The velocity dispersion in Saturn's rings

Abstract

The velocity dispersion in a differentially rotating disk of particles such as Saturn's rings is determined by the details of the collision process. Collisions give rise to a viscous stress that converts orbital energy into random motions. Since the collisions are not perfectly elastic, the energy in random motions is dissipated as heat. With increasing velocity dispersion the latter process becomes more important relative to the former because the collisions become less elastic. The velocity dispersion adjusts so that the effects of these two processes balance. The condition for this balance fixes the coefficient of restitution ϵ of the colliding particles as a function of the disk's optical depth τ. We solve the Boltzmann moment equations to determine ϵ(τ). If the rings are about as old as the solar system then their radial width implies that the velocity dispersion of the ring particles is less than 0.2 cm sec −1. The corresponding vertical thickness is then less than 10 m. We discuss the effects of collisions on the particles in Saturn's rings. If the particles are made of ice they are eroded by collisions and accrete the collisional debris. The time scale for erosion and accretion is probably shorter than the age of the solar system. Furthermore, for ice particles ϵ is likely to be substantially less than unity even at impact velocities as low as 10 −3 cm sec −1. Thus, a ring of ice particles would be a monolayer thick.