Collisions and Gravitational Interactions between Particles in Planetary Rings

Particles in planetary rings orbit the central planet, and undergo collisions and gravitational interactions with other particles. As a result, their orbits become inclined and non-circular. Also, ring particles likely have rough surfaces, and an oblique impact between them leads to rotation. In the case of dilute rings where collision frequency is sufficiently smaller than the orbital frequency, particles’ orbits evolve through successive two-body collisions and/or gravitational encounters, and the evolution can be described by the formulation based on the three-body problem. We describe basic equations for such cases, and derive evolution equations for particle velocity dispersion and spin rates. We also discuss effects of rings’ self-gravity, which become dominant in dense rings. Collisions and gravitational interactions between particles result in angular momentum transfer in planetary rings. We discuss ring viscosity, which determines the rate of angular momentum transfer in rings. Viscosity in dilute rings can be expressed by particles’ velocity dispersion and is proportional to the ring surface density for a given particle size, while the dependence of the viscosity on ring surface density is stronger in dense self-gravitating rings, where angular momentum is transferred by interactions between gravitational wakes.