The role of particle collisions for the dynamics in planetary rings

Abstract

A new model of collisions between particles in planetary rings is proposed, where the bulk friction of the material of the ring particles (mainly ice for Saturn's rings) as well as the friction between their surfaces are taken into account at the collision. The normal and tangential restitution coefficients, ϵ N and ϵ T , respectively, which measure the loss of kinetic energy due to collisions, have been obtained by numerical integration of collision equations developed in this paper. In the case of the normal restitution coefficient ϵ N we have developed a new model for colliding spheres, basing on the Hertz contact theory, which has been extended to plastic, or dissipative deformations. The results of the integration of this model have been compared with those ones of laboratory experiments for ϵ N carried out by Bridges et al. A surprisingly good agreement with the experimental results has been achieved justifying our model. According to this, we have obtained a decreasing normal restitution coefficient ϵ N with increasing impact velocity, i.e. the higher the impact velocity the more mechanical energy is dissipated. For the tangential component ( ϵ T ) of the collision process we try to account for the sub-microscopic properties of the surfaces of the particles. We assume that asperities can be deformed either elastically at low relative tangential velocities, or plastically (sticking) in the case of higher tangential impact velocities. In the latter case a slipping of the surfaces might also take place corresponding to positive values of ϵ T . These behaviours—slipping ( ϵ T > 0), sticking ( ϵ T ≈ 0) as well as a reversal of the tangential motion ( ϵ T < 0)—should also depend on the normal component of the contact force (Coulomb friction) which yields a coupled system of equations describing the dynamics of binary collisions. We have found that the value ϵ T depends sensitively on the properties of the asperities—their size (roughness) and especially their shape—and on the nature of the bulk dissipation assumed for the tangential motion. The common rough tendency of all considered models is a partly elastic reversal for low tangential velocities corresponding to ϵ T < 0, and a slipping behaviour for sufficient high velocities according to ϵ T → 1. Thus, the tangential restitution coefficient takes values of −1 < ϵ T < 1 depending on the normal component of the collision force, mostly neglected in former work. Finally, we briefly discuss the results of our numerical investigations with respect to the kinetics of the particle ensemble in planetary rings. The main problems are the dependence of the kinematic viscosity v and the velocity dispersion c 2—the temperature of the rings—on the particle number density n. Both dependencies are determined by the loss of kinetic energy due to collisions expressed by ϵ N and ϵ T . If there are regions where c 2 and v are decreasing functions of n, a viscous instability can take place which corresponds to an (anomalous) diffusion with negative diffusion coefficients. In that case density fluctuations will be amplified, and not smoothed out, which could be an explanation for the strange irregular structures observed in Saturn's B-ring.