An Inelastic Collision Model for the Evolution of “Planetary Rings”

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Abstract
We show that ringlets of a planet can be formed by inelastic collisions. If the collision law is v ′ =- e v ( v ′ and v are relative velocities after and before the collision), then the configuration of the particles evolves from a 3-dimensional torus to a 2-dimensional disc and finally to 1-dimensional sharp ringlets. The theory is also demonstrated and confirmed by numerical experiments. Velocity vectors of the particles tend equalize at collisions. This causes the angular momenta to equalize, and the orbits draw nearer. The above statement is rigorous in our special model, and applies qualitatively in the general inelastic collision model. It seems apparent that collisions diffuse a spatial pattern. But, on the contrary, ringlets are sharpened by inelastic collisions.

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