Some Observaties on the Fig8 Solution of the 3BP

TL;DR

This paper discusses properties of the figure-8 solution in the three-body problem, highlighting its geometric configurations and historical context, with an emphasis on its unique periodic trajectory and related properties.

Contribution

It provides a detailed analysis of the figure-8 solution's properties, configurations, and historical overview within the context of three-body problem solutions.

Findings

The figure-8 solution has zero angular momentum.

It exhibits specific triangular and collinear configurations.

The solution relates to well-known quartic curves with figure-eight shapes.

Abstract

Some properties of the periodic solution of the three-body problem where three particles of equal mass follow the same trajectory are discussed. This trajectory has the shape of a figure-8. The three particles have a constant separation in time of 1/3 of the period. The angular momentum of the system is zero. The paper by Broucke and Elipe1 gives a thorough overview of history of this solution and places this solution in the general framework of the periodic solutions of the three-body problem. The two most interesting properties are the triangular and collinear configuration. Their occurrence in well known quartic curves with a figure eight shape are collected as an appendix.