The authors present a new numerical method for studying the equilibrium properties of planetary rings, in which a small patch of the ring is simulated with periodic boundary conditions. The results of the simulations compare favorably with the results of kinetic theory of dense fluids so long as the filling factor in the midplane is less than about 0.5. The conclusion of Araki and Tremaine (1986) that the condition for the viscous instability is not fulfilled is confirmed. The authors find that the "standard" ring model, with the particles all of the same size, develops vertically stratified layers at large optical depths (τ ⪆ 1) when mean self-gravity is included.
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Wisdom et al. (1988) studied this question.