A Planetary Theory with Elliptic Functions and Elliptic Integrals Exhibiting no Small Divisors

Abstract

This paper develops a planetary theory in three dimensions with elliptic functions and elliptic integrals. In an earlier treatment, (Williams, Van Flandern, and Wright, 1987) presented a two dimensional planetary theory to the first order of a Picard iteration. The theory did avoid expansions in powers of the ratio of the semi-major axes and it contained only two explicit small divisors, n – n′ and 2n – n′. These advantages are retained in the new theory and in fact no small divisors appear explicitly. Secular terms are removed by adopting an averaging technique rather than continuing the Picard iteration. The Lie series method of (Deprit, 1969) is chosen for the averaging. In order to simplify the Lie operator, the framework for the problem is chosen to be the circular restricted three body problem written in the polar-nodal coordinates of Whittaker. The algorithm is described and a few representative terms are discussed.