Formal Expressions for the Motion of N Planets in the Plane, with the Secular Variations Included, and An Extension to Poisson’s Theorem

Abstract

In Poincarés “Méthodes Nouvelles de la Mécanique Celeste”, pp. 127–135, there is given a formal solution of the problem of three bodies in motion under their mutual gravitational attractions, in the case where one of the bodies (the “primary”) has mass appreciably larger than the others, and where the motion of the others relative to the primary is nearly circular, and nearly coplanar. We may call this the “planetary case” of the general gravitational problem of three bodies. The solution is derived using Von Zeipel type transformations, in a number of stages, making use of infinite series in powers of the ratios of the smaller masses to that of the primary, and, in the treatment of the secular variations, powers of quantities of the order of the orbital eccentricities and orbital inclinations. In the present treatment, the simple extension to n planets is made, and the transformations employing power series are introduced making use of the Lie series method introduced by Hori (Pub. Astron. Soc., Japan, Vol. 18, pp. 287-295, 1966) which gives explicit expressions for the transformed variables in terms of the untransformed, and vice-versa. The short-period terms are removed by a single transformation at the outset, so that the elements of the matrix defining the linear transformation employed in the secular variation theory are functions of the constant transformed major axes, and so do not themselves possess short-period terms as those in Poincare’s solution do.