Hydrodynamic Description of Planetary Rings

Abstract

We demonstrate the suitability of a hydrodynamic description of the dynamics of a dense planetary ring considering as an example the viscous oscillatory instability. For the successful application of hydrodynamics the transport coefficients must be known as functions of surface mass density σ and the granular temperature T of the ring material. We arrange scaling laws for these quantities and use parameters determined from particle simulations of a planetary ring. With such a preparation, the theory predicts an oscillatory instability of a dense ring: waves of radial wavelengths λ ≈ 100m...200m evolve, driven by the combined effects of viscous momentum flux and Coriolis force. These predictions are confirmed by direct particle simulations: wavelengths of the unstable modes, their growth rates, as well as the properties of the eigenfunctions of the perturbations. This success bears the hope to explain a class of so-called fine-structure in Saturn's rings, which are poorly understood to date. The experiments of the Cassini-spacecraft in orbit around Saturn offer the unique chance to identify the predicted density features by analyzing the imaging and occultation data.

Dense planetary rings and the viscous overstability

“…We present three interesting cases. The first, corresponding to v c /v b = 20, represents a 'bimodal' profile; such cases have been discussed by Hämeen-Anttila (1982) and Spahn and Schmidt (2006), and were first revealed in simulations by Salo (1991). For low to middle τ the disk is in the dilute regime; there the angular momentum flux increases sharply and then gently decreases after reaching a turning point.…”

The viscous overstability, nonlinear wavetrains, and finescale structure in dense planetary rings

“…Dσ + σ∂ x u = 0, (49) where D = ∂ t + u∂ x is the Lagrangian derivative, T x and T y are the radial and azimuthal accelerations due to the weak collective effects of pressure, viscosity, and self-gravity, and ǫ is an ordering parameter. Note that the notation here is different to the main body of the paper: u denotes the radial velocity and v the azimuthal velocity.…”

“…Our starting point is the viscous overstability, which is now regarded as a key player in the short scale radial dynamics of Saturn's rings (Schmit and Tscharnuter 1995, Spahn and Schmidt 2006, Latter and Ogilvie 2008. The viscous overstability is an axisymmetric oscillatory instability that afflicts the homogeneous state of Keplerian shear.…”

Coefficient of restitution for wet particles

“…Due to its omnipresence in nature and various industries, granular matter has drawn great attention from both physical and engineering communities in the past decades [9]. Concerning the modeling of granular matter, an appropriate collision model is essential for the successful implementation of kinetic or hydrodynamic theories to granular matter [10][11][12][13], see for example the dynamics of Saturn's rings [14], or the pattern formation under vertical agitation [15]. Despite those successful examples for dry granular matter, a continuum description for wet granular matter, which considers the cohesion arising from the wetting liquid phase, is still far from established [16,17].…”