The Torque at a Lindblad Vertical Resonance

A fairly uniform, rapidly and differentially rotating disk of rarely colliding particles (when the frequency of interparticle collisions is much smaller than the local orbital frequency) in a planet-moon system is considered. A moon causes a number of orbital resonant effects in this continuous, viscous (through ordinary collisions) disk. In the frame of hydrodynamical theory, the gravitational torques exerted by an exterior moon on particles at an inner Lindblad horizontal resonance and corresponding vertical resonance are estimated. It is shown that the torques are negative at these resonances, so gaps in the disk near each resonance may be created. The latter result can be used to provide a viable clue to solving the puzzle of narrow, dense, and widely separated rings of Uranus. The model advocated suggests that the Uranian ring orbits have a close connection with the small moons of the planet that are interior to the orbit of Miranda, from Cordelia to Mab, discovered by Voyager 2 imaging observations in 1986.