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

摘要

The discovery of Pluto's small moons in the last decade brought attention tothe dynamics of the dwarf planet's satellites. Recent work has consideredresonant interactions in the orbits of Pluto's small moons, with thePluto-Charon system apparently inducing rotational chaos in non-spherical moonswithout the need of resonance. However, New Horizons observations suggest thatdespinning due to tidal dissipation has not taken place. Still, a tidallyevolving Styx does appear to exhibit intermittent obliquity variations andepisodes of tumbling, suggesting some form of chaos in the rotational dynamics.With these systems in mind, we study a planar $N$-body system in which all thebodies are point masses, except for a single rigid body. We then present areduced model consisting of a planar $N$-body problem with the rigid bodytreated as a 1D continuum (i.e. the body is treated as a rod with an arbitrarymass distribution). Such a model provides a good approximation to highlyasymmetric geometries, such as the recently observed interstellar asteroid'Oumuamua, but is also amenable to analysis. We analytically demonstrate theexistence of homoclinic chaos in the case where one of the orbits is nearlycircular by way of the Melnikov method, and give numerical evidence for chaoswhen the orbits are more complicated. We show that the extent of chaos inparameter space is strongly tied to the deviations from a purely circularorbit. These results suggest that chaos is ubiquitous in many-body problemswhen one or more of the rigid bodies exhibits non-spherical and highlyasymmetric geometries. The excitation of chaotic rotations does not appear torequire tidal dissipation, obliquity variation, or orbital resonance. Suchdynamics 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 highlynon-spherical.