Central configurations in the planar 6-body problem forming two equilateral triangles

Abstract

A Central Configuration (CC) is a special arrangement of masses in the n-body problem where the gravitational force on each body points proportionally toward the center of mass. A stacked CC is a CC that has a proper subset of the n bodies also forming a CC. In this paper, six bodies are located on two equilateral triangles Δ123 and Δ456. Assume that both triangles are symmetrical about the line connecting m3 and m4. Within these configurations, the six body configuration is not a central configuration if the triangle Δ123 is above or below the triangle Δ456. It is also not a central configuration if more than two of the six bodies are collinear. When the two equilateral triangle configurations have a common centroid, masses on each equilateral triangles must be same respectively and the configuration can form a central configuration only if the ratio of the lengths of the sides between Δ123 and Δ456 falls into one of five intervals. Moreover there are some numerical evidences that, first there are exactly two nested central configurations but there may be one, two, or three twisted nested CCs for a given mass ratio; and second, there exists central configurations other than same centroid.