Journal of Geophysical Research: Planets

Interior structure models and tidal Love numbers of Titan

Authors


Abstract

[1] Interior models of a differentiated Titan with an internal ammonia-water ocean and chondritic radiogenic heat production in an undifferentiated rock + iron core have been calculated. We assume thermal and mechanical equilibrium and calculate the structure of the interior as a function of the thickness of an ice I layer on top of the ocean as well as the moment of inertia factor and the tidal Love numbers for comparison with Cassini gravity data. The Love numbers are linearly dependent on the thickness of the ice I shell at constant rheology parameters but decrease by one order of magnitude in the absence of an internal ocean. Ice shell thicknesses are between 90 and 105 km for models with 5 wt.% ammonia and for core densities between 3500 and 4500 kg m−3. For 15 wt.% ammonia, the shell is 65 to 70 km thick. We use a strongly temperature-dependent viscosity parameterization of convective heat transport and find that the stagnant lid comprises most of the ice I shell. Tidal heating in the warm convective sublayer is of minor importance. The internal ocean is several hundred kilometers thick, and its thickness decreases with increasing thickness of the ice shell. Core sizes vary from 1500 to 1800 km radius with associated moment of inertia factors of 0.30 ± 0.01.

1. Introduction

[2] Titan, the largest satellite of Saturn, is unique among the moons in the Solar System because of its substantial atmosphere and photochemically-produced, dark-orange stratospheric haze layer that precludes direct observations of the surface. Therefore constraints imposed by surface geology on the satellite's interior structure are not available at the present time. However, recent near-infrared data indicate that Titan's leading hemisphere contains a well-defined equatorial bright region about the size of Australia that is also bright at radar wavelengths consistent with icy surface material. The rest of the surface is characterized by dark terrain possibly representing hydrocarbon based solid or liquid deposits [Lorenz and Lunine, 1997].

[3] Titan is intermediate between the Jovian satellites Ganymede and Callisto with respect to its radius of 2575 km and mean density of 1881 kg m−3. The mean densities of Ganymede, Callisto, and Titan indicate that their interiors are composed of ice and silicates at nearly equal shares by mass [Lupo, 1982]. However, Titan's dense atmosphere that is dominated by molecular nitrogen with a substantial component of methane suggests that the satellite's interior is more enriched in volatile ices such as NH3·H2O and CH4·nH2O, which have densities similar to that of water ice [Hunten et al., 1984]. This is likely to be due to the lower temperature of the less massive proto-Saturnian nebula in comparison with the proto-Jovian nebula, which favors the incorporation of a larger amount of volatiles such as ammonia and methane at the distance of Titan (∼20.6 Saturn radii) without significantly affecting the satellite's mean density in comparison to Ganymede and Callisto. Furthermore, the absence of rocky inner satellites in the Saturnian system may indicate lower temperatures of the nebula during the evolution of the gaseous and dusty accretion disk from which the satellite system formed by mutual collisions of icy and rocky planetesimals [Coradini et al., 1995; Owen, 2000]. On the basis of the assumption that all nitrogen was originally supplied as NH3, volatile components could appear in roughly cosmic abundances in the interior of Titan with ratios of NH3/H2O ≃ 1/6 and NH3/CH4 ≃ 1. Corresponding water ice-ammonia mixtures then have an ammonia content of 14 wt.%, a composition more water-rich than the peritectic composition with ∼32 wt.% [Lewis, 1972; Hunten et al., 1984; Stevenson, 1992].

[4] The interior of Titan is likely to be differentiated at least into a rock + iron core and an icy mantle as a consequence of substantial accretional heating accompanied by partial outgassing and atmosphere formation [e.g., Hunten et al., 1984; Grasset et al., 2000]. Whether or not Titan's deep interior is further differentiated like Ganymede's into an iron core and a rock mantle above it [e.g., Sohl et al., 2002], is more speculative. Depending on the amount of volatiles incorporated into the icy mantle and depending on the satellite's thermal evolution, an ammonia-rich liquid water layer may be located between the near surface ice I layer and solid high pressure ice layers at greater depth [Grasset and Sotin, 1996; Grasset et al., 2000]. The ice layers crystallized upon early cooling and freezing of the pristine water-rich ammonia-water ocean that may have formed as a consequence of the blanketing effect of a hot proto-atmosphere, if accretion was completed within 104 to 105 years [Kuramoto and Matsui, 1994]. It has been suggested that the ice I layer on top of the ocean was initially cooling and thickening due to thermal conduction and became unstable to convection about 200 Ma after accretion [Grasset and Sotin, 1996]. Some volatiles will presumably be present in the form of hydrates and clathrates, thereby affecting the rheological and thermal properties of the ice layers [Hunten et al., 1984; Stevenson, 1992]. The presence of volatiles tends to reduce the freezing temperature of the ocean, thereby lowering the temperature of the ice I layer and increasing its viscosity. Larger viscosities due to temperatures far below the melting temperature then would retard convection and slow cooling of the ice I layer, so that the subsurface ocean may have survived to the present day because of the progressing enrichment of ammonia as compared to the primordial ocean [Grasset and Sotin, 1996; Deschamps and Sotin, 2001].

[5] Titan is subject to periodically varying tidal distortions resulting from its significant orbital eccentricity of 0.0292 in addition to its permanent rotational flattening. The quadrupole moments of its gravity field, J2 and C22, will be obtained from Doppler measurements using several flybys of the Cassini spacecraft and the moment of inertia (MoI) factor as a measure for the satellite's concentration of mass toward the center will be inferred. The measurements can be used to test the hypothesis of hydrostatic equilibrium in Titan [Rappaport et al., 1997]. The orbital eccentricity of Titan causes tidal variations of the quadrupole moments from which the second degree potential Love number k2 will be derived. k2 is related to the satellite's interior structure and rheological properties and will reflect the gravitational signature of a putative internal ocean. In order to obtain the value of k2 with high accuracy, four flybys of the Cassini spacecraft will be devoted to measuring Titan's gravity field near both apokrone and perikrone [Castillo et al., 2002]. k2 may be calculated from the gravity data derived from the dedicated flybys with precision radio tracking during which thruster operations are restricted to avoid perturbations. These will be T11, T22, T33 and T38 between February 2006 and December 2007. Some limited gravity information may be gleaned from other flybys.

[6] In the present study, we calculate thermal and mechanical equilibrium models of Titan's interior structure that are consistent with a given ice I shell thickness above an internal ocean. We assume that Titan's silicate portion is chondritic in composition and that the physical parameter values of the individual layers are known. These models may be useful in interpreting Cassini data that constrain the ice shell thickness. We motivate this approach by the fact that there have been a number of independent observations during the Galileo missions that constrained the ice shell thicknesses of the icy Galilean satellites of Jupiter [e.g., Anderson et al., 1996, 2001b]. We calculate the moment of inertia factors for these models for comparison with Cassini observations. The models satisfy the known mass and average density of Titan as summarized in Table 1. In addition, we calculate the second degree tidal Love numbers k2 and h2, the dimensionless and scaled tidal gravity and surface deformation signals, and show that the Love numbers are linearly related to the ice shell thickness. In the following section, we introduce our model of the satellite's interior and thermal structure. Results of this study are presented in section 3, and section 4 summarizes the discussion and conclusions of this study.

Table 1. Global Parameters of Titan
ParameterValueReferencesa
Mean surface radius Rp, km25751,2,3,4
Global mass Mp, kg1.346 × 10232,3,4
Planetocentric constant, GMp, km3 s−28978.21
Surface gravity gp, m s−21.353,4
Mean density equation image, kg m−318812,3,4
Rock to ice ratio (by mass)∼55:455
Orbital semimajor axis a, km1.2218 × 1062
Free orbital eccentricity e0.02922
Mean orbital motion n, s−14.56 × 10−6 
Period of revolution τrev, d15.9452,6
Period of rotation τrot, d15.9452,6
Surface pressure pS, Pa1.496 × 1053
Surface temperature Ts, K943,6

2. Model

[7] Our model of Titan's present interior structure is based on the assumption of thermal and mechanical equilibrium. We assume that the satellite is differentiated into a central rock + iron core which is surrounded by an icy mantle. The core density is calculated from the model but is kept within reasonable bounds for silicates and iron/silicate mixtures. We do not consider models with an iron-rich core underneath a rock layer since our present knowledge of Titan does not suggest that that much detail of the model is justified.

[8] The outermost layer of the icy mantle is assumed to be predominantly composed of pure H2O -ice I (density 917 kg m−3). The phase diagram of the ammonia-water system [see, e.g., Sotin et al., 1998; Leliwa-Kopystyński et al., 2002] implies that the crystallization of the outermost ice I layer was accompanied by the freezing of a high-pressure ice zone at the bottom of the pristine ocean. The density of the high-pressure ice layer that is likely made of ice VI at the prevailing pressure and temperature conditions [Stevenson, 1992] is taken as 1310 kg m−3 in the present study [Grasset and Sotin, 1996]. The density of the ammonia-water subsurface ocean depends only little on the actual ammonia concentration and is taken as 950 kg m−3 representative of the peritectic liquid [Croft et al., 1988].

[9] Using experimental data of the ammonia-water system, Grasset et al. [2000] have constructed pressure-dependent melting curves for various compositions of Titan's pristine ammonia-water ocean. The resultant melting curves as functions of pressure are compared to that of pure water ice in Figure 1. With increasing pressure and depth, the melting curve of an ammonia-water mixture decreases from 273 K, the melting temperature of pure water ice, at the surface to the minimum melting temperature of the peritectic composition at 176 K [e.g., Yarger et al., 1993]. Ammonia-water liquid and solid water ice coexist within this temperature range of about 100 K, thereby giving rise to the formation of an outermost ice I layer above an ammonia-rich subsurface ocean. Upon further cooling below the peritectic temperature of 176 K, two coexisting solid phases, ammonia dihydrate and water ice, would begin to crystallize. In the high-pressure regime, the melting curve of an ammonia-water mixture increases again with increasing pressure (Figure 1).

Figure 1.

Melting temperatures as a function of pressure for various compositions of Titan's primordial ammonia-water ocean. The melting curves for ammonia (NH3) contents of 5 wt.% (long-dashed line) and 15 wt.% (short-dashed line), respectively, have been constructed from the polynomial representations provided by Grasset et al. [2000]. The empirical formulae of Chizhov [1993] have been employed to obtain the melting curve of pure water ice (0% NH3, solid line).

[10] The ice I layer is assumed to be viscoelastic with a temperature-dependent Newtonian viscosity that is calculated in terms of the homologous temperature, the ratio between the temperature T and the melting temperature Tm, according to [e.g., Kirk and Stevenson, 1987]

equation image

where η0 is the melting point viscosity representative of the entire ice layer and l is a dimensionless constant in the range of 18–35. The pressure dependence of the activation enthalpy for creep is then expressed in terms of the pressure dependence of the melting temperature Tm. We choose η0 = 1013 Pa s and l = 25 as suggested by Hunten et al. [1984] which is equivalent to an activation energy for creep of about 60 kJ mol−1 at standard pressure and temperature conditions.

[11] The temperature profile within the ice I shell is presented in Figure 2. As the bottom of the ice I layer is at the melting temperature equation imagem of the subsurface ocean, the two branches of the pressure-dependent ammonia-water melting curve given in polynomial form by Grasset et al. [2000] can be used to determine the bottom temperature of the ice I shell and the transition pressure to the high-pressure ice layer. The ice I layer consists of an elastic, stagnant lid and a viscoelastic, well-mixed sublayer. This is the typical thermal structure of a convecting layer with strongly temperature-dependent viscosity [see, e.g., Davaille and Jaupart, 1993; Solomatov, 1995; Grasset and Parmentier, 1998]. In the stagnant lid heat is transported by thermal conduction and the temperature varies linearly between the surface temperature Ts ≈ 94 K and Ttop at the top of the well-mixed sublayer. In the latter layer the temperature Tint is almost uniform and convection is the major heat transfer mechanism. Most of the viscosity variation in the ice I layer occurs in the stagnant lid. Laboratory studies [Davaille and Jaupart, 1993] and numerical calculations [Grasset and Parmentier, 1998] provide an empirical relationship between the temperature difference driving convection and the viscosity function in terms of a viscous temperature scale. The empirical relation suggests that through the well-mixed convective sublayer the viscosity varies only by about one order of magnitude. This observation can be used to calculate the temperature Ttop at the top of the well-mixed sublayer from equation (1) as outlined in Appendix A:

equation image

where Tm(Dice I)/equation imagem(Dice I) is the ratio of the melting temperature of pure ice I and that of the ammonia-water system at the bottom of the ice I shell. Note that because the melting temperature of ice I is pressure-dependent Tm in the numerator of equation (2) should be taken at the pressure of the bottom of the stagnant lid at the depth Dstag which will be greater than the melting temperature at the bottom of the ice shell at depth Dice I. The temperature of the layer Tint depends on the ratio between internal heating by, e.g., tidal dissipation in the ice shell and bottom heating by radiogenic heating of the rocky core. Sohl et al. [1995] have shown, however, that tidal heating of Titan's interior is at least one order of magnitude lower than radiogenic heating at the present time. Therefore bottom heating dominates and Tint ≈ 0.5 (equation imagem + Ttop). If the ice shell is not convecting, the sublayer will be absent and the shell will consist only of the stagnant lid.

Figure 2.

Schematic diagram showing the structural and thermal profile of Titan. The ice layer is subdivided into a conductive stagnant lid at surface temperature Ts, wherein the temperature increases linearly with depth, and a convective sublayer of uniform temperature Tint ≈ 0.5 (equation imagem + Ttop), where equation imagem is the melting temperature of the underlain ammonia-water ocean and Ttop is the temperature at the top of the convective layer. The thermal boundary layers of the convective sublayer are assumed to be symmetric because the layer is mainly heated from below. Qrad, Qconv, and Qstag are the radiogenic heat flow of the silicate core, the heat flow through the convective layer, and the surface heat flow through the stagnant lid, respectively. Thicknesses of the layers are not to scale.

[12] To calculate the thicknesses of the ice layers and the subsurface ocean, we use the following equilibrium conditions:

equation image

where Qrad, Qconv, and Qstag are the radiogenic heat flow from the rocky core, the heat flow through the convective layer, and the surface heat flow through the stagnant lid, respectively (compare Figure 2). The thickness of an individual layer can be calculated from the heat flow and the temperature difference across the layer. The heat flow through the stagnant lid is

equation image

where Rp is the surface radius and k ≈ 3.3 Wm−1K−1 is the thermal conductivity [Spohn and Schubert, 2003], assumed to be constant within the ice I layer. The convective heat flow out of the well-mixed sublayer for Ra ≥ Rac is

equation image

where Dconv is the thickness of the sublayer, Ra the Rayleigh number, Rac the critical Rayleigh number (of order 103) and β = 0.3 [Schubert et al., 1979]. The Rayleigh number for bottom heated convection is given by

equation image

where ρ is the density of ice I, gp = 1.35 m s−2 is Titan's surface gravity, α = 1.6 × 10−4 K−1 is the thermal expansion coefficient for ice I and κ = 1.47 × 10−6 m2s−1 is the thermal diffusivity of ice I [Kirk and Stevenson, 1987]. If Ra ≤ Rac, the ice shell will transfer heat purely by conduction.

[13] The heat flow from below, Qrad, is assumed to be dominated by the decay of long-lived radiogenic isotopes in the rocky core. Additional contributions due to, e.g., whole ice mantle cooling from 300 to 100 K and ice I crystallization have been estimated to contribute about 10% and less than 3% to the total heat flow, respectively [Grasset et al., 2000]. We assume a chondritic composition with a present day specific heat production rate of H = 4.5 × 10−12 W kg−1 [Spohn and Schubert, 2003] but consider uncertainties in this value by ±10%. Thermal evolution models suggest that, subsequent to a phase of early rapid cooling, almost constant time rates of change of internal temperature prevail during most of the thermal history of planetary bodies operating in the stagnant lid regime [see, e.g., Grasset and Parmentier, 1998]. To assess the possible consequences of secular cooling contributing substantially to the heat flow, we take into account an additional heat source term in the stationary energy balance. Since most of the retained accretional energy and the gravitational energy released during internal differentiation [Schubert et al., 1986] may have been deposited deep in Titan's interior, we consider a case in which we assume that the specific heat production rate of Titan's rocky core is twice as large as the chondritic heat production rate.

[14] In doing the model calculations we proceed as follows: A given thickness of the outer ice I layer fixes the bottom temperature of the layer and the thickness of the ocean by assuming that the ocean is isothermal. The surface heat flow Qstag can be calculated together with the thickness of the stagnant lid from equations (2)(6). Equation (3) then gives the radioactive heat flow from the core, the mass of which is calculated from Mc = Qrad/H. The knowledge of the masses of the ice I layer, the subsurface ocean, and the core allows the calculation of the mass of the high-pressure ice layer. Since the density of the latter layer is assumed to be known, its thickness and thus the radius of the core can be calculated. The radius and the mass of the core finally give its density. We limit the range of permissible core densities to 2500 kg m−3 ≤ ρc ≤ 5000 kg m−3. We consider this range to be generous. At the lower density limit, the value is consistent with hydrated rock and strongly oxidized iron phases such as magnetite [Lodders and Fegley, 1998]; at the higher limit, the value is consistent with the low-pressure density of FeS. We also indicate in our results a more stringent core density range from 3500 kg m−3 to 4500 kg m−3. The density of 3500 kg m−3 is close to the density of anhydrous metal-bearing chondritic matter [Endress et al., 1996; Hutcheon et al., 1998] while 4500 kg m−3 is the density of an anhydrous silicate rock and iron mixture with about 35 wt.% iron.

[15] Once the satellite's radial density structure is obtained, the moment of inertia is calculated from

equation image

where ρi and ri are density and outer radius of the ith layer, respectively, beginning with the innermost layer and r0 = 0. The moment of inertia factor Ip/(MpRp2), the ratio between Ip and the product of the satellite's mass Mp and radius Rp squared, then provides a means to characterize Titan's central mass concentration.

[16] The potential Love number k2 and the radial displacement Love number h2 are obtained by numerical integration of the linearized field equations for small viscoelastic deformations in a spherical and self-gravitating incompressible body subject to certain continuity and boundary conditions [e.g., Segatz et al., 1988; Wieczerkowski, 1999]. In the absence of oceans and seas on Titan's surface and disregarding atmospheric loading effects, the surface will be kept in a stress-free state due to vanishing radial and tangential stress components upon body tide forcing. The second degree tidal Love numbers k2 and h2 depend on the satellite's interior structure and rheological properties and parameterize Titan's tidal response: k2 = ϕie is the ratio of the tidal potential due to the displacement of mass in the satellite's interior ϕi and the external tidal disturbance potential ϕe due to Saturn's gravity. h2 = u/ξ is the ratio of the radial displacement of the surface u and the height of the equilibrium tide ξ = ϕe/gp with gp the surface gravity, that corresponds to the radial displacement of an equipotential with respect to Titan's undisturbed surface. The correspondence principle [e.g., Zschau, 1978] which relates the viscoelastic problem to the well-known problem of small elastic deformations, allows to use standard Runge-Kutta methods in the frequency domain to solve for the radial factors h2 and k2 of radial displacement and potential perturbation, respectively. We use the linear viscoelastic Maxwell rheology for which the complex shear modulus is equation image = inμ/(in + τ−1), with i = equation image, n the mean orbital motion, μ the elastic shear modulus and τ = η/μ the Maxwell time. The viscosity η of the well-mixed sublayer is temperature-dependent according to equation (1), whereas the shear modulus μ of ice I is assumed to be constant at 3.3 GPa [Sotin et al., 1998] for simplicity. The subsurface ocean is considered to be inviscous with vanishing shear modulus. Furthermore, we assume that the stagnant lid, the high-pressure ice layer, and the silicate core respond elastically to the tidal forces exerted by Saturn.

3. Results

[17] Model results obtained for an overall ammonia concentration of 5 wt.% NH3 are summarized in Figure 3. As expected, the heat production rate in the core varies strongly with the ice I layer thickness. While the chondritic heat production rate with an error margin of ±10% is consistent with ice I layer thicknesses in the range from about 90 to 105 km, two times the chondritic rate requires a thickness of about 70 to 75 km, if allowable core densities are those of mixtures of anhydrous rock and iron. These results compare well to present day thicknesses of about 80 km obtained by Grasset et al. [2000]. The stagnant lid constitutes the major fraction of the ice I shell in these calculations. The ice I shell is found to be entirely conductive for layer thicknesses >94 km as a consequence of lower ocean temperatures and hence higher ice viscosities.

Figure 3.

Summary of model results for a primordial ammonia-water ocean composition of 5 wt.% NH3 and different core heat production rates H as a function of ice I layer thickness: (a) Temperatures of convective sublayer Tint and subsurface ocean Tocean, (b) surface heat flow, (c) core density, (d) moment of inertia (MoI) factor, (e) radial displacement Love number h2, (f) potential Love number k2. From left to right, there are four partly overlapping sets of curves in each figure that correspond to H = 9 × 10−12 W kg−1, H = 5 × 10−12 W kg−1, H = 4.5 × 10−12 W kg−1, and H = 4 × 10−12 W kg−1. Solid curve branches indicate model results consistent with core densities of anhydrous rock and iron between 3500 and 4500 kg m−3.

[18] In Figure 3a, the temperatures at the top of the convecting sublayer Ttop and of the subsurface ocean Tocean are shown as a function of the thickness of the ice I layer for different specific heat production rates in the core. Ttop and Tocean increase with decreasing thickness of the ice I layer due to the pressure dependence of the corresponding ammonia-water melting curve (see Figure 1). Furthermore, the smaller ice I thicknesses require larger heat production rates in the core to satisfy the assumption of thermal equilibrium. For chondritic heat production rates and anhydrous rock and iron core densities, Ttop and Tocean range from about 190 to 230 K, whereas for two times the chondritic rate Ttop and Tocean attain values of about 225 and 235 K, respectively. The temperature drop across the convecting sublayer, that can be estimated from the difference between Ttop and Tocean, is comparatively small but slightly more pronounced for elevated core heat production rates and small ice I thicknesses.

[19] In Figure 3b, we show the radiogenic surface heat flow Qstag as a function of the ice I layer thickness for various core heat production rates. Qstag is of the order of several 1011 W and increases considerably with decreasing ice layer thickness and increasing specific heat production in the core. This corresponds to surface heat flows per unit area of about 4 mW m−2 for chondritic core heat production rates and anhydrous rock and iron compositions of the core. Larger surface heat flows are consistent with higher temperatures of the well-mixed convective sublayer (Figure 3a), thereby causing steeper temperature gradients in the stagnant lid that may vary between 1 and 3 K/km. However, the temperature gradient close to the surface could be much steeper in the presence of a porous regolith layer as a possible result of impact gardening that is characterized by thermal conductivities orders of magnitude lower than that of polycrystalline ice.

[20] In Figure 3c, the core density is shown to be a strong function of the thickness of the ice I layer. Mass balance constraints require that smaller core densities correspond to larger core sizes at the expense of the thickness of the high-pressure ice layer. In Figure 3d, the moment of inertia (MoI) factor decreases considerably with increasing thickness of the ice I layer which is caused by the corresponding concentration of mass in smaller but denser cores. Most probable values of the MoI factor consistent with core densities of anhydrous rock and iron range from 0.29 to 0.31.

[21] In Figures 3e and 3f, the tidal Love numbers h2 and k2 of second degree are plotted against the ice I layer thickness. The tidal Love numbers slightly increase with increasing heat production in the core because the subsurface ocean thickness increases at the expense of the high-pressure ice layer and the ice I layer, thereby providing greater flexibility to the latter. While k2 varies between 0.32 and 0.38, h2 ranges from 1.05 to 1.25 in these models. Note that there is a linear relationship between the Love numbers and the ice thickness. The slope changes notably for thicknesses above 94 km corresponding to entirely conductive ice I layers.

[22] Model results obtained for a bulk ammonia concentration of 15 wt.% are summarized in Figure 4. Equilibrium thicknesses of the ice I layer consistent with permissible core densities are found to be considerably smaller than those for a bulk ammonia concentration of 5 wt.%. Ice I layer thicknesses range from about 65 to 70 km if a chondritic heat production rate with error margins of ±10% is assumed. For two times the chondritic rate, ice thicknesses of about 45 to 50 km are obtained. The ice I shell is found to be entirely conductive so that the stagnant lid comprises the entire layer.

Figure 4.

Summary of model results for a primordial ammonia-water ocean composition of 15 wt.% NH3 and different core heat production rates H as a function of ice I layer thickness: (a) Temperatures of convective sublayer Tint and subsurface ocean Tocean, (b) surface heat flow, (c) core density, (d) moment of inertia (MoI) factor, (e) radial displacement Love number h2, (f) potential Love number k2. From left to right, there are four partly overlapping sets of curves in each figure that correspond to H = 9 × 10−12 W kg−1, H = 5 × 10−12 W kg−1, H = 4.5 × 10−12 W kg−1, and H = 4 × 10−12 W kg−1, respectively. Solid curve branches indicate model results consistent with core densities of anhydrous rock and iron between 3500 and 4500 kg m−3.

[23] Figure 4a shows that the temperatures at the top of the convecting sublayer Ttop and of the subsurface ocean Tocean are smaller by about 20 to 30 K in comparison to those shown in Figure 3a. This is due to the fact that the ammonia-water melting curve for 15% NH3 is shifted toward lower temperatures relative to that for 5% NH3 (see Figure 1). For chondritic heat production rates and anhydrous rock and iron core densities, Ttop and Tocean range from about 170 to 190 K. Two times the chondritic rate results in Ttop and Tocean values of about 200 and 210 K, respectively. The temperature drop across the immobile sublayer wherein viscosity varies by one order of magnitude is comparable to that obtained for the 5% NH3 models and is once again slightly more pronounced for large core heat production rates and small ice I layer thicknesses.

[24] Figure 4b shows that the surface heat flows are equivalent to those shown in Figure 3b for equivalent core heat production rates, but are transferred through thinner ice I layers. This implies that the corresponding core masses and sizes are similar in both cases and do not depend much on the choice of the bulk ammonia concentration.

[25] In Figures 4c and 4d, core density and moment of inertia (MoI) factor are shown as functions of the thickness of the ice I layer. For each core heat production rate, the core density strongly increases with increasing ice thickness, whereas the corresponding MoI factor decreases considerably. Larger core sizes are accompanied by smaller core densities and larger MoI factors. The range of most probable MoI factors consistent with densities of anhydrous rock and iron is similar to that shown for bulk ammonia concentrations of 5% NH3. Differences in ice I layer thickness do not affect the MoI factor much because of the comparatively small density contrast between the ice I layer and the underlying ammonia-water ocean.

[26] In Figures 4e and 4f, the tidal Love numbers h2 and k2 of second degree are plotted against the ice I layer thickness. The tidal Love numbers are linearly related to the ice thickness and somewhat larger than those shown in Figures 3e and 3f for ocean compositions of 5% NH3 and various core heat production rates. This is due to the fact that thinner ice I layers are more deformable than thicker layers.

[27] Figure 5 indicates that the mass fractions of anhydrous rock and iron within Titan's interior range from about 50 to 60 wt.% independent of the assumed core heat production rate. This finding is consistent with Titan's bulk rock-to-ice ratio of 55:45 as given in Table 1.

Figure 5.

Mass fractions of rock/iron and ice plus subsurface ocean for various core heat production rates H as a function of ice I layer thickness. From left to right, there are four sets of curves in each figure that correspond to H = 9 × 10−12 W kg−1, H = 5 × 10−12 W kg−1, H = 4.5 × 10−12 W kg−1, and H = 4 × 10−12 W kg−1, respectively: (a) primordial ammonia-water ocean composition of 5 wt.% NH3, (b) 15 wt.% NH3. Solid curve branches indicate model results consistent with core densities of anhydrous rock and iron between 3500 and 4500 kg m−3.

4. Discussion and Conclusions

[28] We have calculated equilibrium models of the interior structure of Titan underneath a water ice I shell of given thickness covering an ammonia-water ocean. We derive the thickness of the ocean as well as the thickness of the high-pressure ice layer below it and the radius and the density of the rock + iron core. We have shown that the Love numbers depend linearly on the ice shell thickness. Unfortunately though, these data will not be unambiguous because the Love numbers depend not only on the chemistry of Titan as shown in Figures 3 and 4 but also on the rheological properties of the satellite. The determination of Titan's MoI factor from the Cassini gravity data can only partly remove the ambiguity. In application to the Jovian moon Europa, Castillo et al. [2000] have shown that the potential Love number k2 also depends to some extent linearly on the size of a liquid metallic core for constant subsurface ocean densities. This is consistent with results reported by Wu et al. [2001] for a core radius of about half the surface radius and may be due to the fact that mass balance constraints require larger core sizes being compensated by lower core densities to account for the satellite's mean density and MoI factor. This effect should be less important, however, in the case of a rocky core overlain by a high-pressure ice layer, as it is assumed in the present study for Titan, because the tidally induced mass redistribution associated with these solid layers are expected to be much smaller. Wu et al. [2001] have shown that Europa's potential Love number k2 is almost linearly related to the thickness of the outermost ice shell, as far as the shell is underlain by a liquid water ocean. The tidal response depends also significantly on the rigidity of the ice and the density of the ocean, whereas the density distribution of the deep interior is less important. Moore and Schubert [2000] have concluded that the tidal response of Europa is controlled by the product of the thickness and rigidity of the decoupled ice shell. Given the uncertainties associated with the rheology of the ice I layer and the composition of the ammonia-water ocean of Titan, these results indicate that the determination of the ice thickness from measurements of the tidal Love numbers will be difficult due to the inherent ambiguity. The gravitational signature of an internal ocean will become apparent, however, as the potential Love number k2 will be determined with an accuracy of a few tenths of a percent from the Cassini gravity science experiment when Titan is near perikrone and apokrone [Rappaport et al., 1997]. This will not be possible from the MoI factor alone because of the small density difference between the ice shell and the ocean. In the absence of an ocean, k2 would be smaller than predicted here by one order of magnitude [e.g., Sohl et al., 1995].

[29] The hypothesis of an internal ocean on Titan is supported by recent findings of the Galileo spacecraft in the Jovian system. Magnetic field measurements indicate that both Ganymede and Callisto may have subsurface oceans underneath ice layers of around 100 km or more thickness in which magnetic fields are induced [Khurana et al., 1998; Zimmer et al., 2000; Kivelson et al., 2002]. Images of tectonic surface units and magnetic field data implying the presence of an induced magnetic field suggest that Europa's icy crust is underlain by a water ocean [Kivelson et al., 2000; Zimmer et al., 2000]. Furthermore, equilibrium models of heat transfer by conduction and thermal convection of strongly temperature-dependent viscosity fluids provide further arguments for subsurface water oceans beneath ice shells of tens of kilometers to more than hundred kilometers thickness in the icy Galilean satellites [Hussmann et al., 2002; Spohn and Schubert, 2003].

[30] Other data that may constrain the thickness of Titan's outermost ice shell are magnetic induction data and geology data such as crater morphologies recently used by Schenk [2002] and impact crater simulations performed by Turtle and Pierazzo [2001] to constrain the thickness of the ice shell on Europa. It should be noted, however, that Titan's thick atmosphere precludes flybys at altitudes lower than about 950 km which limits the fidelity of both gravity and magnetic measurements. Galileo flybys of Jovian satellites were as low as a couple of hundred kilometers. A further significant impediment to finding an ocean via inductive effects is that the Saturnian magnetic field is closely aligned with the rotational pole, unlike that of Jupiter, with the result that there is a much weaker modulation of the field applied to (and thus the induced response in) Titan. Nonetheless, if ionospheric currents and other effects are adequately characterized and compensated for, an inductive signal may be isolated.

[31] We have assumed that Titan is differentiated. Although this is widely accepted as a reasonable hypothesis, the example of Callisto shows that it is not necessarily true. The MoI factor for an undifferentiated Titan with pressure-induced ice phase transitions would be about 0.38. This value has been recently predicted by McKinnon [1997] for an undifferentiated model of Callisto but the estimate can be applied to Titan as well due to its similar size and mean density. Much smaller values of the MoI factor are to be expected for completely differentiated models of Titan's interior as we have shown in the previous section.

[32] We have further assumed that Titan contains a substantial amount of ammonia and possesses a subsurface ocean. The recent models of the icy Galilean satellites by Spohn and Schubert [2003] and earlier models of Titan by Grasset et al. [2000] and Titan-like bodies by Deschamps and Sotin [2001] have shown that oceans are difficult to avoid if ammonia is present even in relatively small quantities. According to the present knowledge of ice phase diagrams, ammonia produces one of the largest reductions of the melting temperature. Recent laboratory experiments by Mousis et al. [2002] suggest that two new high-pressure ammonia hydrate phases may exist above about 0.5 GPa along with substantially elevated peritectic temperatures of ≈232 K relative to those at lower pressures (≈176 K). For primordial ocean compositions of >11 wt.% NH3, high-pressure ammonia hydrate and ice may crystallize, thereby causing ammonia depletion in the subsurface ocean as ocean temperature falls short of 232 K upon cooling. Crystallization of high-pressure ice will proceed at the base of the ocean for pressures of <0.5 GPa. Eventually, the freezing of an ammonia-rich high-pressure phase at elevated temperatures may make the remaining subsurface ocean thinner than our model suggests. However, further experimental studies are required to confirm the existence of the new high-pressure ammonia hydrate phases in the NH3-H2O system [Mousis et al., 2002].

[33] For simplicity, we have used a limited set of constant model parameter values for ice such as the densities and the transport parameters. Sohl et al. [2002] have calculated detailed density models of Ganymede and Callisto that account for self-compression and thermal expansion in the interiors. The hydrostatic pressure in the center of Ganymede, assumed to be completely differentiated, is about 10 GPa, whereas the central pressure of partially differentiated Callisto is about half that of Ganymede. It is seen that the density increase with depth within each chemically homogeneous layer is only small in these models due to the small pressure increase and limited temperature variation with depth. Since Titan is a body of similar size and mass, we can safely neglect the effects of pressure and temperature on the densities.

[34] More relevant uncertainties are in the thermal conductivity k of the stagnant lid and in the well-mixed convective sublayer, in the rheology parameters η0 and l of the ice I shell, in the heat transfer parameters Rac and β, and in the specific heat production rate H of the core. It is an advantage of our model that the uncertain transport parameters only matter in the ice I layer. Transport parameter values for high pressure ices are even more uncertain. The thermal conductivity of compact, hexagonal ice Ih is the highest known so far for solar system ices and is an inverse function of temperature T according to the empirical relation [Klinger, 1980]

equation image

where temperature is given in Kelvins. k(T) may vary from 2 to 6 Wm−1K−1 at the prevailing temperature conditions in the ice I layer, ranging from 250 K at the bottom of the convective sublayer to 94 K at Titan's surface, respectively. Non-crystalline and porous ices have significantly smaller thermal conductivities of less than 1 Wm−1K−1 [Ross and Kargel, 1998] and less than 10−1 Wm−1K−1 [Seiferlin et al., 1996], respectively. In the presence of a porous regolith layer of perhaps a few tens of meters, the temperature difference through the stagnant lid may increase by a few tens of Kelvins. Contamination with ammonia will significantly reduce the thermal conductivity of the ice I layer, too. Recent laboratory experiments indicate that the thermal conductivity of NH3-rich water ice will be two to three times lower than that of pure water ice at 100 to 170 K [Lorenz and Shandera, 2001]. As a consequence, the temperature gradient in the stagnant lid would be steeper than that for pure water ice and the melting isotherm will be reached at shallower depth resulting in thicker subsurface oceans. On the other hand, a hydrocarbon-soaked regolith may permit hydrothermal circulations with a lower near-surface temperature gradient. Contamination of the ice by rock, gas bubbles, and dislocations may also alter the thermal conductivity that locally may deviate significantly from a global average. Studies of a sample of a carbonaceous chondrite (Allende C3) gave a thermal conductivity of about 2 to 3 Wm−1K−1 [Horai and Susaki, 1989]. Thermal conductivities of rock in the temperature range of interest may be even one order of magnitude larger but if the concentration of rock is well below 60 vol.% this will be irrelevant. If we neglect the very small values of k of 10−1 Wm−1K−1 that should be applicable only to the ice regolith layer, then the value of the thermal conductivity that we have used should be representative of the possible range of values but is uncertain by a factor of about two. Given an ice shell thickness, a factor of two uncertainty in the thermal conductivity translates into a factor of two uncertainty in the mass of the core. It will be clearly possible to reduce this uncertainty by using the MoI factor to be obtained from Cassini gravity measurements.

[35] We assume a Newtonian creep law which requires that grain sizes are small enough and convective stresses are generally low in the sublayer, so that stress-independent diffusion creep (n = 1; see equation (9)) is likely to dominate [Schubert et al., 1986]. A general flow law for steady state creep of ice at temperatures above about 0.5 times the melting temperature is [e.g., Durham and Stern, 2001]

equation image

where equation image is the strain rate, P is hydrostatic pressure, d is grain size, σ is differential or deviatoric stress, R is the gas constant, and A, p, n, E*, and V* are flow constants related to the dominant creep mechanism. Laboratory-derived flow laws for pure, polycrystalline ice suggest that both dislocation (n ≥ 3) and grain-size-sensitive creep are appropriate flow mechanisms for ice I, whereas dislocation creep prevails for high-pressure ice phases at laboratory strain rates and stresses [Weertman, 1983; Durham et al., 1998; Durham and Stern, 2001]. On the basis of creep experiments on fine-grained ice and in agreement with previously published laboratory creep data, Goldsby and Kohlstedt [2001] report a constitutive equation for the flow of ice that includes diffusion, dislocation and grain-size-sensitive flow mechanisms. Durham et al. [1997] provide a compilation of laboratory derived steady state flow laws for ice I and high-pressure water ice phases and emphasize that the dominant flow mechanisms at geologic timescales and low strain rates may be different from those which occur under laboratory conditions where strain rates are likely to be orders of magnitude higher. In particular, grain growth over time in the presence of low stresses suggests that grain-size-sensitive creep may be the dominant flow mechanism in many planetary applications [McKinnon, 1998; Durham and Stern, 2001; Goldsby and Kohlstedt, 2001].

[36] The uncertainties in convection heat transfer parameters will induce similar uncertainties. However, the strongly temperature-dependent viscosity parameterization of convective heat transport used in the present study suggests that the conductive stagnant lid comprises most of Titan's ice shell at the expense of the well-mixed convective sublayer. Because of the thermal isolation by the substantial stagnant lid, the convective sublayer tends to be warmer than in the case of a comparable constant viscosity convecting layer and ocean freezing over time is efficiently prevented [Spohn and Schubert, 2003]. As a further caveat we note, that the rheology of the ice I layer may be extremely sensitive to temperature if ammonia dihydrate, NH3·2H2O, is present. Laboratory experiments have shown that pure ammonia dihydrate is about four orders of magnitude less viscous than water ice immediately below the peritectic temperature of 176 K, whereas below 140 K the viscosity of ammonia dihydrate is comparable to that of pure water ice or even larger [Durham et al., 1993]. The substantial reduction of viscosity in the well-mixed sublayer will therefore increase the efficiency of convective heat transfer. On the other hand, the stagnant lid will become thinner and stiffer in the presence of ammonia dihydrate with larger temperature and viscosity gradients which may have a profound effect on impact crater morphology and surface tectonics. It should be noted that while an idealized model of Titan's accretion would have a pure ice I crust, it may be that subsequent evolution superimposes deposits of ammonia dihydrate, as cryovolcanic flows [Lorenz, 1996]. These might be morphologically apparent in surface imaging, and could perhaps also be discriminated on the basis of the higher dielectric constant of that material compared with ice I.

[37] We have assumed mechanical and thermal equilibrium. It is possible to test the assumption of hydrostatic equilibrium by comparing the terms J2 and C22 to be obtained from Cassini gravity measurements the ratio of which should be close to 10/3. In the presence of non-hydrostatic contributions to Titan's gravity field due to, e.g., internal processes such as solid-state convection in the lower part of the ice I layer, the ratio between J2 and C22 will deviate from 10/3 and the separately determined moment of inertia factors would differ from each other [Rappaport et al., 1997]. However, as Anderson et al. [2001a] have pointed out for Io, the equivalence of the MoI factors so determined will not prove hydrostatic equilibrium but will only be consistent with the assumption of equilibrium. Some uncertainty about the MoI factor simply cannot be eliminated with our present means. The assumption of thermal equilibrium may be more critical though. Thermal evolution calculations for satellites and terrestrial planets suggest that secular cooling of their interiors contributes substantially to the energy balance (up to about 50%) in which case another factor of two error will be introduced. The ratio between the surface heat flow and the rate of radiogenic heating (if expressed as a heat flow) could be a factor of 2 for the Earth and the Moon. The value for the Moon, a body closer in size to Titan than the Earth, has been estimated from the numerical results of Konrad and Spohn [1997] and Spohn et al. [2001]. We have modeled this effect by assuming a specific heat production rate twice as large as the chondritic value in some of our models. These calculations show that the effect can be substantial. But, again, we note that the MoI factor will be available as an additional constraint to reduce some of this uncertainty.

[38] A representative model of the interior structure of Titan assuming a chondritic core heat production rate and an NH3 content of 15% for the primordial ocean is shown in Figure 6. The internal ammonia-water ocean extends over a depth range of several 100 km beneath a few tens of kilometers thick ice I layer and is underlain by a substantial high-pressure ice shell. Additional calculations have shown that the thickness of the subsurface ocean moderately decreases with increasing core density, whereas higher heat production rates in the core will result in thicker subsurface oceans at the expense of the ice layers on top and below. It has been concluded that ice viscosity variations by one order of magnitude have only minor effects on the subsurface ocean thickness [Grasset et al., 2000]. Denser cores are generally smaller in size with radii ranging between 1500 and 1800 km for anhydrous rock and iron densities. Grasset et al. [2000] argue that the existence of a metallic iron core may be possible for Titan. Iron melting has been found to occur well before the onset of core convection in a putative EH enstatite chondritic core with an iron content of xFe ∼ 35 wt.%. Therefore the present state of the core may be characterized by either a well convecting silicate sphere or a liquid iron core that is overlain by a slowly convecting silicate layer [Grasset et al., 2000].

Figure 6.

Radial structure and density distribution of Titan. A primordial ammonia-water ocean composition of 15 wt.% NH3 is assumed. The interior structure model is optimized to satisfy the bulk rock-to-ice ratio of 55:45. The ice I layer is 69 km thick and entirely conductive. The subsurface ocean extends over a depth range of 224 km and is underlain by a high-pressure ice layer of 612 km thickness. The rocky core has a radius of 1671 km and a mean density of 3800 kg m−3. The satellite's density distribution is associated with a moment of inertia (MoI) factor of 0.304. The radiogenic heat flow amounts to 3.35 × 1011 W corresponding to a total surface heat flow per unit area of about 4 mW m−2.

[39] Although Titan rotates synchronously with its orbital revolution and has low inclination and obliquity angles, it is exposed to varying gravitational forces due to its orbital eccentricity. This causes both radial and librational tides. Sohl et al. [1995] have found that tidal heating within a decoupled ice shell contributes less than 10% to the total heat production rate. This value is further reduced in the models presented here, because of the derived small thickness of the viscoelastic, well-mixed convecting sublayer in equilibrium with the stagnant lid. The latter is substantially thicker, but due to the steep thermal gradient within this layer it can be considered as elastic and does not contribute to the dissipation of tidal energy. The global tidal dissipation rate equation image is proportional to the imaginary part of the potential Love number Im(k2) and can be calculated from [Segatz et al., 1988]

equation image

where n is mean orbital motion, e is free orbital eccentricity and G is gravitational constant. Assuming a Maxwell model for the viscoelastic part of the ice I shell, we obtain dissipation rates of up to several 1010 W from equation (10) in the presence of a convective sublayer, which is about one order of magnitude below the radiogenic heating rate. If the ice I shell becomes entirely conductive, such as for our nominal model shown in Figure 6, solid tidal dissipation rates will be negligible due to higher ice viscosities. Thus the thermal model is not altered significantly if tidal heating is additionally taken into account. Note that low tidal dissipation rates are more consistent with a primordial origin of Titan's significant orbital eccentricity that otherwise should have been damped to smaller values over the age of the Solar System. Sotin et al. [2002] have recently suggested that tidal dissipation may be particularly important in warm upwelling plumes in the Europan ice shell. Whether or not this mechanism is important in Titan requires further modeling. We note, however, that the tidal dissipation rate in Europa's ice shell is at least one order of magnitude larger than in Titan's ice I layer [Hussmann et al., 2002]. Tidal heating would become more important if the viscoelastic ice I layer with a mean temperature not far from the melting temperature would somehow get very thick. However, such states are unstable because a high heating rate would lead to melting and thinning of the ice layer. This is detailed by Hussmann et al. [2002] in application to Europa.

Appendix A:: Temperature Ttop at the Top of the Well-Mixed Convective Sublayer

[40] The viscosity of ice I (see equation (1)) used in the model calculations is a strong function of the homologous temperature, the ratio between the temperature T and the melting temperature Tm, according to

equation image

with constant rheology parameters η0 and l. Note that the melting temperature of the ice I layer depends on the depth z in that Tm(z) varies from 273 to 251 K as the pressure rises to 200 MPa. The viscosity at the base of the ice I shell, i.e., at the depth Dice I, is therefore given by

equation image

where Dice I is the thickness of the ice I layer. The temperature at the bottom of the ice I layer is determined by the pressure-dependent ammonia-water melting temperature equation imagem. Therefore we get

equation image

The viscosity of ice I at the top of the convecting sublayer, i.e., at the depth Dstag, is given by

equation image

where Ttop is the temperature at the top of the convecting sublayer. As the viscosity is assumed to vary by about one order of magnitude through the convecting sublayer, the viscosity ratio can be written as

equation image

so that we obtain from inserting equations (A3) and (A4)

equation image

Solving for Ttop, we get

equation image

where Tm(Dice I)/equation imagem(Dice I) is the ratio of the melting temperature of pure ice I and that of the ammonia-water system at the bottom of the ice I shell. A ratio of 1 is obtained in the case of pure water oceans (0% NH3), since then equation imagem(Dice I) corresponds to Tm(Dice I). An upper bound of 273 K/176 K = 1.55 is derived from the ratio of the maximum melting temperature of pure ice I and the minimum melting temperature of the peritectic composition in the NH3-H2O system.

Acknowledgments

[41] We have benefited from discussions with Julie Castillo, Ralf Greve, Nicole Rappaport, and Gabriel Tobie. We thank the reviewers for their helpful comments on the manuscript. This work was supported by grants from the Deutsche Forschungsgemeinschaft.

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