Families of periodic orbits in the planar Hill four-body problem

TL;DR

This paper numerically explores families of planar periodic orbits in the Hill approximation of the restricted four-body problem, revealing properties relevant to celestial systems like Sun-Jupiter-asteroid.

Contribution

It introduces a numerical analysis of periodic orbit families in the Hill limit of the four-body problem, connecting classical three- and four-body dynamics.

Findings

Identification of periodic orbit families for various mass parameters.

Analysis of horizontal and vertical stability of these families.

Insights into the Sun-Jupiter-asteroid dynamical system.

Abstract

In this work we perform a numerical exploration of the families of planar periodic orbits in the Hill's approximation in the restricted four body problem, that is, after a symplectic scaling, two massive bodies are sent to infinity, by mean of expanding the potential as a power series in m3^1/3, (the mass of the third small primary) and taking the limit case when m3 tends to zero. The limiting Hamiltonian depends on a parameter mu (the mass of the second primary) and possesses some dynamical features from both the classical restricted three- body problem and the restricted four-body problem. We explore the families of periodic orbits of the infinitesimal particle for some values of the mass pa- rameter, these explorations show interesting properties regarding the periodic orbits for this problem, in particular for the Sun-Jupiter-asteroid case. We also offer details on the horizontal and vertical stability of each family.