Generalization of the Hill problem as an application for the Trojan asteroids of the solar system

TL;DR

This paper generalizes the Hill problem by analyzing the limit case of the restricted four body problem as one mass tends to zero, resulting in a new Hamiltonian with features of three- and four-body dynamics, applicable to Trojan asteroids.

Contribution

It introduces a new Hamiltonian system derived from the restricted four body problem in the limit of a small mass, extending the classical Hill problem to a more general setting.

Findings

Existence of a limit Hamiltonian as the small mass tends to zero.

The new Hamiltonian inherits features of both three- and four-body problems.

Analysis of dynamical properties relevant to Trojan asteroid dynamics.

Abstract

The restricted four body problem studies the dynamics of a massless particle under the gravitational force produced by three masses (primaries) in an equilateral configuration. One primary, say m3, is considered too small compared with the other ones. In a similar way as in the classical Hill problem, we study the limit case when m3 tends to zero in the Hamiltonian of the R4BP. In this paper we prove that such limit exists and the resulting limit problem produces a new Hamiltonian that inherits some basic features of the restricted three and four body problems. We analyze some dynamical aspects of this new system that can be considered as a generalization of the Hill problem.