Sharp Gap Edges in Dense Planetary Rings: An Axisymmetric Diffusion Model

Abstract

One of the most intriguing facets of Saturn's rings are the sharp edges of gaps in the rings where the surface density abruptly drops to zero. This is despite of the fact that the range over which a moon transfers angular momentum onto the ring material is much larger. Recent UVIS-scans of the edges of the Encke and Keeler gap show that this drop occurs over a range approximately equal to the rings' thickness. Borderies et al. (1982Borderies et al. ( , 1989 show that this striking feature is likely related to the local reversal of the usually outward directed viscous transport of angular momentum in strongly perturbed regions. In this article we revise the Borderies et al. (1989) model using a granular flow model to define the shear and bulk viscosities, ν and ζ, and incorporate the angular momentum flux reversal effect into the axisymmetric diffusion model we developed for gaps in dense planetary rings (Grätz et al. 2018). Finally, we apply our model to the Encke and Keeler division in order to estimate the shear and bulk viscosities in the vicinity of both gaps.

Accretion of Ornamental Equatorial Ridges on Pan, Atlas and Daphnis

“…If the lobes vary by a factor of 2 in their linear mass density (mass per unit distance along the equator) across 90 • in equatorial longitude, then the slope of equation 8 gives a distance d edge ∼ 0.3R H over which the surface density drops. This size seems reasonable compared to analytical models of gap profiles (Grätz et al, 2019).…”

“…However, the gaps might have been narrower at some time in the past, allowing accretion. Models taking into account spiral density wave driven torque and diffusion in the gap edge have predicted gap edge surface density profiles (Grätz et al, 2019). The gap edge could display small scale structures, such as the few hundred meter length wispy features near the Keeler gap edge (Porco et al, 2005;Tajeddine et al, 2017b).…”

“…The torque from the moon onto the ring is probably exerted onto the disk over a larger range in semi-major axis ∆a if the moon is on an eccentric orbit, than if the moon is on a circular orbit. Unfortunately, predictions of gap surface density profiles that balance the torque caused by the moon onto the ring against viscous diffusion and accretion usually assume the moon is in a circular orbit (Grätz et al, 2019). This is usually a good assumption because of the short eccentricity damping time (as we discussed in section 3).…”

Density waves and the viscous overstability in Saturn’s rings

“…That being said, hydrodynamic models have been found useful for at least a qualitative description of various structures in planetary rings, mainly those of Saturn. Prominent examples are the propagation of satellite induced spiral density waves (Goldreich & Tremaine 1978b,c, 1979Shu 1984;Shu et al 985a;Shu et al 985b;Borderies et al 1985Borderies et al , 1986Schmidt et al 2016;Lehmann et al 2016Lehmann et al , 2019, the formation of moonlet induced gaps and 'propeller' structures (Goldreich & Tremaine 1980;Showalter et al 1986;Borderies et al 1989;Spahn et al 1992;Hahn et al 2009;Hoffmann et al 2015;Spahn et al 2018;Grätz et al 2019;Seiß et al 2019;Seiler et al 2019) or the small-scale axisymmetric viscous overstability (Schmit & Tscharnuter 1995Spahn et al 2000;Schmidt et al 2001;Salo et al 2001;Schmidt & Salo 2003;Latter & Ogilvie 2009, 2010Lehmann et al 2017Lehmann et al , 2019. It is the latter to which this study is devoted.…”

The radial density profile of Saturn's A ring

“…Grätz et al applied the model to the Encke and Keeler gap to estimate the shear viscosity ν of the ring, and to conclude that tiny icy satellites cannot be the cause for the numerous gaps observed in the C ring and the Cassini division. Grätz et al (2019) extended the diffusion model for circumferential gaps in dense planetary rings to account for the angular momentum flux reversal (Borderies et al 1982) in order to model the extremely sharp Encke and Keeler gap edges.…”