Central configurations of the spatial seven-body problem: two twisted triangles plus one

TL;DR

This paper proves the existence and uniqueness of a specific central configuration in the spatial seven-body problem, where six bodies form a regular octahedron and one is at its center, with no restriction on the seventh body's mass.

Contribution

It introduces a new class of central configurations involving a regular octahedron and a central body, with a proof of existence and uniqueness.

Findings

Existence of a central configuration with six bodies at octahedron vertices and one at the center.

Uniqueness of this configuration under the given geometric constraints.

No restriction on the mass of the central body.

Abstract

In this work we are interested in the central configurations of the spatial seven-body problem where six of them are at vertices of two congruents equilateral triangles belong to parallel planes and one triangle is a rotation by the angle $\pi/3$ from the other one. We show a existence and uniqueness of a class of central configuration with this shape, namely six bodies with equal masses at the vertice of a regular octahedron with the seventh body in its center. There is not restriction to the mass of the seventh body.