Collisionless periodic orbits in the free-fall three-body problem

TL;DR

This paper discovers 234 new collisionless periodic orbits in the free-fall three-body problem, significantly expanding the known solutions and revealing a generalized Kepler's third law for specific mass ratios.

Contribution

It reports a large set of new collisionless periodic orbits and establishes a generalized Kepler's third law, advancing understanding of the three-body problem.

Findings

234 collisionless periodic orbits identified, including 231 new ones.

Existence of a generalized Kepler's third law for specific mass ratios.

Theoretical possibility of obtaining periodic orbits with arbitrary mass ratios.

Abstract

Although the free-fall three-body problem have been investigated for more than one century, however, only four collisionless periodic orbits have been found. In this paper, we report 234 collisionless periodic orbits of the free-fall three-body system with some mass ratios, including three known collisionless periodic orbits. Thus, 231 collisionless free-fall periodic orbits among them are entirely new. In theory, we can gain periodic orbits of the free-fall three-body system in arbitrary ratio of mass. Besides, it is found that, for a given ratio of masses of two bodies, there exists a generalized Kepler's third law for the periodic three-body system. All of these would enrich our knowledge and deepen our understanding about the famous three-body problem as a whole.