Viscosity in keplerian disks: Steady-state velocity distribution and non-local collision effects

A kinetic model based on a numerical algorithm, rather than Boltzman's equation, yields the viscosity and velocity distribution for colliding, finite-size particles in a planetary ring. Results are similar to those of many-particle simulations, and show that non-local effects due to the finite size are dominant in many cases of interest. Only for small particles does the viscosity decrease with increasing optical thickness sufficiently for the standard ringlet instability model to apply. This numerical kinetic theory will allow study of multi-size particle distributions, as well as scattering due to gravitational interactions or alternative collision models.