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

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.