MECHANICS 701) Cakula- 147. We will now indicate, as briefly as possible, the don of nl ore ordinary transformations by which the preceding planet s fannu^ are adapted (for astronomical applications) to .notion. . , , , S numerical calculation. Suppose APA (fig. 47) to be an elliptic orbit described about a centre of attraction in the focus S. Also suppose P to be the posi tion of the particle at any time t. Draw PM perpendicular to the major axis ACA , and produce it to cut the auxiliary circle in the point Q. Let C be the common centre of the curves. Join CQ. When the moving particle is at A, the nearest point of the orbit to S, it is said to be in " peri helion." The angle ASP, or the excess of the particle s longitude over that of the perihelion, is called the " true anomaly." Let us denote it by 6. The angle ACQ is called the "excentric anomaly," and is gene rally denoted by u. And, if 2ir/n be the time of a complete revol ution, nt is the circular measure of an imaginary angle called the " mean anomaly ;" it would evidently be the true anomaly if the particle s angular velocity about S were constant. It is easy from known properties of the ellipse to deduce the fol lowing relations between the mean and excentric, and also between the true and excentric, anomalies : nt = u -e sin it. u /fl-e. -- = A / - n tan . 2 V 1 + e / 2 By far the most important problem is to find the values of and r as functions of t, so that the direction and length of a planet s radius-vector may be determined for any given time. This gene rally goes by the name of Kepler s Problem. Before indicating the systematic development of u, r, and in terms of t from our equations, it may be useful to remark that, if e be so small that higher terms than its square may be neglected, we may easily obtain developments correct to the first three terms. Thus tan - Also = nt + e sin (nt + e sin nt) nearly, = nt + esinnt +^sin2nt. = 1 - ecosu, = 1 - c cos (nt + esinnt), = 1 - ecosnt + 4e 2 (l -cosZnt). And i which may be written e 2 ) 2 (l+ecos0) 2 dt Keeping powers of e lower than the third, ( 1 -2ccos0 + |c 2 cos20 ) f .,- = n, V J dt or nt = 6 whence = /i< + 2esin0-e 2 sin20, = nt + 2csin (nt + Zesinnt) - e 2 siu2)i, = nt + 2csin nt + 4e 2 cos?isin nt - |c 2 sin 2nt, nt + 2e sin n t + c 2 sin 2nt . Kepler s KEPLER S PROBLEM To find r and as functions of t from the problem, equations r = a(l ecosu) (1); e , -tan- 2 - . (2), . . . (3). These equations evidently give r, 0, and t directly for any assigned value of it, but this is of little value in practice. The method of solution which is commonly adopted is that of Lagrange, and the general principle of it is this : We can develop from equation (2) in a series ascending by powers of a small quantity, a function of e, the coefficients of these powers involving u and the sines of multiples of u. Now by Lagrange s theorem we may from equation (3) express u, 1 - c cos it, sin u, sin 2 it, &c., in series ascending by powers of e, whose coeffi- cients are sines or cosines of multiples of nt. Hence, by substitut ing these values in equation (1) and in the development of (2), we have r and expressed in series whose terms rapidly decrease, and whose coefficients are sines or cosines of multiples of nt. This is the complete practical solution of the problem. But we must refer the reader to special treatises, for the full development of this subject. Compare 52. 148. We may take an opportunity here of giving a Stability sketch of a particular case of the important question of of circular " kinetic stability." The general treatment of this subject or is entirely beyond our limits. But we may investigate its conditions, in the case of a central orbit naturally circular, by a very slight modification of our equations. Whatever be the law of central force, provided it depend on the distance alone, we can write the acceleration due to it as Hii-f(u), where u is the now reciprocal of the radius-vector, as in 144. The kinematics of the motion is then entirely summed up in the equations If I/a be the radius of the circle, the first equation becomes simply Now let a slight disturbance be given to the motion, such that h is unaltered, but* that u becomes a + x. Then we have Expanding to first powers of x only, and thereby assuming that x is always exceedingly small, we have the terms independent of x vanishing by the condition for a circular orbit. By eliminating the ratio ju//t 2 we have NN =0. To secure stability, x must not be capable of increasing indefi nitely. This leads to the result that the multiplier of x in the above equation must be positive; i.e., For, if the multiplier were negative, the value of x would consist of two real exponential terms, one of which would increase indefi nitely with the angle 9, and would disappear from the value of a; under special conditions only. If the multiplier were zero, x would be a linear function of 6. Hence, in the only case we need consider, we have The radius-vector is therefore a maximum and minimum (i.e., apses occur) alternately as the angle increases by successive in crements each equal to Suppose the force to vary as the inverse nth power of the distance. Here f(a) oc a"- 2 , and we have 1 - ^^ = 1 - ( - 2) = 3 - n. Thus n must be less than 3 ; i.e.,*. circular orbit, with the centre of force in the centre, is essentially unstable if the force vary as the inverse third, or any higher inverse power of the distance. If?i = 2, which is the gravitation case, the apsidal angle is evi dently IT, 149. A very curious result, due to Newton, may bs indicated here, viz., that, if any central orbit be made to revolve in its own plane with angular velocity propor tional at each instant to that of the radius-vector in the fixed orbit, it will still be a central orbit ; and the additional force required will be inversely as the cube of the radius- vector.
Generally, in a central orbit,