Topological Dependence of Kepler's Third Law for Collisionless Periodic Three-Body Orbits with Vanishing Angular Momentum and Equal Masses

TL;DR

This paper discovers a linear relationship between the topology and period of collisionless three-body orbits with zero angular momentum and equal masses, revealing a fundamental mathematical law.

Contribution

It introduces the first known relation linking topological and kinematical properties of three-body systems, supported by numerical calculations.

Findings

Period depends linearly on topology variables.

Predicts periods of undiscovered orbits.

Shows the set of short-period orbits is countable.

Abstract

We present results of numerical calculations showing a three-body orbit's period's $T$ dependence on its topology. This dependence is a simple linear one, when expressed in terms of appropriate variables, suggesting an exact mathematical law. This is the first known relation between topological and kinematical properties of three-body systems. We have used these results to predict the periods of several sets of as yet undiscovered orbits, but the relation also indicates that the number of periodic three-body orbits with periods shorter than any finite number is countable.