Tree code simulations of planetary rings

A tree code method that incorporates a local shearing disc model and fourth-order integration algorithm is applied to the problem of planetary rings, with particular emphasis on the dynamics of Saturn's B ring. The new code, described in detail elsewhere, allows for particle self-gravity, a distribution of particle sizes, and surface friction (particle spin). Important changes made to the original code, to ensure an accurate treatment of collisions under severe high-density conditions, are described in detail. Comparison with work by Wisdom & Tremaine for the case of equal-size particles and mean self-gravity shows excellent agreement. Similar analysis is performed for the new regimes of selfgravity and particle size distributions, and it is shown that the condition for viscous instability is still not satis ed for these models. Particle spins lie generally within a rotational energy equipartition envelope, and are retrograde on average in the rotating (orbital) frame but exhibit a large spread in obliquity. The mean z-directed spin in the xed frame for most models varies between about 0.2 and 0.4 times the orbital angular velocity, similar to the 0.3 value found by Araki for the equal-size case, while the x and y spin components are generally an order of magnitude smaller. All three spin components have Lorentzian distributions at equilibrium. It is found that aggregates readily form even for conservative size ranges, and the development of gravitational wakes reported by Salo is con rmed. It is proposed that the density variations seen in the models presented, which are the most realistic to date, may account in part for observed non-uniformities in Saturn's outer rings.