Evaluation of collision and stirring rates in circumplanetary particle disks based on three-body orbital integrations

We study collision and velocity stirring rates of particles in a circumplanetary disk with low optical depth where the epicyclic approximations can be applied to particle motions. We use orbital integrations in the three-body problem where both direct collisions and gravitational encounters between particles are taken into account. Using these numerical results together with the previously derived analytic results for nongravitating particles (Ohtsuki, 1999. Icarus 137, 152–177), we examine the dependence of collision and stirring rates on eccentricities and inclinations in the case of gravitating particles to calculate the averaged rates for a Rayleigh distribution of particles’ eccentricities and inclinations. Numerical results show that the dynamical friction rates for nongravitating ring particles can approximate the numerical results of three-body orbital integrations fairly well for a wide range of parameters. However, the effects of self-gravity substantially changes the collision and viscous stirring rates in low velocity cases. On the basis of the numerical results of orbital integrations in such low velocity cases and the analytic results for nongravitating cases, we derive semi-analytic formulae for collision and stirring rates for gravitating particles which can be used for the entire range of velocities. We calculate the evolution of root mean square eccentricities and inclinations of ring particles using these formulae, and find excellent agreement with the results obtained by N-body simulations.