Asymptotic Solutions of the Restricted Three-Body Problem by Use of Perturbation Methods.

Abstract

This work presents a comparison of matching asymptotic solutions for the limiting case of the restricted threebody problem by the use of perturbation methods. The problem is of a singular-perturbation type. We investigate two alternative methods to deal with it: the classical method of matched asymptotic expansions and the improved method of matched asymptotic expansions. Two expansions, outer and inner, are involved. The outer expansion breaks down in the inner region where sharp changes occur, and the inner expansion becomes nonuniformly valid in the outer region. To obtain a uniformly valid composite solution, we need a matching procedure to relate these two expansions. Instead of straightforward matching of the outer and inner expansions to higher-order terms, in the improved technique the higher-order solutions are derived by generating perturbations between the lower-order composite solutions and the exact solutions. The perturbation equations are then integrated in the outer and inner regions, respectively, for a higher-order matching. Improved asymptotic solutions of second order are obtained for the limiting case of the restricted three-body problem. Compared to the solutions obtained by the classical method of matched asymptotic expansions and the pure numerical integration for various values of a small parameter µ, the improved asymptotic solutions are very accurate. Moreover, the asymptotic solutions obtained by use of the improved method give better accuracy than those using the classical method over wide ranges of the small parameter.