A new formulation of the viscosity in planetary rings

We present a new formulation of the viscosity in planetary rings, where particles interact through their gravitational forces and direct collisions. In the previous studies on the viscosity in self-gravitating rings, the viscosity consists of three components, which are defined separately in different ways. The complex definitions make it difficult to evaluate the viscosity in N-body simulation of rings. In our new formulation, the viscosity is expressed in terms of changes in orbital elements of particles due to particle interactions. This makes the expression of the viscosity simple. The new formulation gives a simple way to evaluate the viscosity in N-body simulation. We find that for practical evaluation of the viscosity of planetary rings, only energy dissipation at direct inelastic collisions is needed. For tenuous particle disks (i.e., optically thin disks), we further derive a formula of the viscosity. The formula requires only a numerical coefficient that can be obtained from three-body calculation. Since planetesimal disks are also tenuous, the viscosity in planetesimal disks can be also obtained from this formula. In a subsequent paper, we will evaluate this coefficient through three-body calculation and obtain the viscosity for a wide range of parameters such as the restitution coefficient and the radial location in rings.