Overview and comparison of approaches towards the planar restricted five-body problem with primaries forming an axisymmetric four-body central configuration

Abstract

The aim of this paper is to present recent results on the restricted five-body problem where the primaries are located at a known central configuration having the $x$-axis as symmetry axis, so that two bodies with equal masses are situated symmetrically with respect to this axis. We firstly give an overview of the axisymmetric central configurations computed by Erdi and Czirjak (2016), as well as that where three bodies with equal masses are situated at the vertices of an equilateral triangle, while the fourth body lies at the center of the triangle. This last one bridges the gap between convex and concave four-body central configurations. After that, the characterization of the geometry of the restricted five-body problem families with these configurations are described, and the number and the evolution positions of equilibrium points, which depend on the mass parameters of the primaries, are compared.