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

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.