Studies in the application of recurrence relations to special perturbation methods

Abstract

The procedure of numerical integration of the elliptic three dimensional restricted threebody problem by the use of recurrence relations to evaluate successively higher derivatives of the relative position and velocity vectors of the bodies and of the variational matrix is investigated. A set of recurrence relations is developed which involves the introduction of fewer auxiliary variables than in previous papers of this series, while the recurrence relations themselves are of a simpler form than those in other treatments involving the same number of such auxiliary variables. A technique for automatic adjustment of the integration step-length at each point in the orbit, such that the local truncation error remains close to, but always less than, some specified amount, is incorporated. This technique involves the comparison of pre-integration values with those obtained after consecutive forward and reverse integration steps, and has decided advantages over step-adjustment methods currently in use.