Perturbed particle disks

The Boltzmann moment equations are solved to determine the velocity ellipsoid in a particle disk near an isolated satellite resonance. In a coordinate frame which rotates with the pattern speed of the perturbation potential, the solutions are stationary functions of the azimuthal angle. From the velocity ellipsoid we obtain the stress tensor due to particle collisions and consequently, the viscous angular momentum flux. We show that the magnitude of the rate of deformation tensor in a perturbed particle disk is bounded from above by $\text{KΩ(1 + τ}{}^2\text{)}{}^{\mathrm{<mtext>1</mtext><mtext>2</mtext>}}\text{,}\text{where}\text{Ω}$ is the orbital angular velocity, τ is the optical depth, and K is a dimensionless constant of order unity. It is also found that in sufficiently perturbed regions there are ranges of azimuthal angle within which the radial component of the angular momentum flux is negative. It is even possible for the angular momentum luminosity, the radial flux integrated over azimuth, to be negative. These results are important for understanding sharp edges and the decay of density waves in planetary rings. They are also relevant to the damping of differential precession and eccentricity in narrow ringlets.