journal homepage: www.elsevier.com/locate/icarus N-body simulations of cohesion in dense planetary rings: A study

Dense planetary rings and the viscous overstability

This dissertation is primarily focused on exploring whether weak cohesion among icy particles in Saturn’s dense rings is consistent with observations—and if so, what limits can be placed on the strength of such cohesive bonds, and what dynamical or observable consequences might arise out of cohesive bonding. Here I present my numerical method that allows for N-body particle sticking within a local rotating frame (“patch”)—an approach capable of modeling hundreds of thou-sands or more colliding bodies. Impacting particles can stick to form non-deformable but breakable aggregates that obey equations of rigid body motion. I then apply the method to Saturn’s icy rings, for which laboratory experiments sug-gest that interpenetration of thin, frost-coated surface layers may lead to weak bonding if the bodies impact at low speeds—speeds that happen to be characteristic of the rings. This investigation is further motivated by observations of structure in the rings that could be formed through bottom-up aggregations of particles (i.e., “propellers ” in the A ring, and large-scale radial structure in the B ring). This work presents the implementation of the model, as well as results from a suite

Irregular structure in planetary rings is often attributed to the intrinsic instabilities of a homogeneous state undergoing Keplerian shear. Previously these have been analysed with simple hydrodynamic models. We instead employ a kinetic theory, in which we solve the linearised moment equations derived in Shu and Stewart 1985 for a dilute ring. This facilitates an examination of velocity anisotropy and non-Newtonian stress, and their effects on the viscous and viscous/gravitational instabilities thought to occur in Saturn’s rings. Because we adopt a dilute gas model, the applicability of our results to the actual dense rings of Saturn are significantly curtailled. Nevertheless this study is a necessary preliminary before an attack on the difficult problem of dense ring dynamics. We find the Shu and Stewart formalism admits analytic stability criteria for the viscous overstability, viscous instability, and thermal instability. These criteria are compared with those of a hydrodynamic model incorporating the effective viscosity and cooling function computed from the kinetic steady state. We find the two agree in the ‘hydrodynamic limit ’ (i.e. many collisions per orbit) but disagree when collisions are less frequent, when we expect the viscous stress to be increasingly non-Newtonian and the velocity distribution increasingly anisotropic. In particular, hydrodynamics predicts viscous overstability for a larger portion of parameter space. We also numerically solve the linearised equations of the more accurate Goldreich and Tremaine 1978 kinetic model and discover its linear stability to be qualitatively the same as that of Shu and Stewart’s. Thus the simple collision operator adopted in the latter would appear to be an adequate approximation for dilute rings, at least in the linear regime.

Saturn’s rings are among the most familiar, beautiful, and puzzling ob-jects in the Solar system, if not all of Space. Their complex, striated struc-ture, much like the grooves carved in a vinyl record, inspires equal degrees of aesthetic pleasure and theoretical agitation. My thesis takes this radial stratification as its theme, and examines the physical mechanisms which gen-erate and sustain it. Specifically, I explore structure formation on the finest scales we have observed, those of about 100 m, where recent images show abundant and irregular patterns. These are thought to be the product of a pulsational instability associated with the viscous properties of the system. In order to model the instability I attend to the subtle collective dy-namics of a ‘gas ’ of icy particles — dynamics that the usual tools of fluid mechanics neglect but which in this context are essential. This level of atten-tion can only be supplied by kinetic theoretical models, which have generally been thought too mathematically involved to deploy in detailed dynamical studies. My thesis, however, presents a kinetic formalism that is both acces-

The Cahn-Hilliard variational inequality is a non-standard parabolic vari-ational inequality of fourth order for which straightforward numerical ap-proaches cannot be applied. We propose a primal-dual active set method which can be interpreted as a semi-smooth Newton method as solution technique for the discretized Cahn-Hilliard variational inequality. A (semi-)implicit Euler discretization is used in time and a piecewise linear finite element discretiza-tion of splitting type is used in space leading to a discrete variational inequality of saddle point type in each time step. In each iteration of the primal-dual active set method a linearized system resulting from the discretization of two coupled elliptic equations which are defined on different sets has to be solved. We show local superlinear convergence of the primal-dual active set method and demonstrate its efficiency with several numerical simulations.

Abstract. Planetary rings are found around all four giant planets of our solar system. These colli-sional and highly flattened disks exhibit a whole wealth of physical processes involving dust grains up to meter-sized boulders. These processes, together with ring composition, can help understand better the formation and evolution of proto-satellite and proto-planetary disks in the early solar sys-tem. The present chapter reviews some fundamental aspects of ring dynamics and composition. The forthcoming exploration of the Saturn system by the Cassini mission will bring both high resolution and time-dependent information on Saturn’s rings.

eingereicht an der

variational inequality with a semi-smooth Newton method

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. c ○ 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim 1