Evolution of Particle Velocity Dispersion in a Circumplanetary Disk Due to Inelastic Collisions and Gravitational Interactions

Abstract

We study the evolution and equilibrium values of velocity dispersions of particles in a circumplanetary disk, taking into account both inelastic collisions and gravitational interactions between particles. For a disk of particles with a Rayleigh distribution of orbital eccentricities and inclinations, we derive an evolution equation for mean square eccentricities and inclinations on the basis of the Hill's approximations in the three-body problem. We find that the evolution is governed by the terms of viscous stirring and energy equipartition, the latter tending to equalize the product of mass and mean square eccentricities and inclinations of particles with different sizes. Stirring rates of mean square eccentricities and inclinations of nongravitating particles due to direct collisions are obtained analytically. In the case where both direct collisions and gravitational interactions are taken into account, stirring rates are numerically evaluated by three-body orbit integrations, and their Rayleigh distribution averages are calculated. Using these stirring rates, the evolution of the root mean square eccentricities and inclinations of particles are simulated. We confirm that effects of both finite size and gravity of particles play an important role in maintaining nonzero velocity dispersions. For a system with two particle size components, the relative importance of different terms in the evolution equation is investigated and found to vary depending on the relative abundance of particles of each size component. The evolution of root mean square eccentricities and inclinations of particles calculated on tha basis of the present formalism is also compared with numerical results ofN-body simulations for a disk of particles with low optical depth, and excellent agreement is found between the results of the two methods.