The Self-Similarity of Shear-Dominated Viscous Stirring

Abstract

We examine the growth of eccentricities of a population of particles with initially circular orbits around a central massive body. Successive encounters between pairs of particles increase the eccentricities in the disk on average. As long as the epicyclic motions of the particles are small compared to the shearing motion between Keplerian orbits, there is no preferred scale for the eccentricities. The simplification due to this self-similarity allows us to find an analytic form for the distribution function; full numerical integrations of a disk with 200 planetesimals verify our analytical self-similar distribution. The shape of this non-equilibrium profile is identical to the equilibrium profile of a shear-dominated population whose mutual excitations are balanced by dynamical friction or Epstein gas drag.