Numerical Investigation for Periodic Orbits in the Hill Three-Body Problem

Abstract

The current work performs a numerical study on periodic motions of the Hill three-body problem. In particular, by computing the stability of its basic planar families we determine vertical self-resonant (VSR) periodic orbits at which families of three-dimensional periodic orbits bifurcate. It is found that each VSR orbit generates two such families where the multiplicity and symmetry of their member orbits depend on certain property characteristics of the corresponding VSR orbit’s stability. We trace twenty four bifurcated families which are computed and continued up to their natural termination forming thus a manifold of three-dimensional solutions. These solutions are of special importance in the Sun-Earth-Satellite system since they may serve as reference orbits for observations or space mission design.