§ 13. The Problem of Three Bodies.—We have pointed out that the motion of a planet is determined by the mutual attraction which ex ists between the planet and the sun. It follows, however, as a necessary consequence from the law of universal gravitation, that each planet is attracted not only by the sun but by every other planet, and indeed, so far as we know, by every other body in the universe. The effect of the attraction of the sun preponder ates so enormously over the other attractions that the obedience of the planets to Kepler s laws, which would be perfect were the sources of disturbance absent, is still so nearly perfect that the departure therefrom can only be perceived either by very accurate or by very long-continued observations. The refinements of modern observations have, however, brought to light a very large number of perturbations, as they are called, in the motions of the planets and their satellites, which are due to the interference of other bodies. The explanation of these different perturbations by the law of universal gravitation has proved in the great majority of cases triumphantly successful. In any problem where perturbations are involved, we have at least three bodies, viz., the principal body S, a body P which circu lates around S, and a disturbing body P . If we had only the two bodies S and P to consider, then P would describe around S a conic section, of which S was the focus, and the radius vector SP would sweep over equal areas in equal times. By the introduction of the third body P , which attracts both the former bodies S and P, the motion is deranged, the orbit which P describes is no longer a conic section, and its radius vector has ceased to describe equal areas in equal times. The general case of this problem, as we have here described it, is one of excessive complexity. It is, however, a fortu nate circumstance that, up to the present time, astronomers have had but little occasion to attack the problem in its general form. 1 In the solar system we have a great number cf different problems of perturbations to discuss, but there is one feature common to all these problems. This feature is that the perturbing force is very small in comparison with the primary force, bij which the motion of the disturbed body is controlled. In consequence of this peculiarity, the problems of perturbations in the solar system become greatly simplified. The circumstances of the orbits of the planets and their satellites are also happily such as to furnish additional facilities in the solution of the problems of perturba tions. The eccentricities of the orbits are small, and in nearly every case the inclinations are small also. The problems of perturba tions in the solar system are consequently adapted for the methods of successive approximation ; but we should perhaps warn the reader that, even with the assistance so fortunately rendered by the circumstances of the case, the problems are still among the most difficult to which analysis has ever been applied.
Let S, P, P (fig. 12) denote the three bodies of which the masses are M, m, TO . Let the dis tances SP, SP , PP be denoted by r, r , p respectively; then if e denotes, as before, the gravitation between two units of mass separated by the unit of distance, we have on S the forces e ^ and e ~ along SP and SP respectively ; on P the forces e M and e~ along PS and PP respectively ; on P the forces Mm and e- m along P S and P P respectively. r* p 2 In the problem of the perturbation of a planet which is moving around the sun, the absolute motion of the planet in space is not what we are concerned with ; what we do require is the relative motion of the planet with respect to the sun. Suppose that we apply to each unit of mass of the three bodies, forces equal to tmr- 3 parallel to PS and tm r -" 2 parallel to P S, then it is clear that, from the second law of motion, each unit of mass of the system will receive in a small time, so far as these forces are concerned, equal and parallel velocities, and therefore the rela tive motions of the three bodies will not be altered by the introduc tion of these, forces.
1 There can be little doubt that in the case of multiple stars the problem of three or more bodies would often have to be faced in all its ruggedness, but the observations which would be required are certainly wanting, and probably unattainable.
Rut these forces will, so far as the body S is concerned, amount to eMm -f- r" 2 and eMm -f-r 2 in the directions PS and P S respectively, and will therefore neutralize the attractions of P and P upon S. It follows that, so far as the relative motions only are concerned, we may consider the body S as fixed, while the body P is acted upon by the forces em(M + ?H) r 2 on PS tmm p^ on PI" finm r" 2 parallel to P S, with of course similar expressions for the forces on P . The problem of perturbation is therefore reduced to the determina tion of the motion of P in obedience to these forces, while S regarded as fixed. It is plain that, in order to find the motion of P, we should know the motion of P , which is itself disturbed by P ; but as the effect of the perturbation is small, we may, without appreciable error, use the place of P , derived on the supposition that the motion of P is undisturbed, for the purpose of calculating disturbances caused by P in the motion of P. We shall therefore suppose that the orbit of P is an ellipse which is described around the focus S in consequence of an attractive force e/M (M + m )-^-r 2 directed towards S. In computing the disturbed motion of P we transform the three given forces into three equivalent forces, which will be more con venient for our purpose. We take first a force T directed along the radius vector ; secondly, a force V perpendicular to the radius vector, and lying in the plane of the orbit ; and, thirdly, a force W, which is normal to the plane of the orbit. T is to be regarded as positive when it endeavours to increase the distance between P and S ; V is to be regarded as positive when it tends to move P to the same side of the line PS as that in which the direction of its motion tends to carry it ; and W is to be regarded as positive when it tends to raise P to the north of the plane of its orbit.
Fig. 13.
From P (fig. 13) let fall a perpendicular P Q upon the plane of the undisturbed orbit cf P, and let fall from Q a perpendicular QR upon the line SP. The force along PI" can be resolved into a com ponent parallel to QP and another parallel to QP ; and the latter can again be decomposed into components parallel to RQ and RP. Hence we have, as the equivalents to the force on PP , the following three forces which form parts of T, V, W respectively : o :1 P In a similar manner we find for the components of the force parallel to SP, the following three forces which form parts of T, Y, W respectively: SR emwi + -VT Hence we deduce the following expressions : T em(M + m) _ ^^ pT> _. f^ 1 r* p 3 r 3 W tmm Let the line SO, from which the longitudes are reckoned, be drawn in the plane of the undisturbed orbit of P. Let the longi tudes of P and P be I, I respectively, and let the angle P SQ or the latitude of P be denoted by b , then we have QP = r sin b ; SQ = ?- cos?/. KQ =SQ sin (l -l) = r cos b sin (V-l). SR = SQ cos (I -I)- r cos b cos (I 1 - 1).
PR = r - r cos b cos (I 1 - 1).
