Three-dimensional Dynamics of Narrow Planetary Rings

  title={Three-dimensional Dynamics of Narrow Planetary Rings},
  author={Eugene Chiang and Christopher J. Culter},
  journal={The Astrophysical Journal},
Narrow planetary rings are eccentric and inclined. Particles within a given ring must therefore share the same pericenter and node. We solve for the three-dimensional geometries and mass distributions that enable the Uranian α- and β-rings and the Saturnian Maxwell and Colombo (Titan) rings to maintain simultaneous apsidal and nodal lock. Ring self-gravity, interparticle collisions, and the quadrupole field of the host planet balance each other to achieve this equilibrium. We prove that such an… 
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