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EXPLANATORY SUPPLEMENT TO THE EPHEMERIS

~XPLA

SUPPLEMENT

TO

THE )ASTRONOMICAL EPHEMERIS

----

AND

THE AMERICAN EPHEMERIS

AND NAUTICAL ALMANAC

PREPARED JOINTLY BY THE NAUTICAL ALMANAC OFFICES OF THE UNITED KINGDOM AND
THE UNITED STATES OF AMERICA
Issued by H.M. Nautical Almanac Office by Order of the
Science Research Council

CREATED FOR HIS MaJesto
LONDON
HER MAJESTY'S STATIONERY OFFICE 19 61

a
© Crown copyright 1961
Third impression (with amendments) 1974

AU

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ISBN 0 I I 880578 9*

PREFACE

The purpose of this Explanatory Supplement is to provide the users of The Astronomical Ephemeris (prior to 1960 entitled The Nautical Almanac and Astronomical Ephemeris) and The American Ephemeris and Nautical Almanac with fuller explanations of their content, derivation, and use than can conveniently be included in the publications themselves. A rigorous treatment is given of the fundamental basis of the tabulations; this is supplemented by a detailed derivation, showing how each tabulated quantity is obtained from basic data. The use of the ephemerides is also explained and illustrated, but completeness is not attempted. Auxiliary tables, lists of constants, and miscellaneous data are added, partly for convenience of use with the Ephemeris and partly for reference.
By its nature this Supplement must primarily be a reference book. However, it is hoped that certain sections will come to be regarded as full, connected, and authoritative treatments of the subjects with which they deal, and that the tables and other data will prove of general use in astronomical computing. An account of its origins and much inform~tion of a general nature about the purpose and scope of the unified Ephemeris is given in section I, " Introduction".
Although published in the United Kingdom, the Explanatory Supplement has been prepared jointly by the Nautical Almanac Office, United States Naval Observatory, under the immediate supervision of its Director, Edgar W. Woolard, and by H.M. Nautical Almanac Office, Royal Greenwich Observatory, under the immediate supervision of its Superintendent, D. H. Sadler. It has been edited by G. A. Wilkins, assisted by Miss A. W. Springett.

B. L. GURNETTE, Captain, U.S. Navy, Superintendent, Naval Observatory, Washington.

R. v. d. R. WOOLLEY, Astronomer Royal,
Royal Greenwich Observatory, Herstmonceux Castle, Sussex.

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--

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NOTE ON 1974 REPRINT
It is regretted that it has not been possible to revise this Explanatory Supplement to take account of the many changes that have been made in The Astronomical Ephemeris and The American Ephemeris and Nautical Almanac since the editions for 1960. The Supplement to the A.E. I968 has, however, been reprinted, with change of pagination, before the Index of this volume; it includes a specification of the
lAD system of astronomical constants, an account of its introduction into the
almanacs for 1968, a list of the principal consequential changes in this Explanatory Supplement, and a list of the known errors in the original edition. The errata and corrections listed on pages 520 to 521 have all been carried through, or otherwise noted, on the relevant pages of this edition. Some other amendments have also been made; in particular, some of the reference data given in section 18 have been brought up to date. The changes described on pages 514 to 519 have not been made, although attention has normally been drawn in footnotes to the changes that would be appropriate to the new system of constants.
All changes in the bases of the ephemerides have been mentioned in the Prefaces to the editions in which they were first made, and corresponding changes have been made in the Explanations at the ends of the volumes. Even apart from these changes, this Explanatory Supplement is now out of date in a number of respects, and so should be used with care. In particular, the following points should be noted:
(a) Even where the basis of an ephemeris has not been changed, an improved method of computation may have been used, so that the numerical example may not define precisely the technique used.
(b) For certain purposes the printed fundamental ephemerides are of inadequate precision, but improved ephemerides are now available. Further details can be obtained from the Bureau International d'Information sur les Ephemerides Astronomiques, 3 Rue Mazarine, Paris (6e), France.
(c) The second of the intern::ltional system (51) of units is now defined in terms of the frequency of a particular caesium resonance, and a scale of "international atomic time" is currently available for reference purposes. As from 1972 January I the principal time signals are hased on a scale (UTC) that differs from IAT by an exact number of seconds and from UTI by an amount that does not normally exceed 0'7 seconds.
It is hoped that, in spite of these deficiencies, this Explanatory Supplement will continue to be of value to all who require information in the fields that it aims to cover until such time as a completely revised edition can be prepared. General suggestions concerning the nature of such a revision, as well as notes on specific amendments, should be sent to the Superintendent of H.M. Nautical Almanac Office (G. A. Wilkins), Royal Greenwich Observatory, or to the Director of the Nautical Almanac Office (R. L. Duncombe), U.S. Naval Observatory.
January, 1973
Vl

CONTENTS

(e-

I. INTRODUCTION

al or

A. Origin of the Supplement

~e

B. History of the Ephemerides

~e

C. History of international co-operation

le

D. Scope and purpose of the Ephemeris

lY

E. Scope and purpose of the Supplement

d.

F. Other publications of relevance

ie

G. Summary of notations

o

n

n

2. COORDINATES AND REFERENCE SYSTEMS

t

A. Coordinate systems

B. Precession

C. Nutation

D. Aberration

E. Refraction

F. Parallax

3. SYSTEMS OF TIME MEASUREMENT
A. Introduction B. Astronomical measures of time C. The practical determination of time D. Historical development

4. FUNDAMENTAL EPHEMERIDES
A. Introduction B. The Sun C. The Moon ... D. The planets E. Other members of the solar system F. Ephemerides at transit G. Comparison of observation with theory ...

,). MEAN AND APPARENT PLACES OF STARS
A. Mean places and star catalo~u-:_ B. Apparent places ... C. Reduction from mean to apparent place D. Day numbers E. Pole Star tables ...
V11

PAGE
1
3 7
10 12
1419
24 28 41 46 54 57
66
69 82 88
96 98
106 III
129 134 142

- '....

-

VIII

CONTENTS

6. THE SYSTEM OF ASTRONOMICAL CONSTANTS *

168

7. HISTORICAL LIST OF AUTHORITIES

A. Introduction

175

B. The Nautical Almanac, 1767-19°0

176

C. The American Ephemeris, 1855-19°0

184

D. The Nautical Almanac and The American Ephemeris, 1901-1959 ... 189

E. References ...

194

F. List of appendices and supplements

198

8. CONFIGURATIONS OF THE SUN, MOON, AND PLANETS

A. Introduction

203

B. The Sun and Moon

203

C. The planets...

206

D. Diary '"

210

9. ECLIPSES AND TRANSITS

A. Introduction

2II

B. Solar eclipses-fundamental equations

21 4

C. Solar eclipses-predicted data ...

223

D. Solar eclipses-local circumstances

24 1

E. Lunar eclipses ...

257

F. Transits of Mercury ...

262

10. OCCULTATIONS

A. Introduction

277

B. Prediction ..

279

C. Reduction .

297

D. Analysis and discussion

3°2

E. Occultations of radio sources

3°3

F. Planetary occultations

3°4

II. EPHEMERIDES FOR PHYSICAL OBSERVATIONS

A. Introduction

306

B. The Sun

3°7

C. Phases and magnitudes

3II

D. The Moon ...

316

E. The rotations of the planets

32 7

F. Mercury and Venus ...

32 9

G. Mars, Jupiter, and Saturn

33°

H. Historical notes ...

34 1

*See also reprint of the Supplement to the A.E. I968 at the end of this volume.

CONTENTS

IX

68

12. SATELLITES

A. Introduction

342

B. The satellites of Mars

35°

C. The satellites of Jupiter

5 6

D. The rings and satellites of Saturn

354 362

E. The satellites of Uranus .

4

F. The satellites of Neptune ..

9

G. Historical list of authorities

387 39° 393

13. RISINGS, SETTINGS, AND TWILIGHT
A. Introduction B. Sunrise, sunset, and twilight C. Moonrise and moonset D. Derivation ...

14. THE CALENDAR

A. Introduction

4°7

B. Historical calendars

408

C. The Gregorian calendar

412

D. The week ...

418

E. Ecclesiastical calendars

420

F. Chronological eras

43°

G. Other modern calendars

43 2

H. The calendar and other tables ...

43 6

IS. THE DISTRIBUTION OF TIME

A. Radio time signals

443

B. Instruments and equipment

447

C. List of radio time signals ...

45°

D. List of standard frequency transmissions

45 2

E. List of coordinated time and frequency transmissions ...

453

16. COMPUTATION AND INTERPOLATION

A. Computing techniques

454

B. Interpolation

459

C. Numerical differentiation ...

47 1

17. CONVERSION TABLES

473

18. REFERENCE DATA

SUPPLEMENT TO THE A.E. 1968 ...

497

ADDE DA AND NOTES (1973, 1974)

522

INDEX

LIST OF TABLES
2.1 Equatorial precessional elements ... 2.2 Equatorial precessional elements (for rectangular coordinates) 2·3 Ecliptic precessional elements 2·4 Approximate annual precession 2·5 Series for the nutation ... 2.6 Correction for diurnal aberration ... 2·7 Differential aberration 2.8 Factors for computing geocentric coordinates
Reduction from universal to ephemeris time
Second-order terms in star reductions Corrections for short-period nutation ...
6.1 The system of astronomical constants ...
7. 1 Coefficients in nutation, 1834-1959
9. 1 Extreme and mean values for solar eclipses
11.1 Optical librations of the Moon 11.2 Elements for the physical ephemeris of Mars 11.3 Elements for the physical ephemeris of Jupiter ...
12. I Elements of the Rings of Saturn 12.2 Elements of Triton
14. 1 Equivalent dates in the Julian and Gregorian calendars 14.2 Perpetual calendar ... 14·3 Golden number 14·4 Julian ecclesiastical lunar calendar 14·5 Dominicalletter-Julian calendar 14.6 Julian Paschal table 14·7 Epact-Gregorian calendar ... 14.8 Gregorian ecclesiastical lunar calendar 14·9 Dominical letter-Gregorian calendar ...
x

PAGE 32 36 37
4°
44
5°
52 59
9°
153 159
169
320 33 6 339
41 7
419
422 422
42 3 42 3 42 5
426 42 7

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LIST OF TABLES

Xl

14. 10 Gregorian Paschal table ...

428

14·II Date of Easter Day for the years 1961-2000

429

14· IZ Calendar for common years ...

434

14. 13 Beginning of the Besselian year, 1900-1999

434

14.14 Julian and Greenwich sidereal day numbers

437

14.15 Julian day number, 1900-1999

43 8

14.16 Greenwich sidereal day number, 1900-1999

440

16.1 Bessel second-difference correction

462

16.zA Bessel second-difference coefficient

464

16.zB Bessel third- and fourth-difference corrections

465

16·3 Maximum differences in the fundamental ephemerides

470

16·4 Formulae for plane and spherical triangles ...

472

17.1 Intervals of mean sidere'al to mean solar time

474

17·z Intervals of mean solar to mean sidereal time

476

17·3 Decimals of a solar day to sidereal time

478

17·4 Decimals of a day to hours, minutes, and seconds

479

17·5 Hours, minutes, and seconds to decimals of a day

480

17.6 Minutes and seconds to decimals of a degree

482

17·7 Decimals of a degree to minutes and seconds

483

17.8 Time to arc

484

17·9 Arc to time

485

17.10 Low-precision conversion of angular measures

486

LIST OF GRAPHS

Figure 3.z (a) General trend of Ll T, 1660-1972 (b) Excess length of day... 91

Figure 13.1 Twilight illumination on a horizontal plane ...

399

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1. INTRODUCTION
A. ORIGIN OF THE SUPPLEMENT
The Nautical Almanac for 1931 was completely redesigned; for the first time it included a comprehensive Explanation and a Derivation illustrating the calculation of every quantity tabulated in the Almanac. Although the Derivation was discontinued after that year, the Explanation was continued in full and was gradually expanded. This was the consequence partly of newly-added matter, requiring detailed explanation, and partly of more comprehensive illustrations of the use of the tabulated data, such as, for example, in the case of eclipses. The Almanacs for the years 1937,1938,1939, and 1940 contained (with appendices) 951, 940, 912, and 920 pages respectively. All of this added material was (and much still is) of considerable value, but much was inappropriate for the day-to-day use of the Almanac as an astronomical ephemeris; and much was of permanent rather than ephemeral interest. Many practical astronomers complained of the unwieldy volume and more than one suggested the separation of the permanent tables and explanations from the purely ephemeral data. The omission of most of the apparent places of stars in the edition for 1941, consequent on the introduction of the international volume of Apparent Places of Fundamental Stars, reduced the number of pages to 759. At this juncture a drastic cut was imposed on the overall size of subsequent editions by the exigencies of war. The opportunity was taken of inaugurating a policy that had been under consideration on its own merits. To quote from the Preface to the edition for 1942:
" It is intended that in future, starting with this edition, the Nautical Almanac should in general contain, in addition to the ephemeral data which will continue to be printed in the established form, only such auxiliary tables and explanations as are necessary for the user to extract the ephemeral data in the form he requires. In previous editions considerably more auxiliary tables and more detailed explanations than have been required for this purpose have been given and, although these have been of considerable benefit to some users, they have detracted from the convenience of the Almanac for the majority of routine observers and computers. As it has not been possible to include all the auxiliary tables, illustrations and explanations required in the application of the tabulated ephemeral data, the Almanac has never been completely self-contained; with this in mind, it is further intended to publish a separate supplement, which will be of a permanent character and which will contain all the permanent tables and explanations previously given, together with such added information as can be included in the rather wider scope provided by a separate publication. It is considered that this separation of the ephemeral data from the permanent tables and explanation wili not only lead to a desirable reduction in the size of the Almanac, but will

2

EXPLANATORY SUPPLEMENT

also add to the convenience of the user requiring both books; it is easier to refer to two books at once than to two different places in the same book.
" It is possible that publication of the Supplement will be delayed for some time; in the meantime reference should be made to the relevant portions of previous editions."

It was, unfortunately, not possible to take any active steps towards the preparation of the promised" Supplement" until several years after the end of the war. At one time it was hoped that it would be possible to issue the Supplement to relate to The Nautical Almanac for 1952, and much work was actually done, particularly in the preparation of detailed examples of eclipse calculation; but this hope could not be fulfilled.

With the introduction ofthe concept of ephemeris time at the Paris (1950) Conference on the Fundamental Constants of Astronomy, it became clear that substantial changes in the Almanac could not be long delayed. This view was confirmed at the Rome (1952) General Assembly of the International Astronomical Union, when a series of recommendations involving fundamental changes in the ephemerides was agreed, to become effective as from 1960. The advantages of still further delaying the Supplement were evident; by relating it to the edition of 1960 it could present the new system as a unified whole, without the complication of a detailed explanation of the old. And it was accordingly agreed to introduce the Supplement as from 1960.

In 1954 the first steps were taken to achieve the" conformity" of The Nautical Almanac and The American Ephemeris and Nautical Almanac; and this has eventually led to their complete unification as from 1960. The plans for the publication of the Supplement naturally affected the contents of the unified Ephemeris, particularly in regard to the explanation and auxiliary tables; and as the Supplement would apply equally to The American Ephemeris it was natural that it should become a joint production.

This Supplement has accordingly been prepared jointly by the Nautical Almanac Office, U.S. Naval Observatory, and by H.M. Nautical Almanac Office. Although the latter has perforce accepted editorial responsibility, and the general work of compilation has been shared, the principal authors have been as follows:

In H.M. Nautical Almanac Office:
D. H. Sadler, Flora M. McBain Sadler, J. G. Porter, G. A. Wilkins, and
H. W. P. Richards. H. M. Smith (Time Department) prepared section 15.

In Nautical Almanac Office, U.S. Naval Observatory:

G. M. Clemence, E. W. Woolard, Simone D. Gossner, and A. Thomas.

In both Offices other members of the staff, not named individually, have shared in the work of compilation and proof reading.

The note on page vi indicates the policy that has been adopted in the editing of this reprint of the original edition, and draws attention to its current deficiencies.

lB. INTRODUCTION

3

oks

the

B. HISTORY OF THE EPHEMERIDES

The brief histories that follow are concerned solely with the major changes of

t

form and content, and are intended as a general introduction to the detailed analyses

to

given in section 7.

In

I. The Astronomical Ephemeris

ot

" The Commissioners of Longitude, in pursuance of the Powers vested in them by a late

Act of Parliament, present the Publick with the NAUTICAL ALMANAC and ASTRONOMICAL

EPHEMERIS for the Year 1767, to be continued annually; a \Vork which must greatly con-

tribute to the Improvement of Astronomy, Geography, and Navigation. This EPHEMERIS

1-

contains every Thing essential to general Use that is to be found in any Ephemeris hitherto

d

published, with many other useful and interesting Particulars never yet offered to the

Publick in any Work of this Kind. The Tables of the Moon had been brought by the late
1,
Professor MAYER of Gottingen to a sufficient Exactness to determine the Longitude at Sea,

within a Degree, as appeared by the Trials of several Persons who made Use of them. The

Difficulty and Length of the necessary Calculations seemed the only Obstacles to hinder

them from becoming of general Use: To remove which this EPHEMERIS was made; the

Mariner being hereby relieved from the Necessity of calculating the Moon's Place from the

Tables, and afterwards computing the Distance to Seconds by Logarithms, which are the

principal and only very delicate Part of the Calculus; so that the finding the Longitude

by the Help of the EPHE:\1ERIS is now in a Manner reduced to the Computation of the Time,

an Operation .... "

" All the Calculations of the EPHEMERIS relating to the Sun and Moon were made from ;vIr. MAYER'S last manuscript Tables, received by the Board of Longitude after his Decease, which have been printed under my Inspection, and will be published shortly. The Calculations of the Planets were made from Dr. HALLEY'S Tables; and those of .... "

The above extracts from the Preface to the first edition, for 1767, of The Nautical Almanac and Astronomical Ephemeris were written by Nevil Maskelyne, then Astronomer Royal. The main incentive for, and the main emphasis of, the publication was the determination of longitude at sea using the method of lunar distances. The ephemerides were all given in terms of apparent solar time, for the reasons given in the Explanation.

" It may be proper first to premise, that aU the Calculations are made according to apparent Time by the Meridian of the Royal Observatory at Greenwich."
" What has been shewn concerning the Equation of Time chiefly respects the Astronomer, the Mariner having little to do with it in computing his Longitude from the l\100n's Distances from the Sun and Stars observed at Sea with the Help of the Ephemeris, all the Calculations thereof being adapted to apparent Time, the same which he will obtain by the Altitudes of the Sun or Stars in the Manner hereafter prescribed.
"But if Watches made upon Mr. John Harrison's or other equivalent Principles should be brought into Use at Sea, the apparent Time deduced from an Altitude of the Sun must be corrected by the Equation of Time, and the mean Time found compared with that shewn by the Watch, the Difference will be the Longitude in Time from the Meridian by which the Watch was set; as near as the Going of the Watch can be depended upon."

Apart from many changes in the sources of the data, and in particular the tables from which the Moon's position was calculated, the main pages of the Almanac remained essentially unchanged until 1834. Forthat year, to quote from the Preface:

4

EXPLANATORY SUPPLEMENT

" The NAUTICAL ALMANAC and ASTRONOMICAL EPHEMERIS for the Year 1834, has been constructed in strict conformity with the recommendations of the ASTRONOMICAL SOCIETY of LONDON, as contained in their Report ... ; and will, it is believed, be found to contain almost every aid that the Navigator and Astronomer can require."

The changes were both fundamental and substantial, and involved almost doubling the size. The most fundamental change was to replace apparent time by mean time as the argument of the ephemerides. In the words of the Report:

" The attention of the Committee was, in the first instance, directed to a subject of general importance, as affecting almost all the results in the Nautical Almanac; viz., whether the quantities therein inserted should in future be given for apparent time (as heretofore), or for mean solar time. Considering that the latter is the most convenient, not only for every purpose of Astronomy, but also (from the best information they have been able to obtain) for all the purposes of Navigation; at the same time that it is less laborious to the computer, and has already been introduced with good effect into the national Ephemerides of Coimbra and Berlin, the Committee recommend the abolition of the use of apparent time in all the computations of the Nautical Almanac; excepting .... "

The direction of at least some of the other changes was influenced by the view that was expressed in the Report as:

" And here perhaps it may be proper to remark, that, although in these discussions the Committee have constantly kept in view the principal object for which the Nautical Almanac was originally formed, viz., the promotion and advancement of nautical astronomy, they have not been unmindful that, by a very slight extension of the computations, and by a few additional articles (of no great expense or labour), the work might be rendered equally useful for all the purposes of practical astronomy."

The requirements of the navigator were by no means overlooked; in particular the number and presentation of " lunar distances", including distances from the planets, was greatly improved. However, the Explanation and Use of the previous editions, which had still been based on Maskelyne's, was replaced by a completely new Explanation in which little reference was made to the use of the ephemerides for navigation; tables of refraction were excluded and no example was given of clearing an observed lunar distance for the effects of semi-diameter, parallax, and refraction.
Apart from the omission of lunar distances in 1907 the first part of the Almanac, containing the ephemerides of the Sun and Moon, remained unchanged in form, though of course based from time to time on different data and tables, until 1931. At various dates other matter was added, particularly ephemerides of the Moon and planets at transit on the Greenwich meridian and the apparent places of many more stars; later, ephemerides for physical observations were added and in 1929, anticipating the redesign in 193 I, ephemerides of the Sun referred to the standard equinox of 1950'0 were given for the years 1928 and 1929.
Much of the added matter was of no interest to the practical navigator, and in 1896 " Part I (containing such data as are more particularly required for navigational purposes) " was" also published separately for the convenience of sailors". This consisted of a straight reprint of the monthly pages comprising the first part of the Almanac, with selections from the other data and a few pages specially prepared.

~ -- ------ - - ~~

lB. INTRODUCTION

5

een

In the Preface to the edition for 1914 it was announced briefly that" Part I has been

. of ost

remodelled for the convenience of sailors"; thus was introduced The Nautical

Almanac, Abridged for the Use of Seamen, which was specially designed for its

purpose. This Almanac was redesigned in 1929 and again in 1952, when it was

renamed The Abridged Nautical Almanac; it was rearranged in a different form in

1958 and, as from 1960, it takes on the appropriate portion of the original title,

namely The Nautical Almanac.

Prior to the revision in 1931 a fundamental change, requiring consequential

er

changes in the Almanac, had taken place in the measure of mean solar time.

~r

iY

Before 1925 the astronomical day was considered to start at noon, and the principal

ephemerides had been given for 011, i.e. noon, on each day. As from 1925 January 1

CI

the tabular day was brought into coincidence with the civil day and was

tI e

considered to start at midnight; the ephemerides were still given for Oh, now

indicating midnight.

The revision of 1931 was much more than a rearrangement of the same data in a different form; the changes of page size, of presentation, of provision for interpolation, and of content were less important than the complete break with the centuryold lay-out designed primarily for navigation for which the Almanac had ceased to provide. The new form could be, and was, designed for the astronomer without the necessity for considering the requirements of navigation. Its arrangement has remained basically unchanged, though there has been frequent change in content of the less fundamental matter.

Major changes were introduced in the edition for 1960 when the Almanac was unified with The American Ephemeris, the principal one being the use of ephemeris time, instead of universal time (mean solar time on the meridian of Greenwich), as the argument for the fundamental ephemerides. This change still further emphasized the unsuitability of the volume for navigation, and led to the adoption for its new title of the appropriate part of the original full title, namely The Astronomical Ephemeris. The changes are fully described in the Preface to the edition
for 1960. *

2. The American Ephemeris and Nautical Almanac
During the first half of the nineteenth century, The Nautical Almanac remained in general use on American ships and among astronomers and surveyors in the United States. However, with the continued development of the country, and its growth as a maritime nation, an increasing need for a national almanac was felt and eventually led to the establishment of a Nautical Almanac Office in the Navy Department by an Act of Congress approved in 1849. The Office was set up in Cambridge, Massachusetts, where library and printing facil.ities were available, and began work during the latter part of 1849. The first volume of The American Ephemeris and Nautical Almanac was for the year 1855, and was published in 1852. The Office was moved to Washington in 1866, but was not located at the Naval Observatory until 1893.
*Some additional notes on the history of The Astronomical Ephemeris are given on pages ix-xviii of the volume for 1967, the two-hundredth anniversary edition.

6

EXPLANATORY SUPPLEMENT

For the years 1855-1915 inclusive, the volume was divided at first into two parts, then, beginning with 1882, into three. The first part during this entire period was an ephemeris for the use of navigators that was also reprinted separately, with the inclusion of a few pages from the remainder of the volume, as The American Nautical Almanac. It comprised 12 monthly sections, for the meridian of Greenwich, each containing ephemerides of the Sun, Moon, and lunar distances for the month; following the monthly sections were ephemerides of Venus, Mars, Jupiter, and Saturn for the year, and, beginning with 1882, of Mercury, Uranus, -and Neptune.

The second part of the volume contained ephemerides of the Sun, Moon, planets, and principal stars, for meridian transit at Washington; and data on eclipses, occultations, and a few other phenomena, which in 1882 were formally grouped as a third part with the title" Phenomena". The explanatory sections and a few miscellaneous tables completed the volume.

During the period 1855-1915, few changes were made in the form or content. The nautical part remained virtually unaltered; lunar distances were omitted, beginning with 1912, but a page explaining how to calculate them continued to be included. The principal revisions in the other parts of the volume were in 1882 and 1912-1913. The rearrangement of the 1882 volume was accompanied by some additions and omissions. The principal omission was the ephemeris of Moon-culminating stars for determining longitude. The principal additions were: the physical ephemerides of Mercury and Venus, in place of the former meagre data for the apparent disks, for the reduction of meridian and photometric observations; daily diagrams of the configurations of the four great satellites of Jupiter; and ephemerides for the identification of the satellites of Mars, Saturn, Uranus, and Neptune. In the volume for 1912, the ephemerides of the satellites were extended to include tables for determining the approximate position angle and apparent distance; in 1913, physical ephemerides were added for the Sun, Moon, Mars, and Jupiter. These revisions, and minor additions, omissions, and rearrangements, are described in more detail in section 7.

In the volume for 1916, the first to be issued under the international agreements resulting from the Paris Conference of 19II, extensive revisions were made in the form and arrangement that had been retained essentially unchanged since 1882; but the content remained substantially the same. The arrangement of the Greenwich ephemerides of the Sun and Moon by monthly sections was discontinued, and replaced by annual ephemerides. At the same time, The American Nautical Almanac was no longer a reprint of part of The American Ephemeris, but a separately prepared volume especially designed for the navigator.

In 1925, the astronomical reckoning of time from 011 at noon was replaced by the civil reckoning from midnight.

During the interval from 1916 until the fundamental revisions in 1960 when The American Ephemeris was unified with The Astronomical Ephemeris, the revisions of form and content were mostly only in details; but a few major changes occurred,

lC. INTRODUCTION

7

10

and in the volumes for 1934-1937 a number of further subdivisions and rearrange-

re

ments of the contents were made. In 1937, the volume had become formally

y,

divided into seven parts; the part constituting the ephemeris for Washington had

In

been reduced to only ephemerides of the Sun, Moon, and planets for meridian

l-

transit at Washington, all the other material having been transferred to other parts

Ie

and referred to the Greenwich meridian.

r,

Because of the limited usefulness of the Washington-transit ephemerides except

d

to observers on the Washington meridian, the publication of this part was discon-

tinued beginning with the 1951 volume. Otherwise, the general form and arrange-

1,

ment adopted in 1937 were retained until 1960. The other principal changes in

content during 1916-1959 were the following: In 1919, tables of the rising and

a

setting of the Sun and the Moon were added. In 1941, the number of stars for

which apparent places were given, after having reached 887, was decreased to 212

when Apparent Places of Fundamental Stars was first published; in 1957, apparent

places were omitted entirely, but precise mean places of 1551 stars which had been

given beginning with 1951 were continued. The elements and predictions of

occultations were successively extended to more and fainter stars, and to additional

standard stations, because of their importance for determining the departures of

the Moon from gravitational theory that are due to variations in the rotation of the

Earth. An ephemeris of Pluto was added to the planetary ephemerides in 1950;

and ephemerides of Ceres, Pallas, Juno, and Vesta in 1952.

C. HISTORY OF INTER ATIONAL CO-OPERATION
Formal co-operation may be regarded as dating from the International Meridian Conference held in Washington in October 1884 at the invitation of the Government of the United States. The resolutions of that conference included:
" ... the adoption of the meridian passing through the centre of the transit instrument at the Observatory of Greenwich as the initial meridian for longitude."
" That from this meridian longitude shall be counted in two directions up to ISO degrees, east longitude being plus and west longitude minus."
". . . the adoption of a universal day for all purposes for which it may be found convenient ... "
" That this universal day is to be a mean solar day; is to begin for all the world at the moment of mean midnight of the initial meridian, coinciding with the beginning of the civil day and date of that meridian; and is to be counted from zero up to twenty-four hours."
" That the Conference expresses the hope that as soon as may be practicable the astronomical and nautical days will be arranged everywhere to begin at mean midnight."
Although the other resolutions are now in use, it has been customary for many years in astronomy, but not in all other related sciences, to treat west longitude as positive, and east longitude as negative. This is the convention adopted in the Ephemeris.
At the invitation of the Bureau des Longitudes the directors of the national ephemerides, and other astronomers, met in Paris in May 1896 for the Conference

8

EXPLANATORY SUPPLEMENT

Internationale des Etoiles Fondamentales. In addition to adopting resolutions

concerning the fundamental catalogue, and the calculation and publication of

apparent places of stars, the Conference adopted the following fundamental

constants:

Nutation

9"'21

Aberration 20"'47

Solar parallax 8" ·80

which are still in operation. It also agreed to adopt Newcomb's definitive values (which were not then in final form) of luni-solar and planetary precession.

Active co-operation between the offices of the national ephemerides dates from the Congres International des Ephemerides Astronomiques held at the Paris Observatory in October 1911. This conference was called, on the initiative of the Bureau des Longitudes, by B. Baillaud, Director of the Observatory and President of the Comite International Permanent de la Carte Photographique du Ciel. Its purpose was" d'etablir une entente permettant d'augmenter, sans nouveaux frais, la masse des donnees numeriques fournies annuellement aux observateurs et aux calculateurs ". Although the Conference was primarily concerned in obtaining a greatly increased list of apparent places of stars, it extended its attention to all the ephemerides of bodies in the solar system. Its comprehensive recommendations covered the distribution of calculations between the five principal ephemeris offices (France, Germany, Great Britain, Spain, and the United States), specified standards of calculation and presentation, arranged for publication of additional data, and fixed the values of two further constants to be used in the ephemerides: the flattening of the Earth (1/297) and the semi-diameter of the Sun at unit distance for eclipse calculations (15' 59" ·63). Most of these recommendations are still in force.

Official approval was in some cases necessary for the adoption of these recommendations, as illustrated by the following extract from the Act of Congress of August 22, 1912 (37 Stat. L., 328, 342):
" The Secretary of the Navy is hereby authorized to arrange for the exchange of data with such foreign almanac offices as he may from time to time deem desirable, with a view to reducing the amount of duplication of work in preparing the different national nautical and astronomical almanacs and increasing the total data which may be of use to navigators and astronomers available for publication in the American Ephemeris and Nautical Almanac: Provided . . ."
Here follows a number of provisions, the most important astronomically being the repeal of the proviso in the appropriation Act of September z8, 1850 (9 Stat. L., 513, 515) that " hereafter the meridian of the observatory at Washington shall be adopted and used as the American meridian for all astronomical purposes, and that the meridian of Greenwich shall be adopted for all nautical purposes".
Such exchange agreements have been carried out in spite of international difficulties.

In 1919 the International Astronomical Union was founded; Commission 4 (Ephemerides), which numbers among its members the directors of the national ephemerides, thereafter provided the formal contacts by which the previous agreements could be continued and extended.

Flattening the Earth?

IC. INTRODUCTION

9

The 191 I agreements had been directed almost entirely to the reduction of the total amount of work by the avoidance of duplicate calculation. In 1938 Commission 4 recommended that the principle should be extended to the avoidance of duplicate publication by the collection in a single volume of the apparent places of stars then printed in each of the principal ephemerides. This recommendation, coupled with the adoption of the Dritter Fundamentalkatalog des Berliner Astronomischen Jahrbuchs (FK3), was implemented for 1941 by the publication, under the auspices of the International Astronomical Union, of the international volume Apparent Places of Fundamental Stars. By this means astronomers gained access to the apparent places of stars in one volume, and the individual ephemeris offices were saved the work of the compilation and proof reading, as well as the cost of type setting, of most of the stars which they previously published.

Continuing the precedents of the 1896 and 19II conferences, the Director of the Paris Observatory (Professor A. Danjon) convened a further conference that was held in Paris in March 1950 to discuss the fundamental constants of astronomy. The leading recommendation was" .... that no change be made in the conventionally adopted value of any constant". But the recommendations with the most far-reaching consequences were those which defined ephemeris time and brought the lunar ephemeris into accordance with the solar ephemeris in terms of ephemeris time. These recommendations were addressed to the International Astronomical Union and were formally adopted by Commission 4 and the General Assembly of the Union in Rome in September 1952.

Commission 4 had, at various times, made arrangements for the redistribution of calculations between the ephemeris offices; for example, the Institute for Theoretical Astronomy in Leningrad contributed apparent places of stars to the international volume for the years 1951-1959. With the availability of fast automatic calculating machines it is now both practicable and efficient for large blocks of work, such as the calculation of apparent places of stars, to be done in one office; and at the 1955 General Assembly of the Union in Dublin, a general redistribution of calculations on these lines was agreed by the directors of the national ephemerides and confirmed by Commission 4. Full details of these agreements, of changes in the bases of the ephemerides, and of the discussions leading to the introduction of Apparent Places of Fundamental Stars are given in the reports of Commission 4 in Transactions of the International Astronomical Union.

The logical development of this co-operation would appear to be a single international ephemeris; this is not yet practicable. Following the successful unification of the navigational almanacs, and greatly assisted by the common language, it was however agreed in 1954 to unify the British and American ephemerides as from the year 1960; and this has now been done. In reporting this agreement to Commission 4, it was announced that reproducible material for the whole Ephemeris, with the exception of the short introductory section, would be made available to other ephemeris offices through H.M. Stationery Office at a small fee. And the hope was expressed that use would be made of this facility to effect a considerable saving of type setting and proof reading, while still preserving

10

EXPLANATORY SUPPLEMENT

for each country its own ephemeris with its own language headings and explanations and its own selection of mattrial.

The Berliner Astronomisches Jahrbuch (published annually since 1776) and the Astronomisch-Geodtitisches Jahrbuch (introduced for the year 1947) ceased publication with the years 1959 and 1957 respectively; in Germany either the British or American editions of the unified Ephemeris will be used and there will be no separate German edition.

References

The proceedings, recommendations, and resolutions of the international conferences referred to above have been published as follows:-

Protocols of the Proceedings of the International Conference held at lVashington for the purpose of fixing a Prime Meridian and a Universal Day. October 1884. Washington, D.C., 1884.
Proces-Verbaux of the Conference Internationale des Etoiles Fondamentales de 1896. Paris, Bureau des Longitudes, 1896.
a Congres International des Ephimerides Astronomiques tenu l'Observatoire de Paris du 23
au 26 Octobre 191 I. Paris, Bureau des Longitudes, 1912. A full account, with English translations of the resolutions, is given in M.N.R.A.S., 72, 3~2-345, 1912.

Colloque International sur les Constantes Fondamentales de l'Astronomie. Observatoire de Paris, 27 Mars-ler Avril 1950. Colloques Internationauxdu Centre National de la Recherche Scientifique, 25, 1-131, Paris, 1950. The proceedings and recommendations are
* also available in Bull. Astr., IS, parts 3-4, 163-292, 1950.

The reports and recommendations of Commission 4 of the International Astronomical Union have been published as follows:

Trans.I.A.U., 1,159,207; 1923· 2, 18-19, 178,229; 1926. 3, 18, 224, 300; 1929. 4, 20, 222, 282; 1933. 5, 29-33, 281-288, 369-37 1 ; 1936. 6, 20-25, 336, 355-363; 1939. 7,61, 75-83; 1950. 8,66-68,80-102; 1954. 9, 80-91 ; 1957· 10,72,85-99; 1960. II, A, 1-8; 1962. B, 164-167,441-462; 1962. 12, A, 1-10; 1965. B, 101-105, 593-625; 1966. 13, A, 1-9; 1967. B, 47-53, 178-182; 1968. 14, A, 1-9; 1970. B, 79-85, 198-199; 1971.

Assembly

Rome

1922

Cambridge, England 1925

Leiden

1928

Cambridge, Mass. 1932

Paris

1935

Stockholm

1938

Zurich

1948

Rome

1952

Dublin

1955

Moscow

1958

Berkeley

196 I

Hamburg

1964

Prague

1967

Brighton

1970

D. SCOPE AND PURPOSE OF THE EPHEMERIS

The Astronomical Ephemeris and The American Ephemeris and Nautical Almanac are identical in content and presentation, apart from a few preliminary pages. Except in the few cases where distinction is desirable. they will be referred to collectively as " the Ephemeris" or by the initials A.E.
Scope. Now that other publications provide for the practical requirements of navigators and surveyors, the Ephemeris need no longer do so. Its content is
*See page 174 for references to proceedings, of later conferences on the system of astronomical constants.

lD. INTRODUCTION

II

accordingly restricted to providing fundamental ephemerides of the Sun, Moon, and planets to the highest precision, and ephemerides derived from them for the requirements of the practical astronomer.

Fundamental ephemerides. The main purpose of the fundamental ephemerides *
of the Sun, Moon, and planets is to provide a rigorous continuous reference system, to which observations, if necessary spread over many years, can be referred. In order to achieve this the ephemerides should be calculated strictly in accordance with a self-consistent theory, which can be specified precisely in regard to both form and numerical constants. It will suffice that the adopted constants be close enough to their true values for any possible variations to lead to linear changes in the ephemerides; but it is important that all known physical forces and effects be fully incorporated.

The ephemerides are calculated in accordance with the Newtonian law of gravitation, modified by the theory of general relativity. The values of the adopted constants are given partly in section 6 and partly in section 4 under the individual body concerned; most are collected together in section 18. The independent variable of the ephemerides is ephemeris time, which is independent of the unpredictable variations in the speed of rotation of the Earth. The highest standard of precision in the calculations is achieved for the five outer planets Jupiter, Saturn, Uranus, Neptune, and Pluto; the calculations for the Sun, Mercury, Venus, and Mars do not at present reach the same standard. For the Moon other requirements are very severe; extremely accurate values of the Moon's motion, over short intervals of time, are required for tre determination of relative positions on the Earth through observations of eclipses and occultations; consistent positions of the Moon over long intervals (say 10 years) of time are needed for the practical determination of the length of the fundamental unit of time, the ephemeris second. But the precision of the ephemeris is reasonably adequate for the present.

It is convenient, but not necessary, that the fundamental ephemerides should give positions sufficiently close to the actual positions to provide for the observational astronomer and as a basis for further predictions. The ephemerides are, in fact, amply close enough for this secondary purpose in terms of ephemeris time; but the correction to universal time is large enough to make its application necessary for the ephemeris of the Moon.

Other data. The only other data of a fundamental character given in the Ephemeris are those required for the calculation of apparent places of stars; these include the values of precession and nutation required to specify the observational frame of reference. Apparent places themselves are not included as they are given in Apparent Places of Fundamental Stars. Some deduced data, such as the Besselian elements of eclipses, are of the highest precision; but generally all other ephemeral data are intended to assist observation and are not of adequate accuracy for precise comparison with observations. In particular the theories on which the orbits of the satellites are based are too imperfect to provide ephemerides of a fundamental character.
*In most cases new ephemerides of higher precision and accuracy are now available, but the ephemerides in the A.E. provide a useful common standard of reference.

12

EXPLANATORY SUPPLEMENT

* E. SCOPE AND PURPOSE OF THE SUPPLEMENT
As stated in the Prefau the purpose of this Explanatory Supplement is to provide users of the Ephemeris with fuller explanations of its content, derivation, and use than can conveniently be included in the Ephemeris itself. To a limited extent it also provides the auxiliary tables and reference data required in the application of the data tabulated in the Ephemeris; but, because of the availability of other publications and of changing methods of calculation, these requirements are much less than when the Supplement was first proposed in 1940.
In particular it has been decided not to include a section on observatories, as originally planned. The list of observatories in the editions of The Nautical Almanac prior to 1942 changed little from year to year, and formed one of the motivations for a separate supplement; in recent years, however, the rapid increase in the number of observatories, both optical and radio, together with more frequent changes of position, make any list incomplete and out-of-date in one or two years. The list of observatories in A.E., pages 434-452 in 1960, contains full details of place, description, positions, and certain derived constants for use in the reduction of observations, for some 320 optical and 27 radio observatories; the list includes only major observatories and those specifically engaged on observations requiring an accurate knowledge of position for their reduction. It is necessarily prepared some two years before the year of the Ephemeris in which it is printed, and is out-of-date to that extent. A full description of the list and an explanation of the quantities tabulated are given in the Ephemeris itself.
The data in the Ephemeris will suffice for most requirements for the reduction of current observations. Much more detailed information about the equipment, programmes of ob3ervation, and staff of observatories is given in the publication Les Observatoires Astronomiques et les Astronomes by F. Rigaux, published in 1959 by l'Observatoire Royal de Belgique under the auspices of the International Astronomical Union. As with all such lists the data, particularly as regards individual astronomers, are rapidly becoming out of date. No derived constants are given, and the positional data are not always complete or specific. In any case, users who require precise positions for the reduction of observations should obtain positions for the particular telescope used.
It was originally planned to include a comprehensive list of former observatories, on the lines of the lists published in the editions of The Nautical Almanac for the years 1929 to 1938 inclusive. Changes of position of several observatories have added to this list in recent years. But the small amount of additional data hardly justifies the re-publication of data that must now be rarely, if ever, used.
The Ephemeris does not contain all ephemerides of position. Ephemerides of the stars, of minor planets, of comets, and of other bodies are tabulated in other publications, mainly for the general convenience of users; it is proper to regard these as forming an integral part of the totality of astronomical ephemerides. The
*See also note on page vi regarding the 1974 reprint.

-- --- -~~----

-----~H·

IE. INTRODUCTION

13

scope of the Supplement is accordingly extended to include reference to such ephemerides; but, generally, less detailed explanations and derivations are given for these.

It is a necessary preliminary to the main purpose to define frames of reference and systems of coordinates with some care. In doing so the text-book approach has been deliberately avoided: all elementary definitions and proofs have in general been omitted. An attempt has been made to combine complete rigour of treatment with practical requirements, giving the errors of all approximate procedures; but no attempt has been made to be comprehensive.

The treatment of the main sections varies according to the nature of their content, particularly as to whether they refer to fundamental data, or derived quantities, or to ephemerides in the Ephemeris or elsewhere.

In one section only, that on Systems of Time Measurement, has an attempt been made to give a completely exhaustive, and authoritative, treatment of the subject. This subject is fundamental to the whole purpose of the Ephemeris and is one of extreme difficulty, especially in view of the many recent changes in both conception and practical determination. It is hoped that this section will be regarded as providing authoritative and precise statements as to the definitions of Universal Time, Sidereal Time, and Ephemeris Time and of the relationships between them.

The most important specific function of the Supplement is to define precisely, for each individual ephemeris: the quantity tabulated; the fundamental data on which it is based; and how it is derived from those data. No such definition can be regarded as complete, or as free from possible misunderstanding, until it is illustrated by a numerical example in which every figure is derived from the stated fundamental data by means of the stated procedure and formulae; only by such means can ambiguities of wording be clarified, and procedures and formulae verified. To achieve this purpose fully, numerical examples should be chosen so as together to cover all cases and to avoid accidentally-small contributions in which significant errors of principle might lead to negligible numerical differences. In principle the tabulated values should be reproduced exactly; but in practice there must always be a small, and almost always negligible, area of uncertainty in which a real difference of principle may be masked by legitimate variations of procedure and by accumulation of rounding-off errors owing to differences in computing methods.

Although the "derivation", as understood above, of every ephemeris is illustrated numerically in the Supplement no claim is made to have achieved complete coverage. The single examples given cannot cover every case and may sometimes leave uncertainties due to unsuitable choice of date and time; this is especially so as a fixed epoch (1960 March 7 at Oh E.T.) has been adopted for most of the examples. Moreover, the examples have been calculated on a desk calculating machine one stage at a time, recording intermediate results where necessary; the final results may therefore differ both from the values printed in the Ephemeris, which are calculated systematically on punched-card machines, and from those

EXPLANATORY SUPPLEMENT

obtained by adopting different stages in the calculation. None of these deficiencies

is likely to be serious, or to result in difficulties of interpretation, provided the

24~

limitations are understood. It is intended that every printed figure should be

19~

I'

obtainable directly, correctly rounded off, from the stated formula using the actual

of

printed values of the basic data and intermediate results quoted; however, with a

fOI

calculating machine, there are different methods of accumulating products and of

to

doing continuous multiplications, and in a few cases, by oversight, the rounding-off

of an intermediate or final result may differ from that formally obtained. Similarly,

values of trigonometric functions may differ according to the interval and number

of figures in the tables used.

gr st

The numerical examples are designed primarily to illustrate unambiguously

the formulae quoted, and they do not necessarily indicate either the best method of

(l

calculation or the method actually used. It is not possible to illustrate numerically

II

many of the actual methods used for systematic calculation on punched-card and

rr

CI

electronic computing machines.

tl

Details of methods of calculation are omitted from the numerical examples; a

Y

short note on computing techniques, particularly in regard to the solution of

spherical triangles, is given in section 16A.

t

F. OTHER PUBLICATIONS OF RELEVANCe
For convenience of reference, there are listed below the full titles, descriptions, and adopted abbreviations of British, American, and other publications which are likely to be of interest to astronomers; the British publications may be obtained through H.M. Stationery Office and the American publications through the Superintendent of Documents, U.S. Government Printing Office.
r. Unified publications; British and American editions
The Nautical Almanac (N.A.) (about 276 + xxxv pages) contains data for astronomical
navigation at sea. Of astronomical interest are: the Greenwich Hour Angle (G.H.A.) and Declination (Dec.) to 0"1 for each hour for the Sun, Moon, Venus, Mars, Jupiter, and Saturn; times of sunrise, sunset, and beginning and end of civil and nautical twilights for latitudes N. 72° to S. 60° for every third day; times of moonrise and moonset for latitudes N. 72° to S. 60° for every day.
The Air Almanac (A.A.) (four-monthly edition, about 242 + 90 pages) contains data
°'. for astronomical navigation in the air. Chief astronomical interest lies in the tabulations of
G.H .A. and Dec. of the Sun ( to I ) , and of the Moon and three planets ( to I'), for each lOrn.
Sight Reduction Tables for Marine Navigation, U.S. Naval Oceanographic Office, H.O.
Pub. No. 229, six volumes each covering ISO of latitude, 1970 onwards. Reproduced as (British) Hydrographic Department, N.P. 4°1, 1971 onwards. These tables give altitude to 0"1, with variations for declination, and azimuth to 0°'1, with arguments latitude, hour
angle, and declination, all at I ° interval. They provide all solutions of the spherical triangle,
given two sides and the included angle, to find a third side and adjacent angle.

~--- -----'-"" ..

IF. INTRODUCTION

15

Sight Reduction Tables for Air Navigation, U.S. Naval Oceanographic OfficeH.O.Pub.No. 249, reproduced as (British) Air Publication; A.P. 3270; vol. I, Selected Stars (epoch 1975'0),1973; vols. 2 and 3, Declinations 0°-29°,1953. Volume I contains the altitude to I' and the azimuth to 1 0 for the seven most suitable stars for navigation, for each degree of latitude and for each degree of local sidereal time. Volumes 2 and 3 give similar data for each degree of declination to 29° and for each degree of hour angle; tabulations extend to depressions of at least 5° below the horizon.

2. British publications
The Star Almanac for Land Surveyors (S.A.) (about 90 pages) is designed for topographical surveyors. Its principal interest lies in the apparent places (to 0·' I and I ") of 685 stars, including all stars not fainter than magnitude 4'0.
Planetary Co-ordinates for the years 1960-1980 referred to the equinox of 1950'0 (Planetary Co-ordinates) 180 pages, 1958;I'(Earlier volumes covering the years 1900-1940 and 1940-1960 were published in 1933 and 1939, respectively.) These volumes are intended mainly for the calculation of perturbations of comets and minor planets. They give heliocentric, spherical and rectangular coordinates, referred to the standard equinox of 1950'0, of the planets, together with auxiliary tables, explanations, and illustrations; the volume for the years 1960-1980 also contains a comprehensive collection of formulae.
Interpolation and Allied Tables (I.A.T.) 80 pages, 1956;tis a collection of interpolation tables and formulae of numerical analysis, with explanations and illustrations, designed as a working handbook for the computer.
Subtabulation, 54 pages, 1958, contains descriptions and tables for various methods of subtabulation, many of which .:Ire used in the compilation of the Ephemeris.
Seven-figure Trigonometrical Tablesfor every Second of Time, 101 pages, 1939, reprinted 1961.
Five-figure Tables of Natural Trigonometrical Functions (for every 10"), 123 pages, 1947, reprinted 1969.
Greenwich Observations. A complete list of the appendices and special investigations included in the annual volumes of Observations made at the Royal Observatory, Greenwich, and a list of the separate publications of the Observatory are given in the volume for 1946, published in 1955. In particular:
" Reduced observations of lunar occultations for the years 1943-1947 ", published in 1952, as an appendix to the Observations for 1939.
Royal Observatory Annals (R.O. Ann.). This series of publications includes: Number I," Nutation 1900-1959 ", 1961; values based on E. W. Woolard's series, see section 2C. There are also Royal Observatory Bulletins (R.O. Bull.).
Annals of Cape Observatory. This series includes many papers and much observational data that are also of relevance to the ephemerides.

3. American publications
The Ephemeris, U.S. Department of the Interior, Bureau of Land Management, 30 pages. For surveyors.
Improved Lunar E-phemeris, 1952-1959. A Joint Supplement to The American Ephemeris
and The (British) Nautical Almanac (I.L.E.), xiv + 422 pages, 1954. Extends the lunar
ephemeris in A.E. backwards to 1952, and includes a detailed account of the basic computation from Brown's theory. It also gives revised values of nutation and aberration for 1952-1959 and an account of their calculation.

*Reprinted 1962.
j- Reprinted 1972.

16

EXPLANATORY SUPPLEMENT

Tables of Sunrise, Sunset, and Twilight, Supplement to The American Ephemeris, 1946 (S.S. T.), 196 pages, 1945. Contains permanent and comprehensive tables of the times of sunrise, sunset and twilight for each degree of latitude to 75 0 ; variations are given by which times can be calculated simply for any year and any place.

Astronomical Papers prepared for the use of The American Ephemeris and Nautical Almanac (A.P.A.E.). Introduced in 1882, there are now sixteen volumes, almost every part of which is of direct interest to users of the Ephemeris. A full list of the contents follows:

Volume I.
I. Simon Newcomb. "On the recurrence of solar eclipses, with tables of eclipses from B.C. 700 to A.D. 2300". 1879.
II. Simon Newcomb, aided by John Meier. "A transformation of Hansen's lunar theory, compared with the theory of Delaunay". 1880.
III. Albert A. Michelson. "Experimental determination of the velocity of light made at the United States Naval Academy, Annapolis". 1880.
IV. Simon Newcomb. "Catalogue of 1098 standard clock and zodiacal stars". 1882. V. George W. Hill. "On Gauss's method of computing secular perturbations, with an application to the action of Venus on Mercury". I 88 I. VI. Simon Newcomb. "Discussion of observed transits of Mercury, 1677-1881". 1882.

Volume II.
I. Simon Newcomb and John Meier. "Formulae and tables for expressing corrections to the geocentric place of a planet in terms of symbolic corrections to the elements of the orbits of the Earth and planet". 1883.
II. Truman Henry Safford. "Investigation of corrections to the Greenwich planetary observations, from 1762 to 1830". 1883.
III. Simon Newcomb. "Measures of the velocity of light made under the direction of the Secretary of the Navy during the years 1880-1882". 1885.
IV. Albert A. Michelson. "Supplementary measures of the velocities of white and colored light in air, water, and carbon disulphide, made with the aid of the Bache fund of the National Academy of Sciences". 1885.
v. Simon Newcomb. "Discussion of observations of the transits of Venus in 1761 and 1769". 1890.
VI. Simon Newcomb. "Discussion of the north polar distances observed with the Greenwich and Washington transit circles, with a determination of the constant of nutation ~'. 1891.

Volume III.
I. Simon Newcomb. "Development of the perturbative function and its derivatives, in sines and cosines of multiples of the eccentric anomalies, and in powers of the eccentricities and inclinations". 1884.
II. George W. Hill. "Determination of the inequalities of the Moon's motion which are produced by the figure of the Earth". 1884.
III. Simon Newcomb. "On the motion of Hyperion". 1884.
IV. George W. Hill. "On certain lunar inequalities due to the action of Jupiter and discovered by Mr. E. Neison". 1885.
v. Simon Newcomb. "Periodic perturbations of the longitudes and radii vectores of the- four inner planets of the first order as to the masses". 1 89 I.

Volume IV. G. W. Hill. "A new theory of Jupiter and Saturn". 1890.

IF. INTRODUCTION

17

Volume V.

I. Simon Newcomb. "Development of the perturbative function in cosines of

multiples of the mean anomalies and of angles between the perihelia and common node and

in powers of the eccentricities and mutual inclination". 1895.

7l

II. Simon Newcomb. "Inequalities of long period, and of the second order as to the

rt

masses, in the mean longitudes of the four inner planets". 1895.

III. Simon Newcomb. "Theory of the inequalities in the motion of the Moon produced by the action of the planets". 1895.

s

IV. Simon Newcomb. "Secular variations of the orbits ofthe four inner planets".

v. Simon Newcomb. " On the mass of Jupiter and the orbit of Polyhymnia".

["
Volume VI. Tables of the four inner planets.

I. Simon Newcomb. "Tables of the motion of the Earth on its axis and around the Sun". 1895.
II. Simon Newcomb. "Tables of the heliocentric motion of Mercury". 1895. III. Simon Newcomb. " Tables of the heliocentric motion of Venus". 1895. IV. Simon Newcomb. " Tables of the heliocentric motion of Mars ". 1898.

Volume VII.
I. George William Hill. "Tables of Jupiter, constructed in accordance with the methods of Hansen". 1895.
II. George William Hill. "Tables of Saturn, constructed in accordance with the methods of Hansen". 1895.
III. Simon Newcomb. "Tables of the heliocentric motion of Uranus ". IV. Simon Newcomb. "Tables of the heliocentric motion of Neptune".

Volume VIII.
I. Simon Newcomb. "A new determination of the precessional constant with the resulting precessional motions". 1897.
II. Simon Newcomb. "Catalogue of fundamental stars for the epochs 1875 and 1900 reduced to an absolute system". 1899.
III. Henry B. Hedrick. "Catalogue of zodiacal stars for the epochs 1900 and 1920 reduced to an absolute system". 1905.

Volume IX.
I. Simon Newcomb. "Researches on the motion of the Moon. Part II". 1912.
II. Frank E. Ross. "New elements of Mars and tables for correcting the heliocentric positions derived from Astronomical Papers, Vol. VI, Part IV". 1917.
III. W. S. Eichelberger and Arthur Newton. "The orbit of Neptune's satellite and the pole of Neptune's equator ". 1926.

Volume X.
I. W. S. Eichelberger. "Positions and proper motions of 1504 standard stars for the equinox 1925'0". 1925.
II. James Robertson. "Catalog of 3539 zodiacal stars for the equinox 195°'°". 1940.

Volume XI. I. G. M. Clemence. "The motion of Mercury, 1765-1937 ". II. G. M. Clemence. " First-order theory of Mars". 1949.

1943·

18

EXPLANATORY SUPPLEMENT

III. H. R. Morgan. "Definitive positions and proper motions of primary reference stars for Pluto". 1950.
IV. Paul Herget, G. M. Clemence, and Hans G. Hertz. "Rectangular coordinates of Ceres, Pallas, Juno, Vesta, 1920-1960 ". 1950.

Volume XII.
W. J. Eckert, Dirk Brouwer, and G. M. Clemence. planets, 1653-2060". 1951.

"Coordinates of the five outer

Volume XIII.
I. A. J. J. van Woerkom. "The motion of Jupiter's fifth satellite, 1892-1949 ". 1950. II. Dirk Brouwer and A. J. J. van Woerkom. "The secular variations of the orbital elements of the principal planets". 1950.
III. H. R. Morgan. "Catalog of 5268 standard stars, 1950'0, based on the normal system N30". 1952.
IV. G. M. Clemence. " Coordinates of the center of mass of the Sun and the five outer planets, 1800-2060 ". 1953·
v. G. M. Clemence. "Perturbations of the five outer planets by the four inner ones". 1954.

Volume XIV. Paul Herget. "Solar coordinates 1800-2000". 1953.

Volwne XV.
I. Edgar W. Woolard. "Theory of the rotation of the Earth around its center of mass". 1953.
II. Hans G. Hertz. "The mass of Saturn and the motion of Jupiter 1884-1948 ". 1953·
III. Paul Herget. "Coordinates of Venus 1800-2000". 1955.

Volume XVI.
I. Raynor L. Duncombe. "The motion of Venus, 1750-1949". This list is continued on page 522.
4. Other publications

1958.

The following are international volumes published under the auspices of the International Astronomical Union.
Apparent Places of Fundamental Stars (A.P.F.S.), about xl + 500 pages, contains the
apparent places of the 1535 stars in FK3'!' It contains explanations in English, French, German, Russian, and Spanish. From its inception in 1941 until 1959 it was compiled by H.M. Nautical Almanac Office and published by H.M. Stationery Office, London. It is now compiled and issued by the Astronomisches Rechen-Institut in Heidelberg, and is published by Verlag G. Braun, Karl-Friedrich-Strasse 14, Karlsruhe, Germany.
Ephemerides of the Minor Planets (E.M.P.), about 170 pages, contains elements a~d search ephemerides of all known minor planets. A brief introduction in English is given, and a full translation of the Russian text is also available. It is now compiled by the Institute of Theoretical Astronomy, Leningrad, and is published by the Academy of Sciences of U.S.S.R. From 1898 to 1946 it was prepared by the Astronomisches Rechen-Institut, Berlin, and from 1947 to 1951 Doth by the Minor Planet Center, Cincinnati, and by the Institute of Theoretical Astronomy.
Notes on other publications and circulars giving current ephemerides of minor

* FK4 in A.P.F.S. 1964 onwards.

IG. INTRODUCTION

19

e

planets, comets, and satellites are given in the relevant sections of this Supplement.

The following tables may be used for approximate calculation of astronomical phenomena for dates in the past or future for which no fundamental ephemerides are available.

Schoch, K. Planeten-Tafeln fur Jedermann, Berlin-Pankow, Linser-Verlag G.m.b.H., 1927·
Ahnert, P. Astronomisch-chronologische Tafeln filr Sonne, Mond and Planeten, Leipzig, Barth, 1960.
Neugebauer, P. V. Astronomische Chronologie, 2 volumes, Berlin and Leipzig, Walter de Gruyter, 1929.
Baehr, U. Tafeln Zllr Behandlung chronologischer Probleme, Veroff. Astr. Rechen-Inst. zu Heidelberg, no. 3, 1955·
Neugebauer, P. V. Tafeln zur astronomischen Chronologie, 3 volumes, Leipzig, 19121925. Some of the tables in these volumes have been superseded or corrected by tables in the preceding two references.

5. A note on references

In addition to the abbreviations given above the following are used in this Supplement in references to astronomical journals and publications.

A.J. Ast. Nach. Bull. Astr. J.B.A.A. M.N.R.A.S. Mem. R.A.S. P.A.S.P. Trans. I.A. U.

The Astronomical Journal Astronomische Nachrichten Bulletin Astronomique, Paris Journal of the British Astronomical Association Monthly Notices of the Royal Astmnomical Society Memoirs oj the Royal Astronomical Society Publications of the Astronomical Society of the Pacific Transactions of the International Astronomical Union

G. SUMMARY OF NOTATIONS
In general, notations are defined and explained as they occur, and no attempt is made to adopt a consistent system throughout the Supplement. The adopted symbols may differ from those recommended by the International Astronomical Union (Trans. I.A. U., 6, 345, 1939), and may also differ in different sections.
Symbols are generally used to denote the physical quantities which they represent rather than the numerical expression of those quantities in some particular units. Thus the day numbers C, D are angular displacements which may be expressed in seconds of arc, in seconds of time, or in radians. Where it is desired to use a symbol for the numerical value, this is either specifically stated or the unit used is indicated after the symbol: for example, nS and n" are the numbers of seconds of time and arc in the annual general precession in declination n. Angles are otherwise expressed in radians, so that powers of small angles occurring in expansions do not require to be modified by powers of sin I", as is often done;

20

EXPLANATORY SUPPLEMENT

occasionally the square of a small angle, say 82, may be written as 8 sin 8 to emphasise this point.

The following summary refers to those symbols and notations that are used consistently throughout the Supplement.

I. Subscripts for reference systems

The reference system for equatorial or ecliptic coordinates is defined by the equinox and either the equator or the ecliptic; there are four such systems in general use. In many applications it suffices to specify the reference system in precise terms such as:

" referred to the mean equinox and equator (or ecliptic) of date"

and thereafter to use appropriate symbols without subscripts to denote the reference

system; this specification may be abbreviated in later references in the same

application to:

" for mean equinox of date".

1\

Where necessary to avoid confusion or circumlocution, or merely to assist interpre-

tation, the following subscripts are used consistently to indicate the reference

I

system to which the coordinates are referred. Positions may be geometric,

d

apparent, or astrometric according to the corrections applied for aberration, and

e

subscripts are adopted for all combinations of reference systems and positions

that are in use:

,
Position

Reference system

geometric apparent astrometric

s

Mean equinox of 195°00

S

R

Mean equinox of beginning of year

B

*

Mean equinox of date

M, C

True equinox of date

T

A

C is used as an alternative to M for ecliptic coordinates.
* No symbol is used for this combination, although it is implicitly used as an
intermediate step in the calculation of apparent places of stars.

2. Symbols for heliocentric and geocentric coordinates

Heliocentric:

spherical ecliptic rectangular equatorial rectangular ecliptic, geometric
for mean equinox of date

I, b, r x,y, z

} with appropriate subscripts

Xc, Yc, Zc

Geocentric:

spherical ecliptic spherical equatorial rectangular equatorial rectangular equatorial (Sun)

II, j3, Ll
a, 0, Ll
g, 'Y), ~
X, Y,Z

} with appropriate subscripts

IG. INTRODUCTION

21

aSlse

3. Precession and nutation

ifi = annualluni-solar precession in longitude

used

p = annual general precession in longitude

m = annual general precession in right ascension

n = annual general precession in declination

€ = obliquity of the ecliptic

[ t~e
sm

LJifi = (total) nutation in longitude difi = short-period terms of nutation in longitude

n In

LJ€ = (total) nutation in obliquity

d€ = short-period terms of nutation in obliquity

4. Fundamental epochs and measures of time

. Ephemeris time. The fundamental epoch to which the elements of the Sun,

Moon, and planets are referred is:

1900 January 0 at I2h ephemeris time

re-

= 1900 January 0'5 E.T. = J.E.D. 241 5020'0 E.T.

mce

Ephemeris time is measured conventionally in years, months, days, and sub-

tric,

divisions of a day. The interval T of ephemeris time from the fundamental

and

epoch contains:

IOns

T Julian centuries of 36525 days, each of 86400 ephemeris seconds;

d, or 10000 D, ephemeris days (d = 36525 T; D = 3.6525 T).

When desirable to emphasise that these relate to an interval of ephemeris time, a

ilC

subscript E is added thus: T E, dE' DE'

Universal time. The fundamental epoch which is used in the definition and

derivation of universal time is:

1900 January 0 at I2h universal time

= 1900 January 0'5 D.T. = J.D. 241 5020'0 D.T.

The interval Tv of universal time from this epoch contains:

>an

Tv Julian centuries of 36525 days, each of 86400 seconds of D.T.;
du, or 10000 Du, days of D.T. (dv = 36525 Tv; Du = 3.6525 Tv).

The subscript u is always used, unless the context makes it superfluous.

E.T. - U.T. At any instant the measure of ephemeris time (epoch + TE) is
equal to the measure of universal time (epoch + Tv) + LJ T; thus:
LJT = E.T. - D.T. = TE - Tu
L1 T is most conveniently expressed in seconds of time.

It must be emphasised that the fundamental epochs used for ephemeris time

and universal time, although denoted by the same measure, do not correspond to

the same instant of time; in fact at each epoch LJ To is about - 48, i.e. the epoch of

E.T. is 48 later than that of D.T. The interval of time between two instants, the

later one being indicated by a prime, can be expressed as:

or as:

T~ - TE of ephemeris time
T~ - Tv = (T~ - TE ) - (LJT' - LJT) of universal time

i

.~

--

22

EXPLANATORY SUPPLEMENT

The difference in the two measures involves the values of .1 T at both instants; It IS
only because the two fundamental epochs have the same measure that it is possible to write:

The Besselian solar year. For certain applications it is more convenient to measure time in units of tropical centuries of 36524'21988 ephemeris days, the fundamental epoch being the beginning of the Besselian (fictitious) solar year 19°0'0, or 1900 January Od·813 E.T. In the great majority of such cases the difference in length of the century is not significant: the same symbol T is accordingly used, though always with a specific explanation. The difference between the lengths of the Besselian solar year and the tropical year (os. 148 T) can always be neglected and multiples of ° ·01 in T thus relate to the beginning of the corresponding Besselian year (see section 2B).
The fraction of the tropical year is denoted by T, measured backwards or forwards from the beginning of the Besselian year; a unit difference in T corresponds to a difference of 0'01 in T.
An interval of time measured in tropical years is denoted by t. Initial and general epochs are denoted by to and t respectively. The context will indicate the meaning to be attached to to and t:
t - 1950'0 clearly implies that t is an epoch, e.g. 196o,°
1950'0 + t clearly indicates that t is an interval, e.g. 10'0 In some contexts the epoch to is used for that of 19°0'0 + To, and the epoch t for
that of 19°0'° + To + T; to = 100 To and t = 100 T are both intervals, but are
used conventionally to describe epochs.
Other notations jor time. T is also used to denote light-time in the application of corrections for aberration.
Special notations for time, defined as they occur, are used in the sections on eclipses and occultations, and in respect of some of the satellites. No attempt has been made to adhere to a single uniform notation throughout.

5. Day numbers and star-constants

A,B,C,D,E
j,g, G, h, H, i j ' ,gI, G' a, b, c, d a', b/, c' , d'
I
I'

Besselian day numbers Independent day numbers Independent day numbers (short-period terms) Star constants in right ascension Star constants in declination Second-order day number in right ascension Second-order day number in declination

For derivation and formulae see section 5C.

IG. INTRODUCTION

23

6. Figure of the Earth

<P = geographic, or geodetic, latitude-see special note in section 2F

<P' = geocentric latitude

tan <P' = (I - e2) tan <P

<PI = parametric latitude

tan <PI = (I - f) tan <P

e = ellipticity, or eccentricity, of the Earth's meridian

f = flattening

I - f = (I - e2)~

p = geocentric distance in units of the Earth's equatorial radius

S, C = auxiliary functions such that p sin 4>' = S sin 4> p cos 4>' = C cos cP = cos 4>1

For other relations and formulae see sections 2F and 9B.

2. COORDINATE AND REFERENCE SYSTEMS
A. COORDINATE SYSTEMS
The fundamental astronomical reference systems are based on the celestial equator, coplanar with the Earth's equator, and the ecliptic, the plane*of the Earth's orbit round the Sun. The angular coordinates in these planes are measured from the ascending node of the ecliptic on the equator, or the point at which the Sun in its annual apparent path round the Earth crosses the equator from south to north; and they are measured positively to the east, that is in the direction of the Sun's motion with respect to the stars. The ascending node of the ecliptic on the equator is referred to as " the vernal equinox", "the first point of Aries", or simply as " the equinox". The axes of the corresponding rectangular coordinate systems are right-handed, i.e. the x-axis is directed towards the equinox, the y-axis to a point 90° to the east, while the z-axis is positive to the north.
The position of a point in space may be specified astronomically by reference to a wide variety of coordinate systems; and it may be given by means of (among other less usual systems) either spherical coordinates, consisting of a direction and a distance, or rectangular coordinates, consisting of the projections of the distance on three rectangular axes. The systems are determined by the two following characteristics:
(a) Origin of coordinates-and designation. (i) The observer-topocentric. (ii) The centre of the Earth-geocentric.
(iii) The centre of the Sun-heliocentric. (iv) The centre of mass of the solar system-barycentric. (b) Reference planes and directions-and designation of spherical coordinates.
(i) The horizon and the local meridian-azimuth and altitude. (ii) The equator and the local meridian-hour angle and declination. (iii) The equator and the equinox-(equatorial) right ascension and declination. (iv) The ecliptic and the equinox-(ecliptic or celestial) longitude and
latitude. (v) The plane of an orbit and its equatorial or ecliptic node-orbital longitude
and latitude.
24
"'More strictly, the mean plane of the orbital motion, ignoring periodic perturbations.

2A. COORDINATE AND REFERENCE SYSTEMS

25

Barycentric coordinates are often referred to the centre of mass of the Sun and the inner planets, and less often to other combinations. The equator, the ecliptic, and the equinox are constantly in motion due to the effects of precession and nutation, and must be further specified; this is done in sub-sections Band C. A notation to distinguish the· various systems in current use is introduced in section IG.

The reduction from geocentric to topocentric coordinates depends on the figure of the Earth, and is considered in detail in sub-section F. In most cases of astronomical interest, the differences are so small that they can be applied as firstorder differential corrections.

Positions may be of several kinds, including: the geometric position derived from the actual position at the time of observation; the apparent position in which an observer, situated at the origin of coordinates, would theoretically see the object; and the astrometn"c position, in which corrections have been made for some small terms of aberration in order that it may be directly comparable with the tabulated catalogue positions of stars. The apparent position is derived from the geometric position by the application of corrections for aberration, and where relevant for refraction. However, refraction is dependent on the observer's local reference system and is invariably treated as a correction to the observation rather than to the ephemeris position; exceptions only occur for phenomena that are essentially topocentric, such as rising and setting and (in principle, though the correction is neglected in practice) for eclipses and occultations. For geocentric coordinates the apparent position is the direction in which an observer at the centre of the Earth would see the object, and refraction does not enter. Aberration is dealt with in sub-section D and refraction briefly in sub-section E.

In the present sub-section the effects of precession, nutation, aberration, refraction, and parallax are ignored in order to present the relationships between the coordinate systems. The general notation used is restricted to this purpose and should not be confused with the more detailed notation in section I G necessary to distinguish between the different kinds of position.

Not all combinations of (a) and (b) occur and many are not used in the Ephemeris; (a) (iv), in particular, is therefore not referred to again. Moreover, if corrections for parallax be deferred, there is no difference between (a) (i) and (a) (ii), which can be treated together.

For geocentric spherical coordinates there are thus the four practical reference systems of:

(i) azimuth (A) measured from the north through east in the plane of the horizon, and altitude (a) measured perpendicular to the horizon; in astronomy the zenith distance (z = 90° - a) is more generally used, but the altitude is retained in the formulae for reasons of symmetry;

(ii) hour angle (h) measured westwards in the plane of the equator from the meridian, and declination (8) measured perpendicular to the equator, positive to the north;

26

EXPLANATORY SUPPLEMENT

(iii) right ascension (a) measured from the equinox eastwards in the plane of the equator, and declination (8);

(iv) longitude (,\) measured from the equinox eastwards in the plane of the ecliptic, and latitude (fJ) measured perpendicular to the ecliptic, positive to the north.

The formulae connecting these coordinates are:

Azimuth/altitude

Hour angle/declination

cos a sin A
~a~A ~a

-cos 8 sin h
~8~~-~8~h~~ ~8~~+~8~h~~

cos 8 sin h -cos a sin A

~8~h ~8

~a~~-~a~A~~ ~a~~+~a~A~~

where ~ is the latitude of the observer. Note that the conversion corresponds to a simple rotation of the frame of reference through an angle 90° - ~ in the plane of

the meridian.

Hour angle/declination

Right ascension/declination

The two systems are identical apart from the origin, and direction, of measurement of hour angle and right ascension, which are connected by the relation:

h = local sidereal time - a

since local sidereal time is the hour angle of the equinox.

Right ascension/declination

Longitude/latitude

cos 8 cos a = cos fJ cos ,\

cos 8 sin a = cos fJ sin ,\ cos € - sin fJ sin €

sin 8

= cos fJ sin ,\ sin € + sin fJ cos €

cos fJ cos ,\

cos 8 cos a

cos fJ sin ,\ = cos 8 sin a cos € + sin 8 sin €

sin fJ

-cos 8 sin a sin € + sin 8 cos €

where € is the obliquity of the ecliptic (corresponding to the particular equator and ecliptic used). Geocentric longitude and latitude are used now only for the Sun and Moon. Note that the conversions correspond to a simple rotation round the x-axis through an angle €.

The corresponding equatorial rectangular coordinates and distance are denoted
by X, Y, Z, and R for the Sun and by g, TJ, " and Ll for the planets; they are
derived from the spherical coordinates by the formulae:
X/R or giLl = cos 8 cos a Y/R or TJ/Ll = cos 8 sin a Z/R or '/Ll = sin 8

Geocentric ecliptic rectangular coordinates are rarely (if ever) used.

For heliocentric coordinates there are only the two practical reference systemsthe equatorial and the ecliptic; and in the equatorial system only rectangular coordinates are used. The relationships between the ecliptic rectangular

-

1"-"'..

2A. COORDINATE AND REFERENCE SYSTEMS

27

coordinates (xc, Yc, zc), the ecliptic longitude, latitude, and distance (I, h, r), and the equatorial rectangular coordinates (x, y, z) are:

Xc = r cos b cos I Yc = r cos b sin I Zc = r sin b

x
+y cos € + z sin € .- y sin € + z cos €

X = Xc

= r (cos b cos I)

Y =ycCOs€ - zc sin € = r(cosbsinlcos€ - sinbsin€)

z = Yc sm € + Zc cos € = r (cos b sin I sin € + sin b cos €)

The conversion from heliocentric to geocentric coordinates is performed in terms

of equatorial rectangular coordinates through:

t=x+X

7)=Y+Y t=z+Z

where X, Y, Z are the geocentric coordinates of the Sun.

The calculation of the sph~rical coordinates from the rectangular coordinates,

or from the known direction cosines, typified by:

t Ll cos 0 cos a =

Ll cos S sin a = 7)

Ll sin 8

=t

is performed by:

tan a = 7)lt Ll = (g2 + 7)2 + '2)~

cot a = g/7)
sin 0 = tiLl

t The quadrant of a is determined by the signs of and 7), and that of 0 by the sign

of ~; LI and Ll cos 0 are always positive. The formulae for a and 0 may be written:

a = tan-l 7)lt
= cot-1g/7)
o = sin-ItlLl

or arctan 7)lt or arccot tl7)
or arcsin tiLl

provided that the appropriate values, and not necessarily the principal values, of

the multi-valued functions are taken.

Notes on the technique of practical calculation using these formulae, and on the most suitable trigonometric tables to use, are given in section I6A.

Many of the conversions above correspond to a simple rotation of the frame of

reference about one of its axes. These are special cases of the general conversion

from a set of axes designated by x, y, z to a set designated by x', y', z'; the two

systems are connected by the formulae:

x = II x' + 12 y' + 13 z'
y = mi x' + m2 y' + m3 z' z = ni x' + n2 y' + n3 z'

x' = II X + mi Y + ni z
y' = 12 X + m 2 Y + n2 z z' = l3 X + m3 Y + n3 z

where II, mI, nI ; 12, m2, n2 ; 13, m3, n3 are the direction cosines of x', y', z' referred to

the system x, y, z. The direction cosines satisfy the relations typified by:

Ii + mi + ni = I + + 12 l3 m 2 m 3 n2 n3 = 0

Ii + l~ + l~ = I mi ni + m2 n2 + m3 n3 = 0

EXPLANATORY SUPPLEMENT
These mne quantities can be expressed in terms of the Eulerian angles
e, ep, If by:
11 + cos ep cos () cos If - sin ep sin If 12 - cos ep cos () sin If - sin ep cos If
la + cos ep sin ()
m1 + sin ep cos () cos If + cos ep sin If m2 - sin ep cos () sin If + cos ep cos If
ma + sin ep sin ()
n1 - sin () cos If n2 + sin () sin If
na +cos ()
In this case the conversion corresponds to a rotation ep about the z-axis, () about
the new position of the y-axis, and If about the new (and final) position of the
z-axis. The transformation is equivalent to a single rotation about some line not in general coincident with one of the axes; but such single rotations are not frequently encountered in astronomical practice.
B. PRECESSION
The equator and the ecliptic, and hence the equinox, are continuously in motion. The motion of the equator, or of the celestial pole, is due to the gravitational action of the Sun and Moon on the equatorial bulge of the Earth: it consists of two components, one luni-solar precession being the smooth long-period motion of the mean pole of the equator round the pole of the ecliptic in a period of about 26,000 years, and the other nutation being a relatively short-period motion that carries the actual (or true) pole round the mean pole in a somewhat irregular curve, of amplitude about 9" and main period 18·6 years. The motion of the ecliptic, that is of the mean plane of the Earth's orbit, is due to the gravitational action of the planets on the Earth as a whole and consists of a slow rotation of the ecliptic about a slowly-moving diameter, the ascending node of the instantaneous position of the ecliptic on the immediately preceding position being in longitude about 1740 ; this motion is known as planetary precession and gives a precession of the equinox of about 12" a century and a decrease of the obliquity of the ecliptic of about 47" a century.
In this sub-section the effects of the motions of only the mean poles of the equator and ecliptic, known as general precession, are considered; the effect of nutation is dealt with separately in sub-section C. The treatment is restricted to the development of formulae for the practical application of corrections to coordinates and orbital elements.

2B. COORDINATE AND REFERENCE SYSTEMS

29

Rigorous formulae

The effect of precession on the coordinates of a fixed object is illustrated in figure 2.1, in which the position of a star S is referred at an initial time to to a system of equatorial axes defined by the mean pole of the equator Po and the mean equinox Xo; at this initial epoch the pole of the ecliptic is at Co' P, X, and C are the respective positions of these points at a subsequent time t. Although at any instant P moves, owing to luni-solar precession, in a direction perpendicular to the colure CP, i.e. towards X, the arc PoP is not perpendicular either to CoP0 or to CP; owing to planetary precession C is itself in motion along a curve which is always convex to CP. This complex motion is specified by means of the three angles
'0' z, 8 (where 90° - {o is the right ascension of the ascending node of the equator
of epoch t on the equator of to reckoned from the equinox of to, 90° + z is the right
ascension of the node reckoned from the equinox of t, and 8 is the inclination of the equator of t to the equator of to) together with the corresponding values of the obliquity of the ecliptic E.

Figure 2.1. Precession-polar diagram

In the figure:
'0 = xoPoP
z = 90° - PoPC

{ = 90° - PoPCo
{- CoPC = { - ,\

8 = PoP The great circle PoP is not the actual path taken by the moving pole.

PoCoP = luni-solar precession in longitude in the interval t - to

CoC = planetary precession in longitude in the interval t - to

A = CoPC = planetary precession on the equator in the interval t to

EO = CoP0 = obliquity of the ecliptic at to

E = CP = obliquity of the ecliptic at t

E1 = CoP

z Figure 2.1 has been drawn for an epoch t for which ,\ is negative, i.e. for which

is greater than {.

3°

EXPLANATORY SUPPLEMENT

Positions referred to the reference system specified by the mean pole P, the

mean equinox X, and the pole of the ecliptic C at time t are designated formally as being referred to "the mean equinox and equator (or ecliptic) of epoch t".

Where no confusion can be caused this is abbreviated to " mean equinox of epoch

t". In practice three reference systems are used: the mean equinox of 1950'0

(occasionally referred to as the" standard" equinox), the mean equinox of the

beginning of the Besselian year, and the mean equinox of date (i.e. the epoch of the

reference system is the same as the date and time for which the position is given).

Where necessary quantities referred to these systems are distinguished by subscripts

s or R, B, M or C respectively. (See section I G).

The beginning of the Besselian (fictitious) solar year is the instant when the right ascension of the fictitious mean sun, affected by aberration and measured from the mean equinox, is 18h 40m, This instant always occurs near the beginning of the calendar year and is denoted by the notation '0 after the year; for example, as given in A.E., page 2, the beginning of the Besselian solar year 1960 is January
I d'345 E.T. = 196o,°. Because of the excess of the secular acceleration of the
right ascension of the fictitious mean sun over the mean longitude of the Sun (see section 3B) the Besselian year is shorter than the tropical year by the amount oB'148T, where T denotes the time in centuries after 1900. However, it is usual to ignore this insignificant difference and to regard the length of the Besselian solar year as the same as that of the tropical year.

Newcomb (see references below) gives constants, based partly on theoretical considerations but mainly on observation, from which the following numerical expressions for ~o, z, 8 can be deduced. These depend only to a small extent on the initial epoch.

Initial epoch, to: 19°0'0 + To

Final epoch, t: 19°0'0 + To + T

~o = (23 04"'250 + 1"'396 To)T + 0"'302 T2 + 0"'018 T3

z = ~o + 0"'791 T2

8 = (2004"·682 - 0"·853 To)T - 0"'426 T2 - 0"'°42 T3

where To, T are measured in tropical centuries; the small secular changes in the coefficients of T2 are here ignored.
The series given are for the conversion from the mean equinox of the initial epoch to an epoch T centuries later; it can be verified that ~o, Z, 8, for initial epoch to and interval T, are identically equal to - z, - ~o, - 8, respectively, for epoch
to + T and interval - T. When values are tabulated for reduction from the mean
equinox of to to that of t, the same values can therefore be used for reduction from the mean equinox of t to that of to by replacing ~o, z, 8 (for to) by - z, - ~o, - 8.
Values for the reduction from the mean equinox of the beginning of the current year to the standard equinox of 1950'0 are given in A.E., page 50, and for a selection of years in the three volumes of Planetary Co-ordinates; the reduction from the standard equinox of 1950'0 can be obtained by the simple substitution mentioned above.

2B. COORDINATE AND REFERENCE SYSTEMS

31

'0' Values of z, 0 and related precessional elements M, N are given in table 2.1

for years 1900 to 1980 at intervals of one year. The main tabulation is given for

reduction to the initial epoch 1950'0 but appropriate formulae are given with the

table so that it may be used for reductions between any two epochs. Values for

reduction from the mean equinox of selected years back to 1755 to that of the current year are given in A.E., Table III.

Rigorous formulae for the reduction of positions from one epoch to another are easily deduced from figure 2. I; in triangle PoPS:

PoS=900 - 00 PS = 90" - 8

PPoS=ao+'o

PPo=O

PoPS = 180° - (a - z)

where ao, 80 and n, 8 are right ascension and declination for the initial and final epochs respectively. Then a, 0 are given by:

cos 0 sin (a - z) = cos 00 sin (ao + '0)

cos 0 cos (a - z) = cosO cos 00 cos (ao + '0) - sin 0 sin 00

sin 0

=, cos 0 sin 00 + sin 0 cos 00 cos (ao + '0)

The rigorous formulae for a, 0 may be written in the form:

where

tan (a -

ao -

r)
':>0 - Z

=

I

q sin (ao + - q cos (ao

'0)
+ '0)

io q = sin 0 { tan 00 + tan cos (ao + 'o)}

io tan i (0 - 00) = tan {cos (ao + '0) - sin (ao + '0) tan i (a - ao - '0 - z)}
'0' which permits expansion in terms of the small quantities z, 0, and thus in a series

in the interval T, the coefficients being functions of ao, 00 only. These coefficients

have been tabulated, for various epochs and adopted precessional constants, with

arguments ao and 0o; they are given explicitly in fundamental star catalogues,

where however they generally include the effect of proper motion.

The equatorial rectangular axes defined by the positions of the poles of the

'0 equator and ecliptic (P, C) at the final epoch can be derived from those (defined by
Po, Co) at the initial epoch by rotations of: - about the zo-axis (Po); 0 about the

'0' y-axis; and - z about the z-axis (P). The direction cosines of one set' of axes
referred to the other may be expressed in terms of z, 0 (see sub-section A); m

particular the direction cosines of the initial axes referred to the final axes are:
'0 '0 X", = cos cos 0 cos z - sin sin z '0 '0 Y", -sin cos 0 cos z - cos sin z

Z", Xv Yv

-sin 0 cos z
'0 '0 cos cos 0 sin z + sin '0 '0 -sin cos 0 sin z + cos

cos % cos z

Z v = - sin 0 sin z
'0 X z = cos sin 0 '0 Y z = -sin sin 0

Zz = cos 0

where, for example, Y", is the direction cosine of the initial y-axis referred to the final x-axis.

32

2.1-EQUATORIAL PRECESSIONAL ELEMENTS

FOR REDUCTION TO 1950·0 OR OTHER EPOCHS

Date to
II)00-O I 1)0I I902 I903 I 1)04
I1)05'0 I 906 I 1)07 II)08
I909
I9IO' 0 I9H I9I2 I9I 3 I9I 4
I9I 5'0 I9I6 I9I7 I9I8 I9I 9
I920'0 I92I I922 I923 I924
I925'0 I926 I927 I928 I929
I930.0 I93I I932 I933 I934
I935-0 I936 I937 I938 I939
I940-0 I94I I 942 I 943 I 944
I945'O 1:946 1:947 1:948
1:949. 0

~o
s
+76 .81 4 75-278
73'742 72'206 70·670
+69- 134 67'598 66'062 64'526 62'99°
+61'454 59-9 18 58 '382 56 .845
55'3°9
+ 53'773 52'237 5°'7°1 49'164 47·628
+46'°92 44'556 43'°20 41'484 39'948
+ 38'4 11 36 .875
35'339 33·802 32'266
+3°-73° 29'193 27·657 26'121 24'584
+23-°48 21'5 11
19'975 18'439 16'902
+ 15'366 13·829 12'293 1°'756 9'220
+ 7.683 6'146 4·610
3'°73
+ 1'537

z
+ 7 6 •-82 7
75'291 73'754 72'218 70 -681
+69'145 67. 608 66-072
64'535 62'999
+61'462
59'926 58 '3 89 56 .853 55'3 16 +53'780 52'243
5°'7°7 49'170 47'634
+46 '097 44'5 61 43'°24 4 1-488 39'95 1
+38'415 36 ,878 35-341 33. 8°5 32'268
+3°'732 29- 195 27-659 26'122 24-5 86
+23'°49 21'512 19'976 18-439 16-903
+ 15-366 13. 83° 12'293 10-756 9'220
+ 7, 683 6- 146 4.610
3'°73
+ 1'537

sin 8 cos 8 - I Unit = 10-8

+48 5892
47 61 74 466455
45 6737 44 7018

-1180
1133 1088 1043
999

+4373°0

95 6

42 75 81

9 14

41 7862

873

4081 44

833

398426

794

+388707

756

378989

7 18

36 9270

682

35 955 2

646

34 9834

612

+34°116

579

33°397

545

320679

5 13

31 0961

483

30 1243

453

+29 1525

4 24

28 1807

397

27 2089

37°

26 2371

344

25 2653

3 19

+24 2935

295

23 3217

27 2

223499

25°

21 3781

229

20 4064

208

+ 194346

189

18 4628

17 1

17 49 11

153

16 5193

137

15 5476

121

+ 145758

106

13 604 1

93

12 6323

80

II 6606

68

106888

57

+ 9 7171

47

87454

38

77737

3°

68019

23

58302

[7

+ 4 8585

12

3 8868

8

29 151

4

I 9434

2

+ ° 9717

°

M
+ 153-•640
15°'568 147'495 144'423 141'35°

N=8
+66-•81 5
65'479 64- 142 62·806 61'469

N =8
.
+ 1002-23
982'18
962'14 942'°9 922'°4

+ 138'278 135'206
132'133 129'°61 125'988

+60'133
58 -797 57'460 56 '124 54'787

+ 9°2'00 881'95 861'91 841.86 821·81

+ 122'916 119. 843 116'771 113.698 110·626

+53'45 1 52 '115 5°'778 49'442 48-105

+ 801'77 781'72 761·68 74 1.63 721 '59

+ 1°7'553 +46'769 + 7°1'54

104'480 45'433

681'49

101'4°8 98 -335 95'263

44'°97 42-760
4 1'424

661'45
641'4° 621'36

+ 92'19° 89'117 86- 04 4 82-97 1
79·899

+40'088
38'75 1 37'4 15 36 -079 34'742

+ 601'3 1 581'27 561'22 541'18 521 '13

+ 76.826
73'753 7°·680 67- 607
64'534

+33'4°6 32'070
3°'733 29'397 28-061

+ 5°1'°9
481'°5 461 -00 44°-96 42°'9 1

+ 61'462 58 -3 89 55-3 16
52'243 49- 170

+26'725 25'3 88 24'°5 2 22'7 16
21'379

+ 400 .87 380 .8 2 360 -78
34°'74 320 -69

+ 46'°97 43- 024 39-95 1 36 -878
33 ·805

+2°'°43 18'707 17'371
16'°34 14. 6 98

+ 3°0 .65 280·60
26°-56 24°-52 22°'47

+ 3°'732 + 13'362 + 200'43

27.659 12-026

18°'39

24'586 10·69°

160'34

21'512 18'439

9'353 8'017

14°'3° 120'26

+ 15'366 + 6,681 + 100'21

12'293 9'220 6'146

5'345
4'°°9 2·672

80'17 60'13
4°'°9

+ 3'°73 + 1'336 + 20'°4

_._----.,-

2.1-EQUATORIAL PRECESSIONAL ELEMENTS FOR REDUCTION TO 1950·0 OR OTHER EPOCHS

Date
to
195°.0 1951 1952 1953 1954
1955'° 1956 1957 1958 1959
1960,° 1961 1962 1963 1964
1965. 0 11)66
1967 1968 1969
197°'0 1971 1972 1973 1974
1975'° 1976 1977 1978 1979 1980.0

~o
0'000 1'537 3'°73 4. 610 6'147
7. 683 9'220 10'757 12'294 13. 83°
-15'367 16'9°4 18'441 19'977 21' 514
-23'°5 1 24'5 88 26· 125 27. 662
29'199
-3°'736 32'273 33. 809 35'346 36 .883
-38'420
39'957 4 1'494 43'°3 1 44'568 - 46 '106

z
0·000 1'537 3'°73 4. 610 6'147
7. 6 83 9'220 1°'757 12'293 13.83° - 15'367 16'9°3 18'44° 19'977 21'5 13
-23'°5° 24'587 26' 123 27·660 29'197
-3°'733 32 '27° 33·8°7 35'344 36 .880
- 3 8 ' 4 17 39'954 4 1'49 1 43'°27 44'5 64
.-46 '101

sin 8 Unit
° 97 17 I 9434 2915 I 3 8867
48584 5 8301 68017
77734 87450 97 167 10 6883 I I 6600 12 6316 13 6032
- 145749 15 5465 165181 174897 18 4613
- 19433° 20 4046 21 3762 22 3477 23 3 193
-24 2909 25 2625 26 2341 27 2056 28 1772
-29 1488

cos 8 - I 10-8
° ° 2
4 8
12 17 23
3° 38
47 57 68 80
93 106 121 137 153 17 1
189 208 229 25° 27 2
295 319 344 37° 397 4 24

M

N =8

0'000
3'°73 6'147 9'220 12'293
15'367 18'44° 21'5 13 24'587 27·660
3°'734 33.8°7 36 .881
39'954 43'°28
46. 101
49'175 52'248 55'322 58 '395
61'469 64'543 67.616 7°.690 73'764 76 .837 79'91 I 82'985 86'°59 89' 133
92'206

0'000 1'336 2.672
4'°°8 5'345
6·681 8'017
9'353 10·689 12'025
- 13'361 14.698
16'°34 17'37° 18'7°6
-20'°42 21'378 22'7 14
24'°5° 25'386
-26'722 28'°58 29'394
3°'73° 32'066
-33'402 34'738 36 '°74 37'410 38 '746
-4°'°82

33
N =9
0·00 20'°4 4°'08 60'13 80'17
100'21 120'25 14°'3° 160'34 18°'38
2°°'42 22°'46 24°'5° 26°'55 28~.. 59 3°0 .63 320 .67 34°'7 1 36°'75 38°'79 4°°.83 4 20 . 87 44°'92 46°'96 481 '°0
5°1'°4 521 '08 541' 12 561' 16 581 '20
601'24

These values are for the reduction from the epoch to, in the left-hand argument colUIIUl, to the epoch 1950'0. For reduction from 1950'0 to to enter the table with to as argument, reverse the signs of all respondents except cos 8 - I, and interchange ~o and z.
For reduction from the epoch to + L1t to 1950'0 + L1t, and vice versa, take out values
from the table using argument to, and multiply:
~o, z, M by (I + 0'0000 06 L1t)
and
N, 8, sin 8 by (I - 0'000004 L1t).
Over the range of the table tan t 8 can be taken as t sin 8.

Formulae for the reduction of equatorial spherical coordinates include:

a - ao = M + N sin t (a + ao) tan t (8 + 80)

8 - 80 =

N cos t (a + ao)

where no, 80 are for epoch to, and a, 8 are for epoch 1950'0.

34

EXPLANATORY SUPPLEMENT

Rectangular coordinates x, y, z referred to the final epoch t can thus be
expressed in terms of rectangular coordinates xo, Yo, Zo referred to the initial epoch to by:
x = X,.xo + Y",Yo + Z",zo + + Y = X lI X O Y ll yo ZlI Z O Z = Xzxo + Yzyo + Zzzo

These formulae are precisely equivalent to those connecting the spherical coordin-

ates above.

In systematic computation these and similar formulae are often modified so as to give the reductions (x - xo), (y - Yo), (z - zo) to be applied to the known xo, Yo, Zo to give x, y, z; e.g. the first formula is written as:
x = X o + (X", - I)Xo + Y",yo + Z",zo
F or reduction from the epoch t to the epoch to:
Xo = X"'x + Xyy + Xzz = + X"'X - Y",y Z",z Yo = Y",x + Yyy + Yzz = - XyX + Y ll y + Zyz Zo = Z", x + ZyY + Zzz = - Xzx + Yzy + Zzz
in which the first set of formulae is rigorous, but the second (which has been
largely used) depends on the approximate equality of X y and - Y",; X. and - Z"'; and Y z and ZlI; the approximation is so good that the numerical values are identical.
For reduction from the mean equinox of 1950'0 + To to that of 1950'0 + To + 1':

X. - I = - (29696 + 26 To) T" - 13 ra Y. = -X. = - (2234941 + 1355 To) T - 676 T2 + 221 ra Z. = -X. = - (97 1690 - 414 To) T + 207 T" + 96 T3
Y. - I = - (24975 + 30 To) T" - 15 ra Y. = Z. = - (I 0858 + 2 To) T2
Z. - 1 = - (4721 - 4 To) T"

where To, l' are measured in tropical centuries, and the coefficients on the righthand side are in units of the eighth decimal. Numerical values for reduction from (and to) the equinox of 1950'0 to (and from) the mean equinox of the beginning of the year are given at various intervals for the years 1800 to 1980 in the three volumes of Planetary Co-ordinates.

For reduction from 1950'0 to the mean equinox of date it is more convenient to have these expressions in terms of days (d) measured from some convenient zero near the epoch of the standard mean equinox of 1950'0; this is chosen to be J.D. 243 3°°°'5 (1949 March 25'0), so that 1950'0 corresponds to d = 281'923.

X, - 1 = -

2 + 125'5 D - 2226'0 D2 - 0'3 D3

Yz = -X. = + 17251 - 61 1903.6 D - 51'1 D2 + 4'5 D3 Zz = -X. = + 7500 - 26 6040·8 D + 15'3 D2 + 2'0 D3

Y. - 1

I + 105.6 D - 1872'1 D2 - 0'3 D3

Y. = Z. =

1+

45'9 D - 814'0 D2

Z. - 1 =

+

20'0 D - 353'9 D2

where D = d/IOOOO, and the coefficients on the right-hand side are in units of the
eighth decimal. Numerical values, calculated from the original expressions, are given in table 2.2 with argument Julian date at intervals of 1000 days.

2B. COORDINATE AND REFERENCE SYSTEMS

3S

Annual motions and approximate formulae
The precessional motions during a short interval of time (of the order of a year) are small, and in many cases it is adequate to use first-order corrections equal to the rates of change multiplied by the interval. In figure 2.2 the effect of precession on the mean equator, ecliptic, and mean equinox is illustrated diagrammatically. Qo, Eo, X o are the equator, ecliptic, and equinox respectively at the initial epoch to, and Q, E, X at an epoch t taken to be one tropical year later; the interval is taken as being sufficiently small for the actual displacements to be regarded as annual rates of change.

Eo

,-0 ~

E

00

_toM

x,

o

Figure 2.2. Precession-equatorial diagram

The two equators intersect at M, about 90° from X o, and the two ecliptics intersect at N, about 174° from Xo; M, N are the axes about which the equator and ecliptic rotate. Xl is the intersection of the equator Q of epoch t with the

ecliptic Eo of epoch to' Then:

II = XoN

= longitude of the axis of rotation of the ecliptic,

i.e. of the ascending node of the instantaneous

position of the ecliptic on the immediately

preceding position; it is referred to the mean

equinox of date

7T = XoNX ~o = 90° - XoM

= annual rate of rotation of the ecliptic
z XM - 90°

EO = EoXoQo
E = EXQ

obliquity of the ecliptic at epoch to obliquity of the ecliptic at epoch t

El = EOXIQ

if/ = XOXI

= annual luni-solar precession

A' = XIX

= annual planetary precession on the equator

P = XN - XoN (orXA) = annual general precession in longitude

= t/J' - A' cos EI

m = XM - XoM (or BXo) = annual general precession in right ascension

= t/J' cos EI - A'

n = XoMX (or BX)

= annual general precession in declination
= rate of change of e = t/J' sin EI

The quantity here denoted by t/J' is denoted by t/J in the Ephemeris, and in figure 2.2.

36

2.2-EQUATORIAL PRECESSIONAL ELEMENTS

FOR REDUCTION OF EQUATORIAL RECTANGULAR COORDINATES FROM (AND TO) THE MEAN EQUINOX OF 1950·0

Julian Date
t

X. - I

241 5000 -5 6000
7000 8000
241 f)OOO

-7438 6647 5900 5 198 454°

242 0000-5 1000 2000
3000 4000

-3926
3357 28 33 2353 19 18

242 5000 -5 6000 7000 8000
2421)000

- 1527 118o
878 621
40 8

243 0000 -5

24°

1000

ll6

2000

37

3000

2

4000

II

2435000 '5

66

6000

164

7000

308

8000

495

2439000'

728

244 0000'5 1000 2000
3000 4000

- 1°°5 1326 1692 2102
2557

244 5000 -5 - 3057

Mean o~ + o~ -89

Y. = - X. Z. = - X. Y. - I
In units of the eighth decimal

+ II I 8485
105 7317 99 61 47 934976 873803
+ 81 2629
75 1454 69 0 277 62 9099 56 79 19
+ 506739
44 5557 38 4374 323 189 262004
+ 2008 17
13 9630
78441 + I 7251
43940

+48 6412 45 9804 43 3 197 406589 379982
+35 3375 326768 300161
273554 24 6948
+22 °342 19 3736 16 7130
14 °524 II 3919
+ 8 7314
60709 341°5 + 75°0 I 91°4

-6255
559°
4962
437 1 3818
-3302 2823 23 8 2 1979 161 3
- 1284 993 739 52 2 343
202
97 31
I
10

1°51 32

45707

55

166324

7 2 310

13 8

2275 18

9 8913

259

28 8713

12 5516

417

349908

IS 21I8

612

41 1I05
4723°2 5335°0 594699 65 5899

- 17 8720
20 53 22 23 1923 25 8523 28 5123

845 II IS 1423 1768 2 151

71 7099

-3 1 1723

-2

+1

Z. = Y.
-2720
243 1 2 158 19°1 1660
-1436 1228 1036 860 7°1
55 8 43 2 321 227 149
88 42 13
I
4
24 60 1I2 181 266
367 485 61 9 769 935
-lll8

Z. - I
- 1I83 1°57 938 827 722
624 534 45 1 374 3°5
243 188 14° 99
65
38 18 6
°
2
10 26 49 79 1I6
160 21I 2 69 334 4°7
486

For interpolation to full eight-decimal precision in this table, second differences must be taken into account; they are sensibly constant over the range of tabulation, and mean values of the double second difference are given at the foot of each column for use with the interpolation formula:
f. = (I - p) fo + pfl + B 2 (o~ + oD
If Xo, Yo, %0 are the equatorial coordinates for the mean equinox of 1950'0 and x, y, Z are for the mean equinox of date (t), then the formulae for reduction are:

From 195°-0 to t

x = Xo + (X. - I)Xo +

Y.Yo + Z.zo

Y =Yo +

X-Xo +(Y.-I)Yo + Z.zo

z=zo+

X.xo+

Y.Yo+(Z.-I)Zo

From t to 1950·0

Xo = x + (X.-I)X + X.y + X.z

yo=y+
Zo = Z +

Y.x+(Y.-I)Y+Y.Z

Z.x +

Z.y + (Z.-I)Z

where (X.- I), Y.. ... are obtained by interpolation to the argument t.

2.3-ECLIPTIC PRECESSIONAL ELEMENTS

37

FOR REDUCTION TO (AND FROM) THE MEAN EQUINOX OF 1950,0

Julian Date
to
241 5000'5 6000 7000 8000
241 9000

E
. 23° 26' + 68'28 67'00 65'72 64'44 63'15

2420000'5 1000 2000
3000 4000

61·87
6°'59 59'3 1 58 '°2 56 '74

242 5000 '5 6000
7000 8000 242 9000

55'46 54'18 52·89 5 1.61
5°'33

243 0000 '5 1000
2000
3000 4000

49'°5 47'76 46'48
45'20 43'92

2435000 '5 6000
7000 8000
2439000

42·63 4 1'35 4°'°7 38 '7 8 37'5°

244 0000'5 1000
2000
3000 4000

36 '22
34'94 33·65
32'37 31'°9

244 5000 '5- 29·81

p
.
50 '25 64 257° 2576 25 82 25 88

1T
°'4711 47 II 47 II 4710 4710

n
173 57'0 58'5
174°°'° 01'5 °3'°

a
,.
+41 55.82 3938 '23 3720·63
35 03'02 32 45'42

b

c

c'

+23'57 22'28
2°'99 19'7° 18'41

o

,

6 10'2

08'3 06'4

°4'5 02·6

o

,

528'3

28'7

29. 1

29'S

29'9

50 -2594 2600
2606
261 3 26 19

°'47 10 4710 4710
47°9 47°9

174°4'5 06,0
°7'5 °9'0 1°'5

+ 30 27.82 28 10'21 25 52·60
23 34'99 21 17'38

+17'12 15.83
14'54 13'25 I1'97

60°'7 55 8 .8
57'° 55'1 53'2

53°'3
3°'7 3 1 '1 3 1 'S 31'9

5°'2625 2631
2637 2643 2649

0'47°9
47°9
47°9 4709 4708

174 12'0
13'5
15'° 16'5 18'0

+ 18 59'77 16 42'15 14 24'53 1206'92
949'3°

+10·68
9'39 8'10 6·81
5'52

5 51'3 49'4 47'S 45.6
43'7

532'3 32'7 33'1 33'5 33'9

5o' 2655 2661
2667 2673 2679

°'47°8 4708 4708 4708
47°7

174 19'5 21'0 22'5
24'° 25'5

+ 73 1.68
5 14'05 25 6 '43 + ° 38.80 1 38.83

+ 4'23 .5 4 1.8 534'3
2'94 39'9 34'7 1·65 38 '0 35'° + °'36 36 '1 35'4 °'93 34'2 35,8

5°'2685 26 9 2 2698
2704
2710

°'47°7 47°7
47°7 4707 4706

174 27'0 28'5
3°'0 31'5 33'°

35 6 '46 6 14'09
8 31'72 10 49'36 13 06'99

2'21
3'5° 4'79 6'08
7'37

532'3 3°'4 28'5 26,6
24'7

536'2 36 .6
37'° 37'4 37. 8

5°.27 16 2722 2728
2734 274°

°'47°6 4706 4706 4706 4705

17434'5 36'0
37'5 39'° 4°'5

-15 24.63 1742'27 19 59'91 22 17' 56 2435'20

8·66
9'95 I1'23 12'52 13. 81

522·8 20'9
19'° 17'1 15'2

538'2 38,6
39'° 39'4 39·8

5o'2746 °'47°5 17442'0 -26 52.85 -15'10 5 13'3 540 '2

If Ao, f30 and Qo, wo, i o are ecliptic coordinates and elements for the mean equinox of date (to), and if A, f3 and Q, w, i are for the mean equinox of 1950,o,then over the range of
the above table the formulae for reduction are:

From to to 195°.0

A = Ao + a - b cos (Ao + c) tan f30

f3 = f30

+ b sin (Ao + c)

Q = Qo + a - b sin (Qo + c) cot i o

w = Wo

+ b sin (Qo + c) cosec i o

= io

+ b cos (Qo + c)

From 1950'0 to to

Ao =A - a + b cos (A + c') tan f3

f30 =f3

- b sin (A + c')

Qo = Q -a + b sin (Q + c') cot i

Wo =w

- b sin (Q + c') cosec i

io = i

-bcos(Q + c')

where a, b, c, c' are obtained by linear interpolation to the argument to.

38

EXPLANATORY SUPPLEMENT

The following numerical values are deduced from Newcomb's discussion; they are derivedtfrom fundamental values of the precessional constant, the obliquity, the speed of rotation of the ecliptic, and the longitude of the axis of rotation.

.j/ = 50"'3708 + 0"'0050 T

m = 35 '07234 + 05 '00186 T

,\' = 0"'1247 - 0"'0188 T

11 = 18'33646 - 08'00057 T

P = 50"'2564 + 0"'0222 T

= 20"'0468 - 0"'0085 T

n = 1730 57'·06 + 54"77 T

71' = 0"'4711 - 0"'0007 T

E = 23027' 08"'26 - 46".845 T - 0"'0059 T2 + 0"'00181 T3

where T is measured in tropical centuries from 19°0.0.

Values of these quantities (except if;', A') for the current year are given in A .E.,
n page 50. Values of E, p, 71', are given in table 2.3, which has argument Julian

date and an interval of 1000 days.

.

n P If nil mill' n Ill' m' TT III are the values of the above quantities at an epoch mid-

way between the initial epoch tu and a subsequent epoch t, then:
M = general precession in right ascension = mill (t - to) = So + z
e N = precession in declination = n m (t - to) =

a = general precession in longitude = Pill (t - to)

b = inclination of ecliptic of epoch t to that of epoch to = (t TT III - to)

n + c = 1800 -

III

ta

n c' = 1800 -

t a III -

Values of the above quantities, for reduction to and from 1950'0, are given for
* the current year in A.E., page 50, and for 1800 to 1980 in the three volumes of
Planetary Co-ordinates. Values of M, N are given in table 2.1, and values of
a, b, c, c' are given with the values of E, P, TT, n in table 2.3. They may be used for
the reduction of positions from one mean equinox to another, provided the time
interval is not too long nor the position too close to the pole.

The formulae for the reduction of equatorial coordinates from the mean

equinox of to to the mean equinox of t, or from t to to are:

a - ao = M + N sin t (a + ao) tan t (8 + 80)

8 - 80 =

N cos t (a + ao)

where the right-hand sides are evaluated by successive approximation, if necessary..

The formulae for the reduction of ecliptic coordinates and ecliptic elements are:

From mean equinox of to

to mean equinox of t

A = Ao + a - b cos (Ao + c) tan f3

f3 f30

+ b sin (Ao + c)

Q = Qo + a - b sin (Qo + c) cot i

w = Wo t = to

+ b sin (Qo + c) cosec i + b cos (Qo + c)

From mean equinox of t

to mean equinox of to

Ao = A - a + b cos (,\ + c') tan f30

f30 f3

- b sin (A + c')

Q o = Q - a + b sin (Q + c') cot io

Wo = w

- b sin (Q + c') cosec io

to = i

- b cos (Q + c')

where the final coordinates and elements on the right-hand sides are evaluated by successive approximation, if necessary, although the initial values are usually
sufficiently accurate. Note that when i is small:
Q + w = Qo + W o + a

*Page 11 in A.E. 1972-3, page 9 from 1974·
t See also section 6.

"

& _ _ ..

~_

~

_

_

........ L .

2B. COORDINATE AND REFERENCE SYSTEMS

39

For the transformation of equatorial or ecliptic rectangular coordinates the rigorous formulae obtained above are still the most suitable, though some simplifications may be made in the actual calculations.

Over short intervals of time, of less than a year, or to lower precision the
formulae may be simplified by using constants for the coefficients and by ignoring
second-order terms. Thus the reduction from the mean equinox of the beginning
of one year to that of the next following year may be made through:
a = ao + 38'°73 + 18'336 sin a tan S
S = So + 20"'04 cos a
,\ ='\0 + 5°"'27 - 0"'47 cos (,\ + 6°) tan f3 f3 = f30 + 0"'47 sin (,\ + 6°)

~ = ~o + 0°'°1396 - 0°'°0013 sin (~ + 6°) cot i w = Wo + 0°'°°013 sin (~ + 6°) cosec i i = io + 0°'°°013 cos (~ + 6°)
where it does not matter to which equinox the coordinates are referred.

Approximate annual corrections to right ascension and declination may be taken directly from table 2.4, which has been calculated from the formulae:

in right ascension 38'°73° + 18'3362 sin a tan S

in declination

20" '043 cos a

Coefficients for the approximate reduction from the standard equinox of 1950'0 to the true equinox during the current year are given in the A.E., Table IV.

Differential precession and nutation
The rotation of the frame of reference due to precession (and nutation) causes small changes in the relative coordinates of two adjacent points and, in particular, changes the position angles of a star with respect to others; if to is the epoch to which the angle is to be referred, and t is the epoch of the observation, then the observed position angle must be corrected by applying the angle PSP0 (see figure 2.1) or - (n sec S sin a) (t - to), where n may be taken as 0°'°°56 and the time interval is in tropical years.
Over a small area in the sky the effect of precession (and nutation) varies slowly; thus the corrections for precession and nutation for moving objects will differ little from those for neighbouring stars, to which the positions of the moving object may be referred. Since the positions of the stars for equinox 1950'0, or for the beginning of the year, will be known it is only necessary to apply corrections for differential precession and nutation (and similarly for aberration and refraction) to yield the positions of the moving objects referred to the same equinox.
The effect of differential nutation is always small, and it is convenient to combine precession and nutation in one correction.
If Lla and LIS are the observed differences of coordinates in the sense moving object minus star, then the corrections for differential precession and nutation for reduction to the nearest yearly equinox are, in the notation of section 5:
an in right ascension - {g cos (G + a) tan S} Lla - {g sin (G + a) sec2 S} LIS
in declination + {g sin (G + Lla

40

2.4-APPROXIMATE ANNUAL PRECESSION

Annual precession in right ascension, varies with declination

Ann.

+800 +700 +600 +5°0 +400 +300

+15000 -150

-300 -400 -500

_600 -700 -800

Prec.
In

a

Dec.

h
0·0 +3'1 +3- 1 3- 1 0'5 4- 1 3-6 3-4 1·0 5- 0 4-0 3'7 1-5 6'0 4'5 4-0 2·0 6-9 4'9 4-2
2'5 7'7 5-3 4-5

3- 1 3- 1 3-1 3-3 3-2 3-2
3-5 3-4 3-3
3-7 3-5 3'4 3-9 3-6 3-5 4-0 3·8 3'5

3- 1 3- 1 3- 1 3-1 3- 1 3'0 3-2 3- 1 3-0 3- 2 3- 1 2-9 3-3 3- 1 2-9 3-3 3- 1 2'9

3'1 3- 1 3'1 3- 1 +3- 1 + 3'1 +20 3'0 2'9 2-9 2-8 2-6 2-1 20
2'9 2-8 2'7 2-5 2-1 I-I 19 2·8 2·6 2-5 2-2 1'7 + 0'2 19
2-7 2-5 2-3 1'9 1-2 0'7 17 2-6 2'4 2'1 1-7 0·8 1'5 16

3-0 8'4 5-7 4-7 4-2 3'9 3-6 3'5 9- 1 6-0 4'9 4-3 4- 0 3-7 4. 0 9-6 6-3 5- 1 4-5 4-0 3-7 4'5 10- I 6-5 5-2 4-5 4- 1 3- 8 5- 0 10-4 6-6 5-3 4- 6 4-2 3·8 5'5 10-6 6-7 5-4 4-7 4-2 3-8
6-0 10-7 6-7 5'4 4-7 4-2 3-8 6-5 10·6 6-7 5-4 4-7 4-2 3- 8 7. 0 10-4 6-6 5-3 4- 6 4'2 3-8 7'5 10-1 6-5 5'2 4-5 4-1 3- 8 8,0 9-6 6-3 5- 1 4-5 4-0 ,3-7 8-5 9'1 6'0 4-9 4-3 4-0 3-7

3-3 3- 1 2-8 3-4 3- 1 2-8 3-4 3- 1 2-8 3-4 3-1 2-7 3-4 3- 1 2-7 3-4 3' I 2-7
3-4 3'1 2-7 3-4 3-1 2-7 3-4 3- 1 2-7 3-4 3- I 2-7 3-4 3- 1 2-8 3-4 3- 1 2-8

2'5 2-3 1'9 1-4 0-5 - 2-3 +14 2-5 2-2 1-8 1-2 +0-2 2-9 12 2-4 2'1 1-7 I- I -0-1 3-5 10 2-4 2-0 1-6 0-9 -0-3 3-9 8 2-3 2-0 1-5 0-8 -0-5 4-2 5 2-3 2-0 1-5 0-8 -0-6 - 4-4 + 3
2-3 2-0 1-5 0-8 -0-6 4-5 0 2-3 2-0 1-5 0-8 -0-6 4-4 3 2-3 2'0 1-5 0-8 -0-5 4-2 5 2-4 2-0 1-6 0-9 -0'3 3-9 8 2-4 2-1 1-7 I-I -0-1 3-5 10 2-5 2-2 1-8 1-2 +0-2 2-9 12

9'0 8-4 5'7 4-7 4-2 3-9 3-6 3'3 3- 1 2-8 2-5 2-3 1'9 1-4 0-5 2-3 - 14 9'5 7-7 5-3 4-5 4-0 3-8 3-5 3'3 3- 1 2-9 2-6 2'4 2-1 1-7 0-8 1-5 16 10·0 6-9 4-9 4- 2 3-9 3. 6 3-5 3-3 3-1 2-9 2'7 2-5 2'3 1-9 1'-2 0-7 I7 10-5 6-0 4-5 4-0 3-7 3-5 3'4 3-2 3- 1 2-9 2·8 2-6 2-5 2-2 1-7 + 0-2 19 11·0 5- 0 4- 0 3-7 3-5 3-4 3-3 3-2 3- 1 3-0 2'9 2-8 2-7 2-5 2-1 I- I 19 II'5 4- 1 3-6 3-4 3-3 3-2 3- 2 3' I 3- 1 3-0 3'0 2-9 2-9 2-8 2-6 2-1 20

12·0 3- 1 12'5 2-1 13.0 I - I
13-5 +0'2 14-0 -0'7
14-5 - 1'5

3- 1 3- 1 2·6 2-8
2·1 2-5 1-7 2-2 1'2 1-9 0-8 1-7

3- 1 3- I 3- 1 2-9 2-9 3-0 2-7 2-8 2-9 2-5 2-6 2-8
2-3 2'5 2-7 2-1 2-4 2·6

3'1 3- 1 3- 1 3-0 3'1 3' I 3-0 3-1 3-2 2-9 3- 1 3- 2 2-9 3- 1 3-3 2-9 3- I 3-3

3- 1 3- 1 3- 1 3- 1 3- 1 3-2 3- 2 3-3 3-4 3-6
3-3 3-4 3-5 3-7 4-0 3-4 3-5 3-7 4- 0 4-5 3'5 3-6 3'9 4-2 4-9 3-5 3-8 4-0 4-5 5-3

3- 1 -20 4- 1 20 5-0 19 6-0 19
6-9 17 7-7 16

15.0 -2-3 0-5 1-4 15-5 -2-9 +0-2 1-2 16'0 -3-5 -0-1 I-I 16'5 -3-9 -0-3 0'9 17-0 -4'2 -0·5 0·8 17'5 -4-4 -0-6 0-8
18,0 -4-5 -0-6 0-8 18-5 -4-4 -0·6 0-8 19-0 -4.2 -0-5 0-8 19'5 -3-9 -0'3 0'9 20·0 -3-5 -0-1 I-I 20'5 -2-,9 +0-2 1-2

1-9 2-3 2-5 1-8 2-2 2-5 1-7 2-1 2-4 1-6 2-0 2-4 1-5 2'0 2'3 1-5 2'0 2-3
1-5 2-0 2'3 1-5 2-0 2-3 1-5 2'0 2'3 1-6 2-0 2-4 1-7 2' I 2-4 1-8 2-2 2-5

2-8 3- 1 3-3 2-8 3-1 3-4 2-8 3- 1 3-4 2-7 3-1 3-4 2-7 3'1 3'4 2-7 3- I 3-4
2-7 3- I 3-4 2-7 3' I 3-4 2-7 3- 1 3-4 2-7 3-1 3-4 2-8 3- I 3-4 2-8 3-1 3-4

3-6 3-9 4-2 4-7 5'7 8-4 -14 3-7 4- 0 4-3 4-9 6-0 9- 1 12 3-7 4-0 4-5 5- 1 6-3 9-6 10 3-8 4- 1 4-5 5-2 6-5 10-1 8 3-8 4-2 4-6 5-3 6-6 10-4 5 3-8 4- 2 4-7 5-4 6-7 10-6 3
3- 8 4-2 4-7 5-4 6-7 10·7 0 3-8 4-2 4-7 5-4 6-7 10-6 + 3 3-8 4-2 4- 6 5-3 6-6 10-4 5 3-8 4- 1 4-5 5-2 6-5 10-1 8 3-7 4-0 4-5 5- 1 6-3 9-6 10 3-7 4-0 4-3 4-9 6-0 9- 1 12

21-0 -2'3 21-5 -1-5 22-0 -0-7 22'5 +0-2 23- 0 I-I 23-5 2-1

0-5 1-4 0-8 1'7 1-2 1-9 1-7 2-2 2-1 2-5 2-6 2-8

1-9 2'3 2-5 2-1 2-4 2-6
2-3 2'5 2-7 2-5 2-6 2·8 2'7 2-8 2-9 2-9 2-9 3-0

2-8 3'1 3-3 2-9 3- 1 3-3
2'9 3- I 3-3 2-9 3- 1 3- 2 3-0 3-1 3-2 3-0 3- 1 3- 1

3-6 3'9 4- 2 4-7 5-7 3-5 3. 8 4-0 4-5 5-3 3-5 3. 6 3-9 4-2 4-9
3-4 3-5 3-7 4'0 4-5 3-3 3-4 3-5 3-7 4- 0 3-2 3-2 3-3 3-4 3-6

8-4 +14 7-7 16
6-9 17 6-0 19 5- 0 19 4-1 20

24'0 +3- 1 +3- 1 3- 1 3'1 3- 1 3- 1 3-1 3- 1 3- 1 3- 1 3- 1 3- 1 3. 1 +3- 1 + 3- 1 +20

The precession in right ascension is positive, except where indicated in high declinations_

-

- - ---- ----- -------------,

2C. COORDINATE AND REFERENCE SYSTEMS

41

In.

Tables based on similar formulae are given in A.E., Table VI, for reduction to the

~c.

nearest equinox of the beginning of a Besselian year, and also to that of 195°.0;

c.

the corrections are given in the form:

zo

in right ascension e tan S Lla - f sec2S LIS

zo

in declination

f Lla

19 [9
[7

where e = 101 sin (J + a) j sin Jo = n (t - A) sin I'
f = 101 cos (J + a) j cos Jo = B sin I'

J = Jo - 18 .5 t

[6

and t is the number of years from date to the equinox desired. The small correction

[4

to Jo allows for the fact that it is more accurate to use the right ascension of the

[2

star for the mid-epoch. With j, e, f measured in the units indicated (chosen for

10

convenience of tabulation), and with Lla, LIS in units of I', the corrections are in

8

5

units of 0"'01.

3

References

o

The numerical expressions for the precessional motions are those of S.

3

5

Newcomb; fundamental and derived values are given in:

8

Newcomb, S. The elements of the four inner planets and the fundamental constants of

o

astronomy. Supplement to The American Ephemeris for 1897, Washington, 1895.

2

Newcomb, S. A new determination of the precessional constant with the resulting

4

precessional motions, A.P.A.E., 8, part I, 1897.

6

Newcomb, S. A compendium of spherical astronomy. New York, Macmillan, 1906;

7

reprinted, New York, Dover Publications, 1960.

9

Peters,· J. Priizessionstafeln fur das Aquinoktium 1950.0. Veriiff. Astr. Rechen-Inst.

9 o

zu Berlin-Dahlem, no. 50, 1934.

o

o

9

c. NUTATION

9

7 S

Nutation is essentially that part of the precessional motion of the pole of the

Earth's equator which depends on the periodic motions of the Sun and Moon in

their orbits round the Earth. The progressive long-period motion of the mean

pole has been considered as luni-solar precession in sub-section B; nutation is

the somewhat irregular elliptical motion of the true pole about the mean pole in a

period of about 19 years with an amplitude of about 9". The principal term

depends on the longitude of the node of the Moon's orbit and has a period of

6798 days or 18·6 years; the amplitude of this term, 9"·210, is known as the

constant of nutation. In the complicated theory of the gravitational action of the

Sun and Moon on the rotating non-spherical Earth, other terms arise which

depend on the mean longitudes and mean anomalies of the Sun and Moon and

on their combinations with the longitude of the Moon's node. The resulting

shift of the mean to the true pole can be resolved into corrections to the longitude

(fj!fJ, nutation in longitude) and to the mean obliquity (LIE, nutation in

obliquity), and expressions for these in series constitute the formal speci-

fication of the nutation. The theory and the numerical series upon which the

nutation is now based are developed in full detail by E. W. Woolard in A.P.A.E.,

EXPLANATORY SUPPLEMENT
15, part I, 1953, to which reference should be made for further information.
The terms divide naturally into those not depending on the Moon's longitude, which can be interpolated at intervals of 10 days, and those depending on the Moon's longitude, with periods of less than about 60 days, which cannot be so interpolated. Nutation is therefore conventionally divided into long-period and short-period terms; the latter, consisting of terms with periods less than 35 days, are summed separately as dtf; and dE, being the short-period terms of nutation in longitude and obliquity respectively. In certain special applications, such as the tabulation of the apparent positions of stars at intervals of 10 days, only the long-period terms of nutation are included; and data are provided for the individual application of corrections for the much smaller short-period terms after interpolation.
The terms included in the nutation are given in table 2.5. There are 69 terms in Lltf;, of which 46 are of short period and are summed separately as dif1; for the obliquity there are 40 terms in LlE, including 24 terms in dE. The series include all terms with coefficients of 0" ·0002 or greater. In the table the terms are grouped according to their periods, and are arranged in order of magnitude of the coefficient of the nutation in longitude within each group.
These series may be compared with those used prior to 1960 (with a maximum of 22 terms in longitude and 15 in obl,iquity), which are given in section 7; values were then tabulated only to 0"'0I.
Values of the nutation have been calculated from the series given above for 011 E.T. on each day from 1900 to 2000. Those for 1900 to 1959 are published in Royal Observatory Annals, Number I. The values for 1952 to 1959 have also been included in the Improved Lunar Ephemeris 1952-1959. In each publication there is given a description of the method used for the calculation on punchedcard machines.
The nutation in longitude (Lltf;), to be added to longitudes measured from the mean equinox of date, is tabulated to 0"'001 for 011 E. T. on each day of the year in A.E., pages 18 to 32. The nutation in obliquity (LlE) is not tabulated directly as such, but enters into the obliquity of the ecliptic on pages 18 to 32 and is obtainable immediately as -B, the Besselian day number given in A.E., pages 266 to 285; the short-period terms in both longitude and obliquity, dtf; and dE, are also tabulated on the latter pages, all to 0"'001. The long-period terms Lltf; - dtf; and LlE - dE are not separately tabulated in the Ephemeris, though special values at intervals of 10 sidereal days are incorporated into the day numbers used for the calculation of the apparent places of Io-day stars, published in Apparent Places of Fundamental Stars.
The intersection of the true equator (as affected by both precession and nutation) with the true ecliptic is known as the true equinox of date; and, where distinction is desirable, all coordinates referred to this reference system of the true equinox, true equator, and true ecliptic of date are prefixed by the words "true" or "apparent", the latter being used when th~ direction is affected by aberration.
- - - - - - - - - . " ' ' " ' p..

2C. COORDINATE AND REFERENCE SYSTEMS

43

The equation of the equinoxes, which in editions prior to 1960 was called "nutation in right ascension", is the· right ascension of the mean equinox referred to the true equator and equinox. It is equal to ,1ifi cos E and represents the difference between the mean and true right ascensions for a body on the equator; it is thus the difference between mean and apparent sidereal time. The equation of the equinoxes is tabulated to 0 8 '001 in A.E., pages 10 to 17, and is incorporated into the apparent sidereal time on the same pages.

The simplest and most direct method of converting positions from the mean equinox and the mean equator to the true equinox and the true equator is to add J~ to longitude, since the ecliptic and therefore the latitude is unchanged by nutation. In converting the ecliptic coordinates to equatorial coordinates the
true obliquity (E = Eo + ,1E) must be used. It should, however, be remembered
that coordinates referred to the true equinox cannot be interpolated at intervals longer than one day. First-order corrections ,1a, ,10 to right ascension and declination may be calculated directly from:

Lla = (cos E + sin E sin a tan 0) ,1ifi - cos a tan 0 ,1E LIS = sin E cos a ,1ifi + sin a ,1E

but these are invariably combined with the reduction for precession from the mean equinox of the beginning of the year by means of day numbers. The method, as applied to stars, is described in detail in section 5.
Equatorial. rectangular coordinates referred to the mean equinox can be converted to the true equinox by the application of the corrections:

,1x - (y cos E + z sin E) ,1ifi ,1y = + x cos E ,1ifi - Z ,1E ,1z = + x sin E ,1ifi + Y ,1E

Second-order terms, which are neglected, can only reach one unit in the eighth figure. These formulae are used for the Sun and planets (see sections 4B and 4D). The reduction for nutation can also be combined with that for precession by pre-multiplying the matrix of coefficients Xx, Y x,'" by the matrix whose corresponding elements are:

I

-,1ifi cos E

+,1ifi cos E

I

+,1ifisinE +,1E

It is not sufficient merely to add -,1ifi cos E to Y x , -,1ifi sin E to Z"" ... as second-order terms may be significant.

Differential nutation. For objects within a small area of the sky differential nutation is always combined with differential precession; see sub-section Band A.E., Table VI.

Short-period nutation. Corrections for the short-period terms of nutation may be obtained directly from table 5.2, which is described in section 5D.

44

2.S-SERIES FOR THE NUTATION

* The fundamental argwnents, corrected for amendment to Brown's tables, are:

= 296° 06/16"'59 + 1325 f 198° 50/ 56"'79T + 33"'09T2 + 0",0518T3 = 296°'104608 + 13°'06499 24465d + 0°'00068 90D2 + 0°'00000 0295D3

l' = 358° 28/ 33"'00 + 99f 359° 02/ 59"'10T - 0"'54T2 - 0"·0120T3 = 358°'475833 + 00'9856oo2669d - 0°'00001 12D2 - 0'00000 0068D3

F = llo 15/03"'20 + 1342f 82° 01/ 30"'54T - ll"'56T2 - 0"·0012T3 llo'250889 + 13°'22935 04490d - 0°'00024 07D2 - 0°'00000 0007D3

D 350° 44/14"'95 + 1236f 307° 06/ 51"'18T - 5"'17T2 + 0"·0068T3 = 350°'737486 + 12°'19074 91914d - 0°'00010 76D2 + 0°'00000 0039D3

o = 259° 10/59"'79 -

Sf 134° 08/ 31"'23T + 7"'481'2 + 0"·0080T3

259°'183275 - 0°'05295 39222d + 0°'00015 57D2 + 0°'00000 0046D3

where the fundamental epoch is 1900 January od'5 E.T. = J.E.D. 241 5020'0, and T is measured in Julian centuries of 36525 days, d is measured in days,

D is measured in units of 10 000 days.

Period (days)

ARGUMENT
Multiple of
l' F D 0

6798 3399
130 5 1095 6786 1616
3233
183 365 122 365 178

+1 +2

-2

+2

+1

+2

-2

-2 +2 -2 +1

-2

+2

+2

+1 -I

-I

+2 -2 +2 +1 +1 +2 -2 +2 -I +2 -2 +2
+2 -2 +1

206

+2

-2

173

+2 -2

183

+2

386

+1

+1

91

+2 +2 -2 +2

347

-I

+1

200

-2

+2 +1

347

-I +2 -2 +1

212

+2

-2 +1

120

+1 +2 -2 +1

4 12

+1

-I

"See note on page 523.

LONGITUDE

OBLIQUITY

Coefficient of

Coefficient of

sine argwnent

cosine argwnent

Unit = 0"'0001

-172327 -173'7 T + 2088 + 0·2 T

+92100 +9'1 T 904 +0'4 T

+45

-24

+10

4

+2

-3

+2

-2

-12729 + 1261
497 + 214 + 124

-1'3 T -3'1 T +1'2 T -0·5.T +0'1 T

+55 22 -2'9 T
+ 216 -0·6 T 93 +0'3 T 66

+45 -21 +16
-IS
-IS

-0'1 T +0'1 T

+8 +7

-10

+5

5

+3

5

+3

+4

2

+3

2

3

- - - - - - - - - - - - - - - - - - --- - - - - - - - - - ~ ......

Period (days)
13'7 27·6 13.6 9'1 31.8
27'1 14.8 27'7 27'4 9,6
9'1 7'1 13. 8 23'9 6'9
13,6 27'0 32'0 31'7 9-5
34- 8 13- 2 9. 6 14,8 14- 2
5. 6 12·8 14'7 7'1 23-9
29'S 15'4 29,8 26'9 6'9
9'1 25,6 9'4 13'7 32 .6
13- 8 9·8 7'2 27-8 8'9
5'5

2.S-8ERIES FOR THE NUTATION

45

ARGUMENT
Multiple of
I l' F D n

+2

+2

+1

+2

+1

+1

+2

+2

+1

-2

-I

+2

+2

+2

+1

+1

-I

+1

-I

+2 +2 +2

+1

+2

+1

+2 +2 +2

+2

+1

+2 -2 +2

+2

+2

+2

+2

-I

+2

+1

-I

+2 +1

+1

-2 +1

-I

+2 +2 +1

+1 +1

-2

+1 +2

+2

+1

+2

+2 +1

-I +2

+2

+1

+2 +2 +2

+2

+2· -2 +2

-2 +1

+2 +2 +1

+1

+2 -2 +1

+1

+1

-2

+1 -I

+1

-2

+2

+2

+1 ,

+1

+2

+1 +1

+1 -I +2

+2

-2

+1

-I

+2 -2 +1

+2

+1

-I -I +2 +2 +2

-I +2 +2 +2

+1

+2

+1 +1 +2

+2

+3

+2

+2

LONGITUDE

OBLIQUITY

Coefficient of

Coefficient of

sine argument

cosine argument

Unit = 0"'0001

-2037 -0'2T

+884 -oo5T

+ 675 +o'IT

342 -0'4T

+ 183

261

+II3 -ooIT

149

+ II4 + 60 + 58
57 52

50
31 + 30 + 22

44 32 + 28
+ 26
26

+ 23 + 14
II + II

+ 25 + 19 + 14
13
9

10
7 +7 +5

7 +7 +6
6 6
6 +6
5 5 +5

3
+3 +3
+3 2
+3 +3
3

4
4 +4 +4
4

+2

+3 3
3 2
2

+2 2 2 2
+2

2

EXPLANATORY SUPPLEMENT

D. ABERRATION

Because the velocity of light is finite, the apparent direction of a moving

celestial object from a moving observer is not the same as the geometric direction

of the object from the observer at the same instant. This displacement of the

apparent position from the geometric position may be attributed in part to the

motion of the object, and in part to the motion of the observer, these motions

being referred to an inertial frame of reference. The former part, independent

of the motion of the observer, may be considered to be a correction for light-time;

the latter part, independent of the motion or distance of the object, is referred to as

stellar aberration, since for the stars the correction for light-time is, of necessity,

ignored. The sum of the two parts is called

planetary aberration since it is applicable to planets

and other members of the solar system.

....-

--p(t)

Correction for light-time. Let P and E (see

figure 2.3) be the geometric positions of an object

and a stationary observer at time t, and let P' be the

geometric position of the object at time t - T, where

T is the light-time, i.e. the time taken for the light to

travel from the point of emission, in this case pi,

to the point of observation, E. Then, since E is

regarded as stationary, the direction EP' is the

apparent direction of the object at time t, i.e. the

apparent direction at time t is the same as the geo- Figure 2.3. Planetary aberration metric direction of the object at time t - T.

Stellar aberration. The light which is received at the instant of observation t was emitted, at a previous instant, from the position which the object occupied at time t - T towards the position which the observer was later to occupy at time t; but when the light reaches the observer it appears to be coming, not from this actual direction but from its direction relative to the moving observer. Let the object be considered stationary at p', the position it occupies at time t - T, and let E be moving in the direction EEl with an instantaneous velocity V. Then, according to classical theory, the apparent direction of the object is. that of the vector difference of the velocity of light c in the direction P'E and the velocity V in the direction EEl' The apparent angular displacement is independent of the
distance, but, by definition of the light-time T, PiE = TC so that if EnE is drawn in
the direction of motion and of magnitude T V the apparent direction of the object is EoP'. Thus the apparent direction at time t would be the same as the geometric direction at time t - T were E moving with a constant rectilinear velocity V, i.e. if Eo were identical with E', the position of the observer at time t - T.

The displacement is toward the apex of the motion of the observer; its magnitude (LIB) depends upon the ratio of the velocity of the observer (V) to the

-- - - - - - - --

.... -~---------------------

.,.,.

2D. COORDINATE AND REFERENCE SYSTEMS

47

velocity of light (c), and is given by the solution of the triangle EP'Eo, where 8 is

the angle P'EEl between the direction of P' and the direction of motion.

sin .18 = ~ sin (8 -
c

.18),

or

tan

.18

=

c

+V

~n 8
cos

8

Expanding in powers of Vic:

(V)2 .18

=

~
c

sin

8

-

!--

c

sin 28 + ...

Since Vic is about 0·0001 or 20", the second-qrder term has a maximum of about

0"'001.

The rigorous relativistic theory for stellar aberration gives a different coefficient for the second-order term, but this effect is too small to be observed and is generally ignored.

The motion of an observer on the Earth is the resultant of the diurnal rotation of the Earth, the orbital motion of the Earth about the centre of mass of the solar system, and th,e motion of this centre of mass in space. The stellar aberration is therefore made up of three components which are referred to as the diurnal aberration, the annual aberration, and the secular aberration. The stars and the centre of mass of the solar system may each be considered to be in uniform rectilinear motion; in this case the correction for light-time and the secular aberration are indistinguishable and the aberrational displacement due to the relative motion is merely equal to the proper motion of the star multiplied by the light-time; it is constant for each star, is in general not known, and is ignored.

The term " stellar aberration" is sometimes loosely used in this Supplement in contexts where " annual aberration" should strictly be used.

Annual aberration. In accordance with recommendations of the International

Astronomical Union (Trans. I.A.U., 7, 75, 1950; 8, 67 and 90, 1954) the annual

aberration is calculated as from 1960 from the actual motion of the Earth, referred

to an inertial frame of reference and to the centre of mass of the solar system. The

resulting aberrational displacement .18 may be resolved into corrections to the

directional coordinates by the standard methods. If, for example, - X', - Y',

- Z' are the components of the Earth's velocity parallel to equatorial rectangular

axes, the corrections to right ascension and declination, referred to the same

equator and equinox, in the sense "apparent place minus mean place" are, to

the first order in Vic:

cos 8 .1a X- ' sm. a
C

Y'
-

cos

a

C

.18 = X- ' cos a S.ln"a + -Y's.m a S.In"a -

Z'
-

COS

8

C

C

c

These formulae are usually simplified by the use of the aberrational day numbers C and D (or h, H, i), discussed in detail in section 5; in this simplification the assumption is implicitly made that the direction of motion of the Earth lies in the ecliptic, but the resulting error is negligible. The effect of second-order terms is included in the expressions for the second-order day numbers] and ]'.

EXPLANATORY SUPPLEMENT

Prior to 1960 it was customary, for computational convenience, to approximate to the motion of the Earth by taking:
X' = -ke sin ,\ Y' = +ke cos ,\ cos E Z' = Y' tan E
where k is the constant of aberration and ,\ is the Sun's true longitude. In addition to small periodic terms due to the action of the Moon and planets, this procedure neglects terms depending on the eccentricity and longitude of perihelion of the Earth's orbit (M.N.R.A.S., 110,467, 1950). These terms, of order about 0"'34, are constant throughout the year for any particular star, and change very slowly during the centuries; they are here represented symbolically by E, in the sense of apparent place minus mean place. It is assumed that the observed apparent places of stars have in the past been reduced to give catalogue mean places of stars that already contain the constant part E of the aberrational reduction to apparent place, and so it is desirable to subtract the effect of E from the aberration calculated from the actual motion of Earth; this procedure is recommended by the International Astronomical Union (Trans. I.A.V., 7, 75, 1950; 8,90, 1954). It can be accomplished by applying constant corrections to the components of the Earth's motion.

The sense in which the E-terms are measured can best be appreciated by using symbolic notation; let:

A == apparent place; M == true mean place; M o == catalogue mean place; R == the complete star reduction, including the correction for aberration calcu-
lated from the true motion of the Earth, in the sense apparent - mean;
E == the E-terms of aberration, in the sense here used.

Then:

the true mean place

M =A - R

the catalogue mean place M o = A - (R - E) = M + E

Thus the modified star reduction, to be applied to the catalogue mean place to give

the apparent place, is R - E since:

M o + (R - E) = A

In this sense the E-terms are:

in longitude (,\) +ke sec f3 cos ('lIT - ,\)
in latitude (f3) +ke sin f3 sin ('lIT - ,\) * where k = 20"'47 is the constant of aberration, e = °'°1675 0'00004 T is the eccentricity of the Earth's orbit, and 'lIT = r - 180° = 101°'22 + 1°'72 T is the

longitude of perihelion of the Earth's orbit (see section 4B).

For systematic application to right ascension and declination the E-terms are

best expressed in terms of corrections LlC = +ke cos 'lIT cos E, LID = +ke sin 'lIT

to the day numbers C,D, in such a way that the E-terms are:

in right ascension eLlC + dLlD

in declination

e'LI C + d'LlD

Full details of the practical application to the calculation of the day numbers C and D

are given in section 5D, and numerical expressions are given in section 4G. For

bodies in the solar system the E-terms vary, and so the annual aberration is

calculated, implicitly, without modification from the actual motion of the Earth.

The value that is used for e corresponds to the adopted value of 20"'47 for the

*20"'496 from 1968, corresponding to c = 2'997 925 x 108 m/s.

_ _ _ _ .~_

_.

_

__

_. _ _ 4

----.--

~'\. - - - - - - - - - - - - • •

2D. COORDINATE AND REFERENCE SYSTEMS

49

constant of aberration, and not to the actual velocity of light. It is equivalent to a
light-time of od,0057683 = 4988'38 for unit distance.*

Diurnal aberration. The rotation of the Earth on its axis carries the observer
towards the east with a velocity V o P cos 1>', where Vo (0'46 km/sec) is the equatorial
rotational velocity of the surface of the Earth. The corresponding constant of diurnal aberration is:
-Vco P cos .'f'J .=. '0"'320 P cos '.Jf.'.' = 08 '0213 P cos '.Jf.'.' The aberrational displacement may be resolved into corrections (apparent minus mean) in right ascension and declination:
Lla = 08'0213 P cos 1>' cos h sec 8
Ll8 = 0"'320 P cos 1>' sin h sin 8
where h is the hour angle. The effect is small but is of importance in meridian observations; for a star at transit Ll8 is zero, but
Lla = ± 08'0213 P cos 1>' sec 8
according as the star is above or below the pole. A correction of this amount is usually subtracted from the observed time of transit instead of being added to the right ascension. Values of the correction are tabulated in table 2.6.

Correction for light-time. When a correction for light-time is required, it is usually combined with that for annual aberration; the combined correction for planetary aberration is described in the following paragraph. The correction for light-time alone could be obtained, if desired, by a comparison of the geometric ephemeris at time t with that derived by combining the geometric position of the Earth at time t with the geometric position of the object at time t - T.

Planetary Aberration. The displacement of the apparent position from the geometric position at the same instant by planetary aberration may be obtained from the two independent components due to the instantaneous motion of the Earth and the motion of the body during the light-time; but the practical methods that are used give the planetary aberration directly from the geocentric ephemeris, without explicit separation of the two components. The errors of these methods may be deduced by comparison with the results of using the heliocentric motions (strictly barycentric motions, although the maximum errors due to the motion of the Sun with respect to the centre of mass of the solar system are quite negligible) of the Earth and the body. Such methods are not generally practicable as the geocentric distances must be calculated to give the light-time.
Since the E-terms vary for a moving object, such as a planet, annual aberration must in this case be calculated, without modification, from the actual motion of the Earth. It may be allowed for exactly by displacing the Earth a distance TV in the direction opposite to the Earth's instantaneous velocity V. If the planet's motion in the light-time l' can be regarded as rectilinear and uniform, the position of the planet at time t - l' may be obtained by a displacement of distance TV in the direction opposite to the planet's instantaneous velocity v.

*od, 005 7756 = 499 5 '012 from 1968.

50

2.6-CORRECTION FOR DIURNAL ABERRATION

Lat. 0° 10° 20° 30° 35° 40° 45° 50° 52° 54° 56° 58° 60°

Dec.

Unit 0 8'001

°
°5

21 21 20 18 17 16 15 14 13 13 12 II II 21 21 20 19 18 16 IS 14 13 13 12 II II

10

22 21 20 19 18 17 IS 14 13 13 12 II II

15 22 22 21 19 18 17 16 14 14 13 12 12 II

20

23 22 21 20 19 17 16 15 14 13 13 12 II

25 24 23 22 20 19 18 17 IS 14 14 13 12 12
30 25 24 23 21 20 19 17 16 IS 14 14 13 12 35 26 26 24 23 21 20 18 17 16 15 15 14 13 40 28 27 26 24 23 21 20 18 17 16 16 IS 14 45 30 30 28 26 25 23 21 19 19 18 17 16 IS

50 33 33 31 29 27 25 23 21 20 20 19 18 17 52 35 34 33 30 28 27 25 22 21 20 19 18 17 54 36 36 34 31 30 28 26 23 22 21 20 19 18 56 38 38 36 33 31 29 27 24 23 22 21 20 19 58 40 40 38 35 33 31 28 26 25 24 23 21 20

60 43 42 40 37 35 33 30 27 26 25 24 23 21 62 45 45 43 39 37 35 32 29 28 27 25 24 23 64 49 48 46 42 40 37 34 31 30 29 27 26 24 66 52 52 49 45 43 40 37 34 32 31 29 28 26 68 57 56 54 49 47 44 40 37 35 33 32 30 28

70 62 61 59 54 51 48 44 40 38 37 35 33 31 71 66 65 62 57 54 50 46 42 40 39 37 35 33 72 69 68 65 60 57 53 49 44 43 4 1 39 37 35 73 73 72 69 63 60 56 52 47 45 43 41 39 36 74 77 76 73 67 63 59 55 50 48 45 43 41 39

75

82 81 77 71 68 63 58 53 51 48 46 44 4 1

76 88 87 83 76 72 68 62 57 54 52 49 47 44

77 95 93 89 82 78 73 67 61 58 56 53 50 47

78 103 101 96 89 84 79 73 66 63 60 57 54 51

79 112 110 105 97 92 86 79 72 69 66 63 59 56

Unit 0·'01
0
80 00 12 12 12 II 10 9 9 8 8 7 7 7 6 81 00 14 13 13 12 II 10 10 9 8 8 8 7 7 82 00 IS IS 14 13 13 12 II 10 9 9 9 8 8 83 00 18 17 16 15 14 13 12 II II 10 10 9 9 84 00 20 20 19 18 17 16 14 13 13 12 II II 10

85 00 24 24 23 21 20 19 17 16 IS 14 14 13 12
85 30 27 27 26 24 22 21 19 17 17 16 15 -14 14 86 00 31 30 29 26 25 23 22 20 19 18 17 16 IS 86 30 35 34 33 30 29 27 25 22 22 21 20 19 17 87 00 4 1 40 38 35 33 31 29 26 25 24 23 22 20 87 30 49 48 46 42 40 37 35 31 30 29 27 26 24

88 00 61 60 57 53 50 47 43 39 38 36 34 32 31 88 10 67 66 63 58 55 51 47 43 41 39 37 35 33 88 20 73 72 69 64 60 56 52 47 45 43 41 39 37 8830 82 80 77 71 67 62 58 52 50 48 46 4J 4 1 88 40 92 90 86 79 75 70 65 59 56 54 51 49 46 88 50 105 103 98 91 86 80 74 67 65 62 59 56 52

89 00 122 120 115 106 100 94 86 79 75 72 68 65 61

This correction is to be subtracted from the observed time of transit for transits above pole, and added to the time of transit for transits below pole.

2D. COORDINATE AND REFERENCE SYSTEMS

51

Thus the practical determination of planetary aberration may be based on either of the two principles:

(I) To the order of accuracy that the motion of the object during the lighttime is rectilinear and uniform, the planetary aberration depends upon the instantaneous velocity of the observer relative to the object at the time of obser\"ation in exactly the same wayas stellar aberration depends upon the instantaneous total velocity of the observer.
(2) To the order of accuracy that the motion of the Earth during the lighttime is rectilinear and uniform, the directly observed apparent position at time t is the same as the geometric position that the object occupied at time t - T relative to the position that the Earth occupied at time t - T.

From (I) it follows that the apparent position at time t may be determined by applying to the geometric position at t a correction consisting of the light-time multiplied by the instantaneous velocity of the Earth relative to the object; since this relative motion is the negative of the geocentric motion of the object, the correction to the geometric value of any geocentric coordinate q is -T dqJdt where the instantaneous rate of change dqJdt is obtained by numerical differentiation of the geometric ephemeris. Thus:
Apparent position = geometric position - T (rate of change)
= geometric position - * 0 '0057683 LI (daily motion)

The departure from rectilinear and uniform motion gives rise to errors of order 0"'001 L1Ja2 where a is the mean distance of the planet; this may reach 0"·01 for :\lercury but does not exceed 0"'001 for the outer planets; second-order terms may reach 0"'001 or 0"'002 and are neglected .
.\lternatively from (2) it follows that, if the light-time is not too great, the apparent position at time t may be obtained by interpolating the geometric ephemeris to time t - T; or, conversely, from an observed position, the geometric position at the preceding instant when light left the object is immediately obtained by ante-dating the time t of observation to t - T. The error is generally larger than in using (I) and, for the outer planets, may reach 0" ·001 L1. Corrections for the effect of curvature of the Earth's orbit may be applied if high precision is required; but no formulae for these are given here.
Strictly, the light-time corresponds to the distance from the position of the Earth at time t to the position of the body at time t - T; but, as far as the planets are concerned, the maximum error arising from using the geocentric distance at time t for calculating the light-time is only 0" '0005.
Illustrations of the application of corrections for planetary aberration are given in sections 4B and 4D.

Differential aberration. The differential coordinates of a movmg object with respect to a fixed star will be affected by differential aberration; if ja, L18 are the observed differences of the coordinates in the sense
*0'005 7756 from 1968.

52

2.7-DIFFERENTIAL ABERRATION

The corrections for differential aberration to be added to the observed differences (in the sense moving object minus star) of right ascension and declination to give the true differences are:

I.n ri.ght ascenSI.On

a LIa

+

b

LIS
10

I.n

um.ts

f
0

0"'001

in declination

c Lla + d Ll8 in units of 0"'01 10

where Lla, Ll8 are the observed differences in units of 1m and I' respectively and where a, b, c, d are the coefficients defined by:

n a = -

103 sin 1m h cos (H + a) sec 8

c = + 102 sin 1m h sin (H + a) sin 8

n b = - 104 sin I' h sin (H + a) sec 8 tan 8
d = - 103 sin I' h cos (H + a) cos 8

in which a constant value of 19"·6 is used for h. The values of these coefficients are tabulated
without signs on the opposite page with main argwnents H + a, for the first quadrant only,
and 8, also without a sign; an auxiliary argwnent for the second quadrant is also given on
the extreme right-hand side. Interpolation in this table is,in general, unnecessary. The
day number H is obtained from the critical table below; this table may be used unchanged
for all years. The signs of a, b, c, d, which depend on the quadrant of H + a and the sign
of the declination, and the argument in the first quadrant corresponding to the actual value
of H + a are taken from the second table below; the auxiliary argwnent is also indicated in
the second and fourth quadrants.

Date H

Dec. 26 Jan. 3 23 '5
II 23'0

19 22 '5 26 22 '0

Feb.

3 21 '5
21'0 10 17 20'5

20'0 25
Mar. 4 19'5

II 19'0

Date H

Date H

Date H

Mar. II h

18'5 17
18'0 24

3 1 17'5

Apr.

17'0 7
16'5 IS
16'0 22

29 15'5

May 7 15 '0

IS 14'5

23 14'0

May 23
3 1 13'5
June 9 13'0
17 12.'5

26 12'0

July

5 11 '5 11·0
13 21 10'5

10·0 29
Aug. 6 9'5
9'0 14

Aug. 14 h

22 8'5

8'0 29

Sept. 6 7'5

7'0 13

20 6'5

6'0 27

Oct.

5'5 4

II 5'0

18 4'5

4'0 25

In critical cases ascend

Date H
Oct. 25 Nov. 2 3'5
3'0 9
2'5 17
2'0 24
Dec. 2- 1'5
1·0 10
0'5 18
0·0 26
23'5
Jan. 3

H +a
h
0 6 12 18
24

Signs of a, b, c, d

Positive 8

Negative 8

H +a

ab cd

h ab cd

0

+

6

+

+ + + 12 + + +

+ + + 18 + + +

+

+

24

H +a
h
0 6 12 18 24

Tabular argwnents to be used

Argument on the left

Argwnent on the right

H +a 12h - (H + a) H + a (H + a) - 12h 24h - (H + a) (H + a) - I2h

---_..

o
H+a
h
o
I
2
3 4 5 6
h
o
I 2
3 4 5 6
h
0·0 0'5 1-0
1'5
2-0
2'5
3- 0 3-5 4- 0 4-5 5- 0 5-5 6-0
h
0·0 0-5 1-0 1-5 2-0
2-5
3-0 3-5 4- 0 4-5 5- 0 5-5
6-0

2.7-DIFFERENTIAL ABERRATION

53

30° abc d abc d abc d abc d

6 006 6 006 5 00 5 4 004 3 00 3
001
o 000

6 006 600 5 5 0 I5
4 0I4 3 II3
III
o I I0

6 005 6 0I5
5 II5 4 12 4 3 133 2 I 3I o I 30

7 005

6

I5

6

24

5 2 33

324 2

2 24I

o 340

7 004 7 I I4 623 4
5 343 4 452 2 45I 045 0

H +a
h
12
II
10
9 8
7 6

45°

50°

55°

60°

65°

abc d abc d abc d abc d abc d

h

8 0 0 4 9 0 0 4 10 0 0 3- II 0 0 3 13 0 0 2

12

8 1 2 4 9 2 2 4 10 2 2 3 II 3 2 3 13 5 2 2

II

7 3 3 3 8 4 3 3 9 543 10 7 4 2 12 10 4 2

10

644 3 6 5 5 3 7 7 5 2 8 9 5 2 10 14 5 2

9

4 5 5 2 466 2 586 2 6 II 6 I 7 I7 7 I

8

2 5 6 I 2 ;; 6 I 3 9 7 I 3 13 7 I 3 19 7 I

7

o 5 6 0 o 7 7 0 o 970 o 13 7 0 o 19 8 0

6

62°
abc d 12 0 0 3 12 2 I 3 12 4 2 3 II 6 3 2 II 8 4 2 10 9 5 2
9 II 5 2
7 12 6 2 6 13 7 I
5 14 7 3 IS 7 2 IS 7 0
o IS 8 0

13 0 0 2 13 2 I 2 13 5 2 2 12 7 3 2 II 9 4 2 10 II 5 2
9 13 5 2 8 14 6 2 7 IS 7 I 5 16 7 I 3 17 7 I 2 18 8 0
o 18 8 0

66°
abc d 14 0 0 2 14 3 I 2 14 5 2 2 13 8 3 2 12 10 4 2 II 13 5 2
10 IS 6 2 9 17 6 I
7 18 7 I
5 19 7 I 4 20 8 I 2 21 8 0
o 21 8 0

68°
abc d
IS 0 0 2
IS 3 I 2 IS 6 2 2 14 10 3 2 13 13 4 2 12 IS 5 2
II 18 6 2 9 20 6 I 8 22 7 I 6 23 7 4 24 8 2 25 8 0
o 25 8 0

70°
abc d 17 0 0 2 17 4 I 2 16 8 2 2 IS 12 3 2 14 IS 4 2 13 19 5 2
12 22 6 I 10 24 6
8 26 7 6 28 7 I 4 29 8 I 2 30 8 0
o 31 8 0

b 12-0
11'5
11-0
10-5 10-0
9'5
9- 0
8-5 8-0 7'5 7- 0 6-5
6-0

71°
abc d 18 0 0 2 17 4 I 2 17 9 2 2 16 13 3 2 IS 17 4 2 14 21 5 I
12 24 6 I
I I 27 6 I
9 29 7 I 7 31 7 I 5 33 8 0
2 34 8 0
o 34 8 0

72°
abc d 18 0 0 2 18 5 I 2 18 10 2 2 17 IS 3 2 16 19 4 2 IS 23 5 I

73°
abc d 20 0 0 2 19 6 I 2 19 II 2 2 18 16 3 2 17 21 4 I IS 26 5 I

13 27 6 I I I 30 6 I
9 33 7 I 7 35 8 I 5 37 8 0 2 38 8 0

14 30 6 I 12 34 6 I 10 37 7 I
8 39 8 I 5 41 8 0 3 42 8 0

o 38 8 0 o 43 8 0

74°
abc d 21 0 0 2 21 6 I 2 20 12 2 2 19 18 3 I 18 24 4 I 16 29 5 I
IS 34 6 I 13 38 7 I 10 42 7 I
8 44 8 I 5 46 8 0 3 48 8 0
o 48 8 0

75°
abc d
22 0 0 I
22 7 I 21 14 2 20 21 3 I 19 27 4 I 17 33 5 I
16 39 6 I 13 43 7 I II 48 7 I
8 51 8 I 6 53 8 0 3 54 8 0
o 55 8 0

h 12-0
11-5
11-0
10-5
10-0
9-5
9- 0
8-5 8-0 7-5 7-0 6-5
6-0

54

EXPLANATORY SUPPLEMENT

moving object minus star, then the corrections for differential aberration are:

in right ascension
- h cos (H + a) sec 0 Lla - h sin (H + a) sec 0 tan 0 Llo

in declination
+ h sin (H + a) sin 0 Lla - h cos (H + a) cos 0 Llo

in which a small term i sin 0 Llo has been omitted from the correction in declination;
this may reach 0"·02 near the pole for Llo = 10'; h, H, i are the independent day
numbers defined in section SD.

To the precision required, these corrections may be regarded as independent of the year or of the equinox required, and may be tabulated as functions of position and date. Permanent tables of this kind are given in A.E., Table V, and in table 2.7 of this Supplement; in these tables the coefficients of Lla, Llo in the above equations are tabulated using a mean value of 19"·6 for h.

The corrections should be applied with those for differential precession and nutation (see sub-section B) to give mean positions referred to the same equinox as those of the stars.

Astrometric positions. An astrometric position is obtained by adding the planetary aberration to the geometric ephemeris and then subtracting stellar aberration from which the E-terms have been omitted. The astrometric ephemeris is therefore rigorously comparable with observations that are referred to catalogue mean places of comparison stars, it being only necessary to correct the observations for geocentric parallax. Such positions are discussed in more detail in section 4D.

E. REFRACTION
In the Ephemeris atmospheric refraction enters into only a very few topocentric pp,enomena, such as the times of rising and setting of the Sun and Moon, and, in theory though neglected in practice, the local predictions of eclipses. Consequently all explanations of the theory, and of the practical calculation and application of numerical tables, are omitted; they are adequately covered in the references given at the end of this sub-section.
Rising and setting phenomena. As described in section 13 a constant of 34' is used for the horizontal refraction in the calculation of the times of rising and setting: that is, the zenith distance of the object (upper limb of the Sun or Moon) is 90° 34'·
Eclipses and occultations. Owing to refraction and parallax the geometric direction from an object M outside the Earth's atmosphere to an observer at P is not the same as the initial direction of the ray of light from M to P; the difference is only significant for an object as close as the Moon, and then only at low altitudes. Thus the condition that two objects M and S shall appear to be coincident to an

_______. - - - . ' ' ' l . .

2E. COORDINATE AND REFERENCE SYSTEMS

55

observer at P on the Earth's surface is not precisely the condition that the geomet-

rical direction PMS is a straight line. Let

z - R be the apparent zenith distance at P, R

being the refraction; then the precise condition

is that the geometric direction P/MS is a straight

line, where P' is the point vertically above P

at which the true zenith distance is z (see

M5

figure 2.4). The height (h) of P' above P can

be calculated from the formula given below;

it is independent of the distances of M and S.

Full allowance can therefore be made for

refraction by treating the observer as though he were at P', instead of at P, that is by increasing

Figure 2.4. Refraction

the height of the observer above the spheroid

by h.

It can be shown (Chauvenet, Vol. I, page 516) that, for a spherical atmosph~re:

h P-o sin (z - R)

+ - = I

'----=----;-.-'-------'

z p

sm

where p is the (geocentric) radius and P-o the index of refraction of the atmosphere

at P. To the order of accuracy required the correction can be made either by

increasing the height above sea level by h, or by multiplying the geocentric
rectangular coordinates g, 71, ~ of the observer by I + hlp. Values of h in metres and of I + hip, based on mean values of the quantities concerned, are given for a

point at sea level in the following table. It will be seen that the corrections are

only significant for altitudes less than about 10°, when, however, the refraction

may differ considerably from its mean value.

z

h

I + -h

P

z

h

I + -h

p

0

0

1'000000

m

82

100

1'000016

30

0

1'000000

60

5

1'000001

70

20

1'000003

75

30

1'000005

80

70

l'OOOOll

84

170

1'000026

86

290

1'000046

88

610

1'000095

89

940

1'000148

90

1 540

1'000242

Artificial satellites. Referring to figure 2.4 it will be seen that the refraction correction applicable to the observation of a close object is not R but R - r where r is the angle P'MP. For objects only a few hundred kilometres above the Earth's surface r can be of the order of a minute of arc, and must be allowed for in the reduction of precise observations.

Corrections to right ascension and declination. Refraction affects the observed zenith distance. If an object is observed on the meridian the refraction is a direct correction to the observed declination, and the deduced right ascension is unaffected.

56

EXPLANATORY SUPPLEMENT

If a', 0' are the observed right ascension and declination of an object not on the meridian, the corrections required to give the true values a, 0 are, to first order:

a - a' - R sec 0' sin C
o - 0' = - R cos C

where R is the refraction and C is the parallactic angle, i.e. the angle at the object in the spherical triangle pol~-object-zenith.

Such formulae are rarely used since most observations made out of the meridian are made differentially. The differential refraction of two objects may be calculated directly from the difference between the refractions appropriate to the two altitudes; and this may be resolved into differences of right ascension and declination. It is more usual, however, to consider such corrections as linear over the small area covered by a photographic plate and to allow for them by means of the plate constants determined from the coordinates of the standard stars.

References
Text-books that give the basic theory of refraction and of the corrections for it include:
Chauvenet, W. A manual of spherical and practical astronomy, 2 volumes, Philadelphia, 5th ed. 1892; reprinted, New York, Dover Publications, 1960. (Refraction in the local prediction of eclipses is discussed in Vol. I, pages 515-517.)
Smart, W. M. Text-book on spherical astronomy, Cambridge University Press, 4th ed. 1944·
Theoretical discussions of the calculation of refraction include:
Harzer, P.· Berechnung der Ablenkung der Lichtstrahlen in der Atmosphare der Erde auf rein meteorologisch-physikalischer Grundlage. Publikation der Sternwarte in Kiel, no. 13, 1924.
Willis, J. E. A determination of astronomical refraction from physical data. Trans-
actions of the American Geophysical Union, 1941, part II, 324-336. Garfinkel, B. An investigation in the theory of astronomical refraction. A.J., 50,
169-179, 1944. Astronomical refraction in a polytropic atmosphere. A.J., 72, 235-254, 1967·
Tables used for the calculation of refraction corrections include:
Greenwich. Refraction tables arranged for use at the Royal Observatory, Greenwich, by P. H. Cowell, M.A., Chief Assistant. Greenwich Observations for 1898, appendix I, London, 1900.
Washington. Tables XXVI-XXXIV of " Reduction tables for transit circle observations ", Publications of the United States Naval Observatory, 2nd series, 4, appendix II, Washington, 1904. (The tables" follow the form used in the Refraction Tables published by the Observatory in 1887, but are based upon the Pulkowa Refraction Tables instead of upon those of BESSEL ".)
Harzer, P. Gebrauchstabellen zur Berechnung der Ablenkungen der Lichtstrahlen in der Atmosphare der Erde fUr die Beobachtungen am grossen Kieler Meridiankreise. Publikation der Sternwarte in Kiel, no. 14, 1924.
Pulkovo. Refraction tables of Pulkova Observatory. 3rd ed. (including some supplementary tables taken from Harzer), 1930; 4th ed. (B. A. Orlov) 1956.

- --- --

---------

- --

2F. COORDINATE AND REFERENCE SYSTEMS

57

F. PARALLAX

Introduction
The positions in which the Sun, Moon, and planets are actually observed differ from the geocentric positions tabulated in the Ephemeris by the amount of parallax due to the displacement of the observer from the centre of the Earth. Before comparison with theory, it is thus necessary to correct an observed, or topocentric place, by applying parallax so as to reduce it to a geocentric place. For the Sun and planets the corrections are small and may be treated as first-order quantities whose squares can be neglected. For the Moon the parallax is sufficiently large to require third-order terms in the general expression for the corrections; and it is better to use exact formulae. F or artificial satellites of the Earth the parallax may be so large that exact formulae, based on the actual position of the observer relative to the centre of the Earth, must be used.

The geocentric positions of stars are similarly affected by the annual parallax due to the displacement of the Earth from the centre of the Sun. In this case it is usual to include the parallax in the ephemeris position, so that it is directly comparable with the observed position.

Details of the corrections and the method of calculation follow.

The figure of the Earth
In calculating parallax corrections, the dimensions of the Earth are usually
taken to be thos~ of Hayford's spheroid (see section 6). * This is defined by the
equatorial radius (a) and the flattening (f), for which an exact value of 1/297 is adopted. The adopted value for a is 6378'388 km, from which the polar radius b = a (1 - 1) is derived as 6356'912 km. Otherwise the notation used is:
cP = geographic (or geodetic) latitude cP' = geocentric latitude p = geocentric distance, i.e. the distance of the observer from the centre of
the Earth, in units of the Earth's equatorial radius.

The latitude cP is variously referred to as the geographic latitude, or the
geodetic latitude; on the spheroid the two are identical, but on the actual Earth
they differ on account of gravity anomalies. No significance is to be attached in
this Supplement to the use of one term or the other. t

The position of an observer relative to the centre of the Earth is most readily

expressed in rectangular coordinates; in the meridian section of the Earth through

the observer these are:
p sin cP' = 8 sin cP

p cos cP' = c cos cP

which serve to define the auxiliary functions 8 and C. It should be noted that:

whence

tan cP'

=

b2
11

tan

cP

=

(1

-

1)2 tan cP

a

8 = (1 - 1)2 C C = {COS2cP + (1 - f)2 sin2cP}-~

and

p2 = t (82 + C2) + t (C2 - 8 2) cos 2cP

= C2 { cos2cP + (1 - f)4 sin2 cP }

*From 1968: f = 1/298'25, a = 6378 160 m, b = 6356775 m

tThe difference is however significant for some applications.

58

EXPLANATORY SUPPLEMENT

The following expressions, which contain terms up to f3, may then be derived for

8, C, p, and 4> - 4>' in terms of the geographic latitude (4)) and the flattening (f) ;

they assume that the observer is at sea level.

8 = 1 - !f + 156f2 + ·}d3 - (tf - tF - 654f3) cos 24>

F + (r36 - .l2 f3) cos 44> - 654 f3 cos 64>

C=

1

+

tf

+

5 16

f2

+ '}-'iJ3 -

(if + iF + nf3) cos 24>

+ (136 f2 + 392 f3) cos 44> - 654f3 cos 64>

p

1-

if

+

5 16

f2

+ 352 f3 + (~-f -

-Hf3) cos 24>

U - (156 f2 + 352 f3) cos 44> + f3 cos 64>

4>' = (f + tf2) sin 24> - (tf2 + tf3) sin 44> + ! f3 sin 64>

Inserting the adopted numerical value off = 1/297 leads to the following series:

8 = 0'99495304 - 0'00167783 cos 24> + 0'00000212 cos 44> C = 1'00168705 - 0'00168919 cos 24> + 0'00000214 cos 44>
p = 0'99832005 + 0'00168349 cos 24> - 0'00000355 cos 44> + 0'00000001 cos 64>

4> - ef>' = 695"·66 sin 24> - 1"'17 sin 44>

Values of 8 and C, calculated from the first two of these series, are tabulated

in A.E., Table VII; they may also be found together with p and 4> - 4>' in table 2.8

of this Supplement.* A correction for the height of the observer above sea level is

necessary for the calculation of his actual coordinates p sin 4>' and p cos 4>'. To an

adequate approximation the geocentric radius is increased by o· 1568 h or 0 '0478 H x

10-6 and the angle of the vertical 4> - 4>' is unchanged, where h is the height above

sea level in metres and H is the height in feet. The addition of this correction

leads to the expressions:

p sin 4>' = (8 + 0'1568 h x 10-6) sin 4> = (8 + 0'0478 H x 10-6) sin ef> pcos4>' = (C + 0'1568h X 10- 6) cos 4> = (C + 0'0478H x 10-6) cos ef>
tan 4>' = (0'9932773 + O'OOII h x 10-6) tan 4> = (0'9932773 + 0'0003 H x 10- 6) tan 4>

Values of these three quantities are given for the principal observatories in

the list of observatories in the Ephemeris.

Example 2.1. Geocentric coordinates of Washington

The geographic coordinates of a point at U.S. Naval Observatory, Washington, D.C.,

are'\

. . + 5 h 08m 158 '75, cp = +38° 55' 12"'3, and height = 85m.

,

sin

cos

cP + 38 55 12'3

+0·62823 58

+0'77802 30

tan +0. 8074772

2CP 77 50 25

+0'97756

+0'21064

4CP 155 4 1

+0'412

-0'9 11

From the series:

S 0'99459 77 C 1'0013293

P 0'99867 8
cP - cp' II' 19"·6

It may be confirmed that (to the accuracy of the table) the same values are obtained by

interpolation in table 2.8 to cP = 38° '920.

For a height of 85m a correction of 13'3 x 10-6 must be applied to Sand C, before

forming p sin cP' and p cos cP', and to p. Thus:

p sin cp' + 0·62485 0

P 0'99869 I

P cos cp' + 0'77906 8

cp' + 38° 43' 52"'7

The correction for height to cp - cp' is here negligible.

*The coefficients and table given here are for the Hayford spheroid in use before 1968. See note on page 515.

----------------------- --

2.S-FACTORS FOR COMPUTING GEOCENTRIC COORDINATES

59

8

0-99
o 3277 I 3278 2 3281 3

3

3286 5
8

4 3294 9

5 33°3 II 6 33 14 13 7 33 2 7 I"
8 3342 I~
9 3359 '9

10 3378 21

II 3399 23

12 13

3422 3446

24 27

14 3473 28

15 35°1 30 16 353 I 32 17 35 63 18 359633

19 363 I 35 37

20 21

3668 3706

38

22 3746 441°

23 3787 43

24 3830 44

25 26

3874 46 3920 46

27 28
29

3966 48

4°14 4063

49

5°

30 4 II 3

31 4 164 5'

32 4216 52

33 34

4269 43 23

53
54 55

35 4378
~~ 36 4433 55
37 4489 38 4545 57 39 4602 58

40 4660

41 42
43

47 17 57
~: 4776 59
4834

44 4892 59

45 495 1

C

1 0 00

0000

0001 3
0°°4 5

0°°9 0016 7
10

0026

II

°°37

0050 '3

0065 15

0082 17.

'9

0101 0122

21

01 45 23

01 7° 25
01 97 :~

0225

0 2 55 0287

33°2

0321 34

035 6 35 37

°393 0432

39

0 472

4° 42

°5 14

0557 43

44

0601 46

0647 0694

:~

°742

°79 1 49

5°

084 1

0893 52

0945 52 0999 54 1053 54
55
II 08 1163 55
1220 57

1277 57
1334 ~~

1392 8 11455008 558 1567 59
1626 59 59
1685

P 0-99

9999 3 9996 5 9991 7
9984 9

9975 II 9964 '4 995° '5

9935 17 9918 19

9899 21 9878

9856

22
2-

983 I 2~

9805 29

9776 9746

30 32

97 14 9681

33

9

64

635 37

~~~~ ~~

953 1 41 949° 43 9447 44

4> -4/

° 24

24 48

24 24

72 96

24 24

120 24

1 44 168

24

191

23 23

214 23

237 23

260 282

22

22

30 4 22 .3 26 21

347 21 368 20 388 20
40 8
'9 427 '9

446
464
482

'8 '8 17

499 17

5 1 6 16

940 3 46 9357 46

93 II 926 3

48

921 4 49

5°

9 164 9II2 9060

52 52

9007 53

8953 54

54

8899 6

8843 56 8787 5 8730 57 8673 57
57

8616 8558 8499

58 59

8441 58

83 82 59 58

83 2 4

532 'S 547 '5 562
6 14 57 '3 589 12
601 6 12
13 II 624 635 II 644 9
9
653 8 661 668 7
6
674 6 680
5
685 689 4 692 3 694 2
695
696

8 o 0-99

C
I-00

P 0 0 99

.

45 46
47

495 I S8
5°°9 ' 5068 59

1685 1744

59

1803 59

8324 8265 59 8206 59

696 695 694

48
49

5126 58 5 185 59
57

1862 59
1920 ~:

81 4 8 58 8089 59
58

692
689 3
4

5° 51
52

5242 8 53 0 0 5 5357 57

1978 8 2036 58 2094 5

803 I 58
7973 57 7916

685 681 4 676 5

53 54

5414 57 54705565

21 5 I 2207

57
~~

~~ 7859 57
7803

669 7 662 7
8

55 56
57 58

5525

5580 55

5634 5687

54 53

2263 23 18 55
2373 55
2426 53

7747 55 7692 7638 54
75 84 54

654 8
646 10 636 626 10

59

574053 5'

2479

53 52

7532 52 52

6 II 15 12

60 61 62
63
64

579 1 5° 584 1 5890 49
49 5939 46 5985 46

253 1

2581 5°

263 I 2679
2726

5~
4
:~

7480 7430

50 .

7380 5~

733 2 :8 7284 45

603 12 591 13 578 14 564 15
549 IS

65
66
67 68
69

6031 6076
6II8

45 42

6160 42 6200 40

39

2772 281 7 45 2860 43 2902 42
41
2943 38

7 2 39 45

7 1 94 43 7 I 5 I 42

7 109 7069

40 39

534 16 5 18 16

458°42

8
I
q

467 19

70 71 72

6239 627637
63 I I 35

2981 301 9

38 35

30 54

7°3° 6993 376 6957 3

448 '9 4 2 9 19 410 20

73
74

6345 34 6377 32
3°

308834 3120 32
31

692 3 6891

34 32

31

39° 20 37° 21

75 7·6
77
78
79

6407
643 6 :~

6462

6487 25

65 10

23 21

3151 3 18o 29

3207 27

32 32

25 23

32 55 21

686° 28 6832 6805.27 6780 25
6757 23
22

349 21 328
6 22 3° 22 284 261 23
22

80 81 82
83
84

653 1

6550 19

6

68

8
I

5 15

65 8 3

6596 '3 II

3276
32 95 ~~
33 I 3 '5 3328
3341 1'32

6735 '9

67 16

6699 6683

'

7 6

I

6670

'3
II

239 23

216 192

24

6 23

I 9 24

145 24

85 86
87
88
89

6607 6617 10
6624 7 6629 5 6632 3

3353 9 33 62 3369 7
3374 5
3377 3

6659 6650 9 6642 8 6637 5
6634 3

121 24 97 24
73 24
49 2S 24 2~

9° 6633

3378

6633

°

The above table enables rectangular and polar geocentric coordinates to be calculated
for an observer in geographic (geodetic) latitude 4>, from the formulae:
p sin 4>' = (8 +00I568h x 10-6) sin 4> = (8 +0 0°478 H x 10-6) sin 4> p cos 4>' == (C +0-I568h x 10-6) cos 4> = (C +00°478 H x 10-6) cos 4>
where, h (H) is the height of the observer above the surface of the Earth in metres (feet)_
For reasonable heights, 001568h x 10-6 or 0-0478H x 10-6 can be added to p, and 4> - 4>'
can be used unchanged_

60

EXPLANATORY SUPPLEMENT

Parallax of the Moon

The topocentric hour angle (h), declination (0), and distance (r) of the centre of

the Moon are related to the geocentric coordinates (ho, 00, ro) by the exact equations:

F cos 0 sin h = cos 00 sin ho

== A

F cos 0 cos h = cos 00 cos ho - p cos 4/ sin 7T == B

F sin 0

= sin 00

- p sin 4/ sin 7T == C

where F = r/ro, 7T is the geocentric equatorial horizontal parallax of the centre of

the Moon, and p, 1>' are the geocentric coordinates of the observer. .

If the ephemeris position is known A, B, C can be calculated numerically;

then:

F2 = A2 + B2 + C2

tan h = A/B

sin h = A/(A2 + B2)~

cos h = B/(A2 + B2)~

tan 0 = C/(A2 + B2)~

sin 0 = C/F

cos 0 = (A2 + B2)~/F

Some simplifi~ation of this calculation can be achieved by rounding 00 and ho

to avoid interpolation in the trigonometric tables; if the values actually used are

o~ and h~, giving rise to 0' and h', then:

o = 0' + (00 - o~)

h = h' + (ho - h~)

F = F'

with errors not exceeding 1/60 of the" roundings ".

The reverse problem of deducing the geocentric position from the observed

coordinates is less readily solved. The equations take the form:

G cos 00 sin ho = cos 0 sin h

== A o

G cos 00 cos ho = cos 0 cos h + G p cos (1/ sin 7T == Bo

G sin 00

= sin 0

+ G p sin 1>' sin 7T == Co

where G = rolr = IfF.

These equations may be solved in precisely the same manner if r is known,
possibly by observation, and ro is deduced from the ephemeris, to provide a sufficiently accurate value of G to substitute on the right-hand side of the equations.

The general solution, in which h, 0 are observed and 7T is known (from the
ephemeris), is as follows. Let:
g = p cos ep' cos 0 cos h + p sin ep' sin 0
and substitute:
Go = I + g sin 7T + t (I + g2) sin2 7T
for G on the right-hand side of the equations. The adequacy of this approximation
for G, and the accuracy of the calculation, is checked by comparison with the
value of G determined from (Afi + B3 + q)~. Alternatively Go is taken as unity, G1 is formed from (Afi + Bfi + q)~ and used instead of Go to form G2, and the
process continued until G is known with sufficient accuracy. Two such approxi-
mations generally suffice.

The topocentric distance of the Moon is F times the geocentric distance, 'so that the apparent semi-diameter is greater than the tabulated value, the augmentation being IfF = G.

The formulae may be expressed in alternative forms to give directly the effect
of parallax un the coordinates. If Lla = U o - u, ..10 = 00 - 0 are the corrections

2F. COORDINATE AND REFERENCE SYSTEMS

61

to be applied to the topocentric position to give the geocentric position, then, in

terms of the geocentric position:

where

tan (h - ho) = tan Lla

p cos ep' sin 7T sin ho cos 00 - p cos ep' sin 7T cos ho

tan Llo

p sin ep' sin 7T (cos 00 - m sin 00) I - P sin ep' sin 7T (m cos 00 + sin 00)

m

cot

-1.' 'f'

cos -~ (h cos! (h

+
-

ho) ho)

cot cP' { cos ho - sin ho tan -} (h - ho) }

There are no simple corresponding expressions in terms of the topocentric position.

Example 2.2. Parallax of the Moon at Washington 1960 March 13 at 3h 17m 488'0 D.T.

For the purpose of this example,. the time of observation is taken to be exactly equivalent to 3h 18m 248'0 E.T.; the coordinates of Washington are taken from example 2.1.

The geocentric coordinates of the Moon are obtained by interpolation in the Ephemeris

as:

ao II h 22m 168'16

00 +3° 35' 24"'4

7T 57' 21"'71

The geocentric hour angle is then calculated as follows:

S.T. at Oh (A.E., page I I) D.T. of observation Increment (A.E., Table IX)
-,\

Then
sin 00 +0'06261 83 cos 00 +0'99803 76

-ao
Sum = ho
sin ho -0'46053 42 cos ho + 0.88764 20

A -0'45963 04 B +0.87290 13 C +0'0521926

F 0'9878972 tan h -0'5265548 sin 0 +0'0528320

hm
I I 22 30'237

3 1748 '000

+

32'493

5 08 15'75

-II 22 16'16

22 10 18·82

sm 7T +0'01668 51
P sin if>' sin 7T +0' 01042 57
P cos 4>' sin 7T +0'01299 88
h 22h 08m 558'38 0 + 3° 01' 42"'5

If h~ = 22h 10m 198 and o~ = + 3° 35' 20" had been used, no interpolation would have

been required in the trigonometric tables, and h', 0' would have been obtained as:

h' = 22 h 08m 558'56 giving h - ho = ao - a = - 1m 238'44

0' = + 3° 01' 38"'0 giviIlg

0 - 00 = - 33' 42"'0

For the alternative method five-figure tables and working suffice for a precision of 0"· I;

using h~, o~ as above, to avoid interpolation:
p sin 4>' sin 7T +0'01042 57 sin o~ +0'06260 p cos 4>' sin 7T +0'01299 88 cos o~ +0'99804

sin h~ -0'46052 cos h~ + 0.88765

tan (h _ h) -0'00598 62
o +0'98650
m = 0.88625 cot 4>' = 1'10499

-0'00606 81

where tan! (h - ho) is taken as one-half of tan (h - ho)

whence

h-
o-

ho
00

'4»"" ,? (~
tan 00 -

~) _
0-

++0~ '00S 96841

m 38
-Oh OI 2

• ....p '" .(.,
',~

i _0' 33' "·'0 . ' '.,

" ./

62

EXPLANATORY SUPPLEMENT

The reverse process, assuming the observed position and horizontal parallax to be:

a Il h 23m 398.60

0 +3° 01' 42"'5

7T 57' 21"'71

is as follows: h -Ill 51m 048.62

0 +3° 01' 42"'5

sin h -0'46591 18

sin 0 +0'05283 22

cos h +0.88483 12
sm 7T 0'0166851 sin2 7T 0'00027 84

cos 0 +0'99860 34
P sin 4>' sin 7T +0'0104257 P cos 4>' sin 7T +0'01299 88

The solution of the equations is commenced with:

g = +0'7214

Go = 1'012248

Then A o -0'46526 I I

B o +0·89675 35

Co +0'06338 56

from which G = 1'0122512. With this value of G,no change occurs in A o, Bo,or Co. Hence:

tan ho -0'5188283

ho -Ill 49m 418'17

sin 00 +0'06261 84

00 +3° 35' 24"'4

When the Moon is at transit h = ho = 0, and the parallax applies to the
declination only. The correction Llo to be applied to the topocentric declination

at transit to give the geocentric value is given by:

A" _
tan "-'0 -

I

P
-

s

in 7T sin (cp' -

P

S•In

7T

cos

(..
'f'

J

.0.0')"
-0

0

)

or, by using the observed declination and geocentric latitude of the observatory,

through the equivalent expression:

sin Llo = p sin 7T sin (f - 0)

This may be put in the approximate form:

Llo = 0'999988 p 7T sin (cp' - 0)

with an error not exceeding 0" '04.

Example 2.3. Parallax of the JV100n at transit at Washington

On 1960 August 7 the observed declination at transit is assumed to be - 14° 19' 57"·6. The U.T. of the observation is 5h 16m 488.88 and this is assumed to be equivalent to
5 h 17m 258 E.T. The coordinates of Washington are assumed to be those of example 2.1.

From the Ephemeris:

00 - 13° 3 1' 32 "
4>' - 00 + 52° 15' 25"
sin (4)' - ( 0) +0'79076 cos (4)' - ( 0) +0·61212 tan Ll8 = +0' 01 39390 +0'01409 10
+0'989210

7T 60' 40"·85 P 0'99 86 9 sin 7T 0'01765 04 P sin 7T 0'0 1762 73
Llo +0° 48' 26"'3

From the observation:

4>' - 0 53° 03' 50 "'3
sin (4)' - 8) + 0'7993 I

sin Llo +0'0 1408 97 Ll8 +0° 48' 26"'3

From the approximate formula:

Ll8 = 0'99998 8 x 0'99869 x

= 2906"'3 = 0° 48' 26"'3

The most important use of parallax corrections in the above form, when the Moon is not on the meridian, is in the reduction of observations made with the Markowitz dual-rate Moon camera. In the calculation of eclipses and the reduction of occultations the methods used, involving Besselian elements, do not require parallax corrections in the above form.

~--

-

-

--- .

2F. COORDINATE AND REFERENCE SYSTEMS

63

Parallax of the Sun and planets

For the more distant bodies such as the Sun, the planets, or comets, whose

parallax amounts to only a few seconds of arc, the above formulae may be greatly

simplified. It is sufficient to restrict the expressions to terms of the first order;

expressed as corrections to be applied to the observed positions they are:
LIn = 7T { P cos 4>' sin h sec S } LIS = 7T { P sin 4>' cos S - p cos 4>' cos h sin S }

where h, S may be replaced by ho' So'

When the horizontal parallax of the object is not available, it may be calculated
*from 7T = 8"·80/Ll where LI is the geocentric distance. In preliminary work on

comets and minor planets, where the geocentric distance is unknown, it is con-

venient to calculate parallax factors Pa and Pa for each observation; these may

then be used, once the geocentric distances are determined, to give the parallax

corrections in the form:
LIn = Pa/Ll

LIS = Pa/Ll

The parallax factors are calculated from:

Pa = 8"·80 p cos 4>' sin h sec S = 08'587 P cos 4>' sin h sec S

Pa = 8" ·80 (p sin 4>' cos S - p cos 4>' cos h sin S)

e - e The hour angle h is found from h =

a where is the local sidereal time at

universal time t, given by:
e = S.T. at Oh D.T. + t* - ,\

where S.T. at Oh D.T. is obtained from A.E., pages 10-17t, t* is the sidereal

equivalent of t, and ,\ is the longitude, measured positively to the west. Since t

is usually given in decimals of a day, t* is most readily determined from table 17.3.

Example 2.4. Parallax factors for a minor planet Observation of Vesta 1960 March 7 d 02 h 34m 218 V.T. at Johannesburg

t = Od' I0719 a 17h 57m 218'50 8 -18043' 31"'3

Sidereal time at Oh V.T. (A.E., p. I I) t* (from table 17.3) Correction for longitude, -,\ Local sidereal time, 8 Right ascension, a Local hour angle, h = 8 - a

hm
10 58.8
234. 8
+~
IS 25'9
1757'4 21 28'5

sin 8 -0'3210 cos 8 +0'9471 sec 8 + I '0559

sin h -0.6139 cos h +0'7894

P sin </>' -0'43867 P cos </>' +0.89824

pa -08 '342

pa - 1".65

For a fixed observatory parallax corrections may be further simplified by

forming two permanent tables. The first of these is similar to table 17.3 but gives

t* - ,\ directly in the first part of the table. The second table gives the coefficients

A, B, C in the expressions:

Pa = A sin h

Pa = B - C cos h

where*A 8"·80 p cos 4>' sec S = 08'587 P cos 4>' sec S B 8"·80 P sin 4>' cos S

C 8"·80 p cos 4>' sin S

*8"'794 = 0.'5863 from 1968.

tPages 12 to 19 in A.E. 1972 onwards.

--------.--- --- .-

64

EXPLANATORY SUPPLEMENT

Thus for Copenhagen (A = -oh50m'3, psin4>' = +0.82231, pcos4>' = +°'565°1)
the two tables start as follows:

t* -,\

8

A

B

C

d
0·00
'OI
·02 ·03 ·04

m
° 5°'3 I 04'7 I 19'2 I 33.6 I 48. I

0

+0

+°'33 1

+7'24

+0'00

I

'33 2

7'24

'°9

2

'332

7'23

'17

3

'33 2

7'23

'26

4

'33 2

7'22

'35

0·05 ·06

202'5 2 16'9

5 +°'333

6

'333

·07 23 1 '4 '08 245.8
'09 3 00'3

7

'334

8

'335

9

'336

Then h = S.T. at Oh D.T. + (t* - A) - a

+7'21 7'20 7'18 7' 17 7' 15

+°'43 '52 ·61 ·69 '78

An alternative method of correction when the geocentric distance is not

accurately known, is to modify the solar coordinates X, Y, Z so as to refer them to

the point of observation. These topocentric corrections are:

e e LlX = - a P cos 4>' cos = Ll xy cos

e LI Y = - a p cos 4>' sin = Ll XY sin ()

LI Z = - a p sin 4>'

= LIz

* where a is the Earth's equatorial radius in astronomical units = 426.64 x 10-7,

e and is the local sidereal time. The factors Ll xy and LIz are given for each obser-

vatory in the last two columns of the list of observatories in the Ephemeris.

Example 2.5. Parallax factors for a minor planet (continued)

Using the data of example 2.4:
Ll xy - 383 X 10-7
LIz = + 187 X 10-7

sin () = -0'7823
cos () = -0·6229

LlX +239 X 10-7 LI y +300 X 10-7
LIZ + 187 X 10-7

When the horizontal parallax is small, Sand C may often be taken as unity leading to a simplification of the formulae; this will not be applied to a fixed observatory.

Annual parallax

If 7T is the annual parallax of a star, and X, Y, Z are the solar coordinates, the

star is displaced from its mean place a, 8 by amounts Lla, Ll8 given by:

cos 8 Lla. = 7T (Y cos a - X sin a) Ll8 = 7T (Z cos 8 - X cos a sin 8 - Y sin a sin 8)

These expressions may be simplified by using the star constants c, d, c', d' (see

section 5):

Lla = 7T (Yc - Xd) Ll8 = 7T (Yc' - Xd')

Thus, corrections for annual parallax may be included with the aberration terms

of the reduction from mean to apparent place, as follows:

Lla = (C + 7TY)C + (D - 7TX)d
Ll8 = (C + 7TY)C' + (D - 7TX)d'

*a = 426'35 x 10- 7 a.u. from 1968.

- - - - - - - - - - - - - - - -~--------------------

---

2F. COORDINATE AND REFERENCE SYSTEMS

65

When the annual parallax is small enough, a further simplification can be made by
writing the expressions in the form:
.,1a = C (e + d 7T kI ) + D (d - k C 7T 2)
.,10 = C (e' + d' 7T kI ) + D (d' - e' 7T k 2)
where ki = R sec Elk, k2 = R cos Elk, in which R is the Sun's radius vector and k is *
the aberration constant = 20"'47. The variation in R throughout the year amounts to 1/60 of the mean value, so for a small parallax R may be taken as unity. The
method uses, in effect, modified values of the star constants, and is particularly
valuable in the routine calculation of an ephemeris. The maximum error in the
case of a Centauri is 0"'013.

Example 5.4 illustrates the application of this correction.

It should be noted that, in the above formulae, the corrections for annual parallax are, contrary to normal practice, given in the sense of (observed tabulated); they are included in the apparent places of the stars which are directly comparable with observation.

3. SYSTEMS OF TIME MEASUREMENT
A. INTRODUCTION
The systems of time measurement in use in present-day (post 1960) astronomy are a development of those in use before the variable rotation of the Earth was recognised. A complete appreciation thus requires a full understanding of the earlier concepts. However, consideration of both systems together is necessarily complicated, and it is desirable to have a general understanding of present-day systems before considering how they' have been developed.
In this introductory sub-section there is given a general description of the systems of time measurement in use in astronomy from 1960 onwards. Detailed developments are given in subsequent sub-sections.
A fundamental necessity of any system of time measurement is a one-to-one relationship' between the adopted numerical expression, or measure, of the time (usually in the conventional form of years, months, days, hours, minutes, seconds, and decimals of seconds) and some observable physical phenomenon that is either repetitive and countable, or continuous and measurable, or both. The phenomenon and the precise form of relationship are chosen so that the resulting time-system satisfies some particular requirement; the relationship may be simple and regular, as in a direct count of oscillations, or complex and irregular, as in the motion of the Moon. In all systems there is an additional practical requirement that the time should be free from short-period irregularities to permit interpolation and extrapolation by man-made clocks.
In astronomy there are three such particular requirements, each closely related to some natural observable motion and each leading to a different system of time measurement. The natural motions and the resulting time-systems are:
(i) The alternation of day and night, or the diurnal motion of the Sun: Universal Time.
(ii) The period of rotation of the Earth, or the diurnal motions of the stars: Sidereal Time.
(iii) The orbital motions of the Earth, Moon, and planets in the solar system: Ephemeris Time. Both here and in the following sub-sections, to allow a more logical development, the three time-systems are considered in the reverse order to that above. It is
66
- ------------------------------------- ---

3A. SYSTEMS OF TIME MEASUREMENT

67

emphasised that many complicated details, that do not affect the broad principles, are omitted in this sub-section.
Ephemeris time is theoretically uniform, since the length of the ephemeris second is fixed by definition. The relationship through which ephemeris time is determined in practice is that it is the independent time-argument of the ephemerides of the Sun, Moon, and planets. These ephemerides are thus to be computed in such a way that the ephemeris time determined from them is in accord with its theoretical definition. But, in particular, an observationally-determined value is used for the coefficient of the secular term in the mean longitude of the Moon. Ephemeris time determined from this relationship will depart from the theoretical uniform time in so far as the theory of the motions is inadequate, and the observationally-determined values are erroneous; the possible departure through these causes is small, of the order of two or three seconds in a century.

Sidereal time is directly related to the rotation of the Earth; equal intervals of angular motion correspond to equal intervals of sidereal time.

There is a fundamental difference between the two systems of time measurement: sidereal time reflects the actual rotation of the Earth; ephemeris time is defined to be uniform and is, in practice, determined through the motion of the Moon in its orbit round the Earth. It is thus not possible to express one system in terms of the other; the relation between them must be determined empirically. In fact, the speed of rotation of theEarthisknownto be subjectto unpredictable variations in terms ofephemeristime.

The diurnal motion of the Sun involves both the diurnal rotation of the Earth, related to sidereal time, and the motion of the Earth in its orbit round the Sun, related to ephemeris time. Although it would be possible to define a system of time measurement by means of a relationship to the hour angle of the Sun, this system could never be related precisely to sidereal time and could not, therefore, be determined by observations of star transits.

Universal time, for this reason, is directly related to sidereal time by means of a numerical formula; it contains no reference to ephemeris time and is not precisely related to the hour angle of the Sun. Although it is continuous with uniyersal time as practically determined in the past, it is only since the variable rate of rotation of the Earth was recognised that it has been realised that universal time is not a precise measure of mean solar time as generally understood; it is related to the hour angle of a point moving with the mean speed of the Sun in its orbit by means of an empirical correction, which must be determined by observation.

Universal time and sidereal time are rigorously related in such a way that an expression of time in one system can be converted, by means of the numerical formula, to an equivalent expression of time in the other. A knowledge of one is equivalent to a knowledge of the other. The two systems of time measurement are not independent and the use of one instead of the other is purely a matter of convenience: sidereal time is the more convenient for observations of star transits; universal time is the more convenient for many other purposes.

Uniform time and the laws of motion The astronomical reference systems of position and time are established

68

EXPLANATORY SUPPLEMENT

empirically, by observations of the apparent motions that define them; but these apparent motions reflect actual motions of the Earth and the other celestial bodies, and consequently the reference systems can be constructed on an exact dynamical foundation by means of the gravitational theories of the motions in the solar system. In particular, the measurement of time may be based upon the primary standard that is implicitly defined by the dynamical laws of motion. A clear understanding of the astronomical measurement of time on this basis requires two cardinal principles to be kept in mind:
(a) In astronomy, we are concerned, not with defining time, but only with measuring it. To define a measure of time, it is not necessary to know the ultimate nature of time; we need only devise practicable means for realising a unit of time and for comparing any interval of time with this unit.

(b) A measure of time, like any physical measure, is entirely conventional. Any particular measure may be adopted on the basis of its relative advantages for the specific purposes at hand; no restriction to a unique measure is imposed by physical principles, and no ultimate standard of reference is physically attainable.

For astronomical purposes, the most advantageous fundamental standard for the effective correlation and systematic representation of observed phenomena in terms of the measure of time is the independent variable of the accepted dynamical equations of motion. This measure of time may be characterized as the measure in which observed motions agree with the dynamical theories constructed from the laws of motion; in effect, it is therefore defined by these laws. In the terminology of the traditional formulation of the foundations of dynamics in terms of Intuitive concepts, this independent variable is " uniform time", measured in the invariable unit which, by the law of inertia, would be determined by successive equal rectilinear displacements of a particle moving under no forces. From the preceding principles, however, it follows that a uniform measure necessarily is uniform only by definition. No absolute standard of comparison is accessible, but this is immaterial; an accessible standard that does not lead to any contradiction between theory and observation is all that is required.
The measure of time defined by the laws of motion is not immediately accessible, but the dynamical theory of an observable motion provides a means of obtaining it from the empirical measure determined directly by this motion. Abstractly, uniform time is by definition the independent variable of the equations of motion, inclusive of effects required by relativity; operationally, a uniform measure of time is a measure in terms of which the observed motions of celestial bodies are in agreement with rigorous dynamical theories of these motions.

For designating a measure of time that is defined by the laws of dynamics, ephemeris time has been introduced. It is uniform in the sense that the length of the ephemeris second is defined to be a constant. The dynamical theories of the motions of celestial bodies are developed, in accordance with the fundamental laws of motion, so that the independent variable is ephemeris time as so defined. Beginning with 1960 the designation" Ephemeris Time" is used for the tabular argument in the fundamental ephemerides of the Sun, Moon, and planets.

- - ---------------------- -------------- --

3B. SYSTEMS OF TIME MEASUREMENT
B. ASTRONOMICAL MEASURES OF TIME AND RELATED CONCEPTS
1. Ephemeris time
Ephemeris time is a uniform measure of time depending for its determination on the laws of dynamics. It is the independent variable in the gravitational theories of the Sun, Moon, and planets, and the argument for the fundamental ephemerides in the Ephemeris.
The measure of ephemeris time has been chosen to agree as nearly as possible with that of universal time during the nineteenth century and it is unlikely that the two measures will differ by more than a few minutes in the twentieth century. Ephemeris time is accordingly expressed in the conventional units of centuries, years, months, days, hours, minutes, and seconds. The numerical values of the ephemeris time and the universal time at the same instant differ only slightly; to avoid possible confusion itis essential to indicate unambiguously which measure of time is beingused.
The fundamental epoch from which ephemeris time is measured is the epoch that Newcomb designated as 1900 January 0, Greenwich Mean Noon, but which is now properly designated as 1900 January 0, I2h E.T. The instant to which this designation is assigned is the instant near the beginning of the calendar year A.D. 1900 when the geometric mean longitude of the Sun, referred to the mean equinox of date, was 279° 4-1' 4-8" '04-.
This instant is definitive, but the determination of it depends on observations of the Sun, which are compared with an apparent ephemeris. The observations are themselves definitive, but the apparent ephemeris as deduced from the geometric mean longitude depends on the value adopted for the constant of aberration. All relevant observations and determinations have been made using 20"·4-7 for the constant of aberration; a change in this value will lead to a change in our determination of the instant of the fundamental epoch and thus to a corresponding change in the measures of ephemeris time assigned to all other instants. This particular difficulty could have been avoided by specifying the epoch as the instant when the geometric mean longitude of the Sun, reduced by the constant of aberration and referred to the mean equinox of date, was 279° 4-1' 27"·57; but there are objections
to the implied use of the" apparent mean longitude". *
The primary unit of ephemeris time is the tropical year at the fundamental epoch of 1900 January 0, I2h E.T.; the tropical year is defined as the interval during which the Sun's mean longitude, referred to the mean equinox of date, increases by 360°. The adopted measure of this unit is determined by the coefficient of 1', measured in centuries of 36525 ephemeris days, in Newcomb's expression for the geometric mean longitude of the Sun, referred to the mean equinox of date, namely:
L = 279° 4-1' 4-8".04- + 129602768"'13 T + 1".089 T2
The tropical year at 1900 January 0, 12h E.T. will accordingly contain: 3162096x0267068x.1630 x 36525 x 864-00 = 315 56925.974-7 eph eme·ns seconds
*See note in paragraph 4 on page 502.

7°

EXPLANATORY SUPPLEMENT

The following definition of ephemeris time, in accord with the above concepts, was adopted by the tenth General Assembly of the International Astronomical Union (Moscow, 1958; Trans. I.A.U., 10, 72, 1960) in the following terms (English translation, loco cit. page 500):

" Ephemeris time is reckoned from the instant, near the beginning of the calendar year A.D. 1900, when the geometric mean longitude of the Sun was 2790 41' 48"'04, at which instant the measure of ephemeris time was 1900 January Od 12h precisely."

The ephemeris second had already been adopted as the fundamental invariable unit of time by the Comite International des Poids et Mesures (Proces Verbaux des Seances, deuxieme serie, 25, 77, 1957) in the words:
" La seconde est la fraction 1131 556925,9747 de l'annee tropique pour 1900 janvier 0 a 12 heures de temps des ephemerides". *

As explained in sub-section D, on the historical development of ephemeris time, this definition of ephemeris time makes it equivalent to the system of time measurement used by Newcomb in his theories of the motion of bodies in the solar system. Newcomb considered it to be mean solar time and to be uniform in the sense of sub-section A; but it can be identified directly with ephemeris time so that the ephemerides derived from Newcomb's tables of the Sun and planets can be regarded as having ephemeris time as the independent time argument. The origin and rate of ephemeris time are defined to make the Sun's geometric mean longitude agree with Newcomb's expression; the symbol T in that expression therefore represents the measure of ephemeris time, not only in the theory of the motion of the Earth round the Sun but also in those of the heliocentric motions of the other planets. The first two terms of the Sun's geometric mean longitude are now thus defined to be absolute constants; the corresponding values for the Moon and other planets are, however, subject to possible revision to bring them into accord with observation. The mean longitude of any other planet, or even that of the Moon, could have been so used to define the origin and rate of a uniform time system; and ephemerides of the Sun, Moon, and planets could have been constructed with this time system as independent argument.

The measure of ephemeris time at the instant at which an observation of the Sun, Moon, or planet is made can be obtained by comparing the observed position with the gravitational ephemeris of the body; the ephemeris time will be the value of the argument for which the ephemeris position is the same as the observed position. In practice ephemeris time is obtained by the comparison of observed positions of the Sun, Moon, and planets with their corresponding ephemerides. Observations of the Moon, whose geocentric motion is much greater than those of other bodies, are the most effective and expeditious; but, even so, an accurate determination requires observations over an extended period. In practice universal time, which may be determined very accurately, with little delay, from observations of the diurnal motions of the stars, is used as an intermediary measure
of time; the difference in the two measures of time, ,1 T = E.T. - U.T., which
can be readily formed for each observation, is a suitable quantity for combination

*See additional note on page 95.

- -- --------------------- ---------------- ---

3B. SYSTEMS OF TIME MEASUREMENT

71

over an extended period. The practical determination of ephemeris time is discussed more fully in sub-section C.

Ephemeris time was originally defined (1950 Paris Conference on the Fundamental Constants of Astronomy, Colloques Internationaux du Centre National de la Recherche Scientijique, 25, 1-131, Paris, 1950; reprinted from Bull. Astr., 15, 163-292, 1950) by means of a formula, depending on the observed correction to the lunar ephemeris, for the correction Ll T to be applied to the measure of universal time to give ephemeris time. The use of this formula is precisely equivalent to determining ephemeris time by comparison of observations with the gravitational ephemeris of the Moon. This operational definition has now been superseded by the fundamental definition given above. The latter is independent of possible amendments of either theory or observation; but the former represents, to the best of our present theoretical and observational knowledge, the only practical way of realising the fundamental definition. If in the future a more precise lunar ephemeris is constructed, it will not affect either the definition or the measure of ephemeris time; but it will affect both the operational definiti.on and our determination of ephemeris time.

Julian date
To facilitate chronological reckoning astronomical days, beginning at Greenwich noon, are numbered consecutively from an epoch sufficiently far in the past to precede the historical period. The number assigned to a day in this continuous count is the Julian Day Number which is defined to be 0 for the day starting at Greenwich mean noon on! B.C. 4713 January I, Julian proleptic calendar. The Julian day number therefore denotes the number of days that has elapsed, at Greenwich noon on the day designated, since the above epoch. The Julian Date (J.D.) corresponding to any instant is, by a simple extension of the above concept, the Julian day number followed by the fraction of the day elapsed since the preceding noon.
Although introduced as a continuous count, and measure, of mean solar days the Julian day number and the Julian date can conveniently be applied to ephemeris time, in which case the Julian date will differ from the conventional one by Ll T; the Julian day number will represent the number of ephemeris days that have elapsed, at the preceding I2h E.T., since 12h E.T. on' B.C: 4713 I January I.' It is not necessary in this definition to know to what universal time this epoch corresponds, i.e. to know Ll T at the epoch; in fact the measure may be regarded as conventional, applicable to both systems of time measurement, as in the case of calendar dates. The terminology Julian Ephemeris Date (J.E.D.) may be used when necessary to distinguish the Julian date in ephemeris time with the day beginning at 12h E.T. from the Julian date in universal time with the day beginning at I2h D.T.; such a distinction may be essential in dating orbital elements, or in formulae for light-curves of variable stars, where the time must be given to a large number of decimal places. The fundamental epoch 1900 January Od 12h E.T. is J.E.D. 241 5020'0.
The value of J.D. -240 0000'5 is sometimes used to specify current dates and is known as the Modified Julian Date. It is recommended that the numerical definition be given whenever truncated values are used.

72

EXPLANATORY SUPPLEMENT

2. Sidereal time
In general terms, sidereal time is the hour angle of the (vernal) equinox, or the first point of Aries. Apart from the motion of the equinox itself, due to precession and nutation, sidereal time is thus a direct measure of the diurnal rotation of the Earth. To each local meridian on the Earth there corresponds a local sidereal time, connected with the sidereal time of the Greenwich meridian by means of the relation:

local sidereal time = Greenwich sidereal time - longitude

Sidereal time is conventionally measured in hours, minutes, and seconds, so that longitude in the above equation is measured (positively to the west) in time at the rate of one hour to IS°. An object transits over the local meridian when the (local) sidereal time is equal to its right ascension.

The sidereal time measured by the hour angle of the true equinox, i.e., the intersection of the true equator of date with the ecliptic of date, is apparent sidereal time; the position of the true equinox is affected by the nutation of the axis of the Earth, which consequently introduces periodic inequalities into the apparent sidereal time. The time measured by the diurnal motIon of the mean equinox of date, which is affected by only the secular inequalities due to the precession of the axis, is mean sidereal time. Apparent sidereal time minus mean sidereal time is the equation of the equinoxes due to the nutation; in the ephemerides immediately preceding 1960, it was called the" nutation in right ascension". The period of one diurnal circuit of the equinox in hour angle, between two consecutive upper meridian transits, is a sidereal day; it is reckoned from Oh at upper transit which is known as sidereal noon.

In the practical determination of time (see sub-section C) allowance must be made for the variation in the position of the meridian due to the motion of the geographic poles, and may also be made for short-period irregularities in the rate of rotation of the Earth. With this understanding Greenwich sidereal time may formally be defined as the Greenwich hour angle of the first point of Aries.

Sidereal time is determined in practice from observations of the transits of stars, either over the local meridian or, with a prismatic astrolabe, over the small circle corresponding to a constant altitude.

Owing to precession the mean sidereal day, of 24 hours of mean sidereal time, is about 08'0084 shorter than the actual period of rotation of the Earth; the apparent sidereal day, nominally of 24 hours of apparent sidereal time, differs from the period of rotation by a variable amount depending on the nutation.

Apparent sidereal time, because of its variable rate, is used only as a measure of epoch; it is not used as a measure of time-interval. Observations of the diurnal motions of the stars provide a direct measure of apparent sidereal time; as their right ascensions are measured from the true equinox. But in many practical methods of determining time the right ascensions are diminished by the equation of the equinoxes, so that mean sidereal time is deduced directly from the observations.

- - ---------------------- ------------------ ---

3B. SYSTEMS OF TIME MEASUREMENT

73

Greenwich sidereal date

In order to facilitate the enumeration 'of successive sidereal days the concepts of Greenwich Sidereal Date (G.S.D.) and Greenwich Sidereal Day Number, analogous to those of Julian date and Julian day number, have been introduced. The Greenwich sidereal date is defined as the interval in sidereal days, determined by the equinox of date, that has elapsed on the Greenwich meridian since the beginning of the sidereal day which was in progress at J.D. 0·0. The integral part of the Greenwich sidereal date is the Greenwich sidereal day number; it is a means of consecutively numbering the successive sidereal days beginning at the instants of transit of the equinox over the Greenwich meridian. The zero day is the sidereal day that was in progress at the beginning of the Julian era. The non-integral part of the Greenwich sidereal date is simply the Greenwich sidereal time expressed either in hours, minutes, and seconds, or in fractions of a sidereal day. These concepts can be applied equally well to mean or apparent sidereal time.

There is no direct relationship between Greenwich sidereal date and Julian ephemeris date, as the latter differs from the Julian date (in U.T.) by the unknown difference E.T. - U.T.

The relationships between Greenwich sidereal date, Julian date, and calendar date are considered in section 14H.

The ratio of the length of the mean sidereal day to the period of rotation of the
Earth is 0'99999 99029 07 - 59 x 10-12 T; the period of rotation is 1·0 + (97°93 + 59 T) x 10-12 mean sidereal days. These numbers are not rigorously
constant because the sidereal motion of the equinox due to precession is proportional to the length of the day, that is to the period of the rotation of the Earth, whereas the angular measure of the complete rotation is, of course, constant. However, the conceivable change in the period of rotation is such that the effect of a variation
in the daily precessional motion is inappreciable. The secular variations are almost inappreciable (see sub-section B.3).

3. Universal tiIne
Universal time is the precise measure of time used as the basis for all civil time-keeping; it conforms with a very close approximation to the mean diurnal
motion of the Sun. *
It is, and since the introduction of Newcomb's Tables of the Sun has been,
defined as 12 hours + the Greenwich hour angle of a point on the equator whose
right ascension, measured from the mean equinox of date, is:
R u = 18h 38m 458.836 + 86 40184s'542 Tu + oS,0929 T~
where Tu is the number of Julian centuries of 36525 days of universal time elapsed since the epoch of Greenwich mean noon (regarded as I2h U.T.) on 1900 January o. The expression for R u is identical with that given by Newcomb (Tables of the Sun, A.P.A.E., 6, part 1, page 9, 1895) for the right ascension of the fictitious mean sun, with the exception that Newcomb used T instead of Tu and did not specify in
*See note on page yi regarding the current basis of civil time scales. In general the term "universal time" (U.T.) may be identified throughout this Supplement with the system of U.T.I defined on page 86.

74

EXPLANATORY SUPPLEMENT

what measure of time T was to be reckoned. Newcomb, not recogmsmg the variable rotation of the Earth, considered that T was measured in mean solar time
applicable alike to orbital motions and to hour angles; as explained in sub-section RI, Newcomb's T may now be identified with ephemeris time. The point on the equator whose right ascension is Ru is not identical with the" fictitious mean sun" as defined by Newcomb; the right ascension of the fictitious mean sun is:

R E = I8 h 38m 458.836 + 864°1848'542 TE + 08'°929 Ti
where TE is the number of Julian centuries of 36525 days of ephemeris time elapsed since the epoch of I2h E.T. on 1900 January 0. RE differs from Ru by 0.002738 L1T where L1T is the difference E.T. - V.T.

The implications of this distinction are considered in sub-section B+

The measure of universal time at time Tu, expressed in hours, minutes, and seconds, is thus:

I2h + the Greenwich hour angle of the mean equinox of date - Ru

.The date expressed in the form either of a calendar date or of a Julian date (see sub-section B.I), is that correspondipg to the time Tu.
The Greenwich hour angle of the mean equinox of date is Greenwich mean sidereal time, by definition. At I2h V.T. the Greenwich mean sidereal time will therefore be RUl which may now be described as " the mean sidereal time of I2h V.T. "; it may thus be distinguished from the right ascension of the fictitious mean sun.
Although universal time is no longer definable as " I2h + the Greenwich hour
angle of the fictitious mean sun" it is sufficiently close, compared with the deviation between the mean sun and the true Sun, to justify the retention of the terms" mean solar time" and " mean solar day" in the sense in which they have been used in the past. The continued use of these descriptive terms is not to be regarded as identifying universal time with a precise measure of mean solar time; with this understanding, the danger of confusion is small. In this sense, universal time may be identified with Greenwich mean time.
As with sidereal time, there are local mean solar times corresponding to I2h +
the local hour angle of the point whose right ascension is Ru . These times are connected with universal time (Greenwich mean time) by means of the relation:

local mean time = universal time - longitude

The point whose right ascension is Ru is not observable and practical determinations of universal time are made, through the intermediary of sidereal time, by the

observations of the diurnal motion of the stars. For the practical calculation of

universal time, an ephemeris of sidereal time with argument universal time is

calculated from the relation:

.

Greenwich mean sidereal time = V.T. + Ru + I2h
for Oh V. T. of every day; at V. T. = Oh the value of the right-hand side is obtained

~ ------------------------ ---------------- --

3B. SYSTEMS OF TIME MEASUREMENT

75

by adding I2b to the expression Ru for the mean sidereal time of I2h V.T., and the relation becomes:

G.M.S.T. of Ob V.T. = 6h 38m 458.836 + 86401848'542Tu + 08,0929TJ
where Tu takes on successive values at a uniform interval of 1/36525. The apparent sidereal time is obtained by adding the equation of the equinoxes to the mean sidereal time. The sidereal time at Oh V.T. on successive dates, calculated from this expression, is tabulated in the ephemeris of Universal and Sidereal Times in A.E., pages 10-17. These tabular times are the Greenwich hour angles of the equinox that conventionally define the instants of successive midnights of universal time; they are the means of observationally identifying these instants, and of determining the universal time at any other instant. The instant that is designated as Ob V.T. each day is the moment at which the equinox during its apparent diurnal motion reaches a Greenwich hour angle equal to the value tabulated. At the instant of any observed Greenwich sidereal time, the interval which has elapsed since Ob V.T., expressed in sidereal time, is immediately obtained by subtracting the tabular sidereal time at Ob V.T. from the observed sidereal time at the instant; and the universal time at this instant is the equivalent measure of this interval in mean solar time.

Alternatively, use can be made of the tabulations, also given in A.E., pages 10-17, of the universal times corresponding to the instants of Ob Greenwich (mean and apparent) sidereal times, that is to the instants at which the mean and true equinoxes transit over the Greenwich meridian. An observed sidereal time may be converted to the equivalent interval of mean solar time, which is then added to the tabular universal time to give the universal time at the instant of observation. Examples of the methods of calculation and use of these tables are given in sub-section C.

The mean solar measure of an interval is obtained by multiplying the sidereal

measure by the ratio of the sidereal day to the mean solar day. The mean solar

day, of 24 mean solar hours, is the interval of time between the two instants at

which the equinox reaches the tabular hour angles for two consecutive dates,

corrected for the variations of the meridian due to the motion of the geographic

poles and to variations of the vertical. From this formal definition and the

conventional method of calculating the tabular hour angles of the equinox that

determine Oh V.T. on successive dates, it follows that the hour angle which the

equinox describes during one mean solar day consists of a complete circuit of 24h

plus a further angle equal to the tabular increase in the mean sidereal time of I2h

D.T. for a numerical increase in T of one day. The interval of mean sidereal time

in a mean solar day is therefore:

b
24

+

864°1848'542 + 08' 1858
36525

Tu

=

866368'55536°5

+

08'°000°5°87

Tu

and the ratio of a sidereal day of 864°° mean sidereal seconds to this interval is:

mean sidereal day mean solar day

=

0'997269566414- °'586 Tu

x

10-10

-

---

---

they like to confuse

EXPLANATORY SUPPLEMENT

Inversely, the ratio of the mean solar day to the mean sidereal day is:

+ 8_-6-6"-3-6--B--'5--5"5'-3"-6"'0~~5;-;- 0 8 '00000 =5_0__8__7'___T_=u
864°08

=

1'°027379°9265

+ °'589 T u

x

1 0 -10

Disregarding the inappreciable secular variations, the equivalent measures of the

lengths of the days are:

mean sidereal day mean solar day

2311 56m °48 '°9°54 of mean solar time 2411 03 m 56B'55536 of mean sidereal time

The conversion tables 17.1 and 17.2 are based on these values.

The determination of mean solar time by the established method of converting the elapsed interval since Oil U.T. from sidereal measure to mean solar measure with a fixed conversion factor keeps the ratio of the mean solar day to the sidereal day constant, irrespective of variations in the rate of rotation of the Earth. These variations cause inequalities in mean solar time as conventionally determined from the tabular hour angles of the equinox that formally define Oil U.T., and the length of the mean solar day is slightly variable; but the ratio of the sidereal and the mean solar measures is not altered by variations in the rotation of the Earth. The effect on the length of the mean solar day of the variations in the daily motion of precession is entirely inappreciable, as precession affects the hour angle of the equinox and the right ascension of the mean sun alike. The measure of mean solar time depends only upon the motion of the equinox in hour angle that is due to the rotation of the Earth; the ratio of the mean solar day to the period of rotation is constant to 12 decimals or more.

The numerical value of this ratio is 1'00273 78119 06; the period of rotation
of the Earth in mean solar time is:
od'99726 96632 42 = 2311 56m °48 '°989° 4
and the rate of rotation is 15"'°41067 per mean solar second.

Universal time is obtained, through the intermediary of sidereal time, from observations of the transits of stars. It is thus subject to the same irregularities (divided by the factor 1'°°2738) as those affecting the determination of sidereal time (see sub-section B.2), namely the variation in the local meridian due to the motion of the geographic poles and the short-period variations in the rate of rotation. These irregularities are removed to provide a measure of time which is free of short-period variations (see sub-section C).

4. The ephemeris meridian Ephemeris time is independent of the rotation of the Earth and is consequently unsuitable for the calculation of hour angles, which do depend on that rotation. For facilitating practical calculations of phenomena that depend upon hour angle and geographic location, the concept of an auxiliary reference meridian, known as the ephemeris meridian, has been introduced. The position of the ephemeris meridian in space is conceived as being where the Greenwich meridian would have been if the Earth had rotated uniformly at the rate implicit in the definition of
ephemeris time; it is 1'°°2738 LI T east of the actual meridian of Greenwich on the surface of the Earth, where LIT is the difference E.T. - U.T.

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- _ - - - - - - - - - .. --------~ - -

3B. SYSTEMS OF TIME MEASUREMENT

77

When referred to the ephemeris meridian, phenomena depending on the rotation of the Earth may be calculated ·in terms of ephemeris time by methods formally the same as those by which calculations referred to the Greenwich meridian are made in terms of universal time. The hour angle and the meridian transit of the equinox, which determine the tabular sidereal time at Oh universal time and the universal time at Oh sidereal time, are referred to the actual geographic meridian of Greenwich. The numerical value formally obtained from the same numerical relation as that used to compute the sidereal time at Oh universal time, but with T reckoned expressly in ephemeris time, is the hour angle of the equinox referred to the ephemeris meridian at Oh ephemeris time, and is called Ephemeris Sidereal Time (E.S.T.). Numerically, therefore, the tabular values of sidereal time at Oh universal time are equally the values of ephemeris sidereal time at Oh ephemeris time. Ephemeris transit occurs at the instant when the ephemeris sidereal time is equal to the right ascension.

The hour angle of an object referred to the ephemeris meridian is known as the Ephemeris Hour Angle (E.H.A.) of that object; it may be calculated from the relation:

ephemeris hour angle = ephemeris sidereal time - right ascension

Longitude measured from the ephemeris meridian is distinguished by the term ephemeris longitude; the ephemeris longitude of a place at which the local hour angle has a particular value may be obtained by taking the difference between the local ·and ephemeris hour angles.

All calculations into which the rotation of the Earth enters may be carried out in terms of ephemeris time, referred to the ephemeris meridian, in precisely the same way as in universal time referred to the Greenwich meridian. In the former case, the precise positions of the meridians on the Earth's surface, specified by their ephemeris longitudes, will not be known until L1 T is known; in the latter case a value of L1 T is necessary before the tabulated ephemerides can be interpolated to universal time. The use of the ephemeris meridian enables such calculations to be carried out precisely as far in advance as required; as soon as a sufficiently accurate value of L1 T can be extrapolated, or determined, the longitudes and hour angles can be referred to the Greenwich meridian and the times, in E.T., expressed in terms of V.T. This procedure is followed in predictions of the general circumstances of eclipses.

Apart from its practical advantages the concept of the ephemeris meridian is valuable in providing a clear picture of the relation between ephemeris time and universal time. At any instant:

E.T. -+- I2h = the ephemeris hour angle of the fictitious mean sun, whose
right ascension is RE ephemeris sidereal time - RE
D.T. + I2h = the Greenwich hour angle of the point whose right ascension is Ru
= Greenwich sidereal time - Ru

-

---- -

78

EXPLANATORY SUPPLEMENT

If LlT is the excess of the measure of E.T. over that of D.T., I.e. E.T. - D.T.
LlT, then:
RE = Ru + 0'°°2738 LlT E.T. = D.T. + ephemeris sidereal time - RE
- Greenwich sidereal time + Ru
= D.T. + 1'°°2738 LlT - °'°°2738 LlT = D.T. + LlT
At a time Ll T later the Greenwich meridian will have moved through an angle 1.002738 Ll T and thus will be in the same position as the ephemeris meridian at the earlier time; and the right ascension Ru will have increased by °'°°2738 Ll T and thus will be RE , the same as the right ascension of the fictitious mean sun at the earlier time. The relationship between the ephemeris meridian and the
fictitious mean sun at any instant is precisely the same as that between the Greenwich meridian and the point whose right ascension is R u at a time Ll T later; the two systems of time measurement are identical except that the system of universal
time relates to a time Ll T later than that of ephemeris time.

The speed of rotation of the ephemeris meridian is such that it makes one complete revolution of 360°, relative to the mean equinox, in 23 h S6m °48'°989° 4 of ephemeris time; the ephemeris meridian coincided with the Greenwich meridian at some date between 1900 and 1905.

EPHEMERIS MERIDIAN
I I I J I I I I I
r-

GREENWICH
OR UNIVERSAL
MEFliDIAN

LOCAL MERIDIAN

EPHEMERIS UNIVERSAL

MEAN

MEAN

SUN

SUN

,I

E.S.l:

:
I

12 h + E.l:

-

I I

E.H.A.E.MS. E.H.A. Sun

I I
I E" E.T.
I

EPHEMERIS LONGITUDE I GREENWICH OR UNIVERSAL LONGITUDE

12" +U.T.

-

-

U.S.T. (or S.T.)

L.H.A. Sun

I

U H.A. Sun UH. A.U.M.S

J I

I

E. U.T

I I

I0002738<1T)

TRUE SUN

EQUINOX

R.A.E.M.S =A,

RA. Sun

R.AlJ.M.S.-R u

Figure 3.1. Relations between E.T. and U.T. and related concepts

E.T. = Ephemeris time (T)
E.S.T. = Ephemeris sidereal time E.M.S. = Ephemeris mean sun
R.A.E.M.S. = Right ascension of E.M.S.
E.H.A. = Ephemeris hour angle
Eq. E.T. = Equation of ephemeris time

U.T. = Universal time (T + LlT)
U.S.T. = Universal sidereal time U.M.S. = Universal mean sun
R.A.U.M.S. = Right ascension of U.M.S.
U.H.A. = Universal hour angle
Eq. U.T. = Equation of universal time

U.S.T. and U.H.A. are identical with G.S.T. (Greenwich sidereal time) and G.H.A. (Greenwich hour angle)

The accompanying diagram (figure 3.1), which is intended solely for illustration, shows clearly the relationship between the two systems: E.T. and the ephemeris meridian in the upper part of the diagram, and D.T. and the Greenwich meridian in the lower. In the diagram certain unconventional terminologies and

------------ ---

3B. SYSTEMS OF TIME MEASUREMENT

79

notations have been introduced to facilitate comparison; these are only used in this limited context. A clear distinction is' drawn between the fictitious mean sun (termed the ephemeris mean sun) with right ascension RE, and the point (termed the universal mean sun) with right ascension Ru.
Two distinct concepts, termed -respectively the equation of ephemeris time and the equation of universal time, have been introduced to replace the single concept" equation of time". On the one hand, the equation of ephemeris time is the logical successor to the equation of time regarded as the excess of the hour angle (or defect of right ascension) of the true Sun over that of the fictitious mean sun; and this is the quantity tabulated in the Ephemeris for Oh E.T. under the heading" Equation of Time". On the other hand, the equation of universal time is the logical successor to the equation of time regarded as the excess of apparent solar
time over mean solar time; this is the quantity required to convert I2h + V.T.
into the G.H.A. of the Sun, but it cannot be tabulated without a knowledge of Ll T.

As from 1965 the tabulation in the Ephemeris of the equation of time will be
replaced by the tabulation of the E.T. of ephemeris transit of the Sun. The term "equation of time" will thenceforward be used exclusively for the concept
termed here" the equation of universal time". The equation of time will then be
defined as the correction to be applied to I2h + V.T. to obtain G.H.A. Sun, or more generally the correction to be applied to I2h + L.M.T. to obtain L.H.A.
Sun; it is now so tabulated in the almanacs for navigators and surveyors. The
concept of the equation of ephemeris time will no longer be used.

5. Mean solar time
The purpose of this sub-section is to describe the historical development of the concept of mean solar time, prior to the realization of the variability of the rotation of the Earth, and to discuss the consequences of that variability upon the definition of universal time.
A reckoning of time which conforms more or less closely to the recurrence of daylight and darkness determined by the diurnal motion of the Sun, and which is quickly obtainable with high precision from observation, is a practical necessity. Because of the variations in the rate of motion of the Sun in hour angle, due to the inequalities in the annual motion along the ecliptic and to the inclination of the ecliptic to the equator, the measure of time that is directly defined by the actual diurnal motion of the Sun, known as apparent solar time, is impracticable for the purpose of precise timekeeping. Instead, mean solar time was introduced, determined by the apparent diurnal motion of an abstract fiducial point at nearly the same hour angle as the Sun, but located on the mean celestial equator of date and characterized by a uniform sidereal motion along the equator at a rate virtually equal to the mean rate of the annual motion of the Sun along the ecliptic. Relative to any meridian of longitude, this point has a diurnal motion in hour angle virtually the same as the average diurnal motion of the Sun, and uniform except for variations of the local meridian; the position in hour angle is never more than 16m from the Sun.

80

EXPLANATORY SUPPLEMENT

The precise position of this moving point was abstractly defined by an expression for its right ascension, which fixes its position among the stars at every instant and is a means of determining its diurnal motion from the observable diurnal motions of the stars. The practice in the past has been to adopt for the right ascension, measured from the mean equinox of date, an expression as nearly identical with the expression for the mean longitude of the Sun as is possible, consistent with a sidereal motion at a constant rate. This expression -for the right ascension differs from that for the mean longitude of the Sun by only a slight, progressively increasing, excess of 08 '°2°3 T2 where T is the number of centuries from 1900, due to the secular acceleration of the Sun and to the different rates of the general precession on the ecliptic and the equator. This abstract fiducial point has therefore traditionally been known as the fictitious mean sun; but it has no physical counterpart, and the term is essentially only a name for a mathematical expressIOn.

The system of measuring and determining mean solar time was expressly devised to obtain a measure in agreement with the rotation of the Earth, because, prior to the realization that the rate of rotation is variable, the measure of time that it defines was considered to be uniform. It was for the purpose of obtaining a uniform measure in this way that mean solar time was defined in terms of the diurnal motion of a fictitious mean sun, not by supposing the actual mean sun transferred to the equator, since the mean motion of the Sun in longitude has a secular acceleration.

The definition of the measure of mean solar time was obtained, in the form of the relation to sidereal time, from the formula for the right ascension of the fictitious mean sun. On the Greenwich meridian, in terms of the position of the mean equinox and the position of the fictitious mean sun relative to the mean equinox, mean solar time was defined as:
G.H.A. mean equinox of date - R.A. fictitious mean sun + I2h

For the right ascension of the fictitious mean sun, the numerical formula from whatever tables of the Sun were in current use was adopted. The measure represented by this expression is universal time; it is the mean solar time on the Greenwich meridian reckoned in days of 24 mean solar hours beginning with Oh at midnight, and is the conventional standard measure of mean solar time.

However, because of the variations in the rate of rotation of the Earth, universal time, so defined, does not rigorously conform to the traditional geometric interpretation that originally motivated this method before these variations had been recognized. The right ascension of the fictitious mean sun increased by I2h was taken as the value of the hour angle of the mean equinox to define Oh V.T. in order that mean midnight would be the instant of lower meridian transit of the fictitious mean sun, and the measure of mean solar time at any other instant, reckoned from midnight, would be the hour angle of the fictitious mean sun increased by I2h. In practice, to obtain the tabular values of the hour angle ofthe mean equinox that determine successive intervals of a mean solar day, the right ascension of the mean sun was calculated from successive values of T at uniform

- - - - - - - - - - - - -- --- ~--------~~~--~------

---

3B. SYSTEMS OF TIME MEASUREMENT

81

numerical intervals of 1/36525. The instants at which the equinox reaches these tabular hour angles during its diurnal motion depend on the variable rotation of the Earth, and are at slightly unequal intervals of uniform time; consequently, the actual amount of the sidereal motion of the fictitious mean sun during successive mean solar days is not invariable, and the hour angle of the mean sun at midnight depends on the accumulated departures of the sidereal motion from the tabular amounts.

In contrast, Newcomb's expression was intended to represent a variation of right ascension entirely independent of the rotation of the Earth and due to a rigorously uniform sidereal motion of the mean sun that increases its right ascension by a constant amount per unit increase in the numerical value of T, and to the motion of the equinox that is caused by the general precession in right ascension.

The hour angle of the mean equinox and the actual right ascension of the mean sun increased by 12h do not both reach the tabular value of the mean sidereal time of Oh V.T. at identically the same instant. The tabular value is, by definition, the hour angle which the equinox reaches at mean midnight, but is not precisely equal to the right ascension of the fictitious mean sun increased by 12h at this instant. At this hour angle of the equinox, the fictitious mean sun is not exactly
on the lower meridian; the designation" Right Ascension of Mean Sun + 12h ",
sometimes applied to the sidereal time of Oh V.T. prior to 1960, is inexact when the departure of mean solar time from a uniform measure is explicitly recognized, and was therefore eliminated from the Ephemeris when a formal distinction was made between universal time and ephemeris time. In the expression for the right ascension of the fictitious mean sun, the inequalities are entirely due to the motion of the equinox, and strictly T should be interpreted as denoting a uniform measure of time; but the practical procedure is equivalent to reckoning T in mean solar days. This is immaterial for the purpose of defining a formal measure of time; but it has the consequence that, geometrically, mean solar time is not exactly the hour angle of the fictitious mean sun increased by 12h as it ordinarily has been described, and likewise the mean solar day is not exactly the period of one diurnal circuit of the fictitious mean sun in hour angle as it would be were there no variations in the rate of rotation of the Earth.

The operational procedure used in practice for determining universal time constitutes the actual definition, and supersedes the traditional descriptive characterization. Geometrically, mean solar time and the mean solar day are determined, not by the meridian transit and the hour angle of the fictitious mean sun, but entirely by the diurnal motion of the vernal equinox, in accordance with a conventional formula that specifies a prescribed relation that mean solar time shall have to the observed sidereal time measured by the hour angle of the equinox. The instant-of Oh V.T. is precisely defined by the numerical expression from which the tabular sidereal times of Oh V.T. are calculated; universal time as obtained in accordance with the established practical method, from the observed sidereal time at the instant and the tabular sidereal time at Oh V.T., is essentially a formal measure defined by this abstract expression.

They are trying to say that the addition of the new time is not a direct representation of reality?

- - - - ----------~-------

EXPLANATORY SUPPLEMENT Although this conventional formal measure of time is not the exact equivalent of the traditional geometric representation of mean solar" time, it is numerically identical with the measure of mean solar time that always was actually obtained in practice. Likewise, it is characterized by being strictly in accordance with the measure of time defined by the rotation of the Earth; the mean solar day, when determined from observations of stars and corrected for variations of the meridian, is rigorously proportional to the period of the rotation.
C. THE PRACTICAL DETERMINATION OF TIME
Accurate timekeeping depends upon determining the error of a clock on successive nights by means of determinations of time from astronomical observations. The observed measure of time compared with the reading of the clock at the instant of observation gives the error of the clock; from the successive clock errors, the rate of gain or loss is found, with which the clock error at any intermediate instant may be obtai,ned by interpolation, and over limited periods in advance by extrapolation.
For timekeeping of the highest precision, quartz-crystal clocks have entirely superseded the pendulum clock. A perfect clock, which would run uniformly and have an absolutely constant rate, has not been realized; but the best clocks now available have rates more uniform than the rotation of the Earth. Atomic oscillators are also becoming an important aid in timekeeping, although it is not yet known whether the gravitational and atomic time scales are identical.
Crystal-controlled clocks are more accurate than the individual nightly determinations of time by observation. The clocks are used in practice to smooth out the random errors of observation from night to night, as well as to interpolate between observations; the crystal oscillators that constitute the primary time standards vary in frequency from day to day by only about 2 parts in 1010. However, the length of time over which the clock rate may be extrapolated with confidence is inevitably limited. To maintain a precise standard of time, and to make exact measurements of long intervals, continual direct determinations from astronomical observations are essential.
The determination of sidereal time by observation. To determine the hour angle of the equinox by observations of stars, the location of the equinox among the selected stars is found from ephemerides of their apparent positions. The diurnal motions depend upon the instantaneous rotational motion of the Earth determined by the position of the axis in space and within the Earth, and ,by the 'rate of rotation. The instruments are necessarily oriented with reference to local gravity. Consequently, the measure of time obtained directly from the immediately observed positions of the stars in their diurnal circuits is the apparent sidereal time referred to the instantaneous local meridian. In principle, the time
- - ------------------------- -------------------- ---

3C. SYSTEMS OF TIME MEASUREMENT
may be found from observations of stars at any point of their diurnal arcs, and many different methods have been used,. depending on circumstances and on the precision needed.
For meridian observations, the most precise instrument is the photographic zenith tube, for which no corrections are required for level, azimuth, collimation, or flexure. Each observation gives a measure of both the time and the latitude. Determinations of time by extra-meridian observations, comparable in precision to determinations with the photographic zenith tube, may be made with the Danjon impersonal prismatic astrolabe. With this instrument, the stars are observed when at an altitude of 60°. Each observation of one star gives a linear relation between time, latitude, and declination; two groups of stars are observed, one before midnight and one after midnight. Brief descriptions of these instruments are given in section ISB.
The external probable error of the time determined from the observations on one night by these methods is of the order of ± 4 milliseconds.
The relative positions of the stars observed with these instruments are determined from the observations themselves, and thus are independent of errors in star catalogues. But even though the star places are mutually consistent, they are still dependent on the particular coordinate system (or " equinox ") to which they are referred; different systems would give rise to differing determinations of time. The International Astronomical Union recommended in Stockholm in 1938
(Trans. LA. u., 6, 342, 1939) that the system of the FK3 be used; and the adopted
practice is equivalent to using a zero determined by the average of the FK3 stars in the corresponding declination belt. The FK3 system will be replaced by that of FK4 as soon as it becomes available (Trans. I.A.U., 10,79, 1960).
The varying rate of gain or loss of the clock on apparent sidereal time, and the accumulated error at the times of observation, depend both upon the irregularities of the clock and upon the inequalities in sidereal time. To facilitate the separation of the clock irregularities from the variations in the measure of time, in order to determine accurate clock errors and rates, the transit ephemerides of the stars are often expressed in terms of a more uniform argument than apparent sidereal time, by calculating the mean sidereal time of transit and, for convenience, further converting it to mean solar time.
The mean sidereal time at transit is obtained by omitting from the apparent right ascension the terms of the reduction for nutation that are independent of the coordinates of the star; these terms, common alike to all stars, represent the equation of the equinoxes, which causes the inequality in sidereal time that is due to the nutation of the axis of the Earth. The remaining terms of the reduction for nutation, peculiar to each star, represent the irregularities in the diurnal motion of the star that are produced by the nutation of the axis.
As long as a particular inequality in sidereal time is negligibly small compared to the irregularities of the clock and the inevitable errors of the observations, it may be disregarded in calculating the right ascensions of the stars and in reducing the
- - - - - - - ~--

EXPLANATORY SUPPLEMENT
observations. With the continual increase in the accuracy of observations and the development of more precise clocks, an increasing number of the inequalities have successively become distinguishable from the irregularities of the clock. To obtain a standard of comparison that is as nearly uniform as the running of the clock, successively greater refinements in computation have been necessary, by the inclusion of additional terms of the nutation and, more recently, the application of corrections for the variations of the meridian due to the polar motion. Moreover, the rates of the crystal oscillators now available are so nearly uniform, and the accuracy of the observational comparisons with the stars is so great, that it has also become the practice to include corrections to the observed time for the periodic seasonal variations in the rate of rotation of the Earth.
The calculation of mean solar time. The definition of universal time was left unchanged when ephemeris time was formally introduced into astronomical practice. The practical method of determining universal time that was in established use before 1960 was retained, and the numerical reckoning of universal time was continued without discontinuity except for increased precision resulting from the use of improved values of the nutation.
The sidereal time (hour angle of first point of Aries) at Oh universal time, and the universal time at Oh sidereal time (transit of first point of Aries), which formerly were included in the ephemeris of the Sun, are tabulated in the separate ephemeris of Universal and Sidereal Times in A.E., pages 10-17:both for the mean equinox of date and for the true equinox with the short-period terms of nutation included. This ephemeris also contains the equation of the equinoxes, which in the volumes immediately preceding 1960 was designated as the nutation in right ascension and was included with the ephemeris of the Sun.
In the tabulations for Oh V.T., the argument is the calendar date and the equivalent Julian date. In the tabulations for Oh S.T. the argument is the Greenwich sidereal date (G. S.D.), defined as the number of sidereal days determined by the equinox of date that have elapsed at Greenwich since the beginning of the sidereal day which was in progress at J.D. 0'0. The integral part of the G.S.D., the Greenwich sidereal day number, is a means of consecutively numbering successive sidereal days. (See sub-section B.2.).

Example 3.1. Universal and sidereal times 1960 March 7 at Oh U.T.
Julian date at oh on 1960 March 7 (A.E., p. 2) Julian date at epoch from which Tu is measured Interval in days, d Fraction of Julian century, Tu = d/36525

R u + I2h = 6 h 38m 458.836

+86401848'542 Tu = 236"'55536 049d

+

0"'0929 TJ = 08·00696(d/l0000)2

Sum

= Mean sidereal time at Oh

Equation of the equinoxes (iJif; cos E) = -0"'744 x 0'9174

Sum

= Apparent sidereal time at Oh

*On pages 12 to 19 in A.E. 1972 onwards.

243 7000'5 241 5020'0
2 1980'5 0. 601 79 32922 7
(; 38 45~836
4 20 05'1013 0'0336
10 58 50'971 - 0'046

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3C. SYSTEMS OF TIME MEASUREMENT

85

The universal time of transit of the mean equinox is obtained by: D.T. of transit = 0'99726 95664 (24h - mean S.T. at Oh) = (24 h - mean S.T. at Oh) (I - °'°°273 04336)

24h - mean S.T. at Oh on 1960 March 7 -0'00273 04336 (24h - mean S.T. at Oh) (A.E., Table VIII)
Sum = U.T. of transit of mean equinox Correction to true equinox ( -0'9973 L10/ cos E)
= -0'9973 x -0"'746 x °'9174

h

m

13 01

-2

12 59

01 '056

+ °'°46

Sum = D.T. of transit of true equinox

12 59 01' 102

The nutation in longitude (L10/) is obtamed from the series, and must be interpolated
to the D.T. required; the obliquity (E) is a constant to the precision here required. The D.T. of transit of the mean equinox can be obtained directly from the series:
17h 16m 258.628 - 2358'90946 18 (G.S.D. - 242 1634) - 08'°926 TJ

The practical calculation of universal time from the observed sidereal time with the aid of these tabulations is illustrated by the following example. For full precision it is necessary to use the quantities relating to the mean equinox (e.g. mean sidereal time), interpolating the equation of the equinoxes to the actual universal time concerned.

Example 3.2. Derivation of universal time from observed sidereal time
On 1960 March 7, in longitude 5 h 08m 158'75 west at approximately 2h local mean time, the observed apparent sidereal time was 13 h 05m 378'249; the corresponding U.T. (about 7h on March 7) is obtained as follows:

Observed local apparent sidereal time Equation of the equinoxes (interpolated to 7h U.T.)
Observed local mean sidereal time Longitude (add if west)
Greenwich mean sidereal time Reduction to mean solar time (A.E., Table VIII)
Equivalent interval of mean solar time D.T. of preceding transit of mean equinox (A.E., p. II)
D.T. of observation

hm
13 0S 37'249 - °'°46
13 0S 37'295
+ 5 08 15'75
18 13 53'045 - 2 59'207
18 10 53.838 March 6 d 13 02 56'966 March 7 d 7 13 50·804

Alternatively use can be made of the tabulated sidereal time as follows:

Greenwich mean side-eal time (as above)

18 13

Greenwich mean sidereal time at Oh U.T. on March 7 (A.E., p. I I) 10 58

Difference = mean sidereal time interval Reductio:1- to mean solar time (A.E., Table VIII)

7 15 -I

D.T. of observation

7 13

53'045 5°'971 02'074 11'269
50·805

The apparent sidereal time corresponding to a given U.T. may be calculated directly.
In this case the figures are the same as above; but the reduction from mean solar time to mean sidereal time (1 m 118'27°) is taken from A.E., Table IX, with the U.T. argument 7h 13m 508·805.

The universal time calculated directly from the immediately observed sidereal time referred to the instantaneous meridian is denoted by V.T.o. This measure of universal time contains inequalities due not only to the variations in the rate of rotation of the Earth but also to the variations of the meridian. In practice, the variations of the meridian due to variations of the vertical may be neglected,

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EXPLANATORY SUPPLEMENT

as they are too small in comparison with errors of observation to be significant except in an analysis of a long series of observations; but, because of the high accuracy that has been reached in timekeeping, the inequalities due to the polar motion have become of practical importance. The variations in the rate of rotation of the Earth comprise secular, irregular, and periodic seasonal and tidal inequalities. The tidal variations are almost inappreciable, and the secular variation becomes appreciable only after very long intervals; the irregular variations may reach relatively large magnitudes, but are highly erratic. The seasonal inequality is large enough to be of practical. significance; and, as far as observations have yet shown, it appears to be remarkably stable from year to year. Accordingly, beginning with 1956, in conformity with resolutions of the International Astronomical Union, determinations of universal time by the national time services have been corrected for the annual variation in the rate of rotation, and for the variation in the position of the meridian due to the motion of the geographic poles.

Corrections for the polar motion were first applied in daily practice at the Royal Greenwich Observatory, beginning with 1947. Previously, these corrections had been applied only in the annual analyses of time signals by the Bureau International de l'Heure. In 1955, a special Rapid Latitude Service was established by the International Astronomical Union, for determining the motion of the pole on a nearly current basis in order that accurate corrections to time determinations may be derived. Universal time reduced to an invariable mean Greenwich meridian by correcting U.T.o for the observed polar motion is denoted by the notation U.T.I. The corrections for each time station are issued periodically by the Bureau International de I'Heure; time signals are based on extrapolated values, and definitive time signal corrections on interpolated values.

The correction for seasonal variation is extrapolated a year in advance, and published by the Bureau International de l'Heure for use by all observatories engaged in the determination of time. The measure of universal time obtained by correcting U.T.o for the observed polar motion and for the extrapolated seasonal variation in the rate of rotation of the Earth is denoted by the notation U.T.2. The correction for the annual variation does not wholly eliminate the variability in the length of the mean solar day, but U.T.2 is virtually free of periodic variations. (See section 15A for further details).

The determination of ephemeris time. To determine the correction L1 T for reducing universal time to ephemeris time, an observed position of a celestial body recorded in universal time is compared with a gravitational ephemeris in which the argument is the measure of time defined by Newcomb's Tables of the Sun; by inverse interpolation in the ephemeris, to the value of the argument for which the tabular position is the same as the observed position, the difference of the two measures of time is immediately obtained.
Observations of the Moon are the most effective means for the practical determination of L1 T. However, a direct comparison, in the way just described, with the lunar ephemeris calculated from Brown's Tables of the motion of the Moon

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3C. SYSTEMS OF TIME MEASUREMENT
does not give iJ T immediately, because Brown's theory is not strictly gravitational and his tables are not in complete accord with Newcomb's Tables of the Sun. In terms of the departure of the Moon from Brown's tables, the relation of ephemeris time to' universal time, found from discussions of observations of the Sun, Moon, and planets over periods extending back to ancient times, is represented by:
iJT = +248'349 + 728'318 T + 298'95° T2 + 1.82144 B
where T is rer.koned in Julian centuries from 1900 January ° Greenwich mean noon, and where:
B = (Lo - Lc) + 10"'71 sin (14°°'° T + 240°'7)
- 4",65 - 12"'96 T - 5"'22 T2
in which L o is the tabular mean longitude of the Moon, and L o is the observed mean longitude, referred to Newcomb's equinox, at the observed universal time.
Brown's theory is reduced to a gravitational theory in which the measure of time is the same as defined by Newcomb's Tables of the Sun by eliminating the empirical term from the mean longitude of the Moon, and applying to the tabular mean longitude the further correction:
iJL = -8"'72 - 26"'74 T - II"'22 T2
Consequential corrections are required to some of the periodic terms in longitude, latitude, and parallax. Beginning with 1960, the lunar ephemeris is calculated from this amended theory, directly from the theoretical expressions for the longitude, latitude, and parallax, instead of from Brown's tables as formerly. This improved ephemeris has also been made available for 1952-1959 in the Improved Lunar Ephemeris.
The development of means for photographic determinations of the position of the Moon among the stars, and the introduction of the improved ephemeris of the Moon with which the observed position may be directly compared, enable iJ T to be obtained more expeditiously than by the methods previously available. Formerly, iJ T was determined principally by means of meridian observations of the Moon and observations of occultations of stars, compared with the tabular positions in the lunar ephemeris calculated from Brown's tables; the determination of a definitive value by these methods requires several years. From photographic positions of the Moon obtained with the dual-rate camera devised by Markowitz, accurate values of iJ T should be determined within a relatively brief period.
Strictly a distinction should be drawn between V.T. + iJT and E.T., when .1T is determined as above from observations of the Moon. V.T. + iJT differs
from E.T. in two main respects:
(a) by a quadratic expression in T of the form a + bT + CT2, the coefficients of
which have been observationally determined to be zero, but which almost certainly differ from zero by significant amounts (it should be noted that the term cT2 is of a
more fundamental physical character than a + bT);
(b) by any deficiencies that may be present in Brown's theory of the motion of

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EXPLANATORY SUPPLEMENT

the Moon, including revision of any constants involved; in particular Brown uses 1/294 for the flattening of the Earth.
Thus U.T. + .t1T may differ systematically from ephemeris time as defined by
reference to the Sun's mean longitude. This is of little consequence to astronomy since the values of .t1 T are the best that can be obtained and their significance is fully understood; but it could assume importance in relation to the precise determination of the unit of time. In so far as the use of the Ephemeris is concerned no formal distinction is necessary, and none is made; thus the same symbol .t1 Tis used to denote the actual difference E.T. - U.T., although it is realised that the observations do not relate directly to this quantity.

Only for comparatively recent years can reasonably accurate values of .t1 T be obtained from the available observations; but fairly reliable values may be determined back to the beginning of the nineteenth century, and approximate estimates may be made back into the seventeenth century. Table 3.1 gives the values that
were derived in a comprehensive investigation by Brouwer (A.J., 57, 125, 1952),
supplemented by other determinations for more recent years.

The results of a recent estimation of the variations of L1 T during the past three centuries are illustrated in figure 3.2. The large differences from the general trend of Brouwer's values are due to the use of a different value for the tidal deceleration in the Moon's mean longitude.

The annual values of .t1 T are tabulated for a limited interval ending with the current year in A.E., page vii or viii. For years up to 1948 inclusive, they are taken from Brouwer's smoothed values; for the later years, definitive values available
at the time the Ephemeris is prepared are supplemented by provisional and extra-
poiated values to extend the table to the current year. *

D. HISTORICAL DEVELOPMENT OF SYSTEMS OF TIME MEASUREMENT
Until the introduction of the pendulum clock in the latter half of the seventeenth century, no means of reasonably accurate timekeeping was available. Besides the sundial, methods had been known since ancient times for determining local time by observations of the Sun or stars, within the limits of accuracy of the existing instruments, and the concept of mean solar time together with the principles for determining the equation of time extends back to ancient Greek astronomy; but with the crude mechanical timekeeping devices that were available, satisfactory measurements of intervals of time for interpolating between astronomical observations could not be made. The earliest mechanical clocks introduced during medieval times were not much improvement, and the early pendulum clocks were not highly reliable; not until the late eighteenth century had clocks become sufficiently improved, and watches and chronometers sufficiently perfected, for accurate time to be generally available, especially at sea.
*Current years of the A.E. now show on page vii the relationships between LA.T., E.T., U.T.I and U.T.C. from 1956 onwards.

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3D. SYSTEMS OF TIME MEASUREMENT

89

As long as the only means of obtaining accurate time was by direct astronomical observation, apparent solar time was in general use for practical purposes, and it was the argument in The Nautical Almanac and other national ephemerides until the early nineteenth century. Determinations of local apparent time were commonly made by observing altitudes of the Sun or stars; this is still one of the most generally useful methods, especially at sea. Mean time when needed for any purpose was obtained by applying the equation of time to the apparent time.

The equation of time, in the sense of the correction to be applied to apparent time in order to obtain mean time, had been tabulated in the national ephemerides from their earliest inception, for the express purpose of regulating clocks and of determining the argument for entering astronomical tables. As clocks were improved, and chronometers were perfected and came into extensive use at sea, apparent time was gradually superseded during the late eighteenth and early nineteenth centuries by local mean solar time for general civil use. When apparent time was replaced by mean time as the argument in the national ephemerides, the equation of time was supplemented by the addition of an ephemeris of sidereal time at mean noon to facilitate the determination of mean solar time, independently of the equation of time, by the alternative method of calculating the mean time from sidereal time.

The equation of time has since come to signify the opposite of the original concept. It now denotes the correction for obtaining apparent time from the mean time kept by clocks and chronometers, which are regulated by determinations of mean time from observations of sidereal time.

Previous to 1925, mean solar time was reckoned from noon in astronomical practice. The mean solar day beginning at noon, I2h after the midnight at the beginning of the same civil date, was known as the astronomical day. Mean solar time reckoned from mean noon on the meridian of Greenwich was designated Greenwich Mean Time (G.M.T.); reckoned from mean noon on a local meridian, Local Mean Time (L.M.T.). Beginning with the volumes for 1925, universal time was introduced in the national ephemerides under various names, a .discontinuity of I2h being made in the arguments, so that December 31' 5 in the volumes for 1924 designated the same instant as January 1·0 in the volumes for 1925. In The Nautical Almanac the designation Greenwich Mean. Time (G.M.T.) was still used for the new reckoning, together with Local Mean Time (L.M.T.) where appropriate, whereas in The American Ephemeris the designation Greenwich Civil Time (G.C.T.) was adopted, together with Local Civil Time (L.C.T.). This confusion in terminology was finally removed by dropping both designations and substituting Universal Time (U.T.); it is, however, now called Greenwich Mean Time (G.M.T.) in the navigational publications of English-speaking countries.* Care is necessary to avoid confusion; to distinguish the two reckonings that have both been called Greenwich Mean Time, the designation Greenwich Mean Astronomical Time (G.M.A.T.) should be used for the reckoning from noon. The designation U.T. always refers to time reckoned from Greenwich midnight, even for epochs before 1925.

*In astronavigation the argument G.M.T. implies U.T.I, but in general communications G.M.T. usually means V.T.C. For astronomical purposes the term U.T. is preferable.