CHAPTER VIII. PERTURBATIONS OF THE PLANETS IN LONGITUDE, 532. THE position of a planet in space is fixed when its curtate distance Sp, fig. 77, its projected longitude 7Sp, and its latitude pm, are known. The determination of these three co-ordinates in functions of the time is the principal object of Physical Astronomy; these quantities in series ascending according to the powers of the eccentricities and inclinations are given in article 399, and those following, supposing the planet to move in a perfect ellipse; but if values of the elements of the orbits corrected by their periodic and secular variations be substituted instead of their elliptical elements, the same series will determine the motion of the planet in its real perturbed orbit. 533. The projected longitude and curtate distance only differ from the true longitude and distance on the orbit by quantities of the second order with regard to the inclinations; and when the orbit at the epoch is assumed to be the fixed plane, these quantities as well as those of the latitude that depend on the product of the inclination by the eccentricity are so small that they are insensible, as will readily appear if it be considered that any inclination the orbit may have acquired subsequently to the epoch, can only have arisen from the small secular variation in the elements; besides the epoch may be chosen to make it so, being arbitrary. Hence the perturbations in the longitude and radius vector may be determined as if the orbits were in the same plane, and the latitude may be found in the hypothesis of the orbits being circular, provided the orbit at the epoch be taken as the fixed plane: circumstances which greatly facilitate the determination of the perturbations. The following very elegant method of finding the perturbations, by considering the troubled orbit as an ellipse whose elements are varying every instant, was employed by La Grange; but La Place's method, which will be explained afterwards, has the advantage of greater simplicity, especially in the higher approximations. 534. In the elliptical hypothesis the radius vector and true longitude are expressed, in article 392, by r functions. (a, Y, e, e, w), v = functions . (X, e, ε, ☎), but in the true orbit these quantities become . a + da, 5+85, e + de, e +de, w+dw; and if the values of the periodic variations in the elements in article 529 be substituted instead of da, d, &c., the perturbations in the radius vector and true longitude will be obtained; the approximation extending to the first powers of the eccentricities and inclinations inclusively. 535. The perturbations in longitude may be expressed under a more simple form; for by article 372, an equation belonging both to the elliptical and to the real orbit, since it is a differential of the first order; on that account it ought not to change its form when the elements vary; hence and neglecting the squares of the disturbing forces, the integral is will give the perturbations in longitude when those in the radius vector are known. Perturbations in the Radius Vector. 536. By article 392, r = a(1 + be2 — e cos (nt + e − ☎) — e2 cos 2 (nt + e − ☎)) ; whence dr (da-ase cos (nt+e-w)-acow sin (nt+e-w)) (1+2e cos(nt+e-)) 3eda cos (nt + e-☎)+2aede+ae (ô5+de) sin (nt+e− ̄). If the values of da, de, dw, de, de, from article 529, be substituted in this expression, after the reduction of the products of the sines and cosines to the cosines of multiple arcs, and substitution for Mo, M, N, N, Ns, from article 459, it becomes (149) + m' . e.E.D¡ . cos {i (n't — nt + e' − e) + nt + e − ̄ }, 537. Having thus determined the perturbations in the radius vector, the term 28r r is known; and if substitution be made for la ch will be obtained, and the integral of and de, from article 529, equation (148) will give dv = m′ E.F¡. sin i(n't —nt + e' — e) 2 + m'e. E.G.. sin {i(n't — nt + e' —e) + nt + e−0} + m'e'..H. sin { i(n't — nt + e −c) +nt + c — w' }. 538. In these values of dr and dv, i includes all whole numbers, either positive or negative, zero excepted: dr and do will now be determined in the latter case, which is very important, because it gives the part of the perturbations that is not periodic. 539. If i 0 in the series R in article 449, the only constant term introduced by this value into dr will be Again, in finding the integral da the arbitrary constant a, that ought to have been added, would produce a constant term in dr. In order to find it, let the origin of the time be at the instant of the conjunction of the two bodies m and m', whence cos 0 = 1, and the first term of da in article 529 becomes where extends to all positive values of i from i = 1 to i = ∞. 540. If these values of dr and da be put in equation (148), the And as by article 392 the elliptical parts of r and v that are not periodic, or that do not depend on sines and cosines, are r = a, and vnte: those parts of the radius vector and true longitude that are not periodic are expressed by Thus the perturbations in longitude seem to contain a term that increases indefinitely with the time; were that really the case, the stability of the solar system would soon be at an end. This term however is only introduced by integration, since the differential equations of the perturbations contain no such terms; it is therefore foreign to their nature, and may be made to vanish by a suitable determination of the arbitrary constant quantities. In fact the true longitude of a planet in its disturbed orbit consists of three parts,-of the mean motion, of the equation of the centre, and of the perturbations. The mean motion of the planet is the only quantity in the problem of three bodies that increases with the time: the equation of the centre is a periodic correction which is zero in the apsides and at its maximum in quadratures; and the perturbations being functions of the sines of the mean longitudes of the disturbed and disturbing bodies are consequently periodic, and are applied as corrections to the equation of the centre. All the coefficients of these quantities are functions of the elements of the orbits, which vary periodically but in immensely long periods. The arbitrary constant quantities introduced by integration, must therefore be determined so that the mean motion of the troubled planet may be entirely contained in that part of the longitude represented by v. 541. The values of a, n, e, e, and, in the preceding equations, are for the epoch t = 0, and would be the elliptical values of the elements of the orbit of m, if at that instant the disturbing forces were Let not be the mean motion of m given by observation, then the second of the equations under consideration gives to cease. |