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 1/3 3 , and taking the limit case when m 3 → 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.

Chaotic motions in the restricted four body problem via Devaney's saddle-focus homoclinic tangle theorem

“…The work of [23] considers regularization of collisions with the primary bodies. A Hill's approximation is developed in [24].…”

“…The formal series computations discussed so far focus on the polynomial vector field F : C 7 → C 7 defined in Equation (24). In this section we justify the use of the polynomial problem, and show how to recover the dynamics for the CRFBP.…”

Qualitative Analysis of Some Libration Points in the Restricted Four-body Problem

“…In [12] and in [13] the situation with small mass ν of the planet was considered. The results partially agree with ours from Section 3 (see Remark 2).…”

“…The results partially agree with ours from Section 3 (see Remark 2). In [13] also the Poincaré return map (to a suitably chosen section) was studied numerically.…”

Solar Radiation Pressure Effects on Stability of Periodic Orbits in Restricted Four-Body Problem

“…The network of the families of simple symmetric periodic solutions of the restricted four-body problem was investigated and the effect of radiation on the periodic orbits was studied, their stability, as well as the evolution of the families when the radiation parameter varies, and Poincare sections of the problem are illustrated [9]. The classical lunar Hill problem was extended and the geometry of Poincare sections was investigated, also the direct and retrograde periodic orbits about the infinitesimal mass and their stable and unstable manifolds were studied [10]. Periodic orbits in the photo gravitational restricted problem when the primaries are tri-axial rigid bodies were investigated [11].…”