Abstract
A self-consistent system of the Boltzmann kinetic equation and the Poisson equation is used to study the dynamical evolution of Saturn's main A, B and C rings composed of discrete mutually gravitating particles. The simplified case of relatively rare collisions between identical particles, when the collision frequency is smaller than (compared with) the orbital frequency, is examined. Equations describing the quasi-linear stage of Jeans instability of small-amplitude gravity perturbations in Saturn's rings are derived and solved analytically. Conditions under which the quasi-linearization of the Boltzmann equation can be used to describe a wave-particle interaction are calculated with reference to the excitation of Jeans-type perturbations. The theory, as applied to Saturn's rings, predicts for several features, such as numerous irregular Jeans-unstable wakes, with size and spacing between them of the order of 2πh ≤ 100 m, where h is the typical thickness of the system. The interaction of particles with these almost aperiodically growing gravity perturbations increases both the radial spread of the disc and random velocities of particles in a very short time-scale of only two to three disc orbital revolutions. The latter leads to an eventual stabilization of the system, unless some effective cooling mechanism exists, reducing the magnitude of the relative velocity of particles. It is suggested that inelastic (dissipative) interparticle impacts provide such a cooling mechanism, leading to the recurrent density waves activity. We predict that the high-resolution images from the forthcoming (2004) Cassini spacecraft will reveal this fine-scale recurrent ∼100 m or even less spiral structure in low and moderately high optical depth regions (τ ≤ 1, where r is the normal optical depth) of Saturn's main rings.
Highlights
The rapidly and differentially rotating Saturnian ring disc of mutually gravitating particles is highly dynamic and is subject to various instabilities of gravity disturbances (e.g. Shu 1984)
In the previous subsection we have shown that an oscillation frequency that satisfies equation (36) and the condition ω2J < 0 can have a positive imaginary part, i.e. the gravity perturbations described by this dispersion equation can grow in time
We propose that the classical Jeans instabilities of small-amplitude gravity perturbations effectively generate fine-scale structure in low and moderately high optical depth regions, τ 1, of the system under study
Summary
The rapidly and differentially rotating Saturnian ring disc of mutually gravitating particles is highly dynamic and is subject to various instabilities of gravity disturbances (e.g. Shu 1984). This is because the evolution of the system is primarily driven by angular momentum redistribution (Goldreich & Tremaine 1982). Morozov, Torgashin & Fridman (1985), Willerding (1992), Griv et al (2000a) improved Lynden-Bell and Pringle’s calculation by taking into account the effect of non-uniform rotation in a two-dimensional self-gravitating system.2 It seems that N-body simulations have indicated that the densest parts of the B ring with optical depth τ > 1 can exhibit spontaneous viscous overstability (Salo, Schmidt & Spahn 2001). Abbreviated results have been reported by Griv, Gedalin & Yuan (2003)
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