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

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.