On the degree of accuracy attainable in determining the position of Laplace's invariable plane of the planetary system

T. J. J. See
1903 Astronomical Notes - Astronomische Nachrichten  
I. Theory of the Invariable Plane. In the year I 784 Laplace discovered that in every system of bodies *) revolving freely in space and subjected to the mutual gravitation of its parts, there exists an Invariable Plane, which is a plane of maximum areas. Under the action of the various physical causes which affect our system, this plane is the only geometrical element which is rigorously fixed in space. On that account considerable theoretical interest attaches to it, and several determinations
more » ... eral determinations of its position have been made by astronomers. Unfortunately the elements of the planetary orbits for a long time were not known with extreme precision, and the masses of the planets remained still more uncertain, so that the interest 1 attaching to the Invariable Plane was chiefly a theoretical 1 one. But as the masses of the planets are gradually becoming known with greater and greater precision, while the planetary elements are already quite satisfactory, it appears worth while to inquire into the degree of accuracy at present attainable in the determination of the Invariable Plane of our System. Suppose Ma, nr" m, , m3 . . mn to be the n + I bodies of the Solar System, and let gi, qi, gi denote their coordinates referred to the origin taken at any point in space. Then it may be shown (Cf. Laplace's MCcanique Celeste, Liv. I , Ch. V, 9 2 2 ) that the rigorous integrals of the areas are: Laplace observes that the second terms of the right members of these equations, when multiplied by di, express the Sums of the projections of the elementary areas described by each of the pairs of bodies in the system made up of n + I masses, namely, the Sun, and n planets; one of the bodies in each group being supposed to move about the other considered at rest, and each area being multiplied by the product of the two masses connected by the right line. The functions represented by the two terms of the second members of these equations are similar in character. The first term represents areas described by the radii vectores drawn from the centre of the Sun; and the second, relative areas described by the right lines connecting the other bodies in pairs, Thus when taken for the whole system the two expressions have exactly analogous properties. n(n+ I ) n(n+ I ) right lines connecting the 2 2 n ( n + I ) 2 We shall now put these expressions into a suitable form for computing. the position of the Invariable Plane of the Solar System. Any areal element of the expression occuring in the first term of the second member of (I) may be given in terms of the elements of the orbit of the body about the Sun. Suppose the Semi axis major to be aj, the eccentricity to be ej, and the mass mi; then whatever be the position of the axes Og, O", we shall have If we multiply the second member of this equation by cosxi, where xi is the inclination of the plane of the orbit *) Oeuvres Completes de Laplace, Tome XI, pp. 548-551. II
doi:10.1002/asna.19031641102 fatcat:b6pywq3oujbefjciepnr5jvkru