Collisions in self-gravitating clouds of planetesimals

A theory of partially elastic collisions is constructed for frictionless planetesimals in an arbitrary gravitational field. The non-zero size of the particles and the influence of gravitational encounters are included. The equations for a self-gravitating rotationally symmetric disk or ring are written in an explicit form. Such systems turn out to be bimodal in the same sense as the Keplerian systems, i.e. there are two kinds of stable configurations which may co-exist in adjacent regions without disturbing the mechanical equilibrium. The transitions from one mode to another can also occur at essentially smaller values of the optical thickness than those previously found for Saturn's rings: in one of the numerically studied cases the transition from the dense to the rarefied mode occurred at the optical thickness 3×10^-5 while the reversed process corresponded to a higher value, 10^-2. The difference illustrates the dependence of the transition on its direction. The characteristic S shape which several authors have found for the relation between the viscosity and the optical thickness in Keplerian systems becomes more complicated if the contribution of self-gravitation increases. In some cases the stable solutions also imply a certain minimum value of the optical thickness.