Hill approximation in a restricted four-body problem

TL;DR

This paper extends the classical lunar Hill problem to a restricted four-body scenario, analyzing the motion of a massless particle near a small primary in a three-body system with an equilateral triangle configuration.

Contribution

It introduces a new Hill approximation for a restricted four-body problem, combining features of three- and four-body dynamics, with detailed geometric and dynamical analysis.

Findings

Derived the limiting Hamiltonian for the four-body problem

Analyzed the geometry of Poincare sections and periodic orbits

Identified stable and unstable manifolds and homoclinic intersections

Abstract

We consider a restricted four-body problem on the dynamics of a massless particle under the gravitational force produced by three mass points forming an equilateral triangle configuration. We assume that the mass m3 of one primary is very small compared with the other two, m1 and m2, and we study the Hamiltonian system describing the motion of the massless particle in a neighborhood of m3. In a similar way to Hill approximation of the lunar problem, we perform a symplectic scaling, sending the two massive bodies to infinity, expanding the potential as a power series in m3, and taking the limit case when m3 tends to cero. We show that the limiting Hamiltonian inherits dynamical features from both the restricted three-body problem and the restricted four-body problem. In particular, it extends the classical lunar Hill problem. We investigate the geometry of the Poincare sections, direct and retrograde periodic orbits about m3, libration points, periodic orbits near libration points, their stable and unstable manifolds, and the corresponding homoclinic intersections. The motivation for this model is the study of the motion of a satellite near a jovian Trojan asteroid.