Ephemeris

EPHEMERIS. An ephemeris is a table giving, for stated points of time, the position or other numerical particulars of one or more of the heavenly bodies. Such tables are essential to the navigator, whose observations of the sun, moon, and stars would be valueless without a knowledge of their exact positions; to the astronomer who wishes to direct his telescope to a particular celestial object, or to check the time-keeping of his clock; and to the computer who predicts eclipses or the return of comets. The word ephemeris is frequently applied to a collection of such tables embodied in the form of an almanac, and it is used in that sense in the following remarks.

National Ephemerides.--The oldest of the five principal astronomical ephemerides is the Connaissance des Temps on des Mouvements Celestes, founded by J. Picard in 1679 and published at Paris under the auspices of the Bureau des Longitudes. Its most distinguished superintendent was Le Verrier, immortalized as the co-discoverer with Adams of the planet Neptune.

The

Nautical Almanac, which first appeared in 1766 for the year 1767, owes its existence to the initiative of Nevil Maskelyne, then Astronomer Royal. At that time no wholly satisfactory method of determining longitude at sea existed. The method of Lunar Distances, destined to become the standard method during the nineteenth century, but since discarded, was then impractic able because of the inaccuracy of the existing lunar tables, and because special knowledge and long and tedious computation were necessary in their use. When the tables of Tobias Mayer, of Gottingen, came into the hands of Maskelyne, he quickly realized their value to navigation. To make them practically use ful, however, it was necessary to prepare from them an ephemeris of the moon for every noon and midnight, which should be pub lished and made available to seamen. Maskelyne's representations to the Board of Longitude resulted in the establishment of the Nautical Almanac under his superintendence.

After the death of Maskelyne in 1811 the Nautical Almanac fell into disrepute and became notorious for its errors. As a result of many attacks made on it, principally by Francis Baily and Sir James South, the Admiralty appealed to the Royal Astronom ical Society in 183o. The society appointed a strong committee, whose many recommendations were accepted in their entirety and embodied in the Almanac for 1834. Apart from improvements in the basic data from which the ephemerides are constructed, and a few changes made in at the suggestion of the Royal Astronomical Society, the form of the Almanac remained substan tially unaltered until the issue of that for 1923, when the volume was completely re-cast. Changes necessitated by the alteration in 1925 of the beginning of the astronomical day from noon to midnight were made and consideration was given to the changed requirements of modern astronomers and of computers equipped with calculating machines.

As the greater part of the Nautical Almanac is not of interest to the seaman, the issue was commenced in 1914 of The Nautical Almanac; Abridged for the Use of Seamen, in which the material required for navigation was set up afresh, but only to the lesser degree of accuracy required by that science, Right Ascensions, for instance, being given to the nearest second only.

The third great national ephemeris emanated from Berlin, under the editorship of J. E. Bode. The first volume appeared in 1774, with the title Astronomisches Jahrbuch oder Ephemeriden fur das Jahr 1776. In 183o J. F. Encke changed the title to Berliner Astronomisches Jahrbuch. It is now produced by the Astronom isches Rechen-Institut, and is highly valued because of its presen tation of data in a form adapted to the needs of workers in dynamical astronomy. A supplementary annual volume entitled Kleine Planeten contains opposition ephemerides of over i,000 of the minor planets.

The Almanaque Nautico y Ephemerides Astronomicas, adapted to the Naval Observatory of San Fernando, near Cadiz, in Spain, appeared first in 1827 for the year 183o.

Finally the American Ephemeris and Nautical Almanac was first published at Washington in 1852 for the year 1855, the editor being Admiral C. H. Davis. Later, under the superintend ence of Simon Newcomb, this ephemeris stepped into the front rank and became the best of the national ephemerides.

In addition to the ephemerides already enumerated, nearly every other country publishes one of its own, based on one or more of the above mentioned almanacs. The reason for this is the necessity for having the explanation and directions for use in the language of the country concerned, rather than any special requirements of its astronomers and navigators.

International Co-operation.—Although five extensive na tional ephemerides are printed they are not all computed inde pendently. Conferences of the directors were held at Paris in 1896 and 1911, while more recently opportunities for co-operation have been afforded by the triennial meetings of the International Astronomical Union, one of whose "commissions" is devoted to ephemerides. The Nautical Almanac is responsible for the ephemerides of the sun and planets, using the tables prepared by Newcomb and G. W. Hill for the American Ephemeris, and of the moon, using the tables of E. W. Brown, which, since 1923, have replaced those of P. A. Hansen. The Connaissance des Temps prepares independent ephemerides of the sun and planets from the Le Verrier-Gaillot tables, and of the moon from Radau's tables based on Delaunay's theory. In addition it produces the apparent places of circumpolar stars, predictions of eclipses, and data for the four great satellites of Jupiter, based on the tables of R. A. Sampson. The American Ephemeris is charged with the production of elements of eclipses and occultations from the sun and moon data supplied by the Nautical Almanac, of sun rise, sunset, moonrise and moonset tables, and of tables re lating to the satellites, and to physical observations of the sun, moon, and planets. The Berliner Jahrbuch undertakes the heavy task of preparing asteroid ephemerides, and also tables for the satellites of Saturn, from the elements of H. Struve. The Alma naque Nautico shares with the other ephemerides the work of producing apparent places of over i,000 stars.

Tables of Heavenly Bodies.

Some reference has already been made to the tables used in producing the ephemerides. When a theory of the motion of a heavenly body has been evolved by a master of celestial mechanics, the available observations are used to supply the numerical constants of the theory. Then tables arc formed by means of which the position of the body may be found for any desired date. These tables allow for every known gravitational effect acting on the body ; for in stance, the effect of the attractions of other members of the solar system, and of changes in the size, shape, plane, and orienta tion of the orbit of the body. In Brown's Tables of the Moon no fewer than 1,400 terms are included, and the determination of a single position, which involves the use of 18o tables, is a good day's work for a computer.

Ephemerides of the apparent positions of stars are based on catalogues produced by combining all available observations of these stars and on a knowledge of the motions of the earth, which give rise to precession, nutation and aberration.

In the making of the observations from which these all-import ant tables and catalogues are formed, the national observatories have, of course, played a leading part; and in several of the ephemeris offices much work has been done in the analysis of these observations, in preparation of the numerical material on which the predictions are based, and in the production of working trigonometrical and other tables.

Computation of an Ephemeris.

The production of an ephemeris involves very heavy computing and careful checking, though it may perhaps be a matter for surprise that a calculation is rarely repeated. The principal check employed is known as differencing, and depends on the fact that the quantities are tabulated for equal intervals of time, and should exhibit a certain smoothness of progression. As a simple illustration of checking by differences, a column of numbers and their cubes may be con sidered. The process consists in subtracting each cube from the one following and setting down the result as the first difference. Similarly the second, third and higher differences may be found.

Differences Differences No. Cube First Second Third No. Cube First Second Third I I I I 7 7 2 8 12 2 8 12 19 6 19 6 3 27 18 3 27 18 37 6 37 5 4 4 64 61 6 6o 9 5 5 6 3 6 216 36 6 216 35 127 - 6 127 7 7 343 7 343 169 6 169 6 8 512 48 8 512 48 217 217 9 9 729 In the table on the left the third differences are uniformly 6. In the other table or 125 has been replaced by 124, with the result that the smoothness of the third difference is interrupted. The effect of an error of even a single unit in the last figure of a function is seen to be a considerable distortion of the smoothness of the final order of differences. Not only does the process of differencing provide a most powerful method of detecting acci dental errors but it also enables the computer to locate the erroneous value and to determine the sign and magnitude of the error.

Calculating Machines.

Until the early part of this century tables of logarithms formed the chief tools of the computer, but the tendency now is to employ mechanical calculators whenever possible. The simplest of these is the adding machine, in which addition is performed by the depression of numbered keys. In the lunar and planetary tables, which consist of periodic sine or cosine terms, a constant is usually added to each table so that no negative numbers occur, and after adding perhaps 3o or 4o terms from as many tables a simple subtraction of the sum of the constants is made. Thus a Comptometer or other adding machine relieves the computer of the tedium of much mental addition, and, moreover, reduces the possibility of error.

Still more important are machines for multiplication and divi sion. These are as a rule faster, more accurate, and less fatiguing than logarithms. Electrically driven machines such as the Monroe or Mercedes are particularly efficient in heavy work, while hand machines, such as the Nova-Brunsviga, are no less valuable for casual computations. Two special advantages possessed by calculating machines are that they will add products while ing them, and that they avoid the writing of intermediate figures.

The summation of a large number of the positive terms in Brown's Tables of the Moon is now done very efficiently by means of the Hollerith tabulating machine. Here the same numbers are used repeatedly, but never twice with the same partners. The numbers are, therefore, punched on to cards, which are arranged in the correct order and passed through the machine. The machine performs the necessary additions and prints the required results, while the cards are automatically re-sorted to be used again.

A mechanical integrator can be of the greatest assistance in ephemeris work. Quantities are frequently computed for every fourth or eighth day, and the intervening values obtained by the process known as interpolation. In the form usually employed this process is the reverse of that previously described as differ encing, and in carrying it out the second differences of the new quantities are first obtained and then the column of values is formed by repeated summation. The ideal mechanical integrator would produce and print these new values and their differences. In 1908 a machine was built by Herr Hamann, and successfully used by Bauschinger and Peters for the production of 8-figure logarithm tables; but as this machine was subsequently stolen, and the plans used are lost, it cannot be reproduced.

In 1927 two commercial machines appeared, not specifically designed for mechanical integration, but able to perform the work very satisfactorily. One, the Brunsviga-Dupla, is a hand machine with a capacity of 15 figures, but no printing device. The other, an electric Burroughs machine with two distinct adding mechanisms, will add and print 13 figures, together with four figures of the argument, while all carriage and spacing movements are performed automatically. In addition to its use in the produc tion of ephemerides such a machine lends itself readily to the com putation of extensive mathematical and trigonometrical tables.

(L. J. C.)