Quasi-equilibrium in collisional systems

Quasi-equilibrium states, in which collisional systems do not show the usually observed rapid decrease or increase in orbital eccentricities and inclinations, consist of two major types. In the first type (linear systems) the coefficient of restitution must depend on the velocity of impact to maintain the equilibrium. In the second type (non-linear systems) this is not necessary, but an appropriate perturbation is needed. Drag in a co-moving medium leads to this type, for instance. Two other types (degenerate systems and stochastic systems) can be defined, but they are not studied in this paper. Quasi-equilibrium systems are bimodal. Stable configurations are not possible between certain values of the optical thickness. The accretion of matter in a proto-planet may trigger a jump from one stable region to another in the surrounding medium. If this happens, a dense zone is produced at the distance of 16/9 planetary radii. This phenomenon seems to explain the origin of planetary rings. The distance of 16/9 radii (127 000 km for Jupiter, 107 000 km for Saturn and 44 000 km for Uranus) agrees remarkably well with the observations. In this model the ring is maintained by the pressure of an extremely rarefied medium.