The effect of gravity between particles on the shape of eccentric planetary rings

Abstract

Planetary rings are a very complex system. They are a mixture of different-sized colliding particles. However, in spite of the very small mass of particles, gravitational forces between particles always take place and cannot be neglected. In some cases, effect of mutual ring particle interactions may be significant. On the other hand, how can gravitation be taken into account? It is natural, to consider motion of test particles in the gravitational field of homogenous ring. The perturbation force in this case is calculated as an integral. In the limit, this integral can be substituted by a sum with very large N. At the symmetric circular ring we can apply central configuration (regular N-agon) model. As it is well known, some of planetary rings are elliptical. It is interesting to generalize this model to a case ring with small eccentricity (quasi-central configurations). The stationary distribution of particles along elliptic ring is obtained. The gravitational interaction between particles in rings can change the shape of non-circular rings, so it differs from a Keplerian ellipse. The model of quasi-central configurations gives a way of relating this effect to ring particles properties (mass, size, etc.).