On Stacked Planar Central Configurations with Five Bodies when One Body is Removed

Abstract

In this paper we answer the following question posed by Hampton (Nonlinearity 18:2299–2304, 2005). In addition to symmetric collinear configurations and the square with a mass at its center are there any planar five-body central configurations with a subset forming a four-body central configuration? We prove that, for non-collinear configurations, the only possible strictly planar central configuration for the five-body problem for which it can be removed one body such that the remaining four bodies are already in a central configuration is obtained with four bodies of equal masses at the vertices of a square and the fifth body of arbitrary mass at the center of the square.