Hill’s approximation in a restricted four-body problem

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 \(m_3\) of one primary is very small compared with the other two, \(m_1\) and \(m_2\), and we study the Hamiltonian system describing the motion of the massless particle in a neighborhood of \(m_3\). In a similar way to Hill’s 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 \(m_3^{1/3}\), and taking the limit case when \(m_3\rightarrow 0\). 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 Poincaré sections, direct and retrograde periodic orbits about \(m_3\), 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.