The Set of Degenerate Central Configurations in the Planar Restricted Four-Body Problem

Abstract

In the 1940s, P. Pedersen published an extensive numerical study on the central configurations of the planar restricted four-body problem. His results were later confirmed numerically by C. Simo, in 1978, and J. R. Gannaway, in 1981, who also provided analytic results in several special cases. In this article, classical analysis and exact computations are applied to demonstrate a significant conclusion of Pedersen's work, namely, that the set of degenerate central configurations is a simple closed analytic curve inside the triangle formed by the nonzero masses. These properties of the degenerate set are an important step towards understanding the bifurcations undergone by the central configurations of the planar restricted four-body problem.