Periodic three-body orbits with vanishing angular momentum in the Jacobi-Poincare "strong" potential

TL;DR

This paper systematically discovers 24 topologically distinct periodic three-body orbits with zero angular momentum in a specific potential, revealing sequences with finite asymptotic properties and linear action growth, extending understanding of such orbits.

Contribution

It identifies 24 new periodic three-body orbits with vanishing angular momentum in the Jacobi-Poincare potential, analyzing their topologies, asymptotic behavior, and action growth patterns.

Findings

22 of the 24 orbits are newly discovered.

Orbit actions grow linearly with topology index n.

Initial angles and periods approach finite limits as n increases.

Abstract

Moore and Montgomery have argued that planar periodic orbits of three bodies moving in the Jacobi-Poincare, or the "strong" pairwise potential $\sum_{i>j}\frac{-1}{r_{ij}^2}$, can have all possible topologies. Here we search systematically for such orbits with vanishing angular momentum and find 24 topologically distinct orbits, 22 of which are new, in a small section of the allowed phase space, with a tendency to overcrowd, due to overlapping initial conditions. The topologies of these 24 orbits belong to three algebraic sequences defined as functions of integer $n=0,1,2, \ldots$. Each sequence extends to $n \to \infty$, but the separation of initial conditions for orbits with $n \geq 10$ becomes practically impossible with a numerical precision of 16 decimal places. Nevertheless, even with a precision of 16 decimals, it is clear that in each sequence both the orbit's initial angle $\phi_n$ and its period $T_n$ approach finite values in the asymptotic limit ($n \to \infty$). Two of three sequences are overlapping in the sense that their initial angles $\phi$ occupy the same segment on the circle and their asymptotic values $\phi_{\infty}$ are (very) close to each other. The actions of these orbits rise linearly with the index $n$ that describes the orbit's topology, which is in agreement with the Newtonian case. We show that this behaviour is consistent with the assumption of analyticity of the action as a function of period.