On Kite Central Configurations

TL;DR

This paper investigates convex and concave kite central configurations in the Newtonian four-body problem, proving uniqueness, exploring bifurcations, and analyzing stability with computational and topological tools.

Contribution

It provides a new proof of the uniqueness of convex kite central configurations and explores stability conditions using advanced mathematical techniques.

Findings

Unique convex kite configuration for given masses

Heaviest body must be at least 25 times larger for stability

Discussion of bifurcations and degenerate configurations

Abstract

We study kite central configurations in the Newtonian four-body problem. We present a new proof that there exists a unique convex kite central configuration for a given choice of positive masses and a particular ordering of the bodies. Our proof uses tools from differential topology (e.g., the Poincar\'{e}-Hopf Index Theorem) and computational algebraic geometry (e.g., Gr\"{o}bner bases). We also discuss concave kite central configurations, including degenerate examples and bifurcations. Finally, we numerically explore the linear stability of the corresponding kite relative equilibria, finding that the heaviest body must be at least 25 times larger than the combined masses of the other three bodies in order to be linearly stable.