Planar Symmetric Concave Central Configurations in Four-body Problem

TL;DR

This paper investigates conditions under which symmetric concave four-body configurations can be made central by positive masses, identifying regions where such configurations are impossible or possible.

Contribution

It characterizes the regions in configuration space where symmetric concave four-body setups can or cannot be realized as central configurations with positive masses.

Findings

Certain regions prohibit positive-mass central configurations.

Other regions always admit positive-mass central configurations.

The study delineates the parameter space for feasible configurations.

Abstract

In this paper, we consider the problem: given a symmetric concave configuration of four bodies, under what conditions is it possible to choose positive masses which make it central. We show that there are some regions in which no central configuration is possible for positive masses. Conversely, for any configuration in the complement of the union of these regions, it is always possible to choose positive masses to make the configuration central.