Rings -- from Eric Weisstein's World of Physics

Ring structures are determined by orbital mechanics, collisions among particles (dissipation), sheparding and resonances by moonlets, self-gravitational effects (precession, wave propagation), and drag (Poynting-Robertson, extended atmosphere).

The resonant frequencies caused by satellites are split into horizontal (Lindblad) and vertical resonances. Let m and l be integers, and let the frequencies of the radial and out-of-phase motions be and . Subscripts s and r refer to the satellite and ring particles.

Horizontal (Lindblad) resonances satisfy

and vertical resonances satisfy

Density wave may occur in the plane, producing spokes, as were observed in Saturn's rings. Bending waves involve out-of-plane motions.

To first order in e, the width of a ring is given by

so the inner and outer edges of the ring have

References