The spatial Hill four-body problem I -- An exploration of basic invariant sets

TL;DR

This paper investigates invariant sets in the spatial Hill's four-body problem using analytical and numerical methods, revealing new periodic orbit families and the impact of a second disturbing body on orbit stability.

Contribution

It introduces a combined analytical and numerical approach to study invariant sets in the spatial Hill's four-body problem, discovering new periodic orbit families and stability effects.

Findings

Second disturbing body affects orbit stability and bifurcations

New families of periodic orbits found absent in Hill's three-body problem

Numerical continuation effectively explores parameter space

Abstract

In this work, we perform a first study of basic invariant sets of the spatial Hill's four-body problem, where we have used both analytical and numerical approaches. This system depends on a mass parameter mu in such a way that the classical Hill's problem is recovered when mu = 0. Regarding the numerical work, we perform a numerical continuation, for the Jacobi constant C and several values of the mass parameter mu by applying a classical predictor-corrector method, together with a high-order Taylor method considering variable step and order and automatic differentiation techniques, to specific boundary value problems related with the reversing symmetries of the system. The solution of these boundary value problems defines initial conditions of symmetric periodic orbits. Some of the results were obtained departing from periodic orbits within Hill's three-body problem. The numerical explorations reveal that a second distant disturbing body has a relevant effect on the stability of the orbits and bifurcations among these families. We have also found some new families of periodic orbits that do not exist in the classical Hill's three-body problem; these families have some desirable properties from a practical point of view.