Viscosity and mass transport in nonuniform Keplerian disks

A new quantitative formalism describing the dynamics of a Keplerian particulate disk, based on a heuristic description of viscous transport, permits study of rings with a wide range of ensemble and individual particle properties. Here the formalism is developed and applied to the case of a ring with a radial gradient in optical thickness. A steady-state solution for the velocity distribution directly gives the radial mass transport, as well as the viscosity. The formula for viscosity is identical to that derived a decade earlier by Goldreich and Tremaine for a uniform disk, thus validating the assumption by various workers that it could be applied to nonuniform disks, for example in consideration of ringlet instabilities. Our analytical method involves solving a novel form of the Krook equation by separating the distribution of collisional products in phase space into a symmetrical component and a remainder that can be approximated by delta functions. Unlike most past approaches, a Gaussian form for the solution is not assumed. In the case described here, the model is simplified in common with past work (e.g., small, uniform particles and Krook-type treatment of collisions), but the general approach is extendable to less artificially restricted cases.