Density waves and the viscous overstability in Saturn’s rings

Abstract

This paper addresses resonantly forced spiral density waves in a dense planetary ring which is close to the threshold for viscous overstability. We solve numerically the hydrodynamical equations for a dense thin disk in the vicinity of an inner Lindblad resonance with a perturbing satellite. Our numerical scheme is one-dimensional so that the spiral shape of a density wave is taken into account through a suitable approximation of the advective terms arising from the fluid orbital motion. This paper is a first attempt to model the co-existence of resonantly forced density waves and short-scale free overstable wavetrains as observed in Saturn's rings, by conducting large-scale hydrodynamical integrations. These integrations reveal that the two wave types undergo complex interactions, not taken into account in existing models for the damping of density waves. In particular it is found that, depending on the relative magnitude of both wave types, the presence of viscous overstability can lead to a damping of an unstable density wave and vice versa. The damping of the short-scale viscous overstability by a density wave is investigated further by employing a simplified model of an axisymmetric ring perturbed by a nearby Lindblad resonance. A linear hydrodynamic stability analysis as well as local N-body simulations of this model system are performed and support the results of our large-scale hydrodynamical integrations.

Essentially non-oscillatory and weighted essentially non-oscillatory schemes

“…Euler equations with gravity were computed by high-order well-balanced schemes in Klingenberg, Puppo and Semplice (2019). Density waves and the viscous overstability in Saturn's rings were studied in Lehmann, Schmidt and Salo (2019). A Simflowny-based high-performance 3D code for the generalized induction equation was developed in Vigano et al (2019).…”

Unusual one-armed density waves in the Cassini Division of Saturn's rings

“…More detailed dynamical investigations, numerical simulations, and additional observational constraints on the detailed orbital properties of the nearby ringlets from Cassini occultation measurements late in the orbital tour are all likely to clarify our understanding of these striking features. The recent hydrodynamical simulations by Lehmann et al (2019) of wave propagation through inhomogeneous ring structure represent a significant step in this direction.…”

“…We are grateful to Robert French and Joe Spitale for providing detailed examples of ISS image navigation that guided the development of our own algorithms for image steering, and to Mitch Gordon for assistance with the PDS Ring-Moon Systems Node's OPUS search service. We are grateful to two anonymous reviewers for their helpful suggestions, and for alerting us to the recent dynamical simulations of Lehmann et al (2019). This work was supported in part by the Cassini project and by NASA CDAP grants NNX09SE66G and 80NSSC10K0890.…”

The exterior gravitational potential of toroids

“…This observation may e.g. be of importance in oscillation modes in planetary or other rings (Wisdom & Tremaine 1988;Lehmann et al 2019). It also means that, in a geometrically thin discs where Paczynski's approximation is valid, half of the vertical acceleration comes from the local contribution of matter while the other half comes from the global or long-range distribution of matter (Trova et al 2014).…”

“…The derivation of reliable and compact expressions for the gravitational potential of massive toroids is a longstanding problem of dynamical astronomy, from planetary to galactic scales. This is essential not only to examine the motion of test-particles and fluids orbiting around, in the classical framework as well as in general relativity (Nieto 2005;Šubr & Karas 2005;Semerák & Suková 2010;Tresaco et al 2011;Iorio 2012), but also to understand the conditions for the formation, evolution and stability of toroids themselves (Dyson 1893;Hachisu 1986;Chandrasekhar 1987;Tohline & Hachisu 1990;Woodward et al 1992;Storzer 1993;Christodoulou 1993;Hashimoto et al 1993;Nishida & Eriguchi 1994;Pickett et al 1997;Horedt 2004;Lehmann et al 2019). While it is relatively easy to deduce the mass density corresponding to a given potential (e.g.…”