On the stability of Saturn's rings: a quasi-linear kinetic theory

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 τ is the normal optical depth) of Saturn's main rings.

Fine-scale density wave structure of Saturn's A, B, and C rings

“…2 that the c r < c T case produces rigorous instabilities. One can clearly see that at times t 0.6 a dominant k y is equal about to (2L y /λ J ) tan ψ ≈ 4, which is consistent with the linear stability analysis for the marginally unstable mode ( Griv et al 2003b;Griv & Gedalin 2003). That is, a wavelength of the dominant mode in the streamwise direction λ y = λ crit / tan ψ ≈ 2λ J , where ψ = 20…”

“…In turn, the "hot" model with c r ≡ 2c ϕ = 2c T is expected to be at best only marginally unstable to the growth of spiral waves. See Griv et al (2000Griv et al ( , 2003aGriv et al ( , 2003b and Griv & Gedalin (2003) for an explanation. In all experiments, initially c z = 0.2c r .…”

“…The wakes are created if self-gravity is included; only collisions do not create the structure. Their pattern speed Ω p is zero, because the system is spatially homogeneous (Griv et al 2003b). The Lin-Shu type density wave structure (Lin & Shu 1966;Lin et al 1969;Shu 1970;Lin & Lau 1979;Griv & Gedalin 2003) is time dependent and transient.…”

“…The simulated wake structure rapidly at times t 0.6 reaches nonlinearity, and is thus beyond the scope of our theory (or any corresponding linear analysis of shearing modes). I then simulated a differentially rotating disk which is stable in accordance with the modified stability criterion (Griv et al 2000(Griv et al , 2003a(Griv et al , 2003bGriv & Gedalin 2003). Figure 4 shows the observed evolution of the dynamically hot model with c r = 2c T (or Q = 2, respectively).…”

“…Accordingly, a system of mutually gravitating particles of Saturn's rings exhibits collective, gravitationally unstable modes of motions. A quasi-linear kinetic theory of the almost aperiodic Jeans instability in Saturn's rings has been developed by Griv et al (2000Griv et al ( , 2003aGriv et al ( , 2003b and Griv & Gedalin (2003). The theory predicts that as a direct result of the instability of small-amplitude gravity disturbances the main parts of A, B, and C rings are divided into numerous irregular spiral ringlets of the order of 2π times the local thickness h = 5 − 30 m. Below I describe N -body simulations in order to verify the validities of the theory.…”