Self-Gravity and Dissipation in Polar Rings

We construct models of highly-inclined (``polar'') rings in an external potential including both self--gravity and dissipation due to a drag force. We adopt the oblate spheroidal scale--free logarithmic potential with axis ratio $q=0.85$ and an initial inclination of 80$^\circ$ for the self--gravitating rings. Stellar (dissipationless) rings suffer from mass loss during their evolution which drives a secular change of the mean inclination toward the poles of the potential. As much as half of the ring mass escapes in the process and forms an inner and an outer shell of precessing orbits. If the remaining mass is more than $\sim$ 0.02 of the enclosed galaxy mass, rings remain bound and and are not destroyed by differential precession. The rings precess at a constant rate for more than a precession period $\tau_p$ finding the configuration predicted by Sparke in 1986 which warps at larger radii toward the poles of the potential. We model shear viscosity with a velocity-dependent drag force and find that nuclear inflow dominates over self--gravity if the characteristic viscous inflow time scale $\tau_{vi}$ is shorter than $\sim 25 \tau_p$. Rings with $\tau_{vi}/\tau_p \ \sles \ 25$ collapse toward the nucleus of the potential within one precession period independent of the amount of self--gravity. Our results imply that stars and gas in real polar rings exhibit markedly different dynamical evolutions.