Apse Alignment of Narrow Eccentric Planetary Rings

The boundaries of the Uranian ǫ, α, and β rings can be fitted by Keplerian ellipses. The pair of ellipses that outline a given ring share a common line of apsides. Apse alignment is surprising because the quadrupole moment of Uranus induces differential precession. We propose that rigid precession is maintained by a balance of forces due to ring self-gravity, planetary oblateness, and interparticle collisions. Collisional impulses play an especially dramatic role near ring edges. Pressure-induced accelerations are maximal near edges because there (1) velocity dispersions are enhanced by resonant satellite perturbations, and (2) the surface density declines steeply. Remarkably, collisional forces felt by material in the last ∼100 m of a ∼10 km wide ring can increase equilibrium masses up to a factor of ∼100. New ring surface densities are derived which accord with Voyager radio measurements. In contrast to previous models, collisionally modified self-gravity appears to allow for both negative and positive eccentricity gradients; why all narrow planetary rings exhibit positive eccentricity gradients remains an open question. Subject headings: celestial mechanics — planets and satellites: individual (Uranus, rings ǫ, α, and β)