Angular momentum and topological dependence of Kepler's Third Law in the Broucke-Hadjidemetriou-H\'enon family of periodic three-body orbits

TL;DR

This study reveals a universal relationship between angular momentum and period in certain three-body orbits, supporting the idea of their stability and suggesting a fundamental topological influence on their dynamics.

Contribution

It demonstrates that satellite orbits' angular momentum-period relationship aligns with that of progenitor orbits, indicating a topological and stability connection in three-body systems.

Findings

Satellite orbits follow the same L(T) curve as progenitors.

The standard deviation from the polynomial fit is very low, indicating high regularity.

Supports the conjecture that stable orbits are also KAM stable.

Abstract

We use 57 recently found topological satellites of Broucke-Hadjidemetriou-Henon's periodic orbits with values of the topological exponent $k$ ranging from $k$ = 3 to $k$ = 58 to plot the angular momentum $L$ as a function of the period $T$, with both $L$ and $T$ rescaled to energy $E=-\frac12$. Upon plotting $L(T/k)$ we find that all our solutions fall on a curve that is virtually indiscernible by naked eye from the $L(T)$ curve for non-satellite solutions. The standard deviation of the satellite data from the sixth-order polynomial fit to the progenitor data is $\sigma = 0.13$. This regularity supports Henon's 1976 conjecture that the linearly stable Broucke-Hadjidemetriou-Henon orbits are also perpetually, or Kolmogorov-Arnold-Moser stable.