A family of stacked central configurations in the planar five-body problem

Abstract

We study planar central configurations of the five-body problem where three bodies, (Formula presented.) and (Formula presented.), are collinear and ordered from left to right, while the other two, (Formula presented.) and (Formula presented.), are placed symmetrically with respect to the line containing the three collinear bodies. We prove that when the collinear bodies form an Euler central configuration of the three-body problem with (Formula presented.), there exists a new family, missed by Gidea and Llibre (Celest Mech Dyn Astron 106:89–107, 2010), of stacked five-body central configuration where the segments (Formula presented.) and (Formula presented.) do not intersect.