Stagnant lid convection in the mid-sized icy satellites of Saturn

Icarus 186 (2007) 420–435 www.elsevier.com/locate/icarus

Stagnant lid convection in the mid-sized icy satellites of Saturn Kai Multhaup a,∗ , Tilman Spohn a,b a Institut für Planetologie, Westfälische Wilhelms-Universität, Wilhelm-Klemm-Strasse 10, D-48149 Münster, Germany b Deutsches Zentrum für Luft- und Raumfahrt e.V., Institut für Planetenforschung, Rutherfordstr. 2, D-12489 Berlin, Germany

Received 17 October 2005; revised 5 September 2006 Available online 25 October 2006

Abstract Thermal history models for the mid-sized saturnian satellites Mimas, Tethys, Dione, Iapetus, and Rhea have been calculated assuming stagnant lid convection in undifferentiated satellites and varying parameter values over broad ranges. Of all five satellites under consideration, only Dione, Rhea and Iapetus do show significant internal activities related to convective overturn for extended periods of time. The interiors of Mimas and Tethys do not convect or do so only for brief periods of time early in their thermal histories. Although we use lower densities than previous models, our calculations suggest higher interior temperatures but also thicker rigid shells above the convecting regions. Temperatures in the stagnant lid will allow melting of ammonia-dihydrate. Dione, Rhea and Iapetus may differentiate early and form early oceans, Iapetus only if ammonia is present. Mimas and Tethys with ammonia may differentiate if they accreted in an optically thick nebula with ambient temperatures around 250 K. Our models suggest that the outer shells of the satellites are largely primordial in composition even if the satellites differentiated. In these cases the deep interior may be layered with a pure ice shell underlain by an ammonia dihydrate layer and a rock core. © 2006 Elsevier Inc. All rights reserved.

1. Introduction The saturnian system includes a group of medium-sized, regular satellites in a size range from barely 200 km to well over 700 km. These are in increasing distance to Saturn, Mimas, Enceladus, Tethys, Dione, Rhea and Iapetus. Their low densities suggest that the satellites are mostly composed of water-ice with ammonia as the likely most prominent minor constituent. At the time of this writing Cassini is orbiting Saturn and will image the satellites with improved resolution as compared with Voyager. The first results have already been returned from Iapetus (Porco et al., 2005), Rhea and Enceladus. As a reference for the interpretation of the imaging and other data it is worthwhile to update our understanding of the thermal evolution of the medium size satellites. The most detailed study of the thermal evolution of Saturn’s icy satellites to date is by Ellsworth and Schubert (1983). See also Pollack and Consolmagno (1984) for a review. Assuming homogeneous ice/silicate mixtures for their interiors, * Corresponding author.

E-mail address: [email protected] (K. Multhaup). 0019-1035/$ – see front matter © 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2006.09.001

Ellsworth and Schubert calculated thermal histories considering both cold and hot post-accretion temperature profiles. These authors and other previous workers have assumed the boundary between the stagnant lithosphere and the convecting mantle to be defined by an isotherm at the minimum temperature enabling solid state creep on geological time scales. However, as was demonstrated in laboratory experiments (Davaille and Jaupart, 1993) and verified in theoretical studies (e.g., Solomatov, 1995; Grasset and Parmentier, 1998), convection in a volumetrically heated fluid with strongly temperature-dependent viscosity occurs underneath a stagnant lid the basal temperature of which depends on the rheology parameters and the temperature of the convecting layer. The model of stagnant lid convection as applied to icy satellites has been described with some detail by Spohn and Schubert (2003). The viscosity contrast across the convecting layer has been found to remain constant at about one order of magnitude while the temperature at the bottom of the stagnant lid and the average interior temperature of the layer both vary with time in an evolution calculation. The basal temperature of the stagnant lid, its thickness and the average temperature of the convecting sub-layer can be larger than those predicted by models with fixed boundary temperatures for the lithosphere (e.g., Spohn et al., 2001).

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Fig. 1. A multiplicity of satellites. The medium-sized icy moons in the saturnian system show differing degrees of cratering and surface modifications with the most striking feature on Iapetus being the difference in albedo on its leading and trailing hemispheres. The latter is not believed to be related to internal activities, though.

We apply for the first time the concept of stagnant lid convection to the medium sized icy satellites of Saturn. We study models of thermal evolution for all but one of these regular moons, Enceladus. The heat sources in our models are accretional and radiogenic heat. An additional heat source not included in our models is needed to explain its extensively modified surface (Porco et al., 2006) and the southern hemisphere surface temperatures (Spencer et al., 2006). Tidal heating is the commonly accepted possibility (e.g., Squyres et al., 1983; Wisdom, 2004). Images of the five satellites considered in this work are compiled in Fig. 1. 2. The model We assume the mid-sized saturnian icy satellites to be homogeneous ice/silicate mixture bodies characterized by their orbital distance from Saturn R, surface radius rsat , mean density ρsat , mass Msat and surface gravity acceleration g as listed in Table 1. It is entirely possible that at least the bigger of the midsized satellites are differentiated. Differentiated models have recently been calculated by Husmann et al. (2006). A measurement of the low order terms of the gravity field is necessary to constrain the degree of differentiation of a satellite as has been done in the jovian system for the Galilean satellites (see, e.g., Schubert et al., 2004, for a review). Differentiation is likely to occur when the temperature exceeds the ice melting tempera-

Table 1 Parameter values used for the mid-sized icy satellites of Saturn Satellite

R [103 km]

rsat [km]

Mimas Tethys Dione Rhea Iapetus

185.52 294.66 377.40 527.04 3561.30

199 530 560 764 734

Msat [1020 kg] 0.375 6.22 10.5 23.1 18.0

ρsat [kg m−3 ]

g f [m s−2 ]

TN [K]

1140 1000 1440 1240 1087

0.063 0.148 0.225 0.265 0.223

250 157 123 80 80

0.294 0.121 0.547 0.392 0.234

Values for R, rsat , ρsat , Msat are from Yoder (1995) except for Iapetus. The radius of Iapetus has been adopted from Matson (2005, personal communication); the mass was calculated from the data of Iess et al. (2005). The density and the surface gravity acceleration were calculated from rsat and Msat . The silicate mass fraction f was calculated from ρsat assuming an ice density of 920 kg m−3 and a silicate rock density of 2700 kg m−3 . TN is the assumed ambient temperature in the saturnian sub-nebula at the time of satellite accretion calculated according to Lunine and Stevenson (1982).

ture. We will compare interior temperatures with the melting temperatures of ice and ammonia–ice assemblages. 3. Accretion and initial conditions We assume that the satellites accreted homogeneously and follow Schubert et al. (1986) to construct initial temperature profiles. The model is based on the assumption that in-falling planetesimals deposit a fraction h of their impact energy uniformly in the layer immediately underneath the instantaneous

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surface of the growing satellite. The accretional temperature Tacc (r), where r is the radial distance from the satellite’s center, is then given by:   hGM(r) ru2 1+ + TN . Tacc (r) = (1) cr 2GM(r) In Eq. (1), G is the universal gravitational constant, M(r) is the mass contained in r, c is the specific heat at constant pressure, u is the average velocity of the in-falling planetesimals, and TN is the ambient temperature in the nebula. The first term in Eq. (1) is the potential energy per unit mass of the planetesimals in the gravitational field of the growing satellite divided by the specific heat; the second term is the kinetic energy per unit mass of the planetesimals also divided by the specific heat. We use Eq. (1) to create accretional temperature profiles and set GM(r)/ru2 = 4. For h, we use 0.4 (Ellsworth and Schubert, 1983). The effects of using other values are discussed below. We will show that varying h is not relevant to the long-term thermal evolution. The major drawback in applying this model to satellite accretion is the close proximity of accreting satellites to their primaries as compared with the distances between terrestrial planets and the Sun. Due to the presence of the primary’s gravity field, planetesimals may have stricken the surfaces of the growing planetary bodies at lower velocities resulting in less energy being turned into heat in the process (Schubert et al., 1981). As the mid-sized satellites orbiting Saturn do not have substantial atmospheres, their surface temperatures after the end of bombardment should correspond to the effective surface temperature Teff at the orbit of Saturn, 80 K. (This is the temperature in equilibrium with the solar radiation and assuming an albedo of 1.) The latter temperature would also be applicable if an optically thin nebula is assumed. In an optically thick nebula the temperature would be higher. Following Lunine and Stevenson (1982) we assume that condensation of water ice occurred at the orbit of Mimas and that TN ∝ r −1 : rSaturn , (2) R where rSaturn is the surface radius of Saturn. For orbital distances where TN becomes smaller than the solar ambient temperature, TN is set to Teff . The numerical value of 768.8 results from the assumed sub-nebula temperature of 250 K at the orbit of Mimas. TN (R) is listed for the satellites under study in Table 1. The initial temperature profile Tacc (r) is calculated from Eq. (1) for r = 0 to r = rsat − D and drops linearly from Tacc (rsat − D) to Teff at r = rsat . Following Ellsworth and Schubert (1983) we assume an arbitrary but reasonable value of 30 km for D. We also consider a model for the saturnian sub-nebula as proposed by Mosqueira and Estrada (2003). This model is based on the assumption that the sub-nebulae of giant planets consist of an optically thick inner region and an extended, optically thin outer disk. Mosqueira and Estrada (2003) place Iapetus in the outer disk of the saturnian nebula at TN = 80 K. TN = 768.8 ·

Fig. 2. Heat transfer in the interior of the compositionally homogeneous icy satellites. The conducting core of radius rc is overlain by a convecting sub-layer which underlies the stagnant lid of thickness rsat − rtop . It is possible that the convecting layer extends all the way to the center of the satellite. It is also possible that the satellite is entirely conductive without a convecting layer.

Mimas, Tethys, Dione and Rhea in this model accreted at TN = 250 K. 4. Heat generation and transfer Heat is generated in the silicate fraction of the icy satellites by radiogenic decay. To estimate the silicate mass fraction f , we use the present day satellite densities as listed in Table 1 and reasonable estimates of the densities ρsil and ρice of the silicate and ice I components, respectively. For ρsil , we use 2700 kg m3 and for ρice we take 920 kg m3 . We neglect the pressure and temperature dependencies of the densities since these are small in the pressure and temperature ranges of interest (e.g., Kargel, −ρice ρsil are listed in Ta1991). Calculated values for f ≡ ρρsat sil −ρice ρsat ble 1. The volumetric heat production rate Q is a function of time t and is proportional to the silicate mass fraction: Q(t) = f · Q0 · e−λt ,

(3)

where Q0 is the primordial chondritic heating rate per unit mass of about 45 pW kg−1 . Ellsworth and Schubert (1983) use 1.4 × 10−7 W m−3 . Assuming a rock density of 2700 kg m3 , this translates into 52 pW kg−1 , which is slightly larger than the value we use. Furthermore, λ is the chondritic decay constant of 1.5 × 10−17 s−1 (e.g., Turcotte and Schubert, 1982). We will vary Q0 , however, to discuss its effects on our model results. We subdivide the satellite interior into three layers characterized by modes of heat transfer (see Fig. 2): a conducting and stably stratified primordial core through which temperature T (r, t) at any given time increases with increasing radial distance r from the center, a convecting sub-layer, and a stagnant and conducting lid. In the convecting sub-layer and the lid,

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temperature decreases with increasing r. The radius rc of the primordial core is defined by the maximum temperature Tmax in the satellite interior. Both Tmax and rc are functions of time. Heat flows inward in the core and outward in the layers above the core. Initially, there is just the core and the lid. The core comprises most of the satellite and the initial lid thickness is D. Since the core is a heat trap (i.e., heat cannot escape from the interior of the core because the temperature gradient is positive), its temperature will increase in time. The radius of the core will decrease because of its warming and the cooling of the layers above it. As the core shrinks, the layer immediately above may become unstable to convection and a convecting sub-layer may form between the core and the lid.

found that the relevant parameter values were similar to the ones derived by Grasset and Parmentier for plane layers. Spohn and Schubert (2003) have applied stagnant lid theory to the icy Galilean satellites of Jupiter to discuss the possibility of internal oceans. Through an extensive parameter study they found that reasonable variations of the stagnant lid heat transfer parameter values were uncritical. In applying stagnant lid convection theory we note that the viscosity contrast across the convecting sub-layer is about a factor of 10 (Grasset and Parmentier, 1998). The temperaturedependent viscosity ν at temperature T is   Tm −1 , ν = ν0 exp A (8) T

5. Conduction

where ν0 is the ice I melting point viscosity, A is the activation parameter for solid state creep and Tm is the ice I melting temperature. We set ν0 to 1012 m2 s−1 , A to 24, and choose the melting temperature at zero pressure 273.16 K for Tm (see Spohn and Schubert (2003) for a recent discussion of parameter values for icy satellite convection models). We vary ν0 over plus/minus three orders of magnitude to study its effects on the results. We calculate the minimum viscosity in the potentially convecting layer νmin by using Tmax . Increasing tenfold yields νtop , the viscosity at the top of the potentially convecting region. We then calculate Ttop and search for a match in the temperature profile above rc to identify rtop . With Tmax , Ttop , rc , and rtop we calculate the Rayleigh number Ra and verify that it exceeds the critical value Racr . The critical Rayleigh number for the onset of convection depends on the mode of heating, on boundary conditions, and on the ratio between the bottom and top radii (see, e.g., Spohn and Schubert (2003) and references therein). We adopt a value of 103 following, e.g., McKinnon (1998) and Ellsworth and Schubert (1983), although we note that it may vary between at least 100 and 104 . We will discuss the effects of a variation of Racr further below. The Rayleigh number Ra is defined as

Conduction heat transfer is governed by the spherically symmetric time dependent heat conduction equation for variable thermal conductivity k:   ∂T ∂ ∂T = k(T ) + ρsat Q(t), ρsat c (4) ∂t ∂r ∂r where T is the temperature at time t and radius r. We solve the heat equation by means of explicit time stepping. The time step t is r 2 /2κ with r being the selected spatial grid width thereby satisfying the Courant criterion for numerical stability. In the ice fraction, k is a function of T (Petrenko and Whitworth, 1999): kice (T ) =

651 . T

(5)

For the silicate fraction, we use ksil = 4.2 W m−1 K−1 . The specific heat in the silicate and ice I fractions are csil = 1200 J kg−1 K−1 and cice = 1390 J kg−1 K−1 , respectively. The bulk thermal conductivity and specific heat are   ρsat ρsat f · kice (T ) + f · ksil , k(T ) = 1 − (6) ρsil ρsil c = (1 − f ) · cice + f · csil . (7) We will discuss the effects of varying k and c over wide ranges below. 6. Convection Convection can occur in the layer above the conductive core. We apply, for the first time, the theory of stagnant lid convection (Solomatov, 1995; Grasset and Parmentier, 1998) to the convective heat transport in mid-size icy satellites. Although the theory has been derived from studies of convection in plane layers, the elements of the theory are quite general and have been verified using numerical results for convection in spherical shells with temperature dependent viscosities and largely varying ratios between the inner and outer radii (e.g., Konrad and Spohn, 1997, and Spohn et al., 2001, for the Moon; Conzelmann, 1999, for Mercury). Breuer and Spohn (2003) have derived a stagnant lid parameterization for a spherical shell with a ratio between the outer and inner the radii of 0.5 as applicable for Mars and

αg(Tmax − Ttop )d 3 (9) , κν where α is the thermal expansivity, g is the acceleration due to gravity, and d ≡ rc − rtop . For simplicity, we set g to its surface value. For ν, we use the viscosity at the mid-point temperature of the layer (McKinnon, 1998). α is taken to be 1.6×10−4 K−1 . The convective temperature profile in small satellites is characterized by thermal boundary layers at the top and the bottom and an isothermal core of temperature Tconv . The boundary layer thickness δ is related to Ra by:   Racr β δ = (rtop − rb ) (10) . Ra

Ra ≡

Typically, β takes values close to 0.3 for vigorous convection (Schubert et al., 2004) and may be close to 0.2 for Rayleigh numbers close to critical. Within the boundary layers, temperature varies linearly with radius. The time rate of change dTconv /dt of the temperature of the isothermal core is calculated from an energy balance for

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the convecting layer (e.g., Schubert et al., 1979; Ellsworth and Schubert, 1983): ρsat c

2 ) (qc rc2 − qtop rtop dTconv . = ρsat Q(t) + 3 3 − r3 dt rtop c

(11)

qc is the heat flowing from the conducting core into the base of the convecting layer: k qc = (Tmax − Tconv ). δ qtop is the flux into the stagnant lid: k qtop = (Tconv − Ttop ). δ

(12)

(13)

7. Thermal evolution (a)

We start with the accretional temperature profile defined above. For each new time step, we check for convection in the layer above the core. We solve the heat conduction equation in the core and in the lid and the energy Eq. (11) in the convecting sub-layer (if it is present). The boundary conditions are ∂T =0 ∂r T = Tmax T = Ttop T = Tamb

at r = 0, at r = rc , at r = rtop , at r = rsat .

(14)

If the layer above the core is convectively stable then Eq. (4) is solved for the entire satellite with the above boundary conditions for r = 0 and r = rsat . Once the locations of the top and bottom boundaries of the convecting layer are known and if the newly calculated Rayleigh number continues to exceed the critical value, the thermal boundary layer thickness and the adiabatic temperature are updated. As a consequence of cooling of the core from above and internal heating by radiogenic decay rc will decrease with time. When rc = 0 and Ra  Racr , Tconv = Tmax . There will then be no lower thermal boundary layer and the heat flow into the base of the convecting layer will be zero.

(b) Fig. 3. Thermal evolution of Mimas for TN = 80 K (panel a) and TN = 250 K (panel b). Temperature profiles are shown at selected times after formation of the satellite. The initial temperature profile is at 0 Ma, the present day temperature profile at 4500 Ma. Solid lines indicate layers of conductive heat transfer, dashed lines indicate convection. Also shown in panel b are the ice I melting temperature profile, the liquidi for 5 and 15% ammonia for an ammonia-water mixture, the temperature at which ice I will start to creep, and the ammonia-dihydrate melting temperature.

8. Results

Table 2 Convection in the mid-sized icy satellites as a function of TN

We present model thermal histories in Fig. 3 for Mimas, Fig. 4 for Tethys, Fig. 5 for Dione, Fig. 6 for Rhea, and in Fig. 7 for Iapetus. Temperature profiles for selected points in time are shown. Dashed line segments indicate regions of convection; solid line segments those of conductive heat transfer. We also show the temperatures at which ice I begins to creep which is about 60% of the ice I melting point temperature (Garofalo, 1965), the ammonia-dihydrate melting temperature after Hogenboom et al. (1997), and the liquidi of H2 O–NH3 for 5 and 15 weight-% NH3 , respectively, calculated using the parameterization of Leliwa-Kopystynski et al. (2002). Ammonia is the most effective suppressant of the melting temperature discussed for icy satellites (e.g., Spohn and Schubert, 2003);

Satellite

TN [K]

t1 [Ga]

t2 [Ga]

Mimas Tethys Dione Dione Rhea Iapetus

250.0 250.0 80.0 122.9 80.0 80.0

0.02 0.01 0.23 0.17 0.33 0.65

0.03 0.35 N/A N/A N/A 2.75

The time convection starts is given as t1 . The convective epoch ends at t2 . The time t2 cannot be given for models that lead to differentiation due to ice I melting.

melting of ammonia-dihydrate is widely accepted to be a possible cause of cryovolcanism and resurfacing. Stagnant lid growth and interior cooling are shown in Fig. 8 for Iapetus. Table 2 lists

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(a)

425

(b)

(c) Fig. 4. Thermal evolution of Tethys for TN = 80 K (panel a), TN = 157 K (panel b) and TN = 250 K (panel c). For further explanation see Fig. 3.

all convecting models and the beginning and the end of the convective epoch. Mimas interior evaded convection and melting if it accreted at TN = 80 K (Fig. 3a). The accretion temperature profile in this small satellite is almost isothermal and rc tends to zero rapidly. If we assume TN equal to 250 K, then convection starts soon after accretion but the phase of convective heat transfer is brief and Mimas’ thermal evolution is dominated by conductive cooling (Fig. 3b). A period of less than 100 Ma of very early melting is predicted in this model if ammonia was present. The initial temperature of 250 K is approximately equal to the liquidus temperature for 15 weight-% ammonia in Mimas. It is possible that the satellite differentiated if it contained that much or perhaps even a smaller concentration of ammonia. From about 100 Ma onward, as this model suggests, the satellite should have been geologically dead. Conductive cooling also dominates the thermal history of Tethys. As was the case for Mimas, Tethys evaded convection if TN is taken to be about 160 K and lower (Figs. 4a and 4b). Both present-day Mimas and Tethys should then be cold, conduct-

ing bodies with interior temperatures between 80 and 100 K. With TN = 250 K, however, convection and melting is possible if ammonia is present (Fig. 4c). Convection lasts for about 350 Ma. It starts early in a spherical shell close to the surface. As can be seen in Fig. 4c, the shell quickly expands at the expense of the heat-trap core. Melting is possible for a time a little longer than the convective period. For small ammonia concentrations of a few weight-%, the deep interior would have contained a partial melt. The depth to the melt zone decreases with time which should have frustrated cryovolcanism since the melt would have had to be transported through a thickening lid. An ocean would have been possible even with small amounts of ammonia while whole satellite melting and differentiation may have been possible for ammonia concentrations in excess of 15 weight-%. Dione with a surface radius of 560 km is big enough and has enough radioactive heating to convect even for TN equal to 80 K (Fig. 5a). Convection starts at about 230 Ma after accretion and comprises the entire deep interior. Melting in the presence of ammonia begins shortly after 100 Ma and shortly after 300 Ma,

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(a)

Fig. 6. Thermal evolution of Rhea with TN = 80 K. For further explanation see Fig. 3. Temperature reaches the ice I melting point after 500 Ma. The model of Rhea is remarkably similar to the model of Dione shown in Fig. 5a.

(b) Fig. 5. Thermal evolution of Dione for TN = 80 K (panel a) and TN = 122 K (panel b). For further explanation see Fig. 3. Temperature reaches the ice I melting point after 350 and 200 Ma, respectively. Since ice I melting will most likely lead to differentiation even in the absence of ammonia, the model calculations are not continued beyond this point in time.

temperature has become sufficiently high to even enable melting of pure ice I. Differentiation is thus almost inescapable for Dione. As our model does not account for differentiation, we abstain from presenting temperature profiles beyond this point in time. If the simple Lunine and Stevenson (1982) radial temperature profile for the saturnian sub-nebula (Eq. (2), Table 1) is assumed, Dione starts to convect a little earlier, about 170 Ma after accretion (Fig. 5b). The onset of differentiation will be roughly 100 Ma earlier than in the model with a TN of 80 K. Using 250 K as TN as suggested by Mosqueira and Estrada (2003) will lead to rapid batch melting and differentiation and is not pursued further in this paper. Convection in Rhea (Fig. 6) starts around 330 Ma with TN = 80 K. As was the case with both models of Dione, differentiation occurs before convection ceases. It takes Rhea 500 Ma to heat above the pure ice I melting temperature. In the presence of ammonia partial melting will begin after about 200 Ma.

Fig. 7. Thermal evolution of Iapetus with TN = 80 K. For further explanation see Fig. 3.

Fig. 8. Stagnant lid thickness (bold line) and temperature of the convecting sub-layer (thin line) for the Iapetus model shown in Fig. 7 with TN = 80 K. The stagnant lid grows rapidly early on but then thins as the temperature in the deep interior rises to reach a minimum thickness at about 1.3 Ga. From there on whole satellite cooling leads to renewed stagnant lid thickening.

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Fig. 9. Accretional temperature profiles for selected values of the retention parameter h and for cool nebulas with ambient temperatures of TN = 80 K (arranged at the bottom of each sub-figure) and for hot nebulas with TN = 250 K (at the top). In each group of profiles, the steepest ones are for h = 1. The parameter h decreases in steps of 0.2 from profile to profile. The least steep profile has h equal to 0.2. Note that h = 1 represents the maximum possible value for this parameter. It is not likely to have exceeded a value of 0.8 (Ellsworth and Schubert, 1983). The steepness also increases with the satellite radius.

We find Iapetus to start convecting after about 650 Ma; the maximum interior temperature is reached at 1.4 Ga (Fig. 7). Convection lasts until 2.7 Ga. Thereafter, the process of cooling becomes entirely conductive. The interior is above the ammonia-dihydrate melting temperature for about 3.5 Ga and reaches the liquidus for 15 weight-% ammonia. The stagnant lid thickness as a function of time is shown in Fig. 8 together with the temperature of the convecting sub-layer Tconv . Initial stagnant lid growth quickly turns into thinning the interior temperature increases. At about 1.3 Ga the lid eventually restarts to grow in thickness to reach 550 km after 2.7 Ga.

9. Variation of parameter values As we have discussed above, some of the parameter values used in our calculations are considerably uncertain. These parameters have been varied over broad ranges; in the following, we will discuss the results of these variations. We first vary the retention factor h. With increasing h, an increasingly larger share of heat is deposited underneath the surface of the satellites during accretion (see Fig. 9). The maximum possible value for h is 1, but h is not likely to have been larger than 0.8 (Ellsworth and Schubert, 1983). We have exam-

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(a)

(b) Fig. 10. Early temperature profiles for Iapetus with TN = 80 K for h = 0.2 (panel a) and h = 1 (panel b). The differences in interior temperatures decrease with time and become largely insignificant after about 1 Ga.

ined both hot and cold profiles except for Iapetus, where a hot profile is highly unlikely. The most pronounced effect results if with large values of h and with sufficiently large values of the nebula temperature the melting temperature of ice I is exceeded as may indeed happen. In that case, differentiation can ensue even without ammonia and continue through a runaway process as has been argued by Friedson and Stevenson (1983). If the temperature remains below the melting temperature, varying h may slightly delay or accelerate the onset of convection but the effect on the long term thermal evolution of the model satellites was found to be small. As an example, we show results for Iapetus in Figs. 10a and 10b for two values of h of 0.2 and 1, respectively. Iapetus was chosen as this satellite proved to be particular perceptive to parameter changes. We show temperature profiles up to 1 Ga. Beyond this time, differences are insignificant. Of course, it is possible that for small satellites or for satellites with little radiogenic heating (e.g., Mimas and Tethys), setting h to very low values can prevent convection altogether.

We studied variations of the thermal conductivity, the specific heat, the primordial chondritic heating rate and the ice I melting point viscosity using models that convect assuming baseline parameters. We use Mimas and Tethys at TN = 250 K and Dione, Rhea and Iapetus at TN = 80 K. We do not include the slightly warmer model of Dione, as results are quite similar. A variation of the thermal conductivity of k from 1 to 7 W m−1 K−1 in the silicate fraction did not result in any appreciable difference in most models with respect to the times of onset of convection, cessation of convection, or onset of melting (compare Fig. 11). There is a minor trend in the satellites for a slightly accelerated onset of convection with rising ksil , but it is not of large significance. In Mimas, convection ceases earlier for higher values of ksil . The same is true for Iapetus. Over the full range of parameter values, this difference amounts to 0.5 Ga. To test for effects of varying the ice thermal conductivity while retaining its dependence on temperature we have applied a pre-factor with a value between 0.5 and 1.5 to Eq. (5). The results are shown in Fig. 12. Onset of convection is only little effected by the variation. The same is true for melting and differentiation in both Dione and Rhea. For the other three satellites, however, there is a pronounced decrease of the length in time of the convective epoch with increasing conductivity. Iapetus does not convect with a kice pre-factor larger than 1.4. We also studied the consequences of varying the specific heat c. Results are presented in Fig. 13 for a range of 700– 1900 J kg−1 K−1 . All models start to convect increasingly later with increasing values of c and do convect increasingly longer. Thus, the thermal evolution appears to progress more slowly with increasing values of c. For Iapetus, there is a bending over of the cessation curve with increasing values of c suggesting that there may be a value of c where convection does not occur at all. This value is likely to be beyond reasonable values of c for all satellites. Varying Q(t = 0) from 25 to 75 pW kg−1 (Fig. 14) does not have pronounced effects for Mimas and Tethys although the convection tends to last longer with increasing initial heat source concentration. For pure ice models of Dione and Rhea, differentiation is avoided if the primordial chondritic heating rate is set to values below 30 pW kg−1 . With these values, Iapetus will not even start to convect. Pure ice models of Iapetus differentiate starting at 60 pW kg−1 . With ammonia, Dione and Rhea will differentiate at even smaller values of Q(t = 0), while for Iapetus the 15 weight-% liquidus will be reached for values of Q(t = 0) of 40 pW kg−1 and above. We note that a variation of Q(t = 0), is equivalent to a variation of the silicate mass fraction f and approximately equivalent to a variation of the rock density ρsil . The basic parameter for the ice viscosity is the melting point viscosity ν0 . As a standard parameter value, we use 1 × 1012 m2 s−1 . We have varied this value from 1 × 109 to 1 × 1014 m2 s−1 and show the results in Fig. 15. The time of onset of convection increases generally with increasing values of ν0 but particularly strongly for satellites that are small or rich in ice. The length in time of the convective decreases with increasing values of ν0 . Mimas will not be convecting at all if ν0

Stagnant lid convection in the mid-sized icy satellites of Saturn

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Fig. 11. Results of a variation of thermal conductivity in the silicate fraction. Shown are results for Mimas and Tethys with TN = 250 K and for Dione, Rhea, and Iapetus with TN = 80 K. The solid lines mark the times of onset of convection while the dashed lines mark the end of the convective epoch and the dotted lines mark ice I melting. The reference value of the thermal conductivity is marked with a vertical line at 4.2 W m−1 K−1 . The figure shows that the silicate thermal conductivity has limited influence on the thermal evolution of the satellites.

is above 1013 m2 s−1 . Since the vigor of convective heat transport decreases with increasing values of ν0, pure ice models of Dione and Rhea will start to melt and differentiate for values close to ν0 = 1 × 1011 . If ν0 is set to 3 × 1013 m2 s−1 , then a pure ice model of Iapetus may also differentiate. The presence of ammonia will reduce these values. For most of our models the Rayleigh number is significantly above critical if convection occurs Therefore a value of β = 0.3 seems appropriate. However, for Iapetus Ra remains close to critical for most of the time. Our calculations show that it never exceeds 6000. This suggests that a value for β of 0.2 (e.g., Mitri and Showman, 2005) may be more appropriate. We have tested our model of Iapetus using the lower value but found that differences were insignificant.

Our model results are also not sensitive to variations of the critical Rayleigh number because Ra is substantially overcritical most of the time. In Dione, for example, the Rayleigh number for the convecting region reaches values around 106 before differentiation starts. Even in Iapetus, Racr would have to be increased by several orders of magnitude above its assumed value to prevent the satellite from convecting. 10. Discussion We have calculated thermal history models for the medium sized satellites of Saturn. We assume that the satellites are initially homogeneous and undifferentiated. Depending on their sizes and densities and their initial temperatures we find peri-

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Fig. 12. Results of a variation of the thermal conductivity in the ice I fraction. The conductivity is varied through applying a constant factor between with a value between 0.5 and 1.5 to kice as calculated from Eq. (5). Shown are results for Mimas and Tethys with TN = 250 K and for Dione, Rhea, and Iapetus with TN = 80 K. The solid lines mark the times of onset of convection while the dashed lines mark the end of the convective epoch and the dotted lines mark ice I melting. The reference value of the thermal conductivity is marked with a vertical line. The effects of the variation are stronger than in the case of a variation of the silicate thermal conductivity shown in Fig. 11. The convective period generally increases in length for values smaller than 1 of the pre-factor. Increasing the value beyond 1 will shorten the length in time of the convective epoch. Convection may be avoided altogether in Iapetus if the ice thermal conductivity is augmented by a factor of about 1.5.

ods of convection in the deep interiors underneath stagnant lids that grow in thickness with time. The dependence on size is simply a consequence of the dependence of convective stability on the thickness of the potentially convecting layer. The dependence on the density is a consequence of the content of silicate rock and heat producing elements in the rock increasing with increasing density above the ice I density. We have compared the temperatures in the interiors with the NH3 –H2 O solidus, the liquidus temperatures for 5 and 15 weight-% ammonia, and the ice melting temperature. The models we present in this paper suggest large scale melting of ice I in Dione and Rhea only.

The models for all satellites except for cold accretion models for Mimas and Tethys indicate partial melting of the ice if the satellites are composed of ice and ammonia. The partial melt may cause cryovolcanism if the melt can penetrate through the lid above to the surface. Because the lid will thicken with time, the best chances for cryovolcanism are early in the histories of the satellites. We can assess the potential for differentiation as a consequence of partial melting in the presence of ammonia with the following consideration: separation of the melt from the solid usually requires some minimum degree of melting. Just cross-

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Fig. 13. Results of a variation of the overall specific heat, c. Shown are results for Mimas and Tethys with TN = 250 K and for Dione, Rhea, and Iapetus with TN = 80 K. Solid lines represent the onset of convection, dashed lines the end of convection, and dotted lines the onset of ice I melting. Baseline values of c for ice and silicates are marked with vertical lines.

ing the solidus will likely not suffice although a possible reduction of the ice viscosity upon partial melting may lead to gradual differentiation driven by the difference in density between rock and ice as discussed for Callisto by Nagel et al. (2004). The minimum degree of melting for rapid separation of melt from solid is uncertain and depends on issues such as the dihedral angle between the melt and the solid grains determined mostly by the surface tension between the liquid and the solid. A very simple rule of thumb that is based on the conventional Roscoe–Einstein theory for the close packing limit of an assemblage of solid spheres suggests a degree of melting of 40%. Using the lever rule the degree of melting can be related to the concentration of ammonia in the melt and to the temperature. For a minimum degree of 40% the lever rule would suggest

an ammonia concentration of 60/40 times the original concentration. Cosmochemical arguments suggest concentrations of ammonia of up to 10% for the saturnian satellites (Hersant et al., 2006). Thus the satellite must be heated to the 15% liquidus or to even higher temperatures. An inspection of Figs. 3–8 suggests that Iapetus may in addition to Rhea and Dione differentiate under these circumstances; Mimas and Tethys only if they accreted in an optically thick nebula with nebula temperatures around 250 K. Models of differentiated satellites have been presented most recently by Husmann et al. (2006) who also show that Rhea may have a present day ocean. We like to point out one important difference, though. While Hussmann et al. assume oceans underneath pure water ice shells our models suggest that the near surface ice layers could be largely primor-

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Fig. 14. Results of a variation of the primordial chondritic heating rate. Shown are results for Mimas and Tethys with TN = 250 K and for Dione, Rhea, and Iapetus with TN = 80 K. Solid lines represent the onset of convection, dashed lines the end of convection, and dotted lines the onset of ice I melting. Models of the smaller satellites are relatively insensitive to changes in Q0 . Ice I melting in Dione and Rhea is evaded with heating rates below about 30 pW kg−1 . Above that value, ice I melting can be observed the earlier the larger the assumed value of Q0 . For primordial heating rates below 35 pW kg−1 Iapetus will not be convecting at all. Increasing the value above 60 pW kg−1 leads to ice I melting in that satellite.

dial. Crystallization of oceans may have led to compositional layering with water ice shells underneath shells of primordial composition. The water ice shells may be underlain by shells of ammonia dihydrate. Of course, other chemical components may complicate the issue but the conclusion of a primordial outer shell is largely independent of specific model assumptions. The surfaces of the satellites and their cratering records have been studied using Voyager images. Better resolution images have been obtained recently for Iapetus (Porco et al., 2005) and are soon to be expected for the other moons from the Cassini mission. The Voyager results have been described by Smith

et al. (1981, 1982) and have been discussed in detail in a series of papers by Plescia (1983) and Plescia and Boyce (1982, 1983, 1985), by Neukum (1985) and by Lissauer et al. (1983). A review of the literature on the cratering of planetary satellites as revealed by Voyager and earlier missions has been given by Chapman and McKinnon (1986). Smith et al. identify two cratering regimes that differ in crater size frequency distributions and have been suggested to be due to two differing populations of impactors. Population I would have preceded population II. Timing is difficult but Smith et al. suggest the end of population I cratering at 4 Ga b.p. The reality of the two populations has been questioned by later authors.

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Fig. 15. Results of a variation of the ice I melting point viscosity. Shown are results for Mimas and Tethys with TN = 250 K and for Dione, Rhea, and Iapetus with TN = 80 K. Solid lines represent the onset of convection, dashed lines the end of convection, and dotted lines the onset of ice I melting. The model of Mimas is most susceptible to changes in ν0 . Increasing ν0 by one order of magnitude above the baseline value leads to entirely conductive thermal evolution histories for that satellite. A similar trend is visible in the Tethys model, but the critical viscosity does not lie within the explored range. Dione and Rhea avoid ice I melting for ν0 less than 1011 m2 s−1 . The epoch of convection for Iapetus becomes shorter with increasing values of ν0 and ice I melting occurs for ν0 values larger than 1014 m2 s−1 .

It is widely agreed that there is little if any resurfacing on Mimas and Rhea. Plescia and Boyce (1982) argue for some resurfacing on these satellites but Lissauer et al. (1983) dismiss it and argue that differences in the cratering between various areas on these satellites are artifacts of differing lighting conditions. Our results show the evolution of Mimas to be basically over after 200 Ma which is consistent with the observation. Our calculations for Rhea suggest that the satellite differentiated which could have resulted in surface modifications. This conclusion supports the point of view expressed by Plescia and Boyce (1982).

Resurfacing is unambiguous and agreed upon on Enceladus, Dione and Tethys. Although absolute age determinations are difficult, it has been suggested (Smith et al., 1981, 1982) that Dione was active beyond the end of the population I epoch. For Dione resurfacing may be a consequence of internal differentiation. For Tethys, resurfacing would argue for warm nebula models such as that by Mosqueira and Estrada (2003). The 157 K of the Lunine and Stevenson (1982) model would most likely not suffice. Iapetus has not been resurfaced. This unanimous conclusion from the older Voyager data has been confirmed by Cassini

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observations (Porco et al., 2005; Neukum et al., 2005). The Cassini images of Iapetus show a rift circling the equator of the satellite. The figure of the planet is flattened to an extent that would be in equilibrium with a much higher rotation rate of 17 h than the roughly 80 day synchronous rotation that is observed today. Castillo et al. (2005) have proposed that the satellite was heated by 26 Al during its early evolution. This would have caused widespread melting and differentiation. The problem of 26 Al heating has been discussed extensively for planetesimals and asteroids (e.g., Merk et al., 2002). The problem is that the half-life for the isotope is short, only 700 thousand years, and that the time of formation of the isotope in a super nova is unknown. Iapetus must have accreted very rapidly, within a few half lives of the time of formation of the isotope for the model to be viable. Our model for Iapetus forms a thick lid—so does Tethys in its brief period of convective heat transfer. Even if convection continues underneath the lid it is somewhat difficult to imagine how the convection could feed resurfacing activity unless in the very first few million or perhaps tens of million years. In comparison with the previous models of Ellsworth and Schubert (1983) and Pollack and Consolmagno (1984) our lids are substantially thicker and the deep interiors of the satellites are considerably warmer. It is not surprising, therefore, that resurfacing on undifferentiated satellites, in particular if it extends for a considerable period of time, is best explained by accepting ammonia as a constituent. Our model calculations show that the ammoniadihydrate melting temperature is reached in all models shown except for Mimas and Tethys with TN = 80 K. This will allow partial melting of the H2 O–NH3 ice and ammonia-dihydrate melt to form. Since the latter is buoyant with respect to the ice–rock mixture it may rise to the surface. Ammonia-dihydrate melt occurring in the lids of our models is another difference to the models of Ellsworth and Schubert. These authors had the lid defined using a bottom temperature that was lower than the solidus temperature of the H2 O–NH3 ice. The most extensively convecting satellites are Dione and Rhea. Both of our models for Dione heat up quickly to reach the ice I melting temperature. So does Rhea, but it takes this satellite a few hundred Ma longer to heat up sufficiently. Our model of Rhea starts to convect after roughly 500 Ma at a time when the suggested population I bombardment had probably already ceased. Although Rhea is greater in diameter than Dione, it is far less dense. We also find a very thick stagnant lid with a thickness of at least 350 km for this satellite at the time convection starts. Acknowledgments We thank R. Wagner for critically reading the manuscript and discussing the cratering records of the saturnian satellites with us and Frank Sohl and Hauke Hussmann for discussions about icy satellites in general. We particularly thank Amy Barr and Gabriel Tobie for thoughtful reviews and valuable suggestions.

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