Three‐dimensional Dynamics of Narrow Planetary Rings

Narrow planetary rings are eccentric and inclined. Particles within a givenring must therefore share the same pericenter and node. We solve for thethree-dimensional geometries and mass distributions that enable the UranianAlpha and Beta rings, and the Saturnian Maxwell and Colombo (Titan) rings, tomaintain simultaneous apsidal and nodal lock. Ring self-gravity, interparticlecollisions, and the quadrupole field of the host planet balance each other toachieve this equilibrium. We prove that such an equilibrium is linearly stable.Predictions for the Saturnian ringlets to be tested by the Cassini spacecraftinclude: (1) ringlet masses are of order 1e19 g, (2) surface mass densitiesshould increase from ring midline to ring edges, and (3) rings are verticallywarped such that the fractional variation of inclination across the ring is oforder 10%. Analogous predictions are made for the Uranian rings. Simultaneousapsidal and nodal locking forces the narrowest portion of the ring--its``pinch,'' where self-gravitational and collisional forces are strongest--tocirculate relative to the node, and introduces previously unrecognizedtime-varying forces perpendicular to the planet's equator plane. We speculatethat such periodic stressing might drive kilometer-scale bending waves at afrequency twice that of apsidal precession. Such flexing might be observed overa few weeks by Cassini.Comment: Final revised version, ApJ, in pres