Viscosity in a dense planetary ring with self-gravitating particles

We have investigated the viscosity (the angular momentum flux) in dense, self-gravitating particle disks such as Saturn's main ring, by performing local N-body simulations. Viscosity could play important roles in evolution and structure formation of planetary rings. The ring's viscosity has been investigated with both theoretical and numerical approaches (e.g., Goldreich and Tremaine, 1978, Icarus 34, 227-239; Wisdom and Tremaine, 1988, Astron. J. 95, 925-940). However, these studies mainly considered systems including physical collisions of particles but not mutual gravitational interactions. Local N-body simulations by Salo (1995, Icarus 117, 287-312) and Daisaka and Ida (1999, Earth, Planet Space 51, 1195-1213) showed that a wake-like structure and clumps of particles are formed by the self-gravitational instability and that in such situations coherent motion of particles is dominant rather than random motion, which leads to an increase in radial velocity dispersion of particles. The wake structure and associated coherent motion are considered to affect the ring viscosity significantly. Our simulation in this paper shows that the viscosity is strongly enhanced by the wakes. When the wake structure strongly develops, the coherent motion considerably enhances the translational viscosity, which was usually referred to as a local component in previous studies, and the effective viscosity is dominated by both gravitational torque due to the wake structure and the enhanced translational viscosity. We also find that in the presence of the wakes, the viscosity ν is given as ν ≃ C G²∑²/Ω³₀, where G, ∑, and Ω₀ are the gravitational constant, the surface mass density of a ring, and the angular velocity, respectively. The non-dimensional correction factor C depends on the distance from the central planet. For example, C ≃ 6-20 for Saturn's B-ring. The effects of the enhanced viscosity on the structure and evolution of Saturn's main ring are also discussed.