We model the thermal evolution of Titan's core and search for solutions that are consistent with the mean moment of inertia yielded by the Cassini-Huygens Mission. Like previous studies we assume that Titan's core is enriched in hydrated silicates. However, our modeling accounts for the possible dehydration of these minerals. The resulting models are consistent with Titan's moment of inertia if the inner dry silicate core remains smaller than ∼1300 km in radius. This constraint is met if at least 30% of potassium was leached from the silicate during the hydration event, i.e., the core is depleted in one of its major heat source. In this scenario, the core is currently undergoing dehydration.
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 Multiple gravity flybys by the Cassini orbiter at Titan show that the satellite is mostly relaxed to hydrostatic equilibrium and characterized by a mean moment of inertia MoI = 0.3419 ± 0.0005 [Iess et al., 2010]. We take this value as a reference for this study, but note that these authors quote a value as low as 0.335 in their supplementary material. These authors inferred from that observation that Titan has achieved only limited melting of its ice phase, or, as an alternative, that its core is dominated by low-density hydrated silicates, a model previously proposed by Fortes et al. . Limited melting could be explained by Barr et al.  thanks to the shortage of heat sources, especially accretional energy and short-lived radioisotopes. In this study we explored the alternative scenario in which Titan achieved complete melting of its volatile phase, as suggested by previous models of the satellite's thermal evolution [e.g., Grasset et al., 2000]. Silicate hydration is expected to have been a relatively common process in icy objects affected by ice melting, even at the pressures relevant to large icy satellites [e.g., Ransford et al., 1981]. The long-term stability of hydrated silicates in Titan's large core is in question though. The main caveat of all previous models assuming that Titan's core is dominated by hydrated silicates [e.g., Grasset et al., 2000; Grindrod et al., 2008] is that they neglect the consequences of silicate dehydration when the temperature reaches about 900 K [e.g., Perrillat et al., 2005]. If a large part of the core is subject to dehydration, the resulting density structure may not be consistent with the constraints inferred from the gravity field. This manuscript explores possible evolution models for Titan's core for different initial conditions on the nature of its silicate component, and taking into account silicate dehydration.
2. Density Structure From Geophysical Observations
 We searched for density profiles matching the observed mean moment of inertia following the approach by, e.g., Castillo-Rogez . Because of the size of a putative icy shell, up to 700 km thick, it is necessary to properly model the effect of pressure on the densities of liquid water and ice phases. That calculation has been done by Tobie et al. , which we use as a reference. If Titan's shell is entirely frozen, then the density increases from 0.92 g/cm3 for outer ice Ih to 1.4 g/cm3 for ice VI overlaying the core [Tobie et al., 2006].
 If Titan contains a thick ocean, its density can reach 1.3 g/cm3 at the interface with the high-pressure ice layer and could even be greater if one assumes the presence of solutes and of small silicate particles in suspension [Kirk and Stevenson, 1987]. Thus whether Titan's shell is frozen or contain a thick liquid layer, its density profile is about the same, and its mass fraction 40%.
 Carbonaceous chondrites (Cc) show assemblages of three species of serpentine (antigorite, lizardite and chrysotile), as well as clays, low-density organics, and iron-rich minerals [e.g., Brearley, 2006]. The density of Cc is found to be between 2.4 and 2.9 g/cm3 [e.g., Zolotov, 2009]. In chondritic material iron is primarily in the products of serpentinization, in the form of oxides and sulfides, as the iron-rich ultramafic silicates are very unstable under hydrothermal conditions. The stable form of iron-bearing species at high-pressure and temperature (≥850 K) is primarily in the form of pyrrhotite [Scott et al., 2002]. Part of the iron may also be in soluble form in a putative ocean. For this study, we take a Cc density of 2.75 g/cm3 but track only the evolution of antigorite, which is the most abundant mineral and thus dominates the thermophysical properties of the assemblage.
 Our reference anhydrous silicate density is the average ordinary chondrite density of 3.5 g/cm3 but we considered a range from 3.3 g/cm3 to 3.7 g/cm3. The dependence of the silicate densities on pressure and temperature is:
where K0 is the isothermal bulk modulus and K′0 its derivative with respect to pressure, T0 and P0 are the pressure and temperature at the top of the layer, αav is the volumetric thermal expansion coefficient (Table 1).
Table 1. Thermophysical Parameters for the Endmember Silicate Species Considered in This Modela
Some of which are dependent on temperature T. For mixtures of hydrated and dehydrated silicates, we computed the thermal conductivity of the assemblage from fant kant + (1 − fanh) kanh, and its specific heat capacity by xant Cant + (1 − xanh) Canh, where fi, xi, ki, and Ci correspond to the volume fraction, mass fraction, thermal conductivity, and specific heat, respectively, of antigorite (i = ant) or anhydrous silicate (i = anh).
 The density structures that best match a mean moment of inertia of ∼0.342 require the core to have a radius between 2000 and 2200 km and the inner core of anhydrous silicate to be smaller than 1300 km in radius. The presence of an iron-rich core, less than 500 km in radius, has little affect on the moment of inertia and on the thermal evolution, thus we have chosen not to track that feature in the present study.
3. Modeling Approach
 Experimental investigations of silicate hydration under the high-pressure and low-temperature conditions present in large icy satellites are scarce. However, silicate hydration has been observed in a variety of contexts. It is a very efficient, self-sustained process, responsible for the fully hydrated mineral assemblage of Type I chondrites, which are believed to have formed at temperatures between 273 and 420 K in hydrothermal conditions driven by short-lived radioisotope decay heat [Brearley, 2006]. The rate of serpentinization has been studied experimentally and theoretically at pressures up to 2.7 GPa [e.g., Martin and Fyfe, 1970]. The hydration reaction being surface-controlled, Jakosky and Ahrens  inferred that it can be initiated over a broad range of pressures. Once this reaction has started, the rate at which water molecules diffuse within the silicate crystals is rate-controlled by the temperature. The extent of hydration in a silicate body is then a function of the surface area of material in contact with water, i.e., the surface of defects within the material. From MacDonald and Fyfe  we infer that at 307 K, silicate hydration progresses at a rate of 1 to 80 microns per year. Fine silicate particles in contact with liquid water are likely to be fully hydrated on timescales shorter than the accretion and settling of the core. Serpentinization of larger silicate bodies is also expected to be rapid because that reaction involves large strains and stress resulting in cracks formation and an overall increase of the material permeability as the reaction progresses [MacDonald and Fyfe, 1985]. Since serpentinization is a self-sustained, pervasive process, then the key question is whether or not the thermal conditions in early Titan were suitable for triggering that reaction.
 Recent and ongoing research have been exploring the possibility that outer planet satellites formed within the same timeframe as meteorite parent bodies [e.g., Castillo-Rogez et al., 2007]. As such, these objects would have benefited from the decay heat of 26Al, triggering early and rapid melting of the ice. Barr and Canup  have suggested that the Saturnian system formed after 3.5 My past the production of CAIs. A large satellite like Titan could have accreted on timescales of 105–106 years, during which the impact of 26Al decay heat could have been significant, promoting rapid melting of most of Titan's volatiles.
 In the present study, we assumed that Titan was formed by 4 My after CAIs, with enough accretional heating (15% of the accretional energy) to promote the early melting of a large part of its volume, following the approach previously introduced by Lunine and Stevenson . This approach implies the formation of Titan in a “gas-rich” protosatellite disk. The formation of Titan in a “gas-starved” disk as suggested by Barr et al.  could also yield differentiated models if Titan formed relatively early and rapidly, resulting in the accretion of significant amounts of 26Al.
 Antigorite dehydration occurs in two stages accompanied by the release of water, first with the formation of forsterite and talc, which then combine to form enstatite. Both are endothermic with latent heats equal to 367 kJ/kg and 10 kJ/kg, respectively [Weber and Greer, 1965]. The temperature of these reactions decrease with increasing pressure over the range of pressures relevant to Titan's core. Both reactions span a temperature range from 823 to 973 K at a pressure of 1.5 GPa and 743 to 873 K at Titan's core pressure [e.g., Perrillat et al., 2005]. Relevant thermophysical data are gathered in Table 1.
 We tracked the relative abundances of hydrated and dehydrated silicates as a function of time and updated the thermal properties of the material accordingly (Table 1).
 For the initial concentration in long-lived radioisotopes, we chose a mean ordinary chondrite composition characterized by an average potassium content of ∼800 μg g−1 [Wasson and Kalleymen, 1988], In comparison, Cc present an average potassium content of ∼500–550 μg g−1 [Lodders, 2003]. A similar contrast in the concentrations of other elements very mobile under hydrothermal conditions is notable between these two groups.
 Potential geophysical implications of 40K leaching were first explored by Kirk and Stevenson  for Ganymede, followed by Engel et al.  for Titan. Kirk and Stevenson  assumed 30% of radioisotope present in the ocean, while Glein and Shock  suggested that the extraction of major elements during leaching could be significant, quoting an efficiency of up to 100%, at least in the case of Enceladus. In their modeling of Ceres, Castillo-Rogez and McCord  assumed a mean ordinary chondrite composition for the anhydrous rock and a mean Cc composition for the hydrated silicate component.
 The differentiation of a large icy satellite like Titan is achieved in several stages. Following the end of accretion, a core of ice and rock may remain undifferentiated for a few hundred My until it becomes warm enough for the primordial ice to melt and migrate toward the surface [Kirk and Stevenson, 1987]. The timescale for the core to “overturn” is a function of the melting temperature of the primordial mixture of volatiles and rocks. Assuming pure water ice mixed with rock, the melting temperature (of ice VII) at a pressure of 3–4 GPa is between 400 and 450 K. High-pressure ammonia dihydrate polymorphs would decrease the melting temperature of the primordial core to about 320 K [Loveday et al., 2009]. For an initial temperature of 90 K and a fraction of accretional energy of 15%, that temperature is achieved about 400 My after the end of accretion. Then, the redistribution of material toward a stable density structure may take a few more hundred My to complete [Kirk and Stevenson, 1987].
 In our models, we considered the overturn to be complete by 500 My after the end of accretion, consistent with Lunine and Stevenson . The overturn results in mixing rocky material from various regions of the core, accompanied by hydrothermal circulation, likely to homogenize somewhat the core temperature. The maximum temperature achieved in the rock overlaying the primordial core is primarily a function of the amount of accretional heating. Consistently with several earlier studies, we have assumed the fraction of accretional energy available for heating the material to be small, so that the temperature in that rocky crust at the time of the overturn is only a few hundred degrees warmer than the melting temperature of the primordial core.
 The initial conditions for our modeling are defined by the temperature profile at the time the separation of the rock from the ice was complete. An important feature determining the long-term evolution of the core is the mechanism driving heat transfer. Previous studies assumed the occurrence of convection in the hydrated assemblage of Titan's core at temperatures above 1000 K but did not account for silicate dehydration [e.g., Grasset et al., 2000; Grindrod et al., 2008]. Actually, Scott et al.  pointed out that under high water pressure, hydrated silicate can creep at temperatures as low as 600 K, a situation that should a priori promote the onset of solid-state convection. The Rayleigh number, which determines whether subsolidus convection can become the dominant heat transfer mechanism in a given region, is a function of the contrast in material density across that region. From equation (1) and Table 1 we inferred that the hydrated silicate density increases by ∼3% over the pressure range 1.5–3.5 GPa in the core. The temperature increase over that region would have to be at least 600 K in excess of the adiabatic temperature increase (∼15 K) for thermal expansion to counteract the effect of pressure on density and promote buoyancy. This corresponds to a bottom temperature of at least 900 K, at which antigorite is expected to be dehydrating. As a reference, over the same pressure and temperature ranges, olivine density is expected to decrease by at least 2%. This is the reason why we believe that convection is unlikely in a core dominated by hydrated silicates and should not be assumed a priori.
4. Representative Models Consistent With Titan's Moment of Inertia
 We detail the results of three examples that differ in the assumptions on the initial heat budget. All three models assume that the accretion is complete by 4 My after the production of CAIs and that the accretional heat available to heat the material is 15% of the total accretional energy. The starting time for the modeling is 500 My after the end of accretion. Using these initial conditions, we computed the long-term evolution of the core, leading to the models presented in Figure 1. In Figure 1a no leaching of potassium is assumed. The temperature in the core at the end of the core overturn ranges from 320 to 520 K. Taking one or the other bound as the initial condition for computing the long-term evolution of the core leads to the same result: 65% of the total volume, starts dehydrating after ∼2 Gy, and over a short timescale of ∼500 My. In this core now dominated by anhydrous silicate, the conditions may then become suitable for convection onset (not modeled). However, the moment of inertia corresponding to this model is ∼0.317, and thus significantly departs from the observed value. Dehydration may be avoided if at least 30% of the potassium is removed from the core, as presented in Figure 1b. Dehydration starts about 3.5 Gy after formation and is presently an ongoing process.
 The third model (Figure 1c) is a variant of the latter model, assuming a small inner core of anhydrous silicate following differentiation. Although the region concerned by dehydration may extend up to 1500 km in radius, the material that is fully dehydrated remains limited to the inner 800 km. Accounting for this gradient in density throughout the core, we infer a moment of inertia for that model of ∼0.342.
 Different initial conditions on accretional heating and time of formation affect the timeline of core overturn and dehydration. However, the key parameter determining the duration of the dehydration process is the extraction efficiency of potassium from the rock. Dehydration onset may be delayed until present if the extraction of potassium is ∼30%. If, as suggested by Kirk and Stevenson , 30% of the total mass of silicate were present in the form of thin particles in the ocean, then this sink of all radioisotopes would further limit the thermal evolution of the core. Also, we have assumed that the water of dehydration escapes from the core. However, warm fluids could metasomatize the rock on its way toward the surface [e.g., Facer et al., 2009]. For the pressure and temperature relevant to Titan's core, antigorite may then transform into chlorite, another low-density mineral, very stable at high temperature, and suggested by Scott et al.  as a dominant mineral in the interior of Ganymede.
 Titan's relatively large moment of inertia can be explained by the presence in the core of a thick layer of hydrated silicate. This requires that a fraction of potassium is present in Titan's ocean in order to limit the thermal evolution of the core. Whether that scenario properly represents Titan, or another one assuming limited ice melting is more appropriate, will have to be assessed based on a broader set of observations (e.g., geology).
 Our model results suggest that a large part of the silicate is in the process of dehydrating. Dehydration is a very long process spanning several billion years, primarily a function of the amount of 40K leached as a result of hydrothermal activity. This context supports the suggestion by Niemann et al.  that Titan's atmospheric 40Ar could come from the decay of 40K present in the ocean as a result of leaching. Argon-40 would be stable in the form of clathrate hydrates until thermal destabilization promotes its outgassing. Silicate dehydration that could have been gradually ongoing for the past 1 Gy might be the source for such a heat pulse. This should also encourage very late, even present-day, outgassing of methane to the surface-atmosphere system, as has been suggested by Tobie et al. , but remains to be modeled in the present case.
 This study highlights the importance of better quantifying the extent of hydrothermal activity in icy bodies and characterizing the products of that process and of the subsequent dehydration. This result is relevant to all icy satellites subject to full or partial ice melting, such as Enceladus or Europa. Whether hydrated silicates were supplied by planetesimals or later on, during differentiation, may have consequences on the overall chemistry of the silicates and ocean and implications for the astrobiological potential of these objects.
 The authors are very thankful to Giuseppe Mitri and an anonymous reviewer for their valuable comments that significantly improved this manuscript. JIL's work was financed within the scope of the program Incentivazione alla mobilita' di studiosi straineri e italiani residenti all'estero. JCC's work has been conducted at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with NASA. Government sponsorship acknowledged.