Symmetry, bifurcation and stacking of the central configurations of the planar 1+4 body problem

TL;DR

This paper investigates the symmetry, bifurcation, and stacking properties of central configurations in the planar 1+4 body problem, revealing conditions for symmetry, mass equality, and the number of configurations.

Contribution

It proves symmetry and mass conditions for central configurations and classifies the number of solutions, including bifurcation points based on mass ratios.

Findings

Configurations are necessarily symmetric with equal masses for certain satellites.

Number of configurations varies between one, two, or three depending on parameters.

Bifurcation occurs at specific mass ratios, affecting the count of solutions.

Abstract

In this work we are interested in the central configurations of the planar 1+4 body problem where the satellites have different infinitesimal masses and two of them are diametrically opposite in a circle. We can think this problem as a stacked central configuration too. We show that the configuration are necessarily symmetric and the other sattelites has the same mass. Moreover we proved that the number of central configuration in this case is in general one, two or three and in the special case where the satellites diametrically opposite have the same mass we proved that the number of central configuration is one or two saying the exact value of the ratio of the masses that provides this bifurcation.