Escape and collision dynamics in the planar equilateral restricted four-body problem

TL;DR

This paper investigates the escape and collision behaviors of a test particle in a planar four-body gravitational system with equal masses, classifying orbit types and analyzing basins and times through numerical methods.

Contribution

It provides a detailed numerical analysis of orbit classifications, basins, and escape/collision times in the equal-mass four-body problem, linking results to chaotic scattering and leaking Hamiltonian systems.

Findings

Identified basins of escape and collision in the system.

Correlated orbit types with escape and collision times.

Linked the dynamics to chaotic scattering and leaking systems.

Abstract

We consider the planar circular equilateral restricted four body-problem where a test particle of infinitesimal mass is moving under the gravitational attraction of three primary bodies which move on circular orbits around their common center of gravity, such that their configuration is always an equilateral triangle. The case where all three primaries have equal masses is numerically investigated. A thorough numerical analysis takes place in the configuration $(x,y)$ as well as in the $(x,C)$ space in which we classify initial conditions of orbits into four main categories: (i) bounded regular orbits, (ii) trapped chaotic orbits, (iii) escaping orbits and (iv) collision orbits. Interpreting the collision motion as leaking in the phase space we related our results to both chaotic scattering and the theory of leaking Hamiltonian systems. We successfully located the escape and the collision basins and we managed to correlate them with the corresponding escape and collision times of orbits. We hope our contribution to be useful for a further understanding of the escape and collision properties of motion in this interesting dynamical system.