The velocity dispersion in Saturn's rings

The collisional dynamics of a differentially rotating disk of particles are considered. The investigation is to some extent an analytic counterpart of the numerical studies of Brahic (1977). Following Brahic, it is assumed that the particles are identical, indestructible, imperfectly elastic, smooth spheres. Gravitational interactions between the particles are neglected. The main result of the reported analysis is the derivation of the relation between the coefficient of restitution and the optical depth for a differentially rotating disk of identical, inelastic, smooth spheres. Questions concerning the applicability of the obtained results to Saturn's rings are studied. It is felt that the calculations should give accurate quantitative results if the rings' filling factor is small. If this assumption is not justified, the calculations should be at least qualitatively correct.