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

摘要

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 Delta(123) and Delta(456). Assume that both triangles are symmetrical about the line connecting m(3) and m(4). Within these configurations, the six body configuration is not a central configuration if the triangle Delta(123) is above or below the triangle Delta(456). 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 Delta(123) and Delta(456) 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. (C) 2020 Elsevier B.V. All rights reserved.