A guide to hunting periodic three-body orbits with non-vanishing angular momentum

Abstract

A large number of periodic three-body orbits with vanishing angular momentum have been found in Newtonian gravity over the past 6 years due to a simple search method and to the contribution from practitioners outside the Celestial Mechanics community. Extension of such orbits to non-vanishing angular momentum has been lacking due to inter alia the absence of a sufficiently simple and widely known search method. We present a method, i.e., a general strategy plus detailed tactics (but not a specific algorithm, or a code), to numerically search for relative periodic orbits in the Newtonian three-body problem with three equal masses and non-vanishing angular momentum. We illustrate the method with an application to a specific, so-called Broucke–Hadjidemetriou–Hénon (BHH) family of periodic 3-body orbits: Our search yielded around 100 new “satellite” orbits, related to the original BHH orbits by a topological relation (defined in the text), with infinitely many orbits remaining to be discovered. We used the so-obtained orbits to test the period vs. topology relation that had previously been established, within a certain numerical accuracy, for orbits with vanishing angular momentum. Our method can be readily: (1) applied to families of periodic 3-body orbits other than the BHH one; (2) implemented using various standard algorithms for solving ordinary differential equations, such as the Bulirsch–Stoer and the Runge–Kutta–Fehlberg ones; (3) adapted to 3-body systems with distinct masses and/or coupling constants, including, but not limited to, Coulomb interaction. Our goal is to enable numerical searches for new orbits in as many families of orbits as possible, and thus to allow searches for other empirical relations, such as the aforementioned topology vs. period one.