Aims. We examine the linear stability of the Saturnian ringdisk of mutually gravitating and physically colliding particles withspecial emphasis on its fine-scale mdensity wave structure, thatis, almost regularly spaced, alignedcylindric density enhancements and optically-thin zones with the widthand the spacing between them of roughly several tens particlediameters. Methods. We analyze the Jeans' instabilities of gravityperturbations (e.g., those produced by a spontaneous disturbance)analytically by using the Navier-Stokes dynamical equations of acompressible fluid. The theory is not restricted by any assumptionsabout the thickness of the system. We consider a simple model of thesystem consisting of a three-dimensional ring disk that is weaklyinhomogeneous and whose structure is analyzed by making a horizontallylocal short-wave approximation. Results. We demonstrate that the disk is probably unstable andthat gravity perturbations grow effectively within a few orbitalperiods. We find that self-gravitation plays a key role in theformation of the fine structure. The predictions of the theory arecompared with observations of Saturn's rings by the Cassinispacecraft and are found to be in good agreement. Inparticular,itappears very likely that some of the quasi-periodicmicrostructures observed in Saturn'sA and Brings-both axisymmetric and nonaxisymmetric ones- are manifestations ofthese effects. We argue that the quasi-periodic density enhancementsrevealed in Cassini data are flattened structures, with aheight to width ratio of about0.3. One should analyzehigh-resolution of the order of 10m data acquired for theAand Brings (and probablyC ring aswell) to confirmthis prediction. We also show that the gravitational instability is apotential cluster-forming mechanism leading to the formation of porous100-m-diameter moonlets of preferred mass g each embedded in the outer Aring, although this has yet to be directly measured. Key words: planets and satellites: general - planets andsatellites: rings - planets and satellites: individual: Saturn -instabilities - hydrodynamics
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