Turbulent viscosity and lifetime of Saturnʹs rings

The viscosity (the angular momentum flux) in the disk of mutually gravitating particles of Saturnʹs rings is investigated. The hydrodynamic theory of the gravitational Jeans-type instability of small gravity perturbations (e.g., those produced by spontaneous disturbances) of the disk is developed. It is suggested that in such a system the hydrodynamic turbulence may arise as a result of the instability. The turbulence is related to stochastic motions of “fluid” elements. The objective of this paper is to show that in the Jeans-unstable Saturnian ring disk the turbulent viscosity may exceed the ordinary microscopic viscosity substantially. The main result of local N-body simulations of planetary rings by Daisaka et al. (2001. Viscosity in a dense planetary ring with self-gravitating particles. Icarus 154, 296–312) is explained: in the presence of the gravitationally unstable density waves, the effective turbulent viscosity ν eff is given as ν eff = CG 2 Σ 2 / Ω 3 , where G, Σ , and Ω are the gravitational constant, the surface mass density of a ring, and the angular velocity, respectively, and the nondimensional correction factor C ≈ 10 . We argue that both Saturnʹs main rings and their irregular of the order of 100 m or even less fine-scale structure (being recurrently created and destroyed on the time scale of an order of Keplerian period ∼ 10 h ) are not likely much younger than the solar system.