Equilibrium points and their linear stability in the planar equilateral restricted four-body problem: A review and new results

TL;DR

This paper reviews and extends the understanding of equilibrium points and their stability in the planar restricted four-body problem with three bodies in an equilateral triangle, combining known results with new numerical insights.

Contribution

It unifies existing results on equilibrium points and stability, and provides the first systematic numerical analysis of resonance curves and stability boundaries in this problem.

Findings

Identification of stability and instability regions in mass space

Characterization of the number of stable equilibrium points

Resonance curves delineating stability domains

Abstract

In this work, we revisit the planar restricted four-body problem to study the dynamics of an infinitesimal mass under the gravitational force produced by three heavy bodies with unequal masses, forming an equilateral triangle configuration. We unify known results about the existence and linear stability of equilibrium points of this problem which have been obtained earlier, either as relative equilibria or a central configuration of the planar restricted $(3 + 1)$-body problem. It is the first attempt in this direction. A systematic numerical investigation is performed to obtain the resonance curves in the mass space. We use these curves to answer the question about the existing boundary between the domains of linear stability and instability. The characterization of the total number of stable points found inside the stability domain is discussed.