Dense planetary rings and the viscous overstability
  This paper examines the onset of the viscous overstability in dense particulate rings. First, we formulate a dense gas kinetic theory that is applicable to the Saturnian system. Our model is essentially that of Araki and Tremaine (1986), which we show can be both simplified and generalised. Second, we put this model to work computing the equilibrium properties of dense planetary rings, which we subsequently compare with the results of N-body simulations, namely those of Salo (1991). Finally, we present the linear stability analyses of these equilibrium states, and derive criteria for the onset of viscous overstability in the self-gravitating and non-self-gravitating cases. These are framed in terms of particle size, orbital frequency, optical depth, and the parameters of the collision law. Our results compare favourably with the simulations of Salo et al. (2001). The accuracy and practicality of the continuum model we develop encourages its general use in future investigations of nonlinear phenomena. 
Henrik N. Latter, Gordon I. Ogilvie