Tidal Excitation of Free Oscillations of Icy Satellites

Vedran Lekic and Michael Manga

Motivation

Small bodies, lacking both significant heat of accretion and possessing large surface to volume ratios, were thought to lack significant internal sources of heat, and therefore to be geologically dead. However, exploration of Jovian and Saturnian systems by the Voyager spacecraft revealed that the surfaces of a number of satellites were actively modified by volcanic and tectonic processes. Europa, Jupiter's third-largest satellite, is marked by many fractures, yet few craters, indicative of a geologically young, active surface. Subsequent study by the Galileo probe confirmed the existence of an ocean beneath its icy shell (Kivelson et al., 2000). Images of Enceladus, a smallish moon of Saturn, revealed a highly reflective surface marked by large tectonic features. Recently, the Cassini probe found anomalously high temperatures at ice fractures near Enceladus' south pole (Spencer et al., 2006), and a diffuse plume of water molecules reaching hundreds of kilometers above the surface (Hansen et al., 2006).

Tidal dissipation appears to be an important source of heat for the Galilean satellites, which are locked in orbital resonances which force orbital eccentricities. Eccentric orbits allow transfer of orbital/rotational energy to internal heat even for synchronously rotating satellites. The efficiency of this transfer is related to the anelasticity of the moon in question, which causes a phase lag between the tidal forcing and the resulting deformation, and is on the order of 10-4 (Hussman and Spohn, 2004). Tidal excitation of free oscillations may be a more efficient mechanism of tidal heating, but has heretofore been neglected, due to the assumption that frequencies of free oscillations are incompatible with those of tidal forcing.

Europa's subsurface ocean is a potential habitat for life (e.g. Marion et al., 2003). Orbital modeling that includes the effects of thermal dissipation indicates that heat dissipation within Europa has varied through time, causing the thickness of the liquid ocean to oscillate (e.g. Hussmann and Spohn, 2004). In order to ascertain the potential for life in the Europan ocean, it is important to identify possible tidal dissipation feedback mechanisms that would work to stabilize the ocean on geological timescales. We propose to study the effects of tidal excitation of free oscillations for a variety of ice-shell/ocean models of Europa, and to quantify the resulting heat deposition.

Widespread fracturing and heating of water ice on Enceladus cannot be explained by heat dissipation through tidal deformation (e.g. Poirier et al., 1983; Porco et al., 2006). Furthermore, tidal forcing on Enceladus is comparable to that experienced by other moons that, despite their larger size and radioactive heat budget, do not exhibit signs of geological activity. We propose to investigate the possibility of tidal excitation of free oscillations on Enceladus as a more efficient means of orbital-to-heat energy transfer.

Method

We adopt the approach of Press and Teukolsky (1977), in which the tidal potential is projected onto the normal modes, and the amount of tidal normal mode coupling is quantified in terms of an overlap integral

\begin{displaymath}
A_n(\omega) = \int_V \xi_n \cdot \nabla\tilde{U}(\omega) dv,
\end{displaymath} (36.1)

where the subscript n is the mode identifier, $\xi_n$ is the eigenfunction in question, $\nabla\tilde{U}(\omega)$ is the gradient of the potential in the frequency domain, $\omega$ is the relevant frequency of oscillation, and integration is carried out over the perturbed body. The energy deposited into the mode is then simply
\begin{displaymath}
\Delta E_n = 2 \pi^{2} \vert A_n(\omega)\vert^{2}.
\end{displaymath} (36.2)

We can see that tidal excitation of free oscillations requires that the tidal forces overlap with a given mode both spatially and in the frequency domain. Studies of tidal excitation of stellar oscillations (Wu and Murray, 2003) demonstrate that spatial overlap falls rapidly with increasing radial order of the mode under consideration.

Quantifying frequency overlap requires both calculating the frequency spectrum of tidal forcing and the frequencies of the body's free oscillations. For Europa, we consider the tidal effects of Jupiter, Io, and Ganymede, and plot the dominant frequencies of forcing from these bodies in Figure 1. We expect Saturn, Dione and Rhea to contribute significantly to tidal forces acting on Enceladus, and we plot the relevant dominant frequencies in Figure 2. Next, we proceed to calculate a range of possible frequencies for the $_0S_2$ mode of oscillation ("the football mode") by considering a set of 1D models of elastic velocity and density for Enceladus and Europa. In order to calculate the frequencies and functions describing the free oscillations of a spherically symmetric body, we employ the MINOS code (Woodhouse, 1988). At present, we neglect effects of viscosity.

Preliminary Results

Figure 36.1 shows that the frequency of $_0S_2$ on Europa decreases with decreasing thickness of its subsurface ocean. We use the physical model developed by Cammarano et al (2006) for a chondritic mantle topped by a 137 km thick ice/water layer. Note that tidal forces due to Io excite Europa's $_0S_2$ when the ocean is several kilometers thick, Ganymede's effect becomes relevant when the ocean is a few kilometers thick, and frequency overlap with the dynamical tide due to Jupiter occurs when the ocean is a mere 100 m thick.

Figure 36.1: The dashed line shows the frequency of $_0S_2$ as a function of ocean thickness. Frequencies and amplitudes (in arbitrary units) of tidal forcing due to Ganymede (triangles) and Io (circles) are also shown. The dominant frequency of Jupiter forcing is indicated by the vertical dotted line.
\begin{figure}\begin{center}
\epsfig{file=Lekic06_B_1.eps, width=8.5cm}\end{center}\end{figure}

In Figure 36.2, we show the frequency of $_0S_2$ on Enceladus as a function of the thickness of a hypothetical subsurface ocean. We consider a physical structure of Enceladus which fits the density and shape constraints (Porco et al., 2006). Our model consists of a core with a density of $\rho$ = 1700 kgm$^{-3}$ overlain by a ocean/ice layer that is 10 km thick. The low density of the rocky core is consistent with a 2:3 rock-to-water mixture. We use the upper bounds on seismic velocities in the mantle derived by computing the Voigt average of the rock and water moduli (Watt et al., 1976). We consider a fully differentiated Enceladus - with a non-porous chondritic interior and a $\sim $ 50 km thick ocean/ice layer - to be unlikely, given the paucity of accretionary and radiogenic heating, and inconsistency with the observed shape (Porco et al., 2006). Note that tidal forces due to Dione and Rhea excite $_0S_2$ when the thickness of the hypothetical subsurface ocean $\leq$ 1 km.

Figure 36.2: The dashed line shows the frequency of $_0S_2$ as a function of ocean thickness. Frequencies and amplitudes (in arbitrary units) of tidal forcing due to Rhea (triangles) and Dione (circles) are also shown. The dominant frequency of Saturn forcing is indicated by the vertical dotted line.
\begin{figure}\begin{center}
\epsfig{file=Lekic06_B_2.eps, width=8.5cm}\end{center}\end{figure}

Implications and Future Work

Having identified the interactions that may give rise to tidal excitation of free oscillations, we shall proceed to quantify the effect, and determine whether it is, or may have been, an important source of heat for Europa and Enceladus. Since $_0S_2$ frequencies for very thin ice shells are of the same order as the Maxwell relaxation time, we must modify our purely elastic treatment of the problem to include effects of viscosity.

Recent modeling of Ceres by McCord and Sotin (2005) suggests that it may have a subsurface ocean layer, which would result in very low frequency for $_0S_2$. We will explore the implications that tidal excitation of free oscillations has for Ceres and other bodies suspected of harboring subsurface oceans (e.g. Callisto, Ganymede, etc.). Furthermore, the transfer of angular momentum into heat through excitation of free oscillations - even if insufficient to drastically raise the temperature within the affected body - may significantly affect orbital evolution within the Jovian and Saturnian systems, or perhaps even the asteroid belt.

References

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Press, W.H. and S.A. Teukolsky. On formation of close binaries by two-body tidal capture. Astrophys. J., 213, 183-192, 1977.

Spencer, J.R., J.C. Pearl, M. Segura, et al., Cassini encounters Enceladus: Background and the discovery of a south polar hot spot. Science, 311, 1401-1405, 2006.

Watt, J.P., G.F. Davies and R.J. O'Connell, The elastic properties of composite materials. Rev. of Geophys. and Space Phys., 14:4, 541-563, 1976.

Woodhouse, J.H., The calculation of eigenfrequencies and eigenfunctions of the free oscillations of the Earth and the Sun. Seismological Algorithms, ed. Doornbos, D.J., 321-370, 1988.

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