On the confinement of planetary arcs⋆

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Abstract
We adopt the conventional model for the azimuthal confinement of incomplete planetary rings or arcs to be due to trapping of particles around potential equilibria located at the corotation resonances of hypothetical satellites. Although particle orbits around these equilibria are dynamically stable, they are secularly unstable because the gravitational potential attains local maxima at the equilibria. Energy dissipation due to particle collisions induces a general tendency towards spreading in the radial and azimuthal directions. We show that arc confinement is possible if particles near the corotation resonance are shepherded by a discrete Lindblad resonance of another distant satellite in the following manner: (i) the inner (outer) Lindblad resonance is exterior (interior) to the corotation resonance; (ii) the distance separating the corotation and the Lindblad resonances is comparable to or smaller than the total width of the corotation resonance; (iii) the strength of the Lindblad resonance is sufficiently large to compensate the effect of viscous dissipation of energy. We verify our model with numerical simulations. We discuss the possibility that the non-axisymmetric potentials, required for arc confinement, may be generated by modes of planetary non-radial oscillation as well as by satellites. For relatively low-m modes, planetary oscillations with amplitudes less than 25 m may be sufficient to confine arcs with width of the order of a few km.

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