The Self-Similarity of Shear-dominated Viscous Stirring

Abstract

We examine the growth of eccentricities of a population of particles with nearly circular orbits around a central massive body. Successive encounters between pairs of particles increase the eccentricities in the disk on average. We describe the system in terms of a Boltzmann equation. As long as the epicyclic motions of the particles are small compared to the shearing motion between circular Keplerian orbits, there is no preferred scale for the eccentricities, and the evolution is self-similar. This simplification reduces the full time-dependent Boltzmann equation to two separate equations: one that describes the shape of the distribution and another that describes the evolution of the characteristic eccentricity on which the distribution is centered. We find that the shape of the eccentricity distribution function is a general feature of such systems, and is of the form (1 + x^2)^(-3/2). In particular, bodies evolving under only their own excitations have the same eccentricity distribution profile as bodies whose excitations are balanced by dynamical friction. We find exact expression for the typical eccentricity for these two cases, and allow for time-dependent damping and excitation rates. Full numerical N-body simulations of a disk with 200 planetesimals verify our analytical self-similar distribution.