A uniformly valid asymptotic representation of satellite motion around the smaller primary in the restricted three-body problem

Abstract

This paper summarizes the results of the authors' previous work for the motion of a close satellite of the smaller primary in the restricted three-body problem. The satellite motion is referred to the conventional rotating frame and the central angle between the instantaneous node and satellite is chosen as the independent variable. The solution of the resulting equations is given asymptotically to second order in the small parameter e = μ1/4, where μ is the mass ratio of the smaller primary to the total mass of the system. The method of solution is an extension of an earlier generalized asymptotic expansion procedure using two independent time-like variables. In this problem there are three natural time-scales which differ by one order of magnitude in the small parameter and measure the motions of the satellite, node, and pericenter respectively. Rather large amplitude oscillations in the eccentricity, inclination, and pericenter occur over the longest of these three periods. The theoretical dependence of the bounds of these oscillations upon the initial parameters is analyzed and presented graphically here.