CELESTIAL MECHANICS. That branch of astronomy known as Celestial Mechanics dates its origin from 1687, the year in which Sir Isaac Newton's epoch-making work Philosophiae Naturalis Principia Mathematica was given to the world.
Law of Gravitation.—Bef ore this date, Kepler's three laws epitomized the mechanics of the planetary system; the laws them selves were simply three independent statements based on the observed facts of the planetary motions. (An account of Kepler's Laws and the definitions of the six "elements" which define a planet's orbit are given in the article ORBIT). To Newton was reserved the supreme achievement of discovering a single all embracing law—the law of gravitation—of which the three Kep lerian laws were deducible consequences.
The foundations of Newton's gravitational discoveries were the three dynamical laws of motion which he set forth at the beginning of the Principia and which ever since have borne his name. The first—stated originally by Galileo—asserts that a body will continue to move with uniform velocity in a straight line unless disturbed by some external agency. Now a planet moves round the sun in a nearly circular orbit—the first law of motion requires, therefore, some agency, external to the planet, which prevents the planet going off at a tangent—that is, in a straight line. This agency Newton traced to the sun.
Making use of Kepler's third law, Newton then investigated how the sun's influence varied from planet to planet—he found that it varied inversely as the square of the distances of the planets from the sun. But by Kepler's first law the orbit of any planet is not actually a circle but an ellipse ; Kepler's second law showed Newton that the agency maintaining a planet in its orbit must always be directed towards the sun—in other words, the sun exerted an attraction on the planet—and moreover that the mag nitude of this attraction varied from point to point of the elliptic orbit according to the law of the inverse square of the distance. The second law of motion introduced the conception of mass, the amount of matter in a body.
The third law asserted that action and re-action are equal and oppositely directed ; in particular if the sun attracts a planet with a certain force, the planet attracts the sun with an equal force. The conclusion that Newton reached was that the force of attrac tion exerted by the sun on a planet was proportional to the pro duct of the masses of the sun and planet and inversely propor tional to the square of the distance between them. But the mass of the sun, for example, is the aggregate of the masses of its individual component particles of matter, and Newton was led to conceive that this power of attraction manifested by the sun was the property of the individual particles.
The famous law of gravitation is then expressed in the form: "Every particle of matter attracts every other particle of matter with a force proportional to the product of the masses of the two particles and inversely proportional to the square of the distance between them." As the sun and planets are spherical (or nearly so), then by a theorem due to Newton the mutual attraction of the sun and any planet is equivalent to the mutual attraction of particles with the masses of the sun and planet supposed concentrated at their respective centres.
The motions of the bodies of the solar system are thus, in the main, to be regarded as the motions of material particles without reference to the dimensions of the bodies themselves. A striking consequence of the universal law of gravitation, to be noted in passing, was the calculation of the masses of planets—which are accompanied by satellites—in terms of the sun's mass regarded as the unit of mass ; for example, it is found that the solar mass is 330,00o times the mass of the earth.
The Main Problem.—We approach now the central problem of Celestial Mechanics. In all that has preceded, it has been assumed that the only force acting on a particular planet—let us say Mars—is the gravitational attraction exerted by the sun. But owing to the universality of the law of gravitation, Jupiter and every other planet, together with their satellites, exert each an attraction on Mars which in the case of Jupiter, for example, is proportional to the latter's mass and inversely proportional to the square of the distance between Mars and Jupiter at any given instant. The problem of determining the path of Mars becomes one of tremendous complexity; Mars, as it were, is being tugged by the sun, the planets and their satellites, not by constant amounts and in constant directions, but by varying amounts depending on the changing distances of Mars from these bodies and in directions depending on the ever-altering relative configur ation of all the component bodies of the solar system.
The problem is rescued from intractability owing to the pre ponderating influence of the sun. The mass of the sun is times that of Jupiter—the most massive planet,—and it is mainly because of this stupendous disparity in masses that the sun exercises its decisive gravitational influence. The sun controls the destiny of a particular planet, the remaining planets and the satellites can only introduce comparatively slight modifications. It is clear then that Kepler's laws do not represent accurately the motions of the planets, but owing to the insignificance of the planetary masses in comparison with the solar mass they do represent a close approximation to the truth.
A further simplification follows, for the disturbing effects of all planets on the path of Mars for example can, to a high degree of accuracy, be regarded as the sum of effects produced by the individual planets, each moving, not in its true path, but in the Keplerian ellipse to which it closely approximates. The problem then reduces to the investigation of the motion of a planet under the gravitational attraction of the sun and of one other planet.
The section of Celestial Mechanics which deals with the prob lem just stated is known usually as Planetary Theory. In the same way the motion of a satellite around its primary is approxi mately an ellipse, and the departures from the elliptic orbit are mainly due to the effect of the sun's gravitational attraction— and that of the other satellites (if any). This section of Celestial Mechanics is known as Satellite Theory and in the special case of the earth's satellite, the moon, as Lunar Theory. In either case, the problem is the "Problem of three bodies" celebrated since Newton's day as the field of research in which the great mathematical astronomers of the past have found ample scope for their genius.
Consider now the influence of the planet Q. It also is moving round the sun in a path which, were it not for the effect of the other planets would likewise be a Kep lerian ellipse, defined by six elements. At the particular instant at which the configu ration of S, P and Q in the figure is repre sented, the sun's attraction on P is proportional to the sun's mass divided by the square of the distance S P and the attraction of the planet Q is proportional to the mass of Q divided by the square of the distance P Q; owing to the enormous disparity be tween the mass of the sun and of Q, the latter attraction is only a minute fraction of the solar attraction. Suppose, for a moment, that the attraction of Q on P operates only for an hour. At the end of this time the position of P will be slightly different from the position it would have reached were the planet non-existent and its velocity and direction will also be slightly different. At the end of the hour the planet P will have a definite position and velocity and, as the influence of Q terminates at this moment according to our supposition, P will proceed to describe an elliptic orbit around the sun, the six elements being calculable from the position and velocity of P at the end of the hour. In general, all six elements will be slightly different from the elements of the orbit which was being described by P anterior to the action of Q.
Consider Q to act for another hour; at the end of this time the elliptic elements will again be changed and so the process goes on continuously. The values of the elements can be calculated for any future time by the method devised by Lagrange :—the elliptic orbit to which these elements correspond is clearly the orbit which the planet P would continue to describe were the influence of Q suddenly to cease at the particular future moment considered; this ellipse is called the "osculating ellipse." The changes which the elements undergo owing to the action of the planet Q—the "disturbing" planet—are called "perturbations." In the way indicated, the perturbations due to the action of each of the remaining planets can be calculated, and so the total effect on the elements of P's orbit obtained. It is to be remarked that the path of P computed from the perturbations obtained in this way, although conforming very much more closely to the true path (which may be supposed known from a series of ac curate observations) than to a simple elliptic orbit unaffected by other planets, is yet only an approximation. In order to obtain this approximation, it was assumed amongst other things, that the path of Q was a Keplerian ellipse. But every other planet produces perturbations in Q's motion; when these are cal culated the path of Q can therefore be represented more faith fully ; and consequently when the motion of P is again considered, the perturbations produced by Q can be more accurately repre sented. The process of determining the perturbations is carried on then according to the method of successive approximations.
When the final results have been obtained it is found that the expression for an element, such as the eccentricity, consists of a multitude of terms which can be divided into two classes (i) periodic terms (ii) secular terms.
Any single periodic term, which is itself a contribution to the general aggregate of the perturbations of the element, is such that its value lies between a definite maximum and a definite min imum, these and intermediate values being regularly repeated in cycles. Such terms produce no lasting change on the element. A cork floating on the surface of a lake rises and falls rhythmically with the waves—it is sometimes greater and sometimes less than its average distance from the lake bottom, and its motion is periodic. So it is with each of the periodic terms forming part of the perturbations of any given element.
Secular terms on the other hand are of a different kind. The presence of a positive secular term in the expression for the per turbations of the eccentricity means that the eccentricity would go on increasing year by year, century by century. A cork float ing down a tidal stream is moving from higher to lower levels, which expressed rather unfamiliarly, means that its distance from the center of the earth is gradually but continuously decreasing— this is precisely the nature of a secular change ; but the ebb stream gives way to the flood stream and the cork is carried back to higher reaches of the river, its distance from the centre of the earth now gradually increasing—again in the manner of a secular change. In this way, we can imagine the distance of the cork from the earth's centre to fluctuate between a maximum and a minimum value.
As regards the secular perturbations of the elements there are, first, the truly secular terms which indicate progression in one direction however long into the future we look and, secondly, the secular terms which indicate steady progression for a time, with subsequent reversal—after the manner of the illustration—gen erally of ter very long intervals of time. In the first category are the secular perturbations of the longitude of the ascending node— the orbital plane alters in such a way that the node moves pro gressively around the ecliptic. In the second class are to be found the secular terms defining the progressive changes in the eccen tricities of the planets, which if a sufficiently long interval of time is considered remain between certain definite limits of magnitude.
The importance of the secular terms is seen more particularly as regards the perturbations of the semi-major axis. If there are terms which are truly secular in the sense indicated, the average distances of the planets from the sun either increase and continue to increase, in which event the members of the solar system would in the distant future be scattered in space and cease to function as corporate members of a system; or alternatively, the average distances of the planets from the sun would progressively de crease, the final result being their destruction in the solar furnace. The results of planetary theory indicate that no such secular terms appear in the expressions for the perturbations of the semi major axes—at any rate, up to a very high degree of approxi mation—from which it may be safely concluded that the stability of the solar system is not prejudiced by the gravitational action and interaction of the planets themselves.
Mention may be briefly made of what are known as the "long period inequalities"; they are due to the approximate commen surability of the mean motions of the two planets concerned. For example, twice the mean motion of Jupiter is nearly equal to five times the mean motion of Saturn, or in other words the orbital periods of Jupiter and Saturn are nearly in the ratio of 5 to 2. The consequence is that certain terms in the perturba tions of the elements of Saturn as disturbed by Jupiter, although periodic, have large maximum and minimum values and are con sequently of great importance. Their effect, however, operates very slowly—in the case of the two planets mentioned, the period of such changes is 917 years.
The last word in Lunar Theory is the recently completed "Tables of the Moon" by Professor E. W. Brown in which the gravitational action on the moon's path around the earth of every body in the solar system is fully considered. It might be expected that observation would now confirm theory—in so far as the moon's motion is concerned—with perfect accuracy, but this is not actually so. The beginning and end of a total eclipse of the sun cannot be predicted correctly to a second of time—the error is generally several seconds. It is almost certain that gravi tational theory is not to blame for such discrepancies, but that rather there is some influence, minute no doubt in its effect, which is the ultimate cause of our inability to predict accurately the position of the moon in its orbit. Such an influence is believed to be traced to a very slight irregularity in the earth's rotation; if the day, for example, is gradually getting longer the moon will be seen further ahead in its orbit than calculation predicts from gravitational theory alone. As has been said, the discrepancies are extremely minute and the explanation requires further support before it can be satisfyingly complete.
Other Applications.—The history of gravitational astronomy is rich in discoveries which are subsequently made as the result of discrepancies between observation and prediction. The discovery of the planet Neptune is the most notable example. The planet Uranus had been discovered by Sir William Herschel in the complete machinery of planetary theory was at the service of astronomers to predict its position at any future date. But in the early years of the nineteenth century prediction and observation did not quite agree. Either the Newtonian law of gravitation re quired modification which was only sensible at the great distance at which Uranus was from the sun or the slightly erratic behaviour of the planet was the consequence of the gravitational attraction of an unknown planet yet more remote. Prof. J. C. Adams and M. Le Verrier both independently adopted the latter hypothesis.
The data of the problem consisted of the minute discrepancies between the observed positions of Uranus each year and the cor responding positions calculated according to the known factors involved. Both astronomers, by different methods and independ ently, were able to announce that if the practical astronomer pointed his telescope to a certain part of the sky, he would see an object which would prove to be a planet. On the 23rd Sept. 1846, Neptune was discovered telescopically near the position predicted by Adams and Le Verrier.
The minor planets, whose orbits extend from the orbit of Mars to the orbit of Jupiter, present many interesting problems in Celestial Mechanics. There are certain significant gaps in the distribution of the orbits of these small bodies; in round figures, the mean daily motion of Jupiter is 300" and the gaps occur at distances from the sun corresponding to mean motions for ex ample of 600", 75o" and 900" (planets with these mean motions would have orbital periods 2, s and 3 that of Jupiter). For such planets, if they existed, the principal perturbations due to Jupiter would be "long-period inequalities" similar in character to those found in the case of Jupiter and Saturn. The existence of the gaps has generally been taken to mean that Jupiter's attraction on a minor planet with a mean motion of 600" for example, results in a marked change in the mean motion (and consequently of the mean distance from the sun) and, as it were, sweeps the planet out of the critical orbit (that for which the mean motion is 600"). But whether this is really so or not, Celestial Mechanics has not been able so far to offer an authoritative pronouncement.
An interesting group of minor planets is the "Trojan group," six of which are known. The mean motions of these planets are approximately the same as the mean motion of Jupiter; they are accordingly at approximately the same mean distance from the sun as Jupiter. Also their distance from Jupiter is approximately equal to their mean distance from the sun. At any moment, then, a Trojan planet, Jupiter and the sun are situated at the vertices of a triangle which is nearly equilateral. Accordingly, the longi tude of a Trojan planet is approximately the longitude of Jupiter plus 6o° (this statement characterizes the situation of four of these bodies) or the Trojan's longitude (as for the two remaining members of the group) is roughly the longitude of Jupiter minus 6o°. The effect of Jupiter's attraction on any Trojan is, amongst other things, to limit the range of variation of the mean motion ;. in other words, the extent of the Trojan's orbit varies periodically, being sometimes less and sometimes greater than the orbit of Jupiter. This phenomenon is called "libration." The conclusion is that the orbits of the Trojan planets are "stable"—in contra distinction to the supposed instability of an orbit for which the mean motion, for example, is 600".
As has been demonstrated in several ways, the magnificent ring-system which makes Saturn unique in the heavens is nothing more than a vast assemblage of small bodies, each in fact a satel lite of the planet. The movements of any one of these small bodies are governed by the attraction of Saturn, of the nine satel lites and of the remaining myriads of the constituent members of the rings. The two principal rings—Ring A and Ring B—are separated by a gap of about 2,000 miles in breadth called the Cassini division—analogous to the gaps already referred to in the distribution of minor planet orbits. It has been proved that the emptiness of the Cassini Division is due to the perturbing action of the three satellites nearest Saturn—a beautiful applica tion of the principles of Celestial Mechanics.
BIBLIOGRAPHY.-E. W. Brown, Lunar Theory (1896) ; H. C. PlumBibliography.-E. W. Brown, Lunar Theory (1896) ; H. C. Plum- mer, Dynamical Astronomy (1918, Bibl.) , both for the specialist reader; H. Spencer Jones, General Astronomy (1922)—for the non specialist reader. (W. M. S.)