Solution of the restricted problem of three bodies by power series
Abstract
Solutions in the form of Taylor series are presented using the regularization method of ThieleBurrau. The general terms in the series are found as functions of the mass parameter and of the initial conditions. The argument of the expansion is the regularizing independent variable, the new time. In addition to giving the general formulas, the first eight terms of the series are computed for a collision orbit. The coefficients of the series depend only on the Jacobian constant, on the mass parameter, and on the direction of the initial ve locity vector. Applicability of the results besides the analytic aspects, are in the numerical integration of the restricted problem with Steffensen's method where polynomial predictions are used. The regularized series solution may be either continued analytically from one primary to the next or may he made convergent for all times by Poincare's transformation. In this sense an analytic approach to the finding of "lunar trajectories"is offered, inasmuch as the solution given encompasses all possible initial conditions, and the series is con vergent for all times.
 Publication:

The Astronomical Journal
 Pub Date:
 December 1966
 DOI:
 10.1086/109991
 Bibcode:
 1966AJ.....71..968S