Lévy Flights of Binary Orbits due to Impulsive Encounters

Abstract

We examine the evolution of an almost-circular Keplerian orbit interacting with unbound perturbers. We calculate the change in eccentricity and angular momentum that results from a single encounter, assuming that the timescale for the interaction is shorter than the orbital period. The orbital perturbations are incorporated into a Boltzmann equation that allows for eccentricity dissipation. We present an analytic solution to the Boltzmann equation that describes the distribution of orbital eccentricity and relative inclination as a function of time. The eccentricity and inclination of the binary do not evolve according to a normal random walk but perform a Lévy flight. The slope of the mass spectrum of perturbers dictates whether close gravitational scatterings are more important than distant tidal ones. When close scatterings are important, the mass spectrum sets the slope of the eccentricity and inclination distribution functions. We use this general framework to understand the eccentricities of several Kuiper belt systems: Pluto, 2003 EL61, and Eris. We use the model of Tholen et al. to separate the non-Keplerian components of the orbits of Pluto's outer moons Nix and Hydra from the motion excited by interactions with other Kuiper belt objects. Our distribution is consistent with the observations of Nix, Hydra, and the satellites of 2003 EL61 and Eris. We address applications of this work to objects outside of the solar system, such as extra-solar planets around their stars and millisecond pulsars.