sooner.

In 1967, for instance, the equinox was on March 21, and the Full Moon on March 26 (UT dates). The first Sunday after March 26 was April 2. Nevertheless, Easter Sunday was on March 26.

During the period 1900—2100, the purely astronomical rule yields another date for Easter Sunday than the ecclesiastical rule for the fohowing years: 1900, 1903, 1923, 1924, 1927, 1943, 1954, 1962, 1967, 1974, 1981, 2038, 2049, 2069, 2076,

2089, 2095, and 2096. See also Chapter 60 of my Mathematical Astronomy Morsels

(Willmann-Bell, ed.; 1997).

A period of 5 700 000 years is required for the cyclical recurrence of the Gregorian Easter dates. It has been found that, in the long run, the most frequent Gregorian Easter date is April 19.

 

8. DATE OF EASTER 6 9

Julian E.aster

In the Julian calendar, the date of Easter can be found as follows.

Divide by Quotient Remainder

the year x 4 — a

the year x 7 — b

the year z 19 — c

19c + 15 30 — d

2o + 4 b — d + 34 7

 

d + e + 114 31

 

 

Then f —— number of the month (3 = March, 4 = April),

g + 1 = day of that month upon which Easter Sunday falls.

The date of the Julian Easter has a periodicity of 532 years. For instance, we red April 12 for the years 179, 711, and 1243.

 

 

 

Chapter 9

Jewish and Moslem Calendars

It is not the aim of this Chapter to describe the principles of the Jewish and Moslem calendars. We shall just give some calculation methods which are easily programmable on a computer or on a pocket calculator. The algorithms given here were published by Denis Savoie in 1990 and 1991 in Observations et Travaux, a publication of the Société Astronomique de France.

In what follows we will denote by {a]p the remainder of the division of a by b, a and b being integers. For instance, [16], = 2 and [21] = 0.

INT(z) will mean the integer part of z. It is, in fact, the greatest integer which is not greater than z. For instance, INT(19) and INT(19.95) are both equal to 19. Great care should be taken when the value in negative. Some programming languages have both the INT and the FIX functions. For positive numbers these functions give the same results. But, for instance, INT(—2. 4) = —3, the correct answer, while FIX(—2.4) = —2.

Jewish Calendar

The Jewish (or Hebrew) calendar is luni-solar, being ruled by both the lunation (the synodic lunar month) and the tropical year. The fewish month has 29 or 30 days, and the year has 12 or 13 months. Moreover, both types of years can vary in three ways, so a Jewish common year may contain 353, 354, or 555 days, and an embolismic or leap year 383, 384, or 385 days. The names of the months and their lengths are given in Table 9.A.

The Jewish Easter, or Pesach, always falls on 15 Nisan.

Let A be the year number in the Jewish calendar, and X the year in the Julian or Gregorian calendar. Then the date in year X on which 15 Nisan occurs can be found by the following formulae due to Gauss.

C  =  INT S =  INT

A —- X + 3760 o = [12X + 12] tt  

71

 

T A B LE 9. A

Classification o f Years in the jewish Calendar

Month Common Year Emboli smic (Leap) Year

Deficient Regular Complete Deficient Regulor Complete

Tishri 30 30 30 30 30 30

Heshvan 29 29 30 29 29 30

Kislev 29 30 30 29 30 30

Tevet 29 29 29 29 29 29

Shevat 30 30 30 30 30 30

Adar 29 29 29 30 30 30

Veadar 29 29 29

Nisan 30 30 30 30 30 30

lyar 29 29 29 29 29 29

Sivan 30 30 30 30 30 30

Tammuz 29 29 29 29 29 29

Av 30 30 30 30 30 30

Elul 29 29 29 29 29 29

Sum

  354 355 383 384 385

Q ---- -- 1.904412 361 576 + 1.554 241 796 621 a + 0.25 h

-- 0.003 177 794 022 X + S

j ---- [INT( Q) + 3X + 5 b + 2 -- S] 7

r ---- Q -- INT(@)

1583, or in order to obtain 9 in the Julian calendar, take S -- 0.

One distinguishes the following four cases: 1. if y = 2, 4, or 6, then D -- iNT(Q) + 23 ;

2. if y = 1, a > 6, and r > ---- 0.632 870 370, then D --- INT Q) + 24 ;

3. if y = 0, a > 11, and r > = 0.897 723 765, then D = INT(9) + 23 ;

4. in all other cases, D --- INT(@) + 22.

The Pesach then falls on D March or, if fi > 31, on (D -- 31) April.

Once the date of the Pesach is obtained, just add 163 days to obtain the date of the beginning (1 Tishri) of the nerr Jewish year. The Jewish year A always begins in September or October of the Julian or Gregorian year X = A -- 3761.

IfA is the Jewish year number, then take the remainder [A] §g. IN this remainder is 0, 3, 6, 8, 11, 14, or 17, then that year has 13 months; otherwise it is a common year of 12 months.

 

 

Example 9.a — Calculate the date of 15 Nisan in the Gregorian year A — 1990.

We find successively C -- 19 ; S —— 13 ; a —— 9 ; b -- 2 ; Q —— 19.259 9537042 ; INT (O) = 19 ; y = 3 ; r -— 0.259 953 7042.

We are in the fourth case, so D —- 19 + 22. Hence, the date is 19 + 22 — 31 = 10 April. The Jewish year is A —— 1990 + 3760 = 5750.

Adding 163 days, we find 1990 September 20. This is the Gregorian date corresponding to 1 Tishri 5751. Because [5751] 9 = 13, the Jewish year 5751 is a common year.

To find the number of days (whether 353, 354, or 355) in that year, the s implest way is to search the Gregorian date corresponding to the beginning of the nnr Jewish year, and to make the difference. We find that 1 Tishri 5752 corresponds to 1991 September 9, so the year A -— 5751 has 354 days.

Moslem Calendar

The Moslem (or Islamic) calendar is purely lunar, as it follows the lunar phase cycle without regard for the tropical year.

The year contains twelve months. The months have alternately 30 and 29 days, except the last month which can have 29 or 30 days — see Table 9.B. Consequently, the Moslem year has 354 or 355 days ; it is shorter than the Gregorian year by about 11 days. As a result, the cycle of twelve lunar months regresses through the seasons over a period of about 33 Gregorian years.

TA B LE 9. B

Months o f the Moslem Calendar

The algorithms given below, due to M. Francmur (1841) and modified by Denis Savoie and the present author, will give meaningless results for dates earlier than 622 July 16 of the Julian calendar, corresponding to the beginning of the Islamic era, 1 Muharram A.H. 1 (A.H. = Anno Hegirae).

 

Conversion of a Moslem date to a Gregorian (or JulianJ date

Let H, M, and D be the year, the month number, and the day of the month in the Moslem calendar. Then calculate

N ---- D + INT(29.5001(If -- 1) + 0.99) @ = INT(H/30)

R  --  [If]3

A --  INT((IIIt + 3)/30)

W = 404 Q + 354R + 208 + A Q1 --- INT(l¥/ 1461)

Q2  --- t 1461

G =   621 +   4     INT{7Q + Q1) K -- INT(@2/ 365.2422)

E    --   INT(365.2422N) J --- 92   -- E + N -   1 X = G + K

If > 366 and [X] 4 = 0, then subtract 366 from J, and add 1 to X. If > 365 and [X] 4 > 0, then subtract 365 from /, and add 1 to X.

Then is the number of the day in the Julian year X. To convert to the Gregorian calendar (if the date is later than 1582 October 4), and to find the month and the day of the month, one can proceed as follows.

 

JD ---- INT(365.25 (X -- 1)) + 1721423 + J

= INT ID -- 1867 216.25 

36524.25

 

§ = JD + 1 + a 4

 

However, if ID < 2299 161, then take 9 = JD.

 

b -- Q + 1524

d INT(365.25c)

 

c -- INT

e INT

 

  -- 122.1  

365.25

   b -- d 

30.6001

 

Then the day of the month is b - d -- INT(30.6001 e)

and the month number m is   e -- 1 if e < 14

e -- 13 if e > 13

The year is c -- 4716 if m > 2, or c - 4715 if m < 3.

If [lllt + 3],0 > 18, then H is a leap year of 355 days, otherwise it is a common year of 354 days.

 

 

2fxample 9.b — Find the Julian or Gregorian date corresponding to the first day of the Moslem year 1421.

Here we have If = 1421, If = 1, D -- 1, and we find successively:

N -- 1 ; @ = 47 ;  ft =  11 ;  A -- 4 ;  IY = 23094 ;  Q1 -- 15 ;  Q2 -- 1179 ;

G = 1997 ; K —- 3 ;  E -- 1095,  J -- 84 ;  X = 2000.

This gives the 84th day of the year 2000 in the Julian calendar. Continuing, we

obtain

ID -- 2451641 ; ‹x = 16 ;  ¢ = 2451654 ;  b = 2453 178 ;  c = 6716 ;

d -- 2453 019 ; e -— 5 ;  day = 6 ;  month = 4 ;  year = 2000.

Hence, 1 Muharram 1421 corresponds to 6 April of the Gregorian year 2000.

Because [llJt + 3]30 = 4, which is not larger than 18, the Moslem year 1421 is a common year of 354 days.

Conversion of a Gregorian (or Julian) date to a Woslem date

If the date is given in the Gregorian calendar, we first have to convert it to the corresponding date in the Julian calendar. This can be done as follows.

Let X, If, D be the given year, month number, and day of the month in the Gregorian calendar.

If M < 3, subtract 1 from X, and add 12 to M.

Calculate

 

   x 

100

 

Q = 2 — + INT

 

b —— INT(365.25X) + INT(30.6001 (M + 1)) + D + 1722519 + 9

With this value of b, calculate c, d, e, and the new values (lulian calendar) for the day D, the month number M, and the year X as before tpage 74).

The date being now Julian, proceed as follows.

If [X] 4 = 0, then W = 1, otherwise W = 2.

 

N —— INT 2753f  

9

 

  + 9  + D — 30

12

 

A —— X — 623

 

C —— [A]4 CI  —— 365.2501 C

If CI - C2 > 0.5, add 1 to C2.

 

C2 -- INT {CI )

 

A' = 1461B + 170 + C2

Q -- INT       D' 

10631

fi =  [D ’110631

   R 

354

_ 1IJ + 14  

30

H = 30 @ + I + 1 JJ --- K -- 0 + N - 1

JJ is the number of the day in the Moslem year H. If NJ > 354, we have to look if H is a common year or a leap year, in order to know whether we should subtract 354 or 355 days. This can be done as follows.

-  [ff)so   DL -- [11 CL + 3],d

If DL < 19, subtract 354 from N, and add 1 to IN. If DL > 18, subtract 355 from N, and add l to H.

Finally, if JJ ---- 0, then put JJ --- 355, and subtract l from H.

Now, the day number JJ should be converted to the month number m and the day d of the month:

 

S -   INT JJ - 1

29.5

If NJ = 355, then m = 12 and d = 30.

 

d --- INT(JJ -- 29.55’)

 

 

Example 9.c -- Find the Moslem date corresponding to the Gregorian date 1991 August 13.

Here we have X = 1991, M ---- 8, D -- 13. We find successively:

n = 19 ;  b = --13 ;  b ---- 2450 006 ;  c -- 6707 ;  d ---- 2449 731 ;  e 8 ;

D --- 31 ; If = 7 ;  X = 1991.

So the date in the Julian calendar is 1991 July 31.

IV = 2 ; N  -- 212 ; A -- 1368 ;  B -- 342 ;  C -- CI ---- C2 --- 0 ;  D' -- 499 832 ; @ = 47 ;  A = 175 ; = 0 ; K --- 175 ; 0 =  0 ; D =  1411 ; N =  386.

Because JJ > 354, we calculate CL -- 1 and DL -- 14. Because DL is smaller than 19, we subtract 354 from JJ and add 1 to H, obtaining JI --- 32, If = 1412.

Then m = 2, d -- 2. So the date is 2 Safâr of A.H. i 412.

 

Chapter 10

Dynamical Time and Universal Time

The Universal Time (UT), or Greenwich Civil Time, is based on the rotation of the Earth. The UT is necessary for civil life and for the astronomical calculations where local hour angles are involved. (Universal Time is erroneously called “GreenwlCh Mean Time” in Great Britain and by most navigators. In astronomy, “mean” time has a precise meaning. By definition, mean time is measured from the superior transit of the mean Sun, hence from mean noon. It is the civil time which begins at midnight, so GMT and UT differ by twelve hours.)

However, the Earth’s rotation is generally slowing down. Moreover, this occurs with unpredictable irregularities. For this reason, the UT is not a uniform time.

But the astronomers need a uniform time scale for their accurate calculations (celestial mechanics, orbits, ephemerides). From 1960 to 1983, in lhe great astronomical almanacs such as the Astronomical Ephemeris, use was made oY a uniform time scale called the Ephemeris Time (ET) and defined by the laws of dynamics: it was based on the planetary motions. In 1984, the ET was replaced by the Dynamical Time, which is defined by atomic clocks. The Dynamical Time is, in fact, a prolongation of the Ephemeris Time.

One distinguishes a Barycentric Dynamical Time (TDB) and a Terrestrial Dynamical Time (TDT). These times differ by at most 0.0017 second, the difference being related to the motion of the Earth on its elliptical orbit around the Sun (relativistic effect). Because this very small difference can be neglected for most practical purposes, we will make no distinction between TDB and TDT, and we will name both simply “Dynamical Time”, or TD by dropping the last letter from both TDB and TDT. Hence, our abbreviation TD does nor come from the French “Temps Dynamique”, but should be considered as meaning Time i .

TDT was later shortened to simply TT (“Terrestrial Time"), an odd name because the mean solar time at Moscow, or the sidereal time at New York, are “terrestrial” times too!

The exact value of the difference IT —— TD — UT can be deduced only from observations. Table 10. A gives the value of b T for the beginning of some years. For the years earlier than 1988, they are taken from the miroitnmicnf Almanoc for 1988 [1].  However, the values earlier than 1955 have been sligh tly cnrrected by

77

 

using Chapront’s new value n' = --25.7376 ”/century° for the tidal acceleration of the Moon [2].

For epochs in the neor future, one may extrapolate the values of Table 10.A.

For instance, we can use the provisional values

k T -- +65 seconds in 2000 IT ---- +69  seconds in 2005 IT ---- + 80 seconds in 2015

For other epochs outside the time interval of Table 10. A, an approximate value of IT (in seconds) can be calculated by means of the following expressions due to Chapront and Francou [2] :

Let r be the time measured in centuries from the epoch 2000.0 (r < 0 before 2000), that is,

  -- 2000 

100

Then, before the year +948,

 

k T -- 2177 + 497 r + 44.1 r°

From +948 to + 1600, and after the year +2000,

k T -- 102 + 102 i + 25.3 f 2

 

(10. l)

(10.2)

 

However, to avoid a discontinuity at A.D. 2000, it is advised to add the correction

+0.37 X (year -- 2100) for the years 2000 to 2100.

With these expressions, the uncertainty of UT can reach as much as two hours back to 4000 B.C. Future improvements of the formulae will benefit the user when converting from TD to UT, but will not change the algorithms, programs, ephemerides, or tables given with the uniform time scale of TD.

The quantity IT was slightly negative from A.D. 1871 to 1901. Note that AT

is positive both for the remote past and for the distant future.

Except for the years 1871--1901, an instant given in UT is later than the instant in TD having the same numerical value. For example, 1990 January 27, 0h UT is 57 seconds later than 1990 January 27, 0' TD. We have UT = TD -- k T.

Example 10.a -- New Moon took place on 1977 February 18 at 3'37 40' Dynamical Time (see Example 49.a).

At that instant, IT was equal to +48 seconds. Consequenlly, the corresponding Universal Time of that lunar phase was

3h37"'40' -- 48' = 3'36“52'.

 

10. DYNAMICAL TIME AND UNIVERSAL TIIVIE 79

TA B LE 1 O. A

6 T —— TD — UT (in  seconds)   for the   beginning  of  some  years

year b T year 2Y T year IT year ‹I.T year IT

1620 +121 1700 + 7 1780 +16 1860 + 7.7 1940 +24.3

1622 112 1702 7 1782 16 1862 7.3 1942 25.3

1624 103 1704 8 1784 16 1864 6.2 1944 26.2

1626 95 1706 8 1786 16 1866 5.2 1946 27.3

1628 88 1708 9 1788 16 186& 2.7 1948 28.2

1630 +82 1710 + 9 1790 +16 1870 +1.4 1950 +29.1

1632 77 1712 9 1792 15 1872 — t.2 1952 30.0

1634 72 1714 9 1794 15 1874 —2.8 1954 30.7

1636 68 1716 9 1796 14 1876 —3.8 1956 31.4

1638 63 1718 10 1798 13 1878 —4.8 1958 32.2

1640 +60 1720 +10 1800 +13.1 1880 —S.5 196D ^33.1

1642 56 1722 10 1802 12.5 1&82 —S.3 1962 34.0

1644 53 1724 10 1804 12.2 1884 —5,6 1964 35.0

1646 51 1726 10 1806 12.0 1886 —5.7 1966 36.5

1648 48 1728 10 1808 12.0 1888 —5.9 1968 38.3

1650 +46 1730 +10 1810 +12.0 1890 —6.0 1970 >40.2

1652 44 1732 10 1812 12.0 1892 —6-3 1972 42.2

1654 42 1734 11 1814 12.0 1894 —6.5 1974 44.5

1656 40 1736 11 1816 12.0 1896 —6.2 1976 46.5

1658 38 1738 11 1818 11.9 1898 —4.7 1978 48.5

1660 +35 1740 +l1 1820 +l1.6 1900 —2.8 1980 ISO.5

1662 33 1742 11 1822 11.0 1902 —0.1 1982 52.2

1664 31 1744 12 1824 10.2 1904 +2.6 1984 53.8

1666 29 1746 12 1826 9.2 1906 5.3 1986 54.9

1668 26 1748 12 1828 8.2 1908 7.7 1988 55.8

1670 +24 1750 +12 1830 + 7.1 1910 +10.4 1990 -F56.9

1672 22 1752 13 1832 6.2 1912 13.3 1992 58.3

1674 20 1754 13 1834 5.6 1914 16.0 1994 60.0

1676 18 1756 13 1836 5.4 1916 18.2 199€ 61.6

1678 16 1758 14 1838 5.3 1918 20.2 1998 63.0

1680 +14 1760 +14 1840 + 5.4 1920 +21,1

1682 12 1762 14 1842 5.6 1922 22,4

1684 11 1764 14 1844 5.9 1924 23.5

1686 10 1766 15 1846 6.2 1926 23,8

1688 9 1768 15 1848 6.5 1928 24.3

1690 + 8 1770 +15 1850 + ò.8 1930 -t-24.0

1692 7 1772 15 1852 7.1 1932 23.9

1694 7 1774 15 1854 7.3 1934 23.9

1696 7 1776 16 1856 7.5 193ò 23.7

1698 7 1778 16 1858 7.6 1938 24.0

 

 

£zompie 10.b — Suppose that the position of Mercury should be calculated for February 6 at 6‘ Universal Time of the year + 333.

Here we have T -— (333.1 — 2000)/ 100 = — 16.669, for which formula (10.1) gives the value IT —- + 6146 seconds, or 102 minutes. Hence, TD = 6' + 102 minutes = 7‘42'", and the calculation of the position of Mercury must be performed for 333 February 6 at 7‘42" TD.

The following approximation for b T, valid for the entire time span 1800— 1997, represents the values given in Table 10.A with a maximum error of 2.5 seconds.

IT —— — 1.02 + 91.02 8 + 265.90 82 — 839. 168’ — 1545. 20 84

+ 3603.62 85 + 4385.98 86 — 6993.23 87 — 6090.04 8'

+ 6298.12 8’ + 4102.86 8' 0 — 2137.64 8 '’ — 1081.51 8’2

In this formula, b T is expressed in seconds, and f is the time elapsed since 1900.0 and expressed in Julian centuries (hence 8 < 0 before 1900).

The following formula gives IT for the shorter time span 1800-1899 with a maximum error of 0.9 second:

4 T -— —2.50 + 228.95 8 + 5218.61 82 + 56 282.84 83 + 324 011.78 84

+ 1061 660.75 85 + 2087 298.89 86 + 2513 807.78 87

+ 1818 961.41 8' + 727 058.63 8’ + 123 563.95 8' 0

For the years 1900 to 1997, the following expression gives b T with a maximum error of 0.9 second:

4 T —— —2.44 + 87.24 8 + 815.20 82 — 2637.80 B' — 18 756.3384

+ 124 906.15 8’ — 303 191.19 86 + 372 919.88 8’

— 232424.66 8' + 58 353.42 8’

where 8 has the same meaning as for the first formula.

Note that these three expressions are empirical formulae, and that their use is prohibited outside of their defined validity range! For instance, the second expression would give a value of 70000 seconds for the year 1945 !

REFEREN CES

1. Astronomical Almanac for 1988 (Washington, D. C.), pages K8 and k9.

2. J. Chapront, M. Chapront-Touzé, and G. Francou, Note SSU issued by the Bureau des Longitudes, Paris, in December 1997.

 

Chapter 11

The Earth’s Globe

The actual figure of the Earth’s surface, including all the inequalities of mountains and valleys, is incapable of geometric definition. Therefore, the ideal figure used in geodesy is that of the mean sea level, extended through the continents. This is the geoid, whose surface at every point is perpendicular to the loml plumb line.

However, the heterogeneity of the Earth’s interior and the attraction of mountains are such that the surface of the geoid is not rigorously represented by any definable solid. An approximation sufficient for most geographical and astronomical purposes is obtained by considering it to be an ellipsoid of revolution.

Geoerniric rectangular coordinates of an obsewer

The Figure represents a meridian cross section of the Earth. C is the Earth’s center, N its north pole, S its south pole, EF the equator, ?fA the horizontal plane of the observer 0, and OP the perpendicular to HK. The direction Off, parallel to

SN, makes with OH an angle e which is the geographical latitude

 

M

H

N e o

 

 

of 0. The angle OPF too is equal to p. The latitude is positive in the northern hemisphere, negative in the southern hemisphere.

The radius vector OC, joining the observer to the center of the Earth, makes with the equator CF

F an angle p' which is the geocentric

 

C P latitude of 0.   We have p = p' at

the poles and at the equator; for all

other latitudes

81

 

at    y be the Earth’s flattening, and blo the ratio NCI CF of the polar radius NC —— b to the equatorial radius CF —— a. In 1976 the International Astronomical Union adopted the values

 

a —— 6378.14 km,  

from which we have

b —— a [l — f) -- 6356.755 km

   = 1 —/ = 0.996 647 19

The eccentricity e of the Earth’s meridian is e = f f -— 0.081 819 22

We have the relations

 

  I 298.257

 

_  a —b  

a For a place at sea level,

 

l — e 2 = (I —/)-

 

   

If H is the observer’s height above sea level iR meters, the quantities p sin ‹,o' and p cos e’ , which are needed in the calculation of diurnal parallaxes, eclipses and occultations, may be calculated as follows:

    S'‘ + 6 378 140 SQL

p COS p’ = cos ti+ 6 378 140 cos e

The quantity p sin ‹,o' is positive in the northern hemisphere and negative in the southern one, while p cos p' is always positive.

The quantity p denotes the observer’s distance to the center of the Earth {OC

in the Figure), the Earth’s equatorial radius being taken as unity.

Example 11.a — Calculate p sin p' and p cos p’ for the Palomar Observatory, for which

p = + 33° 21'22”, H = 1706 meters.

 

We obtain

= 33°.356 111

ti = 33°. 267 796

p sin p' = +0.546 861

p cos p' = +0.836 339

Other fomiulae concerning the Earth’s ellipsoid

For a given point on the ellipsoid, the difference between the geographic latitude and the geocentric latitude can be found from

p — p' = 692“.73 sin 2 ‹;o — 1“. 16 sin 4 p

The difference p — p' reaches a maximum value for u —- 45°. Ife d and p0' are the corresponding geographic and geocentric latitudes, we have

whence, for the IAU 1976 ellipsoid,

eo = 45°05'46“.36 r = 44°54' 13“.64 to to' - 11’32“.73

The quantity p (for sea level) can be found from

p — 0.998 3271 + 0.001 6764 cos 2‹¡o — 0.000 0035 cos 4p

The parallel of latitude p is a circle whose radius is

Rp a cos e

1 e 2 S  2

where, as above, e is the eccentricity of the meridian ellipse.

Hence, one degree of longitude, at latitude y, corresponds to a length of

180

The rotational angular velocity of the Earth (with respect to ltte stars, not with

respect to the moving vernal equinox) is

o = 7.292 114 992 x 10 5 radian/second.

Strictly speaking, this is the value at the epoch 1996.5 [I]. 11 decreases slowly with time because the rotation of the Earth is slowing down — see Chapter 10.

 

The linear velocity of a point at latitude p, due to the rotation of the Earth, is oQ per second.

The radius of curvature of the Earth’s meridian, at latitude p, is

_ o (1 — e°)

(1 — e° sin*p)°'°

and one degree of latitude corresponds to a length of 180 '°

Rq reaches a minimum value at the equator, o (l — e’) —— 6335.44 km, and a maximum value at the poles, o / lie = 6399.60 kilometers.

 

ñzompfe 11.b — For p = + 42°, the latitude of Chicago, we find

Rg —— 4747.001 km

1° of longitude = 82.8508 km

linear velocity = mfig = 0.34616 km/second fig = 6364.033 km

1° of latitude = 111.0733 km

Distance between two points on the Earth’s surface

If the geographic coordinates of two points on the surface of the Eanh are known, the shortest distance s between these points, measured along the Earth’s surface, can be calculated. Let the first point having longitude and latitude L, and pt , respectively. Let La and pt be the coordinates of the second point. We will suppose that these points are at sea level.

If no great accuracy is needed, we may consider the Earth as being spherical with a mean radius of 6371 kilometers. Find the angular distance d between the two points by means of the formula

cos d —— sin ‹,ot sin  p 2 + cos ‹,e, cos ‹,r  cos (L, — L ) (11. 1)

which is similar to formula (17.1) for the angular separation between two celestial bodies. Formula (11.1) does not work well when d is very small — see Chapter 17. Then the required linear distance is

 

 6371 c d 

S 180

 

kilometers

 

(11. 2)

 

where d is expressed in degrees.

 

Higher accuracy is obtained by the following method, due lo H. Andoyer [2); the relative error of the result is of the order of the square of the Earth’s flattening.

As before, let a be the equatorial radius of the Earth, and/ the flattening. Then calculate

   + A

2 2

S —- sin2 G cos2h + cos2J sin° k

C -- cos2 G cos2 h + sin2 F sin2 h

R —— where o is expressed in radians

 

D = 2 t» n

 

 3A — 1  

2 C

 

  3A + 1 

2 S

 

and the required distance will be

s —- D (1 + fH 1 sin2 F cos2G - fH cos2 I 5in° G)

Example 11.c — Calculate the geodesic distance between the Observatoire de Paris (France) and the U.S. Naval Observatory at Washington, D.C., adopting the following coordinates:

Paris: L —— 2° 20’14" East = —2° 20’14”

pt = 48° 50' 11" North = +48° 50’11"

Washington: = 77° 03’56“ West =   + 77°03'56“

p2 = 38° 55’17“ North = + 38°55’ 17"

We find successively

F

+43.°878 8889

G + 4.°9575000

k —39.°7013889

S 0.216 426 96

C 0.783 573 04

u› 27.°724 274 = 0.483 879 87 radian

it 0.851 0555

D 6172.507 km

 

and finally s -- 6181.63 kilometers, with a possible error of the order of 50 meters.

If we use the approximate expressions (11.1) and (11.2), we obtain cos d -- 0.567 146

d --- 55.°44855

s 6166 km

REFEREN CES

1. International Earth Rotation Service, Annual Report for 1996 (Observatoire de Paris, 1997).

2. Annuaire du Bureau des Longitudes pour 1950 (Paris), page 145.

 

Chapter 12

Sidereal Time at Greenwich

We shall denote by By the sidereal time at Greenwich at 0' UT of a given date, and by the sidereal time at Greenwich for any given instant UT.

The sidereal time at the meridian of Greenwich, at 0h Universal Time of a given date, can be obtained as follows.

Calculate the JD corresponding to that date at 0' UT (Chapter 7). Thus, this is a number ending on .5. Then find T by

 

T --     JD — 2451 545.0  

36525

 

(12. 1)

 

The mean sidereal time at Greenwich at 0' UT is then given by the following expression which was adopted in 1982 by the International Astronomical Union:

 

B   = 6'41‘50'.54841 + 8640 184‘.812 866 T

+ 0.093 104 F 2 — 0‘.000 0062 T

Expressed in degrees and decimals, this formula can be written B0 = 100.460 618 37 + 36 000.770 053 60g T

+ 0.000 387 933 T2 — r3 / 38 710 000

 

(12. 2)

(12.3)

 

Imporiani: the formulae (12.2) and (12.3) are valid only for those vaiues of T

which correspond to 0‘ UT of a date. All other values would give incorrect results.

To obtain the sidereal time 8 at Greenwich for any instant UT of a given date, multiply that instant by 1.002 737 909 35 and add the result to the sldereal time 8 at 0‘ UT.

The mean sidereal time at Greenwich, expressed in degrees, can at so be found directly for any instant as follows. If JD is the Julian Day corresponding to that instant in UT (not necessarily 0‘), find T by formula (12.1), and then

87

 

80 = 280.460 618 37 + 360.985 647 366 29 (JD — 2451 545.0)

+ 0.000 387 933 T — T3 / 38 710 000

 

(12.4)

 

If high accuracy is needed, this formula requires the use of a computer language working with a sufficient number of significant digits.

The sidereal time obtained by formulae (12.2), (12.3), or (12.4) is the mean sidereal time, that is, the Greenwich hour angle of the mean vernal point (the intersection of the ecliptic of the date with the mean equator of the date).

The apparent sidereal time, or the Greenwich hour angle of the true vernal equinox, is obtained by adding the correction AQ cos e, where AQ is the nutation in longitude, and e the true obliquity of the ecliptic (see Chapter 22). This correction for nutation is called the mutation in right oscension or equation of the equinoxes. Because AQ is a small quantity, the value of e may be taken to the nearest 10” here.

If AQ is expressed in arcseconds (seconds of a degree), the correction in seconds of time is

 AQ cos e 15

Example 12.a — Find the mean and the apparent sidereal time at Greenwich on 1987 April 10 at 0h UT.

This date corresponds to JD 2446 895.5, and formula (12. 1) gives

T -— —0.127 296 372 348

We then find by means of formula (12.2)

8p —- 6‘41*50‘.54841 — 1099 864.18158 seconds

or, by adding a convenient multiple of 86400 seconds (the number of seconds in one

B0 = 6‘41“50'.54841 + 23335 \ 81842

= 6h4l‘ 50'.54841 + 6‘28”55'81842

= 13‘10‘46'.3668

which is the required mean sidereal time.

From Example 22.a we have, for the same instant,   Af = —3”.788 and e = 23° 26’36“.85. [In fact, these values are for 0‘ TD, not for 0' UT, but here we will neglect the very small variation of AJ during the time interval AT = TD — UT.]

—3.788

 

Hence the mutation in right ascension is and the required apparent sidereal time is

 

CO5 23°.44357 = —0°.2317,

1S

 

13‘10‘46 “.3668 — 0‘.2317 = 13‘l D‘46‘.1351

 

12. SIDEREAL TIME 89

Example 12.b — Find the mean sidereal time at Greenwich on 1987 April 10 at l9 h2l"'00' Universal Time.

First, we calculate the mean sidereal time for that date at 0‘ UT. We find 13'10“46'.3668 (see the previous Example). Then

1.002 737 909 35 x 19'21”00’

= 1.002 737 909 35 x 69660 seconds

= 69850.7228 seconds

= l9‘24‘10'.7228

and the required sidereal time is

13'10‘46'3668  + 19‘24‘10'7228  =  32'34‘57‘.0896

=    8‘34*57'.0896

Alternatively, we may use formula (12.4). The Julian Day corresponding to 1987 April 10 at 19‘21‘00‘ UT is

JD = 2446 896.30625

and, by (12.1), the corresponding value of T is —0.127 274 30. Formula (12.4) then gives

80 = — 1677 831.262 1266 degrees

or, by adding a convenient multiple of 360°,

80 = 128°. 737 8734

This is the required mean sidereal time in degrees. We obtain it in hours by dividing it by 15 (since one hour corresponds to 15 degrees) :

80 = 8'.582 524 89 = 8'34'"57'.0896,

the same result as above.

 

 

Chapter 13

Transformation of  Coordinates

We will use the following symbols:

n = right ascension. This quantity is generally expressed in hours, minutes, and seconds of time, and hence should first be converted into degrees (and decimals) and then, if necessary, into radians, before it is used in a formula. Conversely, if n has been obtained by means of a formula and a programming language, it is expressed in radians or in degrees; it may be converted to hours by division of the degrees by 15, and then, if necessary, be converted into hours, minutes, and seconds;

= declination, positive if north of the celestial equator, negative if south;

 

1950

1950 ’2OOO '2OOO

 

= right ascension referred to the standard equinox of B1950.0;

= declination referred to the standard equinox of B1950.0;

-— right ascension referred to the standard equinox of f2000. 0;

= declination referred to the standard equinox of J20£O.0;

= ecliptical (or celestial) longitude, measured from the vemal equinox along the ecliptic;

= elliptical (or celestial) latitude, positive if north of the ecliptic, negative if south;

 

l = galactic longitude;

b =  galactic latitude;

h = altitude, positive above the horizon, negative below;

A = azimuth, measured westward from the Souih. Note lhat navigators and meteorologists count the compass direction, or azimuth, from the North (0°), through the East (90°), South (180°), and West (270°). But astronomers disagree (see the box on next page) and we shall measure the azimuth from the South, because the hour angles too are measured from the South, at least for observers in the northern hemisphere. Hence, a celestial body which is exactly on the southern meridian has A = H = 0°;

91

 

 

e = obliquity of the ecliptic; this is the angle between the ecliptic and the celestial equator. The mean obliquity of the ecliptic is given by formula (22.2). If, however, the apparent right ascension and declination are used (that is, affected by the aberration and the nutation), the true obliquity c + Ae should be used (see Chapter 22). If o and b are referred to the standard equinox of J2000.0, then the value of e for that E poch should be used, namely e2  = 23° 26’21“.448 = 23°.439 2911. For the standard equinox of B1950.0, we have 6{guy ' 23°.445 7889;

p = the observer’s latitude, positive if in the northern hemisphere, negative in the southern one;

H —- the local hour angle, measured westwards from the South.

If 8 is the local sidereal time, 80 the sidereal time at Greenwich, and L the observer’s longitude (positive west, negative east from Greenwich), then the local hour angle can be calculated from

H —— 8 — a or H = 8b — L - a

If n is affected by the nutation, then the sidereal time too must be affected by it (see Chapter 12).

For the transformation from equatorial into elliptical coordinates, the following formulae can be used:

 

13. TRANSFORMATION OF COORDINATES

_   sin n cos e + tan h sin e  

COS CI

sin 9 =   sin 6 cos e — cos é sin e sin a

Transformation from ecliptical into equatorial coordinates: p sin k cos e — tan Q sin e

cos k

sin 6 = sin /S cos 6 + cos g sin c sin X

Calculation of the local horizontal coordinates:

tan A —— sin if

COS H Sirl ‹¡o — tdfl d COS @

sin h — sin e sin fi + cos p cos 6 cos H

 

93

(13. 1)

(13. 2)

(13.3)

(13. 4)

(13.5)

(13.6)

 

If one wishes to reckon the azimuth from the North instead of the South, add 180° to the value of A given by formula (13.5).

Transformation from horizontal into equatorial coordinates:

 

  sin A

cos A sin ‹,r + tan h cos p

sin 6 = sin p sin h — cos p cos h cos

The current galactic system of coordinates has been defined by the International Astronomical Union in 1959. In the standard equatorial system of B1950.0, the galactic (Milky Way) North Pole has the coordinates

1950 12‘49‘ = 192. 25, éttt  =  +27°. 4

and the origin of the galactic longitudes is the point (in western Sagittarius) of the galactic equator which is 33° distant from the ascending node (in western Aquila) of the galactic equator with the equator of B1950.0.

These values have been fixed conventionally and therefore must be considered as crore for the mentioned equinox of B1950.0.

Transformation from equatorial coordinates, referred to the standard equinox of B1950.0, into galactic coordinates:

 

  sin (192.°25  —  n) cos (192°.25  — n)  sin  27°.4  —  tan  é cos 27°.4

l —— 303‘ — x

 

(13.7)

 

sin b —— sin h sin 27°.4 + cos h cos 27.°4 cos (192°. 25 — n) (13.8)

Transformation from galactic coordinates into equatorial coordinates referred to the standard equinox of B1950.0:

  sin (/ - 123°)                             cos (I — 123‘) sin 27°.4 — tan b cos 27°. 4

o =   y + 12°.25

sin 6 = sin b sin 27°.4 + cos b cos 27.°4 cos (I — 123°)

If the 2000.0 mean place of the star is given instead of the 1950.0 mean place, then, before using formulae (13.7) and (13.8), convert n and hip 1950

and 61950 . See Chapter 21.

The formulae (13.1), (13.3), etc. , give tan h, tan o, etc. , and then X, n, etc. , by the function arctangent. However, the exact quadrant in which the angle is situated is then unknown. To remove the ambiguity of 180°, apply the ATN2 function to the numerator and the denominator of the function (instead of performing the actual division), or use another trick. See “The correct quadrant" in Chapter 1.

 

J 3. TRANSFORMATION OF COORDINATES 95

'.xample 13.a — Calculate the ecliptical coordinates of the slar Pollux (# Gem), whose equatorial coordinates are

2 = 7'45‘18'.946, ét = + 28° 01’34“.26.

Using the values n = i 16°.328 942,  6 = + 28°.026 183, and e = 23°.439 2911,

formulae (13.1) and (13.2) give

 

   + 1.034 039 86  

—0.443 523 98

9 = +6°.684 170.

 

whence X = 113°. 215 630 ;

 

Because « and 6 are referred to the standard equinox of 2000.0, h and 9 too are referred to that equinox.

Ex e rcis e. — Using the values of X and Q found above, find n and d again by means of formulae (13.3) and (13.4).

Example 13.b — Find the azimuth and the altitude of Venus on 1987 April 10 at 19‘21*00’ UT at the U.S. Naval Observatory at Washington, D.C. (longitude = + 77° 03'56“ = + 5‘08‘15'. 7, latitude =

+ 38°55’17”).

The planet’s apparent equatorial coordinates, interpolated from an ephemeris, are a = 23h09* 16'.641, é = —6° 43’ l l’.61

These are the apparent right ascension and declination of the planet. So we need the apparent sidereal time for the given instant.

We first calculate the menu sidereal time at Greenwich on 1987 April 10 at 19‘21"00' UT, and find 8‘34"57'.0896 (see Example 12.b).

By means of the method described in Chapter 22, we fi nd for the same instant: nutation in longitude : AQ = —3". 868

true obliquity of the ecliptic: t = 23°26'36". 87

The apparent sidereal time at Greenwich is

 

80 = 8‘34‘57’.0896 + —3.868 CO

 

e seconds   = 8'34°'56'.853

 

15 S

Hour angle of Venus at Washington: If = 8ø — L — a

= 8'34*56'.853 — 5'08*15 \ 7 — 23'09 16'.641

= — 19 42"35'488 = — 19'.709 8578 = —295°.647 867

-- +64.°352 133

Formulae (13.5) and (13.6) then give

 

tan A --

 

  +0.901 4712  

+0.363 6015

 

whence A --- +68°.0337

 

h -- + 15.°1249

so the planet is 15 degrees above the horizon between the southwest and the west.

Note that formula (13.6) does not take into account the effect of the atmospheric refraction, nor that of the planet’s parallax, nor the dip of the horizon. For the atmospheric refraction, see Chapter 16. The correction for parallax is dealt with in Chapter 40.

As an exercise, find the galactic coordinates of Nova Serpentis 1978, whose equatorial coordinates are

O ]g  ' 17‘48‘59'. 74, 1950 --14° 43’08“. 2

Answer:   l -- 12.°9593, b +6.°0463.

REFEREN CES

1. The Nautical Almanac and Astronomical Ephemeris for the year 1835, p. 508 (London, 1833).

2. The American Ephemeris and Nautical Almanac for the Year 1B57, p. 491 (Washington, 1854).

3. 77ie Astronomical Ephemeris for the Year 1960, pp. 434 & fol. (London, 1958).

4. W. Chauvenet, A Manual of Spherical and Practical Astronom:y, Vol. I, pp. 317 & foI. (Philadelphia, 1891).

5. A. Danjon, Astronomie Générale, p. 46 (Paris, 1959)

6. S. Newcomb, A Compendium of Spherical Astronomy, p. 119 (New York, 1906).

 

Chapter ld

The Parallactic Angle, and three other Topics

Suppose that on a bright morning we are looking at the Sun through a piece of dark glass, and that we see a large sunspot near the western (“right”) limb of the Sun (Figure 1, A). At noon, the Sun being near the southern meridian, we notice that the spot is lower (Figure 1, B). And in the afternoon, we see that the spot has moved still farther along the Sun’s limb (Figure 1, C).

The spot did not actually move that much over the solar disk. It is the whole image of the Sun which rotated clockwise. This can be seen easier with the Moon (Figure 2).

This apparent rotation is easily understood when we consider the diurnal motion of the celestial sphere. Each celestial body describes a parallel circle, a diurnal arc (Figure 3). Only when the Sun (or the Moon) is exactly on the southern meridian, will the celestial north be up, in the direction of the zenith.

The constellations show a similar effect. For an observer in the northern hemisphere of the Earth, the constellation of Orion is inclined to the “left” in the southeast, is upright in the south, and is inclined to the “right” in the southwest.

In Figure 4, the circle represents the disk of the Sun (or tha: of the Moon). The arc M is a part of its diurnal arc on the celestial sphere. C is the center of the disk. The direction of the zenith and that of the celestial North are indicated. The latter direction is perpendicular to the arc W. Z is the zenith point of the disk; it is the uppermost point of the disk at the sky as seen by the observer at the given instant.

 

Fig. I : The apparent displacement of a sunspot in the course of the day.- in the morning (A), near noon (B), and in the afternoon (C). fn each of the three sketches, the circle represents the solar disk, and the zenith is at the top.

97

 

Fig. 2 : The First-Quarter Moon for an observer in the northern hemisphere.’ (A) near the sowh, around the time o/ Hauser; and IB) later that evening, The ,•enith is up.

 

zenith

zenith zenith

 

East

 

h o riz on

 

South

Fig. 3

 

West

 

N is the nonh point of the disk; the direction CN points towards the northern celestial pole.

The angle ZCN is called the parallactic angle and is generally designed by q. This parallactic angle has absolutely nothing to do with the parallax! The name arises from the fact that the celestial body moves along a parallel circle. Compare with the “parallactic” mounting of a telescope.

By convention, the angle q is negative before, and positive after the passage through the southern meridian. Exactly on the meridian, we have q = 0°.

The parallactic angle q can be calculated from

q — q cos h — sin I› cos If (14.1)

where, as in the preceding Chapter, ‹,o is the geographical latitude of the observer, 6 the declination of the celestial body, and N its hour angle at the given instant.

Exactly in the zenith, the angle q is not defined. Indeed, in that case we have H = 0° and 6 = e. so formula (14.1) yields tan q = 0/0. This can be compared with somebody who is exactly at the North Pole of the Earth: his geographical

longitude is not defined, because all meridians of the Earth converge to his place. For that special observer, all points of the horizon are in the southern direction!

When a celestial body passes exactly through the zenith, the parallactic angle

q suddenly jumps from —90° to + 90°.

 

1 4.  THE PARALLACTIC  ANGLE 9 9

If the celestial body is on the horizon (hence rising or setting), formula (14.1) simplifies greatly, namely

COS Q ' cos fi

and in that case it is not necessary to know the value of the hour angle.

Ecliptic and Horizon

If e is the obliquity of the ecliptic, p the latitude of the observer, and 8 the local sidereal time, then the longitudes of the two points of the ecliptic which are (180 degrees apart) on the horizon, are given by

 

umh = COS

Sir1 f tdil ‹¡o +     COS e SiR 8

The angle f between the ecliptic and the horizon is given by

 

(14.2)

 

cos f = cos e sin p — sin e cos p sin 8 (14.3)

Note that i is nor the angle which the daily path of the Sun makes with the horizon! In the course of one sidereal day, the angle varies between two extreme values. For example, for latitude 48°00' North, with e = 23° 26’, the extreme values of i are

90° — p + e  =  65° 26' for 8 =  90°

90° — p — e = 18° 34’ for 8 = 270°

Mample 14.o — For e = 23.°44, p = +51°, 8 = 5‘00” = 75°, we find, from formula (14.2), tan X = —0.1879, whence k = 169° 21' and h = 349°21'.

Formula (14.3) gives i = 62°.

 

lOO ASTRONOMICAL ALGORITHMS

Ecliptic and Equator

 

To the North

celestial

 

Let k, Q be the ecliptical longitude and latitude of a star, and e the obliquity of the ecliptic. Then, the angle Q between the direction of the northern celestial pole and the direction of the north pole of the ecliptic, at the star (see Figure at right), is given by

lf in this formula we make Q = 0°, then the formula reduces to

 

To the pole of the

ecliptic

 

pole

 

tAM Q O = -COS l tdd f

and q0 is the angle between the ecliptic (at a given point of longitude k) and the east—west direction on the celestial sphere — see the Figure below. This angle may be of importance when preparing a diagram showing the path of the Moon through the Earth’s shadow during a lunar eclipse.

Diurnal path and t/onzon

The angle I of the diurnal path of a celestial body (not the ecliptic) relative to the horizon at the time of its rising or setting can be found from

B -- tan b e! C —- UB 2 , tan I —— C cos b / e

where 6 is the declination of the body, and ‹ir the observer’s latitude. In these formulae, the declination of the body is supposed to be constant, and the atmospheric refraction is neglected. When b -— 0°, then / = 90° — p.

For example, for the Sun at latitude 40° (north or south), I varies between 50° at the equinoxes and 45°31' at the solstices.

The error in I due to neglecting the variation of the declination will be at most 4’ in the case of the Sun. For the Moon, the error can exceed 1 degree.

 

Chapter IS

Rising, Transit, and Setting

The local hour angle corresponding to the time of rise or set of a celestial body is obtained by putting h -— 0 in formula (13.6). This gives

COS L O = -tdC p tSJ1

However, the instant so obtained refers to the geometric rise or set of the center of the celestial body. By reason of the atmospheric refraction, the body is actually below the horizon at the instant of its apparent rise or set. The value of 0°34' is generally adopted for the effect of refraction at the horizon. For the Sun, the calculated times generally refer to the apparent rise or set of the upper limb of the disk; hence, 0°16' should be added for the semidiameter.

Actually, the amount of refraction changes with air temperature, pressure, and the elevation of the observer (see Chapter 16). A change of temperature from winter to summer can shift the times of sunrise and sunset by about 20 seconds in mid- northern and mid-southern latitudes. Similarly, observing sunrise or sunset over a range of barometric pressures leads to a variation of a dozen seconds in the times. However, in this Chapter we shall use a mean value for the atmospheric refraction at the horizon, namely, the value of 0°34' mentioned above.

We will use the following symbols:

L —— geographic longitude of the observer in degrees, measured positively west from Greenwich, negatively to the east (see Chapter 13);

‹,o = geographic latitude of the observer, positive in the northern hemisphere, negative in the southern hemisphere;

IT —— the difference TD — UT between Dynamical Time and Universal Time, in

seconds of time;

@ = the “standard” altitude, i.e., the geometric altitude of the center of the body at the time of apparent rising or setting, namely,

@ = —0°34’ = —0.°5667 for stars and planets; @ = —0°50' = —0°.8333 for the Sun.

101

 

For the Moon, the problem is more complicated because @ is not constant. Taking into account the variations of semidiameter and parallax, we have @ = 0.7275 z — 0°34’, where z is the Moon’s horizontal parallax. If no great accuracy is required, the mean value @ =  +0.°125 can be used for the Moon.

Suppose we wish to calculate the times, in Universal Time, of rising, of transit (when the body crosses the local meridian at upper culmination), and of setting of a celestial body at a given place on a given date D. We take the following values from an almanac, or we calculate them ourselves with a computer program :

— the apparent sidereal time B0 at 0' Universal Time on day D for the meridian of Greenwich, converted into degrees,-

— the apparent right ascensions and declinations of the body

nl and 6 on day D - 1 at 0' Dynamical Time

n¿ and 62 on day D —

n and fi, on day D + 1 —

The right ascensions should be expressed in degrees, too. We first calculate approximate times as follows.

C  S     0 (15.1)

Attention! First test if the second member is between — l and + l before calculating Off. See Note 2 at the end of this Chapter.

Express ff0 in degrees. f0 should be taken between 0° and + 180°. Then we have:

 

for the transit: =

for the rising: I " 0 "

 

360

360

 

(15.2)

 

for the setting : mt =   m0 +    3

These three values m are times, on day D, expressed as fractions of a day. Hence, they should be between 0 and + 1. If one or more of them are outside of this range, add or subtract 1. For instance, + 0.3744 should remain unchanged, but

—0.1709 should be changed to +0.8291, and + 1.1853 should be changed to

+0.1853.

Now, for each of the three m-values separately, perform the following

calculation.

 

15. RISING, TRANSIT, SETTING

Find the sidereal time at Greenwich, in degrees, from I = Bt + 360.985 647 m

where m is either , mt, or 2

 

103

 

For n —— m + k Tl 8b400, interpolate n from nt, n2, n, and 6 from 6, , 62, fi„ using the interpolation formula (3.3). For the calculation of the time of transit, 6 is not needed.

Find the local hour angle of the body from   If - 8o    L    O, and then the body’s altitude h by means of formula (13.6). This altitude is not needed for the calculation of the time of transit.

Then the correction to m will be found as follows:

— in the case of a transit,

where ff is expressed in degrees and musr be between —180 and + 180 degrees. (In most cases, H will be a small angle and be between —1° and + 1°) ;

— in the case of a rising or a setting,

h - lit

 

360 cos 6 cos ‹,o sin H where h and h are expressed in degrees.

The corrections Am are small quantities, in most cases being between —0.01 and +0.01. The corrected value of m is then m + Am. If necessary, a new calculation should be performed using the new value of m.

At the end of the calculation, each value of m should be converted into hours by multiplication by 24.

Example 15.a — Venus on 1988 March 20 at Boston,

longitude   = +71°05' = +71.°0833,

latitude = +42°20’ = +42.°3333.

From an accurate ephemeris, we take the following values:

1988 March 20, 0‘ UT : B0 = 11 50”58‘. 10 = 177.°74208

Coordinates of Venus at 0‘ TD:

March 19 «; = 2‘42*43'25 = 40°.68021 ô, = + 18°02’51“,4 = + 18.°04761

March 20 n2 = 2 46 55.51 = 41.73129 @ = +18 26 27.3 = + 18.44092

March 21 ri, = 2 51 07.69 = 42.78204 ô3 = + 18 49 38.7 = + 18.82742

 

We take /i0 = —0°.5667, IT -— +56 seconds, and find by formula (15.1) cos ff0 = —0.317 8735, ftp = 108.°5344, whence the approximate values:

transit : zriq =  —0.18035, whence — +0.81965 rising : m = m0 — 0.30148 = +0.51817

setting : nt2 = tub + 0.30148 = + 1.12113, whence m2 = +0.12113 Calculation of more exact times:

rising transit secting

m +0.51817 +0.81965 +0. 12113

’0

  4°.79401

+0.51882 113°. 62397

+0.82030 221.°46827

+0.12178

inter- J n 42°.27648 42.°59324 41°.85927

polation 1 6 + 18°.64229 + 18°.48835

H — 108°.56577 —0.°05257 + 108.°52570

h —0°.44393 —0°. 52711

Am —0.00051 +0.00015 +0.00017

corrected m +0.51766 +0.81980 +0.12130

A new calculation, using these new values of m, yields the new corrections

—0.000 003, —0.000 004, and —0.000 004, respectively, which can be neglected. So we have, finally:

rising : mt = +0.51766, 24‘ x 0.51766 = 12 25‘ UT

transit: = +0.81980, 24h x 0.81980 = l9‘41” UT

setting: = +0.12130, 24a x 0.12130 = 2"55“ UT

Notes

1. In Example 15.a we found that at Boston the time of setting was 2'55 UT on March 20. However, converted to local standard time this corresponds to an instant on the evening of the previous day! If really the time of setting on March 20 is needed in local time, the calculation should be performed using the value m2 = + 1.12113 first found, instead of +0.12113.

2. If the body is circumpolar, the second member of formula (1â.1) will be larger than 1 in absolute value, and there will be no angle Hi. In such a case, the body will remain the whole day either above or below the horizon.

3. If approximate times are sufficient, just use the initial values   , ml , and mt given by (15.2).

 

Chapter 16 Atmospheric Refraction

Atmospheric refraction is the bending of light while passing through the Earth’s atmosphere. As a ray of light penetrates the atmosphere, it encounters layers of air of increasing density, resulting in the continuous bending of the light. As a result, a star (or the Sun’s limb, etc.) will appear higher in the sky than its true position. The atmospheric refraction, which is zero in the zenith, increases towards the horizon. At an altitude of 45°, the refraction is about one arcminute; at the horizon, it amounts to about 35’. Thus, the Sun and the Moon are actually below the horizon when they appear to be rising or setting. Moreover, the rapidly changing refraction at low altitudes gives the rising or setting Sun its familiar oval appearance.

Allowance must be made for atmospheric refraction when determining positions, and one distinguishes two cases:

— the apparent altitude @ of a celestial body has been measured, and one should find the refraction fi to be subtracted from @ to obtain the true altitude h,-

— the true “airless” altitude h has already been calculated from celestial coordinates and formulae of spherical trigonometry, and we want to calculate the refraction R  to be added to h in order to predict the apparent altitude lit.

Almost all refraction formulae we have come across consider the first case only: they are designed for deriving true altitudes from observed ones. But here we will consider both cases.

For many purposes, “average” meteorological conditions may be assumed. However, anomalous refraction near the horizon, exemplified by distoHions of the setting Sun, should remind us that rigorous exactness at very low altitudes cannot be reached.

When the altitude of the celestial body is larger than 15 °, one of the following two formulae may be used, as the case may be:

 

R —— 58“.294 tan (90° J$o) 0“.0668 tan3 (90° — @)

R —- 58”.276 tan (90° — h) - 0”.0824 tan' (90° — h)

105

 

(16. 1)

(16.2)

 

The first formula was given by Smart [1], while the second one has been derived by us from the first formula. For altitudes below 15‘, these expressions will give inaccurate, or even completely meaningless results.

It appears that, at high altitudes, the refraction is proportional to the tangent of the zenithal distance.

A surprisingly simple formula for refraction, with good accuracy at all altitudes from 90° to 0°, was given by G. G. Bennett of the University of New South Wales [2]. If the refraction R is expressed in minutes of arc, Bennett’s formula is

 

1

     7.31

0 + 4.4

 

(16.3)

 

where @ is the apparent altitude in degrees. According to Bennett, this formula is accurate to 0.07 arcminute for all values of 0s The largest error, 0.07 arcminute, occurs at 12° altitude.

Note that for the zenith (@ = 90°) formula (16.3) yields fi = —0”.08 instead of exactly zero. This can be rectified by adding +0.001 3515 to the second member of the formula.

Bennett also showed how his formula can be refined. Calculate R by means of formula (16.3); then a correction to It, expressed in minutes of arc, is

—0.06 sin (14.7fi + 13)

where the expression between parentheses is expressed in degrees. Calculated in this way, the maximum error is stated to be only 0.015 arcminute, or 0“.9, for the whole range 90° —0°. [At the zenith, one finds A = —0”. 89, so expression (16.3), without further correction, is better in this case.]

For the inverse problem, that of calculating the effect of refraction when the true altitude h is known, Smmundsson, of the University of Iceland, proposed the following formula [3]:

 

  1.02

 

(16.4)

 

tan h +

 

      10.3

h + 5. 11

 

This formula is consistent with Bennett’s (16.3) to within 4". Again, it does not give exactly ft = 0 for h —- 90‘. This can be remedied by adding +0.0019279 to the second member.

Formulae (16.1) to (16.4) assume that the observation is made at sea level, when the atmospheric pressure is 1010 millibars, and when the air temperature is 10° Celsius. The effect of refraction increases when the pressure increases or when the temperature decreases.

If the pressure at the Earth’s surface is P millibars, a:nd the air temperature is

 

16. ATMOSPHERIC REFRACTION 10 7

T degrees Celsius, then the values of R given by the formulae (16. l) to (16.4) should be multiplied by

  283

1010 273 + T

However, this is only approximately correct. The problem is more complicated because the refraction depends on the wave-length of the light too! The expressions given in this Chapter are for yellow light, where the human eye has maximum sensitivity.

Example 16.a -- Calculate the apparent flattening of the solar disk near the horizon, when the lower limb is at an apparent altitude of exactly 0°30'. Assume a true solar diameter of exactly 0°32’, and mean conditions of air pressure and temperature.

For hg ---- 0.°5, formula (16.3) gives R 28'754, so the true altitude of the Sun’s

lower limb is

0°30' -- 0°28'.754 = 0°01'.246

and hence the true altitude of the upper limb is

h -- 0°01'.246 + 0°32’ = 0°33’.246 = 0°.5541

For this value of h, formula (16.4) yields R -- 24’.618, so the apparent altitude of the Sun’s upper limb is 33'.246 + 24!618 = 57'.864, and the apparent vertical diameter of the solar disk is 57'.864 -- 30' = 27'864.

Consequently, the ratio of the apparent vertical diameter to the horizontal diameter of the solar disk, under the conditions of this Problem, is 27.864 132 = 0.871.

Note that, while of course the azimuth is unchanged by refraction, the horizontal diameter of the solar disk is very slightly contracted by reason of the refraction. This is due to the fact that the extremities of this diameter are raised along vertical circles that meet at the zenith. Danjon [4) writes that the apparent contraction of the horizontal diameter of the Sun is practically constant and independent of the altitude, and that this contraction is approximately 0“.6.

For heights of a few degrees the results of the formulae should be judged with care. Near the horizon unpredictable disturbances of the atmosphere become rather important. According to investigations by Schaefer and Liller [5], the refraction at the horizon fluctuates by 0°.3 around a mean value normally, and in some cases apparently much more. Remembering our Chapter about accuracy, it should be mentioned here that giving rising or setting times of a celestial body more accurately than to the nearest minute makes no sense.

 

RE FEREN CES

1. W.M. Smart, Text-Book on Spherical Astronost y ; Cambridge (England), University Press (1956); page 68.

2. G. G. Bennett, “The Calculation of Astronomical Refraction in Marine Navigation”,

Journal of the Institwe for Navigation, Vol. 35, pages 255-259 (1982).

3. 1orsteinn Szmundsson, Sky and Telescope, vol. 72, page 70 (July 1986).

4. A. Danjon, Asironomie Générale (Paris, 1959); page 156.

5. B.E. Schaefer, W. Liller, “Refraction near the Horizon”, Publ. Astron. Society of the Pacific, Vol. 102, pages 796—805 (July 1990).

 

Chapter 17

Angular Separation

The angular distance d between two celestial bodies whose right ascensions and declinations are known is given by the formula

Gos d —— sin 6 I sin fi2 + cos é cos by cos (o, — n 2) (17.1)

where n and é; are the right ascension and declination of one body, and O¿ HRd z

those of the other body. This distance is measured along the great circle joining the

two bodies, which is the shortest possible arc between the two points.

The same formula may be used when the ecliptical (celestial) longitudes X and latitudes 9 of the two bodies are given, provided that o, , „ é„ and @ are replaced by h, , , and 9t, respectively.

Formula (17.1) may not be used when d is very near to 0° or to 180° because in those cases I C S d  is nearly equal to 1 and varies very slowly wilh d, so that d cannot be found accurately. For instance,

cos 0°01’00” = 0.999 999 958

cos 0°00'30“ = 0.999 999 989

cos 0°00’15” = 0.999 999 997

cos 0°00’00” = 1.000 000 000

If the angular separation is small, say less than 0°10', then this separation may be calculated by means of the approximate formula

d —— ( cos )2 -I- ( )2 (17.2)

where As is the difference between the right ascensions, Aé the difference between the declinations, while 6 is the average of the declinations of the two bodies. Note that As and A6 should be expressed in the same angular units,

If As is expressed in hours (and decimals), A6 in degrees (and decimals), then

d expressed in seconds of a degree (”) is given by

 

d -— 3600 (1 )2 + ( )

109

 

(17.3)

 

If An is expressed in seconds of time (’), and Ah in seconds of a degree (”), then d expressed in “ is given by

d -- (15 co )2 + (    )2 (17.4)

Formulae (17.2), (17.3), and (17.4) may be used only when d is small.

However, see also the alternative formulae further in this Chapter.

Example 17.a — Calculate the angular distance between the stars Arcturus (n Boo) and Spica (n Vir).

The J2000.0 coordinates of these stars, as taken from a catalogue, are

n Boo : n; = 14 15‘39'7 = 213°. 9154

6t = + 19° 10'57“ = + 19°.1825

n Vir : nt = 13h25‘11‘.6 = 201°. 2983 Q = — 11°09'41“ = — 11°. 1614

Formula (17.1) gives cos d —— +0.840 633, whence d —— 32°.7930 = 32°48'.

Of course, this distance holds only for the epoch for which the stars’ coordinates are given, namely 2000.0. It varies slowly with time, by reason of the proper motions of the stars. It is, however, independent of the precession.

Exercis e. — Calculate the angular distance between Aldebaran and Antares. (Answer: 169°58').

One or both bodies may be moving objects. For example: a planet and a star, or two planets. In that case, a program may be written where first the quantities fi I , fi2, and (m l — nt) are interpolated, after which d is calculated by means of the formulae (17. 1) or (17.2).

Exercise. — Using the following coordinates, calculate the instant and the value of the least angular separation between Mercury and Saturn.

1978

0h TD Me rc u ry 5 atu rn

'2

h m s ° ' “ h m s ' “

Sep 12 10 23 17.65 + 11 31 46.3 10 33 01.23 + 10 42 53.5

13 10 29 44.27 + 11 02 05.9 10 33 29.64 + 10 40 13.2

14 10 36 19.63 + 10 29 51.7 10 33 57.97 + 10 37 33.4

15 10 43 01.75 + 9 55 16.7 10 34 26.22 + 10 34 53.9

16 10 49 48.85 + 9 18 34.7 10 34 54.39 + 10 32 14.9

 

Answer: The least angular separation between the two planets was 0°03'44", on 1978 September 13 at 15‘06*.5 TD = 15'06‘ UT.

As we see, this was a rather close conjunction. We must insist on the fact that, in such a case, first the quantities 6 , 62, and (n t — nt) should be interpolated, not the distances themselves. The distance is to be deduced from the interpolated coordinates.

Suppose that, nevertheless, we try to interpolate the distances themselves. By means of formula (17.1), we find the following distances between Mercury and Saturn, in degrees and decimals, for the five given times:

1978 Sep 12.0 TD d —— 2.°5211

13.0 d2 = 0.9917

14.0 d —- 0.5943

15.0 d4 = 2.2145

16.0 dz —- 3.8710

It is evident that the least separation occurs between 13.0 and 14.0 September, and closer to 14.0 than to 13.0.

If we now use the three central values d2, dz, d   and calculate the value of the minimum by means of formula (3.4), we obtain 0‘.5017 = 0°50'06”. Taking the five values d   to dz, formula (3.9) yields a “better" value for rig, after which (3.8) is used to calculate the value of the function for that value of the interpolating factor n; this gives 0.°4865 = 0°29’11“.

Both results are completely wrong, however. As has been mentioned above, the correct value of the least distance is only 0°03’44”. So, what happened?

The reason is that the conjunction was a close one. Until a short time before the least distance, Mercury was moving almost exactly straight towards 5aturn, and the angular distance between the two planets was decreasing almosi exacUy linearly with time. Similarly, some short time after the least distance, Mercury was moving almost straight away from Saturn.

In the Figure on next page, the solid curve represents the true variation of the angular separation between the two planets. Except very close to the least distance, this curve consists of two almost exactly straight segments (one near B, the other from C to D), and in such a case the interpolation formulae are no longer valid!

Formulae (3.3), (3.4) and (3.5), for instance, suppose that the function, in the considered part of the curve, is a pnrobofn. But the curve is not a parabola, except very close to the minimum, inside the small rectangle.

If we make use of the three points B, C, D, corresponding to the three central distances @, d , dz, then by the interpolation formula (3.3) we in fact draw a parabola through those three points; it is the dashed curve in the Figure. This parabola differs considerably from the true curve, and in particular its minimum is too high.

 

time

And it would be of no help to use the Ave values d to d instead of the three central ones, because the solid curve differs even considerably from a polynomial of the fourth degree!

Hence, performing an interpolation from the distances canriot give accurate results. As we have said, we must interpolate the original coordinates separately, and only then can the accurate distance for an intermediate instant be deduced. Using the interpolation formula (3.8), we so find the value of the distance for several values of the interpolating factor n:

n = —0.50

—0.45 distance = 0.21437 degree

0.14057

—0.40 0.07790

—0.35 0.07028

—0.30 0.12815

The least separation occurs for n between —0.40 and —0.35, so we calculate the angular distance for three more values, at smaller intervals (but, again, from the interpolated coordinates) :

n =   —0.38 distance = 0.06408 degree

—0.37 0.06229

—0.36 0.06448

The tabular interval is now small enough so that formulae (3.4) and (3.5) may be used. We find that the least separation is 0°.06228 = 0°03’44”, for rig =

—0.370 502, corresponding to September 13.629 498 = September 13 at 15'06U5 TD, as mentioned earlier.

 

It is possible, however, to find the least angular separation without trying sev- eral values of the interpolating factor n, namely, by using rectangular coordinates. These coordinates u and v, in seconds of arc, can be calculated as follows [1].

Calculate the auxiliary quantity

_ 206 264.8062

1  +  Sln2 An

where 206 264. 8062 is the number of arcseconds in one radiaii. Then u = — R (1 — tan fit sin A6) cos 6, tan  As

v = K (sin A6 + sin 6t cos 6t tan= '= 2

In the above expressions, o , é are the right ascension and declination of the first planet, and As = nt — n , A6 = h 2 — 6; , where o , fi2 are the right ascension and declination of the second planet.

Calculate the values of ti and v for three equidistant times. For any intermediate time, then, their values can be interpolated by means of formula (3.3), while their variation (in arcseconds per unit of the tabular interval) is given by

U’ — *’ + 7t (Ut  + Uy — 2 )

where n is the interpolating factor, and u I , u2, u, art  the three calculated values of u, and with a similar expression for the variation v'.

Start from any value for the interpolating factor n ; a good choice is it = 0. For this value of n, interpolate u and v by means of formula (3.3), and find the variations u' and v’. Then the correction to n is given by

 uu' + vv’ 

So the new value of n is n + An. Repeat the calculation for the new value of n until the correction An is a very small quantity, for instance less than 0.000 001 in absolute value.

For the final value of n, calculate ti and v again. Then the least distance, in arcseconds, will be 2 + .

Let us apply this method to the above-mentioned conjunction between Mercury and Saturn. The three chosen instants are 13.0, 14.0, and 15.0 September 1978. We find the following values for u and v, retaining one extra decimal to avoid rounding errors:

 

Sept. 13.0 —3322”.44 —1307“.48

14.0 +2088.54 + 463.66

15.0 +7605.36 +2401.71

For n = 0, we have

ii = +2088.54 u' = +5463.90

v = + 463.66 v’ -- + 1854.595

whence  An = —0.368 582, and the corrected value of n is  0 — 0.368 582 =

—0.368 582.

For this new value of n we find

ii = +   81.83 ti' =- +5424.89

v = —208.57 v’ = +1793.07

whence in —— —0.002 142, and the new corrected value of n is

—0.368 582 — 0.002 142 = —0.370 724.

A new iteration gives An = —0.000 003, so the final value of it is

—0.370 724 — 0.000 003 = —0.370 727.

[This value differs from the value n —— —0.370 502 found before, because in the present calculation we used the positions of the planets for only three instants instead of five. But the difference is only 0.000 225 day, or 19 seconds.]

For the value n = —0.370 727, we find u —— +70”.20, v = —212“.42, and consequently the least distance between the two planets is

2 +   v 2   =   224”    =    3’44“,

as found before.

The same method can be used if one of the bodies is a star. The latter’s coordinates are then constant, but it is important to note that the o and b of the star should be referred to the same equinox as that of the moving body.

If the moving body is a major planet whose apparent right ascension and dec- lination referred to the equinox of the date are given, then for the star the apparent coordinates too must be used. If one takes the star’s position from a catalogue, where it is referred to a standard equinox (for instance that of 2000. 0), then the apparent ct and 6 are found by taking into account the proper motion of the star and the effects of precession, nutation, and aberration, as explained in Chapter 23.

If the o and 6 of the moving body are referred to a standard equinox (astrometric coordinates), then the n and 6 of the star should be referred to this same standard equinox, the only correction being those for the proper motion of the star.

 

Alternative cannulae

Although formula (17.1) is truly exact, mathematically speaking, its accuracy is very poor for small values of the angle d, as has been seen at the beginning of this Chapter. For this reason, several alternative methods have been proposed.

One of them (2] consists in using the old haversine (hay) function, which can be a great aid in certain astronomical calculations involving small angles, as it can preserve significant digits. By definition, for any angle 8, we have

hay 8 1 - COS # 

The cosine formula (17. 1) for angular separation is precisely equivalent to hav d —- hav AT + cos 6 cos éj hav An (17.5)

where As = o — o 2 , A6 = él — 62. To use this formula on a computer we can get the help of another identity, namely

hay 8   =   Sifl°   —y

By means of formula (17.5), angular separations can be calculated accurately for angles from nearly 180° all the way down to exactly 0 degree!

V.J. Slabinski [3] offers another approach that can be used:

Slfl 2d ——  (cos 6  S1n )2 +    (sirt     z COS     i COS to COC     z StTl @ j) 2

However, this formula cannot distinguish between supplementary angles, for instance 144° and 36°, and it has a poor accuracy when d is close to 90°.

Mr. Thierry Pauwels, of the Royal Observatory of Belgium, communicated the following method. Calculate

x --  cos fit  sin   —   sin  ht  cos ét  cos (• —2 •  )

y —- cos fi2 sin (a 2 — n t)

 

and then

 

Z = sin bt sin §z + cos §i cos § cos (Oj — Oj)

d —— arctan

 

where d should be taken between 90 and 180 degrees if z is negative.

 

Mathematically speaking, this method is completely identical to formula (17. l), but a computer will yield more accurate results from an arctangent than from an arccosine.

Example 17.b -- Taking again the case described in Example 17.a, we find

z = --0.497 404

y 0.214 303

z =   +0.840 633

from which tan d -- 0.644 283, d -- 32°. 7930, as in Example 17.a.

Relative Position gle

The Position Angle I of a body (nt , h;) with respect to another body (n 2, 6t) can be found from

  sin As                       cos 6z tan 6i -- sin fit cos As

If the denominator of the fraction is negative, then P lies in the range 90° --270°.

REFEREN CES

1. A. Danjon, Astronomie Générale, page 36, formulae 3 bis (Paris. 1959).

2. Sky and Telescope, Vol. 68, page 159 (August 1984).

3. Slcy and Telescope, Vol. 69, page 158 (February 1985).

 

Chapter 18

Planetary Conjunctions

Given three or five ephemeris positions of two planets passing near each other, a program can be written which calculates the time of conjunction in right ascension and the difference in declination between the two bodies at that time. The method consists in calculating the differences An of the corresponding right ascensions, and then calculating the instant when An = 0 by means of formula (3.6) or (3.7) in the case of three positions, or (3. 10) or (3. 11) in the case of five points. When that instant is found, direct interpolation of the differences A6 of the declinations, by means of formula (3.3) or (3.8), yields the required difference in declination at the time of conjunction.

Conjunctions in celestial longitude can be calculated in the same way using, of course, the planets’ geocentric ecliptical (celestial) longitudes and latitudes instead of their right ascensions and declinations.

Note that neither the instant of the conjunction in right ascension, nor that of the conjunction in longitude, coincides with that of the least angular separation between the two bodies. 2ty definition, conjunction is the phenomenon in which two bodies have the same apparent right ascension or celestial longitude as viewed from a third body (generally the Eanh).

Example 18.a — Calculate the circumstances of the Mercury —Venus conjunction of 1991 August 7, using the following apparent positions, for 0' TD of the date, which are taken from an accurate ephemeris:

Date, Me r c u r y

 

h m s ° ’ “ h m s ° ' ”

Aug. 5 10 24 30.125 +6 26 32.05 10 27 27.175 + 4 04 41.83

6 10 25 00.342 +6 10 57.72 10 26 32.410 + 3 55 54.66

7 10 25 12.515 +5 57 33.08 10 25 29.042 + 3 48 03.51

8 10 25 06.235 +5 46 27.07 10 24 17.191 + 3 41 10.25

9 10 24 41.185 +5 37 48.45 10 22 57.024 + 3 35 16.61

117

 

We first calculate the differences of the right ascensions (in seconds of time) and

those of the declinations (in degrees and decimals) :

Aug. 5 An = —177.050 Aö = +2.363950

6 — 92.068 +2.250850

7 — 16.527 +2.158214

8 + 49.044 +2.088006

9 +104.161 +2.042178

Applying formula (3.10) to the values of As, we find that As is zero for the value n = +0.23797 of the interpolation factor. Hence, the conjunction in right ascension took place on

1991 August 7.23797 = 1991 August 7 at 5 42*.7 TD

= 1991 August 7 at 5 42'° UT

With the value of n just found, and applying formula (3.8) to the values of A6, we find Ah = +2°.13940 or +2°08'. Hence, at the time of the conjunction in right ascension, Mercury was 2°08’ north of Venus.

If the second body is a star, its coordinates may be considered as being constant during the time interval considered. We then have

l' ‘2’ ‘3 ' ‘4’ ‘5

The program should be written in such a way that, if the second object is a star, its coordinates must be entered only once.

The important remark given on page 114 does apply here too: i/te coordinates of the star and those of the moving body must be referred to the same equinox.

 

18.   PLANETARY CONJUNCTIONS 119

As an exercise, calculate the conjunction in right ascension between the minor planet 4 Vesta and the star Q Librae in February 1996. The minor planet’s right ascension and declination, referred to the standard equinox of J2000.0, are as follows (from an ephemeris calculated by Edwin Goffin) :

0h TD

  ‘2 ID

1996 Feb. 7 h   m s

15 03 51.937

—°8 57 34 51

12 15 09 57.327 —9 09 03.88

17 15 15 37.898 —9 17 37.94

22 15 20 50.632 —9 23 16.25

27 15 25 32.695 —9 26 01.01

The star’s coordinates for the epoch and equinox of 2000.0, taken from the FK5 star catalogue, are n’ = 15h17'"00'.421 and 6' = —9°22’58".54, and the centennial proper motions (that is, the proper motions per 100 years) are —0'.649 in right ascension and — 1’.91 in declination.

Consequently, from the proper motions during the —3.87 ygars (—0.0387 century) from 2000.0, we find that the star’s position referred to the equinoi of 2000.0, but for the epoch 1996.13, is

‹x’ = 15‘17*00'.446,   6’ = —9° 22’58".47

Now, calculate the conjunction.

Answer: Vesta passed 0°03’38’ north of 9 Lib on 1996 February 18 at 6‘37" Dynamical Time.

Do not confwe conjunction with least angular separation. Two planets are in conjunction when their right ascensions (or their celestial longitudes) are equal. At lefi in the Figure, the motion of planet 1 with respect to planet 2 is depicted. There is conjunction when ihe first planet arrives in A, end this is not the insrnnr of least separation. In the drawing ai right, the least ‹zngubr sepnrnrion occurs in C. Stir it is clear that there is no conjunction here.

 

 

 

Chapter 19

Bodies in Straight Line

In this Chapter and in the next one, we shall deal with two problems which have no importance “scientifically”, but which may be of value to persons interested in nice celestial events or to authors of popular articles.

Let (n; , fit), (nt, fi2), (n„ 63) be the equatorial coordinates of three heavenly bodies. These bodies are in “straight line” — that is, they lie on the same great circle of the celestial sphere — if

tan 6t  sin (n 2 — n  )  + sin (n—  • +  tan  h   sin (n—; •   )  =  0 (19. 1)

This formula is valid for ecliptical coordinates too, provided that the right ascensions n are replaced by the longitudes h, and the decllnations 6 by the latitudes Q.

Do not forget that the right ascensions are generally expressed in hours, minutes, and seconds. They should be converted to hours and decimals, and then into degrees by multiplication by 15.

If one or two of the bodies are stars, then once again the important remark given on page 114 does apply: the coordinates of the star(s) must be referred to the same equinox as that of the planets.

Example 19.a — Find the instant when Mars was seen in straight line with Pollux and Castor in 1994.

From an ephemeris of Mars and a star atlas, it is found that the planet was in straight line with the two stars about 1994 October 1. For this date, the apparent equatorial coordinates of the stars were:

Castor (a  Gem) : n    = 7‘34*16'40 = 113°. 56833

6t  =  + 31° 53’51“.2 = + 3l °.89756

Pollux (Q Gem) : n2 = 7‘45‘00• 10 = 116°. 25042

6 = +28° 02'12“.5 = +28°. 03681

121

 

For our problem, these values of a , 6t , n„ and ét may be considered as being constant for several days.

The apparent coordinates of Mars (a„ fi3 ) are variable. Here are their values, taken from an accurate ephemeris:

TD

h m s ° ° “ °

1994 Sep. 29.0 7 55 55.36 = 118.98067 +21 41 03.0 = +21.68417

30.0 7 58 22.55 = 119.59396 +21 35 23.4 = +21.58983

Oct. 1.0 8 00 48.99 = 120.20413 +21 29 38.2 = +21.49394

2.0 8 03 14.66 = 120.81108 +21 23 47.5 = +21.39653

3.0 8 05 39.54 = 121.41475 +21 17 51.4 = +21.29761

Using these values, the first member of formula (19.1) takes the following values:

Sep. 29.0 +0.0019767

30.0 +0.tD1085l

Oct. 1.0 +0.0001976

2.0 —0.0006855

3.0 —0.0015641

By means of formula (3.10), we find that the value is zero for 1994 October 1.2233 = 1994 October 1, at 5 TD (UT)

In the preceding Example, we made use of geocentric positions of Mars. For this reason the result is, strictly speaking, valid only for a geocentric observer, and for an observer for whom Mars is at the zenith. But for the present problem, it is not worthwhile to take into account the parallax of the planet, which is very small. This is no longer true in the case of the Moon, whose parallax can reach one degree. In this case, the topocentric position of the Moon should be used (see Chapter 40).

Straight lines on the celestial sphere

Once on a winter evening I admired the constellation Orion, when suddenly I thought about the following problem: the three stars of Orion’s “Belt” (6, e, and Orionis) are nearly on a “straight line” on the sky. But how nearly straight, precisely? Then I remembered another nearly-straight-line: when, according lo the well-known rule, the line joining the stars n and fi of Ursa Major is extended northward, we arrive close to the Pole Star (a Ursae Minoris). But exactly how close?

I obtained the following formulae which I give here without proof. Remember that a straight line on the celestial sphere is actually an arc of a great circle.

 

Consider the three stars S , fi2, and fi3 , whose right ascensions and declinations are n , 6 , n2, 62, And n„  6„  respectively, in such a way that 5 2 iS the middle star. The angle S  - S2 — S,  , that is, the angle which the arc fi , S  makes with the arc S2S„ is equal to C + C 2, where the angles C and Ct are given by the following formulae and should be taken between 0° and + 180° :

 

c, =

tnn C

 

 

cos h 2 tan 6 t — sin 62 cos (n 2 — n t)

 

The drawing represents the three stars S , 52, and S . P is the northern celestial pole. The arcs PS I , PS , and P8, If re the celestial meridians (the great circles of constant right ascension) through the three stars. The Figure also illustrates the meaning of the angles Cl and C2.

If the three stars are taken in increasing order of their right ascension (that is, n < n2 < n3), then Cl + C2 is the value of the northern angle at 52.  Of course, this angle can be larger as well as smaller than 180 degrees.

As an example, let us consider the three stars of Orion’s Belt. Their positions for the epoch and equinox 2000.0 are:

fi Ori 5'32*00'.40 —0° l7'56“.9

e Ori 5'36‘12'81 —1°12’07“.0

} Ori 5‘40‘45'52 —1°56’33“.3

 

We find C -- 49°.3622 and C 2 = 123.° 1209. The sum is 172.4831 deg- rees, or 172°29’. So the three stars of Orion’s Belt indeed are not exactly aligned. They form an obtuse angle of 172'/z degrees. Because C + C 2 is smaller than 180°, and we took the stars in increasing order of their right ascensions, the middle star (e Ori) is a little south of the great circle through 6 and Ori.

At what angular distance is e from this great circle? This can be found as follows.

 

ce/es r/a/ eqn  a toc

 

We have the two stars S (nt , ét) and S (« 2, ét), and we wish to calculate the angular distance of a third star fit (at, 60) to the great circle S - 52 . Calculate

 

K — COS b j COS n t

Y —— COS 6 t SiR O

 

 

z = cos   z cos nt A -- Y Zy   Z$ pz pz — cos 62 sin °z B — Njll      ] Zz z = sin   z C — X] Yy pi z

 

   

The required angular distance o is then given by

sin o   = A + Bm + Cn

+ B   + C 1 + + n

where o should be taken between 0° and 90°.

As an example, let us again consider the three stars of Orion’s Belt. Now b On and } Ori are the stars S and S respectively, and we want to calculate the angular distance of e Ori (= star S0) to the line h—}.

Using the stars’ positions mentioned above, we find     — 0.°089 876 = 324”, or a little more than 5 arcminutes.

As an exercise, the reader can calculate the distance of the Pole Star (ct UMi) to the line, extended northward, joining n and Q Ursae Majoris. The 2000.0 positions of these stars are:

o UMa n = 11 03”43'.666 6 = +61°45’03".22

Q UMa 11'01‘50\482 +56°22’56“.65

« UMi 2'31"48‘.704 +89°15‘50“.72

Answer: in = 1°55'. Hence, the line from n to 9 Ursae Major is extended northward misses o Ursae Minoris by almost two degrees.

 

After the preceding formulae were published in the Belgian journal Heelal of May 1988, we received a letter from Mr. Ben Piessens, of Mechelen, Belgium, who gave another way to calculate the angle between two great circles on the celestial sphere. He wrote:

The angle between two planes (or two great circles) as well as the angle between a straight line and a plane can easily be calculated through analytic geometry. For this, only one formula is needed, namely, the expression for the angle between two directions. The angle between two planes is equal to the angle between the perpendiculars to these planes. The angle between a straight line and a plane is the complement of the angle between that straight line and the perpendicular on that plane.

For our problem we then have, using the same symbols (a t , etc.) as before, and

O being the center of the celestial sphere, that is, the observer: Direction numbers of the straight lines OS , 05 2, OS .’

 

a    = cos ö1 Cos u 1

ft2 VOS b2 COS Ct 2

6$ COS Ö$ COS     3

 

b —— cos ö; sin o   bz ' cos z sin ° z

f›3 ' COS b3 SÏR O j

 

Direction numbers of the perpendiculars to the planes OS Sz  OS S , OS 5 :

 

m t = c a2 — ct a

fft2—      C 2 03 C $ t22

 

• - a b  — az b

 

 

   

With these data one can calculate the angle between any two great circles, or the angle between one of the straight lines OS , OS z. OS z and the great circle through the two other points. Let be the angle between the great circles O Sz and O 5253 , and o the angle between O S 2 and the plane OS I S Then we have

 

COS

 

2 2 + 2 z2 + z 2

 

   

If we consider again the case of the stars h, e, } Orionis, we flnd    = 7°3l ', in agreement with our previous result, l72°29'. Indeed, at the crossing point of two arcs there are two angles which are supplementary: 172°29' + 7°31' = 180°.

 

 

 

Chapter 20

Smallest Circle containing three Celestial Bodies

Let A, B, C be three celestial bodies situated not too far from each other on the celestial sphere, say closer than about 6 degrees. We wish to calculate the angular diameter of the smallest circle containing these three bodies. Two cases can occur:

type I :   the smallest circle has as diameter the longest side of the triangle DC,

and one point is inside of the circle;

type II : the smallest circle is the circle passing through the three points d, B, C.

TY   e   l TUp e II

The diameter A of the smallest circle can be found as follows. Calculate the lengths (in degrees) of the sides of the triangle AB C by means of the method given in Chapter 17.

Let a be the length of the longest side of the triangle, and b and c the lengths of the two other sides.

127

 

 

If   o > b2 + c , then the grouping is of type I, and A = o;

if a < b 2 + 2 , then the grouping is of type II, and

  2 abc

(a + b + c) (a + b - c) ‹b + c - a) {a + c - b)

 

(20.1)

 

 

Exampfe 20.a — Calculate the diameter of the smallest circle containing Mercury, Jupiter, and Saturn on 1981 September 11 at 0‘ Dynamical Time. The positions of these planets at that instant were:

 

Mercury Jupiter Saturn

 

= 12 41‘08'.63 é = —5°37'54“.2

12 52 05.21 —4 22 26.2

12 39 28.11 —1 50 03.7

 

The three angular separations are found by means of (17. 1) : Mprcury-Jupiter 3.°00l52

Mercury-Saturn 3.82028

Jupiter—Saturn 4.04599 = n

Because 4.04599 is smaller than (3 152)   + (3 028) , or 4.85836, we calculate A by means of formula (20. l). The result is

A = 4°.26364 = 4° 16'

This is an example of type II.

As an exercise, perform the calculation for the planets Venus, Mars, and Jupiter on 1991 June 20 at 0‘ TD, using the following positions:

Venus n = 9h05‘ 41\44 é = + l8°30’30“.0

Mars 9 09 29.00 +17 43 56.7

Jupiter 8 59 47.14 +17 49 36.8

Show that this is a case of type I, and that A = 2°19’.

A program can be written in which first the right ascensions and the declinations of the planets are interpolated, after which a, b, c, and finally A are calculated. With such a program, it is possible to calculate (by trial) the minimum value of A of a grouping of three planets. Indeed, A varies with time, and the method described in this Chapter provides the value of A for only one given instant.

 

20.   SMALLEST CIRCLE 129

It is important to note that, while the positions of the planets can be interpolated by means of the usual formulae, the values of the circle’s diameter A cannot. The reason is that the variation of A generally cannot be represented by a polynomial. See, for instance, the graph in Example 20.c, on the next page.

Example 2#.h — In September 1981, there was a grouping of the planets Mercury, Jupiter, and Saturn. The positions of these planets were as follows; instead of right ascensions and declinations, we will use ecliptical coordinates (longitudes and latitudes) here.

1981

Oh TD Mercury Jupiter Saturn

long. latit. long. latit. long. latit.

Sep. 7 186.045 —0.560 192.866 +1.117 189.324 +2.226

8 187.482 —0.696 193.069 +1.116 189.439 +2.225

9 188.897 —0.833 193.272 +1.114 189.555 +2.224

10 190.290 —0.971 193.476 +1.113 189.671 +2.223

11 191.661 —1.109 193.681 +1.112 189.788 +2.222

12 193.008 —1.246 193.886 +1.110 189.906 +2.221

13 194.332 —1.384 194.092 +1.109 190.023 +2.220

14 195.631 —1.521 194.299 +1.108 190.142 +2.219

We will not give details here, and leave it as an exercise to the reader. Let us just mention that from September 7.00 to 8.81 the grouping was of type 1, the diameter A of the smallest circle decreasing almost linearly from 7°0l‘ to 5°00'. From September 8.81 to 12.19, the grouping was of type lI, and A reached a minimum value of 4°14' on September 10.53. From September 12.19 on, the grouping was of type I again, A increasing almost linearly with time.

Example 20.c — Let us now consider the following fictitious case. On March 12.0, the ecliptical coordinates (in degrees) of three planets are as follows.

 

longitude

 

latitude

 

daily mol:ion

In /ong‹rude

 

planet P1 214.23 +0.29 >0.11

planet P2 211.79 +0.48 <0. 20

planet P3 208.41 +0.75 1.08

We suppose that the latitudes are constant and that the longitudes increase at the constant rates mentioned in the last column.

Again, we leave the actual calculation as an exercise to the reader. Let us just

 

illustrate the variation of the diameter A of the smallest circle (see the Figure below). Note the discontinuities at the points A and B. Except during two short periods (March 15.87 to 15.91 near A, and March 17.93 to 18.05 near B), where the grouping is of type II, we have type I. The least value of A, namely 1°55’, occurs at B on March 17.94.

12 14 16 is 20 22

da ys  (March)

If one of the bodies is a star, once again the important remark made on page 114 does apply: the coordinates of the star should be referred to the same equinox as that for the planets.

 

Chapter 21

Precession

The direction of the rotational axis of the Earth is not really fixed in space. Over time it undergoes a slow drift, or precession, much like that of a spinning top. This effect stems from the gravitational attraction of the Sun and the Moon on the Earth’s equatorial bulge.

Due to the precession, the northern celestial pole (presently situated near the star o Ursae Minoris, or Polaris) slowly turns around the pole of the ecliptic with a period of about 26 000 years. As a consequence, the vemal equinox, the intersection of equator and ecliptic, regresses by about 50” per year along the ecliptic.

Moreover, the plane of the ecliptic itself is not fixed in space. Due to the gravitational attraction of the planets on the Earth, it slowly rotates around a “line of nodes”, the speed of this rotation being presently 47“ per century.

The plane of the ecliptic and that of the equator, and the vernal equinox, are the fundamental planes and the origin of two important coordlnate systems on the celestial sphere: the ecliptical coordinates (longitude X and latitude Q) and the equatorial coordinates (right ascension n and declination 6). So, due to the precession, the coordinates of the “fixed” stars are continuously challglng. Star catalogues, therefore, list the right ascensions and declinations of stars for a given epoch, such as 1900.0, or 1950.0, or 2000.0.

In this Chapter, we consider the problem of converting the right ascension a and the declination h of a star, given for an epoch and an equinox, to the corresponding values for another epoch and equinox. Only the mean place of a star, and hence the effects of the precession and proper motion, will be considered here. The problem of finding the apparent place of a star will be considered in Chapter 23.

131

 

Loir accuracy

If no great accuracy is required, if the two epochs are not widely separated, and if the star is not too close to one of the celestial poles, the following formulae may be used for the annual precession in right ascension and declination:

An = m + n sin a tan h Ad = dCOSO: (21.1)

where m and n are two quantities which vary slowly with time. They are given by nt = 3'.07496 + 0'.00186 'r

n =   l'.33621 — 0t00057 T n —— 20“.0431 — 0”.0085 T

T being the time measured in centuries from 2000.0 (the beginning of the year

2ooo). nere are the values of m and n for some epochs:

Epoch

s s

1700.0 3.069 1.338 20.07

1800.0 3.071 1.337 20.06

1900.0 3.073 1.337 20.05

2000.0 3.075 1.336 20.04

2100.0 3.077 1.336 20.03

2200.0 3.079 1.335 20.03

For the calculation of Act the value of n expressed in seconds of Sme ‹S ) must be used. Remember that 1‘ corresponds to 15“ (seconds of arc).

In the case of a star, the effect of the proper motion should be added to the values given by formulae (21. 1).

Example 21.a — The coordinates of Regulus (n Leonis) for the epoch and equinox of 2000.0 are

‹x0 = 10h08‘ 22'.3 6t = + 11°58'02“

and the annual proper motions are

—0'0169 in right ascension,

+0“.006 in declination.

Reduce these coordinates to the epoch and the equinox of 1978.0.

 

Here we have o = 152°.093, fi = + 11.°967,  rn = 3’.075,  n = l'.336 = 20“.04.

From the formulae (21.1) we deduce As = + 3'.208, Aâ = — 17“.71, to which we must add the annual proper motion, giving finally an annual variation of + 3'.191 in right ascension, and — 17“.70 in declination.

Variations during —22 years (from 2000.0 to 1978.0) :

in n : +3'.191 x (—22) = —70'.2 = —1°10'.2

in 6 : — 17“.70 x (—22) =  + 389“  =  +6'29“

Required right ascension . n = n0 — 1*10'.2 = 10'07‘12‘.1 Required declination : d = 60 + 6’29“ = + 12° 04'31"

Besselian and Julian Year

The International Astronomical Union has decided that from 1984 onwards the astronomical ephemerides should use the following system.

The new standard epoch is 2000 January 1 at 12‘ TD, corresponding to JDE 2451 545.0. This epoch is designated J2000.0. For purposes of calculating positions of stars, the beginning of a “year" differs from the standard epoch J2000.0 by an integral multiple of the Julian year, or 365.25 days. For example, the epoch J1986.0 is 14 X 365. 25 days before J2000.0, and hence the corresponding JDE is 2451 545.0 — 14 x 365.25 = 2446 431.50.

The letter J, in notations such as J2000.0 or J1986.0, indicates that the unit of time (for star catalogues) is the Julian year. Previously, star position catalogues used for a standard epoch the beginning of a Besselian year. The beginning of the Besselian solar year is the instant when the mean longitude of the Sun, affected by the aberration (—20”.5) and measured from the mean equinox of the date, is exactly 280°. This instant is always near the beginning of the Gregorian civil year. The length of the Besselian year, equal to that of the tropical year, was 365.242 1988 days in A.D. 1900, according to Newcomb.

To distinguish an old epoch, based on the Besselian year, frorri the new system, the letter B is used. For example,

B1900.0 = JDE 2415 020.3135 = 1900 January 0.8135

B1950.0 = JDE 2433 282.4235 = 1950 January 0.9235

but

J2000.0 = JDE 2451 545.00 exactly

J2050.0 = JDE 2469 807.50 exactly

and so on. The notation .0 after a year number (as in 1986.0 or 2000.0) signifies that the start of the year is meant.

 

Rigorous method

Let I be the time interval, in Julian centuries, between J2000.0 and the starting epoch, and let r be the interval, in the same units, between the starting epoch and the final epoch.

In other words, if (JD) and (JD) are the Julian Days corresponding to the initial and the final epoch, respectively, we have

 

T ----   (JD)o -- 2451545.0 

36525

 

  -- (JD) 36525

 

Then the numerical expressions for the quantities {, z and 8 which are needed

for the accurate reduction of positions from one equinox to another are [l] :

 

= (2306”.2181 + 1“.39656 r -- 0“.000 139 T2) r

+ (0“.30188 -- 0“.000 344 T) i* + 0“.017 998 i’

z = (2306”.2181 + 1“.39656 r -- 0”.000 139 T2) r

+ (1”.09468 + 0”.000 066 r) i 2 + 0“.018 203 i°

8 = (2004".3109 -- 0“.85330 T -- 0“.000 217 F*) i

-- (0”.42665 + 0“.000 217 r) r2 -- 0“.041 833 r'

 

(21.2)

 

If the starting epoch is J2000.0 itself, we have T  --- 0 and the eKpressions (21.2) reduce to

} = 2306”.2181r + 0”. 30188r 2 + 0“.017 998 r°

 

z = 2306”.2181r + 1“.09468r* + 0”.018 203 i° 8 = 2004“.3109r -- 0“.42665 i* -- 0”.041 833f^

 

(21.3)

 

Then, the rigorous formulae for the reduction of the given equatorial coordinates

« and fly of the starting epoch to the coordinates « and 6 of the final epoch are:

 

A -- cos o sin (°o + I)

B -- cos 8 cos 60 cos (« 0 + J) -- sin 8 sin 6 C = sin 8 cos 6 cos (nd + }) + cos 8 sin 3b

A B

 

(21. 4)

 

The angle n -- z can be obtained in the correct quadrant by applying the “second” arctangent function ATN2 to the quantities d and B, or by another procedure -- see “The correct quadrant” in Chapter 1.

 

If the star is close to the celestial pole, one should calculate the declination by means of the formula cos h = A2 + B2 instead of sin h = C.

Before making the reduction from n0, 60 to n, é, the effect of the star’s proper motion should be calculated.

2fxample 21.b — The star 8 Persei has the following mean coordinates for the epoch and equinox of J2000.0:

n0 2 44‘ 11'.986 h0 = +49°13’42".48

and its annual proper motions referred to that same equinox are

+0‘.03425 in right ascension,

—0“.0895 in declination.

Reduce the coordinates to the epoch and mean equinox of 2028 November 13.19 TD.

The initial epoch is J2000.0 or JD 2451 545.0. The final one is JD 2462 088.69.

Hence, r = +0.288 670 500 Julian centuries, or 28.867 0500 Julian years.

We first calculate the effect of the proper motion. The variations over 28.86705

years are

+ 0'03425 x 28.86705  =  +0‘.989 in right ascension,

—0‘.0895 x 28.86705 = —2”.58 in declination.

Thus the star’s coordinates, for the mean equinox of J2000.0, but for the epoch 2028 November 13.19, are

= 2 44'°11'986 + 0'.989 =   2'44‘12’.975 =  + 41°.054 063

6 =  + 49°13'42“.48 — 2“.58  =  + 49°13' 39”.90 =  + 49°. 227 750

Since the initial equinox is that of J2000.0, we can use the expressions (21.3).

With the value t —- + 0.288 670 500, we obtain

= +665“.7627 = +0°.184 9341

z = +665“.8288 = +0°.184 9524

8 = +578“.5489 = +0.°160 7080

A —— +0.430 494 05

B -- +0.488 948 49

C = +0.758 685 86

— z = + 41°.362 262

= + 41°.547 214 = 2'46‘l1'.331

6 = +49°.348 483 = +49° 20'54“.54

 

Exercise. — The equatorial coordinates of n Ursae Minoris (the Pole Star), for the epoch and mean equinox of J2000.0, are

a = 2 31‘48‘.704, é = + 89°15'50”.72

and the star’s annual proper motions for the same equinox are

+0'.19877 in right ascension,

—0“.0152 in declination.

Find the coordinates of the star for the epochs and mean equinoxes of B1900.0, J2050.0, and J2100.0.

Answer: B1900.0 n = l 22"33'.90 6 = +88°46’26”.18

J2050.0 3 48 16.43 +89 27 15.38

J2100.0 5 53 29.17 +89 32 22.18

Note that the formulae (21.2) and (21.3) are valid only for a limited period of time. If we use them for the year 32 700, for instance, we find for that epoch that n UMi will be at declination —87°, a completely wrong result!

Using ecliptical coordinates

If, instead of the equatorial coordinates (right ascension and declination) of a body, we use its ecliptical coordinates (longitude and latitude), the following rigorous method can be used [2].

T and r having the same meaning as before, calculate

 

q = (47“.0029 — 0“.06603 F + 0".000 598 F2) r

+ (—0“.03302 + 0”.000 598 T) f 2 + 0“.000 060/’

H = 174.°876 384 + 3289”.4789 T + 0”.60622 T*

— (869".8089 + 0“.50491 T) t + 0“.03536t 2

p —— (5029”.0966 + 2“.22226 T — 0”.000 042 r°) r

+ (1”.11113 — 0“.000 042 T) i 2 — 0 .000 006s

 

(21.5)

 

The quantity q is the angle between the ecliptic at the starting epoch and the ecliptic at the final epoch.

If the starting epoch is J2000.0, we have T —— 0 and the expressions reduce to

 

9 = 47”.0029 r — 0”.03302r° + 0”.000 060r' H = 174°.876 384 — 869”.8089r + 0”.03536r° p -- 5029”.0966s + 1”.11113 i° — 0“.000 006 r^

 

(21.6)

 

Then, the rigorous formulae for the reduction of the given ecliptical coordinates kg and 9 of the starting epoch to the coordinates X and fi of the final epoch are:

A’ —— cos 9 cos   0 sin (H — h0) — sin 9 sin fib

B' —— cos 9 cos (II o)

 

C’ —— cos q sin 90 + sin 9 cos 9 sin (n — x,)

 

(21.7)

 

A’ sin 9 = C'

B’

Exazrtple 21.c — The following astrometric ecliptical coordinates of Venus have been calculated for the instant —214 June 30.0 TD, but in the reference frame J2000.0:

k0 = 149°. 48194,    ßq = + 1.°76549

Reduce them to the mean equinox of that date.

The date corresponds to JDE = 1643 074.5, whence

t -- (1643 074.5 — 2451 545.0) / 36525 = —22.134 716

and we find successively:

9 — 1057“.225 = —0°.293 673

H 180°.22924

p — 110 773 “.167 = —30.°770 324

A’ +0.5111611

B’ +0.859 0225

C’ +0.0281891

p + H — k 30°.75475

X 118°.704

Q + 1°.615

In the case of a star, one should take the proper motion into account. Proper motions, however, are generally given in equatorial, not in celestial (ecliptical) coordinates. The proper motions in longitude p(h) and in latitude p(Q) can be obtained by means of the formulae given at the top of next page, where p(a) and p(6) are the proper motions in right ascension and in declinations, respectively. They should be expressed in arcseconds. Generally, p(n) is given in seconds of time; multiplication by 15 will convert it to arcseconds. The resulting q(X) and p(Q) will be in arcseconds too.

In the formulae, e is the obliquity of the ecliptic, a the star's right ascension, fi its declination, and Q its latitude.

 

p ) p(ó) sin e cos n + p(n) como ecos e cos é + sin e sin fi sin o)  

p($) p(ó) (cos e cos ó + sin e s sin n) — p(n) sin e cos a cos h 

S

T A B LE 2 1 . A

Proper motions o f some  stars in celestial longitude  and latitude

expressed in arcseconds per century  for t/ie epoch 2000.0

Star p(h) p(9) Star y(h) p(fi)

Alcyone(p Tau) + 0.82 — 4.90 Regulus —23.48 — 8.13

Aldebaran + 3.55 — 19.68 Spica — 2.75 — 4.15

Rigel + 0.04 — 0.13 Arcturus —28.10 —226.49

Capella + 4.47 — 42.95 n Lib — 8.17 — 9.48

#Tau + 1.20 — 17.61 asco — 0.60 — 2.73

Betelgeuse + 2.69 +   0.85 9Sco — 0.18 — 1.98

pGem + 5.86 — 10.88 «Sco — 0.67 — 2.21

yGem + 4.51 — 3.87 Antares — 0.63 — 2.15

eCiem — 0.45 — 1.38 esgr + 0.81 — 5.52

Sirius —55.56 —125.50 z Sgr — 0.44 — 3.51

óGem — 2.42 — 1.57 Altair +69.67 + 26.35

Castor —15.57 — 12.52 9Cap + 4.16 — 0.82

Procyon —54.28 —113.08 ó Cap +14.96 — 36.73

Pollux —61.37 — 15.67 Fomalhaut +25.26 — 28.68

 

The old precessional elements

As we have said earlier, for star catalogues and for the purpose of calculating star positions, the standard epoch is now J2000.0 and the unit of time is now the Julian year (365.25 days) or the Julian century (36525 days). Previously the beginning of the Besselian year was taken as reference instant and the unit of time was the tropical year or the tropical century.

However, these are not the only differences between the old system (the FK4) and the new one, the FK5. [“FK" means Juitdamenra/ Katalog.]

Firstly, there is a small error (the “equinox correction") in the zero point of the right ascensions of the FK4.

Secondly, as we shall see in Chapter 23, the aberrational displacements of a star in longitude (Ah) and in latitude (AQ) resulting from the motion of the Earth in its elliptical orbit are given by

+ e«   COS (F k) 

COS

AQ =  — K sin (O — h) sin d + e x sin (r — X) sin 9

where Ö is the longitude of the Sun, w the longitude of the perihelion of the Earth’s orbit, e the eccentricity of this orbit, and x the constant of aberration.

Now, the second terms in the right-hand sides of these expressions are almost eonstant for a given star, because e,r — h, and ß vary extremely slowly with time. For this reason, it has been astronomical practice to leave this part of the aberration (the so-called E-terms) in the mean positions of the stars.

Presently, the terms depending on the ellipticity of the Earth’s orbit are no longer included in the mean places of stars; they are, instead, calculated in the reduction from mean to apparent places (see Chapter 23).

A procedure for performing the conversion of mean positions and proper motions of stars referred to the mean equinox and equator B1950. 0 and based on Newcomb’s expressions for the precession (the FK4) to the new IAU system at J2000.0 (the FKS) can be found, for instance, in the Astronomical Almanac for 1984 [3).

The precessional formulae (21.2) and (21.3) may be used only for the stars referred to the FK5 system. If only FK4 positions and proper rnotions are available, then one should proceed as follows to calculate apparent star positions in the FKS system :

1. use must be made of Newcomb’s precessional formulae (see below);

2. in the reduction from mean to apparent place, the £-terms of the aberration should be dropped;

3. to the final right ascension of the star, add the equinox correction

 

A =   0'.0775  +  05.0800 F

where T is the time in Julian centuries from J2000.0.

Newcomb’s precessional expressions are the following ones.

Let (JD) and (JD) be the Julian Days corresponding to the initial and the final epoch, respectively. Then

 

T ——    (JD) — 2415 020.3135  

36524.2199

 

 (JD) — (JD)q 36524.2199

 

} = (2304”.250 + 1“.396 F) i + 0”.302r° + 0“.018 i 3

z = } + 0”.791r 2 + 0“.001i 3

8 = (2004“.682 — 0”.853 T) r — 0“.426f* — 0”.042i'

If the starting epoch is B1950.0, we have T —— 0.5, and the above expressions become

} = 2304”.948i + 0”. 302r 2 + 0”.018f z = 2304”.948/ + 1”.093 r 2 + 0“.019r 3

8 = 2004“.255 r — 0“.426i* — 0“.042 i3

3fo/ion in space

So far, we have assumed that the proper motion of a star across the sky is uniform. In other words, we considered its proper motions in right ascension and in declination to be constant. This is not correct, however. In fact, the proper motion should be combined with the radial velocity and distance to obtain the star’s true motion through space relative to the Sun. Over thousands of years, the proper motion of a star will vary, as the star is approaching the Sun or is receding from it.

Let us disregard the precession here. That is, we will work in an invariable reference frame, for instance that of J2000.0. Then the method for calculating the effect of proper motion by taking into account the star’s motion in space is as follows.

Let • be the star’s right ascension and declination for the starting epoch, r its distance in parsecs, and Mr its radial velocity in parsers per year (with proper sign!).

If the star’s distance is given in light-years, multiply it by 0.30660 to convert it to parsers. If, instead, the star’s parallax z (in arcseconds) is given, the distance in parsers is 1/z.

 

Radial velocities of stars are generally given in kilometers per second. They should be divided by 977792 in order to have them in parsecs per year.

Let As and A6 be the proper-motion components in radians per year. They are found by dividing the annual proper motion p(n) listed in seconds of time by 13 751, and the annual proper motion p(6) listed in seconds of arc by 206265, respectively. Then calculate [4]

z   =   r cos 6t cos n0 y   =   r cos 60 sin o 0 z = r sin 60

ix —— (xlr) Mr - zbb cos up — yka

AQ ' fl T) IT f,f Y J' SGD °o  +   *   °

= (Qtr) Mr + rib  cos fi0

Then, if r is the number of years from the starting epoch, negative in the past, positive in the future,

 

The final right ascension and declination for time r, but still in the reference frame of the starting epoch, are then given by

tan n = (sin n having the same sign as y’)

* R!* 21.d — Let us calculate the position (mean place) of Sirius for several epochs in the past, but still referred to the equinox of J2000.0, using the following starting values:

nip = 6‘45‘08'. 871 = 101°. 286 962

62 = — 16° 42'57“.99  =  — 16°.716 108

proper motions per year:

—0'.03847 in right ascension

— 1“.2053 in declination distance = 2.64 parsecs

radial velocity = —7.6 km/second

We find Mr —— —0.000 007 773, An = —0.000 002 7976, Aâ = —0.000 005 8435

 

Epoch

t This method (motion in space) I/sing uniform proper motions

h m s ° " h   m s ° ' “

1000.0 — 1 000 6 45 47.16 — 16 22 56.0

6 45 47.34 — 16 22 52.7

0.0 —2 000 6 46 25.09 — 16 03 00.8 6 46 25.81 — 16 02 47.4

— 1000.0 —3 000 6 47 02.67 — 15 43 12.3 6 47 04.28 — 15 42 42.9

—2000.0 —4 000 6 47 39.91 — 15 23 30.6 6 47 42.75 — 15 22 36. 8

— 10 000.0 — 12 000 6 52 25.72 — 12 50 06.7 6 52 50.51 — 12 41 54.4

However, an extreme accuracy cannot be obtained, because the results depend on the values adopted for the distance and the radial velocity of the star. In most cases, these values are not known with high accuracy. In the case of Sirius, if we use a radial velocity (at the epoch 2000.0) of —7.7 km/second instead of —7.6, the declination at — 10 000.0 becomes — 12° 50' 13”. 0 instead of — 12°50'06“.7.

The “classical” method, consisting in adopting a uniform proper motion, is good for modem epochs, for instance for the calculation of occultations of stars by the Moon. Indeed, the difference between the results of the two methods varies approximately as the square of the time elapsed. Between the years 1900 and 2100, the error in the declination of Sirius, due to the fact that a uniform proper motion is adopted, is not larger than 0.04 arcsecond. And note that Sirius is only one of a

/eir stars with large proper motion and close to the solar system. Therefore, the “classical” method will give no appreciable errors for epochs which are not too far from A.D. 2000.

Moreover, even the second method (taking the motion in space into account) is not valid nd infinitum. It will indeed give more precise results than the classical method for time lapses of many millennia, but even its validity is limited in time. Indeed, no star has a truly uniform and linear motion in space with respect to the Sun. All stars, including the Sun, describe orbits in our Galaxy system!

REFEREN CES

1. Astronomical Almanac for the year 1984 (Washington, D. C. ; 1983), page S 19.

2. Connaissance des Temps pour 1984 (Paris, 1983), pages XXX and XL.

3. Astronomical Almanac for the year 1984 (Washington, D.C.; 1983), pages S34 - S35. Note: Page S35 contains an error: Am = 1".037 = 0*06912 (not 0.6912).

4. A. Hirshfeld and R. W. Sinnott, Slcy Catalogue 20tXI.0, Vol. 1, page xiv (Sky Publishing Corporation, Cambridge, Mass. ; 1982).

 

Chapter 22

Nutation and the Obliquity of the

Ecliptic

The nutation, discovered by the British astronomer James Bradley (1693— 1762), is a periodic oscillation of the rotational axis of the Eanh around its “mean” position. Due to the nutation, the instantaneous pole of rotation of the Earth oscillates around a mean pole which advances by the precession around the pole of the ecliptic.

The nutation is due principally to the action of the Moon, and can be described by a sum of periodic terms. The most important term has a period of 6798.4 days (18.6 years), but some other terms have a very short period (less than 10 days).

Nutation is conveniently partitioned into a component parallel to and one perpendicular to the ecliptic. The component along the ecliptic is denoted by AJ and is called the mation in longitude -, it affects the celestial longitude of all heavenly bodies. The component perpendicular to the ecliptic is denoted by Ac and is called the nutation in obliquity, since it affects the obliquity of the equator to the ecliptic. The nutation does not affect the latitude of the heavenly bodies.

The quantities AJ and Ae are needed for the calculation of the apparent place of a heavenly body and for that of the apparent sidereal time. For any given instant, AQ and Ae can be calculated as follows.

Find the time T, measured in Julian centuries from the Epoch 12000.0 (JDE

2451 545.0),

 

T —— JDE — 2451 545 

36525

 

(22. l)

 

where JDE is the Julian Ephemeris Day; it differs from the Julian Day (JD) by the small quantity AT (see Chapter 7). Then calculate the following angles expressed in degrees and decimals. These expressions are those wkich are provided by the International Astronomical Union [1]. They differ slightly from those used in Chapront’s lunar theory (Chapter 47).

143

 

Mean elongation of the Moon from the Sun:

D = 297.85036 + 445 267.111 480 T — 0.0019142 'r* + r’/189 474

Mean anomaly of the Sun (Eanh) :

M -- 357.52772 + 35 999.050 340 F - 0.000 1603 r 2 — 'i"°/300 000

Mean anomaly of the Moon:

3f’ = 134.96298 + 477 198.867 398 T + 0.008 6972 T2 + T /5b 250

Moon’s argument of latitude:

F —- 93.27191 + 483 202.017 538 T — 0.003 6825 F2 + F°/327 270

Longitude of the ascending node of the Moon’s mean orbit on the ecliptic, measured from the mean equinox of the date:

12 = 125.04452 — 1934.136 261 T + 0.002 0708 T* + T3/450 000

The nutations in longitude (AQ) and in obliquity (Ae) are then obtained by making the sum of the terms given in Table 22.A, where the coefficients are given in units of 0“.0001. These terms are those of the 1980 IAU Theory of Nutation”

[2] where, however, we have neglected the terms with a coefficient smaller than 0”.0003. The argument of each sine (for AQ) and cosine (for Ae) is a linear combination of the five fundamental arguments D, M, M’, F, and D. For instance, the argument on the second line is —2D + 2F + 2f1.

Of course, if no great accuracy is needed, only the periodic terms with the largest coefficients can be used.

If an accuracy of 0”.5 in AQ and of 0”.1 in Ae are sufficient, then we may drop the terms in r 2 and in T3 in the above expression for £t, and then use the following simplified expressions:

A/ = —17”.20 sin fl — — 1“.32 sin 2L — 0“.23 sin 2J ' + 0“.21 sin 2£i Ae = +9“.20 cos It + 0”.57 cos 2L + 0“.10 cos 2L — 0“.09 cos 2í2

where L and L’ are the mean longitudes of the Sun and the Moon, respectively:

L — 280°.4665 + 36 000°.7698 T

L ’ = 218.°3165 + 481 267.°8813 T

 

22.  NUTATION AND OBLlCtUITY

TABLE 22.A

Periodic terms for the nutation in longitude (6]) and in obliquity (be). The unit is 0”.000 l.

 

145

 

D

A rg ume

multiple M M’

ni

of

F

R

 

Coeffiicieni of the sine

of the argument

Coefficient of the cosine

of the argument

0 0 0 0 1 — 171996 — 174.2 Z’ + 92025 + 8.9 T

—2 0 0 2 2 — 13187 - 1.6 T +5736 -3.1T

0 0 0 2 2 —2274 -0. 2 r +977 —0.5 F

0 0 0 0 2 +2062 0.2 T —895 +0.5 F

0 1 0 0 0 + 1426 -3.4 T +54 -0.1 '

0 0 1 0 0 + 712 + 0.1 r —7

—2 1 0 2 2 —517 -F I.2 T +224 -0. 6 T

0 0 0 2 1 —386 —0.4 ' +200

0 0 1 2 2 —301 + 129 —0.1 T

—2 — 1 0 2 2 +217 —0.5 T —95 + 0.3 F

—2 0 1 0 0 — 158

—2 0 0 2 l + 129 +0. l T —70

0 0 —1 2 2 +123 —53

2 0 0 0 0 +63

0 0 1 0 1 + 63 +0. l T —33

2 0 —1 2 2 —59 +26

0 0 —1 0 1 —58 -0. i r + 32

0 0 1 2 1 —51 +27

—2 0 2 0 0 +48

0 0 —2 2 1 +46 —24

2 0 0 2 2 —38 + 16

0 0 2 2 2 —31 + 13

0 0 2 0 0 +29

—2 0 1 2 2 +29 —13

0 0 0 2 0 +26

—2 0 0 2 0 —22

0 0 —1 2 1 +21 — 10

0 2 0 0 0 +17 —0.1 V

2 0 — 1 0 1 + 16 —8

—2 2 0 2 2 —16 +0.1 F +7

 

TA B LE 22.   A (cont.I

D A

M rg it in

M e n r

F

R

 

size

 

cosine

—2 0 1 0 1 — 13 +7

0 —1 0 0 1 —12 +6

0 0 2 —2 0 +11

2 0 —1 2 1 —10 +5

2 0 1 2 2 —8

 

0 1 0 2 2 +7 —3

—2 1 1 0 0 —7

0 — 1 0 2 2 —7 +3

2 0 0 2 1 —7 +3

2 0 1 0 0 +6

—2 0 2 2 2 +6 -3

—2 0 1 2 1 +6 -3

2 0 -2 0 1 —6 +3

2 0 0 0 1 —6 +3

0 —1 1 0 0 +5

—2 — 1 0 2 1 —5

 

—2 0 0 0 1 —5

0 0 2 2 1 —5

—2 0 2 0 1 +4

— 2 1 0 2 1 +4

0 0 1 —2 0 +4

— 1 0 1 0 0 —4

—2 1 0 0 0 —4

1 0 0 0 0 —4

0 0 1 2 0 +3

0 0 -2 2 2 —3

— 1 — 1 1 0 0 —3

0 1 1 0 0 —3

0 — 1 1 2 2 —3

2 —1 — 1 2 2 —3

0 0 3 2 2 —3

2 — 1 0 2 2 —3

 

22. NUTATION AND OBLIQUITY

The obliquity of the ecliptic

 

147

 

The obliquity of the ecliptic, or inclination of the Earth’s axis of rotation, is the angle between the equator and the ecliptic. One distinguishes the mean and the rrue obliquity, being the angles which the ecliptic makes with the mean and with the true (instantaneous) equator, respectively. In other words, the adjective mean indicates that the correction for nutation is not taken into account.

The mean obliquity of the ecliptic is given by the following formula, adopted by the International Astronomical Union [1] :

e0 = 23° 26'21”.448 — 46“.8150 T — 0“.00059 r 2 + 0”.001 813 'r° (22. 2)

where, again, T is the time measured in Julian centuries from the epoch J2000.0. The accuracy of formula (22.2) is not satisfactory over a long period of time:

the error in eg reaches 1“ over a period of 2000 years, and about 10” over a period of 4000 years. The following improved expression is due to Laskar [3]. Here, U is the time measured in units of 10 000 Julian years from J2000.0, or U —— r/  100.

 

e0 = 23° 26'21”.448 — 4680”.93 U

— 1 55 Iy

+ 1999. 25 V*

— 51.38 t/ 4

— 249. 67 I/5

— 39.05 t/ 6

+ 7.12 U

+ 27.87 U'

+ 5.79 é/ 9

+ 2.45 U ' O

 

(22.3)

 

The accuracy of this expression is estimated at 0”. 01 after 1000 years (that is, between A.D. 1000 and 3000), and a few seconds of arc after 10000 years.

It is important to note that formula (22.3) is valid only over a period of 10000 years on each side of J2000.0, that is, for I U   < 1. For U —— + 2.834, for example, the formula would yield ^o ' 90°, a completely wrong result!

The Figure on the next page shows the variation of e0 from 10000 years before to 10 000 years after A.D. 2000. According to Laskar’s formula, the inclination of the Earth’s axis of rotation was a maximum (24°14’07“) about the year —7530. And near the year + 12 030 a minimum (22° 36'4l “) will be reached. By a mere chance we are presently approximately half- way between these extreme values, near the middle of the curve in the Figure. Here the curve is almost linear; this is the reason why in (22.3) the coefficient of U 2 is very small.

The rrite obliquity of the ecliptic is c = c + Ac, where ne is the nutation in obliquity.

 

 

Centuries   since   the    year  2000

Example 22.a — Calculate AJ, Ae, and the true obliquity of the ecliptic for 1987

April 10 at 0‘ TD.

This date corresponds to JDE 2446 895.5,  and we find

T —0.127 296 372 348

D —56 383°.0377 = 136°.9623

ñf —4225°.0208 = 94°.9792

M’ —60 610°.7216 = 229°.2784

F —61 416°.5921 = 143°.4079

II 371.°2531 = 11°.2531

A/ —3“.788

Ae +9".443

e0 23°26'27“.407

e 23°26’36“.850

REFEREN CES

1. Astronomical Almanac for the year 1984 (Washington, D. C. ; 1983), page S 26.

2. Ibid. , page S23.

3. J. Laskar, Astronomy and Astrophysics, Vol. 157, page 68 (1986).

 

Chapter 23

Apparent Place of a Star

The mean place of a star at any time is its apparent position on the celestial sphere, as it would be seen by an observer at rest on the Sun (or, more exactly, at the barycenter of the solar system), and referred to the ecliptic and mean equinox of the date (or to the mean equator and mean equinox of the date).

The apparent place of a star at any time is its position on the celestial sphere as it is actually seen from the center of the moving Earth, and referred to the instantaneous equator, ecliptic, and equinox. Note that:

— the menu equinox is the intersection of the ecliptic of the date with the mean

equator of the date;

— the rme equinox is the intersection of the ecliptic with the true (instantaneous)

equator, that is, the equator affected by the nutation;

— there is no “mean” ecliptic, because the ecliptic has a regular motion — the slow rotation mentioned on page 131.

arc MT —— b f

149

 

The problem of the reduction of the place of a star from the mean place at one time (for instance, of a standard epoch and equinox, such as J2000.0) to the apparent place at another time involves the following corrections:

(A) The proper motion of the star between the two epochs. We may assume that by its proper motion each star moves on a great circle with an invariable angular speed — however, see also “Motion in space" in Chapter 21. Except when the proper motion is an important fraction of the polar distance of the star, not only the proper motion itself, but also its components in right ascension and declination with respect to a fixed equinox may be considered as constants during several centuries. Therefore, we start by finding the effect of the proper motion when the axes of reference remain fixed, as in Example 21. b;

(B) The effect of precession. This has been explained in Chapter 21 ;

(C) The effect of nutation (see below);

(D) The effect of annual aberration (see below);

(E) The effect of the annual parallax. Of course, stellar parallaxes are of fundamental importance in astronomy. As George Lovi wrote [1] :

Parallax is the only true geometrical link between us and our nearer neighbors in that vast interstellar void. It has enabled astronomers to create and calibrate procedures to take us much farther out.

However, for the person wishing to calculate accurate star positions, the stellar parallax is a nuisance. Fortunately, stellar parallaxes never exceed 0“.8 and they may be neglected in most cases. According to R. Bumham [2], only 13 stars brighter than magnitude 9.0 are nearer than 13 light-years (4 parsecs) and have a parallax exceeding 0”.25. Thèse stars are ri Centauri, Lalande 21185 (in Ursa Major), Sirius, e Eridani, 61 Cygni, Procyon, e Indi, r•2398 (in Draco), Groombridge 34 (in Andromeda), z Ceti, Lacaille 9352 (in Piscis Austrinus), Cordoba 29191 (in Microscopium), and the Star of Kapteyn (in Pictor). None of thèse stars is near the ecliptic, and so none is involved in occultations by the Moon or in close conjunctions with planets.

For this reason, in what follows we shall neglect the effect of the annual parallax in the calculation of the apparent posiüon of a star.

(F) The gravitational defection of light. The path of light is bent by the gravitational field of the Sun in the direction toward the Sun (Einstein effect). Formulae for calculating this effect are given in [3]. However, for any elongation larger than 15° the effect is smaller than 0''03. For this reason, we will neglect this effect here.

The effect of notation

The simplest and most direct method of applying the effect of nutation to mean position is to add AQ to the ecliptical longitude of the objects. The ecliptic and therefore the latitude of a body is unchanged by nutation.

 

This procedure can profitably be used in the calculation of apparent positions of planets, where ecliptical coordinates are calculated first. Stellar positions, however, are generally given in the equatorial system, so we prefer to calculate the corrections in right ascension and in declination directly.

First-order corrections to a star’s right ascension o and declination fi due to the nutation are

 

A« = (cos e + sin e sin n tan é) AQ — (cos o tan h) Ae Ah = (sin e cos n) AQ + (sin n) Ae

 

(23. 1)

 

These expressions are invalid if the star is close to one of the celestial poles. If this is the case, it is better to work in ecliptical coordinates and just add AJ to the longitude, as mentioned above.

The quantities AQ and Ae can be calculated by means of the method described in Chapter 22, while e is the obliquity of the ecliptic given by formula (22.2).

The effect of aberration

Let h and 9 be the star’s celestial longitude and latitude, x the constant of aberration (20”.49552), O the true (geometric) longitude of the Sun, e the eccentricity of the Earth’s orbit, andr the longitude of the perihelion of this orbit.

O can be calculated by the method described in Chapter 25, while

e = 0.016 708 634 — 0.000 042 037 T — 0.000 000 1267 T

z = 102°.93735 + 1 .°71946 T + 0.°00046 T2

where T is the time in Julian centuries from the epoch J2000.0, as obtained by

formula (22.1).

Then the changes in longitude and in latitude of the star due to the annual aberration are

 

   — K COs (O  — h)  +  e K cos  (r  — h) 

COS

AQ = — x sin 9 (sin (O — k) — e sin (r — k))

 

(23.2)

 

In equatorial coordinates, the changes in the right ascension n and in the declination 6 of the star due to the annual aberration are

 

Ó ¢z 2 —   K

 

cos o cos Ó

COS Ó

 COS O COSF GOS £ + Sifl O Sill T 

 

+ e K

 

cos ó

 

(23.3)

 

A6z = — x [ cos cos c (tan c cos 6 — sin o sin 6)

+ cos o sin ó sin O

+ e K [ cos z cos e (tan e cos ó — sin a sin ó)

+ cos n sin ó sinr ]

The total corrections to n and 6, due to the nutation and the aberration, are therefore An + A« 2 and Aé + Ah , respectively. Calculated from the above formulae, both are expressed in seconds of a degree (if AQ, Ae and K are expressed in the same units).

Important remark. — Formulae (23.2) and (23.3) are the complete expressions for the components of the aberration. They include the so-called E-terns and should be used for the star positions given in the FK5 [4) and in all catalogues based on it.

If, however, FK4 positions are used, those parts of formulae (23.2) and (23.3) that contain the eccentricity e of the orbit of the Earth should be dropped, as explained in Chapter 21.

Example 23.a — Calculate the apparent place of 8 Persei for 2028 Nov. 13. 19 TD.

The mean position of this star for that instant, including the effect of proper motion, was found in Example 21.b, namely

n = 2h46‘ 11’.331 = 41º.5472 ó = + 49°20’54’.54 = +49°.3485

The nutations in longitude and in obliquity, for the same instant, can be found by means of the method given in Chapter 22. We obtain

AQ = + 14“. 861 Ae = +2”.705

Formula (22.2) gives e = 23°.436, while the Sun’s true longitude, calculated by means of the method “low accuracy” of Chapter 25, is O = 231°.328, (An accuracy of 0.01 degree is sufficient in this case.) We further find

T -- +0.288 6705 e —— 0.016 696 49 r = 103°. 434

Putting the values of n, 6, e, AQ, Ae, O , e, andr in formulae (23.1) and (23.3), one finds

 

A t = + 15“.843 A t = + 30“.045

 

A6t = + 6".218 2 +6’.697

 

and the total corrections in right ascension and in declination are A = + 15“.843 + 30“.045 = 45“.888 = +3‘.059

A6 = +6“.218 + 6“.697 = + 12“.91

Hence, the required apparent coordinates of the star are

n = 2 46‘1l’.331  + 3‘.059 = 2h46‘ 14’.390

6 = +49° 20'54“.54 + 12“.91 = +49° 21'07“.45

The Ron-Vondrdk expression for aberration

Expressions (23.2) and (23.3) contain the effect of the eccentricity of the Earth’s orbit and will provide quite accurate results. Nevertheless, these results are not rigorously exact because the said formulae are based on an unperturbed motion of the Earth in its elliptical orbit. Actually, the Earth’s motion is somewhat perturbed by the attraction of the Moon and that of the planets. And the Sun itself is slowly moving around the center of mass of the solar system, mainly due to the action of the giants Jupiter and Saturn.

If a very accurate result is required, stellar aberration must, in fact, be computed from the total velocity of the Earth referred to this barycenter. One method of performing this calculation has been presented by Ron and Vondrâk [5].

If T —— (JD — 2451 545)/36525 is, as before, the time in Julian centuries elapsed since J2000.0, then calculate, for the given instant, the following angles expressed in radians :

L2 = 3.176 1467 + 1021.328 5546 T

L3 = 1.753 4703 + 628.307 5849 T

L4 = 6.203 4809 + 334.061 2431 T

L5 = 0.5995465 + 52.969 0965 T

L6 = 0.874 0168 + 21.3299095 T

L7 = 5.4812939 + 7.478 1599 T

L8 = 5.3118863 + 3.813 3036 r

L’ —— 3.810 3444 + 8399.684 7337 T

D -— 5.198 4667 + 7771.377 1486 T

31’ = 2.355 5559 + 8328.691 4289 T

F —— 1.627 9052 + 8433.466 1601 T

 

TA B LE 2 3. A

Velocity components of  the Earth with respect to the center of mass of  the  solar system

No.

Argum ent X' Y' Z'

sin cos sin cos sin cos

I £3 — 1719914 —2 T —25 25 — 13 T 1578089 + 156 'r 10 +32 r 684185 —358 r

2 2L3 6434 + 141 T 28007 — 107 r 25697 —95 T —5904 — 130 r 11141 —48 F —2559 —55 T

3 L5 715 0 6 —657 — 15 —282

4 L' 715 0 0 —656 0 —285

5 3L3 486 —5 T —236 —4 T —216 —4 T —446 + 5 T —94 — 193

6 L6 159 0 2 — 147 —6 —61

7 F 0 0 0 26 0 —59

8 L' + M’ 39 0 0 —36 0 — 16

9 2L5 33 — 10 —9 — 30 —5 — 13

10 2L3 — <5 31 1 1 —28 0 — 12

11 3L3 — 8L4 + 3L5 8 —28 25 8 11 3

12 5L3 — 8@ + 3L5 8 —28 —25 —8 — l l —3

13 2L2 — L3 21 0 0 — 19 0 — 8

l4 L2 — 19 0 0 17 0 8

15 L7 17 0 0 — l6 0 —7

16 L3— 2L5 16 0 0 15 1 7

17 L8 16 0 1 — 15 —3 —6

18 L3 + LS ll — 1 — 1 — 10 — l —5

19 2L2 — 2£3 0 — 11 — 10 0 —4 0

20 L3 — L5 — I I —2 —2 9 — 1 4

21 4L3 —7 —8 —8 6 —3 3

22 3ñ3 - 2 I.s 10 0 0 9 0 4

23 L2 - 2 L3 —9 0 0 —9 0 —4

24 2L2 — 3L3 —9 0 0 —8 0 —4

 

The quantities L2 up to L8 are the mean longitudes of the planets Venus to Neptune referred to the mean equinox of J2000.0 (the effects of Mercury and Pluto are negligible), while L’ is the mean longitude of the Moon.

Then the components X', Y', Z’ of the velocity of the Earth with respect to the barycenter of the solar system, in the equatorial J2000.0 reference frame, are equal to the sums of the terms given in Table

23.A. Here, the argument of each sine and cosine 1s a linear cOmbina- tion of some of the angles £2, L3, etc. For instance, the terms on line 12 of the table have as argument the angle

N = 5L3 — 8L4 + 3 L5

and the contributions to the velocity components are:

to X’ :        +   8 sin A —28 cos d to Y' :       —25 sin d — 8 cos A to Z’ : —11 sin N — 3 cos A

The values of A’, Y’, Z’ thus obtained are expressed in units of 10*' astronomical unit per day. Let c be the velocity of light in the same units, namely

c = 17 314 463 350.

Then the changes in the star’s right ascension and declination due to the annual aberration are, in radians, given by formulae (23.4).

 

 Y’ cos o — X' sin ce 

c cos ö

   (X' cos ri + Y' sin ri) sin é — Z' cos 6  

 

(23.4)

 

Important: the Earth’s velocity components, as calculated by means of Table

23. A, are given in a rectangular coordinate system based on the jed equator and equinox of FK5 for the epoch J2000.0, not with respect to the mean equinox of the date. Consequently, if the Ron-Vondräk method for the calculation of the aberration is preferred instead of the formulae (23.3), then the corrections (23.4) ahould be performed before the calculation of the effects of precession and nutation. In other words, the sequence of the calculations will be: FK5 position (J2000.0), proper motion, aberration (Table 23.A and expressions 23.4), precession (expressions 21.3 and 21.4), nutation (Chapter 22 and expressions 23. 1).

Example 23.b — Let us again calculate the apparent place of 8 Persei for 2028 November 13.19 TD, but now using the Ron-Vondrñk algorithm.

As in Example 21.b, we find that the star’s coordinates for the epoch 2028 November 13.19, but referred to the mean equinox of J2000.0, are (allowing for proper motion)

= 2'44‘12‘.9747 = +41.°054 0613

6 = +49° 13'39“.896 = + 49.°227 7489

We keep extra decimals here, in order to avoid rounding errors. We further find

T +0.288 670 500 L’ 2428.551 5363 rad.

L y 298.003 5712 rad. D 2248.565 7939

L3 183.127 3350 4f’ 2406.603 0750

L4 102.637 1070 F 2436.120 7984

L5 15.890 1621

L6 7.031 3324 X‘ —1363700

L7 7.640 0181 F‘ + 990286

L8 6.412 6746 Z’ + 429285

Formulae (23.4) then give

Au = + 0.000 145 252 radian = +0°.008 3223

Aô = + 0.000 032 723 radian = +0°.001 8749

so that the new values for a and ô, corrected for aberration but still in the J2000.0 reference frame, are

= 41º.054 0613 + 0º.008 3223 = 41º.062 3836

6 = 49º.227 7489 + 0º.001 8749 = 49º.229 6238

 

The effect of precession is obtained by means of formulae (21.4). The values of

}, z, and 8, for the same instant, were found in Example 21.b. We now find

A -- +0.430 549 036

B -- +0.488 867 290

C = + 0.758 706 993

new n = 41°.555 5635

new â = 49°.350 3415

Finally, the corrections for the nutation are given by (23. l). As in Example 23.a, we have AQ = + 14“.861, Ae = +2“.705, and e = 23°.436. We find

A , = + 15“.844 =  +0.°004 4011

A6t   = +6“.217 = + 0°.001 7270

Hence, the required apparent right ascension and declination of the star are

= 41.°555 5635 + 0.°004 4011 = 41°.559 9646

= 2‘46‘14'.392

6 = 49°.350 3415 + 0.°001 7270 = + 49°.352 0685

= + 49° 21'07 ”.45

Compare these results with those of Example 23.a.

REFEREN CES

1. Stay and Telescope, Vol. 77, page 288 (March 1989).

2. Robert Burnham, Burnham’s Celestial Handbook, Vol. III, page 2126 (Dover Publications, New York; 1978).

3. Astronomical Almanac for the year 1984 (Washington, D.C.; 1983), page S 20,

4. Fifth Fundamental Catalogue (FKS), Verö enilichungen Asironomisches Rechen- Institut Heidelberg, No. 32 (Karlsruhe, 1988).

5. C. Ron, J. Vondrdk, “Expansion of Annual Aberration into Trigonometric Series”,

Bull. Astron. Inst. Czechosl. , Vol. 37, pages 96—103 (1986).

 

ORB IT

 

Chapter 24

Reduction of Ecliptical Elements from one Equinox to another one

For some problems, it may be necessary to reduce the orbital elements of a planet, a minor planet, or a comet from one equinox to another one. Of course, the semimajor axis o and the eccentricity e do not change when the orbit is referred to another equinox, and hence only the three elements

i —— inclination,

o = argument of perihelion,

II = longitude of ascending node

should be taken into consideration here. Let it. • be the known values of these elements at the initial epoch, and i, «›, It their (unknown) values at the final epoch.

In the Figure on the preceding page, TO and J O are the ecliptic and the (mean) vernal equinox at the initial epoch, and E and y the ecliptic and (mean) equinox at the rinal epoch. The angle between the two ecliptics is denoted by 9, and the orbit’s perihelion by P.

As in Chapter 21, let T be the time interval, in Julian centuries, between J2000.0 and the initial epoch, and i the time interval, in the same units, between the initial epoch and the final epoch.

Then calculate the angles 9, H, and p by means of formulae (21.5) or, if the initial epoch is J2000.0, by means of (21.6).

Find     = H + p. Then the quantities i and ft — J, and hence 0, can be found from

 

cos i = cos ig cos 9 + sin il sin 9 cos (HO — II)

sin i sin (fl — J) = sin i0 Sin (fi0 — H)

Slfl  f   COS (Û J)     ' Slf1  'ij  COS  No +     COS  'fj  SÎf1  tp  COS (kg Û)

Formula (24. l) should not be used when the inclination is too small.

159

 

(24.1)

(24.2)

 

Then u› = o 0 -1- Aut, where Au is found from

sin j sin Au = — sin 9 sin (ftp — Il)

sin i cos Au = sifl io cos 9 — cos i sin 9 cos (ftp — II)

 

(24.3)

 

If il = 0, then   is not determined, and we have i = q and Cl —   + 180º .

It is important to note that the method described here reduces the orbital elements i, u›, and II from one equinox to another one, but the new orbital elements remain valid for the same epoch as the initial elements. It is, in fact, the same orbit. The calculation of the orbital elements for another epoch is a completely different problem (celestial mechanics!) which we cannot discuss here.

Example 24.a — In their Catalogue Général des Orbites de Comètes de l’an -466 à 1952 (Observatoire de Paris, Section d’Astrophysique de Meudon; 1952), F. Baldet and G. De Obaldia give the following orbital elements for cornet Klinkenberg (1744), referred to the mean equinox of B1744.0 :

i0 = 47°.1220

0  =  151°.4486

fl0 — 45°.7481

Reduce thèse elements to the standard equinox of B1950.0.

The final epoch is B1950.0, or (JD) = 2433 282.4235 (see Chapter 21), and the initial epoch is 206 tropical years earlier (because both epochs correspond to the beginning of a Besselian year), whence

(JD)0 = 2433 282.4235 — (206 x 365.242 1988) = 2358 042.5305.

We then find

T —2.559 958 097

r +2.059 956 002

9 +97“.0341 = +0°.026 954

H 174°.876 384 — 10205“.9108 = 172°.041 409

p + 10 352“.7137 = +2°.875 754

174°.917 163

Then formulae (24.2) give

sin i sin (fï — J) = —0.5906 3831 = A

sin i cos (8 — J) = —0.4340 8084 = B

from which we deduce   sin i = A 2 + B —— 0.7329 9372, i = 47.°1380

fl — = ATN2 {A, B) = — 126°. 313 473 fl = 48°.6037

 

24. REDUCTION OF ECLIPTICAL ELEMENTS

Formulae (24.3) give sin i sin As = +0.0003 7917

sin i cos As = +0.7329 9362 whence As = +0°.0296, and o = 151°.4782.

 

161

 

In his Catalogue of Cometary Orbits, sixth edition (1989), Marsden gives the values i = 47.°1378, u› = 151°.4783, II = 48°.6030. The discrepancy of 0°.0007 between the values of II results from the fact that the new IAU precession formulae yield for the general precession in longitude a value which is a little larger (+ l”. l per century) than that adopted by Newcomb. The effect over 206 years (from 1744 to 1950) amounts to 0.0006 degree.

If the initial equinox is that of B1950.0, and the rinal equinox that of J2000.0, the formulae simplify to the following ones.

S -- 0.000 113 9788 C —— 0.999 999 9935

lY = fly — 174.°298 782

A —- sin i0 sin W

B —— C sin i0 cos W — S cos i0

 

A + B la:n x —— A

fi = 174°.997 194 + x

and finally o = u› + Am, with

  — S sin lY

C sin i0 — S cos i0 cos W

 

(24. 4)

 

Care must be taken for the correct quadrant of the angles x and As. For safety, they should be calculated by means of the ATN2 function, if the latter is available in the programming language, for instance z = ATN2 {A, B). Except when the orbital inclination is very small, the new value of II should be approximately 0°.7 larger than the initial value fl0, and As must lie near 0°, not near 180°.

Example 24.b — S. Nakano calculated the following orbital elements for the 1990 return of periodic comet Encke (3finor Planet Circular 12577) :

Epoch = 1990 November 5.0 TD = JDE 2448 200.5

T —— 1990 October 28.54502 TD

q = 0.330 8858 i   = 11.°939ll

 

= 2.209 1404 II = 334°.04096

e = 0.850 2196 = 186.°24444

 

1950.0

 

We wish to reduce i, II, and u› to the equinox J2000.0, and we find successively

W + 159°.742 178 z + 159°.752 866

A +0.071 628 4465 It 334°.75006

B

sin i —0.194 187 3149

0.206 9767 —0.°01092

186°. 23352

i 11°.94524

The other orbital elements {T, q, a, e j remain unchanged, and the Epoch is still 1990 November 5.0.

However, formulae (24.4) assume that the elements it, c› , and Gq are given in the FK5 system. To convert elements from B1950.0/ FK4 to J2000.0/ FK5, one may use the following algorithm due to Yeomans (note from D. K. Yeomans, Chairman IAU System Transition Committee, to Richard West, President of IAU Commission 20; 1990 August 10).

L’ —— 4.500 016 88 degrees

L —— 5.19856209 degrees

= 0.006 519 66 degrees IY = L + fl0

Then we have

sin (u› — u›0) Sin i = sin 7 sin W

cos (u› — o 0) sin i = sin ii cos 7 + cos i0 Sin 7 cos II cos i — cos ir cos 7 — sin ii sin / cos II

sin (L’ + fi) sin i = sin ii sin lY

cos (L + fl) sin i = cos ip sin / + sin i0 cos J cos W from which i, It, and o can be deduced.

Example 24.c — Same starting values iq, fl0, and z 0 As in Example 24.b. We obtain

i = 11°.94521

II   = 334°.75043 FK5, J2000.0

= 186°.23327

 

Chapter 2S

Solar Coordinates

Low accuracy

When an accuracy of 0.01 degree is sufficient, the geocentric position of the Sun may be calculated by assuming a purely elliptical motion of the Earth; that is, the perturbations by the Moon and the planets may be neglected. The calculation can be performed as follows.

Let JD be the Julian (Ephemeris) Day, which can be calculated by means of the method described in Chapter 7. Then the time T, measured in Julian centuries of 36525 ephemeris days from the epoch J2000.0 (2000 January 1.5 TD), is given by

 

T ——  JD — 2451 545.0 

36525

 

(25.1)

 

This quantity should be calculated with a sufficient number of decimals. For instance, five decimals are not sufficient (unless the Sun’s longitude 1s required with an accuracy not better than one degree): remember that T is expressed in centuries, so that an error of 0.00001 in T corresponds to an error of 0.37 day in the time.

Then the geometric mean longitude of the Sun, referred to the mean equinox of the date, is given by

 

L0 = 280°.46646 + 36 000°.769 83 'r + 0°.000 3032 P 2

The mean anomaly of the Sun is

 

(25.2)

 

M = 357.°52911 + 35 999°.050 29 r — 0.°000 1537 T* (25.3)

(The mean anomaly of the Sun is the same as the mean anomaly of the Earth.

For the definition of the mean anomaly, see Chapter 30.) The eccentricity of the Earth’s orbit is

 

e = 0.016 708 634 — 0.000 042 037 r — 0.000 000 1267 T'

163

 

(25.4)

 

Find the Sun’s equation of the center C as follows:

C —- + (1°.914 602 — 0°.004 817 r — 0‘.000 014 r 2) sin If

+ (0.°019 993 — 0°.000 101 T) sin 2ñf

+ 0.°000 289 sin 33f

Then the Sun’s true longitude is O = La + C and its true anomaly is v = 3f + C

The Sun’s radius vector, or the distance between the centers of the Sun and the Earth, expressed in astronomical units, is given by

 

 1.000 001 018 (1 — e’) 

1 + e cos v

 

(25.5)

 

The numerator of the fraction is a quantity which varies slowly with time. It is equal to

0.999 7190 in the year 1800

0.9997204 1900

0.9997218 2000

0.9997232 2100

The Sun’s longitude O , obtained by the method described above, is the true geometric longitude referred to the mean equinox of the date. This longitude is the quantity required for instance in the calculation of geocentric planetary positions.

If the apparent longitude k of the Sun, referred to the we   equinox Of the date, is required, O should be corrected for the nutation and the aberration. Unless high accuracy is required, this can be performed as follows.

II = 125.°04 — 1934°. 136 T

h = O — 0°.00569 — 0.°00478 sin II

In some instances, for example in meteor work, it is necessary to have the Sun’s longitude referred to the standard equinox of J2000.0. Between the years 1900 and 2100, this can be performed with sufficient accuracy from

Ot = O — 0°.01397 (year — 2000)

If the Sun’s longitude, referred to the standard equinox of J2000.0, should be obtained with a higher accuracy than 0.01 degree, the method given in Chapter 26 can be used.

Due to the actions of the Moon and the planets, the Sun’s latitude is not exactly zero. Referred to the ecliptic of the date, it never exceeds 1.2 arcseconds. Unless high accuracy is required, this latitude may be put equal to zero. In that case, the

 

Sun’s right ascension n and declination h can be calculated from the following expressions where e, the obliquity of the ecliptic, is given by (22.2).

 

  COS C Sifl O 

sin é = sin e sin O

 

(25.6)

(25.7)

 

If the apparent position of the Sun is required, then in formulae (25.6) and (25.7) one should use h instead of O , and e should be corrected by the quantity

+ 0°.00256 cos II (25.8)

Formula (25.6) may of course be transformed to tan n = cos e tan O   but then it must be remembered that n must be in the same quadrant as O . However, if the ATN2 function is available in the programming language, it is better to leave formula (25.6) unchanged and to apply the ATN2 function to the numerator and the denominator of the fraction: n = ATN2 (cos e sin O, cos O).

Example 25.a — Calculate the Sun’s position on 1992 October 13 at 0‘ TD. This date corresponds to JDE 2448 908.5, and we find successively:

T —0.072 183 436

L —2318°.19280  = 201 .°807 20

M —2241°.00603 = 278 .°993 97

e 0.016 711 668

C — 1.° 897 32

0 199°.909 88 = 199° 54'36“

A 0.997 66

II 264°.65

k 199°.90895 = 199° 54’32“

e0 23° 26'24“.83 = 23.°440 23 [by (22.2)]

e 23°.439 99

a,pp ' 161°.619 17 = + 198°. 38083 = 13 \225 389 = 13 13'° 31'.4

6,gp ' —7°.785 07 = —7° 47’06“

The correct values, calculated by means of the complete VSOP87 theory (see Chapter 32), are:

geometric long., mean equinox of date : O = 199° 54‘26“.18 apparent longitude : k = 199°54‘21".56

apparent latitude : 9 = + 0’.72

radius vector : 2t = 0.997 608 53

apparent right ascension : 13‘13*30'.749

apparent declination : —7°47‘01".74

 

Higher accuracy

In their book Planetary Programs and Tables from —4000 to + 2OOO (Wilmann- Bell, Richmond; 1986), Bretagnon and Simon give a method for the calculation of the longitude of the Sun with an accuracy that is s fricient for many applications. Their method yields an accuracy of 0.0006 degree (2“. 2) between the years 0 and

+ 2800, and of 0.0009 degree (3”.2) between —4000 and + 8000, yet only 49 periodic terms are used.

A very high accuracy, better than 0.01 arcsecond, is obtained when use is made of the complete VSOP87 theory (see Chapter 32), but for the Earth this theory contains 2425 periodic terms, namely 1080 terms for the Earth’s longitude, 348 for the latitude, and 997 for the radius vector. Evidently, this big amount of numerical data cannot be reproduced in this book. Instead, we give in Appendix III the most important terms from the VSOP87, allowing the calculation of the position of the Sun with an error not exceeding 1” between the years —2000 and + 6000. The procedure is as follows.

Using from Appendix III the data for the Earth, calculate the latter’s heliocentric longitude L, latitude B, and radius vector fi for the given instant, as explained in Chapter 32. Don’t forget that the time r is measured from JDE 2451 545.0 in Julian millennia (365 250 days), not in centuries, and that the final values obtained for L and B are in radians.

To obtain the geocentric longitude O and latitude Q of the Sun, add 180° (or z radians) to L, and change the sign of B :

O = L -i- 180°, § =   - B

Conversion to the FK5 system. — The Sun’s longitude O and latitude fi obtained thus far are referred to the mean dynamical ecliptic and equinox of the date defined by the VSOP planetary theory of P. Bretagnon. This reference frame differs slightly from he standard FK5 system mentioned in Chapter 21. The conversion of O and 9 to the FK5 system can be performed as follows, where T is the time in centuries from 2000.0, or T -- 10r.

Calculate

 

h’ = O — 1°.397 T — 0°.000 31 T 2

Then the corrections to O and 9 are

AO = —0“.09033

b|3 -— +0”.03916 (cos h’ — sin k')

 

(25.9)

 

These corrections are needed only for very accurate calculations. They may be dropped when use is made of the abridged version of the VSOP87 given in Appendix III.

 

Apparent place of the Sun. — The Sun’s longitude O obtained thus far is the true (“geometric”) longitude of the Sun referred to the mean equinox of the date. To obtain the apparent longitude h, the effects of nutation and aberration should be taken into account.

For the nutation, simply add to O the nutation in longitude AQ (Chapter 22). To take the aberration into account, apply to the Sun’s geometric longitude the correction

 

20“.4898

R

 

(25.10)

 

where R is the Earth’s radius vector in AU. The numerator of the fraction is equal to the constant of aberration (x = 20”.49552) multiplied by n(1 — e 2), the same as the numerator in formula (25.5). Therefore, the numerator of (25.10) actually varies very slowly with time, from 20“.4893 in the year 0 to 20”.4904 in the year + 4000.

But, more important, formula (25.10) will not give a rigorously exact result, because it assumes an unperturbed motion of the Earth in its elliptical orbit. By reason of perturbations, mainly due to the Moon, the result can be up to 0.01 arcsecond in error.

When a very high accuracy is needed — this is not the case when the data of Appendix III are used for the calculation — the correction to the Sun’s longitude due to the aberration can be obtained as follows. Find the variation Ak of the Sun’s longitude, in arcseconds per day, as explained below. The correction for aberration is then

— 0.005 775 518 fi Ak  

where fi is, as before the Sun’s radius vector in astronomical units. The numerical constant is the light-time for unit distance, in days (= 8.3 minutes).

After the Sun’s longitude has been corrected for nutation and aberration, we have obtained the Sun’s apparent longitude h. The apparent longitude X and latitude Q of the Sun can then be transformed into the apparent right ascension n and declination fi by means of formulae (13.3) and (13. 4), where e is the true obliquity of the ecliptic, that is, affected by the nutation in obliquity Ac.

The variation Ah of the geocentric longitude of the Sun, in arcseconds per day, in the fixed reference frame J2000.0, can be obtained by means of the formula given on the next page, where z is the time in millennia from J 2000.0 (as in Chapter 32), and the arguments of the sines are in degrees and decimals.

In that expression, only the most important periodic terms have been retained. Consequently, the result will not be rigorous, but Ah Wlll not be more than 0”. I in error. If the resulting value of Ah is used to calculate the Sun’s aberration by means of (25.11), the error will be less than 0”.001.

If, for some other application, the value of AX is needed with respect to the mean equinox of the date instead of to a fixed reference frame, the constant term 3548.193 should be replaced by 3548.330.

 

DB!”!Y  variation,  in  arcseconds,  o f  the geocentric  longitude of the Sun in a fixed reference frame

The time r is measured from J2000. 0 (JDE 2451 545.0) in lulian millennia.

The arguments of the sines are in degrees.

Ah = 3548.193

+ 118.568 sin ( 87.5287 +    359 993.7286 r)

+ 2.476 sin ( 85.0561 +    719 987.4571 r)

+ 1.376 sin ( 27.8502 + 4452 671. i 152 z)

+ 0.119 sin ( 73.1375 +    450 368.8564 z)

+ 0.114 sin (337.2264 +   329 644.6718 z)

+ 0.086 sin (222.5400 +    659 289.3436 r)

+ 0.078 sin (162.8136 + 9224 659.7915 r)

+ 0.054 sin ( 82.5823 + 1079 981. 1857 r)

+ 0.052 sin (171.5189 +  225 184. 4282 r)

+ 0.034 sin ( 30.3214 + 4092 677.3866 r)

+ 0.033 sin (119.8105 +    337 181.4711 r)

+ 0.023 sin (247.5418 +   299 295.6151 T)

+ 0.023 sin (325.1526 +    315 559.5560 r)

+ 0.021 sin (155. 1241 +    675 553.2846 r)

+ 7.311 z sin (333.4515 +    359 993.7286 i)

+ 0.305 r sin (330.9814 +    719 987. 4571 i)

+ 0.010 z sin (328.5170 + 1079 981.1857 i)

+ 0.309 z2 sin (241.4518 +    359 993.7286 r)

+ 0.021 z2 sin (205.0482 +  719 987. 4571 i)

+ 0.004 z2  sin (297. 8610 + 4452 671.1152 r)

+ 0.010 sin (154. 7066 +   359 993.7286 r)

The periodic terms where r has the coefficient 359 993.7, 719 987, or 1079 981, are due to the eccentricity of the Earth’s orbit. The terms with 4452 671, 9224 660, or 4092 677 are due to the action of the Moon; those

with 450 369, 225 184, 315 560, or 675 553 are due to Venus; those with

329 645, 659 289, or 299 296 are due to Jupiter; finally, the term with 337 181 is due to the action of Mars.

 

 

Example 2J.Jr — Let us again, as in Example 25.a, calculate the position of the Sun for 1992 October 13.0 TD = JDE 2448 908.5.

Using from Appendix III the data for the Earth, we find by the method explained in Chapter 32,

L ——  —43.634 847 96 radians = —2500.092 628 degrees

= + 19.907 372 degrees

B —- —0.000 003 12 radian = —0.°000 179 = —0“.644

R -- 0.997 607 75

Whence

O = L + 180° = 199°. 907 372

Q = +0“.644

Converting to the FK5 system, we find

h’ = 200°.01 AO — —0“.09033 = —0.°000 025 AQ = —0“.023

whence

O  = 199°.907 347 = 199° 54'26“.449 Q = +0“.62

The nutation is calculated by means of the method described in Chapter 22.

AQ = + 15“.908 Ae = —0“.308 true c = 23°. 440 1443

and by (25.10) the correction for aberration is —20’.539.

Hence, the Sun’s apparent longitude is

h = O + 15“.908 — 20“.539 = 199° 54' 21”.818

Then, by (13.3) and (13.4),

n = 198°.378 178 = 13h13° 30‘. 763

6 = —7°.783 871 = —7° 47'01“.94

Resuming, the final results are

 

O = 199° 54' 26“.45

X = 199°54'21“.82

Q = +0“.62

 

A = 0.997 607 75

n = 13'13”30‘.763

6 = —7° 47’01’.94

 

Compare these results with the correct values mentioned at the end of Example 25.a. Our results are now much better than those obtained with the low-accuracy method.

 

 

 

Chapter 26

Rectangular Coordinates of the Sun

The rectangular geocentric equatorial coordinates X, Y, Z of the Sun are needed for the calculation of an ephemeris of a minor planet (see Chapter 33) or a comet. The origin of these coordinates is the center of the Earth. The X-axis is directed towards the vernal equinox (longitude 0°); the r-axis lies in the plane of the equator too and is directed towards longitude 90°, while the Z-axis is directed towards the north celestial pole.

The values of X, Y, Z are given for each day at 0h TD in the great astronomical almanacs; they are expressed in astronomical units. Generally they are not referred to the mean equator and mean equinox of the date, but to a standard equinox, for instance that of J2000.0.

Reference to the mean equinox of the date

Calculate the geometric coordinates of the Sun by means of the method “higher accuracy" described in Chapter 25, with the corrections (25.9) for reduction to the FK5 system, but without the corrections for nutation and aberration.

If O and # are the geometric longitude and latitude of the Sun, and fi its radius vector in astronomical units, then the required rectangular coordinates of the Sun, referred to the mean equator and equinox of the date, are given by

X = fi cos 9 cos O

 

Y -— fi (cos 9 sin O cos e — sin 9 sin e)

Z = ft (cos 9 sin O sin e + sin 9 cos e)

 

(26. 1)

 

where e is the mean obliquity of the ecliptic given by (22.2).

Since the Sun’s latitude, referred to the ecliptic of the date, never exceeds 1.2 arcsecond, one may safely put cos Q = 1 in the formulae (26. l).

171

 

 

Example 26.a — For 1992 October 13.0 TD = JDE 2448 908.5, we have found in Example 25.b:

O = 199°.907347 9 = +0“.62 A = 0.997 607 75

For the same instant, formula (22.2) gives e = 23°26’24“. 827 = 23 °.440 2297

whence, by (26.1),

X = —0.937 9952

Y —— —0.311 6544

Z = —0.135 1215

Reference to  the  standard  equinox  J2000.0

As explained in Chapter 32, calculate for the given instant the Earth’s heliocentric longitude L and latitude B referred to the equinox of 12000.0, and its radius vector fi. For this purpose, use from Appendix III the data for the Earth, with the following exceptions :

— in section LI, replace the   first   value   of   the   coefficient   “A",   namely 628 331 966 747, by 628 307 584 999;

— sections L2, L3, and L4 should be replaced by those given in Table 26. A (next

page);

— drop section L5;

— for the calculation of the latitude B, use section B0 from Appendix III, but sections B1 to B4 from Table 26.A.

Obtain the geocentric longitude O of the Sun by adding 180° (or z radians) to L, and the Sun’s latitude d by changing the sign of B. That is,

O =  L + 180° and Q = —B

At this stage, if only the Sun’s geometric longitude referred to the standard equinox of J2000.0 is required, subtract 0”.09033 frOiTl O ln order to convert the longitude from the VSOP dynamical equinox to the FK5 equinox, as in (25.9). — Otherwise, do not perform this correction and proceed as follows.

Calculate

 

X = fi cos Q cos O Y —— R cos Q sin O Z —— R sin Q

Of course, these expressions are equivalent to X = —fi cos B cos L, Y —— — R cos B sin L, and Z —— — R sin B, respectively.

 

(26.2)

 

26. RECTANGULAR COORDINATES OF THE SUN

TA BLE 2 6. A

EARTH  J 2000.0 (some terms only)

 

NO.

L2

2

3

4

 

6

7

8

 

10

11

12

13

14

15

16

17

18

19

20

L3 1

2

3

4

5

6

7

L4 1

 

   

8 722 1. 072 5

991 3. 1 41 6

2 95 0. 43 7

2 7 0. 05

16 5. 19

16 3. 69

9 0 . 30

9 2.06

7 0.83

5 4.66

4 1.03

4 3.44

3 5.14

3 6.05

3 1.19

3 6.12

3 0.30

3 2.28

2 4.38

2 3.75

289 5.842

21 6.05

3 5.20

3 3.14

1 4.72

l 5.97

1 5.54

8 4.14

 

 

6283.0758

0

12 566.152

3.52

26.30

155.42

18849.23

77713.77

J7S.52 1577.34

7•11

5573.14

796.30

5507.55

242.73

529.69

388.15

553.57

5 223.69

0.98

6283.076

12 566.15

155.42

0

3. 52

24Z.73

18849.23

 

 

2 3 806 3. 370 6

3 3 620 0

4 72 3. 33

5 8 3. 89

6 8 1.7 9

7 6 5.20

 

12566.151 7

0

18849.23

 

5223.69

2 352.B7

 

 

 

 

B3 27 6 0. 5 95

2 17 3. 14

 

&283.0J6

 

 

 

 

The rectangular coordinates X, Y, Z calculated by means of (26.2) are still defined in the ecliptical dynamical reference frame (VSOP) of J2000.0. They can be transformed into the equatorial FK5 J2000.0 reference frame as follows:

@ = X + 0.000 000440 360 x — 0.000 000 190 919 Z

r =   —0.000 000 479 966 X + 0.917 482 137 087 Y - 0.397 776 982 902 Z

@  = 0.397776 982 902 F + 0.917 482 137 087 Z

(26.3)

Pte/erence to the mean equinox of B1950.0

Proceed as above for J2000.0, except that expressions (26.3) should be replaced by the following ones.

@  = 0.999 925 702634 X + 0.012189 716217 F + 0.000011 134 016Z

r = —0.011 179 418 036 X + 0.917 413 998 946 Y — 0.397 777 041 885 Z

@ =  —0.004 859 003 787 X  + 0.397 747 363 646 P + 0.917 482 111 428 Z

Note that the rectangular coordinates obtained in this way are referred to the mean equator and equinox of the epoch B1950.0 in the FK5 system, not in the FK4 system which is affected by the “equinox error” as mentioned in Chapter 21.

Reference to any other mean equinox

First, calculate the Sun’s rectangular equatorial coordinates @,   Y , @ referred to the standard equinox of J2000.0 as explained above, that is, by means of the expressions (26.2) and (26.3).

Then, if JD is the Julian Day corresponding to the epoch of the given equinox, calculate

 JD — 2451 545.0  

t 36525

and then the angles }, z, and 8 from (21.3).

Then the required rectangular coordinates of the Sun are given by

 

26. RECTANGULAR COORDINATES OF THE SUN 175

where

Xp ——    cos    cos z cos 8 — sin } sin z Ñy   —   sin J cos z +   cos     sin z cos 8 Xi = cos } sin 8

Yy ——  — cos  sin z —  sin } cos z cos 8 Yy COS  Cos z — sin } sin z cos 8 Y¿ -— —sin sin 8

zy —-  — cos z sin 8 2y =  — sin z sin 8 Z$ —- cos 8

Note that the coordinates X', Y’, Z’ are referred to the mean equinox of an epoch which differs from the date for which the values are calculated.

Example 26.b — For 1992 October 13.0 TD = JDE 2448 908.5, calculate the equatorial rectangular coordinates of the Sun referred to

(a) the standard equinox of J 2000;

(b) that of B1950.0 ;

(c) the mean equinox of I 2044.0.

We find successively

z = —0.007 218 343 6003

L ——  —43.633 088 03 radians = —2499.991 791 degrees

= +20.008 209 degrees

B —— +0.000 003 86 radian = +0.°000 221 = +0”.796

fi = 0.997 607 75 (as in Example 25.b, of course)

X = —0.937 395 75 ecliptic,

Y —- —0.341 336 25 dynamical equinox,

z —— —0.000 003 85 J2000.0

Xq =  —0.937 395 90 equatorial,

Yp ——  —0.313 167 93 FK5 frame,

Z0 = —0.135 779 24 J2000.0

 

The correct values, obtained by means of an accurate calculation using the complete VSOP87 theory, are --0.937 397 07, --0.313 167 25, and --0.135 778 42, respectively.

X0 = --0.941 487 equatorial,

rd = --0.302 666 FK5 system,

Z0 =   --0.131 214 B1950.0 frame

JD = 2467 616.0

(since the epoch J2044.0 is 44 X 365.25 days later than J2000.0)

r = +0.440 000

} = + 1014“.7959 = +0.°281 8878

z = + 1014“.9494 = +0.°2819304

8 = + 881“.8106 = +0.°244 9474

 

Xz --- +0.999 9424 Xy -- +0•009 8403 X = +0.004 2751

 

Yg 0.009 8403

y +0•999 9516

Y$ 0.000 0210

 

Zg = --0.004 2751

Zy = --0.000 0210

2$ -- + 0.999 9909

 

X’ = --0.933 680 equatorial,

Y' -- --0.322 374 FK5 system,

Z’ = --0.139 779 J2044.0 frame

 

Chapter 27

Equinoxes and Solstices

By definition, the times of the equinoxes and solstices are the instants when the apparent geocentric longitude of the Sun (that is, calculated by including the effects of aberration and nutation) is an integer multiple of 90 degrees. (Because the latitude of the Sun is not exactly zero, the declination of the Sun is not exactly zero at the instant of an equinox.)

Approximate times can be obtained as follows. First, find the instant of the “mean” equinox or solstice, using the relevant expression in Table 27. A or in Table 27.B, on the next page. Note that Table 27.A should be used for the years — 1000 to + 1000 only, and Table 27.B for the years + 1000 to + 3000. In fact, Table 27.A may also be used for several centuries before the year — 1000, and Table 27.B for several centuries after + 3000; the errors will still be quite small.

Important: in the formula for F, given at the top of each table, “year” is an

iiueger,- other values for “year” would give meaningless results!

Then find

T = — 2451 545.0

36525

W = 35 999.°373 T — 2.°47

Ah = 1 + 0.0334 cos W + 0.0007 cos 2 W

Calculate the sum S of the 24 periodic terms given in Table 27.C. Each of these terms is of the form A cos (B + CT), and the argument of each cosine is given in degrees. In other words,

S -— 485 cos (324°.96 + 1934°. 136 T)

+ 203 cos (337.°23 + 32964°. 467

177

 

TABLE 27.A For the years -1000 to +1000 Y _ year

1000

March equinox (beginning of astronomical spring) :

JDEo ' 1721 139.29189 + 365 242.13740 r + 0.06134 r* + 0.00111 Y - 0.00071 r’

June solstice (beginning of astronomical summer) :

JDEo 1721 233.25401 + 365 241.72562 r — 0.05323 Y + 0.00907 F' + 0.00025 Y

September equinox (beginning of astronomical autumn) :

JDEo 721 325.70455 + 365 242.49558 r —  0.11677 r 2 — 0.00297 Y + 0.00074 r’

December solstice (beginning of astronomical winter) :

JDE0 = 1721 414.39987 + 365 242.88257 r — 0.00769 r 2 — 0.00933 Y - 0.00006 r’

TA BLE 2 7. B For the years +1000 to +3000 Y -- year - 2000  

1000

March equinox (beginning of astronomical spring) :

JDE  = 2451 623.80984 + 365 242.37404 r + 0.05169 r* — 0.00411 F° — 0.00057 r 4

June solstice (beginning of astronomical summer)

JDEq = 2451 716.56767 + 365 241.62603 F + 0.00325 Y + 0.00888 F' — 0.00030 F’

September equinox (beginning of astronomical aummn)

JDEp — 24J1 810,21715 + 365 242.01767 r - 0.11575 Y -r 0.00337 r’ + 0.00078 r‘

December solstice (beginning of astronomical winter) :

JDE0 = 2451 900.05952 + 365 242.74049 F — 0.06223 Y N — 0.00823 Y + 0.00032 Y’

 

27. ECtUINOXES AND SOLSTICES

TA BLE 2 7. C

S -- L [A cos {B + CT)] B and C in degrees !

 

179

 

A B C A B C

485 324.96 1934.136 45 247.54 29929.562

203 337.23 32964.467 44 325.15 31555.956

199 342.08 20.186 29 60.93 4443.417

182 27.85 445267.112 18 155.12 67555.328

156 73.14 45036.886 17 288.79 4562.452

136 171.52 22518.443 16 198.04 62894.029

77 222.54 65928.934 14 199.76 31436.921

74 296.72 3034.906 12 95.39 14577.848

70 243.58 9037.513 12 287.11 31931.756

58 119.81 33718.147 12 320.81 34777.259

52 297.17 150.678 9 227.73 1222.114

50 21.02 2281.226 8 15.45 16859.074

TA B LE 2 7. D

Number of errors

< 20 seconds Number of errors

< 40 seconds Largest error (seconds)

March equinox 76 97 51

June solstice 80 100 39

September equinox 78 99

 

December solstice 68 99 41

 

 

Example 27.a — Find the time of the June solstice of A.D. 1962.

We find successively

F = —0.038

JDEt   = 2437 837.38589

T  -- —0.375 294 021

Ak = 0.9681

S -— +635

 

JDE   = 2437 837.38589 + 0.00635 

0.9681

 

= 2437 837.39245

 

which corresponds to 1962 June 21 at 2I‘25‘08’ TD.

The correct instant, as calculated with the complete VSOP87 theory, is 21'24”42‘ Dynamical Time.

Of course, higher accuracy can be obtained by actually calculating the value of the apparent longitude of the Sun for two or three instants, and then finding by interpolation the time when that longitude is exactly 0°, or 90°, or 180°, or 270°.

One should keep in mind that the motion of the Sun along the ecliptic is only 3548 arcseconds per day, approximately. Hence, an error of l” in the calculated longitude of the Sun results in an error of approximately 24 seconds in the times of the equinoxes or solstices.

Alternatively, one may start from any approximate time. The value obtained from Table 27.A or 27.B is more than sufficient. For thal instant, calculate the Sun’s apparent longitude k as explained in Chapter 25, including the corrections for reduction to the FK5 system, for aberration and for nutation. Then the correction to the assumed time, in days, is given by

 

where

 

+ 58 sin (k.90° — k)

k -- 0 for the March equinox, 1 for the June solstice,

2 for the September equinox, 3 for the December solstice.

 

(27. 1)

 

The calculation is then repeated until the new correction is very small or, equivalently, until the new value for the Sun’s apparent longimde is exactly £. 90°.

Mampfe 27.b  — Let us again calculate the instant of the June solstice in 1962.

In Example 27.a, we found that the “mean” solstice took place at JDE = 2437 837.38589 (from Table 27.B). Let us start from this approximate time, and

 

27. EQUINOXES AND SOLSTICES 181

calculate the Sun’s apparent longitude for this instant, using the “higher accuracy” procedure (Chapter 25). We find

L = —234.048 595 59 radians = 270°.003 272

2t = 1.016 3018

Nutation in longitude : AQ =  — 12“.965 (Chapter 22) FK5 correction : — 0“.09033 (formula (25.9))

aberration : —20“. 161 (formula (25.10))

Apparent longitude of the Sun:

k = 270.°003 272 — 180° — 12“.965 — 0“.09033 — 20".16 l = 89.°994 045

Formula (27.1) then gives the correction to the assumed value of JDE : correction = + 58 sin (90° — h) = +0.00603

and hence the corrected time is

JDE = 2437 837.38589 + 0.00603 = 2437 837.39192

Repeating the calculation for this new instant, we find k = 89°. 999 797,

resulting in the correction +0.00021 day. This gives the improved instant JDE = 2437 837.39213.

A final calculation, performed for this new instant, yields h = 89°. 999 998 and a correction smaller than 0.000 005 day.

Hence, the final instant is JDE = 2437 837.39213, which corresponds to 1962 June 21 at 21 24'"40‘ TD.

This differs by only two seconds from the correct time mentioned at the end of Example 27.a.

In 1962, the difference TD — UT was 34 seconds (see Table 10.A), so our result may be rounded to 21‘24‘ Universal Time.

Table 27.E gives the times of the equinoxes and solstices for the years 1996 to 2005, to the nearest second of time.

Table 27.F gives the durations of the four astronomical seasons for some epochs. About the year —4080, the Earth was in perihelion at the beginning of the autumn, and consequently the summer had the same duration as the autumn, and the winter had the same duration as the spring. In A.D. 1246, the Earth was in perihelion at the time of the winter solstice, and consequently the spring had the same duration as the summer, and the autumn had the same duration as the winter. Since the year + 1246, the winter is the shortest season; it will reach its minimum value by about A.D. 3500, and remain the shortest season till aboul A.D. 6427, when the Earth will be in perihelion at the time of the March equinox.

 

TA B LE 2 7. E

Equinoxes and  Solstices,  1996 -2005,  calculated by  meaos o f

the comp/ere  VSOP87  theory.   Instants are  in Ynamical  Time.

Year March equinox June solstice Sept. equinox Dec. solstice

d h m   s d h m   s d h m   S d h m s

1996 20 804 07 21 2 24 46 22 18 0108 21 14 06 56

1997 20 13 55 42 21 8 20 59 22 23 56 49 21 20 08 05

1998 20 19 55 35 21 14 03 38 23 5 3815 22 1 5J 31

1999 21 146 53 21 19 50 11 23 1132 34 22 7 44 52

2000 20 7 36 19 21 148 46 22 17 28 40 21 13 38 30

2001 20 13 3147 21 7 38 48 22 23 05 32 21 19 22 34

2002 20 191713 21 13 25 29 23 4 56 28 22 115 26

2003 21 100 50 21 191132 23 10 47 53 22 7 04 53

2004 20 6 49 42 21 (157 57 22 16 30 54 21 12 42 40

2005 20 12 34 29 21 6 47 12 22 22 24 14 21 18 36 01

T A B L E 2 7. F

Duration o f the astronomical  seasons,   in   da'ys

Year Spring Summer Autumn Winter

—4000 93.55 89.18 89.07 93.44

—3500 93.83 89.53 88.82 93.07

—3000 94.04 89.92 88.61 92.67

—2500 94.20 90.33 88.47 92.25

—2000 94.28 90.76 88.39 91.81

—1500 94.30 91.20 88.38 91.37

—1000 94.25 91.63 88.42 90.94

— 500 94.14 92.05 88.53 90.52

0 93.96 92.45 88.69 90.13

+ 500 93.73 92.82 88.91 89.78

1000 93.44 93.15 89.18 89.47

1500 93.12 93.42 89.50 89.20

2000 92.76 93.65 89.84 88.99

2500 92.37 93.81 90.22 88.84

3000 91.97 93.92 90.61 88.74

3500 91.57 93.96 91.01 88.71

4000 91.17 93.93 91.40 88.73

4500 90.79 93.84 91.79 88.82

5000 90.44 93.70 92,15 88.96

5500 90.11 93.50 92.49 89,15

6000 89.82 93.25 92.79 89,38

6500 89.58 92.96 93.04 89.66

 

Chapter 28

Equation of Time

Due to the eccentricity of its orbit, and to a much less degree due to the perturbations by the Moon and the planets, the Earth’s heliocentric longitude does not vary uniformly. It follows that the Sun appears to describe the ecliptic at a non- uniform rate. Due to this, and also to the fact that the Sun is moving in the ecliptic and not along the celestial equator, its right ascension does not increase uniformly.

Consider a first fictitious Sun travelling along the ecliptic with a constant speed and coinciding with the true Sun at the perigee and apogee (when the Earth is in perihelion and aphelion, respectively). Then consider a second fictitious Sun travelling along the celestial equator at a constant speed and coinciding with the first fictitious Sun at the equinoxes. This second fictitious Sun is the mean Sun, and by definition its right ascension increases at a uniform rate — that is, there are no periodic terms, but its expression contains small secular terms in N, z’, . . .

When the mean Sun crosses the observer’s meridian, it is mean noon there. True noon is the instant when the true Sun crosses the meridian. The equation of time is the difference between apparent and mean time. In other words, it is the difference between the hour angles of the true Sun and the mean Sun.

Defined in this manner, the equation of time E, at a given instant, is given by

E -- L0 — 0.°005 7183 — a + AQ . cos e (28. l)

In this formula, L0 is the Sun’s mean longitude. According to the VSOP 87 theory (see Chapter 32) we have, in degrees,

L0 = 280.466 4567 + 360 007.698 2779 z

+ 0.030 320 28 z2 +   r’/49931  

— r 4 / 15300 — z5 / 2000 000

where T is the time measured in Julian millennia (365 250 ephemeiis days) from J2000.0 = JDE 2451 545.0. @ should be reduced to less than 360‘ by adding or subtracting a convenient multiple of 360°.

183

 

In the French almanacs and in older textbooks, the equation of time is defined with opposite sign, hence being equal to mean time minus apparent lime.

In formula (28.1), the constant 0.°005 7183 is the sum of the mean value of the aberration in longitude (—20”.49552) and the correction for reduction to the FK5 system (—0”.09033); o is the apparent right ascension of the Sun, calculated by taking into account the aberration and the nutation. The quantity AJ . cos e, where AJ is the nutation in longitude and e the obliquity of the ecliptic, is needed to refer the apparent right ascension of the Sun to the mean equinox of the date, as is the mean longitude L0.

In formula (28.1), the quantities @, n, and AJ should be expressed in degrees. Then the equation of time E will be expressed in degrees, too; it can be converted to minutes of time by multiplication by 4.

The equation of time E can be positive or negative. If 6 > 0, the true Sun crosses the observer’s meridian before the mean Sun.

The equation of time is always smaller than 20 minutes in absolute value. If

E   appears to be too large, add 24 hours to or subtract it from your result.

Example 28.a — Find the equation of time on 1992 October 13 at 0‘ TD. This date corresponds to JDE = 2448 908.5, from which we deduce

 

 JDE — 2451 545.0  

365 250

 

= —0.007 218 343 600

 

Lg —— —2318°. 192 807 = +201°.807 193

For the same instant we have, from Example 25.b,

= 198°.378 178

A/ = + 15“.908 = +0.°004 419

e = 23°.440 1443

whence, by formula (28.1),

E -- + 3°.427 351 = + 13.70940 minutes = + 13‘42'.6

Alternatively, the equation of time can be obtained, with somewhat less accuracy, by means of the following formula given by Smart [l] :

 

 

E —— y sin 2L0  — 2e sin M + 4ey sin M cos 2L

  y 2 sin 4@ —  4 e° sin 2If

 

(28.3)

 

where

 

y = tan2 — , e being the obliquity of the ecliptic, Lt = Sun’s mean longitude,

e = eccentricity of the Earth’s orbit,

M —— Sun’s mean anomaly.

 

The values of e, @, e, and M can be found by means of the formulae (22.2), (28.2) or (25.2), (25.4), and (25.3), respectively.

The value of E given by formula (28.3) is expressed in radians. The result may be converted into degrees, and then into hours and decimals by division by 15.

Example 28.b — Find, once again, the value of the equation of time on

1992 October 13.0 TD = JDE 2448 908.5.

We find successively

 

T -- —0.072 183 436

e = 23°.44023

L0 = 201°.80720

 

e = 0.016 711 668

M —— 278°. 99397

y = 0.043 0381

 

Formula (28.3) then gives 6 = + 0.059 825 572 radian

= + 3.427 753 degrees

-— + 13 minutes 42.7 seconds

The curve representing the variation of the equation of time during the year is well-known and can be found in many astronomy books. Presently, the curve has a deep minimum near February 11, a high maximum near November 3, and a secondary maximum and minimum about May 14 and July 26, respectively.

However, the curve of the equation of time is gradually changing in the course of the centuries, because the obliquity of the ecliptic, the eccentricity of the Eanh’s orbit, and the longitude of the perihelion of this orbit are all slowly changing. The figure on the next page shows the curve of the equation of time at intervals of 1000 years, from —2000 to + 5000. On the vertical scale, the tics are given at intervals of five minutes of time; the horizontal line represents the value E -— zero. The tics on this horizontal line divide the year in four periods of three months each, beginning from January 1 at left. We see, for instance, that ltte minimum of February will be less deep in the future.

 

The curve of the equation of time at intervals of1 0€t0 years, from 2000 B.C. to A.D. 5000. For each curve, the scale is given at lefi, in minutes of time. The tics on the horizontal line divide the year in four quarters.- January1 at left, December 31 at right.

 

Between A.D. 1600 and 2100, the extreme values of the equation of time vary as shown in Table 28.A. These are “mean" values: the calculation is based on a non-perturbed elliptical motion of the Earth, and the nutation has not been taken into account.

In A.D. 1246, when the Sun’s perigee coincided with the winter solstice, the curve representing the annual variation of the equation of time was exactly symmetrical with respect to the zero-line: the minimum of February was exactly as deep as the height of the November maximum, and the smaller May maximum was exactly as high as the value of the July minimum — see the last line of the Table.

T A B L E 2 8. A

The extreme values o f the equation o f time in modern times

Year Minimum of February Maximum of

May Minimum of

July Maximum of November

m s tTt S ITI S

1600 —15 01 +4 19 —5 40 +16 03

1700 —14 50 +4 09 —5 53 +16 09

1800 —14 38 +3 59 —6 05 +16 15

1900 —14 27 +3 50 —6 18 +16 20

2000 —14 15 +3 41 —6 31 +16 25

2100 —14 03 +3 32 —6 44 +16 30

1246 —15 39 +4 58 —4 58 + 15 39

RE FER E N CE

1. W.M. Smart, Text-Book on Spherical Astronomy , Cambridge (U.K.), University Press (1956); page 149.

 

 

Chapter 29

Ephemeris for Physical Obsewations

of the Sun

The formulae given in this Chapter are based on the elements determined by Carrington (1863), which have been in use for many years. For a given instant, the required quantities are:

P —— the position angle of the northern extremity of the axis of rotation, measured

eastwards from the North Point of the solar disk;

BO = the heliographic latitude of the center of the solar disk; L 0 = the heliographic longitude of the same point.

Although position angles are generally counted from 0° to 360‘ (this is the case for the Moon, the planets, double stars, etc.), in the case of the Sun it is customary to keep f•, in absolute value, less than 90°, and to assign to it a plus or a minus sign: P is positive when the northern extremity of the rotation axis of the Sun is tilted to the East, negative if towards the West. Celestial and solar north can differ by up to 26 degrees. P reaches a minimum of —26°.3 about April 7, a maximum of

+26.°3 about October 11, and is zero near January 5 and July 7.

B0 represents the tilt of the Sun’s north pole toward (+) or away (—) from Earth. It is zero about June 6 and December 7, and reaches a maximum value about March 6 (—7°. 25) and September 8 (+7°.25).

Lp decreases by about 13.2 degrees per day. The mean synodic period is 27.2752 days. The beginning of each “synodic rotation” is the instant at which Lt passes through 0°. Rotation No. 1 commenced on 1853 November 9.

Let JD be the Julian Ephemeris Day, which can be calculated by means of the method described in Chapter 7. If the given instant is in Universal Time, add io ID the value IT -- TD — UT expressed in days (see Chapter 10). If A T is expressed in seconds of time, the correction to JD will be + b TI 86400.

Then calculate the following quantities:

189

 

8 = (JD -- 2398 220) x     360°  

25.38

I --- 7.°25 = 7°15’

K --- 73.°6667 + 1.°395 8333 JD -- 2396 758  

36525

where i is the inclination of the solar equator on the ecliptic, and K is the longitude of the ascending node of the solar equator on the ecliptic. In the formula for 8,

25.38 is the Sun’s sidereal period of rotation in days. This value has been fixed conventionally by Carrington. It defines the zero meridian of the heliographie longitudes and therefore must be treated as crack. Strictly speaking, bécause the plane of the ecliptic slowly rotates (presently by 47“ per century) while the rotation axis of the Sun is supposed to be fixed in space, the angle I slowly varies over time. However, it is astronomical practice to assign I the constant value 7.°25.

Calculate the apparent longitude k of the Sun (including the effect of aberration, but not that of nutation) by the method described in Chapter 25, and the obliquity of the ecliptic e (including the effect of nutation) as explained in Chapter 22. Let k’ be h corrected for the nutation in longitude.

Then calculate the angles x and y by means of

tan x --- -- cos h' tan e

trim y -- -- cos (h -- K) trim I

where both x and y should be taken between --90° and +90°. Then the required quantities I, By, and Lt are found as follows:

 

f• =  x + y

sin By = sin (k -- K) sin i

 -- sin (h -- K) cos 

" -- cos (k -- K)

 

= tan (k -- K) cos I

 

9 being in the same quadrant as k -- K + 180°,

L   -- y - 8, to be reduced to the interval 0 --360 degrees.

Example 29.a -- Calculate P, B0, and for 1992 October 13 at 0' Universal Time

= JD 2448 908.5.

We will use the value IT ---- +59 seconds = + 0.000 68 day. Consequently the corrected JD, or Julian Ephemeris Day, is 2448 908.50068 and we find successively

 

29. PHYSICAL EPHEMERIS OF THE SUN

8 = 718 985°.8252 = 65°.8252

f = 7°.25

K -- 75°.6597

From Chapters 25 and 22:

L (Earth) = --43.634 836 22 radians = + 19°.908 045

A = 0.997 608

AJ = + 15“.908 = +0°.004 419

e = 23°.440 144

correction for aberration = 20'!4898 = --0°.005 705

 

191

 

 

whence

 

k = L + 180° -- 0°.005 705 = 199.°902 340

k’ = h + AJ = 199°.906 759

tan z = +0.407 664 z = + 22°.1790

tan y = +0.071 584 y -- + 4°.0945

P = 26.°27

sin Be -- +0. 104 324 By +5°.99

 

_ --0.820 053  

+0.562 699

 

= --55°.5431

 

L = -- 121°. 3683 = 238°.63

As mentioned above, a solar “synodic rotation” begins when is equal to 0°. An approximate time for the beginning of Carrington’s synodic rotation NO. C' lS

Julian Ephemeris Day = 2398 140.2270 + 27.275 2316 C (29. l)

where, of course, C is an integer. The instant so obtained will be at most 0.16 day in error. However, the time obtained from the formula above can be corrected as follows. Calculate the angle M, in degrees, from

M ---- 281.96  + 26.882 476 C

Then the correction in days is

 

+ 0.1454 sin M

-- 0.0085 sin 23f

-- 0.0141 cos 2 M

 

t29.2)

 

Between the years 1850 and 2100, the resulting time will be less than 0.002 day in error.

Of course, a correct value for the time of the beginning of a synodic rotation can be obtained by calculating @ for two instants near the time given by the formula above, and then by performing an inverse interpolation to find when Lp is zero.

Example 29.â — Find the instant of the beginning of solar rotation No. 1699.

For C -— 1699, formula (29. l) gives JDE = 2444 480.8455.

We further find M = 45 955.°287 = 235.°287, and the correction as given by (29.2) is —0.1225 day.

To convert from Dynamical Time to Universal Time, there is a further correction of —0.0006 day, because in 1980 the value of b T -- TD — UT was 51 seconds.

Hence, the final instant is

JD = 2444 480.8455 — 0.1225 — 0.0006 = 2444 480.7224

which corresponds to 1980 August 29.22.

The Astronomical Ephemeris for 1980, page 359, gives the same value.

It is customary to give the times of the commencement of the Sun’s synodic rotations to the nearest 0.01 day, hence in days and decimals, not in hours and minutes.

 

Chapter 30

Equation of Kepler

There are several methods for calculating the position of a body (planet, minor planet, or periodic comet) on its elliptical orbit around the Sun at a given instant:

— by numerical integration, a subject which is outside the scope of this book;

— obtaining the body’s heliocentric coordinates (longitude, latitude, and radius vector) by calculating the sum of periodic terms, as will be explained in Chapter 32;

— from the orbital elements of the body, as explained in Chapter 33.

In the latter case, we need to find the true anomaly of the object. This can be achieved either by solving Kepler’s equation or, when the orbital eccentricity is not too large, by using series expressions (see “The Equation of the Center” in Chapter 33).

Figure I

193

 

In Figure 1 we represent one half of an elliptical orbit PKG. The Sun is situated in the focus S,' the other, empty focus of the ellipse is F. The straight line AP is the major axis of the orbit. The center C of the ellipse is exactly half-way between the perihelion P and the aphelion A, as well as half-way between the foci I and S.

Suppose that, at a given instant, the moving body is at K. The distance SK is the radius vector of the body at that instant; this distance r is expressed in astronomical units. The tme anomaly (v) at the same instant is the angle between the directions SP and SK,- it is the angle over which the object moved, as seen from the Sun, since the previous passage through the perihelion P.

The seinimajor axis, CP in Figure 1, is generally designated by o and is expressed in astronomical units. By definition, the eccentricity e of the orbit is equal to the ratio of the distances CS and CP, or e ---- CS/ CP. The eccentricity of an orbit is a measure of how much that orbit deviates from a circle. It takes values between 0 and 1 for an ellipse, 1 for a parabola, and larger than l for a hyperbola. For a perfect circle, e = 0.

The perihelion and aphelion distances are designated by q and Q, respectively. In the perihelion, v = 0° and r -- q, while in the aphelion we have r = 180° and r ---- Q. It follows that

distance CS -- ae

distance SP --- q -- a (1 -- e) distance SA -- Q -- a (l + e) distance PA -- 2a --- q + Q

Let us now consider (Figure 2) a fictitious planet or comet K' describing around the Sun a circular orbit, hence with a constant velocity, with the some period as the real planet or comet K. Moreover, let us suppose that this fictitious body is at I", on the line SP, at the instant when the real body is at the perihelion P. Some time later, when the true body is at K, the fictitious body is at K‘. As we have seen, the angle v = angle PSK is the true anomaly of the body (at the given instant). The angle PSK’ at the same instant is called the ztiean onomaiy and is generally designated by If.

In other words, the mean anomaly is the angular distance from perihelion which the planet would have if it moved around the Sun with a constant angular velocity. By definition, the angle M increases uniformly with time. The value of M at a given instant is easily found, for 3f = 0° when the planet is at perihelion, and it increases by exactly 360° in the course of one complete revolution of the planet.

f7ie problem consists in finding the true anomaly v when the mean anomaly M and the orbital eccentricity e are known. Unless use is made of series expressions such as those given in Chapter 33, one has to solve Kepler’s equation.

In this connection, it is necessary to introduce an auxiliary angle E, called the eccentric anomaly, whose definition is illustrated in Figure 1. The exterior, dashed circle has diameter IP. We draw K Q perpendicular to AP. The angle PC Q is the eccentric anomaly.

 

30. ECtUATlON OF KEPLER

 

19 5

 

When the planet is at peri- helion the angles v, £, and M are all zero. Near the perihelion, the true planet moves at a greater speed than the mean, fictitious planet. Hence, between perihelion and aphelion, when the planet moves away from the Sun, we have v > 3f and, because E is always between v and ñf, we then have

0° < If < R < v < 180°.

 

Figure 2

v

2

 

In the aphelion, v, E, an6 M are all equal to 180°, and after aphelion passage, on its way back to perihelion, the true planet remains behind the mean planet.

When E is known, v can be obtained from

E (30. 1)

2

 

while the radius vector can be calculated from one of the following expressions:

r —— a (1 — e cos E) (30.2)

 

r ——

r --

 

   o (1 — e°) 

1 + e cos v

    q (1 + e) 

1 + e cos v

 

(30.3)

(30.4)

 

But let us now consider the problem of finding the eccentric anomaly E. The equation of Kepler is

E —— M + e sin E (30.5)

This equation must be solved for E. It is, however, a transcendental function which cannot be solved directly. Hundreds of methods of solution to the equation exist. An account of the history of solving the famous equation can be found lfl Colwell’s book [1]. We will describe three iteration methods for finding the eccentric anomaly E, and finally give a formula which yields an approximate result.

 

Pirst Method

In formula (30.5) the angles If andñ   should be expressed in radians. Hence the calculation should be performed in “radian mode”, which is the case for many programming languages. If the calculation is made in “degree mode”, then in (30.5) one should multiply e by 180/ r, or 57.295 7795, the factor for converting radians into degrees. Let eg be the thus “modified” eccentricity. Kepler’s equation is then

 

ñ = M + e0 sin E

and now we can calculate with ordinary degrees.

 

(30.6)

 

To solve equation (30.6), give an approximate value to E in the right side of the formula. Then the formula will give a better approximation for E. This is repeated until the required accuracy is obtained. This process can be performed automatically in a computer program. For the first approximation, we may use 6 = M.

We thus have

£ 0 = M

E   = M + e sin TO E 2 = M + e sin E E —— M + e sinñ 2

etc.

E , r 2, E , etc., are successive and better approximations for E.

Example 30.a — Solve the equation of Kepler for e = 0.100 and 3f = 5°, to an accuracy of 0.000 001 degree.

We have e0 = 0.100 x 180/z = 5.°729 577 95, and the equation of Kepler becomes

£ = 5 + 5.729 577 95 sin E

where all quantities are in degrees. We must now, of course, work in degree mode. Starting with E —— M —- 5°, we obtain successively

5.499 366

5.549 093

5.554 042

5.554 535

5.554 584

5.554 589

5.554 589

Hence, the required value is E = 5.°554 589.

 

 

Figure 3

 

This method is very simple and does always converge. There will be no problems when e is small. However, the number of required iterations is generally increasing with e. For example, for e —— 0.990 and 3f = 2°, the successive results of the iteration procedure are as follows:

2.000000 15.168909 24.924 579 29.813 009

3.979598 16.842404 25.904 408 30.200 940

5.936635 18.434883 26.780 556 30.533 515

7.866758 19.937269 27.557 863 30.817 592

9.763644 21.341978 28.242 483

11.619294 22.643349 28.841 471

13.424417 23.837929 29.362 399

After the 50th iteration, the result (32°.345 452) still differs from the correct result (32.361 007) by more than 0.01 degree.

Figure 3, due to the Belgian calculator Edwin Goffin, is a three-dimensional representation of the number of iterations needed to obtain an accuracy of 10*9 degree, as a function of the orbital eccentricity and the mean anomaly. We see that the number of required iterations becomes large when the eccentricity approaches 1 and when the mean anomaly is close to 0° or to 180°. — Note that 10* degree (4 millionths of an arcsecond) is an absurdly high accuracy; it has been retained here merely as a mathematical exercise.

At the bottom of the drawing we notice a horizontal straight “valley”. This valley extends from the point e = 0, M -- 90° to the point e = 1, M —- 32°42’. (This latter value is equal to z/2 — 1 radians.) This means that, for any eccentricity e, there is a value 3f0 of the mean anomaly for which the number of iterations (to solve Kepler’s equation by the method described above) is a minimum. This

“particular” mean anomaly is given by    I       (• /2 — e) radians and corresponds to the solution E -- w/2 radians = 90° exactly.

The number of required iterations increases as M differs more fro < on both sides of the “valley”. For instance, for e = 0.75 we have 3f   = 47.03 degrees, and the number of steps needed to obtain € with an accuracy of 0.000 001 degree

is as follows:

M iter. M Iter.

5° 51 60° 11

10° 37 70° 12

20° 23 90° 21

30° 15 110° 32

40° 9 l3fi° 43

47° 5 l5fi° 54

55° 8 170° 59

 

An interesting fact is that, when 3f is between Off and 180°, the results of the successive iterations oscillate while converging to the exact value: they do not constantly vary in the same direction as was the case in Example 30.a. For e = 0.75 and M —— 70°, the results of the successive iterations are

70.°000 000 starting value

110.380 316 larger

110.281 870 smaller

110.307 524 larger

110.300 850 smaller

110.302 587 larger

110.302 135 smaller etc.

Second Method

When the orbital eccentricity e is larger than 0.4 or 0.5, the convergence of the method described above can be so slow that it may be advisable to use a better iteration formula. A better value I I for E is

 

  + e sin TO — E0 

1 — e cos TO

 

(30.7)

 

where Ed is the last obtained value for E. In this formula, the angles If, Eb, and Al are all expressed in radians. If one wishes to work in “degree mode”, then in the numerator only of the fraction the eccentricity e should be replaced by the “modified” eccentricity e0 = 180 elm.

Here, again, the process should be repeated as often as is necessary.

Note the difference between formulae (30.6) and (30.7). The first one directly gives a new approximation for E. While formula (30.7) too gives a new approximation E for the eccentric anomaly, the fraction in the second member is actually a correction to the previous value E0.

Example 30.b — Same problem as in Example 30.a, but now using formula (30.7). We shall work in degree mode, so in this case formula (30.7) takes the following

 

E —— €

 

+  5 + 5.729 577 95 sin 6 0 — Eg  

0 1 — 0.100 cos 6b

 

Starting with jS 0 = M --  5°, we obtain the following values:

 

po correction

 

5.000 000 000 +0.554 616 193 5.554 616 193

5.554 616 193 —0.000 026 939 5.554 589 254

5.554 589 254 —0.000 000 001 5.554 589 253

In this case, an accuracy of 0.000 000 001 degree is obtained after only three iterations.

We solved Kepler’s equation for some values of e and ñf; see Table 30. A, where the successive columns give the orbital eccentricity e, the mean anomaly M, the corresponding value of 6, and the number of iterations needed by using the first

(1) and the second (2) method, starting with E —— M as the first approximation. A computer working with twelve significant digits was used, and iterations were performed until the new value of 6 differed from the previous one by less than

0.000 001 degree.

It appears that, generally speaking, a larger value of e requires a larger number of iterations, for the first method as well as for the second one. But with the second method the number of these iterations is much smaller.

For small values of the eccentricity, say for e < 0.3, the first method still seems the best one: we may prefer to perform 5 or 10 easy iterations instead of two iterations with the more complicated formula (30.7). Only for larger values of the eccentricity is formula (30.7) to be preferred.

In some cases, the first method is disastrous. See the next-to-last line of the table, where no less than 150 iterations are needed to obtain E.

Finally, Table 30. A shows that the number of steps needed

 

TA BLE 3 0. A

 

to obtain a given accuracy does not only depend on the value of e, but on that of M too. See the last line of the table, where the first method requires only six iterations, in spite of the large value of the orbital eccen- tricity, e = 0.99.

Although for large values of the eccentricity formula (30.7) is superior to (30.6), there can still be problems. We performed some calculations with formula (30.7) on the old HP-85 microcomputer, each

 

time taking M as starting value for F. Table 30.B gives the successive “better” values of E (in degrees) for three cases.

TA B LE 3 O . B

e —— 0.99 If = 2° e = 0.999 M = 6° e —— 0.999 M -- 7°

188.700250865 930.362114752 832.86912333

90.0043959725 418.384869795 275,954959759

58.7251974236 —345.064633754 —87.610596019

41.762008288

34.1821261793

32.4485414136

32.361223124

32.3610074734

32.3610074722

32.3610074722 10182.3247508

1840.68260539

—5573.41581953

—2776.37618814

—478.97469399

— 185.902957505

— 86.6958017962

—48.9711628749 —48.5623921307

— 11.225108839

340.962715254

—5996.93473678

—2079. 96780001

511. 49423506

257,391360843

5.969894505

— 14.7148241705 1094.05946279

168.189220986 —336D6.763133

92.1098260913 — 12599.3759885

64.2252288664 11889243.763

52.4123211568 3642203.90477

49.7106850572 —432120. 48862

49.5699983807 — 145379. 711482

49.5696248567 142691.415319

49.5696248539 56806.8295471

In the first example (e = 0.99, M —- 2°) we start with E = 2°. The first iteration gives E —- 188°.7, which is even farther away from the solution! But thereafter come the values 90°, 59°, 42°, and then the procedure converges rapidly: after the eighth iteration the result is reached with an accuracy of 0.000 004 arcsecond.

In the second case (e = 0.999, 3f = 6°), the first iterations give bizarre values, almost as if by a random-number generator! There is no convergence at all, until after the 13th iteration the value 168° is obtained; seven more steps then give us the correct solution.

Third case: same eccentricity, but now 3f = 7°. Here, too, the successive results jump irregularly back and forth, and after 20 steps still nothing reasonable is reached. Not before the 47th iteration (not shown in the table) do we obtain the correct solution, namely 52.°270 2615.

It is truly remarkable that for the same eccentricity 0.999, but for M = 7.°0l instead of 7°.00, the correct value of E ’is reached after only nveJve iterations.

 

 

Figure 4

 

30. E£tUATlON OF KEPLER

Fijtore 5

 

2 03

 

The HP-85 worked with 12 significant digits. If you use another computer with another programming language, the number of iterations can sometimes differ appreciably from those we mention here. When one calculates the second case

{e —— 0.999, M -- 6°) with the HP-67 pockei calculator, which works with 10 significant digits, the successive results (in degrees) are

930.3621195

418.3848584

—345.0649049

10182.69391

1883.665232

—162.6729360

—85.06198931

—47.82386405

— 13.18454655

211.0527629

84.65261970

60.76546811

51.35803706

49. 62703439

49.56968687

49.56962485

49.56962485

It is interesting to compare these values with those of Table 30.B. After the third iteration, the difference with the value obtained with the HP-85 is still 0.00027 degree only. After the next iteration, the difference is 0°.37, and after the next one it is 43 degrees! Nevertheless, convergence to the exact value is eventually achieved.

It is evident that, when e is large, formula (30.7) guarantees only a local convergence. The successive results jump irregularly back and forth, and only when by chance a result falls into the “right domain” do the next results converge rapidly.

Figure 4, due to Goffin, is a three-dimensional representation of the number of steps needed to obtain E with an accuracy of 10*’ degree, as a function of the orbital eccentricity and the mean anomaly, when formula (30.7) is used. As before, M is used as the starting value for E. The left corner, near e = 1 and M —— 0°, is the “dangerous zone”. Figure 5 shows a magnification of that zone: we see a large number of peaks which are close together; the number of iterations needed to obtain the stated accuracy differs considerably even when e or M is changed very little.

Consequently, formula (30.7) is rather worrying for large values of e and small values of M. In some cases, the computer runs the risk of overflowing because the denominator of the fraction becomes almost zero. This trouble can be avoided by choosing, as a starting value for £, a better value than just 3f.

Mikkola [2] proposed a procedure to find such a good starting value. It was reproduced in the first edition of this book [3].

 

However, there are easier ways to avoid the (sometimes) many irregular jumps of the results of the successive iterations when e is large. We note that Kepler’s equation can be written as 6 — M —— e sin E, the second member of which can never exceed 1 in absolute value, and has the same sign as E. Therefore, the fraction term in (30.7) should never be allowed to exceed a magnitude of 1.

One method is to take the arcsine of the size of the fraction. This will result in a value which is always between —90° and +90°. This trick was mentioned to the author by Kurt Leingärtner, of Kassel, Germany.

As an example, consider the case e —— 0.99, M -- 0.2 radian. We will work in radian mode. On the first step, the fraction in formula (30.7) takes the value 6.614 719 035 698 radians which, by taking the arcsine of its sine, changes to

0.331 533 728 518. The successive iterations yield the following results :

correction to E new value of E

0.331 533 728 518 0.531 533 728 518

1.161 431415 069 1.692 965 143 587

—0.455 401 365 518 1.237 563 778 069

—0.150 884 433 942 1.086 679 344 127

—0.019 368 331 549 1.067 311 012 578

—0.000 313 565 645 1.£fi6 997 446 933

—0.000 000 081 651 1.066 997 365 282

< 10* 1.066 997 365 282

Hence, the final result is 1.066 997 365 282 radians, or 61.134 445 78 degrees.

Another interesting trick, which avoids the extra functions sine and arcsine, was devised by John M. Steele, of Bloomfield Hills, Michigan [4]. If the absolute value of the fraction in formula (30.7) is larger than 0.5, it is replaced by 0.5, preserving the sign. In BASIC, w being the value of the fraction:

IF ABS (w) > 0.5 THEN  cor = 0.5 SGN(w) ELSE cor = w

According to Steele, a “limit value” of 1 (instead of 0.5) works, although smaller values in the range 0.4—0.6 seem to work better.

Let us again consider the case e = 0.99, M —— 0.2 radian. On the first step, the fraction in formula (30.7) takes the value 6.614 719 035 698 radians, which is changed to the “limit value” 0.5. The successive iterations yield the following results:

correction (rad) changed to new volue of E (rad)

6.614 719 035 698 0.5 0.7

0.567 429 870 979 0.5 1.2

—0.120 513 681 086 unchanged 1.079 486 318 914

—0.012 361 504 682 unchanged 1.067 124 814 232

—0.000 127 435 465 unchanged 1.066997 378 767

—0.000 000 013 485 unchanged 1, 066 997 365 282

 

Third Method

Roger Sinnott [5] devised a method using a binary search to locate the correct value of F. The binary search was already mentioned at the end of Chapter S. The procedure is absolutely foolproof, it always converges to the most exact value of which the machine is capable, and it works for any eccentricity between 0 and 1. The relevant part of Sinnott’s program, in BASIC, is given below. Here, E is the orbital eccentricity, and M the mean anomaly in radians. The result of the program is the eccentric anomaly E expressed in radians, too.

For a computer language with 10-digit accuracy, 33 steps are needed in the binary search. The number of loops in line 180 should be increased to 53 if you are using a 16-digit BASIC. The number of steps needed is 3.32 x the number of required  digits,  where  3.32 is equal  to  1/logt  0 2.

loo P1 = 3.14159265359

110 F = SGN (M) : M = ABS (M) / (2 » PI)

120 M = (M — tNT (M)) 2 P1 a F

130 IF M < 0 THEN M = M + 2 s {•}

140 F = 1

150 IF M > P1 THEN F = — 1

160 IF M > P1 THEN M = 2 » P1 — M 170 E0 = P1/2 : D = PI/4

180 FOR J = 1 TO 33

190 M1 = E0 — E SIN (E0)

200 E0 = E0 + D » SGN (M — M l) : D = D/2

210 NEXT J

220 E0 = E0   F

Fourth Method

 

The formula

 

tan E —— sin If

cos M - e

 

(30.8)

 

 

gives an approximate

eccentricity.

 

value for E, and is valid only for small values of the

 

For the same data as in Example 30.a, the formula (30.8) gives

 

tan E ——    +0.087 155 74  

+0.896 194 70

 

=  +0.097 250 90

 

whence E -- 5°.554 599, the exact value being 5°.554 589, so the error is only 0”.035 in this case. But for the same eccentricity and ñf — 82°, the error amounts to 35“.

The greatest error due to the use of formula (30.8) is

0.°0327 for e -— 0.15

0.0783 for e = 0.20

0.1552 for e -— 0.25

1.42 for e —— 0.50

24.7 for e = 0.99

For the orbit of the Earth (e —— 0.0167), the error is less than 0“.2. In that case, formula (30.8) can safely be used unless high accuracy is needed.

R EFER E NC ES

1. Peter Colwell, Solving Kepler’s Equation (Willmann-Bell, 1993).

2. Seppo Mikkola, “A cubic approximation for Kepler’s Equation”, Celestial Mechanics, Vol. 40, pages 329—334 (1987).

3. Jean Meeus, Astronomical Algorithms, page 193 (Willmann-Bell, 1991).

4. John M. Steele, personal communication to Jean Meeus, 1994 November 20.

5. Roger W. Sinnott, Stay and Telescope, Vol. 70, page 159 (August 1985).

 

 

Chapter 31

Elements of the Planetary Orbits

Although Appendix III mentions the principal periodic terms needed to calculate the heliocentric positions of the planets (with explanations given in Chapter 32), it may be of interest to have information about the mean orbits of these bodies.

The orbital elements of the major planets can be expressed as polynomials of the form

of + a T + a T 2 + •3 T’

where T is the time measured in Julian centuries of 36525 ephemeris days from the epoch J2000.0 = 2000 January 1.5 TD = JDE 2451 545.0.

In other words,

 

T ---- JDE -- 2451 545.0 

36525

 

(3l.1)

 

This quantity is negative before the beginning of the year 2000, positive afterwards. The orbital elements are:

L -- mean longitude of the planet; a   -- semimajor axis of the orbit; e -- eccentricity of the orbit;

i -- inclination on the plane of the ecliptic; II = longitude of the ascending node;

r = longitude of the perihelion.

Many authors denote the longitude of the perihelion by w, which is a modified form of w. But this may be confusing because the argument of the perihelion has the symbol o. For this reason, we prefer the symbolr for the longitude of the perihelion, and we have z = II + o. (But don’t confuse z with the parallax or with the number 3.14159. ..!)

Note that the angles L and z are measured in two different planes, namely from the vernal equinox along the ecliptic to the orbit’s ascending node, and then from this node along the orbit. See the Figure on next page.

209

 

210 ASTRONOMICAL ALGORITHMS

The arc XNX“ is a pan of the ecliptic as seen from the Sun, and N PXX' is a part of the orbit of the planet (the intersection of the orbital plane with the celestial sphere). y is the vernal equinox (longitude 0°), N the ascending node of the orbit, P the planet’s perihelion. At a given instant, the mean planet is at X, the true planet at X'. Then we have

R ---- arc JN = longitude of the ascending node,

u› -- arc NP = argument of the perihelion,

r ---- ore yN + arc N P = fi + z = longitude of the perihelion.

L  = ore yN + arc N X = II + in + M mean longitude of the planet.

M --- arc PX = planet’s mean anomaly, C -- arc XX' = equation of the center,

v   = arc PX’ = 3f + C planet’s true anomaly,

i -- inclination of the orbit angle bet 'een arcs N P and N X“.

The planet’s mean anomaly is given by

Table 31.A gives the coefficients a0 to a for the orbital elements of the planets Mercury to Neptune. The values for the semimajor axes are in astronomical units. Those for the angular quantities L, i, R, andr are expressed in degrees and decimals; they are referred to the ecliptic and mean equinox of the date.

The values have been deduced from a study by Simon e.o. [1]. However, in the case of the planets Mercury to Mars we added the correction +0“.2766 T to o for the elements L, II, and w in order to bring them in accordance with the VSOP87 theory. The elements L, i, fi, andr are actually referred to the mean dynamical ecliptic and equinox of the date, which differ very slightly from the FK5 system (see Chapter 25).

In some cases, it may be desirable to refer the elements L, i, R, andr to a standard equinox. This is the case, for instance, when one wishes to calculate the

 

31.   ELEMENTS OF PLANETARY ORBITS 211

least distance between the orbit of a comet and that of a major planet, when the elements of the first orbit are referred to a standard equinox.

By means of Table 31.B, it is possible to calculate these elements for the major planets, referred to the standard equinox of J2000.0. The elements a and e are not modified by a change of reference frame, of course. They should be calculated by means of Table 31. A.

For the Earth, in order to avoid a discontinuity in the variation of the inclination and a jump of 180° in the longitude of the ascending node at the epoch J2000.0, the inclination on the ecliptic of 2000.0 is considered as negative before A.D. 2000.

Example 31.a Calculate the mean orbital elements of Mercury on 2065 June 24

at 0‘ TD.

We have (see Chapter 7)

2065 June 24.0 = JDE 2475 460.5

whence, by formula (31.1),

T +0.654 770 704 997

Consequently, from Table 31.A we find:

L --- 252°.250 906 + (149 474°.072 2491 x 0.654 770 704 997)

+ (0.000 303 50) (0.654 770 704 997)2

+ (0.000 000 018) (0.654 770 704 997)'

= 98 123°. 494 701 = 203°.494 701

 

a -- 0.387 098 310

e 0.205 645 10

i --- 7°.006 171

fi = 49°.107 650

 

r = 78.°475 382

from which we deduce

If = L -- r - 125°. 019 319 o   =r -- II = 29.°367 732

 

 

From Tables 31. A and 31.B it appears that the inclination of the orbit of Mercury on the ecliptic of the date is increasing, but that it is decreasing with respect to the fixed ecliptic of 2000.0. The opposite occurs for Saturn and Neptune.

Between T -- --30 and T ---- + 30, Venus’ orbital inclination on the ecliptic of the date is continuously increasing, but with respect to the fixed ecliptic of 2000.0 Venus’ inclination reached a maximum about the year + 690.

Uranus’ orbital inclination on the ecliptic of the date reached a minimum about the year + 1000, but with respect to the fixed ecliptic of 2000.0 its value is continuously decreasing during the time period considered here.

The longitudes of the nodes, referred to the equinox of the date, are increasing for all planets. But with respect to the fixed equinox of 2000.0 these longitudes are decreasing, except for Jupiter and Uranus.

 

T A B L E 3 1 . A

Orbital Elements for the mean equinox o f the date

M E R C U RY

L 252. 250 906 +149 474.072 2491 +0.000 303 50 +0.000 000 018

0.387 098 310

e 0.205 631 75 +0.0£D 020 407 —0.000 0£D 0283 —0.000 00 000 18

i 7.004 986 +0.001 8215 —0.000 018 10 +0.000 000 056

II 48.330 893 +1.186 1883 +0. UD 175 42 +0.000 OSD 215

z 77.456 119 + 1.556 4776 +0.000 295 44 +0.000 000 009

V E N U S

L 181.979 801 +58 519.213 0302 +0.0tXl 310 14 +0.000 000 015

a 0.723 329 820

e 0.006 771 92 —0.000 047 765 +0.0€D OOO 0981 -+0.000 000 000 46

i 3.394 662 +0.001 0037 —0.000 0 D 88 —0.000 000 007

II 76.679 920 +0.901 1206 +0.0fD 406 18 —0.000 000 093

r 131.563 703 + 1.402 2288 —0.001 076 18 —0.000 005 678

EARTH

L 100.466 457 +36 000.769 8278 +0.000 303 22 +0.000 OSD 020

a 1.000 001 018

e 0.016 708 63 —0.000 042 037 —0.000 000 1267 + 0.000 000 000 14

i 0

a 102.937 348 + 1.719 5366 +0.00€l 456 88 —0.000 000 018

M A R S

L 355.433 000 + 19 141.696 4471 +0.000 310 52 >0.000 0€D 016

1.523 679 342

e 0.093 4€D 65 +0.000 090 484 —0.000 HD 0806 —0.000 000 00025

i 1.849 726 —0.000 6011 +0.000 012 76 —0.000 000 007

II 49.558 093 +0.772 0959 +0.0£D 015 57 +0 000 002 267

r 336.060 234 + 1.841 0449 +0.HD 134 77 +0.000 US 5S6

 

TA B LE 3 1 . A (cont. )

J U PI T E R

L 34.351 519 + 3036.302 7748 +0.000 223 30 +0.000 000 037

a 5.202 603 209 +0.000 000 1913

e 0.048 497 93 +0.000 163 225 —0.000 000 4714 —0.000 000 002 D l

i 1.303 267 -0.005 4965 +0.000 004 66 —0.000 000 002

fi 100.464 407 + 1.020 9774 +0.000 403 15 +0.000 000 404

r 14.331 207 + 1.612 6352 +0.001 030 42 —0.000 004 464

SATURN

L 50.077 444 +1223.511 0686 +0.UD 519 08 —0.OSD 000 030

o 9.554 90P 192 —0.000 002 1390 +0.000 000 004

e 0.055 548 14 —0.000 346 641 —0.000 0€D 6436 +0.ADO0>D140

i 2.488 879 -0.003 7362 —0.000 015 19 +0.DOOC00087

II 113.665 503 +0.877 0880 —0.000 121 76 -0.DJO002249

z 93.057 237 + 1.963 7613 +0.000 837 53 +0GDCD4928

U RA N U S

L 314.055 005 +429.864 0561 +0.000 303 90 +0AD000026

a 19.218 446 062 —0.000 000 0372 +0.000 000 0(O 98

e 0.046 381 22 —0.000 027 293 +0.000 000 0789 + 0, 000 000 000 24

i 0.773 197 +0.000 7744 +0.000 037 49 —0.000 000 092

II 74.005 957 +0.521 1278 +0.001 339 47 +0.OSD 018 484

r 173.005 291 +1.486 3790 +0.000 214 06 +0.000 000 434

N E PTU N E

L 304.348 665 +219.883 3092 +0.000 308 82 +0.000 OOO 018

o 30.110 386 869 —0.000 000 1663 +0.000 0€D 000 69

e 0.009 455 75 +0.000 006 033 +0.000 000 0000 —0.000 000 000 05

i 1.769 953 —0.009 3082 —0.000 007 08 +0.0fD 000 027

II 131.784 057 + 1. 102 2039 +0.000 259 52 —0.000 000 637

w 48.120 276 +1.426 2957 +0.000 384 34 +0.0fD (TO 020

 

TA B L E 3 1 . B

Orbital Elements for the  standard equinox J2OO0.0

M E R C U RY

L 252.250906 +149472.674 6358 —0.000005 36 +0.0fD000002

i 7.004986 —0.0059516 +0.00000080 +0.000000043

8 48.330893 —0.1254227 —0.0(D08833 —0.0000€O200

r 77.456119 +0.1588643 —0.0tD01342 —0.0fD0(D007

V E N U S

L 181.979801 +58517.815 6760 +0.0£J000165 —0.000000002

i 3.394662 —0.0008568 —0.00003244 +0.000000009

8 76.679920 —0.2780134 —0.0014257 —0.0CD0€O 164

w 131.563 703 +0.004 8746 —0.00138467 —0.0£D005 695

EA RT H

L 100.466457 +35999.3728565 —0.000005 68 —0.000000 001

i 0 +0.0130548 —0.0£D00931 —0.0£D000034

£t 174.873176 —0.2410908 +0.00004262 +0.000000001

r 102.937348 +0.3225654 +0.OSD 14799 —0.0000£D039

M A R S

355.433000 +19140.2993039 +0.0fD 00262 —0.000000003

i 1.849726 —0.0081477 —0.000 02255 —0.000000029

II 49.558093 —0.2950250 —0.00064048 —0.000001964

r 336.060234 +0.4439016 —0.0(D17313 +0.000 000 518

 

TA BLE 3 1. B (cont. I

J U P IT E R

L 34.351519 +3034.905 6606 —0.00008501 +0. 0€D ISO 016

i 1.303 267 —0.iXl19877 +0.00€103320 +0.000 000 097

II 100.464407 +0.1767232 +0.00090700 -O .0tD (D7 272

r 14.331207 +0.2155209 +0.00072211 —0.000 004 485

SATU R N

L 50.077444 +1222.113 8488 +0.0£D21004 -0.GBO0OO46

i 2.488879 +0.0025514 —0 OD04906 +0%O0WDO17

II 113.665503 —0.2566722 —0.000 18399 +0.ODOOO480

93.057237 +0.5665415 +0.00052850 +0.ODDO04912

U RA N U S

L 314.055005 +428.4669983 —0.00000486 + 0.000 000 006

i 0.773197 —0.0016869 +0.00000349 + 0.000 000 016

II 74.005957 +0.074 1431 +0.00(140539 + 0.SCO OCD 119

w 173.€05291 +0.0893212 —0.IXD09470 + 0.OX1000 414

NEPTUNE

L 304.348 665 +218.4862002 +0.00000059 -0.ADODOO02

i 1.769953 +0.0002256 +0.0£D00023

 

II 131.784057 —0.0061651 —0.00£100219

48.120276 +0.0291866 +0.00£107610

 

RE FER EN CE

1. J. L. Simon, P. Bretagnon, J. Chapront, M. Chapront-Touzé, G. Francos, J. Laskar, “Numerical expressions for precession formulae and mean elements for the Moon and the planets", Astronomy & Astrophysics, Vol. 282, pages 663-683 (1994).

 

Chapter 32

Positions of the Planets

In 1982, P. Bretagnon of the Bureau des Longitudes of Paris published his planetary theory VSOP82. The acronym VSOP means “Variations Séculaires des Orbites Planétaires”. The VSOP82 consists of long series of periodic terms for each of the major planets Mercury to Neptune. When, for a given planet, the sums of these series are evaluated for a given instant, one obtains the values of the following quantities for the osculating orbit. The osculating orbit is the “instantaneous” orbit of the planet; see more about this notion in the next Chapter.

a   -— semimajor axis of the orbit k = mean longitude of the planet h —- e sinr

k —- e cosr

p ——  sin '/z i sin C q = sin '/2 i cos £i

where e is the orbital eccentricity, z the longitude of the perihelion, i the inclination, and 0 the longitude of the ascending node.

Once a, h, e and w (from h and k), i and R ( from p and q) are known, the true position in space can be obtained for the given instant.

The inconvenience of the VSOP82 solution is that one does not know where the several series should be truncated when no full accuracy is required. Fortunately, in 1987 Bretagnon and Francou constructed the version called VSOP87, which gives periodic terms for calculating the planets’ heliocentric coordinates directly, namely

L, the ecliptical longitude

B, the ecliptical latitude

fi, the radius vector (= distance to the Sun)

217

 

Note that L is really the planet’s ecliptical longitude, not the orbital longitude. In the figure on page 210, the orbital longitude of the planet is the sum of the arcs JN and fVX’ (in two different planes). Through the planet’s position X', a great circle X’X“ is drawn perpendicularly to the ecliptic. Then the planet’s ecliptical longitude is the measure of the arc yX“.

Although the methods used for the construction of the VSOP82 and VSOP87 have been described in the astronomical literature (see the References l and 2), these theories themselves are available only on magnetic tape or on CD-ROM. By kind permission of Messrs. Bretagnon and Francou, we give in Appendix III the most important periodic terms from the VSOP87 theory. For each planet, series labelled LO, L1, L2, . . . , BO, B1, . . . , RO, R1, . .. are provided.

The series LO, L1 , .. . are needed to calculate the planet’s heliocentric ecliptical longitude L, the series BO, B1 , . .. are needed for the ecliptical latitude B, e:n6 the series RO, R1, . .. are for the radius vector fi.

Each horizontal line in the list represents one periodic term and contains four numbers:

— the current No. of the term in the series. It is not needed in the actual calculation and is given for reference purpose only;

— three numbers which we shall call here A, B, and C, respectively.

Let JDE be the Julian Ephemeris Day corresponding to the given instant.

Calculate the time z measured in Julian millennia from the epoch J2000.0

 

 JDE — 2451 545.0

T 365 250

 

(32.1)

 

The value of each term is given by

A cos (B  + Cr)

For example, the ninth term of the series L0 for Mercury is equal to 1803 cos (4.1033 + 5661.3320a).

In the lists of Appendix III, the quantities B and C are expressed in radians. The coefficients A are in units of 10*' radian in the case of the longitude and the latitude, in units of 10 ' astronomical unit for the radius vector.

When a coefficient A has less decimals, then less decimals too are given for the corresponding B and C. This is merely done to avoid keypunching extraneous digits which do not influence the result.

To obtain the heliocentric ecliptical longitude L of a planet at a given instant, referred to the mean equinox of the date, proceed as follows. Calculate the sum L0 of the terms of series LO, the sum LI of the terms of the series L1, etc. Then the required longitude in radians is given by

 

Proceed similarly for the heliocentric latitude B and for the radius vector R.

The planet’s heliocentric longitude L and latitude B, obtained thus far, are referred to the mean dynamical ecliptic and equinox of the date defined by Bretagnon’s VSOP planetary theory. This reference frame differs very slightly from the standard FK5 system mentioned in Chapter 21. The conversion of L and B to the FK5 system can be performed as follows, where T is the time in centuries from 2000.0, or T --- 10 z.

Calculate

L’ = L -- 1°.397 T -- 0°.00031 T O

Then the corrections to L and B are

Af.   = --0“.09033 + 0”.03916 (cos L + sin L’) tan B (32.3)

i!sB --- +0”.03916 (cos L -- sin L')

These corrections are needed only for very accurate calculations. They may be dropped when use is made of the abridged version of the VSOP87 given in Appendix III.

How to obtain the geocentric positions of the planets will be explained in Chapter 33.

Example 32.a -- Calculate the heliocentric coordinates of Venus on 1992 Dec. 20 at 0h Dynamical Time.

This instant corresponds to JDE 2448 976.5, from which

z 0.007 032 169 747.

For Venus, series LO has 24 terms in Appendix III (there are many more in the original VSOP87 theory), L1 has 12 terms, L2 has 8 terms, L3 and L4 both have 3 terms, while LS contains just a single term. For the sums of these series, we find

 

IN = + 316 402 122

LI -- + 1 021 353 038 718

L2 = +50 055

 

L3 56

L4 = 109

LS 1

 

Hence, by formula (32.2), we find that the heliocentric longitude of Venus, for the given instant and referred to the mean equinox of the date, is

L = --68.659 2582 radians = --3933 .°88572 = + 26.° 11428

We calculate the heliocentric latitude B and the radius vector R in the same way. Note that, in the case of Venus, the series BS and R5 do not exist. The results are

B -- --0.045 7399 radian = --2°.62070, fi = 0.724 603 AU

 

220 ASTRONOMICAL ALGORITHMS

Accuracy of the results

When high accuracy is desired, it appears that the periodic terms in the VSOP87 solution converge rather slowly. What is the magnitude of the errors in the coordinates if one truncates the list of terms at any point? The following empirical rule has been given by Bretagnon and Francou [3] :

If n is the number of retained terms, and A the amplitude of the smallest retained term, the accuracy of the thus truncated series is about

9 x A, where q is a number smaller than 2.

As an example, consider the heliocentric longitude of Mercury. In Appendix III, series LO for this planet contains 38 terms, and the coefficient of the smallest retained term is 100 X 10*' radian. Therefore, we may expect that the greatest possible error in Mercury’s heliocentric longitude, as calculated by means of that truncated series, is approximately

2 x x 100 x 10*' radian = 2”.54.

Of course, series L1, L2, etc. , are truncated too, which gives rise to additional uncertainties of the order of 0“.41 z, 0“.08 z 2, etc.

Z•ofynomiof Expressions

The giant planets Jupiter, Saturn, Uranus, and Neptune move so slowly on their orbits around the Sun, that it is possible to construct polynomial expressions giving their heliocentric coordinates, each expression being valid for one year.

We choosed polynomials of the fifth degree, so that the required value of the heliocentric longitude, latitude, or radius vector is given by

Ay + A t + A;, t 2 + A t’ + A t 4 + A5 t 5 (32.4)

where r is the time (in the scale of Dynamical Time) measured from January 0.0 of the given year in units of 365 days. In other words, if d is the day of the year (with decimals, if any), then r = d / 365. Note that even in the case of a bissextile (leap) year, the denominator in this formula is still 365.

The constants TO to A5 are given in Appendix IV for the years 1998 to 2025. For each planet there are three polynomials per year: one for the heliocentric longitude (L), one for the latitude (B), and one for the radius vector (R). The

coefficients are expressed in degrees for the longitude and the latitude, in astronomical units for the radius vector.

The coordinates so obtained are geometric, and they are referred to the mean equinox of the date in the FK5 reference frame.

 

For the years 1998 to 2012, January 0.0 corresponds to the following Julian

Days:

Year JD Year JD Year JD

1998 2450813.5 2003 2452639.5 2008 2454465.5

1999 2451178.5 2004 2453004.5 2009 2454 831.5

2000 2451543.5 2005 2453370.5 2010 2455196.5

2001 2451909.5 2006 2453735.5 2011 2455561.5

2002 2452274.5 2007 2454100.5 2012 2455 926.5

Example 32.b — Calculate the heliocentric longitude of Saturn on 1999 July 26 at 0' Dynamical Time, referred to the mean equinox of the date.

July 26 being the 207th day of the year, we have

d -- 207 and i = 207 / 365 = 0.567 123 288

From Appendix IV we take for the longitude (L) of Saturn in 1999:

 

A0 = 32.578 4232

A = 12.966 6139

A2 =   0.129 4965

 

A = —0.010 5762

A4 = 0.007 6613

A5 = —0.003 6652

 

whence, by formula (32.4), / = 39.°972 3901 = 39°58’20”.60

This is indeed the result obtained directly from the VSOP 87 theory.

Calculated by means of these polynomial expressions, the maximum error in the heliocentric longitude will not exceed 0.05 arcsecond in the case of Jupiter, and

0.02 arcsecond for Saturn. For the much slower planets Uranus and Neptune, the error will even be less as compared with the VSOP87 theory.

REFEREN CES

1. P. Bretagnon, “Théorie du mouvement de l’ensemble des planètes. Solution VSOP82”, Astronomy and Astrophysics, Vol. 114, pages 278—288 (1982).

2. P. Bretagnon, G. Francou, “Planetary theories in rectangular and spherical variables. VSOP87 solutions”, Astronomy and Astrophysics, Vol. 202, pages 309-315 (1988).

3. Ibid. , page 314.

 

 

Chapter 33 Elliptic Motion

In this Chapter we will describe two methods for the calculation of geocentric positions in the case of an elliptic orbit. In the first method, the geocentric ecliptical longitude and latitude of a major planet (Mercury to Neptune) are obtained from the heliocentric ecliptical coordinates of the planet and the Earth. In the second method, which is better suited for minor planets and periodic comets, the right ascension and declination of the body, referred to a standard equinox, are obtained directly, and use is made of the geocentric rectangular coordinates of the Sun.

Pirst Method

We will describe how the apparent right ascension and declination of a major planet can be calculated for a given instant.

For the given instant calculate, by means of the appropriate series given in Appendix III and using the method described in Chapter 32, the heliocentric coordinates L, B, R of the planet, and the heliocentric coordinates L0, By, fi0 of the Earth. Do not convert from the dynamical ecliptic and equinox to the FK5 ecliptic and equinox at this stage.

Then calculate

x —— R cos B cos L — A0 cos By cos L0

 

y = fi cos B sin £ — TO cOs B0 Sin L0 z = R sin B — A0 Sin BO

The geocentric longitude k and latitude 9 of the planet are then given by

tan Q =

2 + y 2

 

(33. 1)

(33. 2)

 

Look out for the proper quadrant of k. One may use the “second” arctangent function, k = ATN2 (y, z) or use the fact that, if x < 0, then cos X < 0.

223

 

However, the geocentric coordinates h and d obtained in this way are the planet’s geometric coordinates referred to the mean equinox of the date. If high accuracy is needed, it is necessary to take into account the apparent displacement of the planet from its true position due to the finite velocity of light. This apparent displacement includes:

(a) the effect of light-time, the planet being seen where it was when the light

left it;

(b) the effect of the Earth’s motion which, combined with the velocity of light, causes an apparent displacement of the object, just as the annual aberration in the case of a star.

The combination of the two effects is often called “planetary aberration”. However, we prefer to reserve the term aberration to the effect (b) alone, because this effect is of the same nature as the aberration of the stars. Moreover, for some applications it is not necessary to take effect (b) into account. Suppose we want to calculate occultations of stars by planets. Then the effect of light-time must be taken into account in the calculation of the position of the planet; but we may drop effect

(b) on the condition that the effect of aberration on the star’s position is dropped too. Similarly, the effect of nutation can be neglected for both bodies in that particular case. The reason is evident: because the planet and the star are close together on the celestial sphere, the effects of aberration and nutation will not change their relative positions.

(a) eject of light-time: at time t, the planet is seen where it was at time i — z, hence in the direction obtained by combining the Earth’s position at time r with that of the planet at time i — z, where z is the time taken by the light to reach the Earth from the planet. This time is given by

z = 0.005 77 55 183 A days (33.3)

where A is the planet’s distance to the Earth in astronomical units, given by

2 +y2 + 2 (33.4)

(b) the effect of aberration can be calculated as for the stars, namely, by means of formulae (23.2), where O is equal to L0 + 180°.

However, both effects can be calculated simultaneously. To the order of accuracy that the motion of the Earth during the light-time is rectilinear and uniform, the planet’s apparent position at time f is the same as its geometric position at time i — z. In other words, in this method the Earth’s position at time i — z must be combined with the planet’s position at the same time f — z.

Of course, the value of the light-time z is not known in advance because the planet’s distance A to the Earth is not known. But this distance can be found by iteration, using for instance A = 0 (and hence z = 0) in the first calculation.

 

33. ELLIPTIC MOTION 2 2 5

For very accurate calculations, the planet’s geocentric longitude h and latitude fI can be converted from the dynamical ecliptic and equinox to the FK5 ecliptic and equinox by means of formulae (32.3), replacing L by h, and B by 9.

To complete the calculation of the planet’s apparent position, the corrections for nutation should be applied. This is achieved by calculating the nutation in longitude (AJ) and in obliquity (be), as explained in Chapter 22. Add A/ to the planet’s geocentric longitude, and Ae to the mean obliquity e0 of the ecliptic. The apparent right ascension and declination of the planet can then be deduced by means of formulae (13.3) and (13.4).

The elongation of the planet, that is, its angular distance to the Sun, can be calculated from

cos   = cos 9 cos (X -- @) (33.5)

where h, Q are the planet’s apparent longitude and latitude, and @ the Sun’s apparent longitude. The Sun’s latitude, which is always smaller than 1.2 arcsecond, may be neglected here.

Example 33.a -- Calculate the apparent position of Venus on 1992 December 20 at 0' TD = JDE 2448 976.5.

Because the planet’s distance to the Earth is not known in advance, the value of the light-time is not known. Therefore, we start with the calculation of the true (geometric) position of the planet at the given time. We find the following values for the heliocentric coordinates (see Example 32.a) :

L -- 26°.11428 B -- --2.°62070 A = 0.724 603

The coordinates of the Earth are calculated in the same way:

 

A0 -- 88°.35704 B0 -- +0.°00014 A = 0.983 824

whence, by formulae (33.1), (33.4), and (33.3),

 

(A)

 

 

z =  +0.621 746

y = --0.664 810

z = --0.033 134

 

A = 0.910 845

z = 0.005 2606 day

 

A is the true distance of Yenus to the Earth on 1992 December 20.0. We now repeat the calculation of Venus’ heliocentric coordinates for the instant i -- t, that is, for JDE = 2448 976.5 -- 0.005 2606. We obtain

 

L ---- 26°.10588 B -- --2°.62102 A = 0.724 604

Combining these new values with the values (A I of Lp, B0, fi0, we find x = +0.621 794

 

(B)

 

y --- --0.664 905  (C I

z = --0.033 138

 

A = 0.910 947

z = 0.005 2612 day

 

If we repeat the calculation with this new value of z, we find the same values (B) for L, B, and It again, to the given accuracy.

Hence, the final value for the light-time is z = 0.005 2612 day, and A = 0.910 947 AU is the apparent distance of the planet on 1992 December 20 at 0' TD. It is the distance at which we “see" the planet at that instant. In other words, it is the

distance travelled by the light which left the planet at time t - I to reach the Earth

at time t.

Let us now calculate Venus’ geocentric longitude and latitude. If we put the values (C) of x, y, z in formulae (33.2), we obtain

k = 313°.08102 b = —2.°08474

which are corrected for light-time, but not yet for aberration.

From Chapter 23, we find e —— 0.016 711 589, z = 102°.81644

and formulae (23.2) give, for O = 268°.35704,

AX = — 14“.868 = —0.°004l3 A#  = —0“.531  = —0°.00015

and the apparent longitude and latitude of Venus, not yet corrected for nutation, are X = 313°.08102 — 0.°00413 = 313°.07689

b = -2.°08474 — 0°.00015 = —2.°08489

( Alternatively, we could have corrected for the light-time and the aberration together at once by calculating the coordinates of the Earth for the instant r — z, which gives

Lg -— 88°.35168     Bz -— +0°.00014      A = 0.983 825

We now combine these values with Venus’ coordinates I B). Formulae (33.1) and (33.2) then give

z = +0.621 702 k = 313.°07687

y = —0.664 903 p = —2.°08489

' —0 03313' or nearly the same values as before. )

The corrections for reduction to the FK5 system are, from (32.3), AX = —0“.09027 = —0°.00003

A9 = +0".05535 = +0°.00002

so the corrected values are

X = 313°.07689 — 0.°00003 = 313 .°07686

b = —2.°08489 + 0.°00002 = —2.°08487

 

From Chapter 22, we find

AQ = + 16“.749 Ae = — 1“.933 e = 23°. 439 669

and the value of k corrected for nutation is

k = 313°.07686 + 16“.749 = 313°.08151

Finally, by (13.3) and (13.4),

apparent right ascension: n = 316°. 17291 = 21‘.078 194 = 2l'04”41’.50

apparent declination: 6 =  — 18°. 88801 =  — 18° 53' 16'. 8

The exact values, obtained by an accurate calculation using the complete VSOP87 theory, are n = 21'04‘4l‘.454, é = — 18° 53' 16“.84, true distance = 0.910 845 96.

Second Method

Here we use the orbital elements referred to a standard equinox, for instance 2000.0, and the geocentric rectangular equatorial coordinates X, F, Z of the Sun referred to the same equinox. These rectangular coordinates can be taken from an astronomical almanac, or they may be calculated by the method described in Chapter 26.

In this method, the heliocentric longitude and latitude of the body (minor planet or periodic comet) are not calculated. Instead, we calculate its heliocentric rectangular equatorial coordinates x, y, z, after which the right ascension, declination, and other quantities are derived by means of simple formulae.

The following orbital elements are supposed to be known. They may be taken, for instance, from the Circulars of the International Astronomical Union, from the Minor Planet Circulars of the Minor Planet Center, etc.

a —— semimajor axis, in AU e = eccentricity

i -- inclination

o = argument of perihelion

II = longitude of ascending node

n = mean motion, in degrees/day

where i, o, and ft are referred to a standard equinox.

If a or n are not given, they can be calculated from

a -— 1 — e 0.985 607 6686 (33.6)

 

where q is the perihelion distance in AU. The numerator of the second fraction is the Gaussian gravitational constant 0.017 202 098 95 converted from radians to degrees.

The inclination i can take values from 0° to 180°   If 0°ñ    i < 90°, then the body is said to have direct motion. This means that the body moves counter- clockwise as seen from the north pole of the ecliptic. If i is larger than 90°, the motion is said to be retrograde (*).

Strictly speaking, all these elements are valid only for one given instant, called the Epoch. Away from this time they change under influence of planetary perturbations. See, later in this Chapter, the note about osculating elements. Unless high accuracy is required, the elements may be considered as invariable during several weeks or even months, for instance during the whole apparition of a comet.

Besides the above-mentioned orbital elements, either the value 3f of the mean anomaly at the Epoch, or the time T of passage through perihelion, is given. This allows the calculation of the mean anomaly M at any given instant. The mean anomaly increases by n degrees per day, and is zero at time T.

The orbital elements of a minor planet or a periodic comet being given, the geocentric position for a given instant can be calculated as follows. First, we must calculate the quantities a, b, c and the angles A, B, C, which are constant for a given orbit.

Let e be the obliquity of the ecliptic. If the orbital elements are referred to the standard equinox of 2000.0, one should use the value e2 = 23° 26'21”. 448, from which

sin e = 0.397 777 156

cos e = 0.917 482 062

Then calculate

 

F -— cos fi P —- — sin íl cos i

G = sin fl cos s Q —- cos fl cos i cos e — sin i sin e

H —— sin fl sin e A = cos fl cos i sin e + sin i cos c As a check, we can use the relations

N* + G2 + H 2 = 1, P’ + Q 2 + R’ —— 1,

but of course this is not needed in a program.

 

(33.7)

 

 

(*) Some authors call an orbit with i < 90° a prograde orbit. While retrograde is a current English word, even outside astronomy (it means “going backward"), the word prograde is not. It appeared in some astronomical texts around 1960. I don’t know who invented this neologism, nor why. The classic word, in use since more than two centuries, is direct.

 

Then the quantities a, b, c, A, B, C are given by

 

tan A —— F Wn B ——

c ——

 

 

b —— G + @

c —— H    + 2

 

(33.8)

 

The quantities a, b, c should be taken positive, while the angles d, B, C should be taken in the correct quadrant, according to the following rules:

sin A  has the same sign as cos ft,

sin B and sin C have the same sign as sin ft.

However, once again, one may use the “second” arctangent function if it is available in the programming language: A —— ATN2 {F, P), etc.

Attention: do not confuse the quantity a with the semimajor axis o of the orbit!

For each required position, calculate the body’s mean anomaly M, then the eccentric anomaly E (see Chapter 30), the true anomaly v by means of formula (30.1), and the radius vector r by means of (30.2). Then the heliocentric rectangular equatorial coordinates of the body are given by

 

x ——  r a sin {A + in + v) y —— rb sin {B + in + v) z = r c sin (C + u› + v)

 

(33.9)

 

The convenience of these formulae is seen when the rectangular coordinates are required for several positions of the body. The auxiliary quantities a, b, c, A, B, C are functions only of II, i, and e, and thus are constant for the whole ephemeris; for each position only the values of v and r must be calculated. However, remember that II, i, and o are constant only if the body is in an unperturbed orbit.

For the same instant, calculate the Sun’s rectangular coordinates X, F, Z (Chapter 26), or take them from an astronomical almanac. The geocentric right aseension n and declination h of the planet or comet are then found from

              p = Y + y  

                                        (33.10)

sin 6 = }/ñ or tan 3 =

 

where A is the distance to the Earth and thus is positive. The correct quadrant of n is indicated by the fact that sin a has the same sign as 9; however, once more, the second arctangent function can be used: = ATN2 (9, ().

If o is negative, add 360 degrees. Then transform a from degrees into hours by dividing by 15.

The equatorial coordinates « and 6 of the body will be referred to the same standard equinox as the orbital elements and the Sun’s rectangular coordinates X, Y, Z. However, the values of a and 6 obtained in the way described above refer to the geometric (the true) position of the body in space. Just as in the “First Method” in this Chapter, the effect of light-time should be taken into account. This is performed as follows.

For the given time i, calculate the distance A of the body to the Earth as described above, and then the light-time z by means of (33.3). Then repeat the calculation of M, E, v, x, y, z for the time i — r, btit leave the Sun’s coordinates X, F, Y unchanged. With the new values of x, y, z, formulae (33. 10) will give the corrected values of o and 6.

When allowance is made for the light-time only, that is, if no correction is made for aberration nor for nutation, then the values obtained for a and 6 are the so-called astrometric right ascension and declination of the body at the given instant. The astrometric position of a minor planet or a comet is directly comparable with the mean places of stars as given in star catalogues (corrected for proper motion and annual parallax, if significant). Of course, n and é are geocentric.

Instead of expressions (33.7) and (33.8), one may calculate the constants

P   —— cos o cos fl — sin o sin fl cos i

P —- cos e (cos o sin fl + sin o cos fl cos i) — sin e sin o sin i P     ——  sin e (cos u› sin fl + sin o cos ft cos i) + cos e sin o sin i Q -— — sin m cos fl — cos o sin fl cos i

Q —— cos e (cos u› cos fl cos i — sin m sin d) — sin e cos o sin i

Q   —— sin e (cos o cos fl cos i — sin o  sin fl) + cos c cos o sin i

and then, instead of (33.9), one should use

x —-  r [P cos v + Q sin v j y —— r (I, cos v + Q sin v) z = r P cos v + Q sin v)

The elongation / to the Sun and the phase angle # (the angle Sun—body —Earth) can be calculated  from

 

COST =

 

   R 2 + A2 _   y2  

2JtA

 

(33. 11)

(33. 12)

 

where fi = I N 2 -l- i 2 + Z 2 is the distance Earth —Sun. The angles / and Q are both between 0 and + 180 degrees. Do not confuse this R with the quantity R of expressions (33.1), nor with that of (33.7).

The magnitude of the body is then calculated as follows. In the case of a comet,

the “total” magnitude is generally calculated from

m = g + 5 log A + K log r (33.13)

where g is the absolute magnitude, and x a constant which differs from one comet to another. In general, K is a number between 5 and 15.

For the minor planets, a new magnitude system was adopted by Commission 20 of the International Astronomical Union (New Delhi, November 1985). The formula for the prediction of the apparent magnitude of a minor planet is

 

magnitude = H + 5 log rd - 2.5 log [ (1 — G) 4, + G4•>]

with

 

(33. 14)

 

* - exp [ —3.33 (tan

* - exp [ — 1.87

 

)0.63 ]

1.22 ]

 

where Q is the phase angle, and “exp” is the exponential function, EXP (z) = e*. Formula (33.14) is valid for 0° n b n 120°. H and G are magnitude parameters, which are different for each minor planet. H is the mean absolute vital magnitude, while G is called the “slope parameter”. Here are the values of N and G for the brightest minor planets and for some unusual objects [1] :

ii c H G

1 Ceres 3.34 0.12 15 Eunomia 5.28 0.23

2 Pallas 4.13 0.11 18 Melpomene 6.51 0.25

3 Juno 5.33 0.32 20 Massalia 6 50 0.25

4 Vesta 3.20 0.32 433 Eros 11,16 0.46

5 Astraea 6.85 0.15 1566 Icarus 16.9 0.15

6 Hebe 5.71 0.24 1620 Geographos 15,60 0.15

7 Iris 5.51 0.15 1862 Apollo 16.25 0.09

8 Flora 6.49 0.28 2060 Chiron 6.5 0.15

9 Metis 6.28 0.17 2062 Aten 16.80 0.15

 

In formulae (33. 13) and (33.14), the distance to the Sun (r) and the distance to the Eârth (A) are in astronomical units, and all logarithms are to the base 10. In many programming languages, the only available logarithmic function “LOG” is the natural logarithm (to the base e = 2.71828. . .); it can be converted to the common

logarithm (base 10) by multiplication by 0.434 294 4819, which is l /1og e 10.

Example 33.b -- Calculate the geocentric position of periodic comet Encke for 1990 October 6.0 Dynamical Time, using the following orbital elements (see Example 24.b) :

T -- 1990 Oct. 28.54502 TD i = 11.°94524 ecliptic

a = 2.209 1404 AU 8 = 334.°75006 and equinox

e -- 0.850 2196 = 186.°23352 2000.0

We first calculate the auxiliary constants of the orbit by means of(33.7) and (33.8) :

F ---- +0.904 455 59 + 0.417 330 84

G = --0.391 368 30 +0.729 522 09

H = --0.169 678 93 +0.541 878 67

 

A --  65°.230 615

B 331°.787 680

C ----  342°.613 052

 

a -- 0.996 094 85

b -- 0.827 871 74

c = 0.567 823 42

 

From the value 2.209 1404 for the semimajor axis of the orbit, the second formula (33.6) yields n = 0.300 171 252 degree/day.

For the given date (1990 October 6.0), the time since perihelion is --22.54502 days. Hence, the mean anomaly is

M ---- --22.54502 x 0.° 300 171 252 = --6°, 767 367

We then find

A = --34°.026 714 z = +0.250 8066

v = --94°. 163 310 y -- +0.484 9l7J

r ---- 0.652 4867 z = +0.357 3373

The Sun’s geocentric rectangular equatorial coordinates for the same instant, referred to the same standard equinox (2000.0) and calculated by using the complete VSOP87 theory, are

X = --0.975 6732, Y ---- --0.200 3254, Z = --0.086 8566,

from which A = 0.824 3689, and the light-time is v = 0.00476 day.

Repeating the calculation of the comet’s position for r -- z, that is, for 1990 October 5.99524, we find

 

3f = —6.°768 796 x = +0.250 9310 t = —0.724 7422

E —— —34°. 031 552 y —— + 0.484 9477 9 = + 0.284 6223

v

r = —94°. 171 933

-— 0.6525755 z = +0.357 3712 = +0.270 5146

A =   0.824 2811

from which we deduce the astrometric right ascension and declination, and the elongation from the Sun:

2 = 158°.558 965 = 10‘34‘14'.2

ét = + 19°.158 496 = + 19° 09’31“

= 40°.51

Heliocentric ecliptical coordinates

For some applications, the heliocentric rectangular ecliptical coordinates may be needed. In that case one should use the following expressions instead of (33.9), and it is not needed to calculate the auxiliary quantities N, G, . . . , A, B, etc.

x —— r (cos fi cos ti — sin It sin u cos i) y = r (sin II cos ii + cos 0 sin u cos i) z  = r sin i sin u

When these heliocentric rectangular ecliptical coordinates are known, the heliocentric longitude / and latitude b can be found from

tan I = y /x ‹l being taken between 90° and 270° if z < 0) sin b —— or tan b ——

Notes on the osculating elements

Mean orbital elements, such as those given in Chapter 31 for the major planets, represent the elements of a mean reference, slowly varying orbit.

For the periodic comets and the thousands of minor planets, however, no mean orbital elements are calculated. Instead, orbital elements are available for the

*instantaneous" orbit at a given instant (the Epoch). These are the so-called osculating elements, and the instant for which they are valid is the Epoch of osculation.

 

Osculating elements at a particular epoch are defined as the elements of an unperturbed elliptical orbit, referred to as the osculating orbit, in which the position and velocity of the planet at the epoch are identical with the actual position and velocity of the planet in its perturbed orbit at the same instant. The osculating elements therefore contain the effects of the perturbations due to other planets, so that, unlike the mean elements, they are subject to periodic variations. [2]

While the mean elements vary slowly with time (for instance, the eccentricity of the mean orbit of Mars was 0.093 31 in A.D. 1900 and will be 0.093 49 in 2100), the osculating elements vary rather rapidly. These changes generally do not reflect the real changes of the mean orbit.

As an example, let us give the following osculating elements of minor planet Ceres for two epochs separated by only 200 days. These elements are taken from the yearly Ephemerides of Minor Planets (Institute of Theoretical Astronomy of the Russian Academy of Sciences, St. Petersburg, Russia); the elements i, o, and II are referred to the standard equinox of 2000.0.

Epoch (TD) : 1997 Dec. 18.0 1998 July 6.0

Semimajor axis (AU) : o = 2.767 8380 o = 2.766 1801

Eccentricity e -- 0.077 4119 e 0.077 8872

Inclination (degrees) : i = 10.58086 i = 10.58293

Argument of perihelion (deg.) : o = 73.46016 = 73.79924

Longitude of ascending node (deg.) : II = 80.52954 G = 80.50163

Mean anomaly (degrees) M -- 207.08221 ñf 249.60014

Mean motion (degrees/day) : n  = 0.214 039 08 n = 0.214 231 53

From 1997 December 18 to 1998 July 6, the semimajor axis of the “instant- aneous” orbit decreased by 0.00166 AU. From this, however, we may not deduce that during those 200 days the mean distance of Ceres to the Sun decreased by 248 000 kilometers!

On 1997 December 18, the “instantaneous” revolution period of Ceres was 1681.94 days (which is obtained by dividing 360° by n); 200 days later this had decreased to 1680.42 days.

Neptune provides an even better illustration. While the eccentricity of its mean orbit is presently 0.0095, that of its osculating orbit reached a maximum of 0.0124 in November 1964, a minimum of 0.0039 in October 1970, another maximum (0.0122) in December 1976, and so on. These rather large variations are not surprising: the osculating orbit of Neptune refers to the instantaneous position and velocity of the Sun, which itself oscillates around the barycenter of the solar system, mainly due to the actions of the giant planets Jupiter and Saturn, Orbital elements of Neptune referred to that barycenter (instead of to the Sun) would show much smaller variations.

 

Accurate ephemerides of the periodic comets and the minor planets are obtained by numerical integration, and for these calculations the osculating orbital elements provide starting values. Such a numerical integration takes lnto account the perturbations caused by the attraction of the planets, which tend to change the osculating elements of the orbit over time.

Osculating elements may be used to give the actual position and motion of the body at the epoch of osculation, and they provide a good approximation to its actual orbit over short periods around the Epoch. They may not, however, be used as an unperturbed orbit over a long period!

In order to have an idea of the increasing error of an ephemeris calculated by using an osculating orbit as an unperturbed one, we used the above-mentioned osculating elements of Ceres valid for 1998 July 6. The heliocentric longitude of Ceres, calculated in this way, was then compared with the exact one as obtained with the software package “Ceres” developed at the Institute of Theoretical Astronomy, St. Petersburg, Russia. It appears that until 280 days after the Epoch the error is smaller than 5“. During the first 50 days, the error is smaller than 1“. The error in the calculated heliocentric longitude reaches a maximum (+ 4“) 172 days after the Epoch, but after a few months the error AX quickly reaches large negative values:

Number of days after

1998 July 6: 0 40 80 120 160  200  240  280 320 360 400

Ah (arcsec.): 0 + '/z +2 +3 +4 +4 + 1 —4 —13 —26 —44

The further evolution of the error Ah in the calculated heliocentric longitude of Ceres is shown in Figure 1. The oscillating curve represents the variation of the error as a function of time. So, in this particular case, the error does not increase continually with time, but reaches the following extreme values: -+-4” in December 1998, —304” in early November 2000, + 862“ in early September 2003, —383“ in mid-May 20D5, and + 1105” in mid-September 2007.

The situation is somewhat comparable with the undulating curve shown in Figure 2. The true function (the osculating orbit in the case of a minor planet) is represented by the curve C. The dashed line M is the “mean” curve (the mean orbit). If we use this mean curve, then for a given value x of the argument we obtain point A, which differs from the true value B on the true curve. However, the difference between A and B does not exceed a certain limit. At point P, the tangent T to the true curve is drawn. In the vicinity of P, this tangent gives a much better approximation to the true curve C than does the mean curve 4f. But if we use the tangent T at large distances from P, we obtain the very erroneous point E. In this case, the mean curve would give A, which is a better approximation to the correct value B. Unfortunately, for minor planets no mean orbital elements are available.

 

 

Fig. I : The error bX (in arcseconds) in the calculated heliocentric longitude of Ceres when osculating elements are used and the perturbations by the planets are ignored. The points are given ai intervals of 40 days.

-ig. 2

 

The Equation of the Center

If the orbital eccentricity is small, then instead of solving the equation of Kepler (Chapter 30) and then using formula (30.1), the equation of the center C, or the difference v — M, can be found directly in terms of e and M by means of the following formula.

 

e 3

C -- (2e —

 

e5 ) sin M + ( 5 e’ 11

 

‘) sin 23f

 

4 + 96

 

4 24

 

+ ( 13 e3 — e ’) sin 33f + 103 y sin 43f + 1097e’ sin 53f

 

l2 96

 

960

 

The result is expressed in radians, and thus should be multiplied by 180/z or

57.295 779 51 in order to be converted into degrees. The formula is derived from a series expansion [3] and has been truncated after the term in e5. Therefore it is suitable only for small values of the eccentricity. If the eccentricity is rery small, the terms in e4 and e5 may be neglected.

The greatest error is

The formula The formula with terms

e up to terms in e’ e4 end e’ neglected

0.03 0“.0003 0“. 24

0.05 0.007 1.8

0.10 0.45 30

0.15 5 152

0.20 29 483

0.25 111 1183

0.30 331 2456

There exists a series expansion for the radius vector, too. Its terms up to the fifth power of the eccentricity are as follows:

 

— -— 1 +

a

 

e’ — (e — 3 e’

8

 

+ 92•

 

5 ) cos M

 

e 2 e 4 ) cos 23f — (

2 3

 

3 e’ —

 

45 e ) cos 3M

128

 

e 4

3 COS 4ñf

 

125 e5

384

 

cos 5H

 

Velocity in an elliptic orbit

In an unperturbed elliptic orbit, the instantaneous velocity of the moving body, in kilometers per second, is given by the following formula, where r :s the distance of the body to the Sun, and a is the semimajor axis of the orbit, both expressed in astronomical units:

Y = 42. 1219

If e is the orbital eccentricity, the velocities at perihelion and at aphelion, again in km/second, are respectively

 

_ 29.7847  

pa

 

29.7847 

pa

 

 

Example 33.c — For the 1986 return of periodic comet Halley, we have [4]

a —— 17.940 0782 e —— 0.967 274 26

these osculating values being valid strictly for the Epoch 1986 February 19.0 TD.

For this orbit, the velocities at perihelion and at aphelion are Up — 54.52 km/second and VG —— 0.91 km/second, respectively.

At the distance r — 1 AU from the Sun, the comet’s velocity was V —— 41.53 km/second.

Length  of  the  el!“!Rse

While there is an exact formula giving the area of an ellipse (area = cab), there is no exact expression with a finite number of terms and ordinary functions for the length L (the perimeter) of an ellipse. In what follows, e is the eccentricity of the ellipse, a its semimajor axis, and b its semiminor axis given by

b —— a   l    e .

1. An approximate formula given by Ramanujan in 1914 is

L ——  r  (  3 {a + b)  — (a + 3b) (3a + b) )

The error is zero for a —— b (that is, for a circle), increasing to 0.4155 9» for

e —— 1, that is, for an infinitely flat ellipse.

 

2. Another interesting method for finding the length of an ellipse is as follows. Let A, G, and H be the arithmetic, the geometric, and the harmonic means, respectively, of the semi-axes a and b of the ellipse. That is,

 

A ——    a +b 

2

 

G ——

 

H --

 

   2 ab a + b

 

Then we have

 

L —— (  21 A — 2 G —3H)

 

with an error less than 0.001 7c if e < 0.88, and less than 0.01 $ if e < 0.95. But the error amounts to 1 â for e = 0.9997, and to 3 â for e = 1.

3. A formula with an infinite series expansion is

 

e 2 e6

1  

 

— etc.

 

The expression between square brackets takes the value 0.99937 for e = 0.05,

the value 0.99750 for e -— 0.10, and is equal to 0.63662 = 2r/ for e — 1.

4. More rapid convergence is obtained with the following formula, where m = (o — b) f (o + b),

 

p _ 2 r a 

1 + m

 

1  +  ( 1

2

 

     1   ) 2 4

2 x 4

 

     1 x 3    2 6

2 X 4 X 6'

 

 

  1  3 X 5 

2 x 4 x 6 x 8

 

' m" + etc.

 

 

ñxnznpfe 33.d — Periodic comet Halley. Using the elements for the return of 1986 (see Example 33.c), we find that the lenglh of the orbit is 77.07 astronomical units, or 11530 millions of kilometers.

 

RE FER E N CES

1. 1 inor Planet Circulars 28103 —28116 (1996 November  25).

2. Explanatory Supplement to the Astronomical Ephemeris (London, 1961) ; page 114.

3. Annales de l’Observatoire de Paris, Vol. I, pages 202-204.

4. Minor Planet Circular 10634 (1986 April 24).

 

Chapter 34

Parabolic Motion

In this Chapter we explain how to calculate positions of a comet moving around the Sun in a parabolic orbit. We will assume that the elements of this orbit are invariable (no planetary perturbations) and that they are referred to a standard equinox, for instance that of 2000.0.

We assume that the following orbital elements are given:

T —— time of passage in perihelion q = perihelion distance, in AU

i -- 'inclination

o = argument of the perihelion

II = longitude of the ascending node

First, calculate the auxiliary constants A, B, C, a, b, c as for an elliptic orbit; see formulae (33.7) and (33.8). Then, for each required position of the comet, proceed as follows.

Let i — T be the time since perihelion, in days. This quantity is negative for an instant earlier than the time of perihelion. Calculate

0.036 491 162 45 (t — T) (34. I)

The constant in the numerator is equal to 3k/ , where k is the Gaussian

gravitational constant 0.017 202 098 95.

Then the true anomaly v and the radius vector r of the comet are given by

 

                    r = q (1 + s 2 )

where s is the root of the equation

s’ + 3s — W = 0

241

 

(34. 2)

(34.3)

 

For an instant earlier than the time of perihelion passage the quantity s is negative and v is between — 180 and 0 degrees. After the perihelion, s > 0 and v is between 0° and + 180°. At the instant of passage through perihelion, we have s = 0, v = 0°, and r = q.

There are several ways to solve equation (34.3), which is called Barker’s equation.

1. The equation can easily be solved by iteration; this algorithm has the author’s preference, because the iteration formula is simple, the convergence is rapid, no trigonometric functions or cubic roots are involved, and the procedure is valid for positive as well as negative values of r — T, and for r = T (or i = 0) too.

One may start from any value for s ,- a good choice is s = 0. A better value for

 

s is

 

   2s’ + W 

3 (s 2 + 1)

 

(34.4)

 

This calculation is then repeated until the correct value of s is obtained. Note that in expression (34.4) the cube of s must be calculated. If s is negative, this operation is not possible on some calculating machines. When this is the case, calculate s x s x i instead of S'.

2. Instead of solving equation (34.3) by iteration, s can be obtained directly as follows (J. Bauschinger, Tafeln our Theoretischen Astronomie, page 9; Leipzig, 1934) :

 

tan Q = = 54.807791

tan y =

 

 q    Sq t — T

 

(34.5)

 

  2

S

' tan 2y

The constant 54.807791 is equal to   2 / 3k, where I is Ihe Gaussian gravitational constant.

In this method, no iteration is performed, but two problems can nccur:

— at the time of passage through perihelion, r — T is zero, hence lY i£ zero and 2/W becomes infinite. In thai case we have directly v = 0° and r = q, but the possible occurrence of this case must be anticipated in the computer program;

— before the perihelion we have lY < 0, whence tan # is negative. But in this case tan 9/2 is negative too, and computers cannot calculate the cubic root of a negative quantity. This problem can be avoided by replacing W by its absolute value in the

 

34. PARABOLIC MOTION 2 4 3

first formula (34.5). At the end of the calculation, the sign of s should then be changed accordingly. For instance, in BASIC the formulae (34. I) and (34.5) can be programmed as follows, where T stands for r — T, the number of days elapsed since perihelion:

IF T = 0 THEN   ....

W = .03649116245 T / (Q • SQR(Q)) B = ATN (2/ABS (W))

S = 2/ TAN (2 • ATN (TAN (B/2)“(1/3))) IF T < 0 THEN S = — S

3. The following method is easier and does not use trigonometric functions. All expressions under the root signs are positive.

r   = s = F — (34.6)

When s is obtained, v and r can be found by means of (34.2), after which the calculation continues as for the elliptic motion, formulae (33.9) and (33. 10), with the same precept to take the effect of light-time into account.

The first formula (34.2) will give v/2 between —90 and +90 degrees, the range of the arctangent function of the computer languages. That will give v in the correct quadrant, between — 180° and + 180‘, so no additional check will be required.

In the parabolic motion, e —— 1 while a and the period of revolution are infinite; the mean daily motion is zero and therefore the mean and eccentric anomalies do not exist — in fact, they are zero.

Example 34.a — Calculate the true anomaly and the distance to the Sun of comet Stonehouse (C/1998 HI) for 1998 August 5.0 TD, using the values

T —— 1998 April 14.4358 TD

q = 1.487 469

of a parabolic orbit calculated by B. G. Marsden (Mieor Planet Circular No. 31893, 1998 June 10).

For the given instant (1998 August 5.0), the time from perihelion is r — T ——

+ 112.5642 days. Hence, by formula (34.1),

IV = +2.264 206 862.

Starting from the value s —— 0, we obtain the following successive approximations for s by means of the iteration formula (34.4) :

 

0.000 0000

0.754 7356

0.663 4364

0.659 2441

0.659 2360

0.659 2360

Hence, s —— +0.659 2360, and consequently

v = +66°.78862 r  -- 2.133 911

If, instead of the iteration procedure, formulae (34.6) are used, we obtain successively

G =  1.132 103 431

Y —— 1.382 541 577

s —— Y — l | Y —— 0.659 2360, as before.

 

Chapter 3S

Near-parabolic Motion

An eccentricity of exactly 1 means that the orbit is parabolic; in that case, it is easy to calculate the position of the body for a given instant (see Chapter 34). If the orbit has a high eccentricity (say, 0.98 to 1.1), but different from l, it is more troublesome to deal with. An eccentricity greater than 1 means the orbit is hyperbolic.

The German astronomer Werner Landgraf has given an interesting program in BASIC [1], based on Karl Stumpff’s work Himmelsmechanik, Vol. I (Berlin, 1959). Hereafter we give Landgrafi s program, in a slightly  modified form.

First, calculate

where, as before, k is the Gaussian gravitational constant, e is the eccentricity of the orbit, and q is the perihelion distance in astronomical units.

Then solve the following equation iteratively for s :

 

s -— Qt - (1 — 2y) s’

3

 

+ (2 — 3J) 2 (3 4 J)

7

 

(35. 1)

 

where r is the number of days before (—) or after (+) the perihelion. Begin by inserting into the right-hand side of the equation the value of i obtained for an orbit which would be precisely parabolic, that is, with the value of W of formula (34.1) put equal to 3Qi. This evaluation leads to an improved s, which is used in another iteration, and so on until the value of s ceases to change.

Once the final value of s is found, the true anomaly v and the distance r to the Sun are found from

r —-    q (1 + )  

1 + e cos v

245

 

The calculation of geocentric places can then be performed as for the elliptic and the parabolic motions.

Here is Landgraf’s program in BASIC, slightly modified by us. It is valid for highly eccentric elliptical orbits (e slightly less than l), for slightly hyperbolic orbits (e slightly larger than 1), as well as for an orbit that is exactly parabolic. The computer is assumed to be working in radians.

10 P1  =  4  ATN (1)  :  R1  =  180/Pl

12 K = 0.01720209895

14 D1 = 10000 C = 1/3 ’ D = 1E —9

16 INPUT "PERIHELION DISTANCE — "; Q

18 INPUT "ECCENTRICITY = " ; E0

20 Q1 = K • SQR ((1 + E0)/Q)/(2   Q) : G = (1 — E0)/(1 + E0)

22 INPUT "DAYS FROM PERIHELION = "; T

24 IF T < > 0 THEN  28

26 R = Q- V = 0 : €¡OTO 72

28 Q2 = Q I • T

30 S = 2/ (3 ABS (Q2))

32 S =  2/ TAN (2 • ATN (TAN (ATN (S)/2)"C))

34 IF T < 0 THEN S = —S

36 IF E0 = 1 THEN 66 38 L = 0

40 S0 = S- Z = 1 : Y = S   S G1 = — Y • S

42 Q3 = Q2 + 2 G • S • Y/ 3

44 Z = Z + 1

46 G1 = — G1 G Y

48 Z1 = (Z — (Z + 1) • G)/ (2 Z + l)

50 F  = Z1   G1 52 Q3 = Q3 + F

54 IF Z > 50 OR ABS (F) > D1 THEN 78

56 IF ABS (F) > D THEN 44

58 L = L + 1 : IF L > 50 THEN 78

60 S1 = S S = (2 • S • S • S/3 + Q3) / (S • S + 1)

62 IF ABS (S — S1) > D THEN 60

64 IF ABS (S — S0) > D THEN 40

66 V = 2 ATN (S)

68 R = Q (1 + E0) / (l + E0 • COS (V))

70 IF V < 0 THEN V = V + 2 P l

72 PRINT "TRUE ANOMALY = " ; V * R l

74 PRINT "RADIUS VECTOR (A.U.) = "; R

76 PRINT : GOTO 22

78 PRINT "NO CONVERGENCE"

80 PRINT : GOTO 22

 

35. NEAR— PARABOLIC MOTION 247

Some comments about this program :

Line 10 : the first formula is a trick to obtain the number r. Line 12 : the Gaussian gravitational constant k.

Line 14 :    the number D = 10* 9 adjusts  to suit the computer’s precision.

If necessary, one may use 10 ' or 10* '0.

Line 26 : when r = 0 (the body is exactly in perihelion), then r = q and v = 0°. Line 36 : if the orbit is exactly parabolic, the value of has been found.

Line 54 : if in formula (35. 1) more than 50 terms are needed, or if these terms become too large, there is no convergence.

Line 56 : as long as a term of formula (35. 1) is not small enough, the next term should be calculated.

Line 58 : if after 50 iterations no result has still been found, the calculation must be halted.

Lines 60 and 62 : solving equation (35. 1) by iteration. This is an iteration inside of an iteration !

As an exercise, try to calculate the following cases:

Data

 

perihelion distance

q (AU) eccentricity

e days

t true anomaly

v tdegrees distance to the Sun

r tAU)

0.921 326 1.000 00 138.4783 102.74426 2.364192

0.100 000 0.987 00 254.9 164.50029 4.063777

0.123 456 0.999 97 —30.47 221.91190 0.965053

3.363 943 1.057 31 1237.1 109.40598 10.668551

0.587 1018 0.967 2746 20 52.85331 0.729116

0.587 1018 0.967 2746 0 0 0.5871018

After having calculated some cases, you will notice that the calculation time is longer as | r | is larger, that is, as the body is farther away from the perihelion. The calculation time is longer too as e differs more from unity. The table on the next page mentions some calculation times on the old HP-85 microcomputer, together with a rounded value of the true anomaly v, and the number L of iterations.

 

 

e

  Calculation time in seconds

 

L

0.1 0.9 10

20

30 14

47

no convergence 126

142 17

30

0.1 0.987 10

20

30

60

100

200

400

500 4

5

6

9

14

28

87

no convergence 123

137

143

152

157

163

167 7

8

10

12

16

23

38

0.1 0.999 100

200

500

1000

5000 3

4

5

7

18 156

161

166

169

174 6

7

8

10

18

1 0.99999 100000

10000000

14000000

17000000

18000 000 2

5

6

7

no convergence 172.5

178.41

178.58

178.68

 

8

9

9

For q = 0.1 and e —— 0.9, the calculation took 47 seconds for r = 20 days, and there was no convergence for r = 30 days. However, in this case the calculation could better be made using one of the methods for elliptic motion.

For q = 0.1 and e —— 0.999, there is no trouble up to i = 5000 days.

For q = 1 and e = 0.999 99, there is no trouble even for r = 17 million days. This is 465 centuries after the perihelion time; the object’s distance from the Sun is then 7220 astronomical units — at least in theory!

REFEREN CE

1. Sky and Telescope, Vol. 73, pages 535—536 (May 1987).

 

Chapter 36

The Calculation of some Planetary Phenomena

There are two basically different methods for calculating planetary phenomena such as the greatest elongations of Venus, or the time of an opposition of Mars:

(i) either by comparing accurate positions of the planet with those of the Sun;

(ii) or by using formulae where a mean value is corrected  by a sum of periodic terms.

The first method has the advantage of giving very accurate results, because use is made of very accurate positions of the bodies. It has the inconvenience, however, of requiring the availability or the calculation of these accurate ephemerides.

With the second method, the calculation can be performed easily and rapidly for any year. The results, while not so accurate as those of the first method, are still good enough for many applications such as historical research, or even as a first approximation for a more accurate calculation. Examples of this method are found in Chapters 49 (lunar phases), 50 (perigee and apogee of the Moon), 51 (passages of the Moon through the nodes), and 52 (extreme declinations of the Moon).

In this Chapter, we provide formulae for calculating several configurations involving the planets Mercury to Neptune: oppositions and conjunctions with the Sun, greatest elongations, and stations.

Oppositions and conjunctions with the Sun

From the proper line in Table 36.A, take the values of A, B, My, and 3f;.

Let Y be an appropriate time of the required phenomenon, expressed as yeors and decimals. For instance, 1993.0 means the beginning of the year 1993, 2028.5 denotes the middle of the year 2028, etc.

24 9

 

T A BLE 3 6. A

Pla net Eve nt A B Mg M

Mercury Inf. conj. 2451 612.023 115.877 4771 63.5867 114.208 8742

Mercury Sup. conj. 2451 554.084 115.877 4771 6.4822 114.208 8742

Venus Inf. conj. 2451 996.706 583.921 361 82.7311 215.513 058

Venus Sup. conj. 2451 704.746 583.921 361 154.9745 215.513 058

Mars Opposition 2452 097.382 779.936 104 181.9573 48.7fi5 244

Mars Conjunction 2451 707.414 779.936 104 157.6047 48.705 244

Jupiter Opposition 2451 870.628 398.884 046 318. 4681 33. 140 229

Jupiter Conjunction 2451 671.186 398.884 046 121.8980 33.140 229

Saturn Opposition 2451 870.170 378.091 904 318.0172 12.647 487

Saturn Conjunction 2451 681.124 378.091 904 131.6934 12.647 487

Uranus Opposition 2451 764.317 369. 656 035 213.6884 4.333 093

Uranus Conjunction 2451 579.489 369.656 035 31.5219 4.333 093

Neptune Opposition 2451 753.122 367.486 703 202.6544 2.194 998

Neptune Conjunction 2451 569.379 367.486 703 21.5569 2.194 998

 

Then find the integer k nearest to

 365.2425 F + 1721 060 — A 

B

 

(36. 1)

 

It is important to note that k must be an integer. Non-integer values of £ would yield meaningless results. Successive values of k will provide the data for the successive events (for instance, successive oppositions of Mars), the value k —— 0 corresponding to the first one after 2000 January 1. For years preceding A.D. 2000, k takes negative values.

Then calculate

JDE0 = A + kB  

I DE0 is the Julian Ephemeris Day corresponding to the time of the mean planetary configuration (that is, calculated from circular orbits and uniform planetary motions), and M is the mean anomaly of the Earth at that instant.

 

M is an angle expressed in degrees and decimals. Depending on the type of the calculating machine or the programming language, it may be necessary or desirable to reduce that angle to the range 0—360 degrees by adding or subtracting a convenient multiple of 360, and to convert the result into radians.

Find the time T, expressed in centuries from the beginning of the year 2000, from

T -- JDEo — 2451 545

36525

T is positive after the beginning of A.D. 2000, negative before.

For the planets Jupiter to Neptune, additional angles are required. Expressed in degrees, these angles are:

for Jupiter : o = 82.74 + 40.76 T

for Saturn : a —— 82.74 + 40.76 T

b —— 29.86 + 1181.36 T

c = 14. 13 + 590.68 r

d —— 220.02 + 1262.87 F

for Uranus : e —— 207.83 + 8.51 T

/   = 108.84 + 419.96 F

for Neptune : e —— 207.83 + 8.51 T

g —— 276.74 + 209.98 T

The time JDE of the true configuration is obtained by adding to JDE a correction which is given in Table 36.B as a sum of periodic terms which are functions of the angle 3f. By reason of the secular variations of the planetary orbits, the coefficients of these periodic terms are slowly varying with time, whence the presence of terms in T and T 2 in Table 36.B.

For instance, for an inferior conjunction of Mercury, the correction (in days)

iS

+ 0.0545 + 0.0002 r

+ (—6.2008 + 0.0074 T + 0.00003 T2) sin M

+ (—3.2750 —  0.0197 T + 0.00001 T ) cos M

+ (0.4737 — 0.0052 T — 0.00001 T') sin 2ñJ

+ etc ....

The corrected instant obtained in this way is expressed as a Julian Ephemeris

Day (JDE), hence in the scale of Dynamical Time. This can be reduced to the standard Julian Day, ID, based on the Universal Time, by subtracting the quantity b T expressed in days (see Chapter 10). However, between the years 1500 and 2100, the correction — b T can be neglected for our purposes.

Finally, from the JD the corresponding calendar date can be obtained by means of standard procedures (see Chapter 7).

 

 

Example 36.a -- Calculate Mercury’s inferior conjunction that is nearest to 1993 October 1.

From Table 36.A, for Mercury, Inferior conjunction, we have

A ---- 2451 612.023 3f 0 = 63.5867

B ---- 115.877 4771 M 114.208 8742

October 1 is three quarters of a year since January I, hence 1993 October 1 = 1993.75 = r, and  expression (36. l) yields the value --20.28, whence k 20.

Remember that k must be an integer! Then

JDE0 = 2449 294.473

M -- --2220.°5908 = + 299°. 4092

T 0.06162

The sum of the terms in the relevant part of Table 36.B (Mercury, Inferior conjunction) is + 3.171, whence

JDE '   JDEo + 3.171 '    2449 297.644,

which corresponds to 1993 November 6, at 3‘ TD.

Rounded to the nearest integer hour, this is indeed the correct instant.

Example 36.b -- Find the instant of the conjunction of Saturn with the Sun in 2125.

From Table 36.A, for Saturn, Conjunction, we have

A -- 2451 681.124 3f 0 = 131.6934

B --- 378.091 904 M         12.647 487

For F = 2125.0 (that is, the beginning of the year 2125), expression (36. l) gives the value + 120.39. Because we are searching the first Saturn--Sun conjunction after the beginning of the year 2125, we take k ---- + 121, not + 120. Then

JDE0 = 2497 430.244

M ---- 1662°. 0393 = 222°.0393

T    + 1.25627

and for Saturn we have to calculate the following additional angles:

a = 133°.95,     b --- 73°.97,     c = 36.° 18,     d ---- 6°.53.

The sum of the terms in the relevant part of Table 36.B (Saturn, Conjunction with the Sun) is +7.659, whence

JDE = JDE0 + 7.659 = 2497 437.903,

which corresponds to 2125 August 26, at 10' TD.

The correct instant, calculated with a more accurate method, is 2125 August 26, at 11' Dynamical Time.

 

Greatest elongations of Mercury and Venus

To calculate the times and the values of the greatest elongations of Mercury or Venus, we start from the nearest inferior conjunction. So we calculate k, JDE0, N, and T as explained before. But we do not calculate the instant of the true inferior conjunction; instead, we use the periodic terms given in Table 36. C to find the correction (in days) to Mercury’s or Venus’ mean inferior conjunction, to obtain the time of greatest eastern or western elongation. In the same table, periodic terms are provided to find the value of this greatest elongation.

Do not forget that, if the planet is east from the Sun, it is visible in the evening in the west; if the elongation is west, the planet is visible in the morning in the east.

The value of the greatest elongation from the Sun is expressed in degrees and decimals. It concerns the maximum angular distance from the planet to the center of the Sun’s disk, nor the greatest difference between the geocentric ecliptical (celestial) longitudes of the two bodies. There is no “official” deflnitiOn for the elongation of a planet to the Sun, and two different definitions could be considered:

(a) the angular distance between the object and the center of the solar disk;

(b) the difference between the geocentric longitudes of the object and the center of the solar disk.

Both definitions are used in the astronomical literature. Definition (a) has been used in the Astronomical Ephemeris since its beginning in 1960, and from 1981 onwards in its successor, the Astronomical Almanac. It is this definition we prefer. For example, for the visibility of Venus near its inferior conjunction, the important factor is not the longitude difference with the Sun, but the angular separation.

The French astronomers, however, use definition (b), for instance in their Annuaire du Bureau des Longitudes. On page 275 of the volume for 1990 we read: “Les plus grandes Elongations des planétes inférieures: la dlfférence des longitudes géocentriques de la planéte et du Soleil est maximale.”

Consequently, the results will differ somewhat according as one uses definition

(a) or (b). For example, for Mercury’s greatest elongation of 1990 August 11 : the difference between the geocentric ecliptical longitudes of the Sun and Mercury reached its maximum value (27°22') at 15h UT, as mentioned on page 277 of the Annuaire du Bureau des Longitudes for 1990, but the maximum angular separation took place at 21' and was equal to 27° 25'.

Example 36.c — Find the instant and the value of the greatest western elongation of Mercury in November 1993.

We start from the inferior conjunction of November 1993, for which we found in Example 36.a:

JDEo' 2  9 294.473, M -- 299°. 4092, T - 0.06162.

 

With these values of M and T, we find from the relevant part of Table 36.C (Mercury, greatest western elongation) :

correction = + 19.665 days, elongation = 19°.7506.

Hence, the time of Mercury’s greatest western elongation was

JDE = JDEq + 19.665 = 2449 314.14

which corresponds to 1993 November 22, at 15' TD.

The value of this maximum elongation was 19°.7506 = 19° 45’.

Stations in longitude

To calculate the time when a planet is stationary, we start either from the nearest inferior conjunction (in the case of Mercury and Venus), or from the nearest opposition (in the case of Mars, Jupiter, and Saturn). So we calculate k, JDE , M, and T as explained before. We do not calculate the instant of the true inferior conjunction or that of the opposition; instead, we use the periodic terms given in Table 36.D to find the correction (in days) to the mean inferior conjunction or to the mean opposition, to obtain the time when the planet is stationary.

Note that there are two stations. Station l is that when the planet begins to move westward (retrograde) among the stars, while Station 2 is when the planet resumes direct motion. In other words, Station l precedes the inferior conjunction or the opposition, while Station 2 follows it.

The stations considered here are those in celestial longitude, not in right ascension. The time difference between both types of stations can amount to more than one day. For instance, Mars was stationary in longitude on 1997 April 27 at 19h UT, but its right ascension did not reach a minimum until April 29 at 6'.

Example 36.d — Find the instant of Mars’ station in longitude following the opposition of March 1997.

Starting from the opposition of March 1997, we find

k —- —2, JDE = 2450 537.510, M — 84°.5468, T —— —0.02758.

With these values of M and T, we fi nd from the relevant part of Table 36.D (Mars, Station 2) : correction = +28.745 days.

Hence, the time of Mars’ station in celestial longitude was JDE = JDE0 + 28.745 — 2450 566.255

which corresponds to 1997 April 27, at 18‘. The correct time was 19‘.

 

The accuracy of the results

It is evident that the expressions given in Tables 36.B, 36.C, and 36.D are valid only for a limited period of time, namely for a few millennia before and after A.D. 2000, nor for millions of years! Consequently, do not use the method given in this Chapter before the year —2000, nor after A.D. 4000.

For modern times, say between A.D. 1800 and 2200, the instants obtained for the phenomena involving Mercury and Venus will be less than 1 hour in error. The error can reach 2 hours in the case of Saturn, Uranus, and Neptune, 3 hours for Mars, and 4 hours for Jupiter.

It is expected that the maximum possible error will be somewhat larger near the years —2000 and +4000. On the other hand, if the calculations are performed for epochs near A.D. 2000, say between 1900 and 2100, then the terms in 'r 2 may safely be ignored.

Exercises

Check your computer program with the following cases; all times are in TD.

Mercury inferior conjunction 1631 Nov. 7 7' (a)

Venus inferior conjunction 1882 Dec. 6 17' (b)

Mars opposition 2729 Sep. 9 3' (c)

Jupiter opposition —6 Sep. 15 7' (d)

Saturn opposition —6 Sep. 14 9' (d)

Uranus opposition 1780 Dec. 17 14' (e)

Neptune opposition 1846 Aug. 20 4' (O

(a) the first observed transit of Mercury over the solar disk (by Gassendi, at Paris).

(b) the last transit of Venus before that of A.D. 2004.

(c) a perihelic opposition of Mars.

(d) because Jupiter and Saturn were in opposition with the Sun with a time difference less than one day, there occurred a triple conjunction between these two planets in that year.

(e) three months before Uranus’ discovery by William Herschel.

(f) one month before Neptune’s discovery.

 

TA BLE 3 6. B

Periodic terms in daY*

M ER C UR Y

Inferior conjunction

Sup ME R C UR Y

erior conjunction

+0.0545 + 0.0001T —0.0548 — 0.0002 T

sin M —6.2008 + 0.0074 T + 0.00003 T +7.3894 — 0.0100 F — 0.00003 T2

cos M —3.2750 — 0.0197 T + 0.00001 F2 +3.2200 + 0.0197 F - 0.00001 T

sin 2M +0.4737 — 0.0052 T — 0.00001 TC +0.8383 - 0.0064 T - 0.00001 P 2

cos 23f +0.8111 + 0.0033 T — 0.00002 T2 +0.9666 + 0.0039 T - 0.00003 F 2

sin 3M +0.0037 + 0.0018 T +0.0770 — 0.0026 r

cos 3M —0.1768 + 0.00001 T +0.2758 + 0.0002 T - 0.00002 'r°

sin 43f —0.0211 — 0.0004 T —0.0128 — 0.0008 F

cos 4JU +0.0326 — 0.0003 T +0.0734 — 0.0004 T - 0.00001 T

sin 5M +0.0083 + 0.0001 T —0.0122 — 0.0002 F

cos 53f —0.0040 + 0.0001 T +0.0173 — 0.0002 r

V EN U S V E N U S

Inferior conjunction Superior conjunction

sin

3f —0.0096 + 0.0002 T — 0.00001 T2

+2.0009 — 0.0033 T — 0.00001 T

+0.5980 — 0.0104 F + 0.00001 F2

+0.0967 — 0.0018 T — 0.00003 T°

+ 0.0913 + 0.0009 T — 0.00002 T2

+0.0046 - 0.0002 T

+0.0079 + 0.0001 T +0.0099 — 0.0002 T — 0.00001 P 2

+4.1991 — 0.0121 F — 0.00003 F 2

—0. 6095 + 0.0102 r — 0.00002 r 2

+0.2500 — 0.0028 r — 0.00003 r 2

+0.0063 + 0.0025 r — 0.00002 F 2

+0.0232 — 0.0005 r — 0.00001 T

+0.0031 + 0.0004 F

cos M

sin 23f

cos 23f

sin 3M

cos 33f

M A R S M A R S

Opposition Conjunction with Sun

sin

M —0.3088 + 0.00002 r 2

—17.6965 + 0.0363 T + 0.00005 T2

+ 18.3131 + 0.0467 T — 0.00006 T2

—0. 2162 — 0.0198 'r — 0.00001 T

—4.5028 — 0.0019 T + 0.00007 T°

+ 0.8987 + 0.0058 F — 0.00002 'r°

+ 0.7666 — 0.0050 T — 0.00003 r'

—0.3636 — 0.0001 T +  0.00002 P2

+ 0.0402 + 0.0032 F

0.0737 — 0.0008 F

—0.0980 — 0.0011 T ^0.3102 — 0.0001 T + 0.00001 F2

+9.7273 — 0.0156 T + 0.00001 T2

—18.3195 — 0.0467 T + 0.00009 T2

—1.6488 — 0.0133 r + 0.00001 T2

—2.6117 — 0.0020 T + 0.00004 'r 2

—0.6827 — 0.0026 r + 0.00001 T

+0.0281 + 0.0035 T + 0.00001 F2

—0.0823 + 0.0006 T + 0.00001 T2

^0.1584 + 0.0013 r

*0.0270 + 0.0005 T

^0.0433

cos N

sin 2ñf

cos 2M

sin 3M

cos 33f

sin 43f

cos 43f

sin 53f

cos SM

 

36. PLANETARY PHENOMENA TA   B LE    3 6. B (cont. )

 

2'57

 

4

 

 

TA B LE 3 6. B

 

lcont.l

 

URA N US UR AN US

Opposition Conjunction with Sun

+0.0844 — 0.0006 T —0.0859 + 0.0003 T

sin M —0.1048 + 0.0246 T —3.8179 — 0.0148 'r + 0.00003 F2

cos M —5.1221 + 0.0104 T + 0.00DD3 T +5.1228 — 0.0105 F — 0.00002 F2

sin 2M —0.1428 + 0.0005 r —0.0803 + 0.0011 F

cos 23f sin 3M

cos 3M —0.0148 — 0.0013 Z’

0

+0.0055

—0.1905 — 0.0€D6 T

+0.0088 + 0.0001 V

0

cos e +0.8850 +0.8850

cos / +0. 2153 +0.2153

NEPTUNE NEP TU NE

Opposition Conjunction with Sun

—0.0140 + 0.00001 T2 + 0.0168

sin 3f — 1.3486 + 0.0010 T + 0.00001 T2 —2.5606 + 0.0088 'fi + 0.00002 T2

cos M +0.8597 + 0.0037 Z’ —0.8611 — 0.0037 T + 0.00002 r 2

sin 23f —0.0082 — 0.0002 r + 0.00001 T2 + 0.0118 — 0.0004 F + 0.00001 F2

cos 23f cos e

cos g +0.0037 — 0.0003 T

—0.5964

+0.0728 + 0.0307 — 0.0003 T

—0.5964

+0.0728

 

TA BLE  3 6. C

Periodic terms for greatest elongations

MER C UR Y , greatest eastern elongation (evening visibility)

Correction (days) to the time of mean inferior conjunction

Elongation (degrees)

—21.6101 + 0.0002 r 22.4697

sin M — 1.9803 — 0.00b0 T + 0.00001 T2 —4.2666 + 0.0054 T + 0.00002 T’

cOs 3f + 1.4151 — 0.0072 T — 0.00001 T° — 1.8537 — 0.0137 T

sin 23f + 0.5528 — 0.0005 T — 0.00001 T2 +0.3598 + 0.0008 T — 0.00001 T*

cos 2ñf +0.2905 + 0.0034 T + 0.00001 r 2 —0.0680 + 0.0026 T

sin 3Ivi —0.1121 — 0.0001 'r + 0.00001 r 2 —0.0524 — 0.0003 T

cos 3M —0.0098 — 0.0015 r + 0.0052 — 0.0006 f

sin 43f +0.0192 + 0.0107 + 0.0001 r

cos 4M + 0.0111 + 0.0004 T —0.0013 + 0.0001 T

sin 53f —0.0061 —0.0021

cos 53f —0.0032 — 0.0001 T +0.0003

MERC URY , greatest western elongation (morning visibility)

Correction (days) to the time of mean inferior conjunction

Elongation (degrees)

+21.6249 — 0.0002 T 22.4143 — 0.0001 T

sin M +o.1306 + 0.0065 T +4.3651 — 0.0048 P — 0, 00002 P*

cos Of —2.7661 — 0.0011 'r + 0.00001 'r 2 +2.3787 + 0.0121 T — 0, 00001 F2

sin 23f +0. 2438 — 0.0024 r — 0.00001 T2 +0.2674 + 0.0022 T

c0s 2M +0.5767 + 0.0023 r —0.3873 + 0.0008 F + 0.00001 T2

sin 33f + 0.1041 —0.0369 — 0.0001 T

cos 33f —0.0184 + 0.0007 T +0.0017 — 0.0001 T

sin 4M —0.0051 — 0.0001 T +0.0059

cos 43f +0.0048 + 0.0001 T +0.0061 + 0.0001 T

sin 53f +0.0026 +0.0007

cos 53f -r0.0037 —0.0011

 

TA B LE 3 6. C ( cont. )

V EN US, greatest eastern elongation (evening visibility)

Correction (days) to the time

Elongation (degrees)

of mean inferior conjunction

—70.7600 + 0.0002 T — 0.00001f 2 46.3173 + 0.0001 T

sin M +1.0282 — 0.0010'r — 0.00001T +0.6916 — 0.0024F

cos M +0.2761 — 0.0060T +0.6676 - 0.0045 T

sin 2M —0.0438 — 0.0023 r + 0.00002 F2 +0.0309 - 0.0002 T

cos 2M -r0.1660 — 0.0037'r — 0.00004T2 +0.0036 — 0.0001 T

sin 3M +0.0036 + 0.0001T

cos 3M —0.0011 + 0.00001 ' 2

VENUS , greatest western elongation (morning visibility)

Correction (days) to the time

of mean inferior conjunction Elongation (degrees)

+70.7462 — 0.00001 T2 46.3245

sin Of + 1.1218 — 0.0025 F — 0.00001 T2 —0.5366 — 0.0003 r + 0.00001 T’

cos 3f -r0.4538 — 0.0066 r +0.3097 + 0.0016 F — 0.00001 r 2

sin 23f +0.1320 + 0.0020 T — 0.00003 f° —0.0163

cos 2M —0.0702 + 0.0022 T + 0.00004 T° —0.0075 + 0.W30i T

sin 33f +0.0062 — 0.0001 T

cos 33f +0.0015 — 0.00001 T

 

TABLE 3 6. D

Periodic terms in days

MER C UR Y : corrections to the time of mean inferior conjunction

Station 1 Station 2

— 11.0761 + 0.0003 T + 11.1343 — 0.0001 r

sin M —4.7321 + 0.0023 F + 0.00002 T2 —3.9137 + 0.0073 T + 0.00002 F 2

cos Of — 1.3230 — 0.0156 F —3.3861 — 0.0128 r + 0.00001 F 2

sin 23f +0.2270 — 0.0046 F +0.5222 — 0.0040 F — 0.00002 r 2

cos 21vf +0.7184 + 0.0013 F — 0.00002 T2 +0.5929 + 0.0039 F — 0.00002 T*

sin 3M +0.0638 + 0.0016 r —0.0593 + 0.0018 T

cos 33f —0.1655 + 0.0007 T —0.1733 — 0.0007 r + 0.00001 'r 2

sin 43f —0.0395 — 0.0003 T —0.0053 - 0.0006 T

cos 4M +0.0247 — 0.0006 'r +0.0476 — 0.0001 T

sin 53f +0.0131 +0.0070 + 0.0002 r

cos 53f +0.0008 + 0.0002 T —0.0115 + 0.0001 T

V EN US : corrections to the time of mean inferior conjunction

Station 1 Station 2

—21.0672 + 0.0002 T — 0.00001 T + 21.0623 — 0.00001 T 2

sin M + 1.9396 — 0.0029 T — 0.00001 T2 + 1.9913 — 0.0040 r — 0.00001 T

cos M + 1.0727 — 0.0102 'r —0.0407 — 0.0077 T

sin 23f +0.0404 — 0.0023 T — 0.00001 T2 +0. 1351 — 0.0DO9 T — 0.00004 T2

cos 23f +0.1305 — 0.0004 T — 0.00003 T2 +0.0303 + 0.OD 19 T

sin 33f —0.0007 — 0.0002 T +0.0089 — 0.0002 T

cos 3M +0.0098 +0.0043 + 0.0001 T

M A RS : corrections to the time of mean opposition

Station 1 Station 2

—37.0790 — 0.0009 F + 0.00002 T2 + 36.7191 + 0.0016 + 0.00003 T2

sin 3f —20.0651 + 0.0228 T + 0.00004 T2 — 12.6163 + fi.0417 T — 0.00001 T2

cos 3f + 14.5205 + 0.0504 T — 0.00001 r 2 +20.1218 + 0.0379 T — 0.00006 T2

sin 23f + 1.1737 — 0.0169 F — 1.6360 — 0.0190 T

cos 2M —4.2550 — 0.0075 T + 0.00008 T2 —3.9657 + 0.0045 T + 0.00007 T2

sin 33f +0.4897 + 0.0074 T — 0.00001 T2 + 1.1546 + 0.0029 T — 0.00003 f*

cos 33f + 1.1151 — 0.0021 T — 0.00005 T* +0. 2888 — 0.0073 F — 0.00002 f 2

sin 43f —0.3636 — 0.0020 F + 0.00001 r 2 —0.3128 + 0.0017 T + £i. 00002 'r°

cos 43f —0.1769 + 0.0028 P + 0.00002 r 2 +0. 2513 + 0.002s T — 0 00002 r*

sin 5M +0.1437 — 0.0004 T —0.0021 — 0.0016 T

cos 53f —0.0383 — 0.0016 T —0.1497 — 0.0006 r

 

TA BLE   3 6.  D (cont. )

J U PITER : corrections to the time of mean opposition

StatiOrl 1 StatfOft 2

—60.3670 — 0.0001 T — 0.00009 T2 +60.3023 + 0.0002 T — 0.00009 r 2

sin M —2.3144 — 0.0124 T + 0.00007 T* + 0.3506 — 0.0034 T + 0.00004 T2

cos M + 6.7439 + 0.0166 T — 0.00006 T2 + 5.3635 + 0.0247 T — 0.00007 T2

sin 23f —0.2259 — 0.0010 T —0.1872 — 0.0016 T

cos 2ñf —0.1497 — 0.0014 T —0.0037  — 0.0005 T

sin 3M +0.0105 + 0.0001 T +0.0012 + 0.0001 T

cos 3ñf —0.0098 —0.0096 — 0.0001 T

sin a 0 + 0.0144 T — 0.00008 T2 0 + 0.0144 T — 0.00008 T*

cOs +0.3642 — 0.0019 T — 0.00029 T2 + 0.3642 — 0.0019 F — 0.00029 P2

S A TURN : corrections  to the time of mean opposition

Station l StatiOn 2

—68.8840 + 0.0009 T + 0.00023 T2 + 68.8720 — 0.0007 T + 0.00023 'r 2

sin If +5.5452 — 0.0279 r — 0.00020 T2 +5.9399 — 0.0400 r — 0.00015 T2

c0s M + 3.0727 — 0.0430 T + 0.00007 T° —0.7998 — 0, 0266 F + 0.00014 T2

sin 2M +0.1101 — 0.0006 T — 0.00001 T +0.1738 — 0,0032 r

cos 2ñf +0.1654 — 0.0043 T + 0.00001 T2 —0.0039 — 0. OD24 T 0.00001 r 2

sin 3ñf +0.0o10 + 0.0001 T + 0.0073 - 0. DO02 T

cos 33f +0.0095 — 0.0003 T + 0.0020 — 0. DD02 T

sin a o — 0.0337 T +  0.00018 T° 0 — 0,0337 F + 0.00018 T2

cos a —0. 8510 + 0.0044 r + 0.00068 T2 —0. 8J10 + 0.0044 F + 0.00068 T2

sin b o — 0.0D64 T + 0.00004 T2 0 — 0. DD64 T + 0.00004 T2

cos b + 0.2397 — 0.0012 T — 0.00008 r 2 + 0.2397 — 0.0012 P — 0.00008 T2

sin c o — 0.0010 F 0 — 0,0010 T

cOS C +0.1245 + 0.0006 T + 0.1245 + 0.0006 T

sin d 0 + 0.0024 T — 0.00003 'r 2 0 + 0.0O24 T — 0.00003 r 2

cos d + 0.0477 — 0.0005 T — 0.00006 T2 + 0.0477 — 0.0005 T — 0.00006 F°

 

Chapter 37

As for the numerous minor planets (see Chapter 33), no analytical theory for the motion of Pluto is available. However, we have constructed expressions for an accurate representation of the planet’s motion (2000.0 coordinates) for the years 1885 to 2099. The coefficients of the periodic terms were determined by the least- squares method, on the basis of a numerical integration of Pluto’s heliocentric motion performed by Prof. Aldo Vitagliano, of the University of Naples, Italy [l]. Perturbations by the first eight major planets and the three major asteroids were included. This numerical integration itself was based on a model and a set of starting conditions optimized through a least-squares fit on the DE 405 ephemeris calculated at the Jet Propulsion Laboratory, U.S.A.

For the calculation we used the same method as that used in an earlier investigation [2], but now referring Pluto’s heliocentric longitude and latitude to the new standard equinox J2000.0. The results are given in Table 37. A.

Method of calculation

Calculate, by means of formula (22.1), the time T in Julian centuries from the epoch J2000.0, and then the following angles (in degrees) :

= 34.35 + 3034.9057 T

S —— 50.08 + 1222.1138 T

P —— 238.96 +   144.9600 T

Then calculate the periodic terms given in Table 37.A. On each horizontal line, the argument n is a linear combination of the angles J, S, and P, namely

a —— i I + j S + k P

where i, y, k are small integers, given in the second column of the table. The contribution of each argument is

A sin n + B cos n

263

 

For instance, on the 13th line of the table we read the numbers i = 0, y = 2, k —— — 1, so here the argument is n = 2S — P, and for the latitude the contribution is —122 sin n + 175 cos a.

In Table 37.A, the numerical values of the coefficients A and B are given in units of the sixth decimal of a degree in the case of the longitude and the latitude, and in units of the seventh decimal (astronomical units) for the radius vector.

The heliocentric longitude 1, latitude b bo th in degrees), and the radius vector

r of Pluto are then given by

/ = 238.958 116 + 144.96 T + sum of periodic terms in longitude

b -— —3.908 239 + sum of periodic terms in latitude

r —— 40.724 1346 + sum of periodic terms in radius vector

The longitude and latitude obtained by this method are heliocentric, not barycentric, and they are referred to the standard equinox of J2000.0.

Calculated in this way, / will be less than 0“.07 in error, b less than 0”.02, and the radius vector less than 0.000 006 AU, with respect to Vitagliano’s numerical integration on which this representation of the motion of Pluto is based. It is important to note, as has been said, that ihe method giren here is not valid outside the period 1885 -2099.

To find the geocentric astrometric 2000.0 equatorial coordinates n and fi of Pluto:

— rind the geocentric 2000.0 rectangular equatorial coordinates X, x, Z of the Sun (see Chapter 26);

— find those of Pluto by

 

x -— r cos l cos b

y —- r (sin f cos b cos e — sin b sin e) z = r (sin l cos I› sin e + sin b cos e)

 

(37. 1)

 

where e is the mean obliquity of the ecliptic at epoch J2000.0. We have

sin e = 0.397 777 156

cos e = 0.917 482 062

— find n and 6, and Pluto’s distance A to the Earth, by means of formulae (33.10).

However, the effect of light-time should be taken into account. See Chapter 33 and formula (33.3). Hence, to obtain the geocentric n and fi, the values of /, b, r should be calculated for an instant which is earlier than the given instant by the light-time T.

 

T A B L E   3 7. A

Periodic terms for the   heliocentric coordinates a f Pluto

“‘ Argument Longitude Latitude Radius vector

J S P A B A B A B

1 0 0 1 —19799805 1985£D55 —5452852 —14974862 66865439 68951812

2 0 0 2 897144 —4954829 3527812 1672790 —11827535 —332538

3 0 0 3 611149 1211027 —1050748 327647 1593179—1438890

4 0 0 4 —341243 —189585 178690 —292153 —18444 483220

5 0 0 5 129287 —34992 18650 100340 —65977 —85431

6 0 0 6 —38164 30893 —30697 —25823 31174 —6032

7 0 1 —1 20442 —9987 4878 11248 —5794 22161

8 0 1 0 —4£fì3 —5071 226 —64 4601 4032

9 0 1 1 —6016 —3336 2030 —836 —1729 234

10 0 1 2 —3956 3039 69 —604 —415 702

11 0 1 3 —667 3572 —247 —567 239 723

12 0 2 —2 1276 501 —57 1 67 —67

13 0 2 —1 1152 —917 —122 175 1034 —451

14 0 2 0 630 —1277 —49 —164 —129 504

15 1 —1 0 2571 —459 —197 199 480 —231

16 1 —1 1 899 —1449 —25 217 2 —441

17 1 0 —3 —1016 1043 589 —248 —3359 265

18 1 0 —2 —2343 —1012 —269 711 7856 —7832

19 1 0 —1 7042 788 185 193 36 45763

20 1 0 0 1199 —338 315 807 8663 8547

21 1 0 1 418 —67 —130 —43 —809 —769

22 1 0 2 120 —274 5 3 263 —144

23 1 0 3 —60 —159 2 17 —126 32

24 1 0 4 —82 —29 2 5 —35 —16

25 1 1—3 —36 —29 2 3 —19 —4

26 1 1—2 —40 7 3 1 —i5 8

27 1 1—1 —14 22 2 —1 —4 12

28 110 4 13 1 —1 5 6

29 111 5 2 0 —1 3 1

30 113 —1 0 0 0 6 —2

31 2 0—6 2 0 0 —2 2 2

32 2 0—5 —4 5 2 2 —2 —2

33 2 0—4 4 —7 —7 0 14 13

34 2 0—3 14 24 10 —8 —63 13

35 2 0 —2 —49 —34 —3 20 136 —236

36 2 0—1 163 —48 6 5 273 1065

37 2 0 0 9 —24 14 17 251 149

38 2 0 1 —4 1 —2 0 —25 —9

39 2 0 2 —3 1 0 0 9 —2

40 2 0 3 1 3 0 0 —8 7

41 3 0—2 —3 —1 0 1 2 —10

42 3 0—1 5 —3 0 0 19 35

43 3 0 0 0 0 1 0 10 3

 

The angles 7, S, and P are the mean longimdes of Jupiter, Saturn, and Pluto, respectively, as adopted for our calculation of the periodic terms of Table 37. A. It may seem strange that in our solution the mean longitudes of Uranus and Neptune are not needed. The reason is that the mean motion of Uranus is almost exactly twice that of Neptune, or three times that of Pluto. As a consequence, the argument 2N -- P, for instance, where N is the mean longitude of Neptune, has almost the same period as 1 P. The small difference could not have been detected by our investigation based on the rather short interval of 214 years. Therefore, Table 37. A does not contain the argument 2N -- P ,- the effects of the terms with this argument are included in the terms with argument 2P. For the same reason, there are no terms in S -- 4P, S -- 3P, S -- 2f', I - 5P, I -- 4P, and 25 -- 3P: they have almost the same period as 4P, 5 P, 6P, 2fi -- P, 2S, and J -- S + P, respectively.

Example 37.a -- For 1992 October 13.0 TD = JDE 2448 908.5, find

(1) the geometric heliocentric coordinates of Pluto;

(2) its geocentric astrometric coordinates n and d.

(1) We find

T 0.072 183 4360

J --- -- 184°.719 921

S --- --38.° 136 373

P ---- 228°.496 289

Sum of periodic terms in longitude : +    4246 306

in latitude : + 18 496056

in radius vector : --110 130 236

 

from which

 

I -- 238°.958 116 -- 10.°463 711 + 4°.246 306 = 232.°740 71

b -- --3.°908 239 + 18°.496 056 = + 14°.587 82

r = 40.724 1346 -- 11.013 0236 = 29.711 111 AU

 

(2) For the given instant, the Sun’s 2000.0 rectangular equatorial coordinates are (from Example 26.b)

X = --0.937 3959

Y 0.313 1679

Z = 0.135 7792

Using Pluto’s coordinates 1, b, r found above, formulae (37.1) give z   = 17.407 9141

y = 23.973 0804

z = 2.237 4228

 

whence, by formulae (33.10) and (33.3),

A = 30.528 746 AU and z = 0.17632 day

This value of A is Pluto’s true distance to the Earth.

We now repeat the calculation of the planet’s heliocentric coordinates for 1992 October 13.0 — 0.17632 = October 12.82368. The results are

l -- 232°.73949 b --  + 14°.58801 r -- 29.711094

 

whence

 

z = —17.408 3780

y = —23.972 7452

z = — 2.237 1797

 

A = 30.528 739

z = 0.17632 day

 

We obtain for r the same value as before, so no new iteration is needed.

The 2000.0 astrometric coordinates of Pluto for 1992 October 13.0 TD are then found by means of (33.10) :

= 232°.93231 = 15'31‘43'.8

6 = —4°.45802 = —4°27'29“

Mean orbital elements of Pluto near A.D. 2000:

o = 39.543 AU

e -— 0.2490

i -- 17.° 140

II = 110°.307 2000.0

t» = 113.°768

REFEREN CES

1. A. Vitagliano, “Numerical integration for the real time production of fundamental ephemerides over a wide time span", Celestial Mec/tnnics, Vol. 66, pages 293—308 (1997).

2. E. Goffin, J. Meeus, and C. Steyaert, “An accurate representation of the motion of Pluto", Astronomy and Astrophysics, Vol. 155, pages 323—325 (1986).

 

 

 

Chapter 38

Planets in Perihelion and in Aphelion

The Julian Day corresponding to the time when a planet is in perihelion or in aphelion can be found by means of the following expressions:

Mercury JDE = 2451 590.257 + 87.969 349 63 k -- 0.000 000 0000 t 2

Venus JDE = 2451 738.233 + 224.700 818 8 k -- 0.000 000 0327 k*

Earth JDE = 2451 547.507 + 365.259 635 8 k + 0.000 000 0156 k*

Mars JDE = 2452 195.026 + 686.995 785 7 k -- 0.000 000 1187 k 2

Jupiter JDE = 2455 636.936 + 4332.897 065 k + 0.000 1367 k 2

Saturn JDE = 2452 830.12 + 10764.216 76 k + 0.000 827 # 2

Uranus JDE = 2470 213.5 + 30694.8767 k -- 0.005 41 k2

Neptune JDE = 2468 895. 1 + 60190.33 k + 0.034 29 k2

where k is an integer for perihelion, and an integer increased by exactly 0.5 for aphelion. Any other value for k would give meaningless results!

A zero or a positive value of I will give a date after the beginning of the year 2000. If k < 0, one obtains a date earlier than A.D. 2000.

For example, k ---- + 14 and k ---- --222 give passages through perihelion, while

k ---- +27.5 and k -- --119.5 give aphelion passages.

An approximate value for k can be found as follows, where the “year" should be taken with decimals, if necessary:

Mercury k -- 4.15201 (year -- 2000.12) Venus k -- 1.62549 (year -- 2000.53) Earth k -- 0.99997 (year -- 2000.01) Mars k -- 0.53166 (year -- 2001. 78)

Jupiter k 0.08430 (year -- 2011.20) Saturn k -- 0.03393 (year -- 2003.52) Uranus k = 0.01190 (year -- 2051.1) Neptune k = 0.00607 (year -- 2047.5)

2 69

 

 

ñzompie 38.a — Find the time of passage of Venus at perihelion nearest to 1978 October 15, that is 1978.79.

An approximate value of k is

1.62549 (1978.79 — 2000.53) = —35.34

and, since k must be an integer (perihelion!), we take k -- —35. Putting this value in the formula for Venus, we find

JDE = 2443 873.704,

which corresponds to 1978 December 31.204, or 1978 December 31 at 5 Dynamical Time.

Example 38.b — Find the time of passage of Mars through aphelion in the year 2032.

Taking “year" = 2032.0, we find k = + 16.07. Since k must be an integer increased by 0.5 (aphelion!), the first aphelion of Mars after the beginning of the year 2032 occurs for k —— + I6.5.

Using the formula for Mars, this value of k gives

JDE = 2463 530.456,

corresponding to 2032 October 24.956, or 2032 October 24 at 23' Dynamical Time.

Important : The formulae for the calculation of JDE given on the preceding page are based on unperturbed elliptic orbits. For this reason, the instants obtained for Mars can be a few hours in error.

Due to the mutual planetary perturbations, the instants for Jupiter, calculated by the method described here, may be up to half a month in exon. For Saturn, the error can be larger than one month.

For instance, putting k -— —2.5 in the formula for Jupiter gives 1981 July 19 as the date of an aphelion passage, while the correct date is 1981 July 28. For Saturn, k —— —2 gives 1944 July 30, while the planet actually reached perihelion on 1944 September 8.

The error can be even larger for Uranus and Neptune. For these planets, the formulae are given merely for completeness.

Accurate times can be obtained by calculating the value of the planet’s distance to the Sun for several instants near the expected time, and then finding when this distance reaches a maximum or a minimum. The table on the next page gives the dates when Saturn (in the period 1920—2050) and Uranus (1750-2100) are in perihelion (P) or in aphelion (A). After the date, the distance to the Sun in astronomical units is mentioned. These data have been calculated by means of Bretagnon’s complete VSOP87 theory.

 

  Ura nu s

A 1929 Nov. 11 10.0467 A 1756 Nov. 27 20.0893

P 1944 Sep. 8 9.0288 P 1798 Mar. 3 18.2890

A 1959 May 29 10.0664 A 1841 Mar. 16 20.0976

P 1974 Jan. 8 9.0153 P 1882 Mar. 23 18.2807

A 1988 Sep. 11 10.0444 A 1925 Apr. 1 20.0973

P 2003 July 26 9.0309 P 1966 May 21 18.2848

A 2018 Apr. 17 10.0656 A 2009 Feb. 27 20.0989

P 2032 Nov. 28 9.0149 P 2050 Aug. 17 18.2830

A 2047 July 15 10.0462 A 2092 Nov. 23 20.0994

The case of Neptune is peculiar. This planet has a slow motion and a small orbital eccentricity. On the other hand, the Sun is oscillating around the barycenter of the solar system, mainly due to the actions of Jupiter and Saturn. Consequently, the distance of Neptune to the Sun (not to the barycenter of the solar system) can reach a double maximum or minimum.

For example, we had the following extreme values for Neptune’s radius vector:

 

minimum maximum minimum

 

1876 Aug. 28 r —— 29.8148 AU

1881 Dec. 12 29.8213

1886 July 11 29.8174

 

Half a revolution later, near the aphelion pan of the orbit, we had the following extrema:

 

maximum minimum maximum

 

1959 July 13 r = 30.3317 AU

1965 Oct. 6 30.3227

1968 Nov. 21 30.3241

 

The maximum of 1881 was not z:n aphelion, because at that time Neptune was near the perihelion of its orbit. Similarly, the minimum of 1965 did not correspond to a perihelion. The author has coined the new terms apheloid (= “resembling an aphelion”) and periheloid for these odd maximum and minimum, respectively [1]. See also Chapter 28 in my Mathematical Astronomy MorsefA (Willmann-Bell, ed. ; 1997).

Figure 1 shows the variation of the distance of Neptune to the Sun from 1954 to 1972. Note the principal aphelion (1), the periheloid (2), and the secondary aphelion (3). Half a revolution later, we have the situation pictured in Figure 2; this

 

AU

30.33

30.32

 

Figure I

The variation of the distance of Neptune to the Sun, 1954 to 1972.

Figure 2

The variation of the distance of Neptune to the Sun, 2038 io 2054.

 

will be almost a “limiting case" : the principal perihelion (1’) will occur in 2042, while in 2049—2050 the distance to the Sun will decrease only very slightly from the apheloid (2') to the secondary perihelion (3'), as follows:

 

minimum maximum minimum

 

2042 Sep. 5 r = 29.8064 AU

2049 Oct. 24 29.816711

2050 June 25 29.816696

 

For the Earth, it is important to note that the formula given to calculate JDE is actually valid for the barycenter of the Earth-Moon system. Due to the action of the Moon, the time of least or greatest distance between the centers of Sun and Eärth may differ from that for the barycenter by more than one day [2). For instance, k = — 10 in the formula for the Earth yields JDE — 2447 894.911, which corresponds to 1990 January 3.41, while the correct instant is 1990 January 4, at 17h TD.

The values obtained (for the Earth only) can be corrected as follows. Calculate the following angles, in degrees :

A   = 328.41 + 132.788 585 k

A    = 316.13 + 584.903 153 k

A —— 346.20 + 450.380 738 k

A 4  = 136.95 + 659.306 737 i

A  —- 249.52 + 329.653 368 i

Remember that k must be an integer for a perihelion, or an integer increased by

0.5 for an aphelion. Then we have the following correction terms, in days:

perihelion aphelion

+1.278

—0.055 —1.352

+0.061 x sin A I

—0.091 +0.062

—0.056 +0.029 sin A

—0.045 +0.031 sin A5

Calculated in this way, the times for the years 1980—2019 have a mean error of 3 hours. Exceptionally, the error amounts to 6 hours.

For instance, for k —- - 10, we obtain a correction of + 1.261 day, so the value JDE = 2447 894.911 mentioned above is corrected to 2447 896. 172, which corresponds to 1990 January 4, at 16‘ TD, much closer to the exact value.

Table 38.A gives the times of the passages of the Earth in perihelion and aphelion for the years 1991 to 2010, to the nearest 0.01 hour, together with the distance in astronomical units between the centers of the Sun and the Earth. These data have been calculated accurately, using the complete VSOP87 theory, not the approximate method given in this Chapter.

 

T A B L E 3 8. A

Perihelion and Aphelion of the Earth, l99 1 —20 l0 Instants in Dynamical Time

  Perihelion ApheliDn

1991

Jan.

3 h

3.00

0.983 281

July

6 h

15.46

1.016 703

1992 3 15.06 324 3 12.14 740

1993 4 3.08 283 4 22.37 666

1994 2 5.92 301 5 19.30 724

1995 4 11.10 302 4 2.29 742

1996 Jan. 4 7.43 0.983223 July 5 19.02 1.016717

1997 1 23.29 267 4 19.34 754

1998 4 21.28 300 3 23.86 696

1999 3 13.02 281 6 22.86 718

2000 3 5.31 321 3 23.84 741

2001 Jan. 4 8.89 0.983286 July 4 13.65 1.016643

2002 2 14.17 290 6 3.80 688

2003 4 5.04 320 4 5.67 728

2004 4 17.72 265 5 10.90 694

2005 2 0.61 297 5 4.98 742

2006 Jan. 4 15.52 0.983327 July 3 23.18 1.016697

2007 3 19.74 260 6 23.89 706

2008 2 23.87 280 4 7.71 754

2009 4 15.51 273 4 1.69 666

2010 3 0.18 290 6 11.52 702

REFEREN CES

1. J. Meeus, “Le centre de gravité du système solaire et le mouvement de Neptune”,

Ciel et Terre (Belgium), Vol. 68, pages 288—292 (November-December 1952).

2. J. Meeus, Mathematical Astronomy Morsels, Chapter 27 (Willmann-Bell, ed. ; 1997). First published in /’Astronomie (France), Vol. 97, pp. 294-296 (June 1983).

 

Chapter 39

Passages through the Nodes

Given the orbital elements of a planet or a comet, the times t of passages of that body through the nodes of its orbit can easily be calculated as follows.

We have

at the ascending node : v = — o or 360° — ui at the descending node : v -- 180° — o

where, as before, v is the true anomaly, and o the argument of the perihelion. Then, with these values of v, proceed as follows.

Case oJ an elliptic orbit

Calculate the eccentric anomaly E by

(39.1)

where e is the orbital eccentricity, and the mean anomaly M by

If = E — e sin E (39.2)

In formula (39.2), E should be expressed in radians; the resulting value for M is then in radians too. If, however, E is expressed in degrees and the computer is working in degree mode, then in formula (39.2) one should replace e by its value eg converted from radians into degrees, that is, ed = e x 57.°295 779 51.

Express M in degrees. Then, if T is the time of perihelion passage, and n is the mean motion in degrees/day, the required time of passage through the node is given by

t —- T + p days (39.3)

275

 

The corresponding value of the radius vector is given by

r —— a (1 — e cos E)

where o is the semimajor axis of the orbit expressed in astronomical units.

If a and n are not given, they can be calculated from (33.6).

Case of a parabolic orbit

Calculate

 

(39.4)

 

 

Then

r = T + 27. 403 895 {s’ + 3â) q Sq days

where the perihelion distance q is expressed in AU. The corresponding value of the radius vector is

r -- q (1 + s-)

Note . — The nodes refer to the ecliptic of the same epoch as that of the equinox used for the orbital elements. For example, if the orbital elements are referred to the standard equinox of 2000.0, the above-mentioned formulae give the times of passages through the nodes on the ecliptic of 2000.0, nor on the ecliptic of the date. The difference is generally small, except when the inclination is very small or when the motion is very slow.

Example 39.a — For the 1986 return of periodic comet Halley, W. Landgraf [Minor Planet Circular No. 10634 (1986 April 24)] provided the following orbital elements:

T —- 1986 February 9.45891 TD

= 111°. 84644

e -- 0.967 274 26

n = 0.012 970 82 degrees/day

a -- 17.940 0782

the argument of perihelion a being referred to the standard equinox of 1950.0.

For the passage at the ascending node, we have v = 360° — = 248°. 15356

E = —0. 190 6646

2

 

39. PASSAGES THROUGH   THE NODES 277

A = --21°.589 4332

M --- --21.°589 4332 -- (0.967 274 26 x 57 •.295 77951) sin (--21.°589 4332)

= -- 1°.197 2043

 

f = T + --1.197 2043

0.012 970 82

 

-- T -- 92.2998 days

 

Hence, the comet was at its ascending node (on the ecliptic of 1950.0) 92.2998 days before the perihelion passage, that is, on 1985 November 9.16 Dynamical Time.

Formula (39.4) then gives r --- 1.8045 AU. So, at its ascending node the famous comet was a little outside of the orbit of Mars.

For the descending node, we find similarly: v = 180° -- z = 68.° 15356

F   = +9.°9726067

3f = + 0.°374 9928

r = T + 28.9105 days = 1986 March 10.37 TD

r ---- 0.8493 AU, between the orbits of Venus and Earth

The fact that the comet’s motion (i = 162°) is retrograde, is irrelevant here. Anyway, o is measured from the ascending node in the direction of the motion of the body.

Exampfe 39.b -- For comet Helin-Roman (1989a = 1989 IX), B. G. Marsden and G.

V. Williams (tenth edition of the Catalogue of Cometary Orbits,

IAU, 1995) give the following elements of a parabolic orbit:

T -- 1989 August 20.2910 TD

q = 1.324 502 AU

= 154°.9103 (2000.0)

T is the time of passage through the perihelion, not to be confused with the T of formula (31.1)!

 

For the ascending node, we have

s = --4.494 0577

i = T - 4354.66 days

= 1977 September 17

r -- 28.07 AU

 

For the descending node, we have

v = 180° -- o = + 25°.0897

s --- +0.222 5161

f = T + 28.3454 days

= 1989 September 17.636 TD

r -- 1.3901 AU

 

 

Example 39.c -- Calculate the time of passage of Venus at the ascending node nearest to the epoch 1979.0.

We use the elements given in Table 31. A. There we find for Venus

a -- 0.723 329 820, whence ri = 1.602 137

e -- 0.006 77192 -- 0.000047 765 T + 0.000000 0981 T’

=r -- ft = 54°.883 783 + 0°.501 1082 T - 0°.0014824 T2

The terms in T’ can safely be dropped here. The elements e and o vary (rather slowly) with time. Let us calculate their values for the epoch 1979.0, that is, for T ----

--0.21. We find

 

e 0.006 78195

and then, successively,

 

= 54.°778 485

 

v = --z = --54.°778 485

E = --54°.461 662

3f = --54°.145 467

r   = T - 33.7958 days ( T is the time of perihelion passage)

In Example 38.a, we have found T ---- 1978 December 31.204 for the time of passage of Venus in the perihelion. Therefore, we have

i = 1978 November 27.408 or 1978 November 27, at 10‘ TD.

The algorithms given in this Chapter assume that the body moves in an unperturbed orbit. To obtain full accuracy, the heliocentric latitude of the body should be calculated for three or five instants near the expected time. At the node we have, of course, latitude = zero.

Saturn reached the descending node (on the ecliptic of the date) of its orbit on 1990 September 4, and will be at its ascending node on 2005 January 8.

Uranus was at the descending node on 1984 December 21, and will go through the ascending node on 2029 May 19.

For Neptune we have

 

1920 June    3

2003 Aug. 11

2084 Dec. 30

 

ascending node descending node ascending node

 

Chat ter d0

Correction for Parallax

Suppose we wish to calculate the topocentric coordinates of a body (Moon, Sun, planet, or comet) when its geocentric coordinates are known. Geocentric —- as seen from the center of the Eanh; topocentric —— as seen from the observer’s place on the Earth’s surface (Greek: topos —— place; compare with the word “topology").

In other words, we wish to find the corrections An and Ah (the parallaxes in right ascension and in declination), in order to obtain the topocentric right ascension n' = n + Act and the topocentric declination 6’ = 6 + A6, when the geocentric values a and 6 are known.

Let p be the geocentric radius and e' the geocentric latitude of the observer. The quantities p sin ‹,e’ and p cos ‹,e’ can be calculated by the method described in Chapter 11.

Let z be the equatorial horizontal parallax of the body. Fo7 the Sun, a planet, or a comet, it is frequently more convenient to use the distance n (in astronomical units) to the Earth instead of the parallax. We then have

  8".794

 

or, with sufficient accuracy,

 

8“.794 

 

(40. l)

 

Then, if IN is the geocentric hour angle of the body, the rigorous formulae are:

 

  — p cos e'  sinr sin ff cos â   — p cos p’ sinr cos If

 

(40. 2)

 

In the case of the declination we may, instead of computing A6, calculate 6’ directly from

 

 (sin 6 — p sin ‹,o' sin r) cos An cos 6 — p cos p’ sin x cos U

279

 

(40.3)

 

Except for the Moon, the following non-rigorous formulae may often be used instead of (40.2) and (40.3):

 

  cos p’ sin H 

COS é

Aö =   — z (p sin p’ cos ô — p cos p' cos H sin 6)

 

(40.4)

(40.5)

 

If z is expressed in seconds of a degree (“), the Aa and A6 too are expressed in this unit. To express As in seconds of time, divide the result by 15.

Note that As is a small angle, always lying between —2° and +2° in the case of the Moon. It is, of course, much smaller in the case of a planet.

An alternative method is as follows. Calculate

 

A -— cos 6 sin H

B —- cos 6 cos H — p cos ‹;e’ sinr

C —— sin ô — p sin p’ sin z

q   = + B 2 + C 2 > 0

 

(40. 6)

(40.7)

 

Then the topocentric hour angle H' and declination 6’ are given by

A B

Example 40.a — Calculate the topocentric right ascension and declination of Mars on 2003 August 28, at 3'17‘"00‘ Universal Time at Palomar Observatory, for which (Example I1.a)

p sin p' = +0.546 861,    p cos p' = + 0.836 339,

L —— longitude = +7 47‘27' (West)

Mars’ geocentric apparent equatorial coordinates for the given instant, interpolated from an accurate ephemeris, are

= 22 38‘07‘.25 = 339.°530 208

ô = — 15° 46' 15“.9 = — 15.°771083

The planet’s distance at that time is 0.37276 AU. Hence, by formula (40.1), its equatorial horizontal parallax is z = 23“.592.

We still need the geocentric hour angle, which is equal to H —— 80 — L - a, where 80, the apparent sidereal time at Greenwich, can be found as indicated in Chapter 12. For the given instant, we find 80 = 1‘40‘45'. Consequently,

H = 1‘40‘45' — 7'47”27‘ — 22'38 07'

= —28'44‘49‘ = —431.°2042 = +288.°7958

 

40. CORRECTION FOR PARALLAX

Formula (40.2) then gives

 +0.000 090 557

+0.962 324

 

281

 

whence

 

As =  +0.°005 3917 = + l‘.29 n' = n + An = 22‘38"08'.54

 

Formula (40.3) gives

tan 6' _ —0.271 857 13

+0.962 324 47

 

whence 6’ = — 15° 46' 30“.0

 

If, instead of (40.2) and (40.3), we chose the non-rigorous formulae (40.4) and (40.5), we find

An = + 19“.409 = + 1'29, as above;

A6 =  — 14“.1, whence 6' = é — 14“.1 = — 15° 46’30“.0,  as above.

As an exercise, perform the calculation for the Moon, again for Palomar Observatory, using fictive values, for instance

n = 1h00*00'.00 = 15.°000 000 H —- 4'00‘00‘.00 = +60.°000 000

é   =   +5°.000 000 z =   0°59'00”

First, use the formulae (40.2) and (40.3). Then do the calculation over again with (40.6) and (40.7). You should obtain the same results exactly. Compare the results with those obtained by means of the non-rigorous expressions (40.4) and (40.5).

We can consider the opposite proble.m: from the observed topocentric coordinates n' and h', deduce the geocentric values n and 6. In the case of a planet or comet, the corrections An and A6 are so small that the formulae (40. 4) and (40.5) can be used also for the reduction from topocentric to geocentric coordinates, changing the signs of As and A6, of course.

Parallax in horizontal coordinates

The parallax in azimuth is always very small. It would be zero if the Earth were exactly a sphere. At the horizon, the parallax in azimuth is always less than r/300, where z is the equatorial horizontal parallax of the body.

Due to the parallax, the apparent altitude of a celestial body is smaller than its “geocentric" altitude h. Except when high accuracy is needed, the parallax p in altitude may be calculated from sin p — p sin z cos h.

Except in the case of the Moon, the parallax is so small that we may consider

p andr to be proportional to their sines, and then we have p —— p r cos h.

 

The quantity p denotes the observer’s distance to the center of the Earth, the equatorial radius being taken as unity — see Chapter 11. In many cases we may simply write p —— 1.

Parallax in ecliptical coordinates

It is possible to calculate the topocentric coordinates of a celestial body (Moon or planet), from its geocentric values, directly in ecliptical coordlnates. The following formulae are those given by Joseph Johann von Littrow {Theoretische und Practische Astronomie, Vol. I, p. 91; Wien, 1821), but in a slightly modified form. These expressions are rigorous.

Let   h = geocentric ecliptical longitude of the celestial body, 9 = its geocentric ecliptical latitude,

s —— its geocentric semidiameter,

k', 9', s’ —- the required topocentric values of the same quantities, p = the observer’s latitude,

e = the obliquity of the ecliptic, 8 = the local sidereal time,

z = the equatorial horizontal parallax of the body.

For the given place, calculate the quantities p sin p' and p cos p', as explained on page 82. For short, we shall call these quantities S and C, respectively. Then

N COS h GOS § — C Sirl F  COS 8

$,    _ sin  h  cos d   —r    sin (Shin  e  +   C  cos e  sin 6)  

$, cos h' (sin 9 — sin w ( cos e — C sin e sin f)) 

s n s COS ‘'   cos Q' sin s N

As an exercise, calculate X', Q', s' from the following data:

k = 181°46’22“.5 p = +50°05’07“.8 at sea level

§ = +2° 17'26”.2 e =  23°28‘00”.8

z = 0°59'27”.7 8 =  209°46’07”.9

s —— 0‘  16' 15”.5

 

Answer:

 

h' = 181°48'05“.0

Q’ = + 1°29'07“. 1

s’ -— 0°16'25“.5

 

Chapter 41

Hluminated Fraction of the Disk and Magnitude of a Planet

The illuminated fraction k of the disk of a planet, as seen from the Earth, can be calculated from

k —- 1 + COS i (41. 1)

where i is the phase angle (the angle Sun—planet-Earth), which can be found from

r 2 + A2 — R 2

 

COS Ï —

 

2 rA

 

r being the planet’s distance to the Sun, A its distance to the Earth, and fi the distance Sun —Earth, all in astronomical units. Combining these two formulae, we find

k —— (41.2)

If the planet’s position has been obtained by the “flfst method” of Chapter 33, then we have, using the notations used there,

 

R po COS B COS ML    L o) 

COS :

 

(41.3)

 

 

or

cos i =

 

z cos B cos L + y cos B sin L + z sin B 

 

(41.4)

 

The position angle of the mid-point of the illuminated limb of a planet can be calculated in the same way as for the Moon — see Chapter 48.

283

 

 

Ezompfe dI.a — Find the illuminated fraction of the disk of Venus on 1992 December 20, at 0'  TD.

In Example 33.a we have found, for that instant,

r —— 0.724 604 (called R there) fi = 0.983 824 (called fi there)

A = 0.910 947

whence, by formula (41.2), k —— 0.647.

Or, using from the same Example 33.a the values La and    from (A), L, B, R from (B), z, y, z Tom (C), and A = 0.910 947, formulae (41.3) and (41.4) both give cos i = 0.29312, whence k —- 0.647, as above.

For Mercury and Venus, k can take all values between 0 and 1. For Mars, the illuminated fraction of the disk can never be less than approximately 0.838. In the case of Jupiter, the phase angle i is always less than 12°, whence k can vary only between 0.989 and 1. For Saturn, i is always less than 6'Zi degrees, so for this planet k is always between 0.997 and 1, as seen ftom the Earth.

In the case of Ven us , an approximate value for k can be found as  follows.

Calculate T by means of formula (22.1), then

Y = 261°.51 + 22518.°443 T

M —— 177º.53 + 35999.°050 F

N —— 50º.42 + 58517.° 811 T

W = Y + 1°.91 sin M + 0.°78 sin N

A2 = 1.52321 + 1.44666 cos W (A > 0)

k —— (0.72333 + A)2 — l

2.89332 A

An approximate value of Venus’ elongation   to the Sun is then given by

A2 + 0.4768

COS = 2 A

 

Example 41.b — Same as in Example 41.a, but now using the approximate method described above. We find successively

JD = 2448976.5, T -- —0.070 321697, F = — 1322.°025 = + 117°.975,

M ——  —2353°.984 = + 166°.016, N —— —4064.°652 = >255°.348,

W = V + 0.°462 — 0.°755 = 117°. 682, A = 0.922 575, k —— 0.640.

The correct value, found in Example 41.a, is 0.647.

 

41. ILLUMINATED FRACTION AND MAGNITUDE OF A PLANET 2 8 5

Magnitude of the Planets

As seen from the Earth, the apparent (stellar) magnitude of a planet at a given instant depends of the planet’s distance to the Earth (A), its distance to the Sun (r), and the phase angle (i). For Saturn, the magnitude depends also upon the aspect of the ring.

G. Müller’s formulae, based on observations which he made from 1877 to 1891, are used since many years in astronomical almanacs. The numerical expressions for the visual magnitudes are as follows [1]:

Mercury : + 1.16 + 5 log rd + 0.02838 (i — 50) + 0.000 1023 (i — 50)2

Venus : —4.00 + 5 log rd + 0.01322 i + 0.000 000 4247 i°

Mars : — 1.30 + 5 log rA + 0.01486 i

Jupiter :

Saturn : —8.93

—8.68 +

+ 5 log rd

5 log rA

+

0.044   k U   — 2.60 sin | B   + 1.25 sin2B

Uranus : —6.85 + 5 log rd

Neptune : —7.05 + 5 log rd

in which i is expressed in degrees, r and A are in astronomical units, and the logarithms are to the base 10. For Saturn, the quantities b U and B, pertaining to the ring, are defined in Chapter 45; care must be taken to have A t/ and B positive, and to express b U in degrees. (As an approximation, the phase angle i might be used instead of b U.)

Of course, Müller’s expressions are not perfect. For instance, the effect of the phase is not taken into account in the case of Jupiter. In the formula for Saturn, the Sun’s altitude B’ above the plane of the ring is not considered (it is supposed to be equal to B),- and when B and B' have opposite signs, the dark side of the ring is turned towards the Earth, but this case is not considered by Müller.

In any case, the calculated magnitudes should be rounded to the nearest tenth of a magnitude. Giving them to the nearest hundredth makes no sense. Mars, for instance, can differ by as much as 0.3 magnitude from the brightness it “ought” to have. Some regions of Mars have more dark markings than others, so the planet’s brightness depends on which face is turned towards us; and the varying polar caps and a major dust storm can add to its magnitude. In the case of Jupiter and Saturn, there are varying atmospheric phenomena, etc.

Exazriple d1.c — Magnitude of Venus on 1992 December 20.0 TD.

From Example 41.a, we have

r —— 0.724 604, A = 0.910 947, cos i = 0.29312, whence  i = 72.96 degrees. Müller’s formula for Venus then gives —3.8 for the magnitude.

 

 

Example 41.d — Magnitude of Saturn on 1992 December 16.0 TD.

From Example 45.a, we have

r -— 9.867 882,  A = 10.464 606,  B -- 16.°442, 4 U -— 4.°198.

Müller’s formula for Saturn then gives + 0.9 for the magnitude.

Since 1984, the American Astronomical Almanac uses other formulae for the calculation of the visual magnitudes of the planets. It has been stated [2] that these new expressions “are due to D. L. Harris”. In fñCt, ln his article [3] Harris did not provide new expressions at all. No expression for the magnitudes is “due” to Harris.

For Mercury and Venus, Harris (pages 277 and 278 of his article) just mentions expressions due to the French astronomer A. Darijon. For the outer planets, Harris discusses values of the absolute magnitude and of the phase coefficient found by others, but he himself does not propose or give new expressions.

If r and A (in astronomical units) and i (in degrees) have the same meanings as

above, the new expressions used in the Astronomical Almanac since 1984 are:

Mercury : —0.42 + 5 log rd + 0.0380 i  — 0.000 273 i 2 + 0.000 002 i 3

Venus : —4. 40 + 5 log rd + 0.0009 i + 0.000 239 i 2 — 0.000 000 65 i'

Mars : — 1.52 + 5 log rA + 0.016 i

Jupiter : —9.40 + 5 log rA + 0.005 i

Saturn : —8.88 + 5 log rd + 0.044 | A t/ | — 2.60 sin | B | + 1.25 sin°B

Uranus : —7.19 + 5 log rd

Neptune : —6.87 + 5 log rd

Pluto : — 1.00 + 5 log rd

For the magnitudes of the minor planets, see Chaptes 33.

RE FE REN CES

1. Explanatory Supplement to the Astronomical Ephemeris (London, 1961), page 314.

2. Astronomical Almanac for 1984 (Washington, D.C.), page L8; and later volumes.

3. Daniel L. Harris, “Photometry and Colorimetry of Planets and Satellites”, Chapter 8 (pages 272a in Planets and Satellites, ed. G. P. Kuiper and B. L. Middlehurst (1961).

 

Chapter 42

* Rhemeris for Physical Observations of Mars

In this Chapter, the following symbols will be used:

D E = the planetocentric declination of the Earth. When it is positive, Mars’ northern pole is tilted towards the Earth;

D o — the planetocentric declination of the Sun. When it is positive, Mars’ northern pole is illuminated;

f• = the geocentric position angle of Mars’ northern rotation pole, also called position angle of axis. It is the angle that the Martian meridian from the center of the disk to the northern rotation pole forms (on the geocentric celestial sphere) with the declination circle through the center. It is measured eastwards from the North Point of the disk. By definition, position angle 0° means northwards on the sky, 90° east, 180° south, and 270° west;

q = the angular amount of the greatest defect of illumination; it is expressed in arcseconds;

Q —— the position angle of this greatest defect of illumination;

o =  the (areographic) longitude of the central meridian, as seen from the Earth. The word areographic means that use is made of a coordinate system on the surface of Mars. Compare with geographic for the Earth.

The drawing on the next page shows the appearance of Mars on 1992 Nov. 9. As seen from the Earth, the illuminated fraction of the planet’s disk was 90a {k —— 0.90). UV is the greatest defect of illumination. S is Mars’ South Pole (just behind the limb, hence not visible), A is the northern extremity of the axis of rotation. AS is the central meridian. The arrow shows the direction of the northern celestial pole (on the celestial sphere of the Earth). N is the North Point of Mars’ disk (not the planet’s north pole!). The position angles are measured from N, towards the East. So we have

Q —— arc NESV, P -- arc NESVA.

287

 

In the calculation of these quantities, the effect of light-time should be taken into account. Moreover, to obtain full accuracy the aberration of the Sun as seen from Mars must be taken into account in the

calculation of Do: and in the calculation of f• one should take into account the

effect of nutation and aberration on Mars’ position.

During the years, several positions for the north pole of Mars (that is, the coordinates of the point on the celestial sphere towards which the axis is directed) have been used in the astronomical al- manacs.

According to Lowell and Crommelin [1], the right ascension eg and declination 60 of the north pole of Mars at the beginning of the year i, referred to the mean equinox of the date, are given by

ni   =   21h10‘   + 1'.565 (r — 1905.0) é0 = + 54° 30' + 12”. 60 (i — 1905.0)

This position of the north pole was adopted in 1909. But from 1968 to 1980, the Astronomical Ephemeris used the position obtained by G. de Vaucouleurs [3]: at the beginning of the year r

0 = 316.°55 + 0º.006 750 (i — 1905.0)

60 = +52.°85 + 0.°003 479 (i — 1905.0)

Note the difference of I °39' between the two values of Q, for the same epoch 1905.0. Recently adopted values [4] are

 

0 = 317º.342

ôq = +52°.711

0 =   317º.681

ô = + 52.°886

 

equinox 1950.0 and epoch J1950.0 equinox 2000.0 and epoch J2000.0

 

From these values, we deduce the following expressions for the longitude and latitude of Mars’ north pole, referred to the ecliptic and m€afl eQulnox of the date:

 

@ = 352.°9065 + l °. 17330 T

§ 0 =  +63°.2818 — 0º.00394 T

 

(42.1)

 

where T is the time in Julian centuries from the epoch J2000.0; see formula (22. 1). Formulae (42.1) take into account the precession of the rotaüonal axes of both Earth and Mars.

 

For a given instant i, the values of DE, DS, etc. , can be calculated as follows.

1. Calculate @ and JS by means of (42. 1).

2. Calculate the heliocentric longitude 0, latitude bg, and radius vector fi of the Earth, referred to the ecliptic and mean equinox of the date, for instance by using the relevant data from Appendix III and the precepts given in Chapter 32.

3. Calculate the corresponding heliocentric coordinates I, b, r of Mars, but for the instant i — z, where z is the light-time from Mars to the Earth, as given by (33.3). Because Mars’ distance A is not known in advance, it should be found by iteration — see Step 4. One may use A = 0 as a starting value.

4. Calculate

 

x —- r cos b cos / — R cos lb

y = r cos b sin / — It sin

z = r sin b — R sin bp

Then Mars’ distance to the Earth is

A = + y 2 + z 2 > 0

5. Calculate Mars’ geocentric longitude k and latitude b from

 

(42. 2)

(42.3)

 

 

6. sin DE = — sin J O sin Q — cos J O cos Q cos (@ — X)

7. Calculate the longitude N of the ascending node of Mars’ orbit from

N —— 49.°5581 + 0°.7721 T

Then correct / and b for the Sun’s aberration as seen from Mars:

l’ —— l — 0°.00697/r

b’ —— b — 0°.000 225 cos (/ — N)

r

8. sin DS = — sin J O sin b' — cos TO Cos b' cos (@ — I’ )

9. If JDE is the Julian Ephemeris Day corresponding to the given time, calculate the angle iY, in degrees, from

W = 11.504 + 350.892 000 25 (JDE — — 2433 282,5)

where z is the light-time, in days, found in steps 3 and 4.

10. Calculate the mean obliquity of the ecliptic by means of formula (22.2). Then use expressions (13.3) and (13.4) to find the pole’s equatorial coordinates ay and fi0 from the ecliptical coordinates and 90 .

 

11. Calculate

u = y cos eg — z sin eg

' V Ț SİP1 *o    +   Ę COS  ftp

and the angles o, ò, J from

J   _ sin ó0 cos ó COS (o'g — a')    — S1E å COS åg 

cos fi sin (nd — n)

Note that ó is between —90º and + 90º. But a and } can take all values from 0° to 360º, and hence they should be taken in the proper quadrant.

12. Find ui = IV — J, where is expressed in degrees.

13. Calculate the mutations in longitude (AQ) and in obliquity (Ae) as explained in Chapter 22. Only the most important terms may be used here; an accuracy of, say, 0“.01 is not necessary.

14. Correct h and Q for the aberration of Mars:

correction to k : +0.°005 693 cos (fø — X)

cos ß

correction to 9 : +0.°005 693 sin (/ø — X)  sin fi

15. Add AJ to   and to X. Add Ae to e0 to obtain the true obliquity of the ecliptic e.

16. Transform (@, 90) and (k, 9) to the equatorial coordinates (of , ó ) and (n', fi’) by means of the expressions (13.3) and (13.4), using for e the true obliquity obtained above.

17. The position angle P is given by

 

  COS  Õtj    Sİfl (O £f )

sin ój cos ó' — cos ôj sin ó’ cos (aj — n')

 

(42. 4)

 

18. The position angle  of the mid-point of the illuminated limb can be obtained as for the Moon — see Chapter 48. Then the position angle Q of the greatest defect of illumination is + 180º.

19. Mars’ apparent diameter d is given by d = 9“.36/ ó, If k is the illuminated fraction of the planet’s disk (see Chapter 41), then the greatest defect of illumination is Q  = (1 — k) d.

 

 

2fxample 42.a -- Calculate the quantities concerning the appearance of Mars on 1992 November 9, at 0 UT.

The instant corresponds to JD 2448 935.5. For the difference between Dynamical Time and Universal Time, we use the value IT --- + 59 seconds, or + 0.000 683 day, so that the given instant corresponds to

1992 November 9.000 683 TD = JDE 2448 935.500 683.

Step 1. T ---- --0.071 444 1976, @ = 352°. 82267, Be +63°.28208

Step 2. From an accurate ephemeris, calculated by using the complete VSOP87 theory, we deduce

ft = 46°50' 37“.90 = 4ñ°.843 861

b0 = --0“.60 = --0.°000 167

It = 0.990 413 01

Step 3. Geometric heliocentric coordinates of Mars, referred to the ecliptic and mean equinox of the date, taken from an accurate ephemeris:

TD l b r

1992 Nov. 8.0 77°57’48“.45 +0°52’54”.74 1.5403797

9.0 78 28 24.28 +0 53 46.72 1.5416585

10.0 78 58 57.09 +0 54 38.36 1.5429347

We use A = 0 (hence z = 0) as a starting value. For 1992 November

9.000683 TD we find, by interpolation,

I -- 78°.473759, b ---- + 0°.896 321, r --- 1.5416594 AU.

Step 4. z = --0.369 4199

y = +0.787 8856 A = 0.870 5266

z = +0.024 1192

Step 3. With this value of A we obtain for the light-time the value z = 0.005 028 day. Hence, r -- z is

1992 November 9.000683 -- 0.005 028 = November 8.995 655 TD.

For this instant we find, by interpolation of the tabulated values,

l -- 78.°471 197, b ---- +0.°896 249, r 1.5416529.

Step 4. z = 0.369 3536

y = +0.787 8654 A = 0.8704801

z = +0.024 1172

This new value of A yields for the light-time a value which differs by only

0.02 second from the preceding value, so no new iteration is needed. Step 5. X = 115°.117 321, b = + l.°587 619

Step 6. DE = + 12.°44

 

Step 7.

Step 8.

Step 9.

Step 10.

Step 11.

Step 12.

Step 13.

Step 14.

Step 15.

Step 16.

Step 17.

Step 18.

Step 19.

 

N —- 49°.5029, l' —— 78°. 466 676, b' —— + 0°.896 121

DC —— —2°.76

IY = 5492 522°.4593 = 2°.4593

e0 = 23° 26'24“.793 = 23°.440 220

ct0 = 317°.632606

o ' +52°.860916

ti = +0.713 2537 n = 117.°377075

v = +0.3355335 6 = +22°.672176

} = 250°.9052

= —248°.45 = 111.°55

AQ = + 15“.42 Ae = — l “.00

corrected k = 115°. 119 429

corrected 9 = + 1°.587 472

corrected @ = 352°.826 95 e = 23.°439 942

corrected X = 115°. 123712

n   = 317°.63529 n' = 117.°38380

o ' +52°.86236 6' ' + 22.°67062

P -- 347°.64

The right ascension and declination of the Sun can be obtained with sufficient accuracy from (25.6) and (25.7), with O = ft + 180°. We find 224°.378 and — 16°. 869.

The equatorial coordinates of Mars being n and 6, we find by means of formula (48.5) x = 99°.91, whence Q = 279°.91.

Using the values of R, r, and A found in Steps 2 to 4, formula (41.2) yields k —— 0.9012. The greatest defect of illumination is

q = (1 — k) x 9“.36/A = 1“.06.

Mars’ apparent diameter is 9“.36/A = 10’.75.

 

 

REFER ENCES

1. 3fonthly Notices of the Royal Astron. Soc. , Vol. 66, page 56 (1905). Cited in [2).

2. Explanatory Supplement to the Astronomical Ephemeris (London, 1961), page 334. 3. Icarus, Vol. 3, page 243 (1964).

4. M. E. Davies e.a. , “Report of the IAU Working Group on Cartographic Coordinates and Rotational Elements of the Planets and Satellites: 1982”, Celestial Mechanics, Vol. 29, pages 309—321 (1983).

 

Chapter 43

Ephemeris for Physical Observations of J upiter

For Jupiter three rotational systems have been adopted. 5ystem I applies to features within about 10° of the planet’s equator; it has an adopted sidereal rotation rate of exactly 877.90 degrees in 24 hours of mean solar time. System II, for use in higher latitudes, where the cloud features take about five minutes longer to circle the planet than those at the equator, rotates exactly 870.27 degrees per day. It follows that the planet’s sidereal rotation period is 9‘50‘30'. 003 in System I, and 9'55"40‘. 632 in System H.

System III, rooted deep in Jupiter’s interior, applies to radio emissions of the planet. But in this Chapter we will consider only Systems I and II, which are of interest to the visual observers.

As for Mars (see Chapter 42), DE and DS will denote the planetocentric declinations of the Earth and the Sun, respectively, and P the posltion angle of Jupiter’s northern rotation pole. The longitude of the Central Meridian will be denoted m l for System I, and o 2 for System II.

Because Jupiter’s rotation axis is almost exactly perpendicular to the planet’s orbital plane around the Sun, it is not needed to correct and b for the Sun’s

aberration in the calculation of DS. The error in DS made by neglecting this aberration will never exceed 0“.5.

For a given  instant  t,  the values of  Dg , DSN  ,  ni l ,  o¿,  and  P can  be obtained  as follows.

1. Calculate

d —- JDE — 2433 282.5

T —— d

36525

and then the right ascension n0 and declination Q of the north pole of Jupiter, referred to the mean equinox of the date, by the following expressions:

«0 = 268.°00 + 0°.1061 T

o ' + d•.°50 0°.0164 *i

2 93

 

2. Calculate the angles lY and W2 from

W  = 17°.710 + 877º.900 035 39 d

W2 = 16.°838 + 870.°270 035 39 d

These can be large (positive or negative) angles; they should be reduced to less than 360 degrees. The angles lV1 and lY2 are related to the longitude Systems I and II, respectively. The constant terms 17.°710 and 16.°838 have been chosen in order to maintain consistency with the Jovian longitude systems established at the end of the 19th century. The other two constants are equal to the values 877°.90 and 870.°27 mentioned at the beginning of this Chapter, increased by 0°.000 035 39, the daily variation of the arc of the Jovian equator from its ascending node on the celestial equator to its ascending node on the orbit.

3. Calculate the heliocentric longitude 1 , latitude I and radius vector A of the Earth, referred to the ecliptic and mean equinox of the date, for instance by using the relevant data of Appendix III and the precepts given in Chapter 32.

4. For the same instant, calculate the corresponding heliocentric coordinates 1, b, r of Jupiter. Do nor take the light-time into account here.

5. Calculate x, y, z by means of formulae (42.2), and then Jupiter’s distance A by (42.3).

6. Correct Jupiter’s heliocentric longitude / (in degrees) for the light-time: correction to l —— —0.°012 990 A J r 2

(The correction to the heliocentric latitude can be neglected here.)

7. Using the corrected value of 1, calculate x, y, z, A again, as in Step S.

8. Calculate the mean obliquity of the ecliptic ez by means of formula (22.2).

9. Calculate nS and éS from

p - COS £ 0 SÎf1 1 — sin c tan b 

cos

sin f› = cos * sin b + sin eq cos fi sin / The angle « S Should be taken in the proper quadrant.

10. sin Dz —— — sin ô sin 6S — cos ô0 cos ô cos (œq — o ) The extreme values of DE are + 3°.12 and —3.°12.

11. Calculate ii, v, ri, h, and } as for Mars (see Step 11 of Chapter 42).

12. sin DE = — Sin fig sin h — cos 60 Cos 6 cos (n0 — o) The extreme values of DE are +3.°4 and —3.°4.

 

43. PHYSICAL EPHEMERIS OF JUPITER 295

13. If } is expressed in degrees, and A in astronomical units, then t = IY, — J —5°.07033 A

= W2 — } —5°.02626 A

The last term in each formula is the amount of rotation during the light-time.

14. The values obtained for ot and ot should be reduced to the interval 0° —360° by adding or subtracting a convenient multiple of 360 degrees. The results refer to the geometric (the “true”) disk of Jupiter. The planet actually has a very small phase, and the longitudes of the “central meridian” of the illuminated disk

can be obtained by adding to o I and to ot the correction for phase C which is equal to

C -— + 57°.2958 x 2 rA + R 2 — r 2 — A*

4 rA

and has the same sign as sin (/ — /0). The angle C is always small, never exceeding 0°.61.

15. If an accuracy of 0.1 degree is sufficient for the position angle P, go to Step 18. Otherwise, calculate the nutations in longitude (AJ) and in obliquity (Ae), as explained in Chapter 22. Only the most important terms may be used; an accuracy of 0.01 arcsecond is not needed. Add Ae to e0 tO obtain e.

16. Correct n and 6 for Jupiter’s aberration : correction to n :

+0.°005 693    COS     cos ft cos e + sin o sin If  

cos 6

correction to 6:

+0.°005 693 [ cos ft cos e (tan e cos é — sin a sin â) + cos a sin 6 sin ]

17. Correct n, fi, ct0, and h0 for the nutation, by means of expressions (23. l), giving n’, fi’, nd, and éj.

18. Obtain P by means of formula (42.4).

Example 43.a — Calculate the quantities concerning the appearance of Jupiter on

1992 December 16, at 0' UT.

This instant corresponds to JD 2448 972.5. For the difference between Dynamical Time and Universal Time, we shall use the value b T -— +59 seconds = +0.000 68 day, so that the given instant corresponds to 1992 December 16.00068 TD, or JDE 2448 972.50068.

 

Step 1.

Step 2.

Steps 3-4.

 

d ---- 15690.00068 = 268°.04558

TO    =    +0.429569 é0   =  +64°. 49296

(keeping extra decimals to minimize rounding errors)

IV  = 13774 269°.8622 = 309°.8622

lYt = 13654 554°.2851 = 114°. 2851

From accurate ephemerides, calculated by using the complete VSOP87 theory:

 

Step 5.

Step 6.

Step 7.

Step 8.

Step 9.

 

lb --- 84.°285 703 I --- 181°.882168

two 10°.000 197 h = + 1°.290464

R -- 0.984 123 16 r = 5.446 423 20

z = --5.540 0914

y = --1.158 0704 A = 5.661 1645

z = +0. 1226552

I ---- 181°.882168 -- 0.°002479 = 181.°879 689

x 5.5400991

y = -- 1.157 8350 A = 5.661 1239

z = +0. 122 6552

e0 = 23° 26'24“.745 = 23.°440 2069

S = 182°.237 749

ét = +0.°436 472

 

Step 10. D$ 2°.20

Step 11. « = --1.1110767 n = 191.°340327

v = --0.3480441 â = --3.°524749

= 13°.5238

Step 12. DE = --2.°48

Step 13. t = 267°.63 2 72°.31

 

Step 14.

Step 15.

Step 16.

Step 17.

Step 18.

 

These are the longitudes of the Central Meridian of the geometric disk in Systems I and Il, respectively.

C -- +0°.43. Since sin (/ -- If) is positive, so is C.

The longitudes of the Central Meridian of the illuminated disk are: System I ’ o = 267°.63 + 0.°43 = 268°.06

System II : u›t = 72°.31 + 0.°43 = 72°.74

AQ = + 16“.86 Ae = -- 1“.79 e = 23°.439 710

correction to n : --0°.001627 a = 191.°338 700

correction to fi :  +0.°000 560 â = --3.°524 189

n' = 191°. 34305 nt = 268.°04594 h' '     --3°.5259a o '  +64°. 49339 P -- 24°.80

 

 

 

43. PHYSICAL EPHEMERlS OF JUPITER 297

Lower accuracy

The following, shorter method may be used when high accuracy is not needed.

For the given instant (Dynamical Time!), calculate the JDE (see Chapter 7), and then proceed as follows.

Number of days (and decimals of a day) since 2000 January 1, at 12' TD :

d -- JDE — 2451545.0

Argument for the long-period term in the motion of Jupiter:

V —— 172°.74 + 0°.001 115 88 d

Mean anomalies of Earth and Jupiter:

3f = 357.°529 + 0°.985 6003 d

N   -- 20.°020 + 0°.083 0853 d + 0.°329 sin Y

Difference between the mean heliocentric longitudes of Earth and Jupiter:

= 66.°115 + 0.°902 5179 d - 0°.329 sin V

The angles Y, 3f, N, and J are expressed in degrees and decimals. If necessary, reduce them to the interval 0—360 degrees; this depends on the computing language.

Equations of the center of Earth and Jupiter, in degrees:

A -- 1.915 sin M + 0.020 sin 2ñf

B —— 5.555 sin N + 0.168 sin 2N

and then

K —— I +  A - B

Radius vector of the Earth:

fi = 1.00014 — 0.01671 cos 3f — 0.00014 cos 2ñf

Radius vector of Jupiter:

r = 5.20872 — 0.25208 cos N - 0.0061 l cos 2N

Distance Earth —Jupiter:

A  = r 2 +  R 2 2 rR cos

The distances R, r, and A are expressed in astronomical units, and A should of course be taken positive. The phase angle of Jupiter (that is, the angle Earth —

Jupiter—Sun) is then given by

sin = R sin K

 

The angle always lies between — 12° and + 12°. Because R and A are always

positive, the angle / has the same sign as sin K.

The longitudes of the Central Meridian in Systems I and II are then, respectively,

o,    =   210°.98  +  877.°816 9088   d  —   173)   + —  B

t»2 = 187.°23 + 870°.186 9088 d — 7 ) + —  B

where — A/173 is the correction for the light-time in days. The denominator 173 results from the fact that the light-time for unit distance is 1/ 173 day.

The values obtained for ot and o 2 should be reduced to the interval 0° —360°, by adding or subtracting a convenient multiple of 360 degrees. The results refer to the geometric disk of Jupiter. The longitudes of the “central meridian" of the illuminated disk can be obtained by adding to u›t and ot the correction for phase which is equal to

57°.3 sin2

and the sign is opposite the sign of sin K.

Calculated in this way, o and o 2 can be up to 0.1 or 0.2 degree in error.

Find Jupiter’s heliocentric longitude X referred to the equinox of 2000.0 by the formula

h = 34°.35 + 0°.083 091 d + 0.°329 sin F + B

Then we have, in degrees and decimals,

DS —— 3.12 sin (h + 42.°8)

D E = U  — 2.22 sin cos (X + 22°) — 1.30 r — A sin (X — 100°.5)

In these expressions, 3.° 12 is the inclination of the equator of Jupiter on the orbital plane, 2.°22 its inclination on the ecliptic, and 1°. 30 the inclination of the orbital plane on the ecliptic.

 

43. PHYSICAL EPHEMERIS OF JUPITER

Exompfe 43.Jr — Let us take the same instant as in Example 43.a, 1992 December 16, 0h UT

= JD 2448 972.5

= JDE 2448 972.50068.

We find successively

 

299

 

d = —2572.49932

v = 169.° 87

M = —2177°.927 = + 342°.073

 

N

/ A B K R

b

sin

 

d 173

 

=  — 193°.659

= —2255°.670 = + 264°. 330

= —0°.601

-— + 1°.235

= 262°.494

= 0.98413

= 5.44824

= 5.66151

= —0.17234

= —9.°924

= —2572.53205

 

From this we deduce, for the geometric disk of Jupiter:

t = —2258 012°.31 = 267°.69

u›t =  —2238 407°.64 =  72°. 36

The correct values are 267°.63 and 72°.31 (see Step 13 of Example 43.a).

For the correction for phase we find +0.°43, exactly as in Example 43.a, Step 14.

h = — 178°.11

Ds 2.° 194

DE = —2.°194 — 0.°350 + 0.°048 = —2°.50

 

 

 

Chapter 44

Positions of the Satellites of Jupiter

This Chapter gives two methods to calculate, for any given instant, the positions of the four great satellites of Jupiter with respect to the planet, as seen from the Earth. These apparent rectangular coordinates X and Y of the satellites will be measured from the center of the disk of Jupiter, in units of the planet’s equatorial radius.

X is measured positively to the west of Jupiter, negatively

 

East

 

North

 

Jupiter

 

satellite

West

 

to the east, the X-axis coin- ciding with the equator of the planet. F is positive to the north, negative to the south, the r-axis coinciding with the planet’s rotation axis — see the drawing.

The accuracy of the first method (“low accuracy”) is sufficient for identifying the

 

' satellites at the telescope, or

for drawing a wavy-line diagram showing their posi- tions with respect to Jupiter,

as given in several astronomical almanacs and magazines. The high-accuracy method is needed, for instance, to calculate the classical phenomena of the satellites (eclipses, transits, etc.) and their mutual phenomena.

Low accuracy

First, convert the date and the instant (TD) to the Julian Day, using the method described in Chapter 7. Then, obtain the following quantities as explained in Chapter 43 (“lower accuracy”) : d, Y, M, N, I, A, B, K, R, r, ft , , and the planetocentric declination D C of the Earth.

301

 

For each of the four satellites, we now calculate an angle ti which is measured from the inferior conjunction with Jupiter, so that u = 0° corresponds to the satellite’s inferior conjunction, It = 90° to its greatest western elongation, u = 180° to the superior conjunction, and u —— 270° to the greatest eastern elongation.

it = 163°.8069 + 203.°405 8646   d 173   + — B

o2 — 358.°4140 + 101.°291 6335   d - 17 + — B

 

uz —— 5.°7l76 +  50°.234 5180  d —

 

3 + — B

17

 

4 ' 224.° 8092 +  21.°487 9800  d 173  + — B

If necessary, these angles ti should be reduced to the interval 0° —360°. In order to obtain more accurate values, the results can be improved as follows. Calculate the angles G and H by means of the formulae

G = 331°.18 + 50°.310 482  d — 173

H = 87°.45 + 21°.569 231  d 173

Then we have the following corrections, in degrees :

correction to u l . +0.473 sin 2 (u — u2 ) correction to u, : + 1.065 sin 2 (u — o ) correction to : +0.165 sin G correction to u4 : +0.843 sin H

The first correction is due to a periodic perturbation of satellite I by satellite II. The second correction is a perturbation of II by III. The two last corrections are due to the eccentricities of the orbits of satellites III and IV. (The orbits of I and II are almost circular.)

Note that here we take into account only the largest periodic terms in the motions of the satellites. There are many other (but smaller) periodic terms. For instance, satellite I is perturbed by satellite III too, satellite III by II and by IV, etc. See further the “high accuracy” method in this Chapter.

 

The distances of the satellites to the center of Jupiter, in units of Jupiter’s

equatorial radius, are given by

r l  = 5.9057 — 0.0244 cos 2 (e l — o ) r   —- 9.3966 — 0.0882 cos 2 (u 2 — o,) r3 = 14.9883 — 0.0216 cos G

r4 = 26.3627 — 0.1939 cos H

where the iincorrected values of u , etc, must be used. In these expressions, the periodic terms are again due to mutual perturbations of the satellites or to their orbital eccentricities.

The apparent rectangular coordinates X and Y of the satellites are then given by

— r d sin u and - — r cos o t sin DE with similar expressions for the other three satellites.

Example 44.a — Calculate the configuration of the satellites of Jupiter for 1992 December 16 at 0' UT = JD 2448972.5 = JDE 2448 972.50068.

(The value b T —— +59 seconds is used.) For this instant we have found, in Example 43.b,

 

d —— —2572.49932

B —— + 1°. 235

 

d — 173

 

= —2572.53205

 

= —9°.924

 

DE = —2.°50

 

By means of the formulae given in the present Chapter, we then find

 

tit = —523 115°.458 = 324°.542

= —260 228°.719 =  51°.281

ii = —129235°.349 = 4°.651

ti4 = — 55064°.867 = 15°.133

correction to ti; . —0.°054 correction to ti2 . + 1.°063 correction to u3 . +0°.093 correction to ti4 : +0°.541

 

2 (ut — it2) = 546°.52 = 186°.52

2 (it2 — it ) = 93°.26

G —— — 129 094°.15 = 145.° 85 If = — 55 400.°09 = 39.°9l

correctedm l  = 324°. 488

corrected = 52°.344

corrected u —— 4.°744 corrected u, =   15°.674

 

r —— 5.9057 + 0.0242 = 5.9299 X = —3.44 Y —— +0.21

r2 =  9.3966 + 0.0050 = 9.4016 Xt = +7.44 F = +0.25

r —— 14.9883 + 0.0179 = 15.0062 Xj = + 1.24 Y —— +0.65

rz -— 26.3627 — 0.1487 = 26.2140 X4 = +7.08 F = + 1.10

(It is just a coincidence that all four F-values are positive!)

 

With these values of X and F we can draw the following figure which shows the configuration of the satellites at the given time. In this drawing South is up, and West to the left, as in the field of an inverting telescope for an observer in the northern hemisphere.

Jupiter

The X- and r-values resulting from an accurate calculation are mentioned in Example 44.b. The discrepancies between the r-values are mainly due to the fact that in this simplified method the inclinations of the orbits of the satellites on the equatorial plane of Jupiter have been neglected. Actually, the four satellites can reach extreme latitudes of 0°03’, 0°3l ', 0°20’, and 0°44’, respectively, with respect to the equatorial plane of the planet. As a consequence, mutual occultations cannot be calculated with certainty by means of the simplified method described above. In the case of a very close conjunction, it is even not possible to deduce which of the two satellites passes to the north of the other.

High accuracy

The following method is based on the theory “E5” of the satellites due to Lieske [1].

For the given instant, calculate the following quantities (see Chapter 25) :

O = geocentric geometric longitude of the Sun, Q = geocentric geometric latitude of the San,

A = radius vector of the Sun in astronomical units.

Let z be the light-time from Jupiter to the Earth. Because the distance of Jupiter to the Earth is not known in advance, so T is not known. The distance A should be found by iteration. A good starting value is A = 5, since the extreme values of Jupiter’s distance to the Earth are 3.95 and 6.5 astronomical units. The light-time is given by (33.3); a better value for A will be provided by formula (44.2).

Calculate the following values for the given time decreased by the light-time z (see Chapter 32) :

 

= heliocentric longitude of Jupiter,

b —-  heliocentric latitude of Jupiter,

r -— radius vector of Jupiter, in AU.

In the above, the longitudes and latitudes are referred to the ecliptic and mean equinox of the date.

Calculate the rectangular geocentric ecliptical coordinates of Jupiter z = r cos b cos f -l- R cos O

 

y = r cos b sin l -l- R sin O z = r sin b + fi sin Q

and its distance to the Earth

A = x   2 .1- y + z

Calculate Jupiter’s geocentric longitude kg and latitude 9 by @ = ATN2 (y, z) and d = ATN

 

(44.1)

(44. 2)

 

where, as mentioned earlier in this book, ATN2 is the “second” arctangent function. In other words, @ is equal to ATN (y/z) taken in the proper quadrant.

Let r be the time measured in ephemeris days from 1976 August 10 at 0' TD

= JDE 2443000.5, decreased by the light-time z. In other words, if JDE is the Julian Ephemeris Day corresponding to the given instant,

t -— JDE — 2443000.5 — I

In the following expressions, all numerical values are expressed in degrees and decimals. The longitudes are referred to the standard equinox of 1950.0.

Mean longitudes of the satellites:

t = 106.07719 + 203.488 955 790 i

£2 = 175.73161 + 101.374 724 735 f

£  = 120.55883 +  50.317 609 207 f

t 4 = 84.44459 +    21.571 071177 r

Longitudes of  the perijoves (*) :

zt   = 97.0881 + 0.161385 86i

z2 = 154.8663 + 0.047 263 07 r

r = 188.1840 + 0.007 127 34 f

z4 = 335.2868 + 0.001 840 00 r

(*) The term “periapse”, used by some authors, is incorrect — see page 411.

 

Longitudes of the nodes on the equatorial plane of Jupiter:

, = 312.3346 — 0.132 793 86 i

2 100.4411 — 0.032 630 64 i

= 119.1942 — 0.007 177 03 i

4 ' 322.6186 — 0.001 759 34 i

Principal inequality in the longitude of Jupiter:

P = 0.33033 sin (163.°679 + 0°.001 0512 r)

+ 0.03439 sin (34.°486 — 0.°0l6 1731 r)

There is a small libration, with a period of 2071 days, in the longitudes of the three inner satellites: when satellite II decelerates, I and III accelerate. To take this into account, we need the “phase of free libration”

4 h = 199.6766 + 0.173 791 90 r

Longitude of the node of the equator of Jupiter on the ecliptic:

= 316.5182 — 0.000 002 08 i

Mean anomalies of Jupiter and Saturn:

G = 30.23756 + 0.083 092 5701 r + F

G’ —— 31.97853 + 0.033 459 7339 r

Longitude of the perihelion of Jupiter:

H = 13.469 942 (considered as a constant in the E5 theory)

Periodic terms in the longitudes of the satellites

Satellite   I

+0.°47259 sin 2 (t — £ ) —0.00186 sin G

—0.03478 sin (r 3 —r 4) +0.00162 sin (o — z )

+0.01 081 sin (t, — 2 i, +  ,) +0.00158 sin  4 (r, — t,)

+0.00738 sin 4 h —0.00155 sin (t i — 3)

+0.00713 sin (£2 — 2 t, + z,) —0.00138 sin (J + o — 211 — 2 G)

—0.00674 sin (r l + z,  — 2R — 2G) —0.00115 sin 2 (f — 2 f  + o2)

+0.00666 sin (t z — 2 £3 +F  ) +0.00089 sin (r — z 4)

+0.00445 sin ( f I — 3) +0.00085 sin (f + z — 2 R — 2G)

—0.00354  sin  (  —t ) +0.00083 sin  (c    — u› )

—0.00317 sin (2 — 2H) +0.00053 sin (J — ui2)

 

+0.00265  sin  (A—1

 

•4)

 

Call El the sum of these terms.

 

Sate llite II

+ 1.°06 476 sin 2 (t 2 — £,) —0.00 115 sin (£ — 2 £, + z )

+0.04 256 sin (t z“   *at 0.00094 sin 2 ( z *z)

+0.03581 sin (32 — г З) +0.00086 sin 2 (£ — 232 + mt)

+0.02395 sin (ё — 2 t, + г,) —0.00086 sin (5 G' — 2 G + 52.°225)

+0.01984 sin ( 2 — •4) —0.00078 siП ( 2 — 4)

—0.01778 sin 4, —0.00064 sin (3£ — 7*4 “ 4 z4)

+0.01654 sin (£ —r 2) +0.00064 sin (г — z 4 )

+0.01334  siп ( f 2 — 2 £, +r   2) —0.00063  sin  (£t    — 2 £, +  r4)

+0.01 294 sin (г, —r 4 ) +0.00058 sin (m3 — o 4 )

—0.01142 siп (£ — £,) +0.00056 sin 2 (J — II — G)

—0.01057 siп G +0.00056 sin 2 ( 2 — 4)

—0.00775 siп 2 (/ — Н) +0.00055 sin 2 (£ — f )

+0.00524  sin 2 (31  —  t 2) + 0.00052 sin  (3 £, — 7  34  +  z   +  3r   4)

—0.00460 sin ( — ›) —0.00043 sin (£t — z,)

+0.00 316 sin (J — 2G + о З — 2H) +0.00041 sin 5 (f 2 — f,)

—0.00203 sin (r l + г, — 2H — 2G) +0.00041 sin (г, — Н)

+0.00 146 sin (/ *э) + 0.00032 sin (о — о,)

—0.00145 sin 2G + 0.00032 sin 2 (f, — G — Н)

+0.00 125 sin (/— •4)

Call E2 the sum of these terms.

Satellite  III

+0°.16490 sin (33 — z,) +0.00091 sin (о, — 4)

+ 0.09081 sin  (3—3 4) +0.00080  sin  (3 £ 7 £d +  *з +  3Т#)

—0.06907 sin (32 — t ) —0.00075 sin (232 — 3 £ + z 3)

+ 0.03784 sin (г,—   •4) +0.00072 sin (г; + г — 2H — 2G)

+0.01 846 sin 2 (£— 4) +0.00069 sin (г, — Н)

—0.01340 sin G —0.00058 sin (2 t — 3 ft + г,)

—0.01 014 sin 2 (J — Н) —0.00057 sin (f— 4 + •‹)

+0.00704 sin (£t — 2 t, + • ) +0.00056 sin (f + г — 2 R — 2G)

—0.00620 sin ( z 2 d + *z) 0.00052 sin (f 2 — 2 f, + г )

—0.00541 sin (t — 4) —0.00050 sin (г — г,)

+0.Ф381 sin (£2 — 2 f3 + r4) +0.Ф048 sin (f, — 2 f4 + rj)

+0.00235 siп (J *з) 0.00045 sin (2 Ë2 — 3 f + z )

+0.00 198 sin (J — u›4) —0.00041 sin (z — rq)

+0.00 176 sin 4 —0.00038 sin 2G

+ 0.00 130 sin 3 (t, — 34) —0.00037 sin (г — rq + о — 4)

+0.00 125 sin (£ 1 — i,) —0.00032 sin  (3 f, — 7 f 4 + 2z + 2 z4)

—0.00119 sin {S G’ - 2G + 52.°225) +0.00 030 sin 4 (f — f,)

+0.00 109 sin (d l — f 2) +0.00029 sin (f + z, — 2H — 2 G)

—0.00100 sin  (3 t,  — 7 ll  + 4r   4) —0.00 028 sin  (с,   + — 2ТІ  — 2G)

 

+0.00026 sin (í, — n — c) —o.oo021 sin (i — • )

+0.00024 sin (t 2 — 3 £, + 2 £4) +0.00017 sin 2 (t, — r,)

+0.00021 sin 2 (t — H — G)

Call E3 the sum of these terms.

+0.00 178 sin  (J  +  o4 — 2 z  ) +0.00028 sin  (*—4 •d   +   2    — 2H)

+0.00 134 sin (z4 — H) —0.00028 SIR 2 (#4 Eq)

+0.00 125 sin 2 (£4 — G — H) —0.00 027 S1f1 (z — z 4 + o — e )

—0.00117 sin 2G —0.00 026 sin (5 G’ — 3G + 188°.37)

—0.00112 sin  2 (£, —  £4) + 0.00025 sin  (o—• 3)

+0.00 107 sin  (3 £3  — 7 £4 +  4r   4) —0.00 025 sin  (i,   — 3i3  +  2 i 4 )

+ 0.00 102 sin (i4 — G — H) —0.00 023 sin 3 (I — £ )

+0.00096 sin (2 £ — / — o4) +0.00 021 sin (2 i — 2H — 3G)

+0.00087 sin 2 (J— 4) —0.00021 sin (2 t — 3 I + z4)

—0.00085 sin (3 £ — 7 £4 + r, + 3z4) + 0.00019 sin (1 — z 4 — G)

+ 0.00085 sin (t, — 2 *4 + 4) —0.00 019 sin (2 I, — z  — z,)

—0.00081  sin  2 (*—4 /) —0.00 018 sin  (I 4 + G)

+0.00 071 sin (£4 + •4— * H — 3G) —0.00 016 sin (I, + zj — 211 — 2G)

Call £4 the sum of these terms.

The true longitudes of the satellites are

L 4 — £4 + E4

 

Periodic terms in the latitudes of the satellites

The sum of the following terms gives the tangent of the satellite’s latitude B, with respect to Jupiter’s equatorial plane.

 

Satellite I

Saie llite II

Satellite   III

Satellite   IV

 

+0.000 6393 sin

+0.000 1825 sin

+0.0000329 sin

—0.0000311 sin

+0.0000093 sin

+0.0000075 sin (3L — 412 — 1.9927 S 1 + )

+0.000 0046 sin (L i + / 211 — 2 G)

+ 0.008 1004 sirl (Lz *z)

+ 0.000 4512 sin (L2 — u›,)

—0.000 3284 sin (L — J)

+ 0.000 1160 sin (L2 — x 4)

+ 0.000 0272  sin (Bl — 2 £, + 1.0146 £2 z 2)

—0.000 0144 sin (L2 — m l )

+ 0.000 0143 sin (L2 + — 2H — 2 G)

+0.000 0035 sin (L2 — + G)

—0.0000028 sin (£ — 2 t, + 1.0146 E2 + z )

+ 0.0032402 sin

—0.001 6911 sin

+0.000 6847 sin

—0.000 2797 sin

+0.0000321 sin

+0.000 0051 sin (L3 — + G)

—0.0000045 sin

—0.0000045 sin tL + - 2H)

+0.0000037 sin {L + — 2H — 3G)

+0.000 0030 sin

—0.000 0021 sin

—0.007 6579 sin (L, — /)

+0.004 4134 sin (L4 -0 4)

—0.000 5112 sin (L4 — ‹» )

+0.0000773 sin (L4 + Q- 2M - 2G)

+0.0000104 sin  

—0.0000102 sin (L4 — — G)

+0.000 0088 sin  (L4 + — 211 — 3G)

—0.000 0038 sin (L4 + — 2H — G)

 

Periodic terms for the radius vector

 

Saiellite I

Satellite  II

Satellite   III

Satellite   IV

 

—0.004 1339 cos 2 (£ — £ )

—0.0000387 cos (£, — z3)

—0.000 0214 cos (£  — z 4)

+0.0000l70 cos (s, — i,)

—0.000 0131 cos 4 (£, — é2)

+0.0000106 cos (f/ — 43 )

—0.0000066 cos (f/ -I- x/ - 211 — 2G)

+ 0.009 3848 cos (i — i 2)

—0.000 3116 cos (i — r,)

—0.000 1744 cos (é 2 — z 4)

—0.000 1442 cos (é2 — z 2 )

+0.000 0553 cos (£2 — £ )

+ 0.000 0523 cos (£, — i )

—0.000 0290 cos 2 (t, — t 2)

+ o.ooo 0is4 cos 2 (i, — ,)

+0.000 0107 cos (E l — 2 t 3 + z,)

—0.000 0102 cos (£2 — z )

—o.ooo oo9i cos 2 (r, — i,)

—0.001 4388 cos (IQ — K3)

—0.000 7919 cos (£ — z,)

+0.000 6342 cos (f 2 — f/)

—0.000 1761 cos 2 (f/ — fi)

+0.000 0294 cos ( f 3 — f 4)

—0.0000156 cos 3 (f, — f )

+0.0000156 cos (f — f,)

—0.0000153 cos (f, — f 2)

+0.0000070 cos (2 f 2 — 3£, + r,)

—0.0000051 cos (f, + — 2II — 2G)

—0.007 3546 cos (f 4 — r,)

-t-0.000 1621 cos (f —r )

-1-0.0000974 cos (f/ — f 4)

—0.000 0543 cos (é4 + r, — 211 — 2G)

—0.0000271 cos 2 (é

+0.0000182 cos (é4 — n)

+0.0000177 cos 2 (£ — £ )

—0.0000167 cos (214 — — ,)

+0.0000167 cos (J — ,)

—0.0000155 cos 2 (£q — H — G)

+0.0000142 cos 2 (£4 — J)

 

Satellite IV (cont.)

 

+o.ooooio5 (i, — i,)

+0.000 0092 cos (tt — ft)

 

—0.0000089  cos (  —4 — G)

—0.0000062 cos (th +r 4 — 211 — 3 G)

+ 0.0000048 cos 2 (£, — o )

The radius vector A; of satellite No. i, in equatorial radii of Jupiter, is given by

R, —— a, X (1 + sum of periodic terms) with the following values for the mean distances:

satellite I ol   = 5.90569

satellite II o2 = 9.39657

satellite III a, = 14.98832

satellite IV at = 26.36273

If JDE is the Julian Ephemeris Day corresponding to the given instant, calculate T JDE — 2433 282.423  

36525

Then the precession in longitude from the epoch B1950.0 to the date, in degrees, is given by

P —— 1.396 6626 TO + 0.000 3088 T

Add P to the four longitudes L, and to J.

Inclination of Jupiter’s axis of rotation on the orbital plane:

= 3.° 120 262 + 0.°0006 T

where T is the time in centuries since 1900.0.

For each of the four (i = 1 to 4) satellites, we have found the tropical longitude Li , the equatorial latitude B, , and the radius vector fi, in equatorial Jupiter radii. For each of them, calculate

X; = R, cos (L, — /) cos B,

Y; —— R; sin (L, — J) cos B, Z, —— R, sin B;

Now consider a “fifth, fictitious satellite”, situated at unit distance from the center of Jupiter, above the planet’s north pole:

Xt = 0, Y$ -- 0, Z5 = 1.

This fictitious satellite will be needed later.

 

To obtain the apparent rectangular coordinates of the satellites as they appear on the celestial sphere, as defined at the beginning of this Chapter, several rotations must be performed. So, calculate the following for all five satellites (the four real ones and the fifth, fictitious satellite) :

Rotation towards Jupiter’s orbital plane:

A = X

B   = r cos I — Z sin I

Cl   = Y sin i + Z cos f

Rotation towards the ascending node of the orbit of Jupiter:

A —— AI cos & — B1 sin 4• Bd ——  A1 sin 4 + B cos 4 c —— c

where 4 = / — II, fi being the longitude of the node of Jupiter, referred to the mean equinox of the date. See in Table 31.A, under “Jupiter”, the formula for 0.

Rotation towards the plane of the ecliptic:

' A y A z

B —— B 2 cos i — C 2 sin i

C —— B 2 sin i + C2 cos i

where i is the inclination of the orbit of Jupiter on the ecliptic. See in Table 31. A the expression for i.

Rotation towards the vernal equinox:

A4 = A cos fl — B sin fl Bd = A sin fl + B cos fl C4 = C

Then calculate

A —— A4 sin @ — B cos

B5 — A4 cos @ + B4 sin @

A6 = A

B6 = C5 Sth 9 0 + B5 cos No C6 = C5 cos 9 — Bs sirl No

If ( , 9 are the values of A6 and C6 for the “fifth satellite”, that is, ( = A6 (5), p = C6 (5), then calculate

D —— ATN2 (J, 9)

where, as mentioned earlier in this book, ATN2 is the “second” arctangent function which gives the angle D in the correct quadrant.

 

Calculate

 

X = Ap cos D — C6 sin D Y —- A6 sin D + C'6 cos D Z = B6

 

(44.3)

 

X and Y are the required rectangular coordinates of the satellite, as defined at the beginning of this Chapter. The quantity z is negative if the satellite is closer to the Earth than Jupiter, positive if it is more distant than Jupiter.

However, to obtain full accuracy, the apparent coordinates X and Y just obtained should be corrected for two effects:

J. d’JferentiaI light-time: if a satellite is on the nearer half of its orbit, its light- time is smaller than that of Jupiter; if on the far half, its light-time is larger. The correction to be added to X is

        i ‹x/°)

where

K —— 17295 for satellite I 21819 — II

27558 — III

36548 — IV

This correction is zero at the greatest elongations, and positive in all other cases. It is always very small, being at most 0.0003 for satellite I, and 0.0007 for satellite IV. The correction to Y is negligible. In the formula above, fi is the radius vector of the satellite, while X and Z are the values given by (44.3).

2. the Perspective effect, which is due to the fact that Jupiter is not situated at an infinite distance from the Earth. This is illustrated in the figure at the right, which shows the orbits of two satellites around Jupiter (not to scale!). Although the X-coordinates of satellites A and B are equal in space (distances AA’ and BB’ are equal), they are not exactly in

 

conjunction as seen from the Earth: their apparent X-coordinates are not equal. To correct for this perspective effect, the X and r values obtained thus far should be multiplied by the factor

A + Z/2095

where A is Jupiter’s distance to the Eanh in astronomical units as given by (44.2), while Z is in Jupiter radii (44.3). The constant 2095 is the number of equatorial radii of Jupiter in one astronomical unit.

Example 44.b — Same instant as in Example 44.a.

We shall not give the details of the calculation. Let us just mention the values of the sums

E1 = —0.°00654, E2 = + 1.° 10011, F•3 — +0.°04056, E4 = + 0.°59104,

and the final results:

Saiellite I Satellite ll Saiellite fi/ Satellite iY

X —3.4502 +7.4418 + 1.2011 + 7.0720

Y +0.2137 + 0.2753 +0.5900 + 1.0291

Mutual conjunctions — Two satellites are in conjunction when their X-coordinates are equal. The difference between the F-coordinates then corresponds to the separation of the satellites. Of course, if one satellite (or both) is eclipsed or occulted by Jupiter, the conjunction is inobservable.

Conjunctions with Jupiter — A satellite is in inferior conjunction with Jupiter when its X-coordinate is zero and changing from negative to positive; its Z-coordinate is then negative. Similarly, a satellite is in superior conjunction with Jupiter when its X-coordinate, passing from positive to negative, becomes zero. Its Z-coordinate is then positive.

Exercise. — On 1988 November 23, satellites III and IV were almost simultaneously in conjunction with Jupiter. Confirm this with your program. Take the value of A T from Table 10. A.

Answer: Satellite III was in inferior conjunction with Jupiter on 1988 November 23, at 7 28" UT ; at that instant, its Y-value was —0.8043, so the satellite was in transit over the planet’s disk.

Satellite IV was in superior conjunction that same day at 5h15". Its F-value was then + 1.3991. Since this is larger than the polar radius of Jupiter (0.933), the satellite was not occulted, but was visible above the planet’s northern polar regions.

 

Satellite phenomena — The X and Y coordinates are the basic data for the calculation of the satellite phenomena: occultations behind Jupiter, and transits across the planet’s disk. If the calculations are made for the center of the satellite, then an occultation or a transit begins or ends when the distance d of the satellite to the center of Jupiter’s disk, given by d2 = X° + r 2, is equal to the planet’s radius p at the point of contact. Due to Jupiter’s flattening, p varies between 1 (at the equator) and 0.933 (at the poles). One can avoid working with an elliptical disk by “stretching” the scale vertically: multiply the r-values by the factor 1.071374, leaving the X-values unchanged :

Y —— 1.071 374 F

Jupiter’s disk then becomes exactly circular, and the condition for the beginning or end of an occultation or of a transit becomes X 2 + Ft2 = 1.

In the case of an occultation, it remains to be checked whether the satellite is visible at the time of its immersion or emersion, because it could be eclipsed in the shadow of the planet.

Eclipses and shadow transits can be calculated in the same way, except that one should replace X and Y by the apparent coordinates @ and r0 As seen from the Sun. These coordinates are obtained by putting R —— 0 in expressions (44. 1). Moreover, the light-time z to the Eanh should be added to the true times of the eclipses or to those of the shadow transits, because we on Earth see these events later by the amount r. Finally, in the case of an eclipse it remains to be checked whether the disappearance or the reappearance is visible from Earth: indeed, the satellite could be occulted by Jupiter at that instant.

R E FE RE NCE

1. J. H. Lieske, Astronomy and Astrophysics, Supplement Series, to i. 129, pages 205—217 (1998).

 

 

 

Chapter 4S

The Ring of Saturn

In this Chapter, the following symbols will be used with respect to the ring of Saturn. (Of course, we know that Saturn has many rings. But they form one single, compact, planar system. We shall use the word ring, ’in the singular form, to denote the ring system.)

B    the Saturnicentric latitude of the Earth referred to the plane of the ring, positive towards the north; when B is positive, the visible surface of the ring is the northern one;

B’     the Saturnicentric latitude of the Sun referred to the plane of the ring, positive towards the north; when B' is positive, the illuminated surface of the ring is the northern one;

P      the geocentric position angle of the northern semiminor axis of the apparent ellipse of the ring, measured from the North towards the East (see the Figure). Because the ring is situated exactly in Saturn’s equator plane, P is also the position angle of the north pole of rotation of the planet;

a, b -- the major and the minor axes

of the outer edge of the outer ring, in arcseconds.

In the calculation of these quantities, the effect of light-time should be taken into account. Moreover, to obtain full accuracy, the aberration of the Sun as seen from Saturn must be taken into account in the calculation of B’ ; and in the calculation of P one should take into account the effect of the nutation and Saturn’s aberration.

317

 

G. Dourneau [1] gives the following values for the inclination of the plane of the ring and the longitude of the ascending node referred to the ecliptic and mean equinox of B1950.0:

i   --    28.°0817 + 0.°0035 II = 168.°81l2 + 0°.0089

From these values, we deduce the following expressions to calculate i and II referred to the ecliptic and mean equinox of the date:

 

i   --    28.°075 216 -- 0.°012 998 T + 0.°000 004 r 2

12 =   169‘.508 470 + 1°.394 681 T + 0.°000 412 r 2

 

(45. 1)

 

where T is the time from J2000.0 in Julian centuries, as given by formula (22. 1). In expressions (45. 1), we retained extra decimals in order to avoid loss in accuracy.

For a given instant i, the value of B, B' , etc. , can  be calculated as follows.

1. Calculate i and II by means of (45. 1).

2. Calculate  the heliocentric  longitude  /0 ,  latitude  b0 ,  and  radius  vector  R  of  the Earth, referred to the ecliptic and mean equinox of the date, FK5 system, for instance by using the relevant data of Appendix III and the precepts given in Chapter 32.

3. Calculate the corresponding coordinates 1, b, r for Saturn, but for the instant r -- z, where z is the light-time from Saturn to the Earth, as given by (33.3). Because Saturn’s distance A is not known in advance, it should be found by

iteration -- see Step 4. One may use A = 9 as a starting value, since Saturn’s

distance to the Earth is always between 8.0 and 11.1 astronomical units.

4. Calculate

x   ---- r cos b cos l -- R cos I, y ---- r cos b sin I -- R sin fq z =   r sin b -- R sin bz

Then Saturn’s distance A to the Earth is

5. Calculate the geocentric longitude k and latitude b of Saturn from

 

6. sin B —— sin i cos 9 sin (h — fi) — cos i sin $

a —— 375“.35 b —— a sin | B

Factors by which the axes o and b of the outer edge of the outer ring are to be multiplied to obtain the axes of

Inner edge of outer ring 0.8801

Outer edge of inner ring : 0.8599

Inner edge of inner ring 0.6650

Inner edge of dusky ring : 0.5486

7. Calculate the longitude N of the ascending node of Saturn’s orbit from

N —— 113.°6655 + 0°.8771 T

Then correct f and b for the Sun’s aberration as seen from Saturn:

l' —- l - 0.°01759 / r

b’ —— b — 0.°000 764 cos (/ — N)

r

8. sin B’ —— sin i cos b’ sin (l' — II) — cos i sin b’

9. For the calculation of Saturn’s magnitude (see Chapter 41), we need the quantity A t/, the difference between the Saturnicentric longitudes of the Sun and the Earth, measured in the plane of the ring.

p _   sin i sin b’ + cos i cos b' sin (/’ — II) ' cos b’ cos (/’ — II)

tan 1/2 -

EU —- | t/ l — U , to be expressed in degrees.

At/ is a small angle, equal to at most 7°.

10. Calculate the nutations in longitude (AQ) and in obliquity (Ae) and then the true obliquity of the ecliptic e (see Chapter 22). For the nutation, only the most important terms may be used; an accuracy of, say, 0”.01, is unnecessary.

 

11. Find the ecliptical longitude k0 and latitude 9 of the northern pole of the ring plane from

h0 ' II — 90°, fit = 90°  — i

12. Correct h and 9 for the aberration of Saturn :

 

correction to h :

 

+0.°005 693

 

C S (!o k) 

 

correction to # :    +0.°005 693 sin (/ — h) sin #

13. Add AQ to h and to h.

14. Transform (k0› to) and (h, 9) to the equatorial coordinates (o 0, h ) and (a, 6) by means of the formulae (13.3) and (13.4), using for e the true obliquity

obtained in Step 10.

15. The position angle P is given by

  COS o °   (°o °)

 0 cos 6 — cos é sin 6 cos (n — n)

Example 4S.a — Calculate the quantities concerning the appearance of Saturn’s ring on 1992 December 16, at 0‘ UT.

This instant corresponds to JD = 2448 972.5. For the difference between Dynamical Time and Universal Time, we use the value IT —— + 59 seconds = +0.00068 day, so that the instant corresponds to 1992 Dec. 16.00068 TD = JDE 2448 972.50068.

Step 1.    T —- —0.070 431 193

i  =  28.°076 131

II = 169°.410 243

Step 2. From an accurate ephemeris, calculated by using the complete VSOP87 theory, we deduce

f 0 = 84° l7'08“.53 =  84.°285 703

bo +0“.71 ' +0°.000197

A = 0.984 123 16

Step 3. Geometric heliocentric coordinates of Saturn, referred to the ecliptic and mean equinox of the date, taken from an accurate ephemeris:

TD l b r

1992 Dec. 15.0 319°09'44“.23 —1°O4’26“.52 9.8680846

16.0 319 11 36.61 —1 04 30.92 9.8678690

17.0 319 13 28.99 —1 04 35.31 9.8676534

 

Using A = 9 as a first approximation for Saturn’s distance, formula (33.3) yields z = 0.05198. Hence,

t - r --- 1992 December 16.00068 -- 0.05198

= 1992 December 15.94870 TD.

For this instant we find, by interpolation of the values tabulated above, I = 319°.191900, b ---- -- 1.°075 192, r -- 9.867 8801.

Step 4. z = +7.369 7225 A = 10.464 6006

y = --7.427 0295

z = --0.185 1696

Step 3. With this value for A, we obtain the new value z = 0.06044 day for the light-time; hence,

t - r -- 1992 December 16.00068 -- 0.06044

= 1992 December 15.94024 TD

For this instant we find, by interpolation of the tabulated values,

l ---- 319.° 191636, b ---- -- 1.°075 183, r 9.867 8819.

Step 4. z = +7.369 6942 A = 10.464 6059

y = --7.427 0651

z = --0.185 1681

This new value of A gives z = 0.06044 again, so no new iteration is needed.

Step 5. X = 314°.777 850

d = -- 1°.013 885

Step 6. B ---- + 16°.442

o = 35".87

b --- 10’.15

Step 7. N --- 113°.6037

l’ -- 319°.189 853

b' --- -- l °.075 113

Step 8. B’ --- + 14°.679

Step 9. th  = 153°. 2645

tr2 = 149°. 0663

EU ---- 4°.198

Step 10. AQ = +16“.86

Ae = --1“.79

e = 23° 26’22".96 = 23°.43971

Step 11. k0 = 79°.410243

b0 = 61°.923 869

 

 

Step 12.

Step 13.

Step 14.

Step 15.

 

corrected k = 314°.774 228

corrected b = — 1.°013 963

corrected h0 =  79.°414 926

corrected X  = 314°.778 911

n0 = 40º.36365 = 317°.55421 ó   = + 83.°48486 ó = — 17°.37056

P —- +6°.741

 

 

RE FE R E N CE

1. Gérard Dourneau, Observations et étude du mouvemeni dev hurt premiers satellites de Saturne, These de doctorat d’État, Université de Bordeaux I (1987).

 

Chapter 46

Positions of the Satellites of Saturn

In this Chapter a method is given to calculate, for any given instant, the positions of the eight major satellites of Saturn with respect to the planet as seen from the Earth. These apparent rectangular coordinates X and Y of the satellites will be

measured from the center of the disk of Saturn, in units of the planet’s equatorial radius. X will be measured positively

 

East

 

N o rth

 

to the west of Saturn, neg- ‘ ‘“‘ ' /*e atively to the east, the X-axis coinciding with the equator of

the planet, and hence with the major axis of the ring. F will be measured positively to the north, negatively to the south, the Y-axis coinciding with the planet’s rotation axis — see the drawing.

The calculation method is based on the theory of the satellites due to Doumeau [1].

 

For the given instant, calculate the following quantities (see Chapter 25) :

O = geocentric geometric longitude of the Sun,

9 = geocentric geometric latitude of the Sun,

R —— radius vector of the Sun (Earth), in astronomical units.

Let r be the light-time from Saturn to the Earth. Because the distance A of Saturn to the Earth is not known in advance, so is z not known. The distance A should be found by iteration. A good starting value is A = 9, The light-time is given by formula (33.3) ; a better value for A will be provided by (46.2).

323

 

T A B LE 4 6. A

The eight major sateI/ites a I  Saturn

Satellite Year of

discovery

Discoverer

period of revolution iit dnys

mognintdr at mean opposition Diameter in ifilometers

I Mimas 1789 W.Herschel 0.9425 12.9 400

II Enceladus 1789 W.Herschel 1.3704 11.7 498

III Tethys 1684 J.D.Cassini 1.8881 10.2 1046

IV Dione 1684 J.D.Cassini 2.7376 10.4 1120

V Rhea 1672 J.D.Cassini 4.5194 9.7 4528

VI Titan 1655 Ch.Huygens 15.9691 8.3 5150

VII Hyperion 1848 W.C.Bond 21.3188 14.2 286

VIII lapetus 1671 J.D. Cassini 79.9202 10.2—11.9 1460

Calculate the following values for the given time decreased by the light-time z (see Chapter 32) :

l —— heliocentric longitude of Saturn,

b —— heliocentric latitude of Saturn,

r —— radius vector of Saturn, in AU.

In the above, all longitudes and latitudes are referred to the ecliptic and mean equinox of the date.

Calculate the rectangular geocentric ecliptical coordinates of Saturn z = r cos b cos f + R cos O

 

y = r cos b sin + A sin O z = r sin b + R sin JS

and its distance to the Eanh

Calculate Saturn’s geocentric longitude @ and latitude fi0 by

 

(46.1)

(46.2)

 

“ = ATN2 (y’ ) and § 0 ' ATN

 

  + y

x

 

where, as mentioned earlier in this book, ATN2 is the “second" arctangent function. In other words, @ is equal to ATN (y/x) taken in the proper quadrant.

Because Dourneau constructed his theory of the satellites of Saturn in the reference frame of B1950.0, the quantities @ and Q should be converted to that equinox. If (JD) is the Julian (Ephemeris) Day corresponding to the given instant

 

for which the calculation is performed, and (JD) = 2433 282.4235 corresponding to the epoch B1950.0, calculate T and r as explained in Chapter 21, then use the expressions (21.5) and (21.7) to convert @ and 9 0 to B1950.0. (We will still call @ and 90 the coordinates so converted).

Let (JDE) be the Julian Ephemeris Day corresponding to the given time decreased by the light-time z. Then calculate the following “times” which will be needed in the calculation.

 

r = (JDE) -- 2411093.0

r2 ' t / 365.25

 

’7 ' / 36525

8 ' 6/ 365. 25

 

 (JDE) 25282.423 + 1950.0

36

i4 (JDE) -- 2411368.0 r ' r4 / 365.25

r6 = (JDE) -- 2415020.0

 

g ’IDE’ -- 2442 000.5 

r 365.25

itq = (JDE) 2409 786.0

 

We also need the following angles (in degrees) :

W0 ---- 5.095 (r3 -- 1866.39)

W1 -- 74.4 + 32.39 i2

W2 --- 134.3 + 92.62 r2

W3 ---- 42.0 -- 0.5118 it

W4 ---- 276.59 + 0.5118 i5

W5 ---- 267.2635 + 1222.1136/7

W6 = 175.4762 + 1221.5515 t7

W7 ---- 2.4891 + 0.002435 r7

W8 ---- 113.35 -- 0.2597 i

and the quantities

 

st ---- sin 28.°0817

ct --- cos 28.°0817

e --- 0.05589 -- 0.000346 r7

 

s2 -- sin 168°.8112

c2 -- cos 168°.8112

 

Then calculate the following quantities for the satellites. Of course, drop the data for the satellites you don’t need.

 

Mimas (satellite I}

L = 127°.64 + 381.°994497 r — 43.°57 sin W0 - 0.°720 sin 3W0

— 0°.02144 sin 5 lY0

The last three terms represent a perturbation in longitude due to resonance with Tethys.

p —— 106°.1 + 365°.549 rt

M —— L — p

Equation of the center, in degrees:

C = 2.°18287 sin M + 0°.025988 sin 23f + 0.°000 43 sin 33f h(1) = L + C

r(1) = 3.06879 1 + 0.01905 cos {M + C j

(1) = 1°.563

II(1) = 54.°5 — 365°.072 r2

En celadus (satellite Zf)

L —— 200°.317 + 262°.7319002 il + 0°. 256 67 sin US + 0.°208 83 sin lV2

The last two terms represent a perturbation in longitude due to resonance with Dione.

p —— 309°.107 + 123.°44121 i

M —— L — p

Equation of the center, in degrees: C —— 0°.555 77 sin M + 0.°001 68 sin 2 M

 

k(2) =  L +  C

 

r(2) =

 

  3.94118

1 + 0.004 85 cos (3f + C)

 

(2) = 0°.0262

II(2) = 348° — 151°.95 i2

Tethys {satellite III

h(3) = 285.°306 + 190°.697912 26 il + 2.°063 sin W0 + 0.°03409 sin 3W0

+ 0°.001015 sin 5WO

The last three terms represent a perturbation in longitude due to resonance with Mimas.

The orbital eccentricity of Tethys is zero. r(3) = 4.880998

(3) = 1°.0976

It(3) = 111.°33 — 72°.2441 /2

Dione (satellite TV}

L —— 254.°712 + 131°.534 93193 it — 0°.0215 sin W7 — 0°.017 33 sin W2

The last two terms represent a perturbation in longitude due to resonance with Enceladus.

p —— 174°.8 + 30.°820 r2

M —— L — p

Equation of the center, in degrees: C —— 0°.24717 sin M + 0°.00033 sin 2ñf h(4) = L + C

r(4) = 6.24871 1 + 0.002 157 cos (3f + C)

2(4) = 0°.0139

II(4) = 232° — 30°.27 i2

 

Rhea (satellite V}

p’ —— 342°.7 + 10.°057 it

ol = 0.000265 sin p’ + 0.01 sin W4

o2 = 0.000265 cos p’ + 0.01 cos W4

e = ,2 + a J > 0

p —— ATN (n I / a 2 ) to be taken between 90° and 270° if a 2 < 0

N —— 345° — 10.°057 it

h’ = 359.°244 + 79.°690047 20 rt + 0.°086 754 sin N

i —— 28.°0362 + 0°.346898 cos N + 0.°01930 cos W3

II = 168°.8034 + 0.°736936 sin N + 0.°041 sin W3

a —— 8.725924

Now, use the subroutine given in the box on the next page, after which h(5) = k, 2(5) = y, II(5) = w, r(5) = r.

Titan (catellite YZ)

L —- 261°.1582 + 22.°57697855 i4 + 0‘.074025 sin W3

i’ —— 27.°45141 + 0.°295 999 cos W3

fl’ = 168º.66925 + 0º.628 808 sin W3

al = sin JY7 sin (II' — W8)

o2 = cos W7 sin i' — sin W7 cos i' cos (£i’ — W8) g = 102°.8623

/   =   ATN (at   /o   2 ) to be taken  between  90° and  270°  if  a   <  0

s —— > 0

Calculate successive approximations to w and g as follows: w = 1¥4 + 0°.37515 (sin 2g — sin 2$q)

 

This is repeated until w and g no longer vary, but three iterations are always sufficient.

 

e’ = 0.029092 + 0.00019048 (cos 2g — cos 2g )

q = 2 (US — )

b = sin i’ sin (II’ — W8)

bz = cos W7 sin i’ cos (II’ — W8) — sin W7 cos i’

8 = ATN (b i / bt) + W8

where the arctangent is to be taken between 90° and 270° if b2 < 0

e —— e’ + 0.002778 797 e' cos q

p = o + 0°.159215 sin q u = 2 US — 28 +

h = 0.9375 e'° sin q + 0.1875 s° sin 2 (WS — 8)

 

h’ = L — 0°.254 744 (et sin lY6 + 0.75 et2 sin 2 W6 + /i) i = i' + 0°.031843 s cos u

+ 0°.031843 s sin

sin i'

o = 20.216 193

Now use the subroutine given in the box on page 329 to obtain h, y, w, and r.

Then  h(6) = h, y(6) = 9, II(6) = w, r(6) = r.

Hyperion (satellite VII j

q = 92°.39 + 0°.562 1071 f 6

J = 148°.19 19°. 18 lg

f = 184°. 8 35.°41 lg

8'  = 8 — 7.°5

X S = 176° + 12°.22 rt

XS = 8° + 24.°44 it

*s ' Es + °

= 69°.898 — 18°.670 88 rt

= 2 ( — W5)

x = 94.°9 — 2°.292 lg

a —— 24.506 01 — 0.086 86 cos 9 — 0.00166 cos (} + 9) + 0.00175 cos (J — 9)

e —— 0.103458 — 0.004099 cos 9 — 0.000167 cos (} + 9)

+ 0.000235 cos (J — 9) + 0.02303 cos — 0.00212 cos 2

+ 0.000 151 cos 3} + 0.000 13 cos p

p —— m + 0°.15648 sin y — 0.°4457 sin 9 — 0°. 2657 sin (} + q)

— 0°.3573 sin (} — q) — 12°.872 sin 1 .°668 sin 2}

— 0°.2419 sin 3 — 0‘.07 sin ‹;r

k' =  177.°047 + 16.°9l9938 29 r6 + 0.°15648 sin ¿ 9.° 142 sin q

+ 0°.007 sin 29 — 0°.014 sin 3p + 0.°2275 sin (} + 9)

+ 0°.2112 sin (} — 9) — 0.°26 sin } — 0°.0098 sin 2}

— 0°.013 sin a + 0°.017 sin bS — 0°.0303 sin ‹,r

i —— 27°.3347 + 0°.643486 cos y + 0°.315 cos W3 + 0.°0l8 cos 8 — 0.°018 cos cS II = 168°.6812 + 1.°401 36 cos y +  0.°685 99 sin U3

— 0.°0392 sin cS + 0.°0366 sin 8’

Now use the subroutine given in the box on page 329 io obtain h, y, w, and r.

Then   h(7) = h, 9(7) = y, II(7) = w, r(7) = r.

 

lapetus (satellite VIII j

L —— 261°.1582 + 22.°57697855 r4

с ’ = 91.°796 + 0.°562 i7

= 4°.367 — 0.°195 i

8 = 146°. 819 — 3°.198 i

‹,г = 60.°470 + 1°.521 7

Ф =  205°.055 — 2°.091 г

е' =   0.028298 + 0.001156   ‹ о ' 352.°91 + 11.°7l г] ]

= 76°.3852 + 4.°537951 25 / o

i’ —— 18.°4602 — 0.°9518 i t — 0.°072 i 2 + 0°.0054 t

8’  = 143°.198 — 3°.919 it  + 0.° 116 i + 0.°008 i

 

IT   = L — W4

8т ' 34 4

и = 2 (/ + g — ! — )

uz —— I +  2 (g  — 1$ — gs)

u4 ' т + От g

ti  -  2 ( + г›)

о =  58.935028 + 0.004638 cos u, + 0.058222 cos ut

е -— е’ — 0.0014097 cos (g 8т) + 0.0003733 cos (о, — 2g)

+ 0.000 1180 cos u, + 0.0002408 cos 1

+ 0.0002849 cos (/ + ut) + 0.0006190 cos oq

w = 0.°08077 sin (g- от) + 0°.02139 sin (• — ! g ) — 0.°006 76 sin u,

+ 0°.01380 sin / + 0°.01632 sin (/ + іі2 ) + 0.°035 47 sin u4

h’ = р — 0°.04299 sin • — 0°.007 89 sin и  — 0°.063 12 sin

— 0°.00295 sin 2/S — 0.°02231 sin ti5 + 0°.00650 sin (u + J)

i —— i' +0°.042 04 cos (u, + /) + 0°.002 35 cos (f + $ t +  г + От + 9)

+ 0°.00360 cos (• + е)

 

w’ = 0.°04204 sin (u, + J) + 0.‘002 35 sin (f + s + T + IT + H)

+ 0.°00358 sin (try + p)

II = II’ + w’ /sin i’

Now use the subroutine given in the box on page 329 to obtain X, y, w, and r.

Then   h(8) = h, y(8) = 9, II(8) = w, r(8)  = r.

For each required satellite t/ = 1 to 8), calculate

u —— k(/) —  fit/) w = II(/) — 168°. 8112 X(/) = rfi) [cos u cos w — sin u cos y(/) sin w]

Yfi)  —— rfi) [sin u cos ir cos 9(/) + cos o sin w)

Zt/) = rfi) sin u sin J(/)

Now consider a “ninth, fictitious satellite" situated at unit distance from the center of Saturn, above the planet’s north pole:

X(9) = 0, r(9) = 0, Z(9)  = 1.

This fictitious satellite will be needed later.

To obtain the apparent rectangular coordinates of the satellites as they appear on the celestial sphere as defined at the beginning of this Chapter, several rotations must be performed. So, calculate the following for all nine satellites (the eight real ones and the ninth, fictitious satellite) :

Rotation towards the plane of the ecliptic:

A    = X

B —— ct Y — st Z C -- st Y + ct Z

Rotation towards the vernal equinox:

Then calculate

A3   = Hz sin h   —  B 2 cos +o

B —— * cos + Be sin

a Cz Ci

 

 

B —— B cos ;S + C sin Qg

<4 ' C cos Q — B3 sin 0s

If J,  9 are the values of A4   and  C4 for the “ninth satellite",  that is,  5  =  A4(9),

9 = Cl   (9), then calculate the angle

D —— ATN2 (J, 9)

where, as earlier in this book, ATN2 is the *second” arctangent function, which gives the angle D in the correct quadrant. Then calculate

X = A cos D — C4 sin D

 

Y —— At sin D + C cos D

Z =  B4

 

(46.3)

 

X and Y are the required apparent rectangular coordinates of the satellite, as defined at the beginning of this Chapter. The quantity Z is negative if the satellite is closer to the Earth than Saturn, positive if it is more distant than Saturn.

However, to obtain full accuracy, the apparent coordinates X and r just obtained should be corrected for two effects, just as for the satellites of Jupiter (see page 313) :

1. differential light-time: the correction to be added to X is

 

where R = 20947 for satellite I

23715 — II

26382 — III

29876 — IV

 

35313 for satellite V 53800 — VI

59222 — VII

91820 — VIII

 

2. the perspective effect: the values X and Y obtained thus far should be multiplied

by the factor

A + Z/2475

where A is Saturn’s distance to the Earth in astronomical units as given by (46.2), while Z is in Saturn radii (46.3). The constant 2475 is the number of equatorial radii of Saturn in one astronomical unit.

 

 

Example 46.a -- Configuration of the satellites of Saturn on 1999 September 18, at 0‘ UT = JD 2451 439.5 = JDE 2451439.50074.

(The value b T -- +64 seconds is used.)

Using the complete VSOP87 theory, we find that at the given instant the coordinates of the Sun, referred to the mean equinox of the date, are

O = 174°.655 278,   § = +0.°000 228,   A = I.005 0051,

and that the true distance of Saturn to the Earth is A = 8.557 613 AU, so the light- time is 0.04942 day. Consequently, the geometric positions of Saturn and its satellites must be calculated for the instant

JDE 2451 439.50074 -- 0.04942 = 2451439.45132

For this instant, the heliocentric coordinates of Saturn, referred to the ecliptic and mean equinox of the date, are

l ---- 41.°912 356, b --- --2°.360 096, r -- 9.207 193,

whence, by (46.1),

z = +5.845 2457

y = +6.238 7380

z = --0.379 1464

and the “apparent” distance of Saturn to the Earth, by (46.2), is A = 8.557 599.

Then k0 = 46°.865 071, d 0 = --2.°539 334. Converted to the reference frame of

B1950.0, these values become (Chapter 21) kg = 46.°170 287, fig = --2°.544 441.

We shall not give the details of the calculation. Let us just mention the following values.

Rhea: e ---- 0.0102018, X' = 49°.7917, i = 28°.2962, II = 168°.2640

Titan: e ---- 0.0293386, h’ = 273°.4387, i = 27.°7333, fi = 168°.5439

Hyperion : e ---- 0.1187225, X' = 78°.2068, i = 27.°2076, II = 167°.5721

lapetus : e -- 0.0286422, k' = 97°.7552, i = 17.°2486, II -- 138°.9121

Satellite i

  (i) ( O)

 

 

1 320.0015 1.5630 47.7110 3.1224

2 300.4638 0.0262 123.2135 3.9257

3 347.9653 1.0976 51.0615 4.8810

4 102.6463 0.0139 128.2982 6.2561

5 50.9947 0.3359 118.2589 8.7054

6 270.5289 0.3701 8.4467 19.9254

7 91.9269 1.0461 21.5992 24.1160

8 103.1318 15.5062 22.3756 58.9396

 

and the final results:

Satellite X

 

1 + 3.102 —0.204

2 + 3.823 +0.318

3 + 4.027 —1.061

4 — 5.365 —1.148

5 — 1.122 —3.123

6 +14.568 +4.738

7 —18.001 —5.328

8 —48.759 +4.136

REFEREN CE

1. Gérard Dourneau, Observations et étude du mouvement des huit premiers satellites de Saturne, Thëse de doctorat d’État, Université de Bordeaux 1 (1987).

 

 

 

Chapter 47

Position of the Moon

In order to calculate accurately the position of the Moon for a given instant, it is necessary to take into account hundreds of periodic terms in the Moon’s longitude, latitude, and distance. Because this is outside the scope of this book, we shall limit ourselves to the most important periodic terms. The accuracy of the results will be approximately 10“ in the longitude of the Moon, and 4“ in its latitude. The interested reader can find a more accurate method in Chapront’s Loner Tables and Programs [2].

Using the algorithm described in this Chapter, one obtains the geocentric longitude h and latitude Q of the center of the Moon, referred to the mean equinox of the date, and the distance A in kilometers between the centers of Earth and Moon. The equatorial horizontal parallax z of the Moon can then be obtained from

 

sin z =

 

6378. 14 

q

 

The periodic terms given in this Chapter are based on the Chapront ELP- 2000/82 lunar theory [1]. However, for the mean arguments L', D, M, M’, I the improved expressions given later by Chapront [3] have been used.

For the given instant (in Dynamical Time), calculate T by means of formula (22.1). Remember that T is expressed in centuries and thus should be taken with a sufficient number of decimals — at least nine, since during 0.000000001 century (approximately 3 seconds) the Moon moves over an arc of 1.7 arcseconds.

Then calculate the angles L’, D, M, M’, and F by means of the following expressions. The angles so calculated will be expressed in degrees. In order to avoid working with large angles, reduce them to less than 360°. In QuickBasic, this can be achieved by defining the function

DEF FNRED# (X#) = X# — 360# • INT (X# / 360#)

Moon’s mean longitude, referred to the mean equinox of the date, and including the constant term of the effect of the light-time (—0”.70) :

337

 

L’ —- 218.316 4477 + 481267.881234 21 T

— 0.0015786 T2 + T° /538 841 — r‘/ 65 194 000

Mean elongation of the Moon :

D —— 297.850 1921 + 445 267.1114034 T

— 0.001 8819 T2 + T*/ 545868 — T‘/  113 065 000

Sun’s mean anomaly:

M —— 357.5291092 + 35999.050 2909 T

— 0.0001536 T2 + T* /24 490 000

Moon’s mean anomaly:

M’ -— 134.963 3964 + 477198.867 5055 T

+ 0.008 7414 T2 + T*/ 69 699 — r‘ / 14 712 000

 

(47. 1)

(47. 2)

(47.3)

(47. 4)

 

Moon’s argument of latitude  (mean distance of the Moon from its ascending node) :

 

F —— 93.272 0950 + 483 202.017 5233 T

— 0.003 6539 T — T°/  3526 000 + T4 / 863 310 000

Three further arguments (again, in degrees) are needed :

A = 119°.75 + 131.° 849 T

A   = 53°.09 + 479 264°. 290 T

A = 313°.45 + 481 266°. 484 T

 

(47.5)

 

Calculate the sums F•/ and Z r of the terms given in Table 47. A, and the sum lb of the terms given in Table 47.B. The argument of each sine (for Al and Yb) and cosine (for Zr) is a linear combination of the four fundamental arguments D, M, M’, and F. For instance, the argument on the eighth line of Table 47. A is 2D — M — M’ , and the contributions to Zf z:nd Z r are + 57066 sin {2D - M — M’) and — 152 138 cos (2D — M — M’), respectively.

However, the terms whose argument contains the angle M depend on the eccentricity of the Earth’s orbit around the Sun, which presently is decreasing with time. For this reason, the amplitude of these terms is actually variable. To take this effect into account, multiply the terms whose argument contains M or —M by E, and those containing 23f or —23f by E , where

E -— 1 — 0.002 516 T — 0.000 0074 P2 (47. 6)

The coefficient, not the argument of the sine or cosine, should be multiplied by ñ. For example, the 8th term in the longitude is really + 57066 E sin (2D — M  — M’ ).

 

TA B LE 4 7 . A

Periodic rerms for the longitude  (II) and distance (Zr) o f the Moon.

The unit is O.OOO OOO degree for Al, and O.OOO kilomerer for Zr.

D Argument

Multiple of M M’

F Ll

Coefficient of the sine

of the argument Er

Coefficient of the cosine

of the argument

0 0 1 0 6288774 —2090J355

2 0 —1 0 1274027 —3699111

2 0 0 0 658314 —2955968

0 0 2 0 213618 —569925

0 1 0 0 —185116 48888

0 0 0 2 —114332 —3149

2 0 —2 0 58793 246158

2 —1 —1 0 57066 —152138

2 0 1 0 53322 —170733

2 —1 0 0 45758 —204586

0 1 —1 0 —40923 —129620

1 0 0 0 —34720 108743

0 1 1 0 —30383 104755

2 0 0 —2 15327 10321

0 0 1 2 —12528

0 0 1 —2 10980 79 661

4 0 —1 0 10675 —34 782

0 0 3 0 10034 —23 210

4 0 —2 0 8548 —21 636

2 1 —1 0 —7888 24 208

2 1 0 0 —6766 30 824

1 0 —1 0 —5163 —8 379

1 1 0 0 4987 — 16 675

2 —1 1 0 4036 — 12 831

2 0 2 0 3994 — 10 445

4 0 0 0 3861 — 11 650

2 0 —3 0 3665 14 403

0 1 —2 0 —2689 —7 003

2 0 —1 2 —2602

2 —1 —2 0 2390 10 056

1 0 1 0 —2348 6 322

2 —2 0 0 2236 —9 884

 

TA BLE 4 7. A (cont. I

D Argument

Multiple of

Id M’

F Ll

Coefficient of the sine

of the argument

 

Coefficient of the cosine

of the argument

0 1 2 0 —2 120 5751

0 2 0 0 —2069

2 —2 —1 0 2048 —4 950

2 0 1 —2 —1773 4 130

2 0 0 2 —1595

4 —1 —1 0 1215 —3 958

0 0 2 2 —1110

3 0 —1 0 —892 3 258

2 1 1 0 —810 2 616

4 —1 —2 0 759 — 1 897

0 2 —1 0 —713 —2 117

2 2 —1 0 —700 2 354

2 1 —2 0 691

2 —1 0 -2 596

4 0 1 0 549 —1 423

0 0 4 0 537 —1 117

4 —1 0 0 520 — 1571

1 0 —2 0 —487 — 1 739

2 1 0 -2 —399

0 0 2 -2 —381 —4 421

1 1 1 0 351

3 0 —2 0 —340

4 0 —3 0 330

2 —1 2 0 327

0 2 1 0 —323 1 165

1 1 —1 0 299

2 0 3 0 294

2 0 —1 -2 8 752

 

TA B LE 4 7. B

Periodic terms for the latitude of the Moon f Z b).

The unit is 0. OOO 00 I degree.

D Argument

Multiple of

M M'

F

 

Coefficient of the sine

of the argument

D Argument

Multiple of M M'

F Lb

Coefficient of the sine

of the argument

0 0 0 1 5128122 0 0 1 —3 777

0 0 1 1 280602 4 0 —2 1 671

0 0 1 —1 277693 2 0 0 —3 607

2 0 0 —1 173237 2 0 2 —1 596

2 0 —1 1 55413 2 —1 1 —1 491

2 0 —1 —1 46271 2 0 —2 1 —451

2 0 0 1 32573 0 0 3 —1 439

0 0 2 1 17198 2 0 2 1 422

2 0 1 —1 9266 2 0 —3 —I 421

0 0 2 —1 8822 2 1 —1 I —366

2 —1 0 —1 8216 2 1 0 I —351

2 0 —2 —1 4324 4 0 0 1 331

2 0 1 1 4200 2 —1 1 1 315

2 1 0 —1 —3359 2 —2 0 —1 302

2 —1 —1 1 2463 0 0 1 3 —283

2 —1 0 1 2211 2 1 1 —1 —229

2 —1 —1 —1 2065 1 1 0 —1 223

0 1 —1 —1 —1870 1 1 0 1 223

4 0 —1 —1 1828 0 1 —2 —1 —220

0 1 0 1 —1794 2 1 —1 —1 —220

0 0 0 3 —1749 1 0 1 1 —185

0 1 —1 1 —1565 2 —1 —2 —1 181

1 0 0 1 —1491 0 1 2 1 —177

0 1 1 1 —1475 4 0 —2 —1 176

0 1 1 —1 —1410 4 —I —1 —1 166

0 1 0 —1 —1344 1 0 1 —1 —164

1 0 0 —1 —1335 4 0 1 —1 132

0 0 3 1 1107 1 0 —1 —1 —119

4 0 0 —1 1021 4 —1 0 —1 115

4 0 —1 1 833 2 —2 0 1 107

 

Moreover, add the following additive terms to Ef and to lb. The terms involving A; are due to the action of Venus, the term involving A2 is due to Jupiter, while those involving L’ are due to the flattening of the Earth.

Additive to Al :

+3958 sin A

+ 1962 sin (L’ - F)

+ 318 sin A 2

Additive to lb :

--2235 sin L’

+ 382 sin A

+ 175 sin {A -- F)

+  175 sin (A  + F)

+ 127 sin (L’ -- M’ )

-- 115 sin [L' + M' )

The coordinates of the Moon are then given by

' -- * + »» (in degrees)

 

  lb

1000 000

A = 385 000.56 +

 

(in degrees)

   Mr 

1000

 

(in kilometers)

 

Division of the sums by 106 or by 10" is needed because in Tables 47. A and

47. B the coefficients are given in units of l0 *6 degree or of 10*" kilometer.

Example d7.a -- Calculate the geocentric longitude, latitude, distance, and equatorial horizontal parallax of the Moon for 1992 April 12, at 0' TD.

We find successively:

JDE = 2448 724.5 d = 109°.57

T -- --0.077 221 081451 Nt -- 123.°78

L' -- 134°.290 182 A = 229.°53

D --- 113.° 842304 F = 1.000194

3f = 97.°643 514 E/ = -- 1 127 527 with the ad-

3f' = 5°.150 833 bh = --3 229 126 I ditive terms

F ---- 219.°889 721 Lr ---- -- 16590 875

 

From which we deduce

k = 134°. 290 182 — 1.° 127 527 = 133°. 162655

b = —3.°229 126 = —3° 13'45“

A = 385000.56 — 16590.875 = 368 409.7 km

r = arcsine (6378.14 / 368 409.7) = 0°.991990 = 0°59'31“.2

The apparent longitude of the Moon is obtained by adding to X the nutation in longitude (AQ), which is equal to + 16“.595 = +0°.004 610 (see Chapter 22). Consequently,

apparent k = 133°. 162655 + 0°.004610

= 133°.167 265

= 133° 10'02“

For the given instant, the true obliquity of the ecliptic is (Chapter 22) e = e0 + Ae = 23° 26'26“.29 = 23°.440636

The Moon’s apparent right ascension and declination are then found by means of expressions (13.3) and (13.4) :

= 134.°688 470 = 8'58‘45'.2

6 = + 13°.768 368 = + 13° 46'06"

The exact values, obtained by using the complete ELP-2000/82 theory, are k = 133° 10’00“ = 8'58‘45‘.1

d = —3° l3’45“ 6 = + 13° 46’06”

A = 368 405.6 km z   = 0°59'31”. 2

Lunar node and lunar perigee

According to Chapront [3], the longitude of the (mean) ascending node II and that of the (mean) perigee H of the lunar orbit, in degrees, are given by

 

12 = 125.044 5479 — 1934.1362891 T + 0.002 0754 T2

+ T°/467 441 — T / 60 616000

H = 83°.353 2465 + 4069.013 7287 T — 0.010 3200 T’

- r°/ 80053 + r 4 / 18 999 000

 

(47.7)

 

where T has the same meaning as before. These longitudes are tropical, that is, they are measured from the mean equinox of the date.

From the formula for II we can find the times when the (mean) ascending or descending node of the lunar orbit coincides with the vemal equinox, that is, when It is equal to 0° or to 180°, respectively. During the period 1910—2110, this occurs at the following dates:

 

 

II =  0°

1913 May 27

1932 Jan. 6

1950 Aug. 17

1969 Mar. 29

1987 Nov. 8

2006 June 19

2025 Jan. 29

2043 Sep. 10

2062 Apr. 22

2080 Dec.  1

2099 July 13

 

£1 =   180°

 

1922 Sep. 16

1941 Apr. 27

1959 Dec.   7

1978 July 19

1997 Feb. 27

2015 Oct. 10

2034 May 21

2052 Dec. 30

2071 Aug. 12

2090 Mar. 23

2108 Nov. 3

 

The longitude of the true ascending node (the node of the instantaneous lunar orbit) can be deduced from II by the addition of periodic terms, which are given in the Tables of Chapront [2]. The principal of these terms are

— 1°.4979 sin 2 (D — F)

— 0°. 1500 sin 3f

— 0.° 1226 sin 2D

+ 0°. 1176 sin 2F

— 0°.0801 sin 2 (If' — F)

See also Chapter 1 of my Mathematical Astronomy Morsels {4j.

REFEREN CES

1. M. Chapront-Touzé and J. Chapront, “The lunar ephemeris ELP 2000”, Astronomy and Astrophysics, Vol. 124, pages 50—62 (1983). — This article gives a description of that lunar theory and discusses its accuracy. It does not give, however, the list of the many periodic terms. ‘ELP” means Ephéniérides Lunaires Parisiennes, although this work is not an ephemeris (a list of calculated positions) but rather an analytical theory (a series of periodic terms).

2. M. Chapront-Touzé and J. Chapront, Lunar Tables and Programs from 4000 B. C. to A.D. 8000, Willmann-Bell, 1991.

3. J. Chapront, M. Chapront-Touzé, and G. Francou,1ntroduction dans ELP 2000-82B de nouvelles valeurs des paramétres orbitaux de la Lune et du barycentre Terre- Lune, Paris, January 1998.

4. I. Meeus, Mathematical Astronomy Morsels, Willmann-Bell, 1997.

 

Chapter d8

Illuminated Fraction of the Moon’s Disk

The illuminated fraction k of the disk of the Moon depends on the selenocentric elongation of the Earth from the Sun, called the phase angle (i). Selenocentric

means “as seen from the center of the

Moon". The formula is

k —— +  COS i (48. 1)

and this is the value of both the ratio of the illuminated area of the disk to the total area, and the ratio of the illuminated length of the diameter perpendicular to the line of cusps to the complete diameter (see the Figure).

The phase angle i of the Moon, for a geocentric observer, can be found as follows. First, find the geocentric elongation of the Moon from the Sun by means of one of the relations

COS S1f1 Ö0 SIH Ö

+  CO S CO COS Ö COS (  0 )

(48. 2)

where a 0, 6 , @ and o, b, k are the geocentric right ascensions, declinations, and longitudes of ihe Sun and the Moon, respectively, and 0 is the geocentric latitude of the Moon. Then we have

345

 

 

_ A sin

' A — R cos

 

(48.3)

 

where fi is the distance Earth —Sun, and A the distance Earth —Moon, both in the same units, for instance in kilometers. The angles / and i are always between 0 and 180 degrees. Once i is known, the illuminated fraction k can be obtained by means of formula (48.1).

Of course, for the calculation of k it is not needed to calculate the geocentric positions of the Moon and the Sun with high precision. An accuracy of, say, 1' will be sufficient.

If no high accuracy is required, it will suffice to put cos i = —cos J. The resulting error in k will never exceed 0.0014.

Lower accuracy, though still a good result, is obtained by neglecting the Moon’s latitude and by calculating an approximate value of i as follows:

 

i = 180° — D — 6.°289 sin M’

+ 2.°100 sin M

— 1.°274 sin (2D — M')

— 0°.658 sin 2D

— 0.°214 sin 2N’

— 0.° 110 sin D

 

(48. 4)

 

where the angles D, M, and M’ can be found by means of formulae (47.2) to 47.4). In this case, the geocentric positions of the Sun and the Moon are not needed.

Position Angle of the Moon’s bright limb

The position angle of the Moon’s bright limb is the position angle y of the midpoint of the illuminated limb of the Moon (C in the Figure on page 345), reckoned eastward from the North Point of the disk (not from the axis of rotation of the lunar globe). It can be obtained from

 

  cos 60  sin (n0 —  n) sin  6   cos 6  —  cos 6   sin  6 cos (nd — >)

 

(48.5)

 

where n0, 60, n , and 6 have the same meaning as before.

The angle 2 is in the vicinity of 270° near First Quarter, near 90° after Full Moon. It can be found in the correct quadrant by applying the ATN2 function to the numerator and the denominator of the fraction in formula (48.5) — see “the correct quadrant” in Chapter 1.

 

48. ILLUMINATED FRACTION OF MOON 3 4 7

If x is the position angle of the (midpoint of the) bnght limb, then the position angle of the cusps are y -- 90° and ;t + 90°. The angle x has the advantage that it unambiguously defines the illuminated limb of the Moon.

Note that the angle x is not measured from the direction of the observer’s zenith. The zenith angle of the bright limb is g -- q, where q is the parallactic angle (see Chapter 14).

Finally, note that formula (48.5) is valid in the case of a planet too.

Example 48.a -- The Moon on 1992 April 12, at 0‘ Dynamical Time.

From Example 47.a we have, for that instant, n = 134°.6885

é = + 13°.7684 A = 368 410 km

The apparent position and the distance of the Sun at the same instant are n0 = 1h22" 37'.9 = 20.°6579

â0 = +8° 41'47“ = + 8.°6964

It = 1.0024977 AU -- 149 971 520 km

The first formula (48.2) then gives cos = --0.354 991, whence = 110°. 7929.

Then

tan i = + 2.615 404 by formula (48.3)

i -- 69.°0756

and, by formula (48.1), k = 0.6786, which should be rounded to 0.68.

If we use the approximate relation cos i = -- cos /, we find k -- 0.6775, which again rounds to 0.68.

Let us now use the approximate formula (48.4). In Example 47.a, we have found for the given instant

D 113°. 8423

M --- 97.°6435

M' ---- 5.°1508

Then formula (48.4) gives i = 68.°88 whence, by (48. l), k = 0.6802, which again rounds to 0.68.

Finally, formula (48.5) gives

 

Mn X

 

 --0.90283  

+0.24266

 

whence x = 285°.0

 

 

 

 

 

Chapter 49

Phases of the Moon

By definition, the times of New Moon, First Quarter, Full Moon, and Last Quarter are the times at which the excess of the apparent geocentric longitude of the Moon over the apparent geocentric longitude of the Sun is 0°, 90°, 180°, and 270°, respectively.

Hence, to calculate the instants of these lunar phases, it is necessary to calculate the apparent longitudes of the Moon and the Sun separately. (However, the effect of the nutation may be neglected here, since the nutation in longitude AQ will not affect the difference between the longitudes of Moon and Sun,)

However, if no high accuracy is required, the instants of the lunar phases can be calculated by the method described in this Chapter. The expressions are based on Chapront’s ELP-2000/82 theory for the Moon (with improved expressions for the arguments M, M’, etc., as mentioned in Chapter 47), and on Bretagnon’s and Francou’s VSOP87 theory for the Sun. The resulting times will be expressed in Julian Ephemeris Days (JDE), hence in Dynamical Time.

The times of the mean phases of the Moon, akeady affected by the Sun’s aberration and by the Moon’s light-time, are given by

 

JDE = 2451 550.09766 + 29.530 588 861 I

+ 0.000 154 37 F 2

— 0.000 000 150 T

+ 0.000 000 000 73 T’

where an integer value of k gives a New Moon, an integer increased by 0.25 gives a First Quarter,

by 0.50 gives a Full Moon,

by 0.75 gives a Last Quarter.

 

(49.1)

 

An y  other  value  for  k   w ill   give  meaning less  r esul t s !

The value k = 0 corresponds to the New Moon of 2000 January 6. Negative values of k give lunar phases before the year 2000.

3 49

 

3 S O

For example,

 

ASTRONOMICAL  ALGORITHMS

 

+479.00 and —2793.00 correspond to a New Moon,

+479.25 and —2792.75 correspond to a First Quarter,

+479.50 and —2792.50 correspond to a Full Moon,

+479.75 and —2792.25 correspond to a Last Quarter.

An approximate value for k is given by

k -- (year — 2000) X 12.3685 (49.2)

where the *year” should be taken with decimals, for example 1987.25 for the end of March 1987 (because this is 0.25 year since the beginning of the year 1987). The sign = means “is approximately equal to”.

Finally, in formula (49.1) T is the time in Julian centuries since the epoch 2000; it is obtained with sufficient accuracy from

 

T —— k

1236.85

and hence is negative before the epoch 2000.0.

 

(49.3)

 

Calculate E by means of formula (47.6), and then the following angles, which are expressed in degrees and may be reduced to the interval 0 —360 degrees and, if necessary, to radians before going further on.

Sun’s mean anomaly at time JDE:

M —— 2.5534 + 29.105 356 70 I

 

Moon’s mean anomaly:

 

— 0.0000014 T'

— 0.000000 11 T

 

(49.4)

 

3f' = 201.5643 + 385.816 935 28 k

+ 0.010 7582 T2

+ 0.000 012 38 T

— 0.000 000 058 T

Moon’s argument of latitude:

F —— 160.7108 + 390.670 502 84 k

— 0.001 6118 T2

— 0.000 002 27 T

+ 0.000 000 011 T

Longitude of the ascending node of the lunar orbit: II = 124.7746 — 1.563 755 88 k

+   0.002 0672 T2

+ 0.000 002 15 T

 

(49.5)

(49. 6)

(49.7)

 

Planetary arguments, again in degrees:

A = 299.77 +  0.107 408 k — 0.009 173 T2

A 2 = 251.88 +  0.016 321 k

A —— 251.83 + 26.651 886 k

A4 = 349.42 + 36.412 478 k

 

A5 = 84.66 + 18.206239 k

A6 = 141.74 + 53.303 771 k

A —— 207. 14 +  2.453 732 k

Kg -  154.84 +  7.306 860 k

Aq —- 34.52 + 27.261239 k

 

A' = 207.19 +  0. 121 824 k

A — 291.34 + 1.844 379 k

A = 161.72 + 24. 198 154 k

A = 239.56 + 25.513 099 k

A = 331.55 + 3.592 518 k

 

To obtain the time of the rriie (apparent) phase, add the following corrections (in days) to the JDE obtained above.

New Moon Full Moon

—0.40720 —0.40614 x sin M'

+0. 17241 x E +0. 17302 x E

 

+0.01608 +0.01614 23f'

+0.01039 + 0.01043 2 F

+0.00739 x E +0.00734 x € 3J' — If

—0.00514 x E —0.00515 x E ñf’ + M

+0.00208 x E +0.00209 x E 2M

—0.00111 —0.00111 3f’ — 2F

—0.00057 —0.00057 3f’ + OF

+0.00056 x £ +0.00056 x E 2ñf’ + M

—0.00042 —0.00042 34f’

+0.00042 x E +0.00042 M 2F

+0.00038 +0.00038 M — 2F

—0.00024 —0.00024 13f’ — M

—0.00017 —0.00017 fl

—0.00007 —0.00007 M’ + 2M

+0.00004 +0.00004 2M’ — 2F

+0.00004 +0.00004 33f

+0.00003 +0.00003 M’ M — 2F

+0.00003 +0.00003 PM’ + 2J

—0.00003 —0.00003 M’ M + 2 F

+0.00003 +0.00003 — + 2F

—0.00002 —0.00002 M’ - M — 2F

—0.00002 —0.00002

+0.00002 +0.00002

 

First and  Last  Quarters

—0.62801 x sin 3f’

+0. 17172 x ñ If

—0.01183 x £ M’ + M

+0.00862 23f’

+0.00804 2 F

+0.00454 If' — 3f

+0.00204 24f

—0.00180 M’ - 2F

—0.00070 M' + 2 F

—0.00040 33f'

—0.00034 x E 23f' — 4f

+ 0.00032 x E M + 2F

+ 0.00032 x £ N — 2 F

—0.00028 x E 2 M' + 23f

+0.00027   x ñ 2M’ + M

—0.00017 II

—0.00005 M’ — 3f — 2s

+0.00004 2M’ + 2J

—0.00004 M’ M + 2F

+ 0.00004 M’ — 2M

+0.00003 M’ + M - 2F

+0.00003 3M

+0.00002 23f' — 2F

+0.00002 M’ — M + 2F

—0.00002 33f' + M

Calculate, for the Quarter phases only,

W = 0.00306 — 0.00038 E cos 3f + 0.00026 cos M'

— 0.00002 cos (3f' - Nt) + 0.00002 cos (3f' + M) + 0.00002 cos 2F

Additional corrections: for First Quarter: + W

for Last Quarter : — W

Additional corrections for all phases .

+ 0.000325 + 0.000056 x sin A$

165 047 Aq

164 04s A io

126 040 A n

110 037 A i›

062 035 Ai›

060 023 A4

 

 

Ezonpfe 49.o -- Calculate the instant of the New Moon which took place in February 1977.

Mid-February 1977 corresponds to 1977.13, so we find by (49,2)

k (1977. 13 -- 2000) x 12.3685 = --282.87

whence k --- --283, since for the New Moon phase k should be an integer. Then, by formula (49.3), T ---- --0.22881, and then formula (49.1) gives

JDE = 2443 192.94102

With k -- --283 and T ---- --0.22881, we further find

E -- 1.000 5753

M -- --8 234.°2625 = 45.°7375

M' --- -- 108984°.6278 =  95°.3722

N = -- 110 399°.0416 = 120.°9584

ft   = 567°.3176 = 207°.3176

The sum of the first group of periodic terms (for New Moon) is --0.28916, that of the 14 additional corrections is --0.00068. Consequently, the time of the true New Moon was

JDE = 2443 192.94102 -- 0.28916 -- 0.00068 = 2443 192.65118,

which corresponds to 1977 February 18.15118 TD

= 1977 February 18, at 3'37*42' TD.

The correct value, calculated by means of the ELP-2000/82 theory, is 3‘37”40‘ TD.

In February 1977, the difference b T between Dynamical Time and Universal Time was equal to 48 seconds. Hence, the New Moon of 1977 February 18 occurred at 3h37* Universal Time. See also Example 10.a, on page 78.

Example 49.b -- Calculate the time of the first Last Quarter of A.D. 2044.

For “year” = 2044, formula (49.2) gives k = +544.21, so we shall use the value k +544.75.

Then, by formula (49. l), JDE = 2467 636.88597.

Sum of the first group of periodic terms (for Last Quarter) = 0.39153.

Additional correction for Last Quarter = -- W = 0.00251.

Sum of additional 14 corrections = 0.00007.

Consequently, the time of the Last Quarter is

2467 636.88597 -- 0.39153 -- 0.00251 -- 0.00007 = 2467 636.49186

which corresponds to 2044 January 21, at 23 h48‘ 17* TD.

 

For the period 1980 to mid-2020, we compared the results of the method described in this Chapter with the accurate times obtained with the ELP-2000/82 and VSOP87 theories.

Mean error Maximum error

New Moon : 3.6 seconds 16.4 seconds

First Quarter : 3.8 15.3

Full Moon : 3.8 17. 4

Last Quarter : 3.8 13.0

Mean error of all phases — 3.72 seconds

If an error of a few minutes is not important one may, of course, drop the smallest periodic terms and the fourteen additional terms.

The mean time interval between two consecutive New Moons is 29.530 589 days, or 29 days 12 hours 44 minutes 03 seconds. This is the length of the (mean) synodic period of the Moon. However, mainly by reason of the perturbing action of the Sun, the actual time interval between consecutive New Moons, or lunation, varies greatly. See Table 49.A, taken from [1].

TA B LE 4 9. A

The shortest and the  fongest lunations,  1900 (o 2!00

From the New Moon of to that of Diiration of the lunation

1903 June 25 1903 July 24 29 days 06 hours 35 minutes

2035 June 6 2035 July 5 29 — 06 — 39

2053 June 16 2053 July 15 29 — 06 — 35

2071 June 27 2071 July 27 29 — 06 — 36 —

1955 Dec. 14 1956 Jan. 13 29 days 19 hours 54 minutes

1973 Dec. 24 1974 Jan. 23 29   — l9   — 55 —

REFEREN CE

1. J. Meeus, “Les durées extrêmes de la lunaison”, l’Astronomie (Société Astronomique de France), Vol. 102, pages 288-289 (July-August 1988).

 

Chapter 50

Perigee and apogee of the Moon

In this Chapter a method is given for the calculation of approximate times when the distance between the Earth and the Moon is a minimum (perigee) or a maximum (apogee). The resulting times will be expressed in Julian Ephemeris Days (JDE), hence in the uniform time scale of Dynamical Time. Our expressions are based on Chapront’s lunar theory ELP-2000/82, with improved expressions for the arguments D, If, etc., as mentioned in Chapter 47.

First, calculate the time of the mean perigee or apogee by the formula

 

JDE = 2451 534.6698 + 27.554 549 89 I

—  0.000 6691 T2

—  0.000 001 098 T

+ 0.000 000 0052 T‘

 

(50. 1)

 

where an integer value of k gives a perigee, and an integer increased by 0.5 an apogee. Important : any other value for k will give meaningless results!

The value k = 0 corresponds to the perigee of 1999 December 22.

SO, fOr  irlStaflCe,

k —— +318 and £ = —25 will give a perigee,

k -— +429.5 and k = — 1209.5 will give an apogee, k = +224.87 is an incorrect value.

An approximate value of k is given by

k -- (year — 1999.97) x 13.2555 (50.2)

where the “year* should be taken with decimals. For instance, 2041.33 represents the end of April of the year 2041.

Finally, in formula (50. l) T is the time  in Julian centuries since the epoch 2000.0. It is obtained with a sufficient accuracy from

355

 

T 1325.55 (50.3)

Calculate the following angles; they are expressed in degrees and may be reduced to the interval 0—360 degrees and, if necessary, converted to radians before calculating further.

Moon’s mean eiongation at time JDE:

D -- 171.9179 + 335.910 6046 k

— 0.010 0383 T2

— 0.000 011 56 T

+ 0.000 000 055 T4

Sun’s mean anomaly:

M = 347.3477 + 27.157 7721 k

— 0.000 8130 r 2

— 0.000 0010 T

Moon’s argument of latitude:

F -- 316.6109 + 364.528 7911 k

— 0.012 5053 r 2

— 0.000 0148 T

To the JDE given by (50.1), add the sum of the periodic terms of Table 50.A, taking either those for perigee or for apogee, according to the case.

The Moon’s equatorial horizontal parallax is obtained by making the sum of the terms given in Table 50.B.

From Tables 50. A and 50.B it appears that:

— for the periodic terms for the instant, the sine of the argument should be taken, while for the value of the corresponding parallax the cosine must be used;

— up to a given value of the coefficient, there are more periodic terms for the perigee than for the apogee;

— the successive coefficients in the same “2D ” series (for example the terms in 2D - M, 4D — M, 6D — M, etc.) have alternate Slgns for the perigee, while for the apogee all have the same sign;

— the coefficient of the largest periodic term (the term with argument 2D) is much larger in the case of the perigee than for the apogee. As a consequence, the largest possible difference between the time of the meon and the true passage is 45 hours for the perigee, but only 13 hours for the apogee. Also, the Moon’s perigee distance varies in a larger interval (approximately between 356 370 and 370 350 kilometers) than does the apogee distance (404 050 to 406 720 km).

 

 

-H e SO.a -- The Moon’s apogee of October 1988.

Because the beginning of October corresponds to 0.75 year since the beginning of the calendar year, we put the value year = 1988.75 in formula (50. 2). This gives k -- -- 148.73. We therefore take the value k --- -- 148.5 (apogee!).

Formulae (50.3) and (50.1) then give

 

T 0.112029

Then we find

 

JDE = 2447 442. 8191

 

D  ---- --49 710°.8070 = 329°. 1930

M -- --3685°.5815 = 274°.4185

F  ---- --53 815°.9147 -- 184°.0853

Sum of the terms in Table 50.A (apogee) = --0.4654 day Sum of the terms in Table 50.B (apogee) = 3240.679

Hence, the time of the apogee is

JDE = 2447 442.8191 -- 0.4654 = 2447 442.3537

which corresponds to 1988 October 7, at 20‘29‘ TD. The corresponding value of the Moon’s equatorial horizontal parallax is 3240“.679, or 0°54'00”.679.

The exact values are 20'30” TD and 0°54'00“.671.

 

TA B L E 5 O . A

Periodic terms for the  fíme, in days

For the peri gee

Argument

of sine

Coefficient Argument of sine

Coeffiicient

2D —1.6769 2D - 2M —0.0027

4D +0.4589 4D — 2M +0.0024

6D —0.1856 6D — 2M —0:0021

8D +0.0883 22D —0.0021

2D — If —0.0773 + 0.00019 T 18D — M —0.0021

3f +0.0502 — 0.00013 T 6D + M +0.0019

10D —0.0460 I ID —0.0018

4D — M +0.0422 — 0.00011 T 8D + M —0.0014

6D — 3f —0.0256 4D — 2F —0.0014

12D +0.0253 6D + 2N —0.0014

D +0.0237 3D + M +0.0014

8D — 3f +0.0162 5D + M —0.0014

14D —0.0145 13D +0.0013

2F +0.0129 20D — M +0.0013

3D —0.0112 3D + 2M +0.0011

10D — 3f —0.0104 4D 2F — 2M —0.0011

16D +0.0086 D + 2M —0.0010

12D — 3f +0.0069 22D — M —0.0009

5D +0.0066 4F —0.0008

2D + 2F —0.0053 6D — 2F +0.0008

18D —0.0052 2D — IF + M +0.0008

14D — M —0.0046 23f +0.0007

7D —0.0041 2F — M +0.0007

2D + M +0.0040 2D 4F +0.0007

20D +0.0032 2F — 2H —0.0006

D + M —0.0032 2D — 2F + 2M —0.0006

16D — M +0.0031 24D +0.0006

4D + M —0.0029 4D — 4F +0.0005

9D +0.0027 2D + 2U +0.0005

4D + 2N +0.0027 D — M —0.0004

 

TABLE 50 . A (cont. )

For the apogee

Argument

of sine Coefficient Argmnent of sine Coefficient

2D +0.4392 8D — M +0.0011

4D +0.0684 4D — 231 +0.0010

3f +0.0456 — 0.00011 T 10D +0.0009

2D — M +0.0426 — 0.00011 T 3D + M +0.0007

2F +0.0212 2M +0.0006

D —0.0189 2D + 3f +0.0005

6D +0.0144 2D + 2M +0.0005

4D — M +0.0113 6D + 2F +0.0004

2D + 2F +0.0047 6D — 23f +0.0004

D + M +0.0036 10D — M +0.0004

8D +0.0035 5D -0.0004

6D — M +0.0034 4D — 2F —0.0004

2D — 2N —0.0034 2F + M +0.0003

2D — 23f +0.0022 l2D +0.0003

3D —0.0017 2D + 2F — M +0.0003

4D + 2F +0.0013 D - M —0.0003

 

3 6 O ASTRONOMICAL    ALG ORITHMS

T A B L E 5 O. B

Terms for rhe parallax, in arcseconds

For the peri gee

3629“.215 +0.067 x cos 10D — M

+63.224 x cos 2D +0.054 4D + M

—6.990 4D —0.038 I2D — M

-I-2.834 2D — M —0.038 4D — 2N

—0.0071 T 1 +0.037 7D

+ 1.927 6D —0.037 4D + 2N

—1.263 D —0.035 16D

—0.702 8D —0.030 3D + M

+0.696 M +0.029 D — M

—0.0017 F —0.025 6D + M

—0.690 2F +0.023 23f

—0.629 4D — M +0.023 14A — M

+0.0016 T —0.023 2D + 2M

—0.392 2D — 2F +0.022 6D — 2M

+ 0.297 10D —0.021 2D — 2f — ñf

+0.260 6D — M —0.020 9D

+0.201 3D +0.019 I 8D

—0.161 2D + M +0.017 6D + 2N

+0.157 D + M + 0.014 2F — M

—0.138 12D —0.014 l6D — M

—0.127 8D — M +0.013 4D — 2F

+0. 104 2D + 2F + 0.012 8D + M

+0. 104 2D — 2ñf + 0.01 l l lD

—0.079 5D + 0.010 5D + M

+ 0.068 14D —0.010 20D

For the apogee

3245“.251

—9.147 x cos 2D + 0.052 x cos 6D

—0.841 D + 0.043 2D + M

+ 0.697 2F + 0.031 2D + 2F

—0.656 M —0.023 2D — 2F

+0.0016 T 1 + 0.022 2D — 23f

+0.355 4D + 0.019 2D + 23f

+0.159 2D — M —0.016 23f

+0.127 D + M + 0.014 6D — 3f

+0.065 4D — M + 0.010 8D

 

Using the method described in this Chapter, 600 perigee and 600 apogee passages of the Moon were calculated, namely from June 1977 to August 2022. The results were compared with accurate values obtained with the ELP-2000/82 theory.

The largest errors are

for the time : 31 minutes for the perigee,

3 minutes for the apogee;

for the parallax : 0“.124 for the perigee,

0”.051 for the apogee.

The latter errors correspond to distance errors of 12 and 6 kilometers, respectively. The distribution of the errors of the 600 calculated times is as follows:

 

Number of errors

 

Perigee Apogee

 

The menu time interval between two consecutive passages of the Moon through perigee is 27.55455 days, or 27 days 13 hours 19 minutes. This is the length of the anomalistic period of the Moon. However, mainly by reason of the perturbing action of the Sun, the actual time interval between consecutive perigees varies greatly, between the extremes 24 days 16 hours and 28 days 13 hours. Examples:

 

perigee on 1997 December  9 at 16h. 9

perigee on 1998 January 3 at 8h. 5

perigee on 1990 December   2 at 10‘.8

perigee on 1990 December 30 at 23‘.8

 

diff. = 24 days 16 hours

diff. = 28 days 13 hours

 

The time interval between two consecutive apogees, however, varies between narrower limits, namely between 26.98 and 27.90 days (26 days 23'/i hours and 27 days 21'/z hours).

 

Extreme perigee  and  apogee  distances  of the ñfoon

Between the years 1500 and 2500, fourteen times the Moon approaches the Earth to less than 356425 kilometers, and the same number of times the distance grows to larger than 406710 km. These cases are mentioned in Table 50.C.

For the calculation, use has been made of Chapront’s lunar theory ELP- 2000/82, except that we neglected all periodic terms with a coefficient less than 0.0005 km (50 centimeters).

It appears that, during the time interval of ten centuries considered here, the extreme distances between the centers of Earth and Moon are

356371 km on 2257 January 1

406720 km on 2266 January 7

The smallest perigee distance of the 20th century was that of 1912 January 4, as was akeady found earlier by Roger W. Sinnott, Associate Editor of Sky and Telescope [1). Further, we see that these extreme perigees and apogees all occur during the winter months of the northern hemisphere, the period of the year when the Earth is closest to the Sun. It is evident that the variable Earth —Sun distance somewhat affects the Earth—Moon distance.

T A B LE 5 0. C

Extreme perigees and apogees,  A.D.  f500 to 2500   fUT dates)

perigee < 356425 km apogee > 406710 km

1548 Dec. 15 356 407 km 1921 Jan. 9 406 710  km

1566 Dec. 26 356 399 1984 Mar.   2 406 712

1771 Jan. 30 356 422 2107 Jan. 23 406 716

1893 Dec. 23 356 396 2125 Feb.   3 406 720

1912 Ian. 4 356 375 2143 Feb. 14 406 713

1930 Jan. 15 356 397 2247 Dee. 27 406 715

2052 Dec. 6 356 421 2266 Jan. 7 406 720

2116 Jan. 29 356 403 2284 Jan. 18 406 714

2134 Feb. 9 356 416 2388 Nov. 29 406 715

2238 Dec. 22 356 406 2406 Dec. 11 406 718

2257 Jan. 1 356 371 2424 Dec. 21 406 712

2275 Jan. 12 356 378 2452 I an. 21 406 710

2461 Jan. 26 356 408 2470 Feb.   1 406 714

2479 Feb. 7 356 404 2488 Feb. 12 406 711

REFEREN CES

1. Roger W. Sinnott, letter of 1981 March 4 to Jean Meeus.

2. Jean Meeus, “Extreme Perigees and Apogees of the Moon", S and Telescope,

Vol. 62, pages 110-111 (August 1981).

 

Chapter 51

Passages of the Moon through the Nodes

When the center of the Moon passes through the ascending or through the descending node of its orbit, its geocentric latitude is zero. Approximate times of the passages through the nodes can be obtained as follows. The results will be expressed as a Julian Ephemeris Day, JDE, hence in Dynamical Time.

For a passage through the ascending node, take k = an integer. For a passage at the descending node, take for k rim integer increased by 0.5. Important .- any other value for k will give meaningless results!

Successive values of k will provide successive passages of the Moon through the nodes, the value k --- zero corresponding to the passage at the ascending node of 2000 January 21. Negative values of k yield passages before this date.

For instance, k ---- +223.0 and --147.0 correspond to an ascending node,

+223.5 and --46.5 to a descending node, while +44.76 is not a valid value for k.

An approximate value of k is given by

k -- (year -- 2000.05) X 13.4223 (51. l)

where “year" may be taken with decimals, for instance 2013.25. Then calculate

T k

1342. 23

and the following angles in degrees :

D --- 183.6380 + 331.737 356 82 k + 0.001 4852 T"-

+0.000 002 09 F3 -- 0.000 000 010 T’

M -- 17.4006 + 26.820 372 50 k + 0.000 1186 r 2 + 0.000 000 06 T3

M’ -- 38.3776 + 355.527 473 13 k + 0.012 3499 T’

+0.000 014 627 T’ -- 0.000 000 069 T

363

 

fi = 123.9767 — 1.440 989 56 k + 0.002 0608 r 2

+0.000 002 14 T° — 0.000 000 016 T’

Y = 299.75 + 132.85 T — 0.009 173 r 2

P —— R + 272.75 — 2.3 T

The time of the passage through the node is then given by the following expression, where the terms involving 3f (the Sun’s mean anomaly) should be multiplied by the quantity 6 given by formula (47.6). These terms are indicated by an asterisk.

JDE = 2451565.1619 + 27.212 220 817 k

+ 0.000 2762 T

+ 0.000 000 021 T’

— 0.000 000 000 088 T

— 0.4721 sin 3f’

— 0.1649 sin 2D

— 0.0868 sin (2D — M’)

+ 0.0084 sin (2D + M’)

• — 0.0083 sin (2D — M)

• — 0.0039 sin (2D — M — M')

+ 0.0034 sin 23f’

— 0.0031 sin (2D — 2M')

• + 0.0030 sin (2D + M)

• + 0.0028 sin (N - 3f’)

• + 0.0026 sin 3f

+ 0.0025 sin 4D

+ 0.0024 sin D

• + 0.0022 sin (3f + 3f’)

+ 0.0017 sin II

+ 0.0014 sin (4D — M')

• + 0.0005 sin (2D + M — M’)

• + 0.0004 sin (2D — M + 3f’)

• — 0.0003 sin (2D — 23f)

• + 0.0003 sin (4D — M)

+ 0.0003 sin F

+ 0.0003 sin P

 

51. MOON THROUGH THE NODES 365

Example 51.a — Calculate the instant of the passage of the Moon through the ascending node in May 1987.

Since mid-May corresponds to 0.37 year since the beginning of the current year, we put year = 1987.37 in formula (51.1), which yields the approximate value

— 170.19 for k. For a passage through the ascending node, k should be an integer, so we take k -— — 170. Then we find

T  = —0. 126 655

D  =  —56 211°.71264 = 308°.28736

M   = —4 542°.06272 = 137°.93728

M’  = —60 401°.29263 =  78°.70737

It   = 368°.9449 = 8.°9449

= 282°.92

P   = 641°.99 = 281°.99

E = 1.000319

The final result is JDE = 2446 938.76803, which corresponds to 1987 May 23.26803, or 1987 May 23, at 6h26".0 TD.

The correct value is May 23, at 6h25‘.6 TD.

The table below gives an idea of the accuracy of the results obtained by means of the algorithm given in this Chapter, as compared with the tlmes found by an accurate calculation.

Years (A.D.)

Node Number of instants Greatest error in seconds Number of errors

< 60 sec. Number of errors

> 120 sec.

1980 to 2020 ascending 551 142 487 3

1980 to 2020 descending 551 132 469 2

0 to 40 ascending 551

 

  5

0 to 40 descending 551 l3J 47B

 

 

 

 

Chapter 52

Maximum Declinations of the Moon

The plane of the Moon’s orbit forms with the plane of the ecliptic an angle of 5°. Therefore, in the sky the Moon is moving approximately along the ecliptic, and during each revolution (27 days) it reaches its greatest northern declination (in Taurus, in Gemini, or in northern Orion), and two weeks later its greatest southern declination (in Sagittarius or in Ophiuchus).

Because the lunar orbit forms with the ecliptic an angle of S°, and the ecliptic an angle of 23° with the celestial equator, the extreme declinations of the Moon are between 18° and 28° (North or South), approximately. When, as in 1987, the ascending node of the lunar orbit is in the vicinity of the vernal equinox (see page 344), the Menu reaches high northern and southern declinations, approximately +28'/z and --28'/z degrees. This situation is repeated at intervals of 18.6 years, the revolution period of the lunar nodes.

In this Chapter a method is given for the calculation of approximate times of the maximum declinations of the Moon, and the values of these extreme declinations. These data are geocentric and they refer to the center of the Moon’s disk.

Let k be an integer, negative before the beginning of the year 2000. Successive values of k will give successive maximum northern or southern declinations of the Moon. The value k -- 0 corresponds to January 2000. Important .- a non-integer value of k will give meaningless results!

An approximate value of 1 is given by

 

k -- (year -- 2000.03) x 13.3686

where “year" can be taken with decimals. Then calculate

  k

1336.86

 

(52. 1)

 

and the following angles, in degrees. The quantities between square brackets should be used for southern declinations.

367

 

368 ASTRONOMICAL ALGORITHMS

T A B L E 5 2. A

Periodic  terms (days)  for the  time of  the  Moon ’s maximum declination

Coefficient for

decli- decli-

nation nation

north south Coefficient for

decli- decli-

nation nation

north sowh

d d

+0.8975 —0.8975

—0.4726 —0.4726

—0. 1030  —0.1030

—0.0976 —0.0976

—0.0462 -I-0.0341

—0.0461 +0.0516

—0.0438 —0.0438

+0.0162   +0.0112

—0.0157 +0.0157

+0.0145   +0.0023

+0.0136 —0.0136

—0.0095 +0.0110

—0.0091  +0.0£i91

—0.£D89 +0.1XI89

+0.0£i75 +0.0£l75

—0.0368 —0.0030

+0.0361 —0.0£fi1

—0.0047 —0.0347

—0.0€i43  —0.0343

—0.0340 +0.0340

—0.0337  —0.€D37

+0.0£i31  —0.0031

cos F sin M' sin 2F

sin (2D — M')

cos (M’ — F)

cos (M’ + F)

sin 2D

sin M •

cos 3F

sin (M' + 2f’)

cos ‹2D — F)

cos (2D — ñf' — J)

cos (2D — M' + F)

cos (2D + N) sin 2M'

sin (M' — 2s)

zo' — F)

sin (3f’ + 3F)

sin {2D - M- M')

cos (3f' — 2F)

sin (2D — 2ñf’) sin F d

+0.£D30

—0.0029

—0.íXI29

—0.£D27

+0.IXI24

—0.íXl2l

+0.0019

+0.0018

+0.0fl18

+0.0017

+0.tXi17

—0.£Dl4

+0.tXI13

+0.tXll3

+0.0012

+o.ixin

—o.oon

+0.0010

+0.0010

—0.IXO9

+0.£fO7

—0.0007 d

+0.íXl30

+0.fD29

—0.0029

—0.0327

+0.0024

—0.€D21

—0. €D19

—0.0006

—0.0018

—0.fDl7

+0.0017

+ 0.fDl4

—0.0013

—0.£D13

+0.tX112

+o.lou

+o.ooii

+0.0010

+0.0010

—0.0£O9

—0.0£07

—0.tJ€O7

sin (2D + M')

cos (ñf' + 2F-)

sin (2D — M) sin (3f' + F) sin (M — M')

sin (M’ — 3I) sin (2M' + F)

cos (2D — 2ñf’ — F)

sin 3F

cos (M' + 3F)

cos 23f’

cos (2D — If')

cos (2D + M' + F)

cos M'

sin (3ñf’ + F)

sin {2D - M' + N)

cos (2D — 2M’)

• os {D + f’)

sin (M + M’) sin (2D - 2F) cos {2M’ + F) aos {33f' + F)

D —— 152.2029 + 333.070 5546 k - 0.000 4214 T* + 0.000 000 11 T

[345.6676]

M —— 14.8591 + 26.928 1592 k - 0.000 0355 T - 0.000 000 10 T’

[ 1.3951]

M’ -— 4.6881 + 356.956 2794 I + 0.010 3066 Z' 2 + 0.000 012 51 T’

[186.2100]

F —— 325.8867 + 1.446 7807 k — 0.002 0690 T — 0.000 002 15 T

[145.1633]

 

52. MAXIMUM   DECLINATIONS   OF MOON 369

TA B LE 5 2. B

Periodic terms (degrees) for the   value o f the Moon’s   maximum   declination

Coefficient for

decli- decli—

nation nation

norih south Coe icient for

decli- decli-

nation nation

north south

+5. 1093

—5. 1093

sin F

+0.£D38

—0.0038

cos (2M’ — F)

+0.2658 +0.2658 cos 2J —0.0034 +0.£D34 cos (M' — 2f)

+0.1448 —0.1448 sin (2D — F) —0.0029 —0.0029 sin 23f’

—0.0322 +0.0322 sin 3F +0.0029 +0.0029 sin (3M' + F)

+0.0133 +0.0133 cos (2D — 2F) —0.0€128 +0.0028 cos {2D + M — F)

+0.0125 +0.0125 cos 2D —0.£D28 —0.fD28 cos (M’ — F)

—0.0124 —0.fD15 sin (If' — F) —0.£D23 +0.0023 cos 3 F

—0.0101 +0.0101 sin (M’ + 2F) —0.£D2l +0.fO21 sin (2D + N)

+0.0£i97 —0.0097 cos F +0.£D19 +0.€Dl9 nos { ñf’ + 3 F)

—0.£D87 +0.0087 sin (2D + If — f’) • +0.£D18 +0.0018 zos {D + N)

+0.0074 +0.0074 sin (If’ + 3F) +0.0017 —0.lXJl7 sin (2ñf’ — F)

+0.0067 +0.0067 sin (D + f’) +0.0£i15 +0.£D15 cos (3M' + F)

+0.0063 —0.tXi63 sin (M' — 2F) +0.tX114 +0.£t014 cos (2D + 2ñf’ + F)

+0.0360 —0.OX›0 sin (2D — M — F) —0.0fl12 +0.£D12 sin (2D — 23f' — F)

—0.€D57 +0.£D57 sin (2D — M' — F) —0.0(112 —0.£D12 cos 2M'

—0.0056 —0.fD56 cos (M’ + F) —0.tXl10 +0.£D10 cos M'

+0.0052 —0.0052 cos (M' + 2F) —0.tXl10 —0.0(l10 sin 2f

+0.0041

—0.0040 —0.0041

—0.0340 cos (2M' + N) cos (M' — 3f’) +0.0tXfi +0.ID37 sin (M’ + f’)

The time of greatest northern or southern declination is then

JDE = 2451 562.5897 + 27.321582 247 I + 0.000 119 804 T2

[2451548.9289] — 0.000000 141 T

+ periodic terms of Table 52.A

In Table 52.A, the terms involving 3f, the Sun’s mean anomaly, should be multiplied by the quantity E given by formula (47.6). These terms are indicated by an asterisk.

The value of the greatest declination, in degrees, is

fi = 23.6961 — 0.013 004 T + periodic terms of Table 52.B.

Here, again, the terms indicated by an asterisk should be multiplied by E. Note that the absolute value of the maximum declination is obtained; in the case of a greatest southern declination, this declination thus is not affected by the minus sign.

 

3 7 O ASTRONOMICAL ALGORITHMS

Example 52.a — €ireatest northern declination of the Moon in December 1988.

Inserting the value year = 1988.95 in formula (52. l), we get k = — 148.12, so we take k —— — 148. We then find

T -- —0.110707 M' -- —52 824.°8411 = 95°.1589

D —- —49 142°.2392 = 177°.7608 N = 111.°7631

If = —3 970.°5085 =  349°.4915 E -- 1.000 278

We obtain JDE = 2447518.3347, which corresponds to 1988 December 22.8347

= 1988 Dec. 22 at 20‘02" TD. The correct value is December 22 at 20'01‘ TD.

For the value of that maximum northern declination, we obtain 28°.1562, or

+28°09’22". The correct value is +28° 09' 13“.

ñzompfe 52.b — If we calculate the maximum southern declination for k -- + 659, we get JDE = 2469 553.0834, which corresponds to 2049 April 21 at 14 Dynamical Time, and 6 = 22°.1384, so the greatest southern declination is  —22° 08'.

Example 52.c — To find the Moon’s greatest northern declination of mid-March of the year —4, we have “year” = 0.20 year after the beginning of

the year —4, so “year” = — 4 + 0.20 = —3.80, and not —4. 20 !

This gives for k the approximate value —26 788.40, whence k -— —26 788 (an integer!).

We then obtain JDE = 1719 672.1414, which corresponds to March 16 at 15h TD of the year —4 ;

greatest northern declination = 28°.9739 = +28° 58’.

Using the method described in this Chapter, 600 maximum northern and 600 maximum southern declinations were calculated, from 1977 August to 2022 June. The maximum errors were 10 minutes for the time, and 26" for the value of the maximum declination. For 69 7• of the cases, the calculated time was less than 3 minutes in error, and in 74 $• of the cases the calculated declination was less than 10” in error.

The coefficients of the periodic terms in Tables 52. A and 52.B have been calculated using for the obliquity of the ecliptic its value for the epoch 2000.0. As a consequence, the error resulting from using these terms will increase with time, but the maximum possible error will not exceed half ar hour between the years

— 1000 and +5000.

 

Chapter S3

Ephemeris for Physical Obsewations  of the Moon‘

Optical Vibrations

The mean period of rotation of the Moon is equal to the mean sidereal period of revolution around the Earth, and the mean plane of the lunar equator intersects the ecliptic at a constant inclination, i, in the line of nodes of the lunar orbit, with the descending node of the equator at the ascending node of the orbit.

On the average, therefore, the same hemisphere of the Moon is always turned towards the Earth. However, apparent oscillations known as optical librations, which are due to variations in the geometric position of the Earth relative to the lunar surface during the course of the orbital motion of the Moon, allow about 59 4 of the surface to be observed from the Earth.

The mean center of the Moon’s apparent disk is the origin of the system of selenographic coordinates on the surface of the Moon. Selenographic longitudes are measured from the lunar meridian that passes through the mean center of the apparent disk, positive in the direction towards Mare Crisium, that is, towards the west on the geocentric celestial sphere. Selenographic latitudes are measured from the lunar equator, positive towards the north, that is, they are positive in the hemisphere containing Mare Serenitatis.

The displacement, at any time, of the mean center of the disk from the apparent center, represents the amount of libration, and is measured by the selenographic coordinates of the apparent center of the disk at that time.

The selenographic longitude and latitude of the Earth, as given in the almanacs, are the geocentric selenographic coordinates of the apparent central point of the disk. At this point on the surface of the Moon, the Earth is in the zenith. When the libration in longitude, that is, the selenographic longitude of the Earth, is positive, the mean central point of the disk is displaced eastwards on the celestial sphere, exposing to view a region on the west limb. When the libration in latitude, or the selenographic latitude of the Earth, is positive, the mean central point of the disk is displaced towards the south, and a region on the north limb is exposed to view.

The optical librations in longitude (l’) and in latitude ib’) can be calculated as follows. Let

371

 

I = the inclination of the mean lunar equator to the ecliptic, 1°32’32”.7 or 1.°54242. This is the value adopted by the International Astronomical Union;

k = apparent geocentric longitude of the Moon;

;S = apparent geocentric latitude of the Moon; AQ = nutation in longitude (see Chapter 22);

F —— argument of latitude of the Moon, obtained from (47.5);

II = mean longitude of the ascending node of the lunar orbit, obtained from formula (47.7).

Then we have

w = h — AJ — fi

 

A  -—

 

   Sir1 lY COS Q cos i — sin Q sin i cos lY cos Q

 

(53. 1)

 

l' —— A — F

sin b’ —— -  sin W cos 9 sin — sin 9 cos J

In the calculation of k, the effect of the nutation is supposed to be included, so k — AQ represents, in fact, the “apparent longitude of the Moon without the effect of the nutation".

Physical Vibrations

There is an actual rotational motion of the Moon about its mean rotation; this is called the physical libration. The physical libration is much smaller than the optical libration, and can never be larger than 0.04 degree in both longitude and latitude.

The physical librations in longitude (l“) and in latitude {b”) can be calculated as follows, and the totallibrations are the sums of the optical and physical librations:

l = /' + I“, b = b' + #”.

Calculate the quantities p, o, and z (in degrees) by means of the following expressions from D. H. Eckhardt [1], where the angles D, M, and M’ are obtained by means of expressions (47. 2) to (47. 4); find E by means of (47.6), and the angles K and K, (in degrees) from

KB     =    119.75  +  131.849  T

x 2 =    72.56 +    20.186 T

 

where, as elsewhere in this book, T is the time measured in Julian centuries of 36525 days from the Epoch J2000.0 = JDE 2451545.0.

 

p — —0.02752 cos M’

—0.02245 sin F

+0.00684 cos (3f' — 2 F)

—0.00293 cos 2F

—0.00085 cos (2F — 2D)

—0.00054 cos (M' — ID)

—0.00020 sin (Of’ + F)

—0.00020 cos (M’ + 2F)

—0.00020 cos (Of’ — F)

+0.00014 cos (M’ + 2F — 2D)

« = —0.02816 sin 3f’

+ 0.02244 cos

—0.00682 sin {M’ — 2N)

—0.00279 sin 2F

—0.00083 sin (2F — 2D)

+0.00069 sin (3f’ — 2D)

+0.00040 cos (Of’ + J)

—0.00025 sin 23f’

—0.00023 sin (3f' + 2F)

+0.00020 cos (3f' — F)

+0.00019 sin {M' - F)

+0.00013 sin (Of’ + 2s — 2D)

—0.00010 cos (3f' — 3F)

 

r =   +0.02520 E sin M

+0.00473 sin (2N' - 2F)

—0.00467 sin M'

+0.00396 sin K

+0.00276 sin (2ñf’ — 2D)

+0.00196 sin II

—0.00183 cos (If' — F)

+0.00115 sin (3f' — 2D)

—0.00096 sin (3f' — D)

+0.00046 sin (2F — 2D)

—0.00039 sin (3f' — F)

—0.00032 S1f1 (If' — M — D )

+0.00027 S1f1 (23f’ — M -   2fi)

+0.00023 sift K2

—0.00014 sin 2D

+0.00014 cos (2If' — 2 F)

—0.00012 sin {M’ — 2 F)

—0.00012 sin 2ñf'

+0.00011 sin (23f’ — 23f — 2D)

 

 

Then we have

I” = — r + (p cos A +w sin A) tan b’ b” —- o cos A - p sin A

Position Angle of Axis

 

(53.2)

 

The position angle of the Moon’s axis of rotation, P, is defined as for the planets — see Chapters 42 and 43. lt can be calculated as follows; the effect of the physical libration is taken into account.

i, II, AJ, p, o, and b have the same meaning as before. Let o be the apparent geocentric right ascension of the Moon, and e the true obliqtiity of the ecliptic. Then

 

 

” '   + + sin I

X = min (? + p) sin Y

F = sin (i + p) cos Y cos e -- co• {I + p) sin e

2 + F 2 cos (n -- u›) cos b

The angle o can be obtained in the correct quadrant by using the “second” arctangent function: o = ATN2 (X, Y). If this function is not available, divide X by Y, take the usual arctangent of the result, then add 180° if F < 0.

The position angle f• is to be taken in the first or in the fourth quadrant, that is, either between 0 and 90 degrees, or between 270 and 360 degrees.

Example 53.a -- The Moon on 1992 April 12, at 0h TD. For this instant we have (see Example 47.a) :

 

D 113°.842 304

M   -- 97°.643514

3f'   = 5.°150833

F -- 219°.889 721

A/   =   +0°.004610

E 1.000 194

Then we obtain:

 

=  133.° 167 265

= --3.°229 126

=  133.° 162655

= 23.°440636

= 134.°688 470

 

II IV A

l' =

=

----

-- 274°.400656

218.°761 999

218°.683932

-- 1.°206

 

b"

 

I› =

=

=

---- --0°.025

+0°.006

--1.°23

+4°.20

b’ -- +4°.194

  = 273°.820507

K I = 109°.57 f+p = 1°.53200

K ---- 71°.00

  = --0.026676

p -- --0.01 042 Y = --0.396 022

= --0.01574

  = 183°.8536

z = +0.02673

  = 15°.08

 

Topocentric fiârofions

For precise reductions of observations, the geocentric values of the librations and position angle of the axis should be reduced to the values at the place of the observer on the surface of the Earth. For the librations, the differences may reach 1° and have important effects on the limb-contour.

The topocentric librations in longitude and latitude, and the topocentric position angle of the axis, may be calculated either by direct calculation or by differential corrections of the geocentric values.

a. Direct calculation. — The formulae given before are used, but the geocentric coordinates h, 9, n of the Moon are replaced by the topocentric ones. For this purpose, the topocentric right ascension and declination of the Moon are obtained by means of formulae (40.2) and (40.3); then they are transformed to the ecliptical coordinates h and 9 by the usual conversion formulae (13.1) and (13.2) to obtain the topocentric longitude and latitude.

b. Differential corrections. — Let p be the observer’s latitude, h the geocentric declination of the Moon, H the local hour angle of the Moon (calculated from the local sidereal time and the geocentric right ascension), and r the geocentric horizontal parallax of the Moon. Then calculate

  cos e sin H

cos 6 sin ‹;o — sin fi cos p cos If cos z = sin 6 sin ‹,r + cos é cos p cos H

w' = z (sin z + 0.0084 sin 2z)

Then the corrections to the geocentric librations (I, b) and to the position angle

(P) are

qJ — r’ sin (O — P) 

cos b

lb -— + r'  cos (t2 — P)

IP -- + b l sin (b + lb) — z’ sin 9 tan b

These formulae were given in Reference [2].

 

The selenographic position of the Sun

The selenographic coordinates of the Sun determine the regions of the lunar surface that are illuminated.

The selenographic longitude I and latitude bp of the subsolar point on the lunar surface — the point where the Sun is at the zenith — are obtained by replacing, in the formulae (53.1) for the selenographic coordinates of the Earth, the geocentric ecliptical coordinates h, 9 of the Moon by the heliocentric ecliptical coordinates hH , 9 H of the Moon. With sufficient accuracy we have

kH   =   h   + 180° + x 57.°296 cos fi sin (h — h)

where k0 iS the apparent geocentric longitude of the Sun. The fraction A/It is the ratio of the distance Earth—Moon to the distance Earth —Sun; hence, A and A should be expressed in the same units, for instance kilometers. If, instead, R is expressed in astronomical units, and w is the equatorial horizontal parallax of the Moon expressed in seconds of arc (“), the fraction b/R is equal to 8.794/ rfi.

Hence, to find IO and f›0, first calculate hH and b H . Then use expressions (53.1), replacing h by hH , and 9 by 9p ; this will give I' and b’p. The quantities p, o, and z are found by the unchanged expressions, and finally /”0 and b” by (53.2), using b'o instead of b’. Then

o ' ’o +  ”o dlld bQ- b' + b”

Subtracting ft from 90° or 450‘ gives the selenographic colongitude of the Sun (cp), which is tabulated in some ephemerides.

The quantities (or • ) and be determine the exact position of the terminator on the surface of the Moon. The subsolar point at lb, by is the pole of the great circle on the lunar surface that bounds the illuminated hemisphere. The morning

terminator, where the Sun is rising on the Moon, is at selenographic longitude lb — 90°, or 360° — c The evening terminator, where the Sun is setting, is at longitude ft + 90°, or 180° — c0. When eg = 0°, the Sun is rising at selenographic longitude 0°; this occurs near First Quarter. At Full Moon, Last Quarter, and New Moon, respectively, ct is approximately 90°, 180°, and 270°, and the morning terminator is approximately at selenographic longitudes 270', 180°, and 90°.

Note that, while /0 iS decreasing with time, the colongitude c0 iS increasing.

Their mean daily motion is equal to that of the Moon’s mean elongation D, namely

12.190 749 degrees.

 

At a point on the lunar surface at selenographic longitude 9 (positive towards Mare Crisium!) and latitude 8, sunrise occurs approximately when       -   360‘ — 9 or c0 = — q, noon when c0 = 90° — p, and sunset when ct — 180° — p. The altitude h of the Sun above the lunar horizon at any time may be calculated from

sin h —— sin b0 Sin 8 + cos b0 cos 8 sin (ct + q) (53.3)

To find the time of sunrise or sunset for a given place on the Moon, calculate the Sun’s altitude h for that place for an approximate time. Then the correction to the assumed time, in days, is

 

  h

12.19075 cos 8

 

(53. 4)

 

where the upper sign is to be used for sunrise, the lower sign for sunset. The altitude h of the Sun should be taken with proper sign and be expressed in degrees. If needed, the calculation should be repeated, starting with the new time found. The time so obtained it that of the rise or set of the center of the solar disk.

Exazrtple 53.b — The Moon on 1992 April 12, at 0‘ Dynamical Time.

For this instant we have, from accurate calculations (VSOP 87 and ELP-2000/82), k$ = 22°.33978

A = 368 406 kilometers

fi = 1.002497 69 AU = 149 971500 km

The other relevant quantities were found in Example 53.a. We then find

XH   = 202°.208438 f' 0 = 67.°920 /d = 67.°89

EH ' —0°.007932 b’z —— -A1.°476 t›d = +1.°46

W = 287°.803172 f“ 0 = —0.°026 cd = 22.°11

A -- 287°.809 283 b“0 = —0.°015

Exam ple 53.c — Sunrise for the crater Copernicus in April 1992.

The selenographic coordinates of this crater are (Table 53.A) 9 = —20°. 0, 8 =

+ 9.°7. Sunrise occurs approximately when the Sun’s selenographic colongitude ct is

—9, or + 20.°0 in the present case. This is almost the value found for 1992 April 12 at 0‘ TD in Example 53.b. For this instant we found f›q = + l °. 46 and c = 22°.11.

For these values, formula (53.3) gives /i = + 2.°3253 (keeping extra decimals), whence a correction of —0. 1935 day by formula (53.4), gi ving 1992 April 11.8065.

For this improved time, a new calculation gives bt = + 1°. 46, ct = 19°.75, whence a value of the Sun’s altitude h which is practically zero.

Hence, no other iteration is needed. The required time is 1992 April 11.8065, or 1992 April 11 at 19' TD. This is also 19 h Universal Time.

 

TA B L E 5 3. A

Selenographic coordinates o f some lunar   features

Nome

  8 Name

Archimedes

— 3.9

+29.7

Lansberg

—26.6

— 0.3

Aristarchus —47.5 +23.7 Letronne —43 — 10

Aristillus + 1.2 + 33.9 Macrobius +46.0 +21.2

Aristoteles +17.3 +50.1 Manilius + 9.1 + 14.5

Arzachel — 1.9 —17.7 Menelaus + 16.0 + 16.3

Autolycus + 1.5 + 30.7 Messier +47.6 — 1.9

Billy —50.0 —13.8 Petavius + 61 —25

Birt — 8.5 —22.3 Pico — 8.8 +45.8

Campanus —27.7 —28.0 Pitatus — 13.5 —29.8

Censorinus + 32.7 — 0.4 Piton — 0.8 +40.8

Clavius —14 —58 Plato — 9.2 +51.4

Copernicus —20.0 + 9.7 Plinius +23.6 + 15.3

Delambre + 17.5 — 1.9 Posidonius + 30.0 + 31.9

Dionysius + 17.3 + 2.8 Proclus + 46.9 + 16.1

Endymion +56.4 +53.6 Rolemaeus A — 0.8 — 8.5

Eratosthenes —11.3 +14.5 Pytheas —20.6 +20.5

Eudoxus +16.3 + 44.3 Reinhold —22.8 + 3.2

Fracastorius + 33.2 —21.0 Riccioli —74.3 — 3.2

Fra Mauro — 17 — 6 Schickard —54.5 —44.0

Gassendi —39.9 — 17.5 Schiller —39 —52

Goclenius +45.0 —10. 1 Taruntius + 46.5 + 5.6

Grimaldi —68.5 — 5.8 Theophilus + 26.5 — 11.4

Harpalus —43.4 +52.6 Timocharis —13. l +26.7

Horrocks + 5.9 — 4.0 Tycho —11.0 —43.2

Kepler —38.0 + 8. l Vitruvius + 31.3 + 17.6

Langrenus + 60.9 — 8.9 Walter + 1 —33

REFEREN CEO

1. D. H. Eckhardt, “Theory of the Libration of the Moon”, Soon and Planets,

Vol. 25, page 3 (1981).

2. Explanatory Supplement to the Astronomical Ephemeris (London, 1961), page 324.

 

Ch Rter 54

Eclipses

Without too much calculation, it is possible to obtain with good accuracy the principal characteristics of an eclipse of the Sun or the Moon. For a solar eclipse, the situation is complicated by the fact that the phases of the event are different for different observers at the Earth’s surface, while in the case of a lunar eclipse all observers see the same phase at the same instant.

For this reason, we will not consider here the calculation of the local circumstances of a solar eclipse. The interested reader may calculate these circumstances from the Besselian elements published yearly in the Astronomical Almanac. Besselian elements for all solar eclipses of the years —2003 to + 2526 can be found in the Canon by Mucke and Meeus [1]. For modern times, accurate Besselian elements have been published by Meeus [2). Besides the elements, these works provide the formulae needed for their use, together with numerical examples.

Espenak published a Canon [3] giving data about the paths of total and annular solar eclipses from 1986 to 2035, with beautiful world maps for all eclipses in that period. This work does not contain Besselian elements, however, so it does not provide the possibility to calculate extra data, such as local circumstances for places outside the path of total or annular phase.

Let us also mention the work by Stephenson and Houlden [4], which contains data and chans for the total and annular eclipses visible in East Asia from 1500

B.C. to A.D. 1900.

General data

First, calculate the instant (JDE) of the mean New or Full Moon, by means of formulae (49. 1) to (49.3). Remember that k must be an integer for a New Moon (solar eclipse), and an integer increased by 0.5 for a Full Moon (lunar eclipse).

Then, calculate the values of the angles M, M’, F, and II for that instant by means of expressions (49.4) to (49.7), and the value of E by formula (47.6).

379

 

3 8 O ASTRONOMICAL ALGORITHMS

The value of I will give the first information about the occurrence of a solar or lunar eclipse. If I differs from the nearest multiple of 180° by less than 13.9 degrees, then there is certainly an eclipse; if the difference is larger than 21 .°0, there is no eclipse; between these two values, the eclipse is uncertain at this stage and the case must be examined further. Use can be made of the following rule: there is no eclipse if | sin I | > 0.36.

Note that after one lunation the angle F increases by 30.6705 degrees.

If I is near 0° or 360°, the eclipse occurs near the Moon’s ascending node. If F is near 180°, the eclipse takes place near the descending node of the Moon’s orbit.

Calculate

F = F — 0°.02665 sin Al

A = 299°.77  + 0°.107 408 k - 0°.009 173 T2

Then, to obtain the time of maximum eclipse (for the Earth generally in the case of a solar eclipse), the following corrections (in days) should be added to the time of mean conjunction or opposition given by expression (49.1).

 

—0.4075 x   sin M’ for lunar eclipses, change these

+0.1721   X E M constants to —0.4065 and +0. 1727

+0.0161 23f'

—0.0097 2Nt

+0.0073  x  E M'  -   M

—0.0050 M’ + M

—0.0023 M’ - 2F

+0.0021   x £ 23f

+0.0012 M’ +  2F

+0.0006  x  f’ 23f’ + M

—0.0004 33f’

—0.0003 x E M + 2F

+0.0003 A

—t).QQQ2 x E If — 2Ft

—0.0002 x E 23f’ — 3f

—0.0002 II

 

(54. 1)

 

This algorithm should not be used, of course, if high accuracy is needed. For the 221 solar eclipses of the years A.D. 1951 to 2050, the method gives a mean error of 0.36 minute, and a greatest error of 1. 1 minute in the times of maximum eclipse.

 

Then calculate

 

P —- +0.2S070

 

E   X sin M

 

9 =   +5.2207

 

+0.0024 X E sin 23f

—0.0392 sin M’

+ 0.0116 sin 23f’

—0.0073 X E sin (3f’ + 3f)

+0.0067 sin {M' — M)

+0.0118 sin 2F

 

—0.0048 X E   S cos M

+ 0.0020 x E cos 23f

—0.3299 cos ñf’

—0.0060 x F cos (If' + M)

+0.0041 x E cos (N’ - M)

 

 

y = (I cos Al + Q sin F ) x (1 — 0.0048 W)

ti = 0.0059

+ 0.0046 E cos M

— 0.0182 cos N'

+ 0.0004 cos 23f'

— 0.0005 cos {M + M’ )

Solar eclipses

In the case of a solar eclipse, represents the least distance from the axis of the Moon’s shadow to the center of the Earth, in units of the equatorial radius of the Earth. The quantity y is positive or negative, depending upon the axis of the shadow passing north or south of the Earth’s center. When is between + 0.9972 and

—0.9972, the solar eclipse is central: there exists a line of central eclipse on the Earth’s surface, and for observers on this line the center of the lunar disk passes exactly over the center of the solar disk.

The quantity u denotes the radius of the Moon’s umbral cone in the fundamental plane, again in units of the Earth’s equatorial radius. The radius of the penumbral cone in the fundamental plane is o + 0.5461. The fundamental plane is the plane through the center of the Earth and perpendicular to the axis of the Moon’s shadow.

If | | > 1.5433 + u, no eclipse is visible from the Earth’s surface.

If | y | is between 0.9972 and 1.5433 + ti, the eclipse is not central. In most cases, it is then a partial eclipse. However, when | y | is between 0.9972 and 1.0260, a part of the umbral cone may touch the surface of the Earth (within the polar regions), while the axis of the cone does not touch the Earth. These non- central total or annular eclipses occur when 0.9972 < | y | < 0.9972 + | ti | . Between the years 1950 and 2100, there are seven eclipses of this type:

 

1950 March 18

1957 April 30

1957 October 23

1967 November 2

2014 April 29

2043 April 9

2043 October 3

 

annular, not central annular, not central total, not central total, not central annular, not central total, not central annular, not central

 

In the case of a central eclipse, the type of the eclipse can be determined by the following rules: if ii < 0, the eclipse is total; if u > +0.0047, it is annular; if u is between 0 and +0.0047, the eclipse is either annular or annular-total. In the latter case, the ambiguity is removed as follows. Calculate

= 0.00464 92 > 0

Then, if ii < u›, the eclipse is annular-total; otherwise it is an annular one.

In the case of a partial solar eclipse, the greatest magnitude is attained at the point of the surface of the Earth which comes closest to the axis of shadow. The magnitude of the eclipse at that point is

 

1.5433 + u — | y

0.5461 + 2it

 

(54.2)

 

Lunar eclipses

In the case of a lunar eclipse,    represents the least distance from the center of the Moon to the axis of the Earth’s shadow, in units of the Earth’s equatorial radius. The quantity y is positive or negative depending upon the Moon’s center passing north or south of the axis of the shadow. The radii at the distance of the Moon, again in equatorial Earth radii, are

for the penumbra : p — 1.2848 + if for the umbra : w = 0.7403 — u

while the magnitude of the eclipse may be found as follows:

 

for penumbral eclipses :

for umbral eclipses :

 

1.5573 + u - J 9 J  

0.5450

1.0128 — u — | y

0.5450

 

(54.3)

(54.4)

 

If the magnitude is negative, this indicates that there is no eclipse.

The semidurations of the panial and total phases in the umbra can be found as follows. Calculate

p —— 1.0128 — u i = 0.4678 — u

n = 0.5458 + 0.0400 cos 3f’

Then the semidurations in minutes are

parti al phase : 60 p 2 2 total phase:     — 2 92

For the semiduration of the partial phase in the penumbra, find h —— 1.5573 + u, and then the semiduration in minutes is

The semidurations are the time intervals between the beginning (or end) of the partial phase, the beginning (or end) of the total phase, or the first (or last) contact with the penumbra and the instant of morimiim eclipse. So, for instance, in the case of a total eclipse in the umbra, the semiduration of the partial phase does include half the duration of the phase of totality.

Further, it must be noted that the contacts of the Soon with the penumbra cannot be observed, and that most penumbral eclipses (in which the Moon enters only the penumbra of the Earth) cannot be discerned visually. Only at eclipses occurring deep in the penumbra can a weak shading of the Moon’s northern or southern limb be seen.

In the formulae given above, the increase of the theoretical radii of the shadow cones by the Earth’s atmosphere is taken into account. However, instead of the traditional rule consisting of increasing by 1/50 the theoretical radii, we have preferred the method used since 1951 in the French almanac Connaissance des Temps — see for instance Reference [5]. As compared with the results of the “French rule", the magnitude of a lunar eclipse calculated by using the traditional rule is too large by about 0.005 for umbral eclipses, by about 0.026 for penumbral eclipses.

To obtain the results according to the traditional rule (l/50), the following changes should be made to the constants in the expressions given above:

replace 1.2848 by 1.2985

0.7403 by 0.7432

1.5573 by 1.5710

1.0128 by 1.0157

0.4678 by 0.4707

 

For the predictions of lunar eclipses, such as those published in the various almanacs, it is customary to assume the penumbra and the umbra to be exactly circular, and to use a mean radius for the Earth. In fact, the shadow differs somewhat from a circular cone as the Earth is not a true sphere. By simple geometrical considerations, it is found that the Earth’s shadow, at the Moon’s distance, must be more flattened than the terrestrial globe, the mean value for the flattening of the umbra being 1/214 [6]. The true flattening of the umbra is perhaps even larger still. Soulsby [7] finds a mean oblateness of 1/ 102 from observations made at 18 lunar eclipses in the period 1974 -1989.

Example 54.a -- Solar eclipse of 1993 May 21.

May 21 being the l4lth day of the year, the given date corresponds to 1993.38.

Formula (49.2) then gives k -- --81.88, whence k 82.

Then, by means of formulae (49.3) and (49.1), JDE = 2449 128.5894. Further,

M ---- 135.°9142

M’ ---- 244.°5757

I = 165.°7296

II = 253°.0026

F  -- 165°. 7550

Because 180° -- N is between 13.°9 and 21°.0, the eclipse is uncertain at this stage. We further find

P +0.1842

Q_ = +5.3589

= + 1.1348

u = +0.0097

Because | | is between 0.9972 and 1.5433 + u, the eclipse is a partial one.

Using formula (54.2), we find that the maximum magnitude is

 1.5433 + 0.0097 -- 1.1348     = 0.740

0.5461 + 0.0194

Because F is near 180°, the eclipse occurs near the Moon’s descending node. Because is positive, the eclipse is visible in the northern hemisphere of the Earth.

To obtain the time of maximum eclipse, we add to JDE the terms given by formula (54.1).  This gives

JDE = 2449 128.5894 + 0.5085 = 2449 129.0979

which corresponds to 1993 May 21, at 14'21°'.0 TD.

The correct values, resulting from an accurate calculation [2], are 14'20”14’ Dynamical Time, = + 1.1370, and a maximum magnitude of 0.735.

 

 

Example 54.b -- Solar eclipse of 2009 July 22.

As in the preceding Example, we find:

k -- 118

JDE = 2455034.7071

3f = 196.°9855

if’   = 7.°9628

F ---- 179.° 8301

F - 179°.8531

Corrected JDE = 2455034.6088 = 2009 July 22, at 2'37‘" TD.

P 0.0573

Q - +4.9016

7  =  +0.0695

o = 0.0157

Because | y | < 0.9972, the eclipse is central. Because u is negative, the eclipse is total. Because | y | is small, the eclipse is visible from the equatorial regions of the Earth. Because N is near 180°, the eclipse takes place near the descending node of the Moon’s orbit.

Example S4.c -- Lunar eclipse of June 1973. We find successively:

k 328.5

JDE = 2441 849.2992

If = 161°.4437

M’ ---- 180°.7018

F -- 345°.4505

Corrected JDE = 2441 849.3687 = 1973 June 15, at 20'5l’" TD.

= -- 1.3249 u = + 0.0197

The eclipse took place near the Moon’s ascending node (because F -- 360°) and the Moon’s center passed south of the center of the Earth’s umbra (because y < 0).

According to formula (54.4), the magnitude in the umbra was --0.609. Since this is negative, there was no eclipse in the umbra. Using formula (54.3), we find that the magnitude in the penumbra was 0.462. Hence, the eclipse was a penumbral one.

According to the Connaissance des Temps, maximum eclipse occurred at 20‘50"'.7 Dynamical Time, and the magnitude in the penumbra was 0.469.

 

 

Example 54.d -- Find the first lunar eclipse after 1997 July 1.

For 1997.5, formula (49.2) gives k -- --30.92, so we must try the value k ----

--30.5. This gives F --- 125°.2605, which differs more than 21 degrees from the nearest multiple of 180°, and hence gives no eclipse.

The next Full Moon, k ---- --29.5, gives F --- 155°.9310, hence again no eclipse. But it is evident that the next Full Moon will give F -- 187° and thus give rise to an eclipse. We then find, as before:

k         28.5

JDE = 2450 708.4759

3f = 253.°0507

M’ ---- 5.°7817

F ---- 186.°60l5

Corrected JDE = 2450 708.2835 = 1997 September 16, at 18‘48‘.2 Dynamical Time, or 18‘47‘ UT (if we adopt the value b T ----  TD -- UT = + 63 seconds).

= --0.3791 o = 0.0131

Formula (54.4) then gives a magnitude of 1.187, so the eclipse is total in the umbra.

p ---- 1.0259 i = 0.4809 h ---- 1.5442 n 0.5856

 

Semiduration of partial phase:

0.5856 (1 0259) 2 (0 3791) 2

Semiduration of total phase:

 

= 98 minutes

 

   60

0.5856

 

(0 4809) 2 (0 3791) 2

 

= 30 minutes

 

Semiduration of penumbral phase:

 

     60

0.5856

 

(1 5 )2 (0 3791) 2

 

= 153 minutes

 

Hence, in Universal Time:

first contact with the penumbra :    18‘47” -- 153‘ =   16' 14"

first contact with the umbra : 18 47” --   98‘   =   17 09‘

beginning of total phase- 18‘47” -- 30‘ -- 18'17‘ maximum of the eclipse : l8‘47'"

end of total phase : 18‘47” + 30m =   19‘17" last contact with the umbra : 18 47” + 98” = 20 25" last contact with the penumbra :      18'47” + 153”   =   21‘20"'

 

Notes about the accuracy

The algorithms given in this Chapter are not intended to give highly accurate results. Still, for lunar eclipses the results will be precise enough for historical research, or when high accuracy is not needed. On the other hand, as has been said at the beginning of this Chapter, accurate data for modem solar eclipses can be obtained by using our Elements of Solar EClipseS [2].

The formula given for 2 does not yield rigorously exact results. This is quite evident, if we consider the fact that only twelve periodic terms are used to calculate the quantities P and @, while in fact hundreds of terms are needed to obtain accurate positions of the Sun and the Moon. Even formulae (54.2), (54.3), and (54.4), and the expressions for the quantities p, I, n, and /t are not rigorously exact.

For the 221 solar eclipses of the period 1951 —2050, the mean error of the values of as calculated by using the algorithm of this Chapter is 0.00065, while the maximum error is 0.0024, which corresponds to 15 kilometers. Considering the simplicity of our formulae, this accuracy is quite satisfactory.

From what precedes, it results that in limiting cases the type of an eclipse will still be unknown. In such a case, an accurate calculation is needed to settle the question.

Further, in a search procedure for eclipses, a small safety margin should be considered in order to be sure that no eclipse will be overlooked. For instance,

while the correct condition for a central solar eclipse is indeed I I < 0.9972 (*), a limiting value of 1.000 or even 1.005 should be used in order to find all possible central eclipses when use is made of the value of y obtained with the method described in this Chapter.

Here are some examples.

For the solar eclipse of 1935 January 5 (k -- —804), our method gives 2 =

— 1.5395 and u = —0.00464, whence | 2 | > u + 1.5433 = 1.5387, so we might think there was no eclipse on that date. Formula (54.4) yields the value —0.002 (negt2rive !) for the maximum magnitude. The correct value of y was — 1.5383, however, so there was a very small panial solar eclipse on 1935 January 5, with a maximum magnitude of only 0.001.

For the annular solar eclipse of 1957 April 30 (k = —528), our algorithm yields the value y = +0.9966, so one might think this was a central eclipse. The exact value was 9 = +0.9990, so it was actually a non-central annular event.

For the lunar eclipse of 1890 November 26 (k —— — 1349.5), our algorithm gives a magnitude (in the umbra) of —0.007. In fact, it was a very small partial eclipse in the umbra.

(*) In fact, the “constant” 0.9972 may vary between 0.9970 and 0,9974 from one eclipse to another.

 

Exercises

Find the first solar eclipse of the year 1979, and show that it was a total one visible from the northern hemisphere.

Was the solar eclipse of April 1977 a total or an annular one? Show that there was no eclipse of the Sun in July 1947.

Show that there are four solar eclipses in the year 2000, and that all four are partial eclipses.

Show that there will be no lunar eclipse in January 2008. Show that there were three total eclipses of the Moon in 1982.

Find the first lunar eclipse of the year 1234. (Answer: the partial lunar eclipse of 1234 March 17).

RE FER E N CES

1. H. Mucke, J. Meeus, Canon of Solor Eclipses, —2003 to  +2526; Astronomisches Büro (Wien, 1983).

2. J. Meeus, Elements of Solar Eclipses, 1951 to 22£O (Willmann-Bell, ed.; 1989).

3. F. Espenak, Fifty Year Canon of Solar Eclipses.’ 1986-2035 ,- NASA Reference Publication 1178 (Washington, 1987).

4. F.R. Stephenson, M.A. Houlden, Atlas of Historical Eclipse Maps ; Cambridge University Press (1986).

5. A. Danjon, “Les éclipses de Lune par la pénombre en 1951“, l’Astronomie, Vol. 65, pages 51—53 (February 1951).

6. J. Meeus, “Die Abplattung des Erdschattens bei Mondfinsternissen", Die Sterne,

Vol. 45, pages 116—117 (1969).

7. B. W. Soulsby, Journal of the British Aslronomical Association, Vol. 100, page 297 (December 1990).

 

Chapter 55

Semidiameters of the Sun, Moon, and Planets

Sun and Planets

The angular semidiameters s of the Sun and planets are calculated from

s — SO

where â0 is the body’s semidiameter at unit distance (1 AU), and A the body’s distance to the Earth in AU.

For the Sun, the value adopted in the calculations is [1]

i 0 = 15’ 59“.63 = 959”.63

For the planets, the following values of â0 have been used for many years [2) :

 

Mercury 3.34

Venus 8.41

Mars 4.68

Jupiter :

equatorial 98.47

polar 91.91

 

Saturn :

equatorial 83.33

polar 74.57

Uranus 34. 28

Neptune 36.56

 

(A)

 

Later, the following values have been adopted [3]:

389

 

Mercury 3.36 Satum :

Venus 8.34 equatorîal polar 82.73

73.82

Mars 4.68 ( B I

Jupiter : Uranus 35.02

equatorial 98.44 Neptune 33.50

polar 92.06 Pluto 2.07

Note that, according to the latter values, Neptune is smaller than Uranus.

For Venus, the value 8”.34 refers to the planet’s crust, not to the top cloud level as seen from the Earth. For this reason, we prefer to use the older value 8”.41 for Venus when calculating astronomical phenomena such as transits and occultations.

In the case of Saturn, let o and b be the equatorial and the polar semidiameters at unit distance. Then, while the apparent equatorial semidiameter sE is given by sE = a/b, the apparent polar semidiameter should be calculated from

sp ' •E 1 k cos B

where k —— 1 — {b/a)2 , and B is the Satumicentric latitude of the Earth (see Chapter 45).

If the older values (A) are chosen, namely a = 83“.33 and h — 74”.57, then

k —— 0.199197. If one adopts the values from (B) , then k —— 0.203 800.

Strictly speaking, this procedure should also be used in the case of Jupiter. But for this planet the angle B (called DE in Chapter 43) can never exceed 4°, so it will generally be sufficient to put sz —— blb here.

Moon

Let A be the distance between the centers of Earth and Moon in kilometers, z the equatorial horizontal parallax of the Moon, s its geocentric semidiameter, and k the ratio of the Moon’s mean radius to the equatorial radius of the Earth. In the Astronomical Ephemeris for the years 1963 to 1968, the value k —— 0.272 481 was used in eclipse calculations, and we have used thls value ever slnce.

Then we have rigorously

sin 6378.14 and sin r = k sinr

 

55. SEMIDIAMETERS OF SUN, MOON, PLANETS 391

but in most cases it will be sufficient to use the formula

 

s (in arcseconds) =

 

 358 473 400 

 

which gives an error less than 0.0005 arcsecond, as corripared with the result obtained by the rigorous expressions given before.

Computed in this way, the Moon’s semidiameter is geocentric, that is, it applies to a fictitious observer located at the center of the Earth. The observed, topocentric semidiameter s’ will be slightly larger than the geocentric semidiameter, because the observer is somewhat closer to the Moon than is the center of the Earth (except when the Moon is on the horizon). It is given by

k

—   — SiTt

 

while the topocentric distance of the Moon (that is, the distance from the observer to the center of the Moon) is A’ = qb, q being given by formula (40.7).

Alternatively, the topocentric semidiameter s' of the Moon can be obtained, with an accuracy which is sufficient for many purposes, by multiplying the geocentric value s by

1 + sin h sin z

where h is the altitude of the Moon above the observer’s horizon.

The increase in the Moon’s semidiameter, due to the fact that the observer is not geocentric, is zero when the Moon is on the horizon, and a maximum (between 14” and 18”) when the Moon is at the zenith.

Asteroids

The diameter d of an asteroid, in kilometers, can be calculated from [4] log d —— 3.12 — 0.2 If — 0.5 log N

where H is the absolute magnitude of the body (see page 23l), and A the albedo, or reflective power. The logarithms are to base 10.

If the logarithms are to base e, as in most programming languages, then

z = 3.12 — H f 5 — 1 log A  

2 log 10

 

OT x —— 3.12 — H/ 5 — 0.217 147 log A

and  then   d  (kilometers)   =   10x’

Many asteroids have an albedo of only about 0.04 (4 percent). According to Tedesco [5], the albedo of the first four asteroids are: Ceres 0.10, Pallas 0.14, Juno 0.22, and Vesta 0.38. Asteroid 437 Rhodia has the very high albedo 0.56.

Because many asteroids have an irregular shape, the expressions given above can yield only an approximate value of the “diameter”.

If d is the diameter of an asteroid expressed in kilometers, and if its distance to the Earth is A astronomical units, the apparent diameter of the body, in arcseconds, is

0.001 3788 d f 4

REFEREN CES

1. A. Auwers, Astronomische Nachrichten, Vol. 128, No. 3068, column 367 (1891).

2. See, for instance, the Astronomical Ephemeris lot 1980, page 550.

3. Astronomical Almanac for 1984, page E43.

4. Sky and Telescope, Vol. 85, No. 6, page 84 (June 1993).

5. E.D. Tedesco, pages 1093 and 1098 of Asteroids II (University of Arizona Press,

1989).

 

Chapter 56

Stellar Magnitudes

Adding stellar magnitudes

If two stars have magnitudes m l and mt, respectively, their combined magnitude in can be calculated as follows:

x —— 0.4 ( — )

m = — 2.5 log (10* + 1) where the logarithm is to the base 10.

Example S6.a —  The magnitudes of the components of Castor (n Gem) are 1.96 and

2.89. Calculate the combined magnitude.

One finds

z = 0.4 (2.89 — 1.96) = 0.372

Sri = 2.89 — 2.5 log (100 " 2 + I) = 1.58

If more than two stars are involved, with magnitudes m; , m , , ... , m„ ,

the combined magnitude m can better be found from

m = — 2.5 log 10 0'4

where, again, the logarithm is to the base 10. The symbol L indicates that the sum

 

must be made of all quantities

 

io —0.4 f8¡

393

 

 

Eznzapfe 56.b — The triple star Q Mon has components of magnitudes 4.73, 5.22, and 5.60, respectively. Calculate the combined magnitude.

 

m = — 2.5 log (

 

0 4)(4.73) + 10'*0’  4)(5.2 2) +   10  *  0’  4 )(5’  60))

 

= —2.5 log (0.01282 + 0.00817 + 0.00575) = 3.93

Exazriple 56.c — A star cluster consists of

4 stars of (mean) magnitude 5.0

14 — — 6.0

23 — — 7.0

38 — — 8.0

Calculate the combined magnitude.

4 x   io( 0.4)(5) —— 0.04000

14 x 10 0‘ 4” 6’ = 0.05574

 

23 x 10 *

 

0’  4(

 

7

) = 0.03645

 

38 x   lo( 0.4)(8) -  0.02 398

Sum L    =   0.15617

Combined magnitude = —2.5 log 0.15647 = + 2.02

 

56. STELLAR   MAGNITUDES 395

Brightness ratio

If two stars have magnitudes mt and m2, respectively, the ratio /1 / i 2 of their

apparent luminosities can be found from

x -— 0.4 ( — ›) — = 10‘

2

If the brightness ratio i l / i 2 is given, the corresponding magnitude difference Am = m2 — m can be calculated from

Am = 2.5 log

2

ñzompfe 56.d — How many times is Vega (magnitude 0.14) brighter than Polaris (magnitude 2.12)?

z = 0.4 (2.12 — 0.14) = 0.792

10 = 6.19

Hence, Vega is 6.19 times as bright as the Pole Star.

Example 56.e — A star is 500 times as bright as another one.

The corresponding magnitude difference is

Am = 2.5 log 500 = 6.75

 

Distance and Absolute Magnitude

Ifr is a star’s parallax expressed in seconds of a degree (“), this star’s distance to us is equal to

 

or 3.2616 

 

light-years

 

 

If z is a star’s parallax expressed in seconds of a degree (”), and m is the apparent magnitude of this star, its absolute magnitude M is given by

M -- m + 5 + 5 logr where, again, the logarithm is to the base 10.

If d is the star’s distance in parsecs, we have

N = m + 5 — 5 log d

Unlike the parallaxes within the solar system (see Chapter 40), the parallax considered here is, of course, the stellar, annual parallax resulting from the orbital motion of the Earth around the Sun; so it is not the parallax related to the dimensions of the Earth’s globe !

The parsec is the unit of length equal to the distance at which the radius of the Earth’s orbit (1 AU) subtends an angle of 1” (parallax = 1“). The name is a contraction of parallax and second.

1 parsec = 3.2616 light-years

= 206 265 astronomical units

=   30.8568 x 10'° kilometers

The absolute mngnirude of a star is the apparent magnitude of this star if it were located at a distance of 10 parsecs.

 

Chapter 57

Binary Stars

The orbital elements of a binary star are the following ones:

P —— the period of revolution expressed in mean solar years;

T —— the time of periastron passage, generally given as a year and decimals (for instance, 1945.62);

e = the eccentricity of the true orbit;

o = the semimajor axis expressed in seconds of a degree (“);

i =   the inclination of the plane of the true orbit to the plane at right angles to the line of sight. For direct motion in the apparent orbit, i ranges from 0‘ to 90°; for retrograde motion, i is between 90 and 180 degrees. When i is 90°, the apparent orbit is a straight line passing through the primary star;

II = the position angle of the ascending node;

o =   the longitude of the periastron; this is the angle in the plane of the true orbit measured from the ascending node to the periastron, taken always in the direction of motion.

When these orbital elements are known, the apparent position angle 8 and the angular distance p can be calculated for any given time r, as follows.

 

n = p360°

 

M —— n {t — T)

 

where t is expressed as a year and decimals (just as T) -, n is the mean annual motion of the companion, expressed in degrees and decimals, and is always positive. M is the companion’s mean anomaly for the given time t.

Then solve Kepler’s equation

E —— M + e sin E

by one of the methods described in Chapter 30, and then calculate the radius vector

r and the true anomaly v from

3 97

 

r -- a (1 -- e cos E)

E

Then find (8 -- It) from

 

• _(•

 

• ) - "‘i ,i   ?   lOs'

 

(57. 1)

 

Of course, this formula can be written

tan (8 -- ft) = tan (v + ui) cos i

but in this case the correct quadrant for (8 -- l2) is not determined. As in previous cases mentioned in this book, one may apply the ATN2 function, if it is available in the programming language, to the numerator and the denominator of the fraction in (57. 1). This will place the angle (8 -- 0) at once ln the correct quadrant.

When (8 -- II) is found, add II to obtain 8. If necessary, reduce the result to the interval 0° --360°.

Remember that, by definition, position angle 0° means northward on the sky, 90° east, 180‘ south, and 270‘ west. Consequently, if 8 is between 0° and 180°, the companion is “following” the primary star in the diurnal motion of the celestial sphere; if 180° < 8 < 360°, the companion is “preceding” the primary star.

The angular separation p is found from

 r cos (v + o)

 

However, the possibility exists of the denominator of the fraction being equal to zero. This risky division by zero can be avoided by using the following formula for the same calculation, mentioned by Greaney [1]

p -- r sin (   +    ) co + cos ( +   )

Note that the two terms under the square root sign are the squares of the numerator and the denominator, respectively, of the fraction in formula (57. 1).

Example 57.a -- According to E. Silbernagel (1929), the orbital elements of 9 Coronae Borealis are:

P ---- 41.623 years, T ---- 1934.008, e -- 0.2763, a = 0”. 907,

i = 59°.025, It = 23°.717, o = 219°.907

 

57. BINARY STARS

Let us calculate 8 and p for the epoch 1980.0. We find successively:

n 8.64906

f -- T ---- 1980.0 -- 1934.008 = 45.992

3f = 397.°788 = 37.°788

E --- 49°.897

r -- 0“.74557

v = 63°.416

tan (8 -- II) = 0.500 813

+0.230 440

8 -- II = --65°.291

8 = --41 .°574 =  318.°4

p -- 0“.411

 

399

 

 

As an exercise, calculate an ephemeris for Virginia, using the following elements [2]:

 

P -- 168.68 years

T --- 2005.13

e --- 0.885

= 3“.697

 

i = 148.°0

II = 36.°9 (2000.0)

= 256.°5

 

Answer. -- Here is an ephemeris with an interval of four years, starting at 1980. The position angle 8 decreases with time, since i is between 90 and 180 degrees. Least apparent separation (0“.36) occurs at the epoch 2005.21. The position angles 8 refer to the mean equinox of 2000.0, the same as for the angle Cl.

year = 1980.0 8 = 296°.65 p -- 3“.78

1984.0 293.10 3.43

1988.0 288.70 3.04

1992.0 282.89 2.60

1996.0 274.41 2.08

2000.0 259.34 1.45

2004.0 208.67 0.59

2008.0 35.54 1.04

2012.0 12.72 1.87

 

Eccentricity of the apparent orbit

The apparent orbit of a binary star is an ellipse whose eccentricity e' is generally different from the eccentricity e of the true orbit. It may be interesting to know e', although this apparent eccentricity has no astrophysical significance.

The following formulae have been derived by the author [3] :

A --- (1 -- e2 cos°ui) cos2 i B ---- e° sin o cos u› cos i C = 1 -- e2 sin 2u›

D ---- (A -- C)2 + 4 B2

e'* =

It should be noted that e’ is independent of the orbital elements a and l2, and that it can be smaller as well as larger than the true eccentricity e.

Example 57.b -- Find the eccentricity of the apparent orbit of 9 Coronae Borealis.

The orbital elements are given in Example 57.a.

We find

A -- 0.25298 B ---- 0.01934 C -- 0.96858 D 0.51358

e' 0.860

Hence, for this binary the apparent orbit is much more elongated than the true orbit.

REFEREN CES

1. M. P. Greaney, “The Orbit of a Binary Star”, Sky and Telescope, Vol. 74, No. 1, pages 71-72 (July 1987).

2. W.D. Heintz, “Orbits of 15 visual binaries”, Astronomy and Astrophysics, Sup- plement Series, Vol. 82, pages 65--69 (1990).

3. J. Meeus, “The eccentricity of the apparent orbit of a binary star”, Journal of the British Astronomical Association, Vol. 89, pages 485--488 (August 1979).

 

Chapter S8

Calculation of a Planar Sundial

by R. Sagot and D. Savoie (*)

One wishes to draw a planar sundial of any given orientation and inclination, provided with a straight stylus of length o perpendicular to its surface. Hence, this stylus generally is not directed towards the celestial pole. This sundial has the following principal parameters:

— the latitude e of the place;

— the gnomonic declination D, that is, the azimuth of the perpendicular to the sundial’s plane, measured from the southern meridian towards the west, from 0 to 360 degrees. So, if D = 0°, the sundial is “due south" ; if D —— 270°, it is “due east" ; and so on;

— the zenithal distance z of the direction defined by the straight stylus. If z = 0°, the sundial is horizontal; in this case, fi is meaningless — but see the special case later in this Chapter. If z = 90°, the sundial is vertical.

The coordinates x and y of the tip of the shadow of the straight stylus of length o are measured in an orthogonal coordinate system situated in the sundial’s plane. The origin of this system coincides with the footprint of the stylus. The z-axis is horizontal, while the y-axis coincides with the line of greatest slope of the sundial. In all cases, z is measured positively towards the right, while y is posltiVe upwards.

The Sun’s hour angle Z-I is measured from the upper meridian transit (true noon); it increases by 15 degrees per hour. For example, if = —45° corresponds to 9 hours a.m. (true solar time), if = + 15° to 1 hour p.m. , etc.

(*) Robert SAGOT and Denis SAvoœ are former president and president, respectively, of the “Commission des Cadrans Solaires" (Sundials Section) of the Société Astronomique de France. Denis SAvoœ is the author of Gnomonique moderne (in French), published by the Société Astronomique de France (1997), in which the interested reader can find a mathematical theory of sundîals.

401

 

In the following formulae, for each hour angle H the declination 6 of the Sun will take the successive values (in degrees) —23. 44, —20. 15, — 11. 47, 0, + 11.47,

+20.15, and +23.44, which correspond to the dates when the longitude of the Sun is a multiple of 30°.

In the course of a day, the tip of the shadow of the stylus will describe on the sundial’s plane a curve which is a conic (a circle, an ellipse, a parabola, or an hyperbola). However, if h = 0° the curve is always a straight line.

Calculate

P = sin p cos z — cos p sin z cos D

Q —— sin D sin z sin H + (cos p cos z + sin p sin z cos D) cos If + P‘tan b Nq —- cos D sin H — sin D (sin p cos H - cos p tan 6)

Nd — COS z sin D sin If — (cos ‹,o sin z — sin ‹;e cos z cos D) cos H

— (sin ‹,o sin z + cos p cos z cos D) tan b

Then the coordinates z and y are given by

x —— a Nq / Q  

For each hour angle, one obtains a series of points. By connecting these points, an hour line is created on the sundial. The point (if it exists) to which the hour lines converge, is called the center of the sundial; it is also the point of fixation of the polar sfyfur, which is parallel to the Earth’s axis of rotation. Its coordinates z and

y are given by

z 0 = Cos p sin D, y0 = —p (Sin p sin z + cos p cos z cos D)

The length u of the polar stylus, from its point of fixation to the tip of the perpendicular stylus of length o, is

     a 

while the angle which the polar stylus makes with the sundial’s plane is given by sin = | P

The formulae for the position of the polar stylus become meaningless when P —— 0, that is, when cos D tan z -- tan ‹p. This means that the polar stylus is then parallel to the plane of the sundial.

 

 

The plane represents the plane of the sundial. OP is the perpendicular s5lus, of length a, while IP is the polar stylus, length u. P’ is rir shadow éx. yJ of the tip of the stylus. The point I is called the center of the sundial, while O is the origin of the x - y system.

It is proper to limit the drawing of the sundial to the useful lines. For example, a vertical sundial oriented “due north” {D —— 180°), at latitude + 40°, can never show 11' a. m. , true solar time. At the same latitude, a vertical sundial oriented “due south” cannot indicate 19‘ (= 7‘ p.m.) near the June solstice.

In order to make sure that the sundial really works, two conditions should be fulfilled: the Sun must be above the horizon, and the plane of the sundial must be illuminated. Consequently, it is necessary, for each calculated point (x, j), to verify whether these two conditions are satisfied simultaneously.

 

In practice, for a given arc of declination, the calculation should start at the moment of the geometric rise of the Sun, or at the first integer hour following that rise, and stop at the moment of the geometric sunset. The Sun’s hour angle Mfg at the time of sunrise or sunset is given by

with Hi < 0 for sunrise,  ff0     0 for sunset.

For each value of H, one should look at the sign of i2: if this quantity is negative, this means that the Sun does not illuminate the plane, and in that case one passes over to the next declination. Hence, only those values for which @ is positive must be retained.

It is possible that, on a given date, Q is at first positive, then becomes negative, and later is positive again.

Example 58.a -- Consider an inclined sundial at latitude 40° North, with D 70°,

z = 50°, and a = 1. For 6 = +23°.44 (summer solstice), we

have ff 0 = -- 111°.33 (or 4 35‘ a.m., true solar time).

Beginning the calculations with If = -- 105°, we find Q < 0. This quantity is negative again for H = --90°, --75°, and --60°. Only from H -- --47° on is the sundial illuminated, and it will remain illuminated till sunset. Hence, if a step of 15 degrees has been chosen, the values of z and y should be calculated for If = --45° to + 105°.

For H -- + 30° and 6 = +23°.44, we find z = --0.0390, y 0.3615.

For Z-f =  --15° and 6 = -- 11.°47, we find z = --2.0007, y 1.1069.

The coordinates of the center are zt = + 3.3880, yp = --3.1102, and we have

= 12°.2672.

Example 58.b -- Consider a vertical sundial at latitude p = --35°, with D -- 160°, z = 90°, and a = 1.

For 6 = 0° (equinox), we have H --- --90° and Q < 0. Q becomes positive for H = --57°, so the calculations will be made for H -- --45° till sunset (> + 90°).

For H = +45° and 6 = 0°, we find z = --0.8439, y = 0.9298.

For H ---- 0° and 6 = +20.° 15, we find z = +0.3640,   y 0.7410.

The coordinates of the center are z = +0.3640, yb = +0.7451, and we have = 50°.3315.

 

 

Example 58.c — Inclined sundial at latitude 40° N, with D - 160° and z = 75°. For fi = + 23°.44, this sundial will be illuminated from sunrise (when If =

— 111°) until if = —84°. Then it will be illuminated again from H = +2° until sunset (if = + 111°). So, if a step of 15° has been chosen, the calculation will be made for If = — 105°, —90°, and then for + 15° to + 105°.

The formulae given above form the most general case which can occur in gnomonics. They allow the calculation of the classical hour lines of true solar time, but also the declination curves, the lines for mean time (when introducing the equation of time in the calculation of IN), the lines for Universal Time or of zone time, azimuth and altitude lines, etc.

The formulae simplify greatly for some special cases, which we shall now examine briefly.

Special cases

(1) Equatorial sundial

The plane of this sundial is parallel to the plane of the equator and hence there are two sides: the northern side serves for the positive declinations (spring and summer), the southern side for the negative declinations of the Sun (autumn and winter). At a place of latitude p, we have

for the northern side: z = 90° — e d D — 180°

for the southern side: z = 90‘ + p and D —- 0°

The line of 12 hours (H = 0°) coincides with the line of greatest descending slope. Further,

                       *o ' 0

cos££ 90

where the upper sign is to be taken for the northern side, the lower sign for the southern side.

 

The plane of this sundial is horizontal, so z = 0°. The angle D is not defined and the direction of the z-axis can be chosen at will. We shall consider the case fi = 0°, where the z-axis is directed towards the east, the y-axis towards the north. The formulae simplify to

 

t2 = cos p cos If + sin p tan h

y Sln ‹¡o cos h - COS p tdh d

 

z 0 = 0

 

 

(3) Vertical sundial

The plane of this sundial is vertical, so z = 90°. The z-axis is horizontal; the y-axis is directed towards the zenith. The formulae simplify to

t2 = sin D sin If + sin p cos D cos H — cos p cos D tan h

  sin IN — sin e sin D cos If + cos p sin D tan h  

z = — o tan D

CoS p Cos D

y   - + a tan p / cos D

General Remarks

In the case of a sundial with a perpendicular stylus, as considered here, it is the extremity of the umbra of that stylus which indicates the time, while in the case of a sundial with a polar stylus it is the entire umbra which gives the time.

Because we give the coordinates x0, y0 of the center of the sundial, it is always possible to construct the polar stylus IP, if this is wanted: the polar stylus is the straight line connecting that center with the extremity of the perpendicular stylus. See the Figure on page 403.

The advantage of the system of axes x -y used in this Chapter is that the per- pendicular stylus does always exist; this is not always the case for the polar stylus.

 

Appendix I

3Yathematical constants

z  =  3.14159 26535 89793 23846 .....

e   -- 2.71828 18284 59045 23536 .....

1 radian = 180/z degrees = 57.295 779 513 082 degrees

= 206 264.806247 arcseconds

1 degree = r/ 180 radian = 0.017 453 292 519 943 radian

logt  0o   =  log, o /log,10  =   0.434 294 481 903 loge o

Distances

1 astronomical tinii (AU) = 149 597 870 kilometers = 499.0048 light-seconds

= 8.32 light-minutes = 0.005 77 55 183 light-day

1 parsec -- 30.8568 x 10'° kilometers = 3.2616 light—years = 206 264.8 AU

= the distance at which the length of one astronomical unit subtends an angle of 1”.  The name is a contraction of parallax and second

1 light-year -- 9.4607 X 10'° kilometers = 0.30660 parsec = 63 241 astron. units

= the distance that light travels in one year (in vacuo)

Distance Earth —Moon (mean) = 384 400 kilometers

Earth : equatorial radius = 6378.14 km, polar radius = 6356.76 km

Diameter of Sun = 1392 000 km

Diameter of Moon = 3476 km

407

 

nme

1 sidereal day = 23 hours 56 minutes 04.0905 seconds of mean solar time

= 0.997 269 5663 mean solar day

1 mean solar day = 1.002 737 909 35 sidereal days

Length of the year in mean solar days (*), for epoch 2000.0:

Tropical (equinox to equinox) 365.242 190

Sidereal (fixed star to fixed star) 365.256 363

Anomalistic (apse to apse) 365.259 636

Julian 365.25

Length of revolution period of Moon, in

mean solar days (*) :

Tropical (equinox to equinox) 27.321 582

Sidereal {Rxek star to fixed star) 27.321 662

Anomalistic (apse to apse) 27.554 550

Draconic (node to node) 27.212 221

Synodic (New Moon to New Moon) 29.530 589

Vari‘a

 

Mean obliquity of the ecliptic: in 1900 :  23° 27' 08“

in 1950:   23° 26’ 45“

in 2000: 23° 26' 21“

in 2050: 23° 25' 58”

Eccentricity of Earth’s orbit: in 1900: 0.016 751

in 2000:   0.016 709

in 2100:  0.016 666

General annual precession (in 365.25 days) :

in 1900: 50”.269

in 2000:   50”.291

in 2100: 50“.313

 

Mean parallax of Sun = 8“.79415 Constant of aberration = 20“.4955 Flattening of the Earth = 1/298.257

Gaussian gravitational constant: k = 0.017 202 098 95

or, converted from radians to degrees, 0.985 607 6686

Speed of light in vacuo

= 299 792.458 km/second

Earth -Moon mass ratio = 81.3007 Sun—Earth mass ratio = 332 946

 

 

(*) Or, more precisely, ephemeris days, in the uniform time scale of Dynamical Time. One ephemeris day is approximately equal to one mean solar day at epoch 1900.0.

 

Appendix II

Some Astronomical Terms

The following notes may be found helpful by those who are not familiar with the technical terms used in this book, but further guidance should be sought from textbooks on astronomy.

The celestial equator is the great circle that is the projection of the Earth’s equator onto the celestial sphere. Its plane is perpendicular to the axis of rotation of the Earth.

The celestial poles are the poles of the celestial equator, or the intersections of the axis of rotation of the Earth with the celestial sphere.

The ecliptic is defined to be the plane of the (undisturbed) orbit of the Earth around the Sun.

The equinox or, better, the vernal equinox, which is the zero point of both right ascension and celestial longitude, is defined to be in the direction of the ascending node of the ecliptic on the equator. It is that intersection of equator and ecliptic where the ecliptic runs (eastwards) from negative to positive declinations. The other intersection, which is diametrically opposite, is the a tumzinf equinox.

The eguiziozes  are the instants when the Sun’s apparent longitude is 0°  or 180°.

Solstices : both the points on the ecliptic 90 degrees away from the equinoxes, and the instants when the apparent longitude of the Sun is 90° or 270°.

Celestial longitude, or ecliptical longñude, often called simply longitude, is measured (from 0° to 360°) from the vernal equinox, positive to the east, along the ecliptic.

Celestial latitude, or ecliptical latitude, or simply ku'ifude, is measured (from 0° to +90° or to —90°) from the ecliptic, positive to the north, negative to the south.

Right ascension is measured (from 0 to 24 hours, sometimes from 0° to 360°) from the vernal equinox, positive to the east, along the celestial equator.

Declination is measured (from 0° to +90°) from the equator, positive to the north, negative to the south.

409

 

Owing to the effects of precession and nutatiozt , the ecliptic and equator, and hence the equinoxes and the poles, are continuously in motion, and so the current celestial coordinates of a “fixed" direction change continuously. The motion of the equator is primarily due to the action of the Sun and the Moon, while the (much slower) motion of the ecliptic is primarily due to the perturbing action of the planets.

Mean equator: the instantaneous celestial equator exclusive of the periodic perturbations of the nutation.

Mean equator and equinox, or simply mean equinox : an expression used to denote that the reference system takes into account the precession (secular effects) but not the nutation (periodic effects).

Coordinates : two (or three) numbers which define the position of a point on a surface (or in space). Examples: longitude and latitude are the two geographical coordinates of a point on the surface of the Earth; right ascension and declination; the rectangular coordinates X, Y, Z of a point in three-dimensional space.

heliocentric : referred to the center of the Sun, for instance a heliocentric orbit, heliocentric coordinates.

Geocentric : referred to the center of the Earth, for instance a geocentric observer, geocentric coordinates.

Topocentric : referred to the observer on the Earth’s surface, for example the topocentric right ascension and declination of the Moon.

Aberration is the apparent displacement of the position of an object due to the finite speed of light. The annual aberration of a star is due to the orbital motion of the Earth around the Sun (or, more exactly, around the barycenter of the solar system).

Aâmuth : the angular distance measured from the South, positive to the West, along the horizon, to the vertical circle through the point in question. Navigators and meteorologists measure the azimuth from the North, positive to the East.

Ascending node : that intersection of the orbital plane with the reference plane where the latitudinal coordinate is increasing (going north). The other intersection is the descending node.

Conjunction : that configuration of two celestial objects such that either their right ascensions or their celestial longitudes are equal.

Opposition : that configuration of two celestial objects such that their celestial longitudes differ by 180°. Most frequently used when one of the objects is the Sun. Heliographic coordinate system : a coordinate system on the surface of the Sun.

Planetographic coordinate system : a coordinate system on the surface of a planet. In the case of Mars, the term areographic is generally used. For the Moon, the term is selenographic. Compare with geographic for the Eanh.

Epoch : a particular fixed instant used as a reference point on a time scale, such as B1950.0 or J2000.0.

A .Julian century is a time interval of 36525 days.

 

SOME  ASTRONOMICAL   TERMS 411

An ephemeris dog is equal to 86400 seconds in the uniform time scale known as Dynamical Time.

The sidereal fime is the measure of time defined by the motion of the vernal equinox in hour angle; it is the hour angle of that equinox (at a given place and for a given instant). The true solar time is the local hour angle of the Sun. The mean solar time is the hour angle of the mean Sun, and thus is measured from mean noon. The civil time is the mean solar time increased by 12 hours, and thus is measured from mean midnight. — The expression “mean time measured from midnight” is a contradictio in terminis, since the mean (solar) time by definition is measured from noon. Many people erroneously use the expression “Greenwich Mean Time”, when in fact Greenwich Civil Time is meant.

Universal Zime is the civil time on the meridian of Greenwich.

The astronomical unit (AU) is a unit of length used to measure distances in the solar system. It is often called the “mean distance of the Earth to the Sun”. But, rigorously, one AU is the radius of the circular orbit which a particle of negligible mass, and free of perturbations, would describe around the Sun with a period of 2z/k days, where k is the Gaussian gravitational constant, 0.017202098 95.  As a consequence, the semimajor axis of the elliptical orbit of the Earth is not exactly 1 AU, but 1.000001 018 AU.

Radius vector: the straight line connecting a body to the central body around which it revolves, or the distance between these bodies at a given instant. The radius vector of a planet or comet is generally expressed in astronomical units.

Apsides (plural of apse) : the points of intersection of the major axis with the orbit of a planet, a minor planet, a satellite, or a comet. These are the points of the orbit that are closest (perihelion, perigee, etc.) and farthest (aphelion, apogee, etc.) from the central body.

Perihelion : the point of the orbit (of a planet, minor planet, or comet) which is nearest to the Sun. For the corresponding point of the Moon’s orbit with respect to the Earth, the term is perigee. For a satellite of Jupiter with respect to this planet, the traditional term is perijove (*). For a double star, one says periastron.

(*) The term perijove was already used by Laplace (1749- 1827) and has become a classical term in astronomy. The word “periapse”, used by some authors, is incorrect. The word perihelion means the point of the orbit that is closest to the Sun (from the Greek: peri -- near + helios -- Sun). Similarly, perigee is the point closest to the Earth (ge —— Earth). Therefore, “periapse” would really mean the point closest to the apse; but this is ridiculous, because what is meant is the apse itself!

For the Moon, the terms periselene and aposelene seem the most appropriate; compare with selenographic and selenocentric. One should not create more neo- logisms, however. It would be absurd to speak of “peri flore” for an orbit around minor planet Flora, or “perikosmodemyanskaya” for an orbit around minor planet 2072 Kosmodemyanskaya. For an orbit around another body than the Sun, Earth, Moon, or Jupiter, the best terms seem periastron and opastron, as for double stars.

 

The geometric position of a planet is the “true” position of that body at the given instant; that is, no allowance is being made for the effects of aberration and light-time.

Astrometric position : see page 230.

Anomalies. — The mean anomaly ‹M) of a planet is the angular distance, as seen from the Sun, between the perihelion and the mean position of the planet. The angular distance measured from the perihelion to the true position of the planet is called the true anomaly {v). The eccentric anomaly is an auxiliary quantity needed to obtain the true anomaly through solving Kepler’s equation. The emotion o/ the center is the difference between the true and the mean anomalies (C = v — M) ; it is the difference between the actual position of the body in its elliptic orbit and the position the body would have if its angular motion were uniform.

An ephemeris is a table of positions or other calculated data of a celestial body (Sun, Moon, planet, comet, etc.) for a series of (generally equidistant) instants. From the Greek ’e‹pypcpoq, ephemeros —— daily.

Parallax : the difference in apparent direction of an object as seen from two different locations. For objects in the solar system (Sun, Moon, planet, asteroid, comet), the parallax is the difference in direction between a topocentric observation (by the actual observer at the Eanh’s surface) and a hypothetical geocentric observation. For the stars, the (annual) parallax is the difference between geocentric and heliocentric positions.

Arcminute (’) and arcsecond (”) are 1/60 and l/3600, respectively, of a degree. Not to be confused with minute and second of time (1/60 and 1/3600 of an hour).

 

Appendix IM

Planets : Periodic Terms

In this Appendix, pages 414—454, the most important periodic terms from the French planetary theory VSOP87 are given. The successive columns contain the following data:

— the name of the planet;

— the label of the series (L for the heliocentric longitude, B for the latitude, R for

the radius vector);

— the current No. of the term in the series;

— the quantities A, B, and C, which all are positive (or zero).

In each series, the terms are sorted by decreasing values of A. For example:

P1 a net Ser   es No . A B

 

 

VENUS

VENUS

 

R0 l

2

3

4

5

6

7

8

9

10

11

12

R1 l

2

3

 

72 354 821 0

489 824 4. 021518

1 658 4, 902 I

1 632 2. 845 6

1 378 1 . 128 5

498 2. 587

374 I . 425

264 5.529

237 2.551

222 2.013

126 2.728

119 5.020

34551 0.89199

254 J . 772

254 3.142

 

0

10213.285546

20426.5711

7860.4194

11790.6291

9683.595

3950.210

9437.765

15720.839

19367.189

1577.Z4A

10 404. 734

10 2t3. 285 $5

20 436.571

0

 

 

For more explanation about the use of these terms, see Chapter 32.

413

 

M E R C UR Y L 0 1 440 250 710 0 0

2 40989415 1.48302034 26 087.903 141 57

5 5 046 294 4. 477 854 9 52 175.806 283 1

4 855 547 1 . 165 205 78 263.709 425

5 165 690 4. 119 692 104 351.612 566

6 84 562 0. 779 5l 130 439.515 71

7 7 583 5. 713 5 156 527.418 8

8 3 560 I .512 0 I 109.378 6

9 1803 41033 5 661.332 0

10 1726 0.3583 182 615.322 0

1I 1 590 2.995 1 25 028.521 2

12 1 365 4.599 2 27 197.281 7

15 1 017 0. 880 8 3I 749.235 2

14 714 1.541 24 978.525

I5 644 5.303 21 535.950

16 451 6.050 51116.424

17 404 3.282 208703.225

18 352 5.242 20 426.571

I9 345 2.792 I5 874.618

20 343 5.765 955.600

21 339 5.863 25 558.212

22 325 I.337 53 285.185

23 273 2.495 529.691

24 264 3.917 57 837.138

25 260 0.987 4 SSI .953

26 239 0.Y1S 1 059.382

27 235 0.267 11 522.664

28 217 0.660 15 521 . 751

29 209 2.092 47 625. 855

30 183 2.629 27 043. 505

31 182 2.434 25 661.30S

32 176 4.5S6 51066.428

33 173 2.452 24 498.830

34 142 3.360 37 410.567

35 138 0.291 IO 213.286

36 125 3.721 39 609.655

37 118 2.781 77 204.327

38 106 4.206 19 804.827

MyR CURY L 1 1 26 08 814 706 223 0 0

2 1 126 008 6.217 039 7 26 087.903 1416

3 303 471 3.055 655 52 175.806 283

4 80 538 6.104 55 78 263.709 42

5 21 245 2.835 32 104 351.612 57

6 5 592 5.826 8 130 459.515 7

7 1472 2.5185 156 527.418 8

8 388 5.480 182 615.322

9 352 3.052 1 109.379

10 103 2.149 208 703.225

11 94 6.12 27 197.28

12 91 0.00 24 978 . 52

13 52 5.62 5 661. 33

14 44 4.57 25 028.52

15 28 3.04 51066. 45

16 27 5.09 234 791 .15

ifIERCURY L2 1 53 050 0 0

2 16 904 4.690 72 26 087.903 14

3 7 397 1.347 4 52 175.806 3

4 3 018 4.456 4 78 263.709 4

5 1 107 I .262 3 104 351.612 6

6 878 4. 320 130 439.516

7 123 1.069 156 527.419

 

 

MERCURY L2 8

(cont.) 9

10

MERCURY  L3 1

2

 

4

5

6

7

8

 

39 4.08

15 4.63

12 0.79

188 0.035

142 3.125

97 8 . 00

44 6.02

35 0

18 2.78

7 5.82

3 2.57

 

182 61 S .52

1109, 38

208 705 . 23

52 175.806

26 087.903

78 265.71

104 351.61

 

ISO 489. 52

156 527. 42

182 615. 32

 

MERCURY  L4 l 114 31416

2 3 2.03

3 2 1.42

4 2 4.50

5 l 4.50

6 1 1.27

 

0

26 087. 90

78 265, 7t

52 I75. 81

104 351. 61

ISO 459. 82

 

MERCURY L5 l 1 314 0

MERCURY BO l 11737529 1.98357499 26 087. 903 141 57

2 2 388 077 5.057 389 6 52 l75 . 806 283 l

3 1 222 840 3. 141 592 7 0

4 543 252 1 .796 444 78263.709425

5 129 779 4.832 325 104 35l .612 566

6 3I 867 I . 580 88 130439.51571

7 7 963 4.609 7 156 527. 418 8

8 2 014 I .353 2 182 615.322 0

9 514 4.378 208 703.225

10 209 2.020 24 978. 525

11 208 4. 918 27 197. 282

12 132 I . 119 25479t.128

13 121 1.813 532Bâ.185

14 100 5.657 20 426. 571

MERCURY BI I 429151 3.501698 26 087. 905 142

146234 3.141593

 

22 675 0.015 15 52 175. 806 28

4 10 895 0. 485 40 78 268 . 709 42

5 6 353 3.429 4 104 551 . 612 6

6 2 496 0. 160 5 l5O4ZP.515T

7 860 3.185 I56 527.419

8 278 6.210 182615.322

86 2.95 208 705.23

10 28 0.29 27 197.28

11 26 5.98 24791.13

MERCURY B2 1 11831 4.79066 26 087.905 14

2 1914 0

 

5 1 045 I .212 2 52 175.806 5

4 266 4.454 78263.709

5 170 1.623 104351.613

6 96 4.80 150439.52

7 45 1.61 156 527.42

8 18 4.67 IB2 615.32

9 7 1.45 208 703.23

MERCURY B8 1 235 0. 554 26 087.900

2 161 0

 

5 19 4.36 S2 175.81

  6 2.51 7B 263.71

 

MER C UR Y B3 5 S 6. 14 1 ß4 851 .61

(cont.) 6 3 3.12 130439.52

7 2 6.27 156 527.42

MERC UR Y B4 1 4 1 . 75 26 087. 90

2 1 5. 14 0

MER C UR Y R 0 1 59 528 272 0 0

2 7 834 132 6.192 537 2 26 087. 903 141 6

5 795 526 2. 959 897 52 175. 806 288

4 121 282 6. 010 642 78 265. 709 425

5 21 922 2. 778 20 104 851 . 612 57

6 4 8s4 5. 828 9 150 459. 515 7

7 918 2. 597 156 527. 419

8 290 l . 424 2F 028. 52J

9 260 3.028 27}97.282

10 202 5.64T 182615.322

II 201 5.592 3IT49,235

12 142 6. 255 24 978. 525

13 100 3.734 21 535.950

MER C UR Y R I I 217 348 4.656 172 26 087. 903 42

2 44142 1.42386 52 \ 75. 806 28

  10094 4.47466 78 263.709 42

4 2453 1.2423 104 351.612 6

5 1624 0 0

6 604 4.293 130 439.516

7 153 1.061 156 527.419

8 3Q 4.11 182 615.32

MERCURY R2 1 3 118 3.082 3 26087.9031

2 1 245 6. ISO 8 52175.8063

5 425 2.926 78265.709

4 136 5.980 104351.613

5 42 2.75 150459.52

6 22 314 0

7 13 5.80 156 527, 42

MERCURY R3 1 58 1.68 26 087.90

2 24 4.63 52 175.81

12 1.39 78 263.71

5 4.44 104351. 61

5 2 1 21 130 439.52

VENUS

Ł 0

1

517 614 667

0

0

2 1 353 968 5.593 133 2 20 2J3.285 546 2

3 89 892 5.306 50 20 426.571 09

4 5 477 4.416 5 7 860.419 4

5 3 456 2.699 6 11 790.629 1

6 2 372 2.995 8 3 950.209 7

7 ł 664 4.250 2 1 577.343 5

8 1 438 4.157 5 9 683.594 6

9 1 517 5. 186 7 26.298 3

DO 1 201 6.153 6 30 639.856 6

11 769 0. 816 9 4Z7. 765

12 761 1.950 529.691

13 708 t.065 775.523

14 585 3.998 191.448

15 500 4.125 15720.859

16 429 3.586 t9367.189

17 527 5.677 S 507. 553

18 326 4.591 10404.734

 

 

V EN US L 2 1 54 127 0

2 3891 0.345t S0 215.285 5

3 1338 2.0201 20 426. 571 1

4 24 2.05 26.30

5 19 3.54 30 639.86

6 10 3.97 775.52

7 7 1.52 1 577. 54

8 6 1.00 191 . 45

V EN US L3 1 136 4.804 10 2t3.286

2 78 367 20 426.57

3 26 0 0

V EN US L4 1 114 5.141 6 0

2 5 5. 21 20 426. 57

8 2 2. 51 10 215. 29

V EN US L 5 1 1 8 . 14 0

VENUS B0 1 5923638 0.2670278 10213.2855462

2 40108 114737 20426.57109

  32815 3.141 59 0

4 1011 1.0895 506B9.8566

5 149 6.254 18075.705

6 538 0.860 1577.344

7 130 3672 9437.763

8 120 3. 705 2 252.866

9 108 4. 539 22 003 . 915

V EN US Bl l 513 348 I .803 643 10 213, 285 546

2 4 380 3,38ó 2 20 426. 571 I

3 t99

4 197 0 0

2.530 50 639.857

 

V EN US B4 1 14 0.32 10213.29

V EN US R 0 l 72 334 821 0

 

2 489 824 4. 021 518 10 213.285 546

  1 658 4. 902 I 20 426. 571 I

4 1632 2.845 5 7860.4194

5 1 378 1 . 128 5 I 790.629 1

498 2. 587 9 ó8J.595

374 1 . 428 3 950.210

8 264 5.529 9 437.763

237 2.PSI 15 720.839

lO 222 2.0I3 19 367.189

11 126 2.728 I 577.344

12 119 3.020 10 404. 734

VENUS R I S 34 551 0.891 99 10 213.285-55

2 234 I . 772 20 426.571

  234 3. 142 0

V EN US R 2 1 1 407 5.063 7 10 213.285 5

2 16 5.47 20 426. 57

  13 0

 

VENUS R3 l 50 3.22 10 213.29

VENUS R4 J 1 0. 92 10 213. 29

EARTH

L0

1

175347046

0

0

3 341 656 4.669 256 8 6 285. 075 850 0

34 894 4.626 10 12 566. 151 70

4 3 497 2. 744 l 5 758 . 584 9

5 3418 2.8289 S. 525 1

6 3 136 3.627 7 77 713. 771 5

7 2 676 4. 418 1 7 860. 419 4

8 2 343 6 135 2 8 950. 209 7

  I 324 0.7425 11 506. 769 8

10 1273 2.0571 529. 691 0

11 I 199 1.109 6 1 577. 345 5

12 990 5.233 5884.927

I3 902 2.045 26.298

14 857 3.508 398 149

15 780 1.179 5223 694

16 755 2. 558 5 507. 553

17 505 4.583 t8 849.228

18 492 4.205 775.523

19 357 2.920 0.067

20 317 5.849 I I 790.629

21 284 1.899 796.298

22 271 0.315 10 977. 079

23 243 0.345 5486.778

24 206 4.80ó 2544.314

25 205 1. 869 S 573. 145

26 202 2.458 6 069. 777

27 1 56 0.833 213.299

28 152 3.41I 2 942.463

29 126 I .085 20.775

30 115 0.645 0.980

31 103 0.636 4694.003

32 102 0.976 15720.839

33 102 4.267 7.114

34 99 6.21 2146.17

35 98 0. 68 t55.42

36 86 5. 98 161000.69

 

 

EARTH LO 57 85 1.50

(cont.) 38 85 5.67

39 80 1.81

40 79 8.04

41 75 1.76

42 74 5.50

45 74 4.68

44 70 0.85

45 62 3.98

46 61 1.82

47 57 2.78

48 56 4.39

49 56 5.47

50 62 0.19

51 52 1.33

52 51 0.2B

53 49 0.49

54 41 5.37

55 41 2.40

56 39 6.17

57 57 6.04

58 37 2.57

59 36 I.71

60 36 1.78

61 33 0.59

62 30 0.44

63 30 2.74

64 25 316

EAR TH L 1 1 628 331 966 747 0

2 206 059 2.678 255

5 4 303 2.656 1

4 425 l . 590

5 119 5.796

6 109 2. 966

7 93 2. 59

8 72 l . l4

9 68 1 . 87

10 67 4. 41

11 59 2.89

12 56 2. 17

13 45 0. 40

14 36 0. 47

15 29 2. 65

16 21 5. 54

17 19 1. 85

18 19 4. 97

19 17 2. 99

20 16 0. OF

21 16 l . 43

22 15 l . 21

23 12 2.85

24 12 5. 26

25 12 5. 27

26 12 2. 08

27 11 0. 77

28 10 1 . 50

29 10 4. 24

30 9 2. 70

31 9 5.64

32 8 S. 50

38 6 2.65

54 6 4.67

 

6 275.96

71 450 .70

17 260. 15

12 056. 46

5 088.65

3 164. 69

801. 82

9 487.76

8 827. 39

7 084. 90

6 286.60

14 143.50

6 279.55

2 T 39.  55

1 748. 02

5 856. 48

1 194. 45

8 429.24

19651.05

10447.39

10 213.29

I 059.38

2 552. 87

6 812.77

17 789.85

83 996.85

l 549.87

4 690.48

6 283.075 850

12 566 . 151 7

3.523

26.298

I 577.344

18 849.23

529.69

398.15

5 507. 55

S 225. 69

l6 S. 42

796. 50

775. 52

7.11

0. 98

5 486. 78

21330

6275.96

2544.3t

214617

1097708

174%02

5088.65

1194.45

4694.00

S53. S7

6 286.60

I 549.87

242.73

95t .72

2 352.B7

9437.76

4 690. 48

 

EARTH L2 1 52 919 0 0

2 8 720 1.072 I 6 283.075 8

  309 0.867 12 566. 152

4 27 0.05 3.52

  16 5.19 26.30

6 16 3. ó8 155.42

7 10 0.76 18 849.23

9 2.06 77 715. 77

7 0.83 775. 52

10 5 4.66 1 577. 34

11 4 1.03 7. 11

12 4 344 5 575. 14

13 3 514 796. 80

14 3 6.05 5 507. 55

15 3 1.19 242.73

16 3 6.12 529.69

1 7 3 0.31 398.15

18 3 2.28 553.57

19 2 4.38 5 223.69

20 2 3.75 0.98

EAR TH L 8 1 289 5. 844 6 283.076

SS 0 0

17 5. 49 12 566.15

4 8 5. 20 155. 42

1 4. 72 3./i2

6 1 S. 30 18849.23

7 1 S. 97 242. 75

EAR TH L 4 1 114 5. 142 0

2 8 4. 13 6 283.08

  5. 84 12 566. 15

EA R TH L 5 1 1 314 0

EARTH BO 1 280 3.199 84334.662

2 102 5.422 5 507.553

3 80 388 5223.69

4 44 3.70 2 352.87

5 32 4.00 1 577. 34

EA R TH B 1 1 9 3. 90 5 507. 55

2 6 I .73 5 223.69

EARTH R 0 1 100 018 989 0 0

2 1 670 700 5. 098 463 5 6 288. 075 850 0

  15 956 5. 0s5 25 12 566. 151 70

4 3084 S. 198 5 77 718. 771 5

5 1 628 1 . 17Z 9 S 753. 384 9

6 1 576 2.846 9 7 860. 419 4

7 926 5.45Z 11 506. 770

8 542 4. 564 8 950. 210

  472 5. 661 5 884. 927

10 546 0. 964 5 507. 555

11 329 5. 900 5 223. 694

12 507 0. 299 6 575. 148

13 243 4.275 1I 790.629

14 312 5.847 1577.344

IS 186 5.022 10977.079

16 175 5.0t2 18849.228

17 110 5.055 S486.778

18 98 0.89 6 069.78

19 86 5.69 15 720.84

 

 

EA R TH R 0 20

(cont.) 21

22

23

24

25

26

27

28

29

50

3I

32

33

34

35

36

37

38

39

40

 

86 1.27

65 0.27

63 0.92

57 2.01

56 5.24

49 5. 25

47 2.58

45 5.54

43 6.01

39 5.36

38 2.89

37 0.85

57 4.90

56 1.67

55 1.84

33 0 24

32 018

32 1.78

28 1.21

28 1.90

26 4.59

 

161 000.69

17 260.15

529.69

83 996.85

71430.70

2 544.31

775.52

9 437.76

6 275.96

4 694.00

8 827.59

19 651. 05

12 159.55

12 056.46

2 942.46

7 084.90

5 088.63

398.15   6 266.60 6 279.55 10 447.39

 

EARTH Rl l 103019 1.107490 6 283.075 850

2 1721 1.0644 12 566. L5L 7

3 702 3.142 0

32 1.02 18 849.23

5 31 2.84 5 507. 55

6 25 1.32 5 225.69

7 18 1.42 1 877.34

8 10 5.91 10 977. 08

9 1.42 6 275. 96

10 9 0.27 5 486. 78

 

EA R TH R 2 1

2

 

4

5

6

EARTH R3 l 2

 

4 559 6. 784 6

124 5. 579

12 3. 14

9 3.63

6 1.87

3 5.47

145 4.273

7 5. 92

 

6 283.0’75 8

12 566.152

0

77 715. 77

5575. 14

18 849. 25

6 283.076

12 566. IS

 

 

 

 

MARS L0 l 2

3

4

 

6

7

8

9

10

11

12

13

14

15

16

 

620347712 0 0

18 656 368 5.050 37100 3 340.612 426 70

1 108 217 5.400 998 4 6 681.224 853 4

91 798 5.754 79 t0 021. 837 28

27 745 5.970 50 3.523 12

12 316 0.8d9 56 2 810.921 46

10 610 2.939 59 2 281. 230 50

8 927 4.157 0 0.017 3

8 726 6.110 1 1S 362.449 7

7 775 3.339 7 5 621.842 9

6 798 0.564 6 398.149 0

4161 0.2281 g942.4634

5 575 1 . 6ó1 9 2 544. 314 4

3075 0. 857 0 191. 448 5

2 938 6. 078 9 0. 067 3

2 628 0.648 1 3337. 059 5

 

MARS L0 17 2580 0.0300 3544.135 5

( 0 o n t . ) 18 2389 5.0390 796.298 0

19 1799 0.6563 529.691 0

20 1546 2.9158 1 751 .539 5

21 1528 1.1498 6 151.533 9

22 1 286 3.068 0 2 146.165 4

23 2 264 3.622 8 5 092.152 0

24 1 025 3.693 3 8 962. 455 S

25 892 0.183 16 703.062

26 859 2.401 2914.014

27 833 4.495 5 340.630

28 833 2.464 5 34O. 595

29 749 3.822 155. 420

30 724 0. 675 3 738.761

31 713 3.665 I 059.382

52 655 0.489 3127.313

33 636 2.922 8 432. 764

34 553 4.475 1 748.016

35 5S0 3.810 0.980

36 472 3.625 1 194.447

37 426 0.554 6 283.076

38 415 0.497 213.299

39 312 0.999 6 677.702

40 307 0.381 6 684.748

41 302 4.486 3 552.061

42 299 2.783 ó 254.627

43 293 4.221 20.775

44 284 5.769 3 149.164

45 281 5.882 I 349.867

46 274 0.542 3 340.545

47 274 0.134 3 340.680

48 239 5.572 4 136.910

49 236 5.755 3 333.499

50 231 1.282 3 870.303

51 221 3.505 582.897

52 204 2.821 1 221.849

53 193 3.557 3.590

54 189 1.491 9 492.146

55 179 1.006 951.718

56 174 2.414 553.569

57 172 0.439 5 486.778

58 160 3.949 4 562. 461

59 144 1.419 135.065

60 140 3.526 2 700.715

61 138 4.501 7.I I4

62 131 4.045 12 303.068

65 128 2.208 1 592.596

64 128 1.607 5 088.629

65 117 3.128 7 903. 075

66 113 3.701 1 589.073

67 110 1.052 242. 729

68 105 0.785 8 827.390

69 100 3.245 11 773.377

MAR S L 1 1 334 085 627 474 0 0

2 1 458 227 5.604 260 5 3 340.612 426 7

3 164 901 3.926 3IS 6 681. 224 853

4 19 965 4.265 94 10 021.837 28

5 3 452 4.732 1 3.523 1

6 2 465 4,612 8 13 362. 449 7

7 842 4.459 2 281.230

8 538 5.016 398.149

 

 

MARS

(conL)

MA R S

 

L 1 9

10

l l 12 IS 14

15

16

17

18

I9

20

21

22

23

24

25

26

27

28

29

30

31

32

35

 

35

36

37

38

39

40

41

42

43

44

45

46

L2 l

2

 

5

6

7

8

9

10

11

12

15

14

15

16

17

18

19

20

21

22

23

 

521 4. 994

433 2. 561

430 5. 516

382 3.559

314 4.963

283 3.160

206 4.569

169 1.329

158 4.185

134 2.233

134 5.974

118 6.024

117 2.213

114 2. 129

114 5. 428

9I 1.10

85 3.91

83 5.30

81 4.43

80 2.25

75 2.50

73 5.84

71 3.86

68 5.02

65 1.02

65 3.05

62 4.15

57 3.89

48 4.87

48 1.18

47 1.31

41 0.71

40 2.73

40 S. Z2

33 5.41

28 0.05

27 3.89

27 5.11

58 016 2.049 79

54188 0

13 908 2.457 42

2465 2.8000

398 3 14l

222 3.194

121 0543

62 3.49

54 3.54

34 6.00 ’

32 4.14

30 2.00

23 4.33

22 3.45

20 5.42

16 0.66

16 6.11

16 1.22

15 6.10

14 4.02

14 2. 62

15 0. 60

12 3.86

 

3 344. 136

191. 448

155.420

796.298

16 703.062

2 544.314

2 146.165

3 337.089

1 751.540

0.980

1 748.016

6 151.554

1 059.582

1 194.447

3 738.761

j 349.87

553.57

6 684. 75

529.69

8962.46

951 .72

242.73

2914.01

382.90

3 340.60

3 340.63

3 149. 16

4136.91

213 30

3 333. 50

3 185.19

1 592.60

7.11

20 043.67

6 283. 08

9 492.15

1221.85

2700.72

5340.61243

0

6681.22485

10021.8375

13 362. 450

3.523

155. 420

16 705. 06

5 344. 14

2 281 . 25

191 . 45

796. 80

242.73

598. I5 553.57 0. 98

2 146. 17

1 748.02

3 185. 19

951.72

1349.87

1 194.45

6 684.75

 

 

MARS

(conL)

MARS

MAR S

 

L2 24

25

26

27

28

29

30

31

32

33

L 3 1

2

3

 

5

6

7

8

9

l0 11

12

L4 1

2

 

4

5

6

7

8

 

II 4.72

10 0.25

9 0.68

9 3.83

9 3.88

8 5. 46

7 2.58

7 2. 38

6 5.48

6 2.34

1 482 0. 444 3

662 0. 885

188 1. 288

41 1.65

26 0

23 2. 05

10 i . 58

8 2. 00

5 2. 82

4 2. 02

3 4. 59

3 0.65

114 5. 1416

29 5.64

24 5.14

11 6.03

5 0.15

3 3.56

I 0.49

I 1.32

 

2 544.31

382. 90 059.38 20 043.67 3 738. 76 751.54

3 149.16

4136.91

1 592.60

3 097. 88

3340.6124

6681.225

10021.837

13362.45-

0

155.42

3.52

16 703. 06

242.73

3 344.14

3 185.19

553. 57

0

6 681. 22

5 540. 61

10 021.84

l3 562.45

155. 42 l6 705. 06

242. 73

 

MARS MARS

MARS

 

L5 1

2

B0 l

2

 

4

5

6

7

8

9

10

11

12

I3

14

15

16

Bl 1

 

1 5. 14 0

1 4.04 6 681.22

3 197 135 3.7ó8 520 4 3 340.612 426 7

298 053 4.106 170 6 681. 224853

289 105 0 0

31366 4.446 51 10 021. 837 28

3 484 4.788 I I3 362.449 7

443 5.026 3344.136

443 5.652 3337.089

399 5.13I 16703.062

293 3.795 2281.230

182 ó.136 6151.534

163 4.264 529.691

160 2.232 1059.582

149 2.16s 5621.845

145 1.182 5340.595

145 5.213 8340.630

139 2.418 8962.455

350069 5.368478 3340Bl2427

 

 

 

 

MAR S

MAR S

MARS

MARS

 

16 727 0.602 21 3 540.612 45

4 987 3.1416 0

302 S.559 6 681.22S

26 1.90 13 362. 45

21 0.92 10021.84

12 2.24 3337.09

8 2.25 16705.06

607 1.981 3 340.612

43 0 0

14 I . 80 6 681.22

3 3.45 10 02t .84

13 0 0

I I 3.46 3 540.61

1 0.50 6 681.22

153 033 488 0 0

14 184 953 3.479 712 84 3 340. 612 426 70

660 776 3.8I7 834 6 681.224 853

46 179 4. 155 95 10 021. 837 28

8 110 5.559 6 2 810.921 5

7 485 1.772 4 5 621.842 9

5 523 1.364 4 2 281. 230 5

3 825 4.494 1 13 362. 449 7

2 484 4.925 5 2 942. 463 4

2 307 0.090 8 2 544.514 4

1 999 5.360 6 3 337.089 3

1 960 4.742 5 3 344. 135 5

167 2.112 6 5 092. ï 52 0

1 103 5.009 J 398.149 0

992 5.839 6 151.534

899 4.408 529.691

807 2.t02 1059.582

798 3.448 796.298

741 1.499 2146,165

726 1.245 8432.764

692 2.134 8962.455

633 0.894 3540.595

633 2.924 3340.630

630 1.287 I751.540

574 0.829 2914.0\4

526 5.383 3738.761

473 5.199 3127.3I3

548 4.852 16 705. 062

284 2.907 3532.061

280 5.257 6285.076

276 1.218 6254.627

275 2.908 1748.016

270 5.764 £884.927

259 2.057 1194.447

234 5.105 5486.778

228 5.255 6 872.673

223 4.199 3149.564

219 5.585 191.448

208 5.255 5540.545

208 4.846 5540.680

18fi 5.699 6677.702

183 5.081 6684.74B

179 4.184 Z558.499

176 5.955 5870.303

164 3.799 4156.910

 

MA R S RI I 1 107 453 2.032 505 2 3 340.612 426 7

2 103176 2.370 718 6 681.224 853

3 12 877 0 0

4 10816 2.708 88 IO 021.837 28

5 1 195 3.047 0 133ó2. 449 7

6 459 2.888 2 281.230

7 396 3.423 5544.156

8 185 1. 584 2 544. 514

9 156 3.385 16 703. 062

10 128 6.043 3 337.089

ll 128 0.630 1 059.382

12 127 I .954 796.298

13 1GB 2.998 2 146.165

14 88 3.42 398.15

15 83 3.86 3 738.76

16 76 4. 45 6 III . 55

17 72 2,76 529.69

18 ó7 2,55 1751.54

19 66 4.41 1748.02

20 58 0.54 1194.45

21 54 0.68 8962.46

22 61 3.75 6684.75

23 49 5. 73 8 340. 60

24 49 1.48 3540.63

25 48 2.58 3149.16

26 48 2,29 2914.01

27 59 2.32 4136.91

MAR S R2 l 44 242 0. 479 51 3340. 612 48

8 158 0.870 0 6 681. 224 9

1 275 I . 225 9 10 021 . 837 8

4 187 ł.573 13362.450

5 52 3.14 0

6 41 ł.97 3344.14

7 27 1.92 16703.06

8 18 4.43 2281.23

9 12 4. S5 8 185. 19

10 10 5.39 1059.38

11 10 0.42 796.30

 

MARS

 

R3 1

2

S

4

/› 6

 

I I13 5.149 9 S 340.612 4

424 5.613 6 681. 225

100 S. 997 10 021 . 837

20 0. 08 13 362. 45

5 5. 14 0

5 0. 45 16 703. 06

 

MARS

 

R4 l

2

 

4

 

20 3.58

16 4.05

6 4.46

2 4.84

 

3 340. 61

6 681. 22

10 021 . 84

13 362. 45

.. .... ... .

 

JUP1TER L0 l

2

 

4

5

6

7

8

9

10

 

59 954 691

9 695 899

573 ó 10

306 389

97 178

72 903

64 264

39 806

38 858

27 965

 

0 0

5. 061 917 9 529. 690 965 1

1 .444 062 7. U3 547

5.417 547 1 059. 581 930

4.142 6S 652. 785 74

5.640 45 522. 577 42

5.411 45 105. 092 77

2. 29Z 77 419. 484 64

1 . 272 52 316. 391 87

l . 7ß4 55 S36. 804 51

 

JUPITER L0 11

(cont.) 12

13

14

15

16

17

18

19

20

21

22

25

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

 

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

JUP IT E R L l I

2

 

4

5

6

7

8

 

13 590 5. 774 81

8769 3.ó500

8246 3.582 S

7368 5.0810

6265 0.0250

6114 4.5132

5505 4.Iß63

5505 1.306 7

4905 1.3208

4647 4.6996

5045 4.3168

2610 1.5667

2028 1.0638

1921 0.9717

1765 2.1415

1723 5.8804

1633 3.5820

1432 4.296 8

973 4.098

884 2. 487

733 6. 085

751 8 . ß06

709 1.293

692 6.134

614 4109

582 4.540

495 3.756

441 2.958

417 1.036

390 4.897

376 4.705

341 5.715

330 4.740

262 1.877

261 0.820

257 5.724

244 5.220

235 I . 227

220 1.651

207 1 . 856

202 1.807

197 5.293

ż75 3.750

175 3.226

175 5.9J0

158 4.365

l 5l 3. 90ó

149 4. 377

141 3.UB

138 1.318

 

117 2.500

117 5.889

106 4.554

52 993 480 757 0

489 741 4.220 667

228 919 6.026 475

27 655 4.572 66

20 721 5.459 S9

12 106 0. I69 ßfi

6 068 4.424 2

5434 3.9848

 

1 589.072 90

949. 175 6

206.185 5

735. 876 5

213.299 I 1 162. 474 7 1 052.268 4

14.227 I

10.206 3 3.952 2

426.598 2 846.0828 5, 181 4 659. 897 5 1 066. 495 5 1 265, 667 5 515. 465 9 626. 670 2

95. 979

412.571

838. 969

1581.959

742.990

2 118.764 l4T8.867 309.278

523.505

454.909

2.448

1692.ł66

1368.660

533.625

0. 048

0. 963

580 1 28

199.072

728.763

909.819

543918

525.759

375.774

I 155.361

942.062

1 898. 351

956.289

1 795 . 258

74.782

1 685. 052

491.558

I 169.588

1 OÆ5.155

1 596.186

0.521 526.510

0

529.690 965

7.113 547

I 059.381 93

522.577 42

556. 804 51

103, 092 8

419. 484 6

 

 

JUPITER Ll 9

( oont. ) 10

11

12

I3

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

 

4238 5.8901

2212 5.2677

l 74fi 4. 926 7

1296 5.5513

1173 5.8565

1 163 0.5145

1099 5.5070

10D7 0.4648

1004 3.1504

848 5.758

827 4.803

816 0. 586

725 5.5I8

568 5.Q89

474 4.tT2

413 5.737

345 4.242

336 3.732

234 4.035

234 6.243

199 1.505

195 2.219

187 6. 086

184 6.280

17I 5.4I7

131 0.626

115 0.680

1 U 5.286

108 4.495

80 5.82

72 5. 34

70 5.97

67 5.73

66 0.13

65 6.09

59 0. S9

58 0. 99

57 5. 97

57 1 . 4l

55 5. 43

52 5. 73

52 0. 23

50 ó . OB

47 3 63

47 0. 51

40 416

34 010

33 5.04

32 5.S7

29 5.42

29 3.5ó

29 0. 76

25 1.61

 

14. 227 1

206.185 5 1 589. 072 9

31814

1 052.268 4

3. 932 2

515. 463 9

755. 876 5

426.598 2 110.206 213. 299

1 066. 495

639.897

625.670

412.371-

95.979

632.784

1 162. 475

949. 176

309.278

858. 969

323. 505

742. 990

543. 918

199.072

728.763

846.083

2 118. 764

956.289

1 045. 15

942.06

532. 87

21.34

526.51 1 581.96 1 155.36 l 596.19

I 169.59

533.62

0.29

117.32 1 368.66 525.76

1 478. 87

I 265.57

1692.17

302.16

220. 41

508.35

I 272.68

4.67

88.87

831. 86

 

JUPITER L2 1

2

3

4

5

6

7

 

47 254

38 96ó

30 629

3 189

2 729

2 723

1721

 

4.32 t 48

 

2. 930 21

1. 055 0

4. 845 5

3.414 1

4.187 3

 

7.113 55 0

529.690 97

522.577 4

536. 804 5

1 059. 3819

14. 227 1

 

 

JUPlTER L2 10

(cont.) ]l

12

15

14

15

16

l7

18

19

20

21

22

23

24

25

26

27

28

29

30

SI

52

 

34

3S

36

37

38

 

40

41

42

43

44

45

46

47

48

49

50

5l

52

53

54

5S

56

57

JUPlTER L3 l 2

] 4

 

6

 

8

 

10

11

12

13

14

 

367 6.055

337 5.786

308 0.ó94

218 3 814

199 5. 340

197 2.484

156 1.406

146 3.814

142 1.654

130 5.837

117 1.414

97 4.03

91 1.11

87 2.52

79 4.64

72 2.22

58 0.83

57 3.12

49 1.67

40 4. 02

40 0.62

36 2.53

29 3.61

28 5.24

26 4.50

26 2.51

25 1.22

24 8. 0l

19 4.29

18 0.81

17 4.20

17 1.83

I5 5.81

15 0.68

15 4. 00

i4 s.qs

14 1.80

I3 2.52

13 4.37

11 4.44

10 1 . 72

9 2.18

9 3.29

9 3.32

8 5. 76

8 2.71

7 2.18

6 0.50

6 502 2.598 6

1 357 1.546 4

471 2.475

417 3.245

353 2.974

156 2. 076

87 2.51

44 0

54 3.83

28 2.45

24 . 28

23 2.98

20 2.10

20 1.40

 

103.093

3.18I

206.186

1 589.073

1 066.495

3.932

052.268

639.897

426.598

412.371

625.670

10.21

95.98

632.78

543. 92 735.88 199.07 213.30 309.28 21. 34 323.51 728.76 10.29 838.97 742.99 1 162.47 1 045.15 956.29 532.87 508.35 2 118.76

526.51

I S96. 19

942.06

17.32

316.39

302.16

88.87

I 169.59

525.76

1 581.96

I 155.36

220.41 831. 86 846.08 553.62 1 265.57 949.18

7. 115 5

529.691 0

14.227

556.805

522.577

2 059.382

515.46

0

I 066.50

206.19

412.37

543. 92 659.90

419. 48

 

JUPITER L3 15 I9 1.59 108. 09

(cont.) 16 17 2.30 21 . 34

17 17 2.60 1 589.07

18 16 315 625.67

19 16 3.56 1 052.27

20 13 2.76 95.98

21 13 2.54 199.07

22 13 6.27 426.60

23 9 1.76 10.29

24 9 2.27 110.21

25 7 3.43 309.28

26 7 4.04 728.76

27 6 2.52 508.55

28 5 2.91 I 045.15

29 5 5.25 323.51

30 4 4.30 88. 87

31 4 5.52 302.16

32 4 4.09 735.88

33 5 1.45 956.29

34 3 4.86 1 596.19

35 5 1.25 21330

36 3 5.02 838.97

37 3 2.24 117.32

38 2 2.90 742.99

39 2 23 6 942.06

JU PITER L 4 1 669 0. 853 7. 114

2 114 3142 0

100 0.743 14.227

4 50 1.65 536. 80

  44 5.82 529.69

6 32 4.86 522.58

7 15 4.29 515. 46

8 9 0.71 1 059. 38

5 1.30 543.92

10 4 2.32 1 066.50

11 4 0.48 21 . 34

12 3 300 412.37

13 2 0.40 639.90

14 2 4.26 199.07

15 2 4.91 625. 67

16 2 4.26 206.19

17 1 5. 26 I 052. 27

18 1 4.72 95. 98

19 1 1.29 1 589.07

JU P IT E R L 5 1 50 5.26 7.11

2 16 5.25 14.23

4 0.01 536.80

4 2 1.10 522.68

5 1 314 0

JU PI TE R B 0 1 2 268 616 3.558 526 I 529.690 965 1

2 110 090 0 0

109 972 5. 908 093 1 059.581 930

8 101 3.605 1 522.577 4

5 6 438 0.306 3 536.804 5

6 6 044 4.258 8 I 589.072 9

7 1107 2.985 3 1162.474 7

8 944 1.675 426.598

9 942 2.956 1 052.268

10 894 1.754 7.114

11 836 5.179 103.093

 

 

JUPITER B0 12

(oont) 13

14

15

16

17

18

19

20

21

22

23

24

25

26

JUPITER Bl 1

2

 

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

JUPITER B2 1

2

3

4

5

6

7

8

 

10

 

12

I3

14

 

767 2.155

684 3.678

629 0.643

559 0.014

532 2.703

464 1.173

431 2.608

551 4.611

132 4.778

123 3.350

116 1.387

115 5.049

104 3.701

103 2.319

102 3.155

177 352 5.701 665

3 230 5.779 4

3081 5.4746

2212 4.7848

1694 3.1416

546 4.746

234 5.189

196 5 186

150 3 927

114 1439

97 2.91

82 5.08

77 2.51

77 0.61

74 5. S0

61 545

50 3.95

46 0.54

45 I .90

57 4.70

36 6.11

32 4. 92

8094 1.4632

813 51416

742 0. 957

399 2.899

342 I .447

74 0.41

46 3.48

30 I .95

29 0.99

23 4.27

\4 2.92

12 5.22

II 4.88

6 6.21

 

632.784

213.299

1 066. 495

846. 083

110. 206

949.176

419. 485

2 118. 764

742. 990

1692.166

323.505

516.392

515. 464

1478.867

1 581. 959

529.690 965

1 059.3819

522. 577 4

536.804 5

0

052.268

1 066. 495

7. 114

1 589.073

632. 784

94918

I 162.47

103.09

419.48

515.46

213.30

735.88

1 J 0.21

846.08

543.92

316.S9 1581.96

529.691 0

0

522.577

556.805

059.382

052.27

I 066.50

1589.07

515.46

7.11

543.92

632.78

949 18

1 045. t5

 

JUPtTER B3 1 252 5.381 529.691

2 122 2.733 522.577

3 49 1.04 536.80

11 2.31 I 052.27

5 8 2.77 515.46

6 7 4.25 1 059.38

7 6 1.78 1 066.50

8 4 1.15 543.92

9 3 3.14 0

 

JU P ITE R B4 l 15 4.55 522.58

2 5 4.47 529.69

  4 5.44 536.80

4 3 0 0

5 2 4.52 515.46

6 l 4.20 1 052.27

JU PITER B5 1 I 0.09 522.58

JU PIT ER R 0 1 520 887 429 0 0

2 25 209 327 3. 491 086 40 529.690 965 09

3 610 600 3.841 154 1 059.381 930

4 282 029 2.574 199 652.783 739

5 187 647 2.075 904 522.577 418

6 86 793 0.710 01 419. 484 64 -

7 72 065 0.214 66 536.804 51

8 65 517 5. 979 96 516. 591 87

9 30 135 2.161 32 949. 175 61

10 29135 1.67759 103.09277

11 23947 0.27458 7.11355

12 23453 3.54023 735.87651

13 22284 4.19363 1589.07290

14 13033 2.96043 1162.47470

15 12 749 2.715 50 1 052.268 58

16 9 703 1.906 7 206.185 5

17 9 161 4,413 5 213.299 1

18 7 895 2.479 1 426.598 2

19 7 058 2.1818 1 265.567 5

20 6 138 6.264 2 846.082 8

21 5 477 5.657 3 639.897 3

22 4 170 2.016 1 515. 463 9

23 4 157 2.722 2 625.670 2

24 3 503 0.565 5 1 066. 495 5

25 2 617 2.009 9 1 581.959 3

26 2 500 4.551 8 838.969 3

27 2 128 6.127 5 742.990 1

28 1 912 0.856 2 412.371 1

29 1611 3.088 7 1 368.660 3

30 1 479 2.680 3 I 478.866 6

31 1 231 1.890 4 323.505 4

32 1 217 1.801 7 110.206 3

33 1015 1.586 7 454.909 4

34 999 2.872 309.278

35 961 4.549 2 118.764

36 886 4.148 533.623

37 821 I . 593 1 898.351

38 812 5.94t 909.819

39 777 3.677 728.763

40 727 5. 988 1 155. 361

41 655 2.791 1685.052

42 654 3.582 1692.166

43 621 4.823 956.289

44 615 2.276 942.062

45 562 0.081 543.918

46 542 0.284 525.759

JUP ITE R R 1 1 1271802 2.649 375 1 529.690 965 1

2 61662 3.000 76 1 059.381 93

53 444 3.897 18 522.577 42

4 41 390 0 0

5 31185 4.882 77 536. 804 51

6 II 847 2.413 30 419.4B4 64

7 9 166 4.759 8 7.113 5

 

 

JUP IT E R R 1 8

( cont. ) 9

10

11

12

I3

14

J 5

16

17

18

19

20

22

22

23

24

25

26

27

28

29

30

3I

32

33

34

35

36

37

38

39

40

41

42

43

JU PITER R 2 I

2

 

4

5

6

7

8

9

10

1

12

13

14

15

16

17

38

19

20

21

22

23

24

25

26

 

3 404 3.346 9

8203 5.2108

5176 2.7930

2806 5.7422

2677 4.5505

2600 8.6844

2412 1.4695

2 101 5.9276

1646 5.3095

1641 4.4163

1050 8.161 I

1 025 2.554 5

806 2.678

741 2.171

677 6.250

567 4.577

485 2. 469

469 4, 710

445 0.403

416 5.368

402 4.605

347 4.681

338 3.168

261 5.343

247 3.923

220 4.842

203 5B0O

200 4.4Z9

lQ7 3.706

196 3. 759

184 4. 265

180 4.402

170 4.8b

146 6130

153 1.322

152 4.512

79645 1.35866

8 252 5. 777 7

7 050 5. 274 8

5 314 1.838 4

1 861 2.976 8

964 5.480

836 4.199

498 5142

427 2.228

406 3.783

377 2.242

363 5.368

342 6.099

339 6.127

3Z3 0.003

280 4.262

257 0.963

250 0.705

201 3.069

200 4.429

139 2.932

114 0.787

95 1.70

B6 5.14

83 0.06

80 2.98

 

589.072 9 735.876 5

103.092 8

515.463 9 1 052.268 4 206 .185 5

426.598 2

659. 897 0

I 066.495 5

625.6702

2132991

412.3711

632.784

l lb2.475

838.969

742.990

949.176

543.918

323.505

728.763

509.278

14.227

956.289

846.083

942.062

1 368. 660

1 155.361

1 045.155

2 118. 764

199. 072

95. 979

532. 872

526. 510

533.623

110.206

525.759

529. 690 97

522. 577 4

536.804 5

059.381 9

7.113 5

515.464

419.485

0

639.897

066.495

1 589.073

206.186

I 052.268

625.670

426.598

412.371

652.784

735.B77

543.918

105.093

14.227

728.763

838.97

525.52

309.28

742.99

 

 

J UPITER R 2 27

(cont.) 28

29

30

31

32

33

J4

3S

36

JU PITER R 5 1

2

 

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

JU PIT ER R 4 l

 

5

6

7

8

9

lO  1 l

12

13

14

15

JU P IT E R R 5 1

2

 

4

 

6

7

 

75 1.60

70 ï . 51

67 S. 47

62 6.10

56 0.96

52 5.58

50 2.72

45 5.52

44 0.27

40 5. 95

3519 6.0580

1073 1.6732

916 1.413

342 0.523

255 1.196

222 0. 952

90 314

69 2.27

58 1.41

58 0.53

51 5.98

47 1 . 58

43 6.12

37 118

54 1.67

34 0.85

51 1 . 04

30 4.63

21 2.50

15 0.89

14 0.96

l3 I . 50

12 2.61

12 356

11 1.79

ll 6.28

10 6.26

9 345

129 0.084

ll3 4.249

83 330

38 2.73

27 5.69

18 5.40

13 6.02

9 0.77

8 5.68

7 1.43

6 5.12

5 3.34

3 340

3 4.16

3 2.90

11 4. 75

4 5.92

2 5.57

2 4.30

2 369

2 4.13

2 5.49

 

956.29

215.30

199.07

1 045. 15

I 162.47

942.06

532. 87

508. 35 526.51 95.98

529.6910

536.804 5

522. 577

059.382

7. 114

515. 464

0

1066.50

545.92 639.90 412.37 625.67

419. 48

14.23

I 052.27

206. I9 I 589.07 426.60 728.76 199.07 508.35

I 045.15

735.88

325. 51 309.28 956.29 05.09 838.97

536.805

529.691

522.58

515.46

7.11

059.38

S45.92 I 066.50

14.25

412.37

639.90

625.67

1 052.27

728.76

426.60

536. 80 522.58 515.46

548. 92 7. 11

1 059.38

1 066.50

 

SAT URN L 0 1 87 401 354 0 0

2 11 107 660 8. 962 050 90 218. 299 095 44

3 1 414 151 4. 585 815 2 7. 113 547 0

4 398 379 0. 521 120 206. 185 548

5 550 769 5. 303 299 426. 598 191

6 206816 0.246584 105.092774

7 79271 3.B4007 220.41264

8 23990 4.66977 110.20632

9 16574 0.43719 419.48464

10 15820 0.93809 632.78374

11 15054 2.7I670 639.89729

12 14907 5.76903 316.39187

13 14610 1.56519 3.93215

14 13 160 4. 448 91 14. 227 09

15 13005 5.98119 11.04570

16 10725 3.12940 202.25340

17 6126 1.7633 277.0350

18 5863 0.2366 529.6910

19 5228 4.2078 3.1814

20 5 020 3. 177 9 453. 711 7

21 4593 0.6198 199.0720

22 4006 2.2048 63.7359

23 3874 3.2228 138.5175

24 3269 0.7749 949.1756

25 2Q54 0.9828 95.9792

26 2461 2.0316 735.8765

27 1 758 3. 265 8 522. 577 4

28 1 640 s . 505 0 846. 082 8

29 1 581 4.372 7 309. 278 5

30 1 391 4.025 3 323. 505 4

51 1 124 2.837 8 415.552 5

32 1 087 4. 183 4 2. 447 7

88 1 017 5. 717 0 227. 526 2

34 957 0. 607 1 265. 567

35 853 3. 421 175. 166

36 849 3. 191 209. 367

57 789 5. 007 0. 968

38 749 2. 144 858. 196

39 744 5.253 224. 345

40 687 1.747 1052.268

41 654 1.599 0.048

42 634 2.299 412.371

43 625 0.970 210.118

44 580 3.093 74.782

45 546 2.127 550.532

46 543 1.5t8 9.561

47 530 4.449 117.320

48 478 2.965 137.035

49 474 5.475 742.990

50 452 1.044 490.334

51 449 1.290 127.472

52 372 2.278 217.231

53 355 3.01 858.969

54 347 1.539 340.771

55 343 0.246 0.521

56 330 0.247 1581.959

57 322 0.961 203.738

58 322 2.572 647.011

59 309 3.495 216.480

60 287 2.370 351.817

61 278 0.400 211.815

62 249 1.470 1368.660

63 227 4.910 t2.530

 

SATURN LO 64 220 4.204 200.769

(cont.) ó5 209 1.345 625.670

66 208 0.483 11ó2.475

67 208 1.283 39.357

68 204 6. 011 265. 989

69 185 3.503 149.563

70 184 0. 978 4.198

71 182 5.491 2.921

72 174 1.863 0.751

73 165 0.440 5.417

74 149 5.756 52.690

75 148 1.535 5.629

76 146 6.231 195.140

77 140 4.295 21.341

78 131 4.068 10.295

79 ł25 6.277 1898.351

80 122 l . 976 4.666

81 118 5.541 554.070

82 117 2.679 1155.361

83 114 5.594 1059.582

84 112 I.105 191.208

85 110 0.166 1.484

86 109 3.438 536.805

87 107 4.012 956.289

88 104 2.192 88.8ó6

89 108 1 . 197 1 685. 052

90 101 4.965 269.921

SA TUR N L 1 1 21 854 295 596 0 0

2 1 296 855 1. 828 205 4 213.299 095 4

3 56434B 2.B8500l 7.113547

4 107679 2.277699 206.t85 548

5 98325 1.08070 426.59819

6 40255 2.04128 220.41264

7 19942 1.279 55 103.09277

8 10 512 2.748 80 14.227 09

9 6 939 0.404 9 639.897 3

10 4 803 2.4419 419.484 6

Il 4 056 2.921 7 110.206 3

12 3 769 3.649 7 3.932 2

I3 5 585 2.416 9 5.18I 4

14 3 302 1.262 6 433. 711 7

15 8 071 2.527 4 199. 072 0

16 1 953 3.563 9 11.045 7

17 1 249 2.628 0 95.979 2

18 922 I .961 227.526

ł 9 706 4.4I7 529.691

20 650 6.174 202.253

21 628 6.111 309.278

22 487 6.040 853.196

28 479 4. 988 522. 577

24 4ó8 4.617 63.736

25 417 2.117 323.505

26 408 1.299 209.367

27 552 2. 317 652. 7ß4

28 344 3. 959 412. 571

29 540 5. 684 316. 892

30 556 5. 772 735. 877

31 332 2.8ó1 210.118

32 289 2.753 117.320

33 28t 5.744 2.448

34 266 0,543 647.011

 

S

 

SATURN L2 1

2

 

4

 

6

7

8

 

10

Il 12 I3 14

15

16

 

116 441 1,179 879

91 921 0.074 25

90 592 0

15 277 4.064 92

10 631 0.257 78

10 605 5.409 64

4 265 1 . 046 0

1 216 2.916 6

1 165 4.ó09 4

1082 5.6913

1045 4.0421

1020 0.6857

634 4.588

549 5.573

457 1.268

425 0.209

 

7. 113 547

213.299 10

 

206.185 55

220.412 64

426.598 I9

14. 227 l

105.092 8 659.897 8 430 . 711 7

199.072 0

3.181 4 419.485

3.932

110. 206

227 . 526

 

 

SATURN L2 17

(e ont. ) l8

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

SATURN L3 1

2

 

4

5

6

7

8

9

10

11

12

13

14

 

274 4.288

162 I .381

129 I .566

117 3.881

105 4.900

101 0.893

96 2. 9J

95 5.63

85 5.75

83 6.05

82 L02

75 4.76

67 0.46

66 0.48

64 0.35

61 468

53 2.75

46 5.69

45 1.67

42 5.71

32 0.07

32 1.67

31 4.16

27 0.83

25 5.66

20 5.94

18 4.90

17 1.63

16 0.58

14 0.21

14 3.76

12 4. 72

12 0.13

12 3 12

1l 5.92

11 5. 60

11 5. 20

10 4.99

10 0.26

10 4.15

9 04é

8 2. 14

8 525

8 4.03

7 5.40

6 4.46

6 5.93

16 039 5.759 45

4 250 4.585 4

1907 4.7608

1466 5.9133

1162 5.6197

1067 3.6082

239 3.861

237 5.768

166 5.116

151 2.736

131 4.743

63 0.23

62 4.74

40 5.47

 

96. 979

11. 046

509. 278

855. 196

647. OH

21 . 841

316. 39

412.57

209. 37

216. 48

117. 32 21012 522.58 10.29 525.51 632.78 529.69 440.83 202.25 88.87 6374 302.16 191.96 224.34 735.88 217.23 625. 67 742. 99 515.46 838.97 195.14 203.00 234.64 846.08

556. 80

728.76

I 066.50

422.67

330.62

860.31

956.29

269.92

429.78

9.56

1052.27

284.15

405.26

7.113 55

213.299 1

220. 4J2 6

206.185 5

14.227 I

426.598 2 433.712 199.072

3 181

639.897

227.526

41948

105. 09

21.34

 

 

SATURN L8 15

(cont.) 16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

59

40

41

42

45

44

45

46

47

 

SATURN L4 1

2

 

4

5

6

7

8

 

10

1 1

12

I3

14

15

16

17

18

i9 20

21

22

23

24

25

26

27

 

40 5.96

39 5.83

28 301

25 0.99

l9 1. 92

18 4. 97

18 1.03

18 4.20

18 3.32

16 3.90

16 5.62

13 1.}8

II 5.58

11 5.93

I0 3.95

9 3.39

8 4.88

7 0.38

6 2,25

6 1.06

5 4,64

4 314

4 2.31

3 2.20

3 059

3 4.93

3 0.42

2 4.77

2 3.55

2 320

2 1.19

2 1.35

2 4.16

2 3.07

1662 3.9983

257 2.984

286 3902

149 2.741

114 5. 142

110 1 . 516

68 1.72

40 2.05

58 1.24

31 3.01

I5 0.8S

9 5.71

6 2.42

6 I.Ió

4 1.45

4 2.12

3 4.09

3 2.77

3 301

3 0.00

3 0.39

2 3.78

2 2.83

2 5.O6

2 2.24

2 5.19

1 1.55

 

95.98

110.21

647.01

3 93

855.20

10.29

412.37

216.48

309. 28

440.83

117. 52 88. 87 11.05

 

209. 57

302 16

325.51 652.78 522.58

210. 12

234. 64

0

SIS.46

860.31

529.69

224. 54 625 . 67 530.62 429.78 202.25

1 066.50

405.26

223.59

654. 12

7.113 5 220.413 14.227

2 J 3.299

 

206.186

426.60

433.71

199.07

227.53

659.90

21. 34

419.48

647.01

95.98

440.83

110.21

412.37

B8. 87

853. 20

10309

I 17.32

234.64

309 28

2t6.48

302. 16 191 . 96

 

S

 

 

SATURN BI 16

(cont.) 17

18

19

20

21

22

23

24

25

26

27

28

29

30

 

32

SATURN B2 1

2

3

 

5

6

7

8

9

10

Il

12

13

t4

15

16

17

IB 19

20

21

22

23

24

25

26

27

28

29

SA TUR N B3 l

 

4

5

6

7

8

9

10

11

L2

13

14

I5

 

166 2.444

158 fi. 209

128 1.207

110 2. 457

82 2.76

81 2. Bb

69 1.66

65 I . 26

61 1 . 26

59 I . 82

46 0.82

36 1.82

34 2.84

33 I.31

32 1.19

27 4.65

27 4.44

20630 0.5048Z

3720 3.9983

1627 6, 1819

1546 0

706 3 . 039

565 5.099

530 5.279

219 3.828

139 1.043

104 6.157

93 1.98

71 4.U

52 2.88

49 4.43

41 316

29 4. 55

24 I . 12

21 4.35

20 5.Zl

18 0.85

t7 5.68

16 4.26

14 3.00

12 2.53

8 352

7 556

7 0.29

6 1.16

6 361

666 I . 990

632 5.ó98

398 0

188 4.338

92 4.84

52 3.42

42 208

26 4.40

21 5.85

18 1.99

11 5. 57

IQ 2.55

7 3.4ó

6 4.80

6 0.02

 

199.072

I 10.206

529.ó91

2T 7.231

210. t2 14.23 202.25

216. 48 209.37 323.51 440.83 224.34 117.32 412.37 846.08 1 066.50

1.05

gl3.29910

20ó . 185 5

220.412 6 0

419. 485

426.598

433712

639.897

7114

227.526

316.39

199.07

632.7B

647.01

85150

210.12

14.23

217.25

440.83

110.21

21648

I03CP

412.37

529.69

202.25

209. 37

323.51

Iï7.32 &60.31

218.299

206.1B6

 

220.413

419.4B

433.71

426.6D

227,53

199. 07 639. 90 7, 11 647, 01 316, 39 632, 78 220, 12

 

SAT URN B5 16 6 5. 52 440. 85

(cont.) 17 5 5.64 14.25

18 5 1.22 853.20

19 4 4.71 412.37

20 5 0.65 ì03.09

21 2 3.72 yl6.48

SATURN B4 1 80 1.12 206.19

2 32 3.12 213.30

3 17 2.48 820.41

4 12 3.14 0

6 9 0. 38 419. 48

6 6 t . 56 433.71

7 5 2.65 227.53

8 5 1.28 199.07

9 1 I . 43 426.60

10 I 0.67 647.01

I I I . 72 440.83

12 I 6.18 639.90

SA T UR N B5 1 8 2.82 206. 19

2 1 0. 51 220. 41

SA T UR N R 0 1 965 758 156 0 0

2 52 921 382 2.392 262 20 215. 299 095 44

3 1 873 680 5.235 496 1 206.185 548 4

4 1 464 664 I .647 630 5 426.598 190 9

5 821891 5.935 200 516.391 870

6 547 507 5.015 326 103.092 774

7 371 684 2.2'7 1 L48 220.412 642

8 561 778 3. 159 043 7. 115 547

9 t40 618 5.704 067 632.783 739

10 108975 3.293136 110.206321

11 69007 5.94100 419.48464

12 61058 0.94058 659.89729

13 48913 1.557ZS 202.2s540

l4 34144 O,I95I9 277.03499

15 52402 5.47085 949.17561

16 20957 0.46849 785.876 SI

17 20839 1.52103 433.71174

18 20747 8.332 56 199.07200

19 15298 3.0s944 529.69097

20 14 296 2.6@ 54 323. SOS 42

21 12 8B4 1.648 92 138.517 50

22 I1 993 5. 980 51 846.082 83

23 1I 380 I .731 06 522.577 42

24 9 796 5. 204 B I 265.567 5

25 7 753 5.851 9 95.979 2

26 6 771 5. 004 3 14, 227 l

27 6466 0.1773 1052.2684

28 5850 1.4552 415.5525

29 5507 0.5974 ó3.7559

30 4696 2.1492 227.5262

3I 4044 1.6401 209.3669

32 3688 0,7802 412.3711

33 5461 1.8509 175.1661

34 3420 4.9455 1581.9595

35 3401 0,5539 350.5321

36 3376 5.6953 224.3448

37 2976 5.6847 210.1177

38 2885 1.3876 8T8.9693

39 2B8l 0.1796 853.1964

40 2 508 3.558 5 742. 990 I

 

 

SATURN R0 41

(eont.) 42

45

SAT UR N R 1 1

2

 

4

5

6

7

8

9

10

1l 12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

58

 

2448 6.1841 1368.6603

2406 2,9656 117.3199

2174 0.0151 340.7709

2024 5.054t 11.0457

6 182 981 0. 258 456 2 215. 299 095 4

506 578 0. 711 147 206. 185 548

34J 394 5.796 558 426. 598 191

188 491 0. 472 157 220. 412 642

186 262 3. 141 595 0

143 891 1 . 407 449 7. 115 547

49 621 6. 017 44 105. 092 77

20 928 5. 092 46 639.897 29

19 953 1 . 175 60 419. 484 64

18 840 1 . 608 20 110. 206 52

15 877 0. 758 86 199. 072 00

12 893 5. 945 30 453. 711 74

5 597 1 . 288 5 14. 227 1

4 869 0. 867 9 325. 505 4

4 247 0. 598 0 227. 526 2

5252 1 . 258 5 95. 979 2

3 081 3. 456 6 522. 577 4

2 909 4.606 8 202. 255 4

2 856 2. 167 5 755. 876 5

1 988 2. 450 S 412. 571 I

1 941 6. 025 9 209. 566 9

1 681 I . 291 9 210. 117 7

1 340 4. 808 0 853. 196 4

1 816 l . 258 0 117. 319 9

1 203 1 . 866 5 316.591 9

1 091 0.075 5 216. 480 S

966 0. 480 652.784

954 5, 152 647. 011

898 0.985 529. 691

882 I . 885 1 052. 268

874 1 . 402 224. 545

785 3. 064 858. 969

740 1 . 582 ó25.670

658 4, 144 509. 278

650 1, 725 742.990

613 z. 035 63. 736

599 2. 549 217. 251

508 2. 150 .9$2

 

SAT UR N R 2 1

2

 

4

5

6

7

8

9

10

I I

12

13

14

15

lt 17

18

19

 

436 902 4. 786 71 7

71923 2.50070

49 767 4,97168

43221 3.8ó940

2q 646 5.96310

4 721 2.475 5

4142 4,106 7

3 789 3.097 7

2 964 1.372 I

2 556 2.850 7

2 327 0

2 208 6.275 9

2 188 5. 855 5

1957 4.9245

924 5,464

706 2.971

546 4.129

431 5.178

405 4, 175

 

213.299 095

206. 185 SA 220, 412 64

426.598 19

7.lI355

199. 072 0

4]37ll7

659.8973

303.0928

4194846

0

110.2063

14.2271

227. S26 2

325. 50S 95 . 979 412. 571 522. 577 209. 367

 

 

SATURN R2 20

(cont) 21

22

25

24

25

26

27

28

29

30

31

32

SATURN RS 1

2

5

4

5

6

7

8

9

10

ll 12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

SAT URN R 4 1

2

 

4

5

6

7

8

9

10

11

12

I3

14

15

16

17

18

19

20

 

391 4.481

374 5.854

361 3.277

356 3.192

326 2.269

207 4.022

204 0.088

180 8. 597

178 4. oq7

154 5.155

148 0.136

133 2.594

132 5.933

20 315 3.02187

8 924 3.191 4

6 909 4. 551 7

4 087 4.224 1

3 879 2.010 6

1 071 4. 208 6

907 2.2B5

606 3.175

597 4.135

483 1.173

393 0

229 4.698

18B 4.590

150 3.202

121 3.768

102 4. 710

101 5.Bl9

93 lè4

84 2.63

73 4.15

62 2.3l

55 0.31

50 2.39

45 4.37

41 0.69

40 1.84

38 s.94

32 4.01

1 202 I .415 0

708 1.162

516 6.240

427 2. 469

268 0. 187

170 5. 959

150 0.480

145 1 . 442

121 S . 405

47 5. 57

19 5. 86

17 0. 53

16 2. 90

15 0. SO

14 l . 50

13 2.09

ll 0.22

lI 2.46

10 314

9 1.56

 

216.480

117. 320

647. 011

210.1 \8 853.196 785. 877 202. 255 632. 784 440. 825 625 . 670 302. 165 191 . 958 809. 278

213.299 10 220. 412 6 206. 185 5

7. lls 5 426. 598 2 199. 072 0 458. 712 227. 526 14. 227 689. 897

0

419.485

110.206

103.093

323.505

95.979

412.371

647.01

216.48

117.32

440.83

853.20

209.37

S91.96

522.58

302.16

88.87

21.34

220.4126

213.299

206.186

7114

426.598

199.072

488. 712

227.526

14. 22?

659.90

647.01

440.83

110.21

419.48

4t2.37

523.51

9/ 98

117.32

0

88.87

 

 

SATURN R4 21

( o ont . ) 22

23

SATURN R5 1

2

5

4

5

6

7

8

9

10

11

12

15

14

15

16

17

18

UR AN US L 0 1

 

9 2.28

9 0.68

8 l . 27

129 5.913

32 0.69

27 5.91

20 4.9t

20 0.67

14 2.67

14 146

13 4.59

7 4.63

5 361

4 4.90

3 4.07

3 4.66

3 0.49

3 318

2 37O

2 332

2 0.56

548 129 294 0

 

2134

21648

234.64

220.4t3

7.11

227.53

433.71

14.23

206.19

199.07

426.60

213.30

639. 90

440.85

647.01

19t . 96

325.51

419.48

88.87

9598

117.32

 

 

 

5 272 528 5.358 257

6 70 828 5. 592 54

7 68893 6.09292

8 61 999 2.26952

9 61951 2.85099

10 26469 3.14152

11 25 711 6.It380

12 21079 4.56059

15 17819 l.74A]7

14 14 613 4. 757 52

 

149. S68 197

65. 755 90

7ó .266 07

2.968 95

11 . 045 70

71.812 65

454.909 37

148.078 72 56.648 56 3. 952 15

 

 

1 372 4. 196 4

34 1 284 3. 18 5

35 1 282 0.542 7

86 I 244 0.916 1

37 1 221 0 199 0

38 1151 4. 179 O

39 I 150 0.933 4

 

111.4802

202.2534

222.8603

2.4477

I084dl2 35.6796

3.1814

 

 

URANUS L0 40 1 090 I .775 0

(cont. ) 41 1 072 0.235 6

42 946 1. 192

43 708 5.185

44 653 0. 966

45 628 0. 182

46 607 5. 432

47 559 3.358

48 524 2. 013

49 483 2.106

50 471 1.407

51 467 0.415

$2 434 5.521

55 405 5. 987

54 599 0. 538

55 396 5.870

56 379 2.350

57 310 5.833

58 300 5.644

59 294 5.839

60 252 1.637

61 249 4.746

62 239 2.550

63 224 0.516

64 223 2.843

65 220 1.922

66 217 6.142

67 216 4.778

68 208 5.580

69 202 1.297

70 199 o.qs6

71 194 1.888

72 193 0.916

73 187 1.319

74 182 3.536

75 I7Z 1.559

76 172 5.680

77 170 3.677

78 169 5.879

79 165 1.424

80 165 5.050

81 168 0.758

82 147 1.265

83 148 1.500

84 159 S.386

85 129 4.260

86 124 1.374

87 110 2.027

88 109 5.206

89 104 5.028

90 104 1.458

91 108 0.681

URAN US L 1 1 7 502 545 122 0

2 154 458 5.242 017

3 24 456 1.712 56

4 9 258 0. 428 4

5 8 266 1.502 2

6 7 842 1.319 8

7 5 899 0. 464 8

8 2284 4.1737

9 1 927 0.530 I

10 1 235 1,586 3

 

12.530 2 62. 251 4 127.472 213.299 78.714 984.600 529.691

0.521

299.126

0. 963

184.727

145.110

183.243

8.077

415.552

35I .817

56.622

145. 631

22.091

39.618

221.376

225. 829

37.035

84.343

0.261

67.668

5. 938

340.771

68.844

0.048 t52.532 456.394

455. 425

0.160

79.235

160.ó09 219.891 5 417 18. 159 106. 977 112. 915 54. 175 59.804 35.425 32.195 los.819 7. 114 554. 070 77. 963 0. 751 24.379 14. 978

0

74.781 599

1. 484 47

11.045 7 63.755 9 149.563 2 3.932 2

76.266 I 2.968 9 70.849 4

 

 

URANUS U 11

(cont.) 12

13

14

15

16

L7 18 I9 20

21

22

23

24

25

26

27

28

29

30

3I

32

33

34

35

36

57

58

59

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

URANUS L2 1

2

5

4

5

6

7

8

9

10

11

12

I3

14

15

 

791 5. 456

767 1.996

482 2.984

450 4.138

446 5.723

427 4.731

354 2.583

348 2.454

5l 7 5. 579

206 2.363

189 4.202

184 0.284

180 5.684

171 3.001

158 2.909

155 5.591

154 4.652

152 2.942

143 2.590

121 4.148

116 3.732

102 4.188

102 6.034

88 5.99

88 6.16

81 2.64

72 6.05

69 4.05

59 3.70

47 354

44 5.91

43 5.72

39 4.92

36 5.90

36 3.29

36 3.33

55 5.08

31 5.62

31 5.50

31 5.46

30 1.66

29 1.15

29 4.52

27 5.54

27 615

26 4.99

25 5.74

53 033 0

2 358 2.2601

769 4.526

552 3.258

542 2.276

529 4.923

258 3.69t

239 5.858

182 6.218

54 1.44

49 6. 05

45 3 91

45 0.81

38 I . 78

87 4. 46

 

3.181

73.297

85.827

138.517

224.345

71 . 813 t48.079 9.561

52.690

2.448

56.622

151.048

12.530

78.714

0.965

4.453

35.164

77.75t

62.251

127.472

65.220

145.631

0.112

18.16

202.25

22.09

70.53

77. 96

67.67

351.82

7.11

5. 42

222.86

33.68

8.08

71. 60

38.I3 984.60 59.80 160.61 447.80

462. 02

84.34

I31 .40

299.IS l57.05 3B0 \5

0

74.781 6

11 .046

63.736

3. 952

1.484

3. 181

149.563

70.849

76.27

56.62

2.45

85.83

52.69

2. 97

 

 

URANUS L2 16

(cont.) 17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

 

33 0.86 9.56

29 5. 10 78 . 30

24 2. I1 18. 16

22 5.99 158.52

22 4.82 78.71

21 2.40 77.96

21 2.17 224.54

17 2.54 145.63

17 3.47 12.53

12 0.02 22.09

11 0.08 127.47

10 5. 16 71 . 60

10 4.46 62.25

9 4.26 7.11

8 5. 50 67.67

7 1.25 5. 42

6 3.56 447.80

6 5.45 65.22

6 4.52 151.05

6 5.75 462.02

 

URANUS L3 1 121 0.024 74.782

2 68 4.12 3.93

55 2.39 11 . 05

4 46 0 0

5 4S 2.04 5. 18

6 44 2.96 1 . 48

7 25 4.89 63.74

8 21 4.55 70.85

20 2.31 149.56

10 9 1.58 56.62

11 4 0.23 18.16

12 4 5.59 76.27

13 4 0.95 77.96

14 8 4. 98 85. 85

15 3 4.13 52.69

16 3 0.37 78.71

17 2 0.86 145.63

l8 2 5.66 9. 56

URANUS L4 l ll4 3142 0

2 6 4.58 74.78

3 0.35 11.05

4 1 342 56.62

URANUS B0 1 1 346 278 2.618 778 1 74.781 598 6

2 62 341 5.081 I1 149.563 20

3 61 ó01 3.141 59 0

4 9 964 1.616 0 76.266 1

5 9 926 0.576 3 73.297 1

6 3259 1.2612 224.3448

7 2972 2.2437 1.4845

8 2010 6.0555 148.0787

9 1522 0.2796 63.7359

10 924 4.038 151.048

11 761 ó. 140 71 . 818

12 522 3.32t 158.517

I3 463 0.743 85.827

14 437 5.58t 529.691

15 485 0. 341 77. 751

16 431 3.554 213.299

17 420 5.213 11.046

18 245 0.788 2.969

 

URANUS B0 19 233 2.257 222,860

(cont.) 20 216 1.591 38.133

21 180 5.725 299.126

22 175 1.256

23 174 1 . 957 380.128

24 160 5.336 l1t.450

25 144 5.962 3L164

26 116 5.759 70.849

27 106 0.941 70.328

28 102 2 6I9 78. 714

UR AN US B1 1 206 366 4.123 945 74. 781 599

2 8 563 0.338 2 149. 563 2

  1726 2.121'? 73 .2P7 I

4 1374 0 0

5 1369 3.068 6 76.266 1

6 451 3.777 1.484

7 400 2.848 224.346

8 307 1. 255 148.079

9 154 5.786 63.736

10 112 5.573 151.048

11 11I 5.329 l3B.517

12 83 3.59 71.81

13 56 3.40 8583

14 S4 1.70 77. 75

15 42 1.21 11.05

16 41 4. 45 78. 71

17 32 3.77 222,86

18 30 2.56 2. 97

19 27 5.34 213.30

20 26 0. 42 380. t3

URANUS B2 l 9212 5.8004 7À78l6

2 557 0 0

3 286 2. 177 149.563

4 95 3.84 7360

S 45 4.88 7&27

6 20 5.46 1.48

7 15 0.88 13852

8 14 2.85 148.08

9 14 5.07 63.74

10 10 5.00 224.34

1 1 8 6.27 78.71

URANUS B3 1 268 1 .2S1 74. 782

2 11 3. 14

 

3 6 4.01 149.56

3 5. 78 73.30

URANUS B4 1 6 2. 85 74. 78

URANUS R0 1 \ 92\ 264848 0 0

2 88 784 984 5.603 775 27 74.781 598 S7

  3 440 836 0.328 56I 0 73.297 125 9

4 2 055 653 1.782 951 7 149.563 197 1

S 649 322 4.522 473 76.266 071

6 602 248 5.860 038 63.735 898

7 496 404 1.401399 454.909 567

8 358 526 1 580 027 1S8.517 497

  243 508 I .570 866 7I .812 653

10 t90 522 I .998 094 1 .484 475

11 16I 858 2.791 379 148.078 724

12 143 706 1.585 686 11 , 045 700

 

 

URANUS RO 15

(cont.) 14

15

16

17

18

19

20

21

22

28

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

URANUS R1 1

2

3

 

5

6

7

8

9

10

11

12

13

14

15

 

93 192 0.174 57

89 806 3. 661 OS

71 424 4.246 09

46 677 l .599 77

39026 3,36235

39010 1.66971

36755 3.88649

30349 0.70100

29156 3.18056

25 786 3.78538

25620 5.25656

22637 0.72519

20473 2.79640

20 472 I . 555 89

17901 0.55455

15503 5.35405

14702 4.90434

12897 2.ó2I54

12328 5.96039

I1 959 1.75044

11853 0.99343

11696 3.29826

11495 0.45774

10793 1.421 OS

9 l11 4. 996 4

8421 5.2535

8402 5.0388

7449 0.7949

7 529 3. 972 8

6046 5.6796

5 524 3.1150

5445 5.1058

5238 2.6296

4 079 3. 220 6

3 919 4.250 2

5802 6.1099

3781 3.45&4

5687 2.4B72

5102 4.1405

2963 0.8298

2 942 0.425 9

2940 2.1464

2938 3.6766

2865 0.3100

2538 4.8546

2364 0.4425

2183 2.9404

1479896 3.6720571

71212 6.22601

68627 6.13411

24060 3.14159

21468 2.60177

20857 5.24625

11405 0.01848

7497 0.4236

4244 1.4169

5927 5.I5S 1

3578 2.5116

3506 2.5835

8 229 5.2£S0

3060 0.1532

2 564 0.9808

 

36.648 56

109.945 69

224.44 80

35.164 09 277. 054 99 70.849 45 t46.594 25

I51 .047 67

77. 750 54

85.827 30

380.127 77

529.690 97

70.328 18

202.255 40

2.968 95

38.133 04 108. 461 22 111. 430 16 127. 47180 984.600 33 52.690 20 3.932 15

65.220 37

213.299 10

62.251 4

222.860 3

415.552 5 351.816 6 183.2428 78.7138

9.561 2

145.109 8 35.679 6 840. 770 9 59.617 5 184. 727 5 456. 5938 453.4249 219.8914 56.6224 299.1264 137.0330 140.0020

12.530 2 l5l .405 9 554.0700

305.346 2

74.781598 6

63.735 90

149.563 20 0

76.266 07 11 .045 70 70. 849 45

73.297 1 85.827 3 71.812 7 224.3448

138.517 5 3. 932 2

I . 4845

148.078 7

 

 

URANUS Rl 16

(com) 17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

URANUS R2 l 2

 

4

5

6

7

8

 

10

1 1

12

13

14

15

16

17

18

URAN US R 3 1

2

 

4

5

6

7

8

9

lO

URAN US R4 1

2

 

2429 3.9944

1645 2.6555

584 1.4305

1508 5.0600

1490 2.6756

1413 4.5746

1403 1.3699

1228 1.0470

1 033 0.264 6

992 2. 172

862 5.055

744 3.076

687 2. 499

647 4.473

624 0.863

604 0.907

575 3.231

562 2.718

530 5.917

528 5.151

22 440 0.699 $S

4 727 I .699 0

1682 4648 5

1 650 3 09ó 6

1434 3.5212

770 0

500 6.172

461 0.767

390 4.496

390 5.527

292 0.204

287 3.534

273 3.847

220 l.9&4

216 0.848

205 3.248

149 4.898

129 2.081

1 164 4. 734 5

212 3. 84Z

196 2, 980

105 0. 958

73 1 . 00

72 0.03

55 2. 59

36  1 65

34 3 82

32  1 60

53 3 01

10 1.91

 

52.690 2

127.471 8 78.713 8

151.047 7 56.622 4

202.253 4 77.750 5

62.251 4 13a .403 9 65.220 55l . 817 35.164 77. 963 70. 328

9.561

984.600

447.796

462.023

213.299

2. 969

74.78t 60

68. 785 9

7O. 849 4

11 . 045 7

149. 563 2

0

76.266

3. 932

56.622

85.827

52.690

73.297

138.517

13I .404

77. 965

78.714

127.472

3. 181

74. 781 6

65. 756

70.849

11.046

149.56

56.62

3.93

77.96

76.27

131.40

74.78

5ó.62

 

 

NEPT U NE L 0 l

2

 

6

7

 

9

10

 

531188 633 0

1 798 476 2.901 012 7

1 019 728 0.485 809 2

124 532 4.830 081

42 064 5.410 55

37 715 6.092 22

33 785 1.244 89

16 483 0.000 08

9 199 4.937 5

8 994 0.274 6

 

 

38.I33 035 6

1.484 472 7

36.648 563

2.968 95

35.164 09

76.266 07 491.5S'7 93 39.617 5 175.166 1

 

 

 

14

15

16

17

lg

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

NEPTUNE L1 1

2

8

4

 

 

1 454 2. 785 4

900 2. 076

745 3. 190

506 5.748

400 0.350

345 3.462

540 3. 504

323 2.248

506 0.497

287 4.505

282 2.246

267 4.889

252 5.782

245 1.247

233 2.505

227 1.797

170 3.324

151 2.192

150 2.997

148 0.859

119 3.6‘77

109 2.416

105 0.04L

103 4.404

102 5.705

3 837 687 717 0

16604 4.86319

15807 2.27923

3385 36B20

1306 Z6732

 

74.781 6

109.946

71 . 815

114.399

1021.249

4Ll02

77.751

32. )95

0.521

0. 048

14fi. 594

0. 965

588. 46S 9. 561 137.033 465. 425 108. 461 33.940 5. 938 111 . 480 2448 183.243 0.261 70.328 0.112

0

I .484 47

58.133 04

76.266 1 2.968 9

 

 

8 107 2.451

9 106 2.755

10 73 5, 49

LI 57 I .86

12 57 5. 22

I3 35 4.52

14 S2 5.90

15 30 5.G7

16 29 5. 17

17 29 5. t7

18 26 5.25

 

4.453

33.680

36.65

L 14.40

0.52

74.78

77.75

388.47

9.56

3.45

168.05

 

 

6

27 624 0 0

6 2 000 1 . 510 0 74. 781 6

16 206 4.257 529.691

2 1 803 I . 975 8 76. 26G 1

 

6 148 3.858 74. 782

 

52 5.05

 

7Z.50

 

 

12 37 5.76

 

2.97

 

 

                          59 314

6 6 5.61

 

0

74.78

 

 

2 2 0

2 5. 53

 

0

76. 27

 

NEPTUNE R0 1 3 007 013 206 0 0

2 27 062 259 I .329 994 sq :s8. I:s:s o35 64

3 1 691 764 3.251 861 4 36.648 562 9

4 807 831 5.185 928 1 . 484 473

5 537 761 4.521 139 35.164 090

6 495 726 I .5‘71 057 491.557 929

7 274 572 I .845 523 175.166 060

8 135 134 3.372 206 39.617 508

9 121802 5.797 544 76.266 071

10 100 895 0.377 027 73.297 126

11 69 792 3.796 17 2.968 95

12

l3 46688

24 594 5.74938

0. 508 02 33.67962

109. 945 69

14 16 959 l . 594 22 71 . 812 65

15 14 250 1 . 077 86 74. 781 60

16 12 012 1 . 920 62 1 021 . 248 89

17 8 895 0. 678 2 146. 594 5

18 7 572 l . 071 S 588. 46s 2

19 5 721 2. 590 6 4. 458 4

20 4 840 1. 906 9 41 . 102 0

21 4 483 2. 905 7 529. 691 0

22 4 421 l . 749 9 108. 461 2

23 4 554 0.679 9 32. 195 1

24 4 270 8. 415 4 455. 424 9

25 8 881 0.848 1 185. 242 8

26 2881 1 . 986 0 137. 033 0

27 2 879 5.674 2 550. 332 1

28 2 636 8. 097 6 213. 299 1

29 2 580 5. 798 4 490. 078 5

30 2 523 0.486 3 495. 042 4

31 2 806 2.809 6 70. 528 2

32 2 087 0.618 6 53. 940 2

NEPTUNE RI 1 236 359 0.704 980 38.133 036

2 13 220 3.320 15 1.484 47

5 8 622 6, 21 ó 3 35.164 1

4 2 702 1 . 881 4 39.617 5

5 2 155 2, 0943 2. 968 9

6 2 153 5. 168 7 76.266 I

7 1 603 0 0

8 1 464 1 . 184 2 83. 679 6

9 1 156 3. 918 9 56.648 6

10 898 5, 241 588.465

1 1 790 0.533 168.053

12 760 0,021 182.280

13 607 1.077 1021.249

14 572 5,401 484.444

15 561 2.887 498.671

NE PT UNE R 2 1 4 247 5. 899 1 38. I33 0

2 218 0 348 1 4B4

\65 2.2'59 168 053

4 156 4.594 182.280

5 127 2.848 35. 164

NE PT UNE R 8 l 166 4. 552 38. 135

 

Appendix IV

Coefficients for the Heliocentric Coordinates, Jupiter to Neptune, 1998-2025

On the following pages coefficients are given for the calculation of the heliocentric coordinates (ecliptical longitude L, latitude B, and radius vector fi) of the giant planets Jupiter to Neptune for any instant during the years 1998 to 2025. The formula is

A + A t + A 2 f 2 + A 3 t’ + A i’ + A t’

where i = d/365, and d is the number of the day in the year. In other words, the time is measured from the given Epoch in units of 365 days. The uniform time scale of Dynamical Time is used, and the heliocentric longitude and latitude are referred to the mean equinox of the date (FK5 system). Each expression is valid for one

See more explanations on pages 220—221.

455

 

  A A 2

  x4 A '5

JUPITER

Year 1998

Epoch = JDE 2450813.5

L 329.6821899 32.5373311 0.6251361 —0.0898353 —0.0386157 0.0183980

B —0.9881379 —0.4836213 0.1496847 0.0343504 —0.€D427I2 —0.tXD8253

R 5.0212516 —0.0962003 0.0267709 0.€D46167 0.HD1739 —0.0€O3582

JUPITER

Year 1999

Epoch = IDE 2451178.5

L 2.7246041 33.4053704 0.2040247   —0.1756398 —0.0305659 0.0€D0648

B —1.2928206 —0.1024089 0.2189535 0.€D88476 —0.0085174 0.0000270

R 4.9562546 —0.0298757 0.0376516 O.€D32669 —0.0026228 0.0€O5117

JUPITER Year 2tXD Epoch = JDE 2451 543.5

L 36.1578582 33.2848762 —0.3202376 —0.1607197 0.0101019 0.0048032

B —1.1759189 0.3281328 0.1949128 —0.0233461 —0.1X176376 0.£D15479

R 4.9651863 0.0472665 0.0371120  —0.1X137156 —0.0£D 1438 —0.0€i02643

JUPITER Year 2(D1 Epoch = JDE 2451 909.5

L 69.0649681 32.2226338 —0.6998931 —0.0829214 0.0310692 —0.0040156

B —0.6805959 0.6256131 0.0938187 —0.0391518 —0.iXD3911 0.0007610

R 5.0457384 0.1086124 0.0221681 —0.€046960 —0.€019856 0.0006987

JUPITER Year 2002 Epoch = JDE 2452274.5

L 1€O.5318411 30.6783457 —0.7959767 0.€D46134 0.0241846 —0.0027664

B 0.0€O0541 0.6980207 —0.0179458 —0.0339429 0.0€i40248 —0.(JO0 1891

R 5.1705360 0.1343765 0.€D32467 —0.0€I76620 0.0D16344 —O.(XD5326

JUPITER Year 2£D3 Epoch = JDE 2452639.5

L 130.4402417 29.1831608 —0.6703649 0.0749318 0.007 2343 —0.0006340

B 0.6500218 0.5754659 —0.0977839 —0.0188353 0,0024263 0.0001555

R 5.3015990 0.1218037 —0.0151896 —0.0040044 —0.(ID 6998 0.(XJ€i4889

JU PIT ER

Year 2004

Epoch = JDE 2453034.5

L 159.0345698 28.0927741 —0.4025854 0,0863880 0.0149846 —0.O€13 9962

B 1.1114503 0.33386D6 —0.1383051 —0.£085903 0.0D27221 —0.O€O3263

R 5.4039978 0.0790223 —0.0266881  —0,CD38764 0.0D19314 —0.0005949

 

  A

  A 3 A4

 

JUPITER

Year 2€D5

Epoch = JDE 2453 370.5

L 186.8977139 27.5863948 —0.0972861 0.1182676 —0.0111922 0.0041159

B 1.3£D9218 0.0399162 —0.1506563 —0.IXD5039 0.0017402 0.0000343

R 5.4538435 0.0186227 —0.0322694 —0.0€D3512 0.0(DO610 0.tXD2636

JUPITER Year 2iXi6 Epoch = JDE 2453735.5

L 214.4980139 27.7220245 0.2346710 0.0921343 0.0115682 —0.0055378

B 1.1914523 —0.2557633 —0.1417826 0.€D75267 0.0€O9788 0.D€O4777

R 5.4401702 —0.0454318 —0.0307570 0.€i012036 0.0012608 —0.0003290

JUPITER

Year 2007

Epoch = JDE 2454100.5

L 242.5528742 28.4867192 0.5229238 0.1039235 —0.0236677 0.0055436

B 0.8028896 —0.5104669 —0.1083669 0.0148232 0.0€i37680 —0.0€D3019

R 5.3661167 —0.0999313 —0.0222721 0.0033557 0.0€O7824 —0.OCOD057

JUPITER

Year 2€D8

Epoch = JDE 2454465.5

L 271.6483166 29.7769198 0.7477661 0.0397082 —0, 0€120776 —0.0067762

B 0.2023449 —0.6691526 —0.0441376 0.028£D87 0.0026931 —0.OtO0486

R 5.2480457 —0.1313144 —0.€D81191 0.€D61518 0.OSD 1181 —0.0000020

JUPITER

Year 2€D9

Epoch = JDE 2454831.5

L 302.2897525 31.3542£D1 0.7873094 —0.0101742 —0.0408975 O.%67920

B —0.4821079 —0.6625902 0.0554601 0.0371385 0.H114699 —0.0£J1 1593

R 5.1145278 —0.1285893 0.0115895 0.€D57933 0.0(D5733 —0.0002814

JUPITER

Year 2010

Epoch = JDE 2455 196.5

L 334.3869822 32.7683636 0.5760783 —0.1238230 —0.0142657 —0.%1 2614

B —1.0517888 —0.4401708 0.1643785 0.0313220 —0.1 142798 —0.0010832

R 5.£D36133 —0.0871279 0.0290746 0.0061802 —0.0019050 0.0€O2955

JUPITER

Year 2011

Epoch = JDE 2455561.5

L 7.5920741 33.4861111 0.1105793 —0.1707292 —0.0141162 0.0€186056

B —1.3016221 —0.0399523 0.2216422 0.€D51585 —0.0103989 0.(D09I79

R 4.9501306 —0.0166052 0.0395130 0.D001809 —0.0004479 —0.0002501

 

  A A A A 4

 

JUPITER

Year 2012

Epoch = JDE 2455926.5

L 41.0125248 33.1813654 —0.4045897 —0.1565926 0.027 4250 —0.DD29575

B —1.1242548 0.3817953 0.1837984 —0.0278906 —0.iXi55891 0.0DI0414

R 4.9725213 0.0599571 0.0345019 —0.fD22206 —0.€D24916 0.0€D6584

JUPITER

Year 2013

Epoch — JDE 2456292.5

L 73.7448347 31.9934973 —0.7339396 —0.0653781 0.0248588 —0.€i€i04753

B —0.5893218 0.6489931 0.0771672 —0.0397064 0.iXD66l3 0.1XD639l

R 5.0632434 0.1156997 0.0196696 —0.£D73820 0.D€D9734 —0.0€D3958

JUPITER Year 2014 Epoch = JDE 2456657.5

L 104.9633978 30.4264402 —0.7914405 0.0238249 0.0197071 —0.0€13 2518

B 0.0984324 0.6900469 —0.0320762 —0.0306204 0.0£i27843 0.0€D2I63

R 5.1918082 0.1348517 —0.0€D7139  —0.£D50416 —0.€D11251 0.0€D5752

JUPITER Year 2015 Epoch = JDE 2457022.5

L 134.6386778 28.9775308 —0.6276824 0.0669581 0.016820£1 —0.€D3 2219

B 0.7287832 0.5462341 —0.1048855 —0.0184840 0.€D39686 —0.0005057

R 5.3203546 0.1166386 —0.0168448  —0.0€158563 0.0£120150 —0.0D06241

JUPITER Year 2016 Epoch = JDE 2457387.5

L 163.0690823 27.9743024 —0.3634612 0.1078701 —0.£D3 7752 0.D019072

B 1.1551108 0.2943723 —0.1414807  —0.CD66130 0.0015662 0.0001482

R 5.4156829 0.0703597 —0.0283421 —0.0€118D86 -0.0003632 0.0004003

JUPITER Year 2017 Epoch = JDE 2457 753.5

L 190.8614472 27.56d8846 —0.0376795 0.0950355 0.0125018 —0.€D44412

B 1.3030989 —0.€D22487 —0.1508309 0.0fD9584 0.0€113231 0.0€O1389

R 5.4559510 0.€D85834 —0.0322787  —0.0€110422 0.(XI1 6684 —0.0€O4893

JUPITER Year 2018 Epoch = JDE 2458 118.5

L 218.4917484 27.8027013 0.2742538 0.1168917 —0.0176575 0.0051412

B 1.1524397 —0.2950583 —0.1382465 0.0070141 0.0027396 —0.ID 1175

R 5.4323925 —0.0548542 —0.0297567 0.0018530 0.0€D4233 0.tXD 1513

 

x A A 2 A 3 A 4

 

JUPITER

Year 2019

Epoch = JDE 2458 483.5

L 246.6730789 28.6564988 0.5720391 0.0728729 D.fD58592 —0. €066096

B 0.7287711 —0.5401173 —0.1020927 0.0181279 0.€018647 0.0€D4885

R 5.3502092 —0.1063772 —0.0206549 0.1X141448 0.tXD7904 —0.0301698

JUPITER

Year 2020

Epoch = JDE 2458 848.5

L 275.9737393 30.0100340 0.7588199 0.0547962 —0.0346193 0.0058199

B 0.1070422 —0.6800467 —0.0317680 0.0287437 0.€041069 —0.0€DgO34

R 5.2279426 —0.1329443 —0.£i046128 0.€D54240 0.0€D7122 —0.0€D 1295

JUPITER

Year 2021

Epoch = JDE 2459 214.5

L 306.8551230 31.5865271 0.7714122 —0.0472867 —0.0137630 —0.0043244

B —0.5744886 —0.6445307 0.0717685 0.0374925 0.0£D 6703 —0.SDI 1078

R 5.0960533 —0.1236156 0.0141271 0.D071851 —0.0009036 0.0€D1141

JUPITER

Year 2022

Epoch = JDE 2459579.5

L 339.1476882 32.9113174 0.5060695 —0.1187728 —0.0354495 0.1XJ95128

B —1.1101959 —0.3913655 0.1767019 0.0296471 —0.0D6 4692 —0.D0D4809

R 4.9929604 —0.0768694 0.0318479 0.£D36877 —o.ooicg4 —o.0€D2646

JUPITER

Year 2023

Epoch = JDE 2459944.5

L 12.4203656 33.4726048 0.0288015 —0.1829167 0.0(19 9944 —0.001 €D78

B —1.3021626 0.0227011 0.2221304 —0.€D12170 —0.0087196 0.0€O4485

R 4.9511336 —0.£D43210 0.0384200 0.OD15527 —0.0026066 0.0€D5278

JUPITER

Year 2024

Epoch = JDE 2460 309.5

L 45.7478419 33.0167364 —0.4649779 —0.1353636 0.0149485 0.0036406

B —1.0668191 0.4306968 0.1709431 —0.0302545 —0.0055970 0.0014732

R 4.9847066 0.0693641 0.0330463 —0.€D50572 0.OtD0980 —0.I3€i02446

JUPITER Year 2025 Epoch = JDE 2460 675.5

L 78.2698292 31.7543549 —0.7507255 —0.051114ß 0.0303432 —0.f 148466

B —0.4977303 0.6671174 0.0605776 —0.0387153 0.0010356 0.1XD52I7

R 5.0822406 0.1195829 0.0156830 —0.0049544 —0.0016537 0.0006311

 

’O A A 2

  A 4 A

SATURN

Year 1998

Epoch — JDE 2450813.5

L 19.7448573 12.6969726 0.1368882 0.0062460 —0.€D99952 0.£D34523

B —2.4797134 —0.0370230 0.0601488 0.0020497 —0.0£D7432 0.0001794

R 9.3669872 —0.1015298 0.0037320 0.IXD7303 —0.IXJ02364 0.tXD 1522

SATURN Year 1999 Epoch = IDE 2451178.5

L 32.5784232 12.9666139 0.1294965 —0.0105762 O.€D76613 —0.€03 6652

B —2.4551018 0.0873406 0.0635537 0.D003573 —0.fXD 1129 —0.0€D 1191

R 9.2698355 —0.0920465 0.0052931 0.€D19792 —0.001 3122 0.(XD4223

SATURN Year 2tXD Epoch — JDE 2451 543.5

L 45.6679535 13.2064010 0.1101519 —0.0€I47180 —0.€D30779 0.€Dl2521

B —2.3040822 0.2144789 0.0629352 —0.0(D8924 —0.0(D5166 0.0000635

R 9.1841714 —0.0786804 0.€D80428 —0.0€D4246 0.€DI 1535 —0.0€D4310

SATURN Year 2€Dl Epoch = JDE 2451909.5

L 59.0146928 13.4069581 0.0860321 —0.€066438 —0.0£D9758 —0.0€D5570

B —2.0270929 0.3362430 0.0575093  —0.€D21434 —0.0007976 0.0€D1291

R 9.1136636 —0.0613166 0.0091754 0.0023443 —0.0D18€O9 0.0€O6689

SATURN Year 2€D2 Epoch = JDE 2452274.5

L 72.4995064 13.5523675 0.0590572 —0.0190902 0.007 2511 —0.0026972

B —1.6361526 0.4422784 0.0477640 —0.0045641 0,0€D 1316 —0.000 1680

R 9.0627346 —0.0398191 0.0118501 0.0£D0014 0.0£D 8616 —0.0€D4634

SATURN Year 2€D3 Epoch = JDE 2452639.5

L 86.0963948 13.6289016 0.0157981 —0.0062851 —0.0070380 0.£D25844

B —1.1507107 0.5238144 0.0331432 —0.0Ci48639 —0.0CO8078 0.(XD2564

R 9.0351652 —0.0149521 0.0129€D3 0.0€O6775 —0.(XD7201 0.I3€O2840

SATURN

Year 2£D4

Epoch = JDE 2453004.5

L 99.7303559 13.6261946 —0.0183448 —0.0223133 0.0115296 —0.€D45045

B —0.5991684 0.5735498 0.0161432 —0.(D61982 0.0£D2143 —0.HD 1050

R 9.0333550 0.0114227 0.0127670 0.0€D6€DJ —0.0£D4786 0.1XO0398

 

  A z4z A A 4 A

S ATURN

Year 2£i05

Epoch = JDE 2453 370.5

L 113.3600304 13.5460315 —0.0593246 —0.€088776 —0.0040233 0.fO22l06

B —0.0139545 0.5875677 —0.€D20420 —0.0060833 0.0001101 0.1XD0440

R 9.0578081 0.0371069 0.0127598 —0.0013820 0.OOt05l2 -0.OOO42l7

SATURN Year 2€D6 Epoch — JDE 2453735.5

L 126.8360470 13.3955444 —0.0911468 —0.0122376 0.0 J49237 —0.€D20233

B 0.5656419 0.5658951 —0.0193918 —0.€D50723 —0.0€D0684 0.0£D1223

R 9.1069223 0.0606146 0.0103141 0.0€D7824 —0.€D14604 0.CfD5055

SA TURN

Year 2€D7

Epoch = JDE 24541D0. 5

L 140.1311075 13.1861124 —0.1140389 —0.0128927 0.0055090 —0.€D1 2513

B 1.1071268 0.5122260 —0.0337146 —0.0045624 0.0fO6098 —0.tXD 1060

R 9.1776786 0.0802443 0.€D90515 —0.0020322 0.001 6445 —0.tXD7113

SATURN Year 2£D8 Epoch = JDE 2454 465.5

L 153.1945460 12.9351827 —0.1357811 0.D€D9022 —0.%48274 0.€D20824

B 1.5815797 0.4330239 —0.0447726 —0.€D28207 0.0€D0966 0.0€D10I6

R 9.2658754 0.0953134 0.€D60668 —0.0fD2510 —0.0tO7978 0.1XD3521

SATURN Year 2€D9 Epoch = JDE 2454831.5

L 166.0267815 12.6564216 —0.1389283 —0.0113369 0.0111521 —0.0(J3 847g

B 1.9681285 0.3356251 —0.0517770 —0.0015747 0.0€D1886 0.0000333

R 9.3668471 0.1052642 0.1XI34728 —0.0€D9956 0.(ID 5031 —0.tXO2889

SATURN Year 2010 Epoch = JDE 2455196.5

L 178.5402422 12.3700943 —0.1442983 0.€D62540 —0.0€i70508 0.%34336

B 2.2506239 0.2282659 —0.0549763 —0.0€D6504 0.0€D 3647 —0.0€JO0512

R 9.4748026 0.1097941 0.€013751 —0.0018436 0.€D1O€D8 —0.(003215

SATURN

Year 2011

Epoch = JDE 2455561.5

L 190.7686750 12.0889793 —0.1358846 —0.tX111339 0.€D56670 —0.0022230

B 2.4235765 0.1175682 —0.0552441 0.0£D5608 —0.DOO0541 0.0fO0983

R 9.5848075 0.1094277 —0.€D18981 0.0(I02135 —0.00D9596 0.0fD3480

 

  A A 2 A A 4 A

SATURN

Year 2012

Epoch = JDE 2455 926.5

L 202.7240798 11.8254081 —0.1238258 0. 025325 0.OCO2032 0.0007134

B 2.4865057 0.€D90374 —0.0530558 0.€Dl2504 —0.0CD0236 0.0£D0373

R 9.6919390 0.1041370 —0.DD3 2098  —0.€D23756 0.0019690 —0.0€D7545

SATURN

Year 2ß13 Epoch = VDE 2456292.5

L 214.4608626 11.5891127 —0.1119922 0.0108332 —0.0055£D4 0.€020909

B 2.4434955 —0.0935032 —0.0489102 0.€D13073 0.0303353 —0.0€D0939

R 9.7919646 0.0946956 —0.€D58810 0.0000588 —0.1XD9202 0.0004417

SATURN

Year 2014

Epoch = JDE 2456 657.5

L 225.9454068 11.3858943 —0.088 9779  —0.€01 1738 0.D074964 —0.0025684

B 2.3026308 —0.1865194 —0.0440229 0.0023516 —0.0€O5050 0.0002148

R 9.8803594 0.0816029 —0.€07 2784  —0.SDI 1157 0.0£JD9451 —0.0€O4068

SATURN

Year 2015

Epoch = JDE 2457022.5

L 237.2460773 11.2217378 —0.0741577 0.0144744 —0.(XI97920 0.0041482

B 2.0741498 —0.2684658 —0.0379173 0.D018377 0.IXD3680 —0.0£D 1549

R 9.9541066 0.0654485 —0.0082795 —0.€Dl 1954 0.tD06345 —0.0€i0 1263

SATURN

Year 2016

Epoch = JDE 2457 387. 5

L 248.4024880 11.0981886 —0.0498654 0.tX136715 0.0045755 —0.0019976

B 1.7698174 —0.3380817 —0.0315546 0.€D22754 —O.0€D1424 0.0€D0873

R 10.0105884 0.0472140 —0.0099693 0.IXD5838 —0.IXI08777 0.0€iD3398

SATURN

Year 2017

Epoch — JDE 2457 753.5

L 259.4872465 11.0177308 —0.0281315 0.iX182957 —0.0£129458 0.€D1 6845

B 1.4013204 —0.3946323 —0.0249619 0.0026334 —0.O(D2708 0.0€D0867

R 10.0479536 0.0271211 —0.0096701 —0.0018104 0.IXtI7667 —0.0Lf06334

S ATURN Year 2018 Epoch = JDE 2458118.5

L 270.4838802 10.9829864 —0.IXf84II3 0.0134246 —0.0058676 0.0fll9517

B 0.9841755 —0.4373125 —0.0176187 0.0019J8S 0.0004897 —0.0€O 1736

R 10.0647254 0.tO62681 —0.0108303 0.DA)8458 —0.0013031 0.0€D5968

 

  A

  A A 4 A

SATURN

Year 2019

Epoch = JDE 2458483.5

L 281.4679638 10.9926030 0.0191397 0.CDI9659 0.1X156652 —0.0319957

B 0.5315188 —0.4655717 —0.0106033 0.€D29422 —0.iXD5030 0.0€D2128

R 10.0603027 —0.0151233 —0.0105719 —0.0€D6684 0.0€D8559 —0.0€D3573

SATURN

Year 2020

Epoch = JDE 2458 848.5

L 292.4853419 11.0496805 0.0373569 0.0174504 —0.0108844 0.0042565

B 0.0579959 —0.4789099 —0.0027643 0.0024092 0.1XD3160 —0.0€D126D

R 10.0344379 —0.0366309 —0.0103356 —0.1XD3887 0.0€OI631 0.0€D0851

SATURN Year 2021 Epoch = JDE 2459214.5

L 303.6137622 11.1546585 0.0657521 0.0347931 0.0£J46632 —0.€D23548

B —0.4223848 —0.4765430 0.C053044 0.€D27169 0.0CDO218 0.I3€D0308

R 9.9871735 —0.0574429 —0.0104089 0.€D13747 —0.€D10359 0.IXD4033

SATURN Year 2022 Epoch = JDE 2459579.5

L 314.8412743 11.3076017 0.0873280 0.€D99681 —0.0049667 0.€D21224

B —0.8908538 —0.4575420 0.0137280 0.0031794 —0.0€D2048 0.0€O087O

R 9.9200638 —0.0763029 —0.1XI80128 —0.0fD8734 0.0014653 —0.13€D5126

SATURN Year 2023 Epoch = JDE 2459944.5

L 326.2433278 11.5029149 0.1044171 0.0111777 —0.0044895 0.€D10461

B —1.3316062 —0.4209363 0.0229542 0.€D29128 0.0€D2134 —0.0€D0749

R 9.8358273 —0.0916158 —0.€D70946 0.0€I2OXD —0.0€117546 0.0€D7482

SATURN

Year 2024

Epoch = JDE 2460309.5

L 337.8583941 11.7324983 0.1249764 —0.0(D3094 0.0052573 —0.€021326

B —1.7265370 —0.3658077 0.0322689 0.0€132320 —0.1XD1439 0.0€J€i0585

R 9.7381105 —0.1031246 —0.€D45061 0.0fD1035 0.IXD8145 —0.0€D3540

SATURN Year 2025 Epoch = JDE 246D675.5

L 349.7515396 11.9928729 0.1320617 0.0131542 —0.0107448 0.0337657

B —2.0577285 —0.2916285 0.0415890 0.€D32586 —0.1XD2831 0.0€O0751

R 9.6307415 —0.1103354 —0.IJ022551 0.0£D8725 —0.IXD4414 0.0€D2705

 

A A A 2 A A 4 A

URANUS

Year 1998

Epoch — JDE 2450813.5

L 308.3946411 4.0234022 —0.009%74 0.0013037 —0.€D21907 0.CD10334

B —0.6281267 —0.0316524 0.0015991 0.0€D1139 —0.tXD0869 0.(XD0281

R 19.8439293 0.0443881 —0.£D17243 —0.£D13796 0.0013722 —0.0004733

URANUS

Year 1999

Epoch = JDE 2451178.5

L 312.4082792 4.€D39196 —0.0113243 0.£039059 —0.O336D98 0.€D11560

B —0.6581250 —0.0283220 0.£017436 —0.0€D1090 0.0001411 —0.0D00552

R 19.8861125 0.0399427 —0.€D24498 0.0£111943 —0.£D16195 0.0€D7214

URANUS

Year 20€D

Epoch = JDE 2451543.5

L 316.4023267 3.9842795 —0.0f181955 —0.€D29436 0.€D35933 —0.€D1 4074

B —0.6847265 —0.0248706 0.€D17191 0.0£D1€D5 —0.0€D1187 0.0€D0503

R 19.9239016 0.0357276 —0.fD20273 —0.0€D0038 0.0€D 1878 —0.0€D 1152

URANUS

Year 2tXil

Epoch = JDE 2451909. 5

L 320.38852€D 3.9664687 —0.€D97794 0.%49444 —0.0054151 0.€D21564

B —0.7079044 —0.0213463 0.€017552 0.(XD0082 0.0€D0I80 —0.0€O0143

R 19.9577580 0.0318173 —0.€D13643 —0.0007121 0.(Xi04838 —0.0€O0g77

URANUS Year 2€O2 Epoch =  JDE 2452 274.5

L 324.3468951 3.9508084 —0.€D74756 0.0006290 0.0€O72l8 —0.0€D5g44

B —0.7274836 —0.0178115 0.0018137  —0.0€D0903 0.0€O0868 —0.0€D0288

R 19.9878950 0.0284669 —0.0f121314 0.0014865 —0.0017320 0.0006781

URANUS Year 2003 Epoch = JDE 2452639.5

L 328.2909943 3.9377576 —0.€D51693 0.0€D0154 0.0fD0545 0.0€D0916

B —0.7435137 —0.0142493 0.0017331 O.0CD1269 —0.0CO1435 0.IXD0554

R 20.0146632 0.0250860 —0.€D12394 —0.€D1O780 0.0011495 —0.IXD4444

URANUS Year 2€D4 Epoch = JDE 2453004.5

L 332.2237441 3.9282198 —0.0059965 0.0053021 —0.0048520 0.€D16624

B —0.7559908 —0.0107012 0.0017997 —0.0€D0888 0.0€O 1159 —0.0CD0498

R 20.0381369 0.0217743 —0.0£J17986 0.0£D7608 —0.0012460 0.tXD5746

 

Aq A A 2 A 3 A 4 A

URANUS

Year 2£D5

Epoch = JDE 2453 370.5

L 336.1588225 3.9209574 —0.EDI 8016 —0.0026331 0.0039271 —0.€01 6943

B —0.7649346 —0.£D71424 0.EDI 7870 —0.0£D0051 —0.0fD0l73 0.0€D0138

R 20.0582522 0.0183255 —0.€D20410 0.0€O5349 —0.0€D540l 0.0€O 1568

URANUS Year 2€D6

Epoch = JDE 2453735.5

L 340.0775781 3.9168193 —0.0023459 0.0037264 —0.0€141381 0.€DI6941

B —0.7702986 —0.€D35834 0.fDl7364 0.0£D0929 —0.0000886 0.0000294

R 20.0746882 0.0144546 —0.€D1 4695  —0.tXl10956 0.0009112 —O.0€O2909

URANUS Year 2€O7

Epoch = JDE 2454100. 5

L 343.9933339 3.9152077 —0.00130£O 0.€D21909 —0.0€110142 0.0000326

B —0.7721119 —0.0€D0415 0.£018199  —0.OSD 1263 0.0€D 1434 —0.0€D0556

R 20.0871980 0.0104493 —0.£D26204 0.CD15434 —0.0019845 0.0€O8009

URANUS Year 21Xl8

Epoch = JDE 2454465.5

L 347.9084509 3.9152947 0.0016034 —0.€D20211 0.fD24l12 —0.0£D9125

B —0.7703719 0.€D35180 0.£D17552 0.0€D0880 —0.1XD1148 0.0000494

R 20.0953867 0.€D58701 —0.€D21983  —0.0€D 6627 0.1XD7453 —0.0€O 3326

URANUS Ymr 2009

Epoch = JDE 2454831.5

L 351.8355595 3.91764O€i —0.1Xi04279 0.€D51042 —0.0€152751 0.0019119

B —0.7650567 0.€D70885 0.£017676 0.0€DO365 0.iXD0153 —0.0000132

R 20.0988108 0.0€D8070 —0.€025051 0.0€D0724 —0.IXD5166 O.0€D2948

URANUS Year 2010

Epoch — JDE 2455196.5

L 355.7545126 3.9204849 0.€D21446 —0.€D22638 0.003 4613 —0.001 6180

B —0.7561921 0.0106378 0.€DI8I61 —0.0€D0961 0.0€DO892 —0.0€D0301

R 20.0969632 —0.tD45723 —0.£D32576 0.OD12002 —0.0012976 0.0€D4622

URANUS Year 2011

Epoch = JDE 2455 561.5

L 359.6767218 3.9238438 0.£DI4779 0.tX114628 —0.D020231 0.0€D9043

B —0.7437752 0.0141%2 0.£D17203 0.IXD120D —0.0001449 O.0€O0558

R 20.0894980 —0.0103940 —0.€D23728 —0.0€I11701 0.€D1 2464 —0.0€O4592

 

Ag A A 2 A x4 As

URAN US

Year 2012

Epoch = JDE 2455926.5

L 3.6023875 3.9276448 0.tXD3642 0.tXl32l50 —0.0024109 0.0£D6078

B —0.7278338 0.0176871 0.€D17578  —0.1XD0982 0.D£D 1124 —0.0£D0490

R 20.0763484 —0.0159219 —0.fD30759 0.€D15680 —0.0019150 0.0008013

URANUS

Year 2013

Epoch — JDE 2456292.5

L 7.5425796 3.9313963 0.€D34892  —0.D03402O 0.004 3364 —0.(Xi1 736D

B —0.7083658 0.0211235 0.fD17049 —0.0fD0243 —0.0000146 0.(XO0I25

R 20.0577472 —0.02106€D —0.£D25105 0.0€D2189 O.1XD0573 —0.(XiO0987

URANUS

Year 2014

Epoch = JDE 2456657.5

L 11.4766635 3.9369705 0.€D16703 0.€D45735 —0.004 6517 0.€D1 7806

B —0.6855638 0.0244651 0.0016CD3 0.1XD0765 —0.0000918 0.0 €i0307

R 20.0343542 —0.0256833 —0.€D17734  —0.0€D4195 0.IXt028S1 —0.0€€I0423

URANUS

Year 2015

Epoch = JDE 2457022.5

L 15.417€D66 3.9442719 0.€D41430 —0.1XD6173 0.0023110 —0.CD1 2324

B —0.6594830 0.0276789 0.€D16264  —0.1XD1459 0.OOO 1436 —0.0€D0559

R 20.0067208 —0.0295335 —0.€D24078 0.tXil7729 —0.(Xi19111 0.0CD7160

URANUS Year 2016 Epoch = JDE 2457387.5

L 19.3658827 3.953g550 0.%59123   —0.0(D2440 0.000 5465 —0.OSD 1220

B —0.6302359 0.0307918 0.€D15029 0.iXD0655 —0.0001126 0.0€O0487

R 19.9753573 —0.0331294 —0.tXil2957 —0.iXD938l 0.D011617 —0.0tD4855

URANUS Year 2017 Epoch = JDE 2457 753.5

L 23.3366976 3.9666314 0.fD5I813 0.0(l43644 —0.0034992 0.€D10928

B —0.5978470 0.0337938 0.0014529 —0.iXD01(O 0.0€D0l2D —0.0€D0117

R 19.9405709 —0.0362844 —0.€D17189 0.€D10807 —0.0014608 0.0€O6325

URANUS Year 2018 Epoch = JDE 2458 118.5

L 27.3104684 3.9814942 0.€D94084 —0.€D35588 0 0051513 —0.0021644

B —0.56261€D 0.0366587 0.€014472 —0.0€D1155 0.0€D0929 —0.0000313

R 19.90282€D —0.0391742 —0.0£117216 0.0£O6973 —0.0fD5305 O.(XJO1099

 

  A z4 z

 

 

 

URANUS

Year 2019

Epoch = JDE 2458 483.5

L 31.3€D7991 3.9995492 0.0087290 0.0€132347 —0 0032874 0.EDI 3338

B —0.5245579 0.0394244 0.£D13056 0.0€D1O70 -0.0@ tWO 0WO03O

R 19.8622039 —0.0420977 —0.0€i09882 —0.iXD8669 0.@08068 -0.0002803

URAN US Year 2020 Epoch = IDE 2458 848.5

L 35.3103585 4.0202030 0.0101963 0.(fD8671 0.0tXl6986 —0.IXi06386

B —0.4838089 0.0420579 0.€D13107 —0.0€D1041 0.OSD 1112 —0.0(D0485

R 19.8187746 —0.0448163 —0.0019781 0.€D17314 —0.0020253 0.0007742

URANUS Year 2021 Epoch = JDE 2459214. s

L 39.3527615 4.0428802 0.0129105 —0.%25246 0.0030871 —0.001 1659

B —0.4403595 0.0445777 0.€D12393 —0.0€O0335 —0.0£D0llO 0.0€O0109

R 19.7723296 —0.0478387 —0.£D14164 —0.0tD5968 0.0008328 —O.0CD4044

URANUS Year 2022 Epoch = JDE 2459579.5

L 43.4079488 4.0677519 0.0109318 0.0341260 —0.1X139755 0.%13748

B —0.3945761 0.0469666 0.€D11124 0.0£D0701 —0.1XD0958 0aXD0318

R 19.7229060 —0.0511244 —0.€D16867 0.0003213 —0.0006636 0.tXD3205

URANUS

Year 2023

Epoch = JDE 2459944.5

L 47.4881579 4.0928916 0.0134868 —0.€D35210 0.0048241 —0.€D21126

B —0.3464911 0.0491749 0.Li011049 —0.IXD1601 0.0fD 1422 —0.0€D0562

R 19.6700732 —0.0545761 —0.0023104 O.€D11868 —0.0010745 0.1XO3239

URANUS

Year 2024

Epoch = JDE 2460 309.5

L 51.5937268 4.1181401 0.0122956 0.0€O8849 —0.0(l13419 0.0006425

B —0.2962854 0.0511951 0.0€D9292 0.0fD042O —0.0€D1l18 0.DCD0481

R 19.6136229 —0.0583266 —0.0013417 —0.0011207 0.0€113155 —0.0€O5128

URANUS

Year 2025

Epoch = JDE 2460 675.5

L 55.7356990 4.1433080 0.0111440 0.0018686 —0.1XD9665 0.(XD0679

B —0.2440376 0.0529757 0.OtD8056 —0.1XiO0329 O.OOOOO75 -0.OOO0102

R 19.5534677 —0.0616438 —0.€D20415 0.€D16863 —0.0018582 0.OOO7321

 

  A A A 3 A 4 A 5

NEPTUNE

Year 1998

Epoch = JDE 2450813. 5

L 299.54D7539 2.1902131 0.0€D2I59 —0.1XD0I01 —O.1XD7024 0.0£D4022

B 0.3746913 —0.0657516 —0.0€D3038 0.IXD1425 —0.HD 1276 0.0€D0450

R 30.1452426 —0.0114799 0.0€D0256 —0.IXi16443 0.1X117726 —0.0€D6393

N EPTUN E

Year 1999

Epoch = JDE 2451178.5

L 301.7308726 2.1898447 —0.€D14772 0.£D25844 —0.€D27128 0.0D09371

B 0.3086958 —0.06622€D —0.0€D1651 —0.tXD1133 0.IXD1518 —0.0€D0615

R 30.1332773 —0.0124411 —0.0€D5349 0.0010464 —0.0014170 0.0tO6547

NEPTUNE Year 2tXD

Epoch = JDE 2451543.J

L 303.9200489 2.1884417 0.0€D0758 —0.€D19024 O.€022339 —0.0fO8929

B 0.2422877 —0.0665872 —0.0€D1798 0.1XD0780 —0.0€D0946 0.0€D0426

R 30.1205855 —0.0128018 0.0€D€D27 —0.1 D0216 O.0€D2692 —0.0'CD 1503

NEPTUNE

Year 2€D1 Epoch = JDE 2451 909.5

L 306.1139979 2.1874304 —0.fD1£D30 0.0025367 —0.€D29947 0.€D12256

B 0.1753634 —0.0668803 —0.0€D1554 0.0CD0482 —0.0€D0188 0.iXD0€D2

R 30.1078496 —0.0125395 0.tXD8452 —0.0€D9250 0.IXD7678 —0.iXD 1989

NEPTUNE Year 2€D2 Epoch = JDE 2452274.5

L 308.3011927 2.18716€D —0.0fDl927 0.0£D6350 0.0CD€D66 —0.0(D 1904

B 0.1083572 —0.0671220 —0.0tD0564 —0.1XD0988 0.0€D1133 —0.1XD0401

R 30.0957992 —0.0115299 0.0€D0961 0.0015161 —0.0018195 0.0€D7336

NEPTUNE Year 2003 Epoch = JDE 2452639.5

L 310.4886113 2.1877683 0.€D12570 —0.0fD8609 0.0£D8138 —0.iXD2414

B 0.0411532 —0.0672759 —0.0€D10I9 0.O£D1349 —0.0€D1418 0.0€D0566

R 30.0847956 —0.0104405 0.0€O9560 —0.(D10546 0.00116D7 —0.O3D4588

NEPTUNE Year 2€D4 Epoch = JDE 2453tO4.5

L 312.6773480 2.1898129 —0.0000225 0.0035948 —0.&37652 0.€D13826

B —0.0261749 —0.0673619 —0.0fD€D62 —0.0fD0484 0.0tD0897 —0.1XDO402

R 30.0749585 —0.£D93289 0.IXD6114 0.0€D26l5 —0.0€D7761 O.0CD4204

 

COEFFlCIENTS FOR HELIOCENTRIC COORDINATES 469

A

  A 2

  A4 A

NEPTUNE

Year 2€D5

Epoch = JDE 2453370.5

L 314.8743572 2.1923691 0.€D19779 —0.£D14836 0.0021474 —0.0009828

B —0.0937264 —0.0673602 0.tXO0461 —0.0€D0126 0.IXD0113 0.0£D£D24

R 30.0661238 —0.£D83386 0.1XD0697 0.0€D7833 —0.0€D9169 0.0fD3301

NEPTUNE

Year 2€O6

Epoch = JDE 2455735.5

L 317.0683853 2.1956170 0.0016596 0.£D10143 —0.0015197 0.0€D6827

B —0.1610394 —0.0672478 0.0€D0391 0.0€D1174 —0.0001019 O.0€O0363

R 30.0580514 —0.€D78999 0.0€i07450 —0.€D14202 0.0D12833 —0.0CO4389

NEPTUNE

Year 20fI7

Epoch = JDE 24541€D.5

L 319.2658391 2.1993302 0.0€O8806 0.OD22165 —0.€D20477 O.0€O6061

B —0.2281964 —0.0670460 0.0€D1690 —0.0€D0905 0.0001259 —O.0£O0496

R 30.0503208 —0.fD77131 —0.0€D3349 0.0010779 —0.0015939 0.0fO6915

NEPTUNE

Year 2€D8

Epoch = JDE 2454465.5

L 321.4668247 2.2025604 0.€D23563 —0.0D20D28 0,€D21281 —0.0CD8550

B —0.2950876 —0.0667215 0.0€D1770 0.0€D0796 —0,0€D0857 0.0€DO387

R 30.0424482 —0.€D81002 —0.0001917 —0.0tD5047 0,0€D5724 —0.0002591

NEPTUNE

Year 2iXl9

Epoch = JDE 2454831 . 5

L 323.6770543 2.2055920 0.0€i03305 0.0027771 —0.0034717 0.0£J13507

B —0.3617811 —0.0662780 0.0€D2330 0.0€D0464 —0.0€O0133 —0.0CO0003

R 30.0339404 —0.009£D77 —0.0€D1287 —0.0fD6463 0.0002930 0.0£D0011

NEPTUNE Year 2010 Epoch = JDE 2455196.5

L 325.8836328 2.2074198 0.0€D5140 —0.0fD4733 O.0£D8905 —0.IXJ05519

B —0.4277933 —0.0657282 0.0€D3431 —0.0€D0674 0.OfD0842 —0.1XO0292

R 30.0244517 —0.010£D91 —0.€D10648 0.£D15176 —0.0017270 0.fXD6758

NEPTUNE Year 2011 Epoch = JDE 2455561.5

L 328.0914319 2.2078669 0.0£D4037 —0.0fD7458 0.0LO3201 —0.1XD0318

B —0.4931907 —0.0650512 0.1XD3298 0.0€D1162 —0. tD1192 0.0(D0476

R 30.0138443 —0.0111563 0.iXD0344 —0.tXJl2071 O.&14028 —0.0(D5328

 

Aq A A 2

 

  A

NEPTUNE

Year 2012

Epoch = JDE 2455926.5

L 330.2992450 2.2076168 —0.€D16I78 0.0€i32781 —0.0033868 0.€D11936

B —0.5578675 —0.0642839 0.0€D4196 —0.OXI0338 0.0fD0656 —0.D€D0294

R 30.€D23853 —0.0117358 —0.IXD2383 0.0€D9535 —0.0013463 0.1XO6305

NEPTUNE Year 2013 Epoch = JDE 2456292.5

L 332.5123744 2.2065963 0.0€D1903 —0.0018869 0.€D25870 —0.€D11286

B —0.6219032 —0.0634276 0.0£D4650 —0.0€D0083 0.0€D%41 0.0€O0041

R 29.9906169 —0.0116051 —0.0£i006l3 0.0(D7836 —0.0(D6659 0.0£O2076

NEPTUNE

Year 2014

Epoch = JDE 2456 6J7.5

L 334.7187325 2.2061042 —0.0€D4344 0.€D17465 —0.€020850 0.0(JO8747

B —0.6848660 —0.0624850 0.0€D4570 0.0€D0952 —0.0€D0878 0.0€D0322

R 29.9792758 —0.0110295 0.€010366 —0.0011668 0.£D11044 —0.0€O3637

NEPTUNE Year 2015 Epoch = JDE 2457022.5

L 336.9249386 2.2065131 —0.0€D3495 0.iXI16632 —0.0310552 0.0€D1916

B —0.7468544 —0.0614776 0.0€D5549 —0.0000662 0.0€D0906 —0.0€D0355

R 29.9688569 —0.fD98315 0.0€D1907 0.1XI15577 —0.LD19486 0.0€D7998

NEPTUNE Year 2016 Epoch = JDE 2457 387.5

L 339.1319019 2.2075407 0.€D15261 —0.1X115291 0.€D18420 —0.0€D7186

B —0.8077883 —0.0603801 0.0€D5615 0.1XD0596 —0.1XD0680 0.0300312

R 29.9596250 —0.€D86143 0.tXO8367 —O.0€D6548 O.1Xifi8027 —0.0€D3576

NEPTUNE Year 2017 Epoch = JDE 2457753.5

L 341.3466171 2.2098703 0.0€D1906 0.0033414 —0.0035370 0.€D13188

B —0.8677463 —0.0591918 0.0€D5982 O.(XD0475 —0.IXO0266 0.0€D€D68

R 29.9516174 —0.€D74637 0.0€D7698 —0.tXD1344 —0.0CO2736 0.0€D2032

NEPTUNE

Year 2018

Epoch — JDE 2458 118.5

L 343.5578014 2.2126870 0.£017473 —0.0€D9114 0.€D17482 —0.0€D8936

B —0.9263123 —0.0579263 0.0€D6869 —0.OfD0502 0.1XD0639 —0.0000221

R 29.9447188 —0.0063999 —0.0€D0426 0.€D12655 —0.1XJ14809 0.IXO5456

 

COEFFICIENTS FOR HELIOCENTRIC COORDINATES 471

x A

’2

  ’4 A$

NEPTUNE

Year 2019

Epoch = JDE 2458483.5

L 345.7721788 2.2160304 0.€D19492 0.0tD1302 —0.0€O4086 0.OCD225O

B —0.9835601 —0.0565564 0.0£D6838 0.0£O0946 —0.0fO0980 0.0£D0402

R 29.9386064 —0.£D59103 0.0006087 —0.0011338 0.0010820 —0.0€D4058

NEPTUNE

Year 2020

Epoch = JDE 2458848.5

L 347.9901050 2.2198549 0.1XD6474 0.0£J27839 —0.(D26574 0.IXJO8337

B —1.0393959 —0.0550979 0.0€D7576 —0.0£D€D38 0.0fO0301 —0.0000148

R 29.9328472 —0.f057668 —0.0003583 0.0011605 —0.0017321 0.IXO7507

NEPTUNE

Year 2021

Epoch = JDE 2459214.5

L 350.2176582 2.2230216 0.0022203 —0.0023881 0.0028083 —O.D011985

B —1.0938714 —0.0535429 0.0€D8173 —0.1XD0087 0.0tD0111 —0.0000€O1

R 29.9268841 —0.€D62024 —0.0€D5211 0.1XD2793 —0.0CD2570 0.1XD0426

NEPTUNE Year 2022 Epoch = JDE 2459579.5

L 352.4421218 2.2256340 0.0€D2719 0.0021329 —0.0t128021 0.€D11l15

B —1.1465947 —0.0518896 0.0€D8209 0.iXD0933 —0.0CDO896 0.0£D0341

R 29.9202255 —0.£D72339 —0.0€D1205 —0.€008035 0.0CD5720 —0.0001362

NEPTUNE Year 2023 Epoch — JDE 2459944.5

L 354.6684701 2.2269149 —0.0002141 0.1XD3069 0.0CD1917 —0.0003173

B —1.1976256 —0.0501577 0.0€D9063 —0.1XD0366 0.tXDO579 —0.0000238

R 29.9125033 —0.€D826€D —0.0010453 0.1X116981 —0.0019454 0.0007683

NEPTUNE Year 2024 Epoch = JDE 2460 309.5

L 356.8953523 2.2266074 0.0£D3203 —0.€D17139 0.€D16708 —0.0€D6067

B —1.2468797 —0.0483417 0.0€D9265 0.1XD0376 —0.OD0455 0.0£D0218

R 29.9037191 —0.0092326 —0.0€D0327 —0.0€D7969 0.0010798 —0.0004546

NEPTUNE

Year 2025

Epoch = JDE 2460675.5

L 359.1277282 2.2258347 —0.0018769 0.0034605 —0.1XI36112 0.0€i13105

B —1.2944081 —0.0464433 0.iXO9423 0.1XD06S7 —O.I D056D 0.0fD0191

R 29.8942558 —0.€D96242 0.0€D0726 0.IXD3999 —0.00D6964 0.0€D3620

 

 

 

In dex

The numbers refer to the pages

 

Abbreviations, S

Aberration, 410 ; constant of, 151 ; of planets, 224 ; Ron-Vondrâk expres- sion, 153 ; of stars, 151

Absolute magnitude, of minor planets, 231 ; of stars, 396

Accuracy, needed for a problem, 15 ; of a computer, 16

Altitude, 93

Angles, large, 7 ; modes, 7 ; negative, 9

Angular separation, 109,115 ; least, 111 Anomalistic period of Moon, 361 Anomaly, mean, true and eccentric, 194,

412 ; mean, 210 ; true in parabolic motion, 241 ; true in near-parabolic motion, 245

Aphelion of planets, 269 Apheloid, 271

Apogee of Moon, 355 ; extreme, 362 Apparent place, of a planet, 225 ; of a

star, 149

Apsides, 411

Arcminute, 412

Arcsecond, 412

Areographic, 287, 410

Ascending node, 410

Ascension, right, 91, 93, 409

ASCII characters, 58

Asteroids, diameters, 391 ; magnitudes, 231

 

Astrometric position, 230

Astronomical Unit, 411

ATN2 function, 9

Autumnal equinox, 409

Azimuth, 91, 92, 93, 410

Barker’s equation, 241

BCD, 18

Besselian year, 133

Binary arithmetics, 19

Binary search, 52, 206

Binary   stars, 397 ;  eccentricity of apparent orbit, 400

Bodies in smallest circle, 127 ; in straight line, 121

Brightness ratio of stars, 395 Calendar, Jewish, 71 ; Moslem, 73 Calendar date from JD, 63

Carrington, synodic rotation of the Sun, 189, 191

Center, equation of, 237, 412

Central Meridian of Jupiter, 293 ; of Mars, 287 ; of Sun, 189

Century, Julian, 410

Circle, smallest of three bodies, 127 Colongitude, selenographic of Sun, 376 Comets, magnitude, 37, 231

Conjunctions, 410 ; planetary, 117 ; planets with Sun, 249 ; and least angular separation, 119

Constants, 407

 

473

 

Coordinates, 410 ; galactic, 91, 94 ; geocentric ecliptic, 223 ; geocentric equatorial, 227 ; geocentric rectan- gular of an observer, 81 ; heliocentric ecliptic, 217, 220, 233 ; transformation

of, 91

Correlation, coefficient of linear regres- sion, 38

Curve fitting, 35 ; general, 44 ; linear, 36 ; quadratic, 43

Date, of Easter, 67 ; from JD, 63 ; scientific form, 6

Day, Julian, 59 ; of the week, 65 ; of the year, 65

Declination, 91, 93, 409 ; maximum of

Moon, 367

Defect of illumination, 287, 290 Delta T (A r), 77

Diameters, see Semidiameters Direct, 228

Distance, angular, 109, 115 ; least angular, 111 ; between points on Earth’s surface, 84 ; of stars and absolute magnitude, 396

Diurnal path and horizon, 100 Double stars, see Binary stars Dynamical Time, 77

Earth, eccentricity of orbit, 163 ; globe, 81, 83 ; distance between points, 84 ; orbital elements for mean equinox of date, 212 ; for equinox 2000.0, 214 ;

perihelion and aphelion, 269, 273, 274 Easter, date of, Gregorian, 67 ; Julian,

69

Eccentricity, 194

Eclipses, 379 ; accuracy, 387 ; lunar, 382 ;

solar, 381

Ecliptic, 409 ; dynamical, 166 ; and equator, 100 ; and horizon, 99 ; obliquity of, 92, 147, 228

Ecliptical coordinates, 91, 233

Elements, osculating, 228, 233 ; of planetary orbits, mean equinox of date, 212 ; equinox 2tXD.0, 214 ; reduction to another equinox, 159

Ellipse, length of, 238

Elliptic motion, first method, 223 ; second method, 227

 

Elongation, definitions, 253 ; greatest of Mercury and Venus, 253 ; of planet, 225, 231 ; of Venus (approx.), 284

Ephemeris, 412 Ephemeris day, 41 I Epoch, 228, 410

Equation, of Barker, 241 ; of the center, 237, 412 ; of the equinoxes, 88 ; of

Kepler, 193 ; of time, 183

Equator, celestial, 409 ; mean, 410 Equinox, 409 ; correction, 139 ; mean,

149, 410 ; true, 149

Equinoxes, 177

E-terms, 139

Extremum from three values, 25 ; from five values, 29

FK4, 139

FK5, 139

Fraction illuminated, of Moon, 345 ; of planet, 283 , of Venus (approx.), 284

Galactic coordinates , 94

Gaussian gravitational constant, 228, 241, 41 l

Geocentric, 410 ; latitude, 8 l ; rectan- gular coordinates of an observer, 81

Geographical latitude, 81

Geoid, 81

Geometric position, 224, 412 Greenwich, civil time, 411 ; mean time,

77, 41 l

Haversine, 115

Heliocentric, 410

Heliographic coordinates, 189, 410 Hour angle, 92, 94 ; at rise and set, 101,

102

Illuminated fraction, of Moon, 345 ; of planet, 283 ; of Venus (approximate), 284

INT, 60, 71

Interpolation, from three values, 23 ; from five values, 28 ; remarks, 30, 31 ; to halves, 31 ; with Lagrange’s for- mula, 32 ; extremum, 25, 29 ; zero

value, 26, 29

Iteration, 47

Jewish calendar, 71

Julian century, 410

Julian Day, 59 ; modified, 63

 

Jupiter, coefficients for heliocentric coordinates, 220, 455 ; magnitude, 285, 286 ; orbital elements for mean equinox of date, 213 ; for equinox 2000.0, 215 ; perihelion and aphelion,

269 ; phenomena, 249 ; physical

ephemeris, 293, 297 ; positions of

satellites,  301, 304 ; semidiameter,

389, 390

Kepler, equation of, 193 ; first method, 196 ; second method, 199 ; third method, 206 ; fourth method, 206

Lagrange interpolation formula, 32 Latitude, celestial, 409 ; geocentric, 81

Least squares, 36

Librations of Moon, optical, 371 ; physical, 372 ; topocentric, 375

Light-time, 224

Light-year, 396, 407

Linear regression, 36

Longitude, celestial, 409 ; geographical, 93 ; orbital and ecliptical, 218

L.unation, 354

Magnitude, of a lunar eclipse, 382 ; of a solar eclipse, 382

Magnitude, absolute of stars, 396 ; adding, 393 ; of comets, 37, 231 ; of minor planets, 231 ; of planets, 285, 286

Major axis, 194

Mars, magnitude, 285, 286 ; orbital elements for mean equinox of date, 212 ; for equinox 2i3(D.0, 214 ; peri- helion and aphelion, 269 ; phenomena, 249 ; physical ephemeris, 287 ; semi- diameter, 389, 390

Mercury, magnitude, 285, 286 ; orbital elements for mean equinox of date, 212 ; for equinox 21XD.0, 214 ; peri- helion and aphelion, 269 ; phenomena, 249, 253 ; semidiameter, 389, 390

Minor planets, diameters, 391 ; magni- tudes, 231

Moon, mean anomaly, 338 ; anomalistic period, 361 ; maximum declinations, 367 ; eclipses, 382 ; mean elongation, 338 ; illuminated fraction ot disk, 345 ; optical librations, 371 ; physical

 

librations, 372 ; topocentric librations, 375 ; mean longitude, 337 ; longitudes of node and perigee, 343 ; longitude of true node, 344; extreme lunations, 354 ; passages through nodes, 363 ; correction for parallax, 279, 280 ; perigee and apogee, 355 ; extreme perigees and apogees, 362 ; phases, 3A9 ; physical ephemeris, 371 ; position, 337 ; position angle of axis, 373 ; position angle of bright limb, 346 ; position angle of cusps, 347 ; selenographic position of Sun, 376 ; of some liinar features, 378 ; semidiameter, 390 ; sunrise and sunset on Moon, 377 ; synodic period, 354

Moslem calendar, 73

Motion,  elliptic,  223,    227 ;    near-

parabolic, 245 ; parabolic, 241 Neptune, coefficients for heliocentric

coordinates,  220, 455 ; magnitude,

285, 286 ; nodes, 278 ; orbital ele- ments for mean equinox of date, 213 ; for equinox 2000.0, 215 ; variation of osculating elements, 234 ; perihelion and aphelion, 269, 271 ; phenomena,

249 ; semidiameter, 389, 390 New Moon, see Phases of Moon

Newton’s method for solving an equa- tion, 50

Nodes, passages through, 275, 276 ; longitude of lunar, 343, 344 ; of Moon, 363

Nutation, 143 ; effect of, 150 ; in right ascension, 88

Obliquity of ecliptic, 92, 147

Opposition, 249, 410

Orbital elements, see Elements Osculating elements, 228, 233

Parabolic motion, 241

Parallactic angle, 97

Parallax, 412 ; correction for, 279 ; in ecliptical coordinates, 282 ; in hori- zontal coordinates, 281 ; of stars, 150, 396

Parsec, 396, 407 “Periapse”, 30S, 41 I Periastron, 411

 

Perigee of Moon, 355, 411 ; extreme, 362 ; longitude of, 343

Perihelion, 411 ; argument of, 209 ; of planeu, 269

Periheloid, 271

Perijove, 305, 411

Pesach, 71

Phase angle, 231, 283 Phases of Moon, 349 Phenomena, planetary, 249

Place, apparent of a star, 149 Planetographic, 410

Planets, elements of orbits, 212, 214 ; elongation, 225, 231 ; geocentric posi-

tions, 223, 227 ; heliocentric positions, 217, 220 ; illuminated fraction of disk, 283 ; magnitude, 285, 286 ; passages through nodes, 275 ; perihelion and aphelion, 269 ; phase angle, 231, 283 ;

phenomena, 249 ; semidiameters, 389,

390 ; principal periodic terms, 413 Pluto, magnitude, 286 ; position, 263 ;

semidiameter, 390

Poles, celestial, 409 ; galactic, 94 Position angle, relative, 116 Powers, avoiding, 10 ; of time, 10

Precession, 131, 410 ; low accuracy, 132 ; rigorous method, 134 ; in ecliptical

coordinates, 136 ; old elements, 139

“Prograde”, 228

Program, debugging, 12 ; shortening, l l Proper motion, 150 ; in ecliptical coor-

dinates, 137

Quadrant, correct, 8

Quicksort, 55

Radius vector, 194, 411 ; in elliptical motion, 195, 217, 223, 229 ; in near- parabolic motion, 245 ; in parabolic motion, 241 ; series expansion, 237

Rectangular coordinates, geocentric of observer, 81 ; of a planet or comet, 223, 229 ; of the Sun, 171, 172, 174

Refraction, atmospheric, 105

Regression, linear, 36

Retrograde, 228

Right ascension, 91, 93, 409 Ring of Saturn, 317

Rising, 101

 

Rounding, errors, 17 ; the final results, 21 ; right ascension and declination, 22 Satellites of Jupiter, conjunctions, 314 ; phenomena, 315 ; positions, 301, 304 ;

of Saturn, positions, 323

Saturn, coefficients for heliocentric coordinates,  220, 455 ; magnitude,

285, 286 ; nodes, 278 ; orbital ele- ments for mean equinox of date, 213 ; for equinox 2000.0, 215 ; perihelion

and aphelion, 269, 271 ; phenomena, 249 ; ring, 317 ; positions of satellites, 323 ; semidiameter, 389, 390‘

Seasons, 177 ; durations, 181

Selenographic, coordinates, 371, 410 ; of some lunar features, 378 ; position of Sun, 376

Semidiameters of Sun, Moon, planets, 389

Separation, angular, 109 ; least, 111 Series, accuracy ot truncated, 220 Setting, 101

Sidereal time, 41 l ; apparent, 88 ; at Greenwich, 87

Solar coordinates, see Sun Solstices, 177, 409

Sorting, 55

Star, apparent place, 149 ; binary, see Binary stars ; distance and absolute magnitude, 396 ; magnitudes, 393, 395 ; motion in space, 140 ; parallax, 150 ; proper motion, 150

Stations of planets, 254

Stellar magnitudes, adding, 393 ; bright- ness ratio, 395

Straight line, bodies in, 121 ; on celestial sphere, 122

Sun, mean anomaly, 163 ; coordinates, low accuracy, 163 ; higher accuracy, 166 ; daily variation in longitude, 168 ; eclipses, 381 ; mean longitude, 163 ; physical ephemeris, 189 ; rectangular coordinates, 171, 172, 174 ; semidiam-

eter, 389 ; synodic rotations, 189, 191

Sundial, planar, 401

Symbols, 5, 6

Synodic month, 354

Tests, safety, 12 ; on “smaller than”, 51

 

Time, civil,  411 ; Dynamical and Venus, elongatiOIl, 284 ; illuminated Universal,  77 ; equation  of, 183 ; fraction of disk, 284 ; magnitude, 285, Greenwich, 411 ; sidereal, 87, 411 ; 286 ; orbital elements for  mean solar, 411 equinox of date, 212 ; for equinox Topocentric, 410 ; positions, 279 20€D. 0, 214 ; perihelion and aphelion,

Transformation of coordinates, 91 269 ; phenomena, 249, 253 ; semi-

Transit, time of a body, 101 diameter, 389, 390

Universal Time, 77, 411 Vernal equinox, 409

Uranus, coefficients for heliocentric VSOP, 166, 217 ; principal terms, 413 coordinates,  220, 455 ; magnitude, Week, day of, 65

285, 286 ; nodes, 278 ; orbital ele- X, Y, Z, coordinates of Sun, 171, 172, ments for mean equinox of date, 213 ; 174

for equinox 2000.0, 215 ; perihelion Year, Besselian and iulian, 133 ; day of, and aphelion, 269, 271 ; phenomena, 65 ; leap, 62 ; tropical, 133

249 ; semidiameter, 389, 390 Zero of a function, from three values, 26 ; Velocity in elliptic orbit, 238 from five values, 29

 

 

 

hlollt the £lltlior

Jean Meeus, born in 1928, studied mathematics at the University of Louvain (Leuven) in Belgium, where he received the Degree of Licentiate in 1953. From then until his retirement in 1993, he was a meteorologist at Brussels Airport. His special interest is spherical

and mathematical astronomy. He is a member of several astronomical associations and the author of many scientific papers. He is the

co-author of Canon of solar Eclipses (1966), the Canon ofLunar Eclipses (1979) and the Canon of Solar Eclipses (1983). His Astronomical Formulae for Calculators (1979, 19g2, 1985 and 1988)

has been widely acclaimed by both amateur and professional astronomers. Further works, published by Willmann-Bell, Inc., are Elements ofsolar Eclipses 1951-2200 (1989), Transits (1989), Astronomical Tables of the Sun, 3foon and Planets (1983 and 1995) and Mathematical Astronomy Morsels (1997). For his numerous contributions to astronomy the International Astronomical Union announced in 1981 the naming of asteroid 2213 Meeus in his honor.

In the field of celestial calculations, Jean Meeus has enjoyed wide acclaim and respect since long before microcomputers and pocket calculators appeared on the market. When he brought out his Astronomical Formulae for Calculators in 1979, it was practically the only book of its genre. It quickly became the “source among sources,” even for other writers in the field. Many of them have warmly acknowledged their debt (or should have), citing the unparalleled clarity of his instructions and the rigor of his methods.

And now this Belgian astronomer has outdone himself yet again! Virtually every previous handbook on celestial calculations (including his own earlier work) was forced to rely on formulae for the Sun, Moon, and planets that were developed in the last century —or at least before 1920. The past 10 years, however, have seen a stunning revolution in how the world’s major observatories produce their almanacs. The Jet Propulsion Laboratory in California and the U.S. Naval Observatory in Washington, D.C., have perfected powerful new machine methods for modeling the motions and interactions of bodies within the solar system. At the same time in Paris, the Bureau des Longitudes has been a beehive of activity aimed at describing these motions analytically, in the form of explicit equations.

Yet until now the fruits of this exciting work have remained mostly out of reach of ordinary people. The details have existed mainly on reels of magnetic tape in a form comprehensible only to the largest brains, human or electronic. But Astronomical Algorithms changes all that. With his special knack for computations of all sorts, the author has made the essentials of these modem techniques available to us all.

There are times when an amateur astronomer wants to perform the computations that support his or her observations. Astronomical Algorithms is the reference to have for this. Xeon Meeus concise volume collects most of the algorithms and computational techniques an observer might want—covering coordinate transformations, the apparent place of a star, the positions of solar system bodies, eclipse predictions, and much more. Discussions are complete enough to make the equations fully understandable to the novice, and virtually every algorithm includes a fully worked numerical exampl This is a very handy reference, well worth owning, even if you never have to perform a specific calculation. The text along is helpful for under- standing how the theories of celestial mechanics are applied in practice.

Sky & Telescope magazine

...There is no doubt that the book is very good value for money .. . computer-minded astronomers will never want to be without it.

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