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

Formulation and analytic calculation for the spin angular momentum of a moonlet due to inelastic collisions of ring particles

“…In this paper, we consider the case where mutual gravity between the moonlet and particles can be neglected. Ohtsuki (1992Ohtsuki ( , 1999 demonstrated that analytic expressions of collision and velocity stirring rates derived for non-gravitating ring particles agree well with results of three-body orbital integration that takes into account mutual gravity when particles' random velocity is larger than a few times their escape velocity. Such a high velocity would be realized in rings perturbed by other larger satellites.…”

“…On the other hand, such a correction is not necessary for P col , L x 2 +y 2 , L 2 , and L z2 , where such cancellation does not occur. In order to obtain the analytic expression for L x 2 +y 2 , we introduce a new coordinate system (x ,ỹ ,z ) defined as (Ohtsuki, 1992(Ohtsuki, , 1999…”

“…Following Ohtsuki (1999), who calculated the Rayleigh distribution averages of the viscous stirring and dynamical friction rates of ring particles, we can evaluate the Rayleigh distribution average of L z as…”

“…Note that β 1/2 is expected in such high-velocities that ẽ 2 1/2 > ∼ 2, where the random velocity dominates particles' relative velocity. On the other hand, β can take on smaller values if the velocity stirring of particles occurs in the lower-velocity regime with ẽ 2 1/2 < ∼ 1, where Kepler shear dominates the relative velocity (Ohtsuki, 1999).…”

“…In planetary rings with low optical depth, particles' velocity dispersion can be approximated by σ x ∼ max{v e , (2 − 3)R 2 } in the gravitating case and σ x ∼ (2 − 3)R 2 in the non-gravitating case (v e and R 2 are the escape velocity and the radius of a particle), if the moonlet is small enough to neglect its perturbation on particles' velocities (e.g., Salo, 1995;Ohtsuki, 1999). Using Eq.…”