The Dynamics of Elliptical Rings

We investigate the evolution of eccentric rings under the influence of ( 1) differential precession due to the planetary quadrupole moment; (2) self-gravity; (3) viscous forces due to interparticle collisions; and (4) eccentricity excitation by shepherd satellites. The principal conclusions are that: (a) Uniform precession can be enforced by self-gravity (Goldreich and Tremaine 1979b); the resulting configuration is both dynamically and secularly stable. (b) Due to viscous forces the line of apsides at the inner ring edge is not exactly aligned with the line of apsides at the outer edge; the apse shift may be detectable in the a and /3 rings of Uranus. (c) The mean eccentricity is determined by a balance between viscous damping and excitation by shepherds. (d) We expect the dimensionless eccentricity gradient a..1e/ ..1a to be positive and of order unity in most eccentric rings, as observed.