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Jun 8, 2016 - 1 Astronomy Research Unit, University of Oulu, Finland. 2 Department of Physics, University of Central Flo
The Astrophysical Journal, 824:33 (5pp), 2016 June 10

doi:10.3847/0004-637X/824/1/33

© 2016. The American Astronomical Society. All rights reserved.

ON THE LINEAR DAMPING RELATION FOR DENSITY WAVES IN SATURN’S RINGS Jürgen Schmidt1, Joshua E. Colwell2, Marius Lehmann1, Essam A. Marouf3, Heikki Salo1, Frank Spahn4, and Matthew S. Tiscareno5 1 Astronomy Research Unit, University of Oulu, Finland Department of Physics, University of Central Florida, Orlando, FL 32816-2385, USA 3 San Jose State University, One Washington Square, San Jose, CA 95192, USA 4 Institut für Physik und Astronomie, Universität Potsdam, Karl-Liebknecht-Str. 24/25, D-14476 Potsdam-Golm, Germany 5 Carl Sagan Center, SETI Institute, 189 Bernardo Avenue, Mountain View CA 94043, USA Received 2016 February 23; accepted 2016 April 6; published 2016 June 8 2

ABSTRACT We revisit the equation for viscous damping of density waves derived from linearized theory and show that the damping is not only determined by the magnitudes of shear and bulk viscosity. Modifications arise from the dependence of the viscosity on the ring’s surface mass density. This was noted more than 30 years ago by Goldreich & Tremaine (1978b). Still, to date the consequences have not been explored. In the literature these terms have been neglected throughout when fitting the rings’ viscosity from observations of wave damping. Therefore, one must suspect that these viscosities, as well as the dispersion velocities inferred from them, suffer from systematic bias, which might be small or significant, depending on the local conditions in the ring. We show that the modified damping formula, to linear order, is related to the stability threshold for viscous overstability and argue that the appearance of density waves may be altered by this instability. Key words: planets and satellites: rings 1. INTRODUCTION

behavior of density waves deviates from the one predicted by the classical linear theory (GT78).

Numerous resonances with external satellites excite density waves in Saturn’s rings (Lissauer & Cuzzi 1982; Esposito et al. 1987). A theoretical relation for the amplitude damping of these waves, when they propagate away from the resonance location, was derived from a linearized fluid model (Goldreich & Tremaine 1978b, hereafter GT78; Shu 1984, pp. 513–561); for nonlinear models see Borderies et al. (1985, 1986) and Shu et al. (1985a, 1985b). From this expression the magnitude of the shear viscosity of Saturn’s rings, which is believed to be the main agent of damping, was inferred by comparison to the observed damping length in various data sets (e.g., Esposito et al. 1983; Tiscareno et al. 2007). However, in the full formula for the damping length the fluid’s bulk viscosity should enter (Shu 1984, pp. 513–561), as well as the derivative of shear viscosity with respect to the surface mass density of the ring. In practical work the bulk viscosity and the effect of the density dependence have thus far been neglected, even though GT78 pointed out that this neglect lacks adequate justification. We find that the full expression for viscous damping, in the framework of the linear fluid model, is closely related to the stability criterion for viscous overstability. This is an oscillatory instability leading to a spontaneous development of axisymmetric wavetrains in the rings, from an interplay of inertial forces and viscous stress (Schmit & Tscharnuter 1995; Spahn et al. 2000; Salo et al. 2001; Schmidt et al. 2001; Schmidt & Salo 2003; Latter & Ogilvie 2008). From theory and simulations the typical wavelengths of viscous overstability should be on the range of 100 m. Such microstructure, with wavelengths 150–200 m, was indeed observed in Saturn’s rings by Cassini instruments (Colwell et al. 2007; Thomson et al. 2007; Hedman et al. 2014). But if the damping of density waves and the criterion for the appearance of spontaneous overstability are related, then the very detection of overstability bears the implication that in certain ring regions the damping

2. THE LINEAR DISPERSION RELATION Let the ring’s surface mass density be decomposed into a constant background σ and a perturbation Δσ in the form of a tightly wound density wave in the WKB approximation

{ ò dr k (r)},

Ds µ exp i

(1 )

where k ? r−1. Then the linear fluid model (GT78) gives for the complex wavenumber (see Table 1 for notation) k»

9 ¶ log h 3x ⎤ x 2 x k 2 7 ⎡ 1 +i h + ⎥ . ⎢ 2pGs 7 ¶ log s 7h ⎦ s (2pGs )3 3 ⎣ (2 )

The real part of k describes the dispersion relation of the density wave, with the typical decrease of wavelength as the wave propagates away from the resonance location. It can be used to infer the ring’s surface mass density from observations (Esposito et al. 1983; Tiscareno et al. 2007). The imaginary part represents the viscous damping. The coefficient η is the dynamic shear viscosity and ξ is the dynamic bulk viscosity. Both depend, in principle, on the surface mass density of the ring. Equation (2) results from linearization of a fluid model of the ring flow. Here, η and ξ are constants, with values depending on the constant background surface mass density σ. The functional dependence of the viscosities on variations of σ in the wave is expressed in terms of the derivative ¶ log h ¶ log s . This derivative is then treated as a constant, depending on the background density σ. Often the kinematic viscosity ν = η/σ is used in ring dynamics. Usually, viscosity is derived from fitting the damping part of Equation (1) to the observed damping of a density wave, but neglecting the bulk 1

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Table 1 Nomenclature σ Δσ k τ κ r rL m Ω Ωp x = (r - r L ) r L D = k2 - m2 (W - Wp )2  = (r dD dr )∣r = r L G η ν = η/σ ξ

Unperturbed surface mass density Perturbation of surface mass density Complex wavenumber Optical depth Epicyclic frequency Radial location in the ring Location of a Lindblad resonance Azimuthal wavenumber Mean motion Pattern speed Distance from resonance location Distance in frequency space Derivative of D Gravitational constant Dynamic shear viscosity Kinematic shear viscosity Dynamic bulk viscosity

Figure 1. The factor F from Equation (3) appearing in the linear damping relation. Numbers are derived from N-body simulations of a dense planetary ring (Salo et al. 2001), where effects of self-gravity were approximated by an enhanced vertical oscillation frequency, applicable to systems where selfgravity does not yet lead to a significant strength of gravity wakes. This approximation should, roughly, be applicable to the locations of the Janus 2:1 and 3:2 resonances in Saturn’s B ring, excluding regions of strong self-gravity (see the text). Importantly, for plausible values of the ratio ξ/η the factor F should cross zero at a certain value of optical depth, lying in the range from 1–3, roughly.

viscosity and the density dependence of the shear viscosity. As a result, the thus measured kinematic shear viscosity nˆ differs from the actual viscosity ν by a factor F, i.e., nˆ = n F ,

where F º 1 -

9 ¶ log h 3x + . 7 ¶ log s 7h

simulations of dense rings, but not yet dominated by selfgravity, Salo et al. (2001) inferred ¶ log h ¶ log s in the range of 1.5–2 for optical depths between 0.5 and 2. Strong selfgravity, however, leads to the formation of gravitational wakes in the rings (Salo 1992, 1995). In numerical simulations it was shown that in this case the torques exerted by the wakes should dominate the ring viscosity (Daisaka et al. 2001) with a dependence as n µ G 2s 2 W3 (Ward & Cameron 1978), so that one would expect ¶ log h ¶ log s = 3 (see also the discussion in Section 14.1.3 of Schmidt et al. 2009, pp. 413–458). But it is also known that fully developed wakes tend to suppress overstability, a behavior not captured in the simple hydrodynamic picture (Salo et al. 2001). Thus, for ring regions with strongly developed self-gravity wakes, Equations (2) and (3) should not be expected to represent an accurate damping relation, at least not when using ¶ log h ¶ log s = 3, as one might naively expect from the known dependence of gravitational ring viscosity on the surface mass density. In Figure 1, estimates for the factor F from Equation (3) are shown versus optical depth. This diagram must be taken with a grain of salt, given the approximations applied to derive formula (2) and the uncertainties in the parameters entering (3). We use values for ¶ log h ¶ log s from Salo et al. (2001, Figure 15), derived from simulations that approximate the action of self-gravity by an enhancement of the vertical frequency of oscillations Ωz/Ω = 2 (Wisdom & Tremaine 1988) and a Bridges et al. (1984) elasticity law, bearing in mind that the result will not be applicable where the rings are in a state with strong self-gravity wakes. We find that there is a critical optical depth, depending on the precise value of ξ/η, separating between regimes where the density waves should damp ( F > 0, low τ) or grow ( F < 0, high τ). The damping length inferred from Equation (2) is proportional to (nF )-1 3. Thus, if F ≈ 0 the damping length can be very large, regardless of the value of the shear viscosity ν. However, because of the strong cubic root dependence the effect of nonzero, positive F will in many cases be hard to distinguish from the case F = 1.

(3 )

What value for F do we have to expect in Saturn’s rings? First, we note that F < 0 is identical to the condition for viscous overstability (Schmit & Tscharnuter 1995), in which case linear waves should not damp but amplify. From the detection of overstable waves we might indeed expect F to be negative in some parts of the rings and in others not. In hydrodynamics bulk viscosity ξ accounts for the irreversible transfer of kinetic energy of the mean flow to the random motion of particles, or some other degree of freedom, when the flow is compressed. However, for the rings the concept is less clear. Salo et al. (2001) derived values for the bulk viscosity of a planetary ring from non-self-gravitating N-body simulations, which led to a quantitative match of the growth rates of viscous overstability determined from N-body simulations and a fluid model (Schmidt et al. 2001). It was noted later (see the related discussion in Schmidt et al. 2009, pp. 413–458) that out-ofphase oscillations of individual components of the stress tensor might bias the derivation of ξ, especially at low optical depth, a behavior that is not accounted for by the hydrodynamic model. Nevertheless, it is possible to renormalize ξ/η (Schmidt & Salo 2003), in such a manner that an isothermal linear theory can effectively include (stabilizing) thermal effects. (Use Equation (24) of Schmidt et al. 2001 and the tables given in Salo et al. 2001 to evaluate the factors F2 and F3 of that equation.) That said, effective values for ξ/η in non-selfgravitating and dense rings might be in the range from 2–5 (Salo et al. 2001; Schmidt & Salo 2003). The density dependence of the shear viscosity appears here in the form of the slope ¶ log h ¶ log s . An expression for η (σ), valid for dilute and non-self-gravitating rings, was derived by Goldreich & Tremaine (1978a), but the importance of nonlocal (finite particle size) effects in dense systems was pointed out later (Hameen-Anttila 1975; Shukhman 1984; Araki & Tremaine 1986; Araki 1988; Wisdom & Tremaine 1988). From 2

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Figure 2. (a) Profile of slant ring optical depth inferred by Cassini UVIS from an occultation of the star β-Centauri in revolution 77 of the Cassini mission. Shown is the B ring region near the 2:1 inner Lindblad resonance of Janus. The first order Janus resonances are among the strongest resonances in the rings. A remarkably long density wave propagates outward from the 2:1 resonance radius. The wave shows a characteristic decrease in wavelength as it propagates away from resonance. (b) No clear wavetrain is visible at the 3:2 resonance in the outer B ring, neither in the UVIS data (upper panel, same occultation as in (a)), nor in the slant optical depth derived from an RSS radio occultation (middle panel), nor in the ring reflectance derived from lit-side images taken by ISS. Pronounced but shorter wavetrains emerge from the Janus 4:3 and 5:4 resonances (panels (c) and (d)) in the A ring.

resonance locations (Tiscareno et al. 2006), when Janus swaps the semimajor axis in the interplay with its co-orbital moon Epimetheus. It is the longest one among all Janus waves, traveling at least 500 km outward from the resonance. There is no clear wave signature in the data at the Janus 3:2 resonance in the outer B ring. The Janus 4:3 and 5:4 waves in the A ring again appear prominently, although they clearly damp more quickly than the 2:1 wave. Given the strength of the observed waves, the nonappearance of the 3:2 wave appears puzzling, even if we take

3. JANUS DENSITY WAVES IN CASSINI DATA In Figure 2 we show ring profiles inferred from Cassini data, displaying the regions near the 2:1, 3:2, 4:3, and 5:4 inner Lindblad resonances of the co-orbital satellites Janus and Epimetheus. The torques associated with the Janus resonances are among the largest within Saturn’s main rings and they are systematically increasing radially outward. The innermost wavetrain (Janus 2:1) shows a pattern emerging from the superposition of Janus waves launched at periodically changing 3

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into account that the ring segments in which these resonances lie differ in surface mass density, distance to the planet, and possibly other relevant parameters, like particle size distribution, elasticity, or proximity to a perturbing ring edge. This is not an issue of insufficient resolution or low signal-to-noise. The upper panel of Figure (2(b)) shows the optical depth inferred from a high resolution occultation obtained by the Cassini Ultra Violet Imaging Spectrograph (UVIS). Large fluctuations in optical depth appear around the resonance location. The peak optical depth is large enough that the star light of this occultations is entirely blocked at about 30% of the data points (responsible for the cut-off at τ ≈ 8). There is also no wavepattern recognizable in the optical depth inferred from an occultation obtained by the Cassini Radio Science Subsystem (RSS) and in reflectance profiles from the lit side of the rings, obtained from images taken by the Cassini Imaging Sub System (ISS) (Figure 2(b)). A very weak wave signal might be obtained, however, from the combined analysis of multiple occultations obtained by the Visual and Infrared Mapping Spectrometer (Hedman & Nicholson 2016).

The remarkable length of the Janus 2:1 wave could arise as a consequence of the ring being in a state that corresponds to marginal stability F ≈ 0 in the model. This means that either the wave damps very slowly (case (ii)) in accordance with the linear damping relation, or the amplitude saturates in the weakly nonlinear regime (case (iii)). In the latter case the wave damping may be affected or even dominated by nonlinear effects, as the wave propagates away from resonance with ever decreasing wavelength ∝ x−1. Another example for case (ii) may be the Pandora 5:4 wave in the inner A ring (Tiscareno et al. 2013). The Janus 4:3 wavetrain (Figure 2(c)) appears to be an intermediate case between the very long 2:1 wave and the much shorter 5:4 wavetrain and it might actually correspond to case (ii). The Janus 5:4 wave might correspond to the case (i), as would the Pandora and Prometheus resonances seen in the lower panels of Figure 2, and most of the waves in the A ring. Qualitatively, such behavior is expected, if F is decreasing with increasing τ (Figure 1). We close the paper with a few critical remarks. First, in addition to the forgotten terms in the damping relation reported here, we expect other physical processes to quantitatively influence the wave damping. Among these are dense packing in the wave crests, vertical splashing, and self-gravity. For instance, one should note that naturally the Toomre parameter changes periodically with the wave phase, which alters the wake state (see, e.g., the simulations by Lewis & Stewart 2005 for the response of gravitational wakes on the periodic forcing by a moon). This also implies that the one-to-one correspondence of the instability of density waves reported here with the condition for viscous overstability does not hold quantitatively. Our hope is that the effect helps to explain qualitatively the strong diversity of damping lengths, which has to be verified by improved future modeling. Ultimately, an N-body simulation of a whole density wavetrain might be best suited to settle open problems of density wave theory. Second, one might suspect that the highly perturbed outer B ring edge (located more than 1500 km radially outward from the resonance) still influences the region around the Janus 3:2 resonance, hampering the wavetrain forming, or generating a noisy structure that hides the wave in the data.

4. DISCUSSION We suggest that the instability discussed in Section (2) can influence or even dominate the damping behavior of density waves in Saturn’s rings. From the idealized mathematical model we may distinguish several classes of behavior, depending on the sign and magnitude of the factor F, which in turn depends on the background properties of the ring region under consideration (Figure 1): i. F > 0 and ∣F∣ is O(1): The wave is damping due to viscous effects and the modification arising from F ¹ 1 remains small. In particular for linear density waves, the traditional damping formula should be applicable in this case. ii. F  0: The wave is marginally stable and the damping length is increased significantly by the instability. A fit to the traditional linear damping formula will underestimate the ring viscosity. iii. F  0: The wave is mildly unstable. The exponential growth of amplitudes predicted by Equations (1) and (2) should, however, be saturated by nonlinearities (Cross & Hohenberg 1993) so that the wave eventually damps nonlinearly. iv. F < 0 and ∣F∣ is O(1): Here the growth rates are large and perturbations should grow rapidly to the strongly nonlinear regime. It is not clear what the steady response of the ring on the resonant forcing by the satellite would look like in this case.

We acknowledge funding by the Academy of Finland, the University of Oulu Graduate School, and Deutsches Luft und Raumfahrtzentrum. We thank Matt Hedman for useful discussions. We also would like to acknowledge the work of the Cassini science and engineering teams that allowed the data used in this paper to be obtained. REFERENCES Araki, S. 1988, Icar, 76, 182 Araki, S., & Tremaine, S. 1986, Icar, 65, 83 Borderies, N., Goldreich, P., & Tremaine, S. 1985, Icar, 63, 406 Borderies, N., Goldreich, P., & Tremaine, S. 1986, Icar, 68, 522 Bridges, F., Hatzes, A., & Lin, D. 1984, Natur, 309, 333 Colwell, J. E., Esposito, L. W., Sremčević, M., Stewart, G. R., & McClintock, W. E. 2007, Icar, 190, 127 Cross, M., & Hohenberg, P. 1993, RvMP, 65, 851 Daisaka, H., Tanaka, H., & Ida, S. 2001, Icar, 154, 296 Esposito, L. W., Harris, C. C., & Simmons, K. E. 1987, ApJS, 63, 749 Esposito, L. W., Ocallaghan, M., & West, R. A. 1983, Icar, 56, 439 Goldreich, P., & Tremaine, S. 1978a, Icar, 34, 227 Goldreich, P., & Tremaine, S. 1978b, Icar, 34, 240 (GT78) Hameen-Anttila, K. A. 1975, Ap&SS, 37, 309 Hedman, M. M., & Nicholson, P. D. 2016, Icar, in press (arXiv:1601.07955)

One might speculate whether in the last case (iv) the rings react in a turbulent manner and if such behavior may be responsible for the absence of a coherent Janus 3:2 wave. The input of energy and angular momentum by the resonance might then lead to the irregular ring profiles seen in the data. A rudimental wave might still form (Hedman & Nicholson 2016), and the observed change in spectral properties of the ring particles in the region of this resonance (Hedman et al. 2013) may be due to the enhanced collision rates induced by the resonance. Moreover, the ring might behave similarly at the nearby Pandora and Prometheus 3:2 resonances (Thiessenhusen et al. 1995; Hedman et al. 2013). 4

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Hedman, M. M., Nicholson, P. D., Cuzzi, J. N., et al. 2013, Icar, 223, 105 Hedman, M. M., Nicholson, P. D., & Salo, H. 2014, AJ, 148, 15 Latter, H. N., & Ogilvie, G. I. 2008, Icar, 195, 725 Lewis, M. C., & Stewart, G. R. 2005, Icar, 178, 124 Lissauer, J. J., & Cuzzi, J. N. 1982, AJ, 87, 1051 Salo, H. 1992, Natur, 359, 619 Salo, H. 1995, Icar, 117, 287 Salo, H., Schmidt, J., & Spahn, F. 2001, Icar, 153, 295 Schmidt, J., Ohtsuki, K., Rappaport, N., Salo, H., & Spahn, F. 2009, in Dynamics of Saturn’s Dense Rings, in Saturn from Cassini-Huygens, ed. M. K. Dougherty, L. W. Esposito, & S. M. Krimigis (New York: Springer Science+Business Media), 413 Schmidt, J., & Salo, H. 2003, PhRvL, 90, 061102 Schmidt, J., Salo, H., Spahn, F., & Petzschmann, O. 2001, Icar, 153, 316 Schmit, U., & Tscharnuter, W. 1995, Icar, 115, 304 Shu, F., Yuan, C., & Lissauer, J. 1985a, ApJ, 291, 356 Shu, F. H. 1984, in Planetary Rings, ed. R. Greenberg, & A. Brahic (Tucson, AZ: Univ. Arizona Press), 513

Shu, F. H., Dones, L., Lissauer, J. J., Yuan, C., & Cuzzi, J. N. 1985b, ApJ, 299, 542 Shukhman, I. 1984, SvA, 28, 574 Spahn, F., Schmidt, J., Petzschmann, O., & Salo, H. 2000, Icar, 145, 657 Thiessenhusen, K.-U., Esposito, L. W., Kurths, J., & Spahn, F. 1995, Icar, 113, 206 Thomson, F. S., Marouf, E. A., Tyler, G. L., French, R. G., & Rappoport, N. J. 2007, GeoRL, 34, 24203 Tiscareno, M. S., Burns, J. A., Nicholson, P. D., Hedman, M. M., & Porco, C. C. 2007, Icar, 189, 14 Tiscareno, M. S., Hedman, M. M., Burns, J. A., Weiss, J. W., & Porco, C. C. 2013, Icar, 224, 201 Tiscareno, M. S., Nicholson, P. D., Burns, J. A., Hedman, M. M., & Porco, C. C. 2006, ApJL, 651, L65 Ward, W. R., & Cameron, A. G. W. 1978, Disc Evolution Within the Roche Limit, in Lunar and Planetary Institute Conference Abstracts, Lunar and Planetary Inst. Tech. Rep., Vol. 9, 1205 Wisdom, J., & Tremaine, S. 1988, AJ, 95, 925

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