Chaotic dynamics in the planar gravitational many-body problem with rigid body rotations

TL;DR

This paper investigates chaotic dynamics in a planar N-body system with one rigid, asymmetric body, demonstrating analytically and numerically that chaos is common, especially with non-circular orbits, without needing tidal effects.

Contribution

It introduces a reduced planar N-body model with a rigid continuum body and analytically proves the existence of homoclinic chaos using the Melnikov method.

Findings

Chaos exists near circular orbits as shown analytically.

Numerical evidence indicates chaos persists in more complex orbital configurations.

Chaos extent correlates with deviations from circularity.

Abstract

The discovery of Pluto's small moons in the last decade brought attention to the dynamics of the dwarf planet's satellites. With such systems in mind, we study a planar $N$-body system in which all the bodies are point masses, except for a single rigid body. We then present a reduced model consisting of a planar $N$-body problem with the rigid body treated as a 1D continuum (i.e. the body is treated as a rod with an arbitrary mass distribution). Such a model provides a good approximation to highly asymmetric geometries, such as the recently observed interstellar asteroid 'Oumuamua, but is also amenable to analysis. We analytically demonstrate the existence of homoclinic chaos in the case where one of the orbits is nearly circular by way of the Melnikov method, and give numerical evidence for chaos when the orbits are more complicated. We show that the extent of chaos in parameter space is strongly tied to the deviations from a purely circular orbit. These results suggest that chaos is ubiquitous in many-body problems when one or more of the rigid bodies exhibits non-spherical and highly asymmetric geometries. The excitation of chaotic rotations does not appear to require tidal dissipation, obliquity variation, or orbital resonance. Such dynamics give a possible explanation for routes to chaotic dynamics observed in $N$-body systems such as the Pluto system where some of the bodies are highly non-spherical.