of the nebula. On the other hand, structure I should be the most likely form of clathrate produced in the nebula since H₂S, Xe, CH₄ and CO clathrates are all of structure I (Lunine and Stevenson, 1985). Figure 2.2 also shows that, depending on the considered structure, the Kr/CH₄ ratio in CH₄-dominated clathrate is order of or greater than the value derived from the nebula gas phase composition used in our model. This implies that Kr can be enclathrated at a temperature of ~55 K instead of condensing in the 20–30 K range in the solar nebula (see Fig. 2.1).

## 2.3.2 Formation of Saturn and the circumplanetary disk: Consequences for the delivery of volatiles

### 2.3.2.1 Saturn's formation and envelope enrichment

Favored theories of giant planet formation center around two main paradigms, namely the core accretion model (Safronov, V. S., 1969; Goldreich and Ward, 1973; Pollack et al., 1996) and the gravitational instability model (Cameron, 1978; Boss, 1997). In the frame of the core accretion model, a solid core forms from the accretion of planetesimals and becomes massive enough (~5–10 $M_⊕$) to initiate runaway gravitational infall of a large gaseous envelope in which gas-coupled solids continue their accretion (Alibert et al., 2005b; Hubickyj et al., 2005; Mordasini et al., 2009). This model provides the large amount of heavy elements necessary to explain the supersolar metallicities observed in Jupiter and Saturn via the accretion of planetesimals in their envelopes (Gautier et al., 2001; Saumon and Guillot, 2004; Alibert et al., 2005b; Mousis et al., 2006, 2009c). In the frame of the gravitational instability model, gas giant protoplanets form rapidly through a gravitational instability of the gaseous portion of the disk and then more slowly contract to planetary densities (Boss, 1997, 2005). In this scenario, due to the limited efficiency of planetesimals accretion during the planet formation, its metallicity should be almost equal to that of the parent star (Helled and Bodenheimer, 2010).

By using the sequences of ice formation described in Section 2.3.1, it is possible to predict the abundances of major species (C,N, O, S, P) and noble gases (Ar, Kr, Xe) in Saturn's atmosphere in the context of the core-accretion model (Hersant et al., 2008; Mousis et al., 2009c), and to compare to the observed abundances when available. For instance, adjusting the model to the observed carbon abundance provides constraints on the minimum mass range of ices required in the envelope (estimated between 14.3 and 15.6 $M_⊕$, Mousis et al. (2009c). Such an approach also provides information on the typical composition of the planetesimals in the feeding zone of Saturn, which in turn can be used to constrain the composition of the building blocks of Titan. The analysis of Hersant et al. (2008), based on a simplified description of volatile trapping by clathration and condensation, suggests that the planetesimals that reached Saturn's envelope contained significant amounts of CO₂, NH₃, CH₄ and H₂S, and were depleted in CO and N₂. This

also seems consistent with the composition of Enceladus' plumes, where CO₂, NH₃, CH₄ and H₂S have been detected by the Cassini Ion and Neutral Mass Spectrometer (INMS) (Waite et al., 2009). If this is correct, it would imply that the solids that enrich Saturn's envelope in heavy elements have a composition close to the ones that formed the regular satellites.

### 2.3.2.2 Structure and evolution of Saturn's subnebula

At the beginning of Saturn's formation process, the total radius of the planet equals its Hill's radius, and no subnebula can exist. At the end of the formation process, the cooling rate of the planet's envelope is such that its radius shrinks faster than the disk can supply additional gas. Accretion of gas now proceeds through streamers which eventually collide and form a circumplanetary disk (Lubow et al. 1999). When the radius of proto-Saturn becomes small enough, a subnebula emerges from the contracting atmosphere. This subnebula is fed by gas and solids accreted from Saturn's feeding zone. The accretion rate of gas from the solar nebula to the subnebula is therefore directly driven by the planet's accretion rate of gas, which is derived from giant planet formation models (e.g. Alibert et al., 2005a).

Different models of subdisks have been proposed in the literature (e.g. Canup and Ward, 2002; Mosqueira and Estrada, 2003a; Estrada and Mosqueira, 2006; Sasaki et al., 2010). Here we describe in more details the so-called gas-starved disk model initially introduced by Canup and Ward (2002) and then used by Alibert et al. (2005a) and Alibert and Mousis (2007). Alibert and Mousis (2007) described a model of Saturn's subnebula whose evolution proceeds in two phases. During the first phase, when the solar nebula is still present, the subnebula is hot and dense, and fed through its outer edge (fixed to 1/5 Hill's radius – see Alibert and Mousis (2007) for details), at a rate controlled by Saturn's formation. This justifies a posteriori the use of an equilibrium model as an initial model of the subnebula. This phase lasts until the solar nebula has disappeared and the temporal evolution of the subnebula is therefore directly governed by the last phase of Saturn's formation. The accretion rate as a function of the distance to Saturn is no longer constant and the temperature and pressure conditions of the subnebula decrease rapidly.

Figure 2.3 represents the temporal evolution of the temperature profile within the Saturn's subnebula depicted by Alibert and Mousis (2007). The solid lines are plotted, from top to bottom, at $t = 0$ Myr (corresponding to the time when Saturn has accreted 70% of its final mass), 0.1 Myr, 0.2 Myr, 0.3 Myr, 0.35 Myr, 0.4 Myr, 0.5 Myr, 0.6 Myr and 0.7 Myr. The switch from phase 1 to phase 2 of the subnebula evolution occurs at 0.36 Myr. The dotted line gives the temperature 2000 years after the beginning of phase 2, and illustrates the fast evolution of the subnebula during this stage.
<figure><img src="image_362.png" /><figcaption>**Figure 2.3** Temperature inside the subnebula, as a function of the distance from Saturn (expressed in Saturn's radii $R_{Sat}$) and represented at different epochs of its evolution (figure adapted from Alibert and Mousis (2007) – see text).</figcaption></figure>

### 2.3.3 Delivery of solids and conditions of Titan's formation

Regardless the considered formation mechanism of Saturn's satellites within the subnebula (accretion at their present orbit or migration and growth starting from the edge of the subnebula), the delivery of solids to the disk is an essential step, because it determines the amount of volatiles that can be retained and ultimately incorporated in the interiors of these bodies. Different processes have been advocated (see for instance Estrada et al. (2009) and Canup and Ward (2009) for a detailed discussion): (1) direct transport of small particles into the disk with the inflowing gas, (2) ablation and gas drag capture of planetesimals orbiting the Sun through the gas-rich circumplanetary disk, (3) break-up, dissolution and recondensation of planetesimals in the extended envelope of the forming planet, (4) collisional capture of planetesimals.

Mechanism (1) is valid for micron- to millimeter-sized particles that remain coupled to the subdisk's gas while mechanism (2) requires larger size particles (~1 m) to allow their capture and inward drift due to gas drag. Mechanism (3) is probably not valid for Saturn's subnebula because the D/H measurement performed at Enceladus by the Cassini spacecraft (Waite et al., 2009), which is close to the cometary value, suggests that the building blocks of this satellite were formed in the solar nebula. In contrast, planetesimals formed via this mechanism would have a protosolar D/H ratio. Mechanism (4) applies for gas-poor or no gas environments and may have taken place at epochs subsequent to the subnebula phase. In any case, the relative contribution of each of these mechanisms to the delivery

of solids to the disk is still under debate (see for instance Estrada et al. (2009) and Canup and Ward (2009)). In terms of volatile delivery, each of these processes has very different consequences. In mechanism (1), and to a lower extent in mechanism (2), the particles may preserve the volatile content they acquired in the solar nebula during the condensation sequence, while in mechanism (3) a large fraction of volatile would be likely released when incorporated into the disk. Even in mechanism (4), a significant fraction may be lost during collisions at high velocity.

As a consequence, the volatile content in the building blocks of Titan may be significantly smaller than their content in the planetesimals initially formed in the solar nebula in the vicinity of Saturn. Even for small particles entrained by the gas inflow, they could have been altered once embedded in the subdisk, if they encountered gas temperature and pressure conditions high enough to generate a loss of volatiles during their migration, and if they remained relatively porous, which seems a reasonable assumption. Mousis et al. (2009b), for instance, favored this mechanism to explain the carbon monoxide and argon deficiencies in the atmosphere of Titan. This scenario is based on the fact that CO-dominated clathrate, CO, $N_2$ and Ar pure condensates are not stable at temperatures greater than ~50 K in the gas phase conditions of the nebula or the subnebula (see Fig. 2.1). This implies that if porous planetesimals ultimately accreted by Titan experienced intrinsic temperatures of ~50 K during their migration and accretion in Saturn's subnebula, they are expected to release most of their CO, Ar, Ne and $N_2$ while $CH_4$ and $NH_3$ are expected to remain trapped, or at least to partly re-condense if released. Similarly, if planetesimals are captured by gas drag or collisional capture, the most volatile species, CO, $N_2$, Ne, Ar should be released while the most stable one should be preferentially retained.

Such a volatile loss scenario cannot however account for the deficiency of Titan's atmosphere in Kr and Xe because Kr is significantly trapped in $CH_4$-dominated clathrate (see Fig. 2.2) and Xe is also enclathrated at temperatures higher than ~50 K. However, as is further discussed in Section 2.4.2.2, both Xe and Kr may be sequestred in Titan's icy shell and interior owing to clathration process, partly dissolved in the hydrocarbon seas of Titan (Cordier et al., 2010) or trapped in aerosols (Jacovi and Bar-Nun, 2008), thus potentially explaining their deficiency while predicted to be present in significant amounts in the building blocks.

This scenario seems also problematic regarding the tentative detection of $^{22}$Ne by the *Huygens* GCMS (Niemann et al., 2010). According to this scenario, the building blocks should be strongly depleted in Ne with $^{22}$Ne/$^{36}$Ar << 1, while the ratio estimated in the atmosphere from the *Huygens* GCMS measurements is close to 1 (Niemann et al., 2010). Note, however, that the GCMS detection of $^{22}$Ne is only tentative, and the errors are still very large. In situ measurements of Ne, Ar, Kr and Xe abundances in Saturn's atmosphere by a future probe as well as in surface
sampling on Titan could constrain the origin of their unexpected abundances in Titan's atmosphere. This would help in understanding the process of noble gas trapping in the Saturnian system, thus providing key information on the composition of Titan's building blocks.

Another issue concerns the satellite growth from the solids present in the disk. In a gas-rich disk, the growing solid objects should migrate inward as a result of gas drag for small objects and then of type I and/or II migration for largest objects (~1000 km) (e.g. Estrada et al., 2009; Canup and Ward, 2009). A growing satellite may survive if it grows large enough to open a gap in the disk, limiting the migration (Mosqueira and Estrada, 2003b), or if the gas disk dissipates on a timescale comparable to the satellite formation timescale (Canup and Ward, 2002; Alibert et al., 2005a). In the case of Saturn, only one big moon has survived this migration process.

Nevertheless, several large moons may have existed originally and been lost before the disk dissipates (Canup and Ward, 2006; Sasaki et al., 2010). The difference of satellite architecture between Jupiter and Saturn may thus result from differences in disk evolution and satellite migration (e.g. Sasaki et al., 2010). By considering two different disk evolution/structure models for Jupiter and Saturn, Sasaki et al. (2010) proposed that in the case of Jupiter, the satellite migration may have stopped due to quick truncation of gas infall caused by Jupiter opening a gap in the solar nebula, while the subdisk was still relatively hot. In contrast, in the case of Saturn, the gas infall would have never stopped abruptly, and rather slowly decayed. Most of the moons that formed collided with the planet, and only one last big moon would have survived. This surviving satellite would have formed on long timescales (~Myrs), owing to reduced mass infall. This is obviously only one scenario among others, and several alternative scenarios may be envisioned.

The accretion by Saturn of early formed satellites with sizes analogous to that of Titan may also provide an explanation for the formation of the ring and the inner regular moons. Numerical simulations performed by Canup (2010) suggest that the tidal removal of mass from a differentiated, Titan-sized satellite as it migrates inward towards Saturn may be the origin of the giant planet's ring and inner moons. These simulations show that planetary tidal forces preferentially strip material from the satellite's outer icy layers, while its rocky core remains intact and is lost by collision with the planet. The result of these computations is a pure ice ring much more massive than Saturn's current rings. As the ring evolves, its mass decreases and icy moons are spawned from its outer edge with estimated masses consistent with Saturn's ice-rich moons interior to and including Tethys.

The latter scenario has been further investigated by Charnoz et al. (2011), who confirmed that all the moons interior to Titan's orbit may have been formed by these accretion processes, if Saturn has a sufficiently low dissipa-

tion factor Q (< 2000). This latter condition is required to provide sufficient tidal expansion in order for the satellites to move from the edge of a massive ring system to their present orbital position. They also show that the accretion of the satellites from the outer edge of such a ring system may explain the diversity of rock/ice ratios among the mid-sized moons. Alternatively, the variation in rock/ice fraction among the mid-sized satellites may be attributed to the stochastic nature of late-stage accretion, characterized by giant impacts, as suggested by Sekine and Genda (2011). Unfortunately, the different proposed models lack strong observational discriminants. Estimation of Saturn's Q factor from astrometric measurements is probably one of the most constraining observables. Preliminary analysis by Lainey et al. (2011) indicates that the Q factor may be lower than 2000. Future observations will be needed to confirm this unexpectedly low dissipation factor, which still remains puzzling.

### 2.3.4 Accretion of Titan

There is some analogy between the formation of the terrestrial planets and that of the icy moons. As described above, icy moons probably accreted within the first million years of the Solar System history in circumplanetary disks from aggregates of ice and rock (e.g. Canup and Ward, 2002; Estrada et al., 2009). Among the major icy satellites, Ganymede and Callisto seem to exhibit a strong dichotomy in their degree of differentiation. This has often been attributed to differences in the accretion processes. Titan may represent an intermediate case. However, the presence of its massive atmosphere suggests that Titan formed from different materials than did its Galilean cousins, which may have significant consequences for the final assemblage.

#### 2.3.4.1 Assemblage sequence: Role of impactor distribution

A number of models have been developed for accretional heating of icy moons (e.g. Schubert et al., 1981; Squyres et al., 1988; Coradini et al., 1989; Barr et al., 2010)). Although some ingredients change from one model to another, all these models follow the same approach which was initially proposed by Safronov (1978) and Kaula (1979) for the accretion of the Earth. The main differences between the different models concern the assumption for the impact energy partitioning as well as for impactor size and velocity distributions. The most recent study based on this approach was developed by Barr et al. (2010). In their approach, they assumed that during the satellite growth the impactors are small enough that the impact energy is deposited at shallow depth and is efficiently radiated away. In this condition, they showed that if Titan accreted slowly from numerous cold and small embryos more than 5 Myrs after CAI formation, the melting of the outer layer may have been avoided and the undifferentiated proto-core may extend up to the surface.
<figure><img src="image_363.png" /><figcaption>**Figure 2.4** Assemblage sequence and consequences for thermal evolution of the growing planet</figcaption></figure>

In reality, the size and velocity distribution of the impactors have probably varied during the accretion phase. Two reservoirs of bodies may have contributed to the satellite growth: bodies in orbit around the planet, formed or captured within the circumplanetary disk, and bodies in orbit around the Sun, colliding directly with the growing satellite (e.g. Squyres et al., 1988). The contribution of each of these two reservoirs probably varied as the circumplanetary disk evolved. When the accretion sequence started, centimeter-sized planetocentric particles were probably predominant, while kilometer-sized and larger bodies became more and more frequent during the late-stage of accretion (see Figure 2.4).

The temporal evolution of the size of impactors played an important role in the differentiation of icy satellites since it determined the accretion rate and the volume of the heated region after an impact. The increase of temperature upon impact in the outer region of the satellite is mainly controlled by the efficiency of post-impact cooling. The cooling is determined by radiation at the surface, subsurface thermal conduction as well as local heat advection associated with melting, excavation and impact-induced convection (e.g. Senshu et al., 2002). If the impact rate is too high or if the impact energy is buried at depths larger than a few kilometers, radiation and conduction are not efficient enough to cool the subsurface and the temperature increases.

As noted above, the averaged size of impactors is expected to increase as a function of time during the accretion, and large impactors will be more and more frequent, possibly leading to giant impacts during the late-stage of accretion (Sekine and Genda, 2011). This has major consequences for the efficiency of cooling of the satellite and the possible occurrence of melting during accretion. Heliocentric impactors with high velocities may also become more and more frequent (Agnor et al., 1999) resulting in larger increases of temperature. High velocity impacts may locally

induce significant melting and vaporization (Artemieva and Lunine, 2003, 2005). As the volume of melted material scales with the impact velocity and the impactor volume (Tonks and Melosh, 1992), giant impact events occurring at the end of planetary accretion may generate oceans of melt (e.g. Sekine et al., 2011). Even for impact with moderate velocities, their cumulative effect (especially for large impactors) may lead to progressive increase of temperature up to the melting point of water ice.

### 2.3.4.2 Energy budget during the early stage of Titan's evolution

During the early stages of Titan's history, in addition to impact heating, three main sources of energy may have contributed to the internal thermal budget: radiogenic heating by short-lived radioactive isotopes (mainly $^{26}$Al), tidal heating associated with despinning and viscous heating due to ice-rock separation. In this section, we provide some simple estimates of these sources of energy in order to evaluate their potential impact on the thermal budget of early Titan.

Impact heating results from the deposition of impactor kinetic energy during satellite accretion. It therefore depends on the velocity and the mass of the impactor as well as the way kinetic energy is converted into heat. The impact creates a shock wave that compresses the satellite beneath the impact site. Below the impact site, the peak pressure is almost uniform in a quasi-spherical region, called the isobaric core (e.g. Croft, 1982; Senshu et al., 2002). As shock compression is an irreversible process, the entropy below the impact site increases, leading to a temperature increase. Most of existing shock simulations have been performed at high impact velocity ($> 10 \text{ km.s}^{-1}$, Pierazzo et al. (1997); Barr and Canup (2010)), so that they cannot be used directly to constrain the shock heating in the low velocity regime expected during the satellite growth. The heating and associated temperature increase can however be estimated from energy balance considerations. The temperature increase is determined from the peak pressure within the isobaric core, which is directly related to the impactor velocity, and from the intrinsic properties of the target materials (density, specific heat, thermal expansion, etc.) (e.g. Senshu et al., 2002).

The impact velocity $v_{imp}$ is a contribution of the infinite velocity $v_{\infty} \propto \sqrt{e^2 + i^2}$ with e the eccentricity and i the inclination of the planetary embryos and of the escape velocity $v_{esc} = \sqrt{8/3\pi G\rho R^2}$ of the growing icy satellite with R its radius, $\rho$ its density and G the gravitational constant.

$$v_{imp}^2 = v_{esc}^2 + v_{\infty}^2 \qquad (2.5)$$

The contribution of infinite and escape velocities to the impact velocity can be expressed under the form of the gravitational focusing $F_g$ (Kaula, 1979; Schubert et al., 1981; Barr et al., 2010):

$$F_g = 1 + \left( \frac{v_{esc}}{v_{\infty}} \right)^2 \qquad (2.6)$$
During the early stages of planetary formation, dynamical friction reduces eccentricities and inclinations of growing bodies. Hence, early impacts on Titan probably occurred near the two-body escape velocity (i.e. $v_∞$ was negligible) (Kokubo and Ida, 1996; Agnor et al., 1999).

In this case and assuming that no significant melting occurs when the icy satellite is small, energy balance between kinetic energy and impact heating leads to a local temperature increase $\Delta T_0$ that is proportional to the square of the radius of the growing icy moon (Monteux et al., 2007):

$$ \Delta T_0 = \frac{4\pi}{9} \frac{\gamma}{f(m)} \frac{\rho GR^2}{C_p} \quad (2.7) $$

where $\gamma$ is the percentage of kinetic energy retained as heat deep below the surface, $f(m)$ is the volume effectively heated over the volume of the isobaric core (typically equal to 2.5-3, Pierazzo et al. (1997); Monteux et al. (2007)) and $C_p$ is the averaged heat capacity (typically varying between 1000-1500 J.kg$^{-1}$.K$^{-1}$ for ice-rock mixtures). As long as the radius of the growing satellite is below 1000 km, the local temperature increase is less than 10-15 K after each impact. Above 2000 km, the increase of temperature is larger than 50 K.

Radiogenic heating by short-lived radioisotopes ($^{26}$Al, $^{60}$Fe, $^{53}$Mn) may also significantly contribute during the accretion period. For simplicity, we consider here only $^{26}$Al, which is the principal contributor (e.g. Robuchon et al., 2010; Barr et al., 2010). The increase of temperature due to the decay of $^{26}$Al at a radius $r$ depends on the time, $t_f$, at which the layer at radius $r$ has been formed and it is given by (e.g. Barr et al., 2010):

$$ \Delta T_{26}(r) = \frac{m_r q_{26}(0)}{C_p} \frac{\tau_{26}}{\ln 2} \exp \left( -\ln 2 \frac{t_f(r)}{\tau_{26}} \right), \quad (2.8) $$

where $m_r$ is the rock mass fraction in the accreting material (around 0.5 in the case of Titan), $\tau_{26}$ is the half-life time of $^{26}$Al ($\tau_{26} = 0.716$ Myr) and $q_{26}(0)$ is the initial $^{26}$Al heating rate at the time of the calcium-aluminium-rich inclusions (CAI's). $t_f$ denotes the time after the formation of CAIs. The initial heating rate is determined by the specific heat production of $^{26}$Al $H_{26}(0)=0.341$ W.kg$^{-1}$. (Robuchon et al., 2010), the Al content in rocks, [Al] and the initial $^{26}$Al/$^{27}$Al. Assuming a rock composition similar to CI chondrites, [Al] = 0.865 wt% and initial $^{26}$Al/$^{27}$Al of $5.85 \times 10^{-5}$ (Thrane et al., 2006; Barr et al., 2010) leads to $q_{26}(0) = 1.725 \times 10^{-7}$ W.kg$^{-1}$. This is about 20 million times larger than the radiogenic heating rate due to the decay of long-lived isotopes expected at present within Titan. However, this huge heat production rapidly decays. For an accretion time of 2 Myr after CAI formation, the temperature increase due to $^{26}$Al decay would be of the order of 100 K, but it becomes less than 10 K for accretion times above 5 Myrs. Radiogenic heating by short-lived radioisotopes may have been of importance for the first materials to accrete, i.e. for the materials that formed the inner part of the satellite. By contrast, impact heating be-

comes significant for layers above a radius of about 1000 km.

An additional source of heating corresponds to dissipation due to enhanced tidal friction before the satellite reaches synchronous rotation. It is indeed likely that Titan has an initial spin period much shorter than its present-day period of 15.95 days. Initial spin periods as low as 6-10 hours are possible (Lissauer and Safronov, 1991; Robuchon et al., 2010). In this condition, the tidal bulge raised by Saturn rotates relative to the satellite body frame. Owing to the anelastic properties of the interior, the fast-rotating satellite does not respond perfectly to the tidal forcing, resulting in a misalignment of the tidal bulge with respect to Saturn's direction. This acts as a torque that tends to slowdown the satellite spin, until the satellite is locked in spin-orbit resonance. The gradual decrease of the spin rate $\omega$ as a function of time $t$ is determined from the tidal torque acting on the satellite and is given by:

$$ \frac{d\omega}{dt} = \frac{3 k_2 G M_S^2 R^5}{2 D_{ST}^6 Q C}, \quad (2.9) $$

with $k_2$ tidal Love number, $M_S$ Saturn's mass, $D_{ST}$ mean distance between Titan and Saturn, $Q$ dissipation factor of Titan and C its polar moment of inertia ($C \simeq 0.4 \times M_T R^2$ at the time of accretion). As the despinning rate scales with $R^5$, it rapidly increases when the satellite reaches its final size. For $k_2 \simeq 0.07$, the expected value for an homogeneous rigid interior (Sohl et al., 1995) and $Q=100-500$ and initial rotation period of 10 hours, the time needed to reach the 16-days period range between 2 and 10 Myrs, suggesting that despinning is achieved shortly after the accretion. Note that once large-scale melting occurs, a global liquid layer may form, thus increasing the Love number and the despinning rate, so that the current period is reached even faster. The total amount of energy dissipated during the despinning stage is given by the difference of body rotation energy between the initial and final periods and leads to the following temperature increase:

$$ \Delta T_{desp} = \frac{1}{2} \frac{C(\omega_0^2 - \omega_c^2)}{C_p M_T}, \quad (2.10) $$

with $\omega_0$ and $\omega_c$ the initial and current angular rotation rates, respectively, and $M_T$ Titan's mass. For initial periods varying between 10 and 6 hours, the averaged temperature increase would range between 25 and 75 K, respectively. The localization of the dissipation is very sensitive to the internal structure (Tobie et al., 2005a). For a homogeneous interior, the dissipation will rather uniform, with a maximum of dissipation at about mid-radius. For a differentiated structure, the dissipation will be mainly located in the outer regions (liquid water and ice I) and would be more efficiently transported to the surface.

Ice-rock separation, which may be triggered already during the accretion phase, may also significantly contribute to the energy budget. The total increase of internal temperature due to ice-rock differentiation, $\Delta T_{diff}$, can be estimated from the change of gravitational energy, $\Delta E_g$, between a homogeneous interior and a differentiated interior
consisting of a rock core and a thick H₂O mantle (e.g. Friedson and Stevenson, 1983):

$$ \Delta E_g \approx \frac{3 G M_T^2}{5 R_T} \left[ 1 - \left(\frac{\rho_r}{\bar{\rho}}\right)^2 X_c^5 + \left(\frac{\rho_i}{\bar{\rho}}\right)^2 (1 - X_c^5) + \frac{5}{2} \frac{\rho_i}{\bar{\rho}} \left(\frac{\rho_r}{\bar{\rho}} - \frac{\rho_i}{\bar{\rho}}\right) X_c^3 (1 - X_c^2) \right] \quad (2.11) $$

with $\rho_r$ the density of rock, $\rho_i$ the averaged density of ices and liquid water, $X_c$ the ratio between the rock core radius, $R_c$ and Titan's radius, $R_T$. For $\rho_r = 2700 \text{ kg.m}^{-3}$ and $\rho_i = 1250 \text{ kg.m}^{-3}$, $X_c = 0.75$ and $\Delta T_{diff} = \Delta E_g/c_p M \approx 100-150 \text{ K}$ (for $c_p = 1000-1500 \text{ kg.m}^{-3}$). If the differentiation process is slow (> 1 Gyr), the convective heat transfer should be able to transport this additional energy. If the ice-rock separation is fast enough (< 0.5 Gyr), the dissipation of potential energy may induce runaway melting and thus may create a catastrophic differentiation (Friedson and Stevenson, 1983; Kirk and Stevenson, 1987). In particular, giant impact events occurring at the end of planetary accretion (e.g. Sekine et al., 2011) may generate locally large volume of melts, leading to the formation of rock diapir migrating toward the center (Figure 2.4). The viscous heating associated to the migration of large rock diapirs may generate significant melting in the surrounding ice, potentially triggering differentiation runaway in a way similar to what is expected in terrestrial planets (e.g. Monteux et al., 2009; Ricard et al., 2009).

### 2.3.4.3 Possible post-accretional interior structure

Depending on the conditions under which Titan accreted, different post-accretional interior structures are possible just after accretion as illustrated in Figure 2.5. If the accretion is very slow (> 1 Myr) and the impactors remain small all along the accretion period (< 100 m), the surface should be able to cool down between successive impacts, and differentiation may be avoided. A fully undifferentiated structure also requires that the accretion starts about 3-4 Myr after the CAI formation, in order to limit the increase of internal temperature due to short-lived radiogenic elements, and that the initial spin period of Titan was larger than 8-10 hours. In such a cold accretion case, no significant atmosphere would be generated, and the formation of a significant atmosphere should occur later during the satellite evolution (for instance during the LHB event).

If the growing satellite is hit by larger impactors at the end of accretion, a surface water ocean may be generated ($R > 2300-2400 \text{ km}$) (Figure 2.5, middle). Even if only 10-20% of the total volume of ice melts, release of gases (mainly CO₂ and CH₄, Tobie et al. (2011)) trapped in the ice phase may generate an atmosphere up to 5-10 bars. Even if a global ocean is generated, most of the mass of Titan would still be undifferentiated. Finally, if the accretion is faster and/or if the impactors are large even during the earliest stage of accretion, ice melting rapidly occur during

the satellite growth. In this case, a very massive ocean in equilibrium with a very massive atmosphere ($P > 50 \text{ bars}$) would be generated. Rock blocks resulting from the melting of icy impactors would sediment at the base of the ocean forming a thick layer of unconsolidated rock (Kirk and Stevenson, 1987; Lunine and Stevenson, 1987). The subsequent evolution of these different possible initial structure is discussed in Section 2.4.1.

### 2.3.5 Summary

At the start of this Section we posed three questions; we will conclude by providing some provisional answers.

1. What was the composition of the primordial building blocks that built Titan and the Saturnian system? Most of the solids that formed the Saturnian system probably originate from the surrounding solar nebula. They are predicted to be composed of rock and ice and to contain significant amounts of volatiles (NH₃, CH₄, CO₂, H₂S) trapped in the form of hydrate and pure condensate. The apparent deficit in primordial CO, N₂, Ar in Titan relative to proto-solar composition may be explained by partial devolatilization of building blocks during their migration in Saturn's subnebula.

2. What make the Jupiter and Saturn systems so different? The difference in satellite architectures may be explained by different evolutions of the circumplanetary disk. In the case of Jupiter, the feeding of the disk may have stopped abruptly, while in the case of Saturn, the satellite accretion in the disk may have lasted longer owing to reduced mass infall. During the accretion process, several Titan-size moons may have been formed and lost by migration into proto-Saturn. Titan would be the only surviving big moon of Saturn, while Jupiter was able to retain four big ones.

3. Can Titan accrete undifferentiated?

It seems extremely difficult to avoid ice melting during the accretion phase, owing to combined effects of impact heating, radiogenic heating by short-lived isotopes and tidal despinning. Melting may be prevented only if Titan accreted slowly (> 1 Myr) from a swarm of small building blocks (< 100 m). Even for faster accretion with larger impactors, large-scale melting is only reached toward the end of the accretional growth, so that the inner part of the interior structure (< 1500 km) remains undifferentiated. In any case, a totally undifferentiated structure after accretion is not needed to explain the present-day moderately differentiated structure.
<figure><img src="image_364.png" /><figcaption>**Figure 2.5** Possible evolution scenarios for the interior of Titan for different initial states. Depending mostly on the efficiency of heat transfer in the interior, different bifurcations in the evolutionnary path may have occurred.</figcaption></figure>
## 2.4 COUPLED EVOLUTION OF THE ATMOSPHERE, SURFACE AND INTERIOR OF TITAN

During and after the accretion, exchanges between the interior, surface and atmosphere are likely to have occurred. The icy shell has probably played a crucial role all along Titan's evolution by storing and transferring volatile compounds from the internal ocean to the surface. These exchanges have probably influenced the composition of the atmosphere during most of Titan's evolution, and they are maybe still operating at present. In this section, we discuss the coupled evolution between the different envelops of Titan, and try to address the three questions listed below.

1. To what extent did the interior of Titan influence the evolution of its atmosphere? Evidence for exchange with the interior is provided by the detection of a significant amount of $ ^{40}Ar $ in the atmosphere. However, the degree and timing of internal outgassing still remain unconstrained. In this section, we review the different internal processes that may have occurred all along Titan's evolution and discuss their implications for the evolution of the atmosphere.

2. Did Titan always possess a massive $N_2-CH_4$ atmosphere? The composition and the mass of Titan's atmosphere have probably varied since the satellite accretion. Depending on the initial state acquired during the accretion and on the efficiency of internal outgassing, we will discuss the different evolution scenarios that can be envisioned.

3. Where is the missing hydrocarbon inventory? Hydrocarbon lakes has been identified at the surface of Titan (Stoferl et al., 2007). However, the estimated volume is much smaller than what is expected from continuous photolysis of methane since the formation of Titan (Lorenz et al., 2008a). This suggests that either a large fraction of hydrocarbon products has been removed from the surface or that the photochemical production has not been constant through time. We will discuss the implications of these two different hypotheses for the long-term evolution of Titan.

### 2.4.1 Differentiation and long-term evolution of the interior

#### 2.4.1.1 Ice-rock segregation and formation of a rocky core

As already discussed in Section 2, the moment of inertia ($C/M_T R_T^2 = 0.33-0.34$) inferred from Cassini gravity measurements suggest that Titan's differentiation was not complete. Titan probably does not have an iron core, and a layer consisting of a rock-ice mixture may still exist between a rocky core and a high-pressure ice layer. This possible uncompleted ice-rock separation has been proposed to be the result of cold accretion process (Barr et al., 2010). How-

ever, limited differentiation requires not only slow accretion but also implies that the convective heat transfer was efficient enough to remove heat from the interior (Friedson and Stevenson, 1983; Mueller and McKinnon, 1988; Nagel et al., 2004). This is possible if the viscosity of the ice-rock mixtures is sufficiently low. Nagel et al. (2004) showed that, in the case of Callisto, if the average size of rocks was of the order of meters to tens of meters, the interior may have experienced a gradual, but still incomplete unmixing of the two components. Significant lateral heterogeneities among the ice/rock mixtures existing after accretion could also have enhanced the differentiation processes (O'Rourke and Stevenson, 2011). In these conditions, convective motions associated with ice-rock unmixing processes may be efficient enough to avoid melting. Following this scenario, the interior of Titan would be relatively similar to Callisto, but with a thinner layer of ice-rock mixed materials, as suggested by its smaller MoI factor.

Nevertheless, uncompleted ice-rock separation is not necessarily needed, as the MoI factor may also be consistent with a highly hydrated silicate interior, possibly dominated by antigorite serpentine (Castillo-Rogez and Lunine, 2010; Fortes, 2011). The formation of a highly hydrated core may result from a rapid differentiation process, implying large-scale melting and efficient water-rock interactions (e.g. Fortes et al., 2007; Castillo-Rogez and Lunine, 2010; Fortes, 2011). The combined effect of accretional heating, radiogenic decay by short-lived isotopes, tidal heating associated to despinning, and viscous heating associated with negative rock diapirs may increase the internal temperature above the melting point of ice. The presence of freezing point depressing agents such as ammonia or methanol (Lunine and Stevenson, 1987; Grasset and Sotin, 1996; Grasset and Pargamin, 2005; Deschamps et al., 2010) would further favor the generation of melts as they strongly decrease the melting point of ice. The aqueous melt, which is less dense than ice-rock mixtures, at least at pressure above 100 MPa, will percolate upward, thus interacting with surrounding rock particles. During their ascent, aqueous solutions in proto-Titan may have been able to interact chemically with silicate minerals (Engel and Lunine, 1994) and with other substances in solution, such as sulfates (Fortes et al., 2007). Interactions of aqueous fluids with initially anhydrous silicate minerals at low temperature may result in the formation of hydrated silicate minerals, called serpentine (Fortes et al., 2007; Castillo-Rogez and Lunine, 2010; Fortes, 2011). The hydrated minerals may remain stable as long as the rocky core temperature remains below approximately 900 K (Grindrod et al., 2008; Castillo-Rogez and Lunine, 2010). Above this temperature, dehydration would be expected, releasing warm supercritical water fluids which may interact again with overlying rocks during their ascent (Castillo-Rogez and Lunine, 2010).

As pointed out by Iess et al. (2010), the main challenge with this hypothesis is to keep most of the core hydrated (i.e. < 900 K until present). Thermal convection is unlikely to start in a cold core dominated by hydrated silicates
(Castillo-Rogez and Lunine, 2010). As thermal diffusion is not efficient enough to extract heat, the core temperature should rise up to the dehydration temperature. The temperature increase can however be limited if the core is depleted in radiogenic elements. Castillo-Rogez and Lunine (2010) showed that if at least 30% of potassium are leached into the ocean from the silicate minerals during the earlier hydration event, the internal temperature would slowly increase and the outer part of the core would remain hydrated. As shown by Castillo-Rogez and Lunine (2010), a rock core consisting of only antigorite serpentine is not required to explain the present-day moment of inertia. An inner anhydrous core with radius up to 1300 km overlaid by a hydrated mantle is compatible with the MoI factor of 0.33-0.34 (see Figure 2.5 bottom right). The scenario of a fully hydrated core initially investigated by (Fortes et al., 2007) must be considered as a very extreme case, maybe not fully realistic. Nevertheless, the idea that a large fraction of the core remain hydrated during most of Titan's evolution should be considered as a possible scenario. More complete chemical and thermal evolution models are probably needed in the future to test in more detail this hypothesis.

As illustrated on Figure 2.5, different evolutionnary paths are possible, some of them combining both solid-state un-mixing and ice melting processes. For instance, for an initially warm and gravitationally unstable interior (Fig. 2.5, bottom), the rock core may form rapidly owing to run-away melting of the undifferentiated proto-core, resulting in a catastrophic core overturn (Kirk and Stevenson, 1987), or more slowly with progressive ice-rock separation and limited ice melting (e.g. Estrada and Mosqueira, 2011). By contrast, an initially cold undifferentiated interior may slowly differentiate with no melting and no formation of ocean during the entire evolution. Any intermediate evolution is also possible.

### 2.4.1.2 Water-rock interactions in Titan's conditions

On Titan, the crystallization of the internal liquid water results in the formation of a low-pressure ice layer at its top and of a high-pressure ice layer at its base. On smaller icy moons, such a high-pressure ice layer does not exist due to reduced pressure (high-pressure phases only exist above about 200 MPa). On Europa and Enceladus for instance, the internal water ocean is permanently in contact with the rocky core, thus favoring water-rock interactions all along the satellite evolution. By contrast, on Titan, a thick layer of high-pressure ice rapidly forms during the interior evolution (Figure 2.5). The water ocean is expected to be in direct contact with underlying rocks only during a brief period of time (< 100 Myr) just after accretion, and only in the case of relatively warm accretion (Figure 2.5).

However, even if there is no direct contact between the water ocean and the rocky core, water-rock interactions may have occurred at different periods during Titan's evolution, especially during the differentiation. As discussed in

Section 2.3.1, non-water ice components such as CH₃OH, NH₃, CO₂, H₂S are likely to have been incorporated inside Titan. In this condition, eutectic solutions at temperatures well below 273 K would have formed (e.g. Sohl et al., 2010). These highly concentrated eutectic solutions should facilitate dissolution of mineral phases and chemical reactions. In the case of ammonia-rich fluids, Engel and Lunine (1994) indicated that NH₄⁺-ions contained in the aqueous solution may efficiently replace ions such as K⁺, Na⁺, Rb⁺ and Ca²⁺ contained in the mineral phase. Ammonia-rich liquids are also expected to react with magnesium sulfate, contained in the chondritic phase and with carbon dioxide brought in the ice phase to form ammonium sulfate and ammonium carbonate, respectively (Kargel, 1992; Fortes et al., 2007).

During the migration of the enriched fluids toward the surface, their composition should vary as new elements are extracted from primary minerals and other precipitate in secondary phases (e.g. Zolotov, 2010). In turn, the circulation of aqueous solutions leads to oxydation and hydration reactions, analogous to aqueous processes on chondrite parent bodies (e.g. Sohl et al., 2010). These reactions are well documented at low pressure conditions and have been used for instance to constrain the possible composition of Europa and Enceladus' ocean (e.g. Zolotov, 2007; Zolotov and Kargel, 2009), but they are still poorly known at the high pressure range of Titan's proto-core (> 1 GPa). Only one study by Scott et al. (2002) tried to investigate the aqueous alteration of chondritic materials at high pressure. However, Scott et al. (2002) did not take into account the probable presence of ammonia, methanol or other compounds, which may significantly affect the chemical equilibrium. Future experimental and thermodynamical investigations will be needed to better understand the complex chemistry that probably occurred during the differentiation.

After the complete separation of rock and ice, water-rock interactions are limited to the ice-rock interface at the base of the high-pressure layer. Even though water is mostly in the form of high-pressure ice phase at this interface owing to the elevated pressure (> 0.8 – 1 GPa), liquid water may locally exist and potentially circulate in the upper part of the rock core. In order to efficiently transfer the heat coming from the rocky core through the high-pressure ice layer, the temperature profile in the high-pressure ice is probably close to the melting curve of water ice (e.g. Sotin and Tobie, 2004). In this context, the formation of a transient liquid reservoir at the rock-ice interface is likely. These liquids may potentially interact with the underlying rocks before percolating upward to the ocean. Warm liquid water may also be released from the core when the dehydration temperature is reached (Castillo-Rogez and Lunine, 2010). This hydration event may occur relatively late during Titan's evolution (> 3–3.5 Gyr after formation), provided sufficient amounts of radiogenic potassium were leached from the core and that thermal diffusivity in the core is not too low (Castillo-Rogez and Lunine, 2010) (see Section 2.4.1.1). In summary, in warm evolution scenarios, intensive water-rock interactions
were likely during two main sequences of Titan's evolution: before 1 Gyr during the differentiation and after 3-3.5 Gyr during core dehydration. Moreover, reduced water-rock interactions may also exist all along Titan's evolution owing to circulation of high-pressure ice melts at the rock-ice interface.

It has been proposed that water-rock interactions may lead to efficient production of methane from carbon dioxide, carbon grains or organic matters by serpentinization (Atreya et al., 2006). However, this mode of production seems incompatible with the D/H ratio observed at present in the atmospheric methane, which is about three times greater than the value expected in hydrothermally produced methane (Mousis et al., 2009a). It has also been suggested that the oxidation of primordial NH₃ to N₂ may also occur in hydrothermal systems on early Titan (Glein et al., 2009), as well as on early Enceladus (Matson et al., 2007; Glein et al., 2008). The endogenic production of N₂ from NH₃ is possible only if the hydrothermal systems were sufficiently oxidized with C existing predominately in the form of CO₂, not CH₄. It also implies that primordial NH₃ was enriched in ¹⁵N in order to explain the elevated ¹⁵N/¹⁴N observed in atmospheric N₂. Although water-rock interactions and associated chemical reactions are thermally plausible, there are essentially no constraints available to assess their likelihood. For instance, constraints on the possible primordial value of ¹⁵N/¹⁴N in NH₃ or D/H in water are still missing. Future measurements in comets by Rosetta for instance will bring new bear on these ideas.

Circumstantial evidence for water-rock interactions is, however, provided by the detection of ⁴⁰Ar in the atmosphere (Niemann et al., 2005; Waite et al., 2005; Niemann et al., 2010). ⁴⁰Ar is one of the decay products of ⁴⁰K initially contained in silicate minerals (e.g. Engel and Lunine, 1994). The mass of ⁴⁰Ar in the atmosphere is estimated between 4.0 and $4.6 \times 10^{14}$ kg (Niemann et al., 2010), which corresponds to about 7.5-9% of the total mass that may be contained in the rocky core, assuming a CI chondritic composition (Tobie et al., 2011). As a comparison, on Earth, Mars and Venus, the fraction of ⁴⁰Ar outgassed from the silicate mantle is estimated to about 50% on the Earth (Allègre et al., 1996) and about 25% on Mars and Venus (Hutchins and Jakosky, 1996; Kaula, 1999). The outgassing process is probably very different on Titan. On Earth and terrestrial planets, ⁴⁰Ar is progressively outgassed from the silicate mantle through volcanism (e.g. Xie and Tackley, 2004). ⁴⁰K decays into ⁴⁰Ar in the silicate minerals and the latter is released when the minerals are brought near the surface by convective motions and melt. On Titan, as suggested by the gravity data, Titan's rocky core is probably relatively cold during most of its evolution (< 1000 K, Castillo-Rogez and Lunine (2010)), therefore it has probably never reached magmatic temperatures (at least on a global scale). Moreover, even if silicate melting has occurred, outgassing of erupting silicate magmas should be limited due to the high pressure conditions expected at the

surface of the rocky core (1 ? 1.5 GPa) (McKinnon, 2010).

On Titan, the extraction of ⁴⁰Ar is more likely the result of a three step mechanism. ⁴⁰K may have been extracted first from the silicate minerals by water interactions (e.g. Engel and Lunine, 1994). Then dissolved ⁴⁰K⁺ ions in the ocean would have decayed into ⁴⁰Ar. Finally, the ⁴⁰Ar contained in the ocean would be extracted through direct outgassing of the ocean or progressive incorporation in the outer shell in the form of clathrate hydrate and then destabilization near the surface by cryovolcanism (e.g. Tobie et al., 2006) or impact (e.g. Barr et al., 2010). Assuming that 30% of potassium has been leached into the ocean implies that about 25-30% of the produced ⁴⁰Ar has been outgassed through the outer icy shell, which seems a reasonable outgassing efficiency.

The presence of significant amount of ²²Ne in the atmosphere, if confirmed, may also be indicative of water-rock interactions and efficient outgassing processes (Tobie et al., 2011). Indeed, a chondritic core may contain ²²Ne in sufficient amounts to explain the tentatively observed abundance in the atmosphere (Niemann et al., 2010). As further discussed in Section 2.4.1.4, neon is the least stable gas compound in the form of clathrate and the less soluble in liquid water, and therefore it is expected to be very efficiently extracted during methane-driven outgassing events. The accurate determination of the abundance of the different isotopes of Ne (²⁰Ne, ²¹Ne and ²²Ne) by a future mission will permit confirmation of the tentative detection of neon by *Huygens*, and it will provide pertinent tests on the origin of neon and on its extraction mechanism from the interior.

### 2.4.1.3 Volatile storage in the H₂O mantle

A major issue on Titan concerns the storage and transport of gas compounds in the H₂O mantle, which constitute about 60-70% of the satellite volume. The amount of gas compounds that can be stored in the ocean and in the ice layers is determined by their solubility in water and their stability relative to the clathrate phase. NH₃ and CH₃OH, for instance, are highly soluble in liquid water and do not form clathrate structures. At low temperature (T < 177 K), ammonia combines with water molecules to form ammonia hydrate (not clathrate) and methanol solidifies at relatively similar temperatures. Such low temperature conditions can be reached only near the cold surface, and both ammonia hydrates and methanol ice become unstable below a depth of 5-10 km (depending on the subsurface thermal gradient). As a consequence, ammonia and methanol should be nearly exclusively stored dissolved in the internal ocean.

Although the solubility of most of the other gases, for instance methane, in liquid water is low, large quantities of gas can be stored dissolved in the liquid water as the ocean is very massive on Titan and it is confined at relatively elevated pressures (> 25 MPa). For example, more than
<figure><img src="image_365.png" /><figcaption>**Figure 2.6** Possible present-day structure of Titan's interior and estimated volatile distribution within the thick H₂O mantle. The radius, temperature and pressure of each interface are only indicative.</figcaption></figure>

2000 times the present-day mass of atmospheric methane may be stored dissolved in the ocean (Tobie et al., 2011).

Depending on their stability relative to the clathrate phase, a fraction of the gases dissolved in liquid water can however be incorporated in the form of clathrate hydrate (Choukroun et al., 2010; Tobie et al., 2011; Choukroun et al., 2011). The most stable species expected in the ocean, such as H₂S and Xe, should be preferentially incorporated in the form of clathrate hydrate, whereas the least stable species such as CO and Ne remain mainly in the liquid phase (Tobie et al., 2011). The fraction incorporated in the clathrate phase should also increase as the ocean cools down and crystallizes. Clathrate hydrates which have a composition dominated by H₂S have a density larger that liquid water, so that they are expected to sink to the base of the ocean. As an example, clathrate structures containing 75% of H₂S, 20% of CO₂ and 5 % of CH₄ have a density of about 1050 kg.m⁻³. These H₂S-rich sinking clathrates will also preferentially incorporate Xe owing to its very high stability in clathrate phase. As a consequence, H₂S and Xe are expected to be mostly stored in the form of clathrate hydrate within the high-pressure ice layer.

During the clathration process, the dissolved fraction of H₂S progressively reduces, and CH₄ and CO₂ eventually become the dominant clathrate former. When CH₄ becomes the dominant clathrate former gas, the clathrate density gets lower than the density of liquid water, and clathrate hydrates accumulate at the top of the ocean. Differentiation of Titan's interior during the early stage of its evolution may also lead to massive release of methane-rich clathrate hydrates, and hence accumulation in the outer shell above the ocean (Tobie et al., 2006). Both early differentiation and

late ocean crystallization may result in the accumulation of methane-rich clathrate hydrates in the outer shell. However, even in these conditions, the total thickness of methane-rich clathrates incorporated in the outer ice shell should not exceed 5-10 km, corresponding to about 20% of the total mass of methane that is potentially contained in the interior (Fig. 2.6, Tobie et al. (2011)). A fraction of CO₂, CO and Ar should also be incorporated in the outer shell in the form of clathrate, but to a lesser extent than methane due to differences in solubility and clathrate stability.

#### 2.4.1.4 Volatile release from the H₂O mantle

Even though the interior may potentially contain large quantities of volatiles, their release into the atmosphere is limited owing to their solubility in the internal ocean and owing to clathration processes within the H₂O mantle. Outgassing from the interior may occur through two different mechanisms: (1) destabilization of clathrates stored in the outer ice shell by thermal convective plumes or by impacts, (2) mobilization of dissolved gas molecules contained in the water ocean and subsequent bubble formation if the ocean gets saturated (Tobie et al., 2006, 2009). Both processes are controlled by methane, since it is the dominant gas in the form of clathrate hydrate in the outer shell and since it is the only gas compound that may reach the saturation point in the ocean during periods of reduced ice shell thicknesses. The other dominant compounds, CO₂, NH₃ or H₂S, are more soluble in liquid water, so that they will never reach saturation. For noble gases such Ne or Ar, although they have a low solubility, their concentration is very low and always remain below the saturation point. However, when methane reaches the saturation point, Ne and Ar owing to their low solubility in water may be preferentially incorporated in CH₄-rich bubbles formed at the ocean-ice interface and transported upward with the percolating fluids. Following this extraction mechanism, the detection of ⁴⁰Ar (and potentially ²²Ne) suggests that the atmospheric mass of methane has been renewed at least 20-30 times during Titan's evolution (Tobie et al., 2011).

## 2.4.2 Formation and evolution of the atmosphere

### 2.4.2.1 Origin of the massive N₂-CH₄ atmosphere

The formation of Titan's atmosphere is certainly an event which occurred relatively early in its history, otherwise we should see a number of impact craters too small to have been formed under the dense atmosphere of today. However, whether it is an accretionally generated atmosphere (e.g. Lunine and Stevenson, 1987; Kuramoto and Matsui, 1994) or a late veneer generated during a subsequent late heavy bombardment (Zahnle et al., 1992; Sekine et al., 2011) is an issue that cuts to the heart of the question of whether Titan's formation environment was chemically and thermally distinct from that of Jupiter (see above). If the disk around Saturn was colder than that at Jupiter, rich in volatiles like
methane and ammonia, then the atmosphere most likely was primordial. The likely presence of an interior water ocean, outgassing through time of methane needed to replace that destroyed by photolysis, and the elevated nitrogen isotopic ratio in Titan's atmosphere (Niemann et al., 2010) all hint at the likelihood that Titan acquired its volatile inventory throughout accretion, the present atmosphere being an outgassed remnant of a volatile-rich Titan.

A volatile rich Titan formed under all but the coldest accretional scenarios would have had an initially warm ammonia-rich atmosphere, if indeed the dominant carrier of nitrogen was ammonia. This seems to be required by the measurement of ³⁶Ar to nitrogen (Niemann et al., 2010), which can be interpreted as requiring that molecular nitrogen was initially in the form of ammonia (Owen, 1982). Arguments to the contrary—that somehow the argon is being sequestered by a condensed or crustal reservoir—don't seem to make sense in light of the presence of radiogenic argon at two orders of magnitude larger mixing ratio. Thus we can assume that ammonia was in abundance on early Titan, initially in the form of an atmosphere and surface ammonia-water ocean supported by accretional heating and a strong greenhouse effect (Lunine et al., 2010a). Conversion of ammonia to molecular nitrogen by photolysis (Atreya et al., 1978) and shock heating (Jones and Lewis, 1987; McKay et al., 1988) all provide a plausible story in which the ammonia is rapidly consumed, the atmosphere cools and condenses out, and nitrogen takes its place. Ishimaru et al. (2011), however, demonstrated that efficient conversion of NH₃ into N₂ by impact shock requires on oxidizing CO₂-rich proto-atmosphere on Titan. This is consistent with CO₂ as the dominant gas compound in the building blocks that formed Titan as predicted by Hersant et al. (2008); Mousis et al. (2009c); Tobie et al. (2011).

Sekine et al. (2011) also proposed, based on laser-gun experiments, that ammonia ice contained in the outer icy shell may have been efficiently converted to N₂ by high-velocity impacts during the LHB. The generation of the N₂-rich atmosphere by this process requires the presence of several percents of ammonia in the outer shell, which implies that Titan's outer layer has remained undifferentiated and unmelted until the LHB era. As already discussed, it is very difficult to avoid melting and partial differentiation during the accretion, so that it is unlikely that Titan's icy shell contained ammonia in sufficient amounts at the time of the LHB. While the LHB event has probably affected the evolution of the atmosphere, notably by inducing a significant atmospheric erosion (Korycansky and Zahnle, 2011; Ishimaru et al., 2011), it seems unlikely that it has been the main contributor for the atmosphere generation, a massive atmosphere probably already existed prior to that event.

Outgassing of methane during the accretion and possibly during the LHB would have been vigorous. However, the present-day methane in the atmosphere is unlikely to be the remnant of this primordial outgassing. If it was the case, the

$C^{13}/C^{12}$ in CH₄ should be much larger than the observed value, which is close to the solar value (Mandt et al., 2009; Niemann et al., 2010). This nearly unmodified ratio suggests that methane released during the early stage of evolution should have been totally lost or rapidly re-trapped in the subsurface without being affected by atmospheric fractionation processes. Today's methane has probably been injected much later during the evolution, possibly due to the destabilization of crustal reservoir of CH₄-rich clathrate (Tobie et al., 2006). The only record of the accretional time lies in the nitrogen isotopic ratio, modestly heavier than the terrestrial value (Mandt et al., 2009), suggesting a source distinct from the Earth, or loss of nitrogen to space, or both. However, a single isotope ratio is a poor constraint on the rate and volume of the loss because isotopic enrichment is a function of the rate and mechanism of escape. Further extensive discussion of the early Titan atmosphere may be found elsewhere (Lunine et al., 2010a).

### 2.4.2.2 Interactions with the icy shell

All along Titan's evolution, interactions with the solid icy shell has likely influenced the composition of the atmosphere. During the post-accretional cooling leading to the crystallization of the primordial water ocean, part of the atmospheric gases, notably CO₂ and CH₄, may be re-trapped first by clathration and then by condensation (Choukroun et al., 2010; Lunine et al., 2010a). Large quantities of CO₂ and CH₄ were probably incorporated in the primitive icy shell. Additional CH₄-rich clathrates released during the differentiation of the deep interior may have also accumulated at the base of this primitive outer shell, leading to the formation of a thick (> 10 km) CH₄-rich clathrate layer (Tobie et al., 2006). This shell would then thin in response to the release of heat associated to the core evolution (Tobie et al., 2006), resulting in the release of large amounts of methane which would be mainly stored dissolved in the ocean, only the gas in excess may have been outgassed (Tobie et al., 2009).

According to this scenario, several kilometers of CH₄-rich clathrates may still be present in the outer icy shell at present, and recent thermal destabilizations of this reservoir may explain the present-day atmospheric methane (Tobie et al., 2006). CH₄-rich clathrates are probably not the only crustal reservoir. Large volumes of liquid hydrocarbons may also be potentially stored in a porous crust (Kossacki and Lorenz, 1996; Voss et al., 2007). The morphology and distribution of the hydrocarbon lakes on Titan suggest that they are connected to a subsurface reservoir (Hayes et al., 2008) (see also Aharonson et al. this book). However, no constraints exist on the extension of this subsurface reservoir as well as on the methane content. The lakes seems dominated by ethane (Mitri et al., 2007; Brown et al., 2008; Cordier et al., 2009), which suggests that the methane fraction in a subsurface liquid reservoir is limited at present. Based on the available observations, it is impossible to constrain the contribution of this potential reservoir to
the methane inventory. Additional observations, such as subsurface sounding by a future exploration mission, will be needed to better constrain the nature of the subsurface reservoir.

The mysterious absence or rarity of the primordial no-
ble gases in Titan's atmosphere may also be indicative of
interactions of the atmosphere with the solid icy shell. It
has been shown recently that the apparent deficit in noble
gases could be essentially due to the formation of multiple
guest clathrates at the surface-atmosphere interface of the
satellite (Mousis et al., 2011). These clathrates would prefer-
entially store some species, including all heavy noble gases,
over others. These results have been obtained from the
use of a statistical-thermodynamic model based on recent
determinations of intermolecular potentials and supersede
the previous ones that suggested a limited trapping of argon
(Osegovic and Max, 2005; Thomas et al., 2008). In fact, the
clean water ice needed for the formation of these clathrates
may be delivered by successive episodes of cryovolcanic
deposits. In this scenario, the formation of clathrates in
the porous lavas and their propensity for trapping Ar, Kr,
and Xe would progressively remove these species from the
atmosphere of Titan over its history. A global clathrate
shell with an average thickness not exceeding a few meters
may be sufficient on Titan for a complete removal of the
heavy noble gases from the atmosphere. Note, however, that
there has been no convincing and unambiguous evidence
for cryovolcanic activities on Titan so far (see for instance
Moore and Pappalardo, 2011), so that the proposed trap-
ping mechanism still remains hypothetic.

If clathrate formation is an efficient process as hypothe-
sized for the noble gases, then it is possible that clathrates
are also a sink for the prodigious amounts of ethane, and
lesser amounts of propane, expected to have been produced
over the history of Titan by photochemical processing of
methane. Episodes of crustal volcanism could encourage for-
mation of ethane clathrate (Mousis and Schmitt, 2008),
as could impacts (Lunine et al., 2010b), or simple per-
colation through a heavily fractured crust might progres-
sively allow the substitution of ethane for methane in crustal
clathrate hydrate, and reconversion of water ice that once
was clathrate prior to losing methane by outgassing. Re-
gardless of the process, while methane clathrate hydrate
is slightly less dense than the underlying water-ammonia
ocean, ethane clathrate is not. Hence any ethane clathrate
will have a tendency to move downward in a predominantly
methane clathrate shell, and once at the bottom will sink in
the ocean-to exist stably at the bottom. Timescales might
be an issue here, leaving most of the ethane in the outer shell,
but sequestered away from the surface-atmosphere system. If
this view is correct, Titan's outer shell is laden with ethane
clathrate from billions of years of methane photolysis-the
lakes and seas being only the most recently produced ethane.

2.4.2.3 Long-term climate stability of Titan's
atmosphere.

Titan's atmosphere has been unstable on timescales of be-
tween tens and hundreds of millions of years unless large
areas of methane seas and oceans, an order of magnitude
or more larger than the present-day lakes and seas, have
existed on the surface throughout geologic time. In the
absence of such sustaining bodies of liquid methane, or
an implausibly finely tuned outgassing rate that keeps the
atmospheric methane content within a factor of several of
the current abundance for billions of years, the methane in
Titan's atmosphere must have been periodically depleted.
For the first half of Titan's history, when the Sun's lumi-
nosity was less than 80-85% of its current value, loss of
methane may have caused the nitrogen to rain out onto
Titan's surface (McKay et al., 1993; Lorenz et al., 1997a),
producing a global sea of liquid nitrogen under a thinner
clear atmosphere.

This scenario, bizarre in its contrast with the present-day
state, raises all sorts of issues: (1) Is there geologic evi-
dence for a long-standing ancient global sea of either liquid
nitrogen or (to avoid this catastrophe) a standing ancient
global sea of methane and ethane? (2) How could a thicker
atmosphere be restored, even with massive outgassing of
methane (why wouldn't the methane simply freeze out at
the lower global temperatures for which liquid nitrogen is
stable)? (3) What would prevent under a thin atmosphere
the formation of massive ice caps of nitrogen at cold poles,
possibly rendering as in point (2) a permanently frozen
Titan? It is possible that freeze out did not occur even
under a fainter young Sun, and indeed for values of the
solar luminosity within 15-20% of the present-day, it seems
that a nitrogen atmosphere not much different from that at
present could have been sustained in the absence of methane
(Lorenz et al., 1997a).

As crustal and deeper sources of methane continue to be
depleted by photolysis, it seems inevitable that a state will
be reached in which Titan will lose its surface methane and
revert to a nearly pure nitrogen atmosphere, largely clear of
aerosols, but with a surface temperature not much different
from today or even larger as the Sun continues to grow in
luminosity. Eventually, when the Sun becomes a red giant,
Titan will have lost much of its atmosphere thanks to a
strongly enhanced solar wind, but will for a time support a
surface water ocean until the red giant's luminosity begins
to fade (Lorenz et al., 1997b). Perhaps any life extant in the
deep water interior might then adapt to the surface, develop
a form of photosynthesis, etc., if time is not too brief. Such
speculations would be in the realm of science fiction were it
not for the likelihood that Titan is representative of a class
of planets that would stably exist within 1 AU of M to K
dwarfs, with a spectrum of environments from the present-
day state to an ancient nitrogen sea state, to the ultimate
warmer version discussed here, simply by virtue of adjust-
ing the stellar spectral type from late M to K (Lunine, 2010).
### 2.4.3 Summary

At the start of this Section we posed three questions; we will conclude by providing some provisional answers.

1. To what extent did the interior of Titan influence the evolution of its atmosphere? The probable existence of an internal ocean at present suggests that the differentiation of Titan's interior has been accompanied by significant water-rock interactions and massive releases of gas compounds initially incorporated in clathrate structure. Although most of gas species should be stored dissolved in the massive internal ocean and in the form of clathrate in both the outer ice layer and the high-pressure ice layer, a significant fraction of CH₄ was likely outgassed through time.

2. Did Titan always possess a massive $N_2-CH_4$ atmosphere? Titan likely acquired its atmosphere during its first hundred millions of years, at the latest during the LHB. The atmospheric composition and mass have probably varied along Titan's evolution. The early atmosphere may have been more massive and dominated by $CO_2$. Following the post-accretional cooling phase, nitrogen probably rapidly became the dominant gas. It is still unclear how nitrogen has been produced from ammonia and how atmospheric escape has influenced the nitrogen inventory. The methane abundance has probably significantly varied through time depending on the balance between replenishing and destruction. Long periods without atmospheric methane may even be envisioned.

3. Where is the missing hydrocarbon inventory? Large volumes of hydrocarbon may be stored in the form of liquid or clathrate hydrate in the solid icy shell. However, there are still no direct constraints on the volume of these putative reservoirs. Alternatively, the lack of extensive hydrocarbon reservoirs may indicate that the production of hydrocarbons from methane photolysis has been less efficient than previously anticipated. This may suggest that methane has been present in the atmosphere during limited periods of time.

## 2.5 CONCLUSION AND QUESTIONS FOR FUTURE MISSIONS

Since the arrival of the *Cassini-Huygens* mission to the Saturn system, a large amount of data has been collected on Titan and various models have been developed to interpret them in an evolution context. In spite of considerable progresses in our understanding of Titan's origin and evolution, many questions remain, and they will probably not be answered by the end of the *Cassini-Huygens* mission. A series of investigations *in situ* and from the orbit will be needed in the future to address all the mysteries of Titan that *Cassini-Huygens* just started to unveil. Specifically,

some of the remaining problems include:

1. What was the composition of the building blocks that formed Titan? Theoretical models of planetary formation suggest that the building blocks that formed Titan contained a few percent of various gas compounds, mainly $CO_2$, $NH_3$, $CH_4$, as well as sulfur compounds, methanol and traces of noble gases. Most of the gas compounds initially expected have not been detected so far. Xe and Kr, for instance, appear to be depleted relative to the predicted abundance. By contrast, Ne, which was not expected, has been tentatively detected. $CO_2$, which is predicted to be the dominant carbon-bearing species, has only been identified at *Huygens'* landing site (Niemann et al., 2010) and tentatively in some other specific areas (McCord et al., 2008). Constraining the $CO_2$ fraction on Titan as well as the noble gas content will provide key constraints on the conditions under which the building blocks formed. Atmospheric sampling by a future exploration mission will permit us to determine more accurately the noble gas content of Titan's atmosphere, in particular to confirm the detection of neon. Determination of noble gas abundance in Saturn's atmosphere by an entry probe will also be extremely useful to determine the efficiency of noble gas trapping in planetesimals embedded in the feeding zone of Saturn. Moreover, *in situ* analysis in different materials collected at the surface will also provide pertinent information on the possible $CO_2$ content in the outer shell as well as on the possible sequestration of noble gases in the subsurface.

2. How did Titan's $N_2$-dominating atmosphere form and evolve? It is likely that Titan's atmosphere has been acquired relatively early during the satellite evolution. However, it is still unclear how the $N_2$-dominating atmosphere has been produced and how its composition has evolved through time. The elevated $^{15}$N/$^{14}$N ratio observed in Titan's nitrogen is commonly attributed to enhancement due to massive atmospheric escape. However, we are still missing constraints on the initial value in Titan. Determination of $^{15}$N/$^{14}$N ratio in $NH_3$ in comets, in particular by the *Rosetta* mission on Comet 67P/Churyumov-Gerasimenko, will provide the first direct constraint. Such measurements together with determination of isotopic ratios in non-radiogenic isotopes of argon ($^{38}$Ar/$^{36}$Ar) and neon ($^{22}$Ne/$^{20}$Ne) by *in situ* sampling will allow us to determine how much mass Titan's atmosphere have lost since its formation, thus providing constraints on the early state of Titan's atmosphere.

3. How efficient was the interaction between rock and $H_2O$ in the interior? The moment of inertia inferred from *Cassini* gravity data as well as the detection of $^{40}$Ar in the atmosphere is indicative of prolonged interactions between silicate minerals and $H_2O$ (liquid water or water ice). Such an interaction has key implications for the chemical evolution of Titan's interior, in particular
for the astrobiological potential of its internal ocean. Its efficiency may be constrained by future exploration missions (i) by constraining the degree of differentiation and ocean characteristics (size and composition) from tidal and magnetic monitoring as well as from seismic detection using a fixed lander network at the surface; (ii) by detecting salts and ammonia hydrates at the surface from spectroscopy and surface samplings and (iii) by measuring the ratio between radiogenic and non-radiogenic isotopes in noble gases other than Ar from *in situ* mass spectrometry in the atmosphere and/or in hydrocarbon lakes.

4. *Are Titan's ice shell and interior currently active?* In spite of the Earth-like landscape of Titan, there is still some doubt regarding the internal activity of Titan. Although there is some evidence of interactions with the interior in the past, it is uncertain if Titan's interior has still an influence on the evolution of the atmosphere and surface. Precise determination of the topography and gravity field by a future mission may provide key information on the structure of the icy shell, thus on its internal dynamics. Identification of recent (unambiguous) cryovolcanic flows using high-resolution images and IR spectra would hint at the likelihood of recent activities. The detection of helium, which has a very short residence time in the atmosphere, would be indicative of contemporaneous internal outgassing. Finally, direct observations of eruption would be the most convincing evidence.

5. *How long has the methane cycle been active on Titan?*
At present, Titan has a very active methane cycle, characterized by photochemical production of more complex hydrocarbon species, formation of a haze layer, cloud formation and rain fall, accumulation of hydrocarbon liquids in near-polar lakes and seas. However, it is still unclear if this cycle has existed during most of Titan's evolution or if it was triggered only relatively recently. Determining the possible volume of hydrocarbon stored under the surface by subsurface sounding with ground penetrating radar by a future mission will provide pertinent constraints on the duration of CH₄-driven chemistry on Titan. Identification of clathrates of methane, ethane and others at the surface by *in situ* sampling at the surface will also permit us to evaluate the role of clathration in the storage of hydrocarbon in the subsurface. Determination of ¹³C/¹²C in various carbon-bearing materials collected at the surface will also provide key information on the atmospheric fractionation process, and potentially on the duration of active methane cycle. High-resolution mapping of the surface may also provide information on different climatic conditions in the past, such as evidences for more extended methane seas associated with wetter climate or for N₂ ice caps associated to dryer and colder climate.

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<figure><img src="image_367.png" /></figure>

# Can large icy moons accrete undifferentiated?

J. Monteux<sup>a,*</sup>, G. Tobie<sup>a</sup>, G. Choblet<sup>a</sup>, M. Le Feuvre<sup>b</sup>

<figure><img src="image_368.png" /></figure>

CrossMark

## ARTICLE INFO

**Article history:**

Received 14 November 2013

Revised 22 April 2014

Accepted 23 April 2014

Available online 6 May 2014

**Keywords:**

Thermal histories

Accretion

Satellites, formation

Impact processes

## ABSTRACT

The apparent moments of inertia of Callisto and Titan inferred from gravity data suggest incomplete differentiation of their interior, commonly attributed to slow and cold accretion. To understand whether such large icy moons can really avoid global melting and subsequent differentiation during their accretion, we have developed a 3D numerical model that characterizes the thermal evolution of a satellite growing by multi-impacts, simulating the satellite growth and thermal evolution for a body radius ranging from 100 to 2000 km. The effects of individual impacts (energy deposition, excavation) are simulated and integrated for impactor sizes ranging from a few kilometers to one hundred kilometers, while for smaller impactors, a simplified approach with successive thin uniform layers spreading all over the satellite is considered. Our simulations show that the accretion rate plays only a minor role and that extending the duration of accretion does not significantly limit the increase of the internal temperature. The mass fraction brought by large impactors plays a more crucial role. Our results indicate that a satellite exceeding 2000 km in radius may accrete without experiencing significant melting only if its accretion is dominated by small impactors (<a few kilometers) and that the conversion of impact energy into heat is unrealistically inefficient (<10–15%). Based on our simulations, if more than 10% of satellite mass was brought by satellitesimals larger than 1 km, global melting for large bodies like Titan or Callisto cannot be avoided.

© 2014 Elsevier Inc. All rights reserved.

## 1. Introduction

Differences in composition and internal structure exist between the major icy satellites of Jupiter and Saturn, suggesting distinct accretion and differentiation histories (e.g., Kirk and Stevenson, 1987; Mueller and McKinnon, 1988; Mosqueira and Estrada, 2003a; Barr and Canup, 2008). The high moment of inertia factor inferred from *Galileo* gravity measurements (C/MR² = 0.346) (Anderson et al., 2001) suggests that ice-rock separation may be incomplete in the interior of Jupiter's moon Callisto. By contrast, Ganymede has a much smaller moment of inertia (C/MR² = 0.31) (Anderson et al., 2001) and shows signs of past endogenic activity (Pappalardo et al., 2004). A full separation of ice and rock is suggested for Ganymede together with the formation of a metallic core, which is at the origin of a relatively intense intrinsic magnetic field (Kivelson et al., 1998).

With similar size and mass, Saturn's moon Titan may be an intermediate case between Callisto and Ganymede. Its moment of inertia factor, C/MR² estimated to ~0.33–0.34 from Cassini

gravity measurements (less et al., 2010, 2012) suggests that Titan's interior is more differentiated than Callisto but probably much less than Ganymede. Like Callisto, Titan might still possess a layer of ice-rock mixture between a rocky core and an outer ice-rich mantle, unless the rocky core is mostly composed of highly hydrated minerals (Sohl et al., 2010; Castillo-Rogez and Lunine, 2010). The fact that the interior of Callisto and possibly Titan may still contain a layer of ice-rock mixture suggests that the satellite may have avoided significant melting during accretion and subsequent evolution.

The accretion of giant planet's moons is intimately linked with the evolution of the circumplanetary disk that formed during the transition stage of the planet's accretion, when the planet became massive enough to contract and accrete gas and dust from the circumsolar disk (e.g., Estrada et al., 2009). The timescale of the satellite accretion is therefore mostly controlled by the disk structure, the mass inflow rate, and the lifetime of the circumplanetary disk. Two main categories of circumplanetary disk models have been proposed: the solids-enhanced minimum mass (SEMM) model (Mosqueira and Estrada, 2003a,b; Estrada et al., 2009) and the gas-starved disk model (Canup and Ward, 2002, 2006; Ward and Canup, 2010). In the gas-starved disk model, the disk is assumed to be continuously supplied by ongoing inflow of gas

<footnote><sup>a</sup>Laboratoire de Planétologie et de Géodynamique, LPG Nantes, CNRS UMR6112, Université de Nantes, France</footnote>

<footnote><sup>b</sup>Laboratoire Auscultation et Imagerie, IFSTTAR, Nantes, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail address: julien.monteux@univ-nantes.fr (J. Monteux).</footnote>
and dust particles from the surrounding proto-planetary disk while in the SEMM model, solid components of the disk are supplied by ablation and capture of planetesimal fragments passing through the disk. These two approaches result in different characteristic impactor sizes, ranging typically from a few meters to a few kilometers in the gas-starved approach (Barr and Canup, 2008), while a significant fraction of impactors with radii above 1 km size and up to 100–200 km is envisioned in the SEMM model (Estrada and Mosqueira, 2011). The impactor size is crucial to determine whether the impact energy is buried deep beneath the surface or efficiently released to the space. Hence these two formation models can potentially lead to different early thermal evolutions of growing icy moons.

Previous studies showed that it was possible to avoid melting if the accumulation of accretion energy was inefficient, i.e. if the energy was radiated away at a rate comparable to the accretion rate (e.g., Schubert et al., 1981; Squyres et al., 1988; Kossacki and Leliwa-Kopystyński, 1993; Coradini et al., 1995; Grasset and Sotin, 1996; Barr and Canup, 2008; Barr et al., 2010). Based on these models, the accretion timescales $t_{acc}$ should be longer than 1 Myr to avoid significant melting and hence differentiation of Callisto while an accretion timescale as short as $10^{3–4}$ yr may be possible for Ganymede. However, these timescales are dependent on the way heat deposition and cooling are treated. These studies used a one-dimensional approach initially developed for the accretion of terrestrial planets (Safronov, 1978; Kaula, 1979). In this approach, the evolution is parameterized by deposition of successive material layers. The thermal effect of an impact is not considered individually, but is averaged over the entire surface and integrated. This approach is valid as long as the impactors remain small ($\le 1$ km) and are randomly distributed at the surface. This might be the case during the very early stage of the accretion process, but impactors larger than 1 km probably became more and more abundant at the end of the accretion stage (e.g., Estrada et al., 2009). Impactors larger than 100 km might also be expected (e.g., Sekine and Genda, 2012; Dwyer et al., 2013). For such large impacts, a detailed description of each impact including energy deposition and transfer is required.

For this purpose, we have developed a three-dimensional model that characterizes the thermal evolution of a satellite growing by multi-impacts. The satellite growth and thermal evolution are simulated for a radius ranging from 100 km to 2000 km from different populations of undifferentiated icy impactors, by assuming different accretion rates and conversion rates of impact energy into heat. The effects of individual impacts are simulated and integrated for impactor sizes ranging from a few kilometers to one hundred kilometers. For each impact event, we consider the thermal effects due to the dissipation of the impactor's kinetic energy as heat as well as the topographical effect associated to excavation process. For impactor sizes smaller than a few kilometers, we do not treat the impact individually because the number of impacts to simulate will be extremely time consuming. The small and numerous impactors are modeled by successive thin uniform layers spreading all over the moon. As the icy moon grows, gravitational forces increase and impacts become more and more violent. Due to this, as well as the accumulation of warmed icy material, melting events may occur once the icy moon reaches a critical size. As the main objective of our work is to determine the maximum radius reached by a growing satellite before significant melting occurs (>5%), we make some simple assumptions corresponding to the least efficient scenario for initiating ice melting. The impacts are assumed to occur with the smallest possible velocity corresponding to the escape velocity determined by the mass of the growing satellite. Hence, the accretion efficiency is assumed to be 100% and all impacted mass remains on the growing satellite

(Asphaug, 2010). With these assumptions, we minimize the energy accumulated in the satellite during the growth, and therefore we provide an upper limit for the radius that the satellite can reach without experiencing significant melting. In Sections 2 and 3, we present the details of our model. We first describe the process associated to a single impact event and then we present our multi-impact approach. The results of our simulations for different accretion parameters are provided in Section 4. Finally, in Section 5, we briefly discuss the implications of our results for the post-accretional structure of large icy moons and the subsequent differentiation processes.

## 2. Single impact model

Following an impact and the formation of a crater, a significant amount of heat is buried deep below the impact site. In the following section we describe the scaling laws used to model the thermal and topographical consequences of a large single impact on a growing icy moon.

### 2.1. Impact heating

During an impact event, the initial kinetic energy of the impactor is converted into internal energy produced by shock heating of the satellite and of the impactor, internal energy produced by plastic work, and kinetic energy of ejected material (e.g. O'Keefe et al., 1977; Squyres et al., 1988). O'Keefe et al. (1977) estimated that the fraction, $\gamma_{li}$, of the impactor kinetic energy going into shock heating of the satellite ranged from 0.2 for low-velocity impacts to about 0.6 for very high velocities. As this parameter is difficult to constrain, especially for large impacts, we consider here that it is a free parameter.

During the impact, a shock wave propagates from the impact site. Following the adiabatic pressure release, the peak pressure being almost independent of impactor size, a thermal anomaly remains below the impact site. The heat deposition is nearly uniform in a hemispherical (for $v_{imp} < 1$ km/s) to spherical region next to the impact (the isobaric core), and strongly decays away from it (Croft, 1982; Squyres et al., 1988; Senshu et al., 2002) (see Fig. 1). For simplicity, we consider in our models that the shape of the isobaric core is spherical and that it does not depend on the impact velocity. Energy balance calculations and shock simulations suggest that, for impact velocities lower than $10 \text{ km s}^{-1}$, the radius of the isobaric core $r_{ic}$ is comparable or slightly larger than that of the impactor $r_{imp}$ (Pierazzo et al., 1997; Senshu et al., 2002; Kraus et al., 2011). Considering the extreme case in which all of the impact energy is perfectly transferred to the internal energy within the isobaric core and impactor itself gives an estimation of the maximum value for $r_{ic}/r_{imp} = 3^{1/3}$ (Senshu et al., 2002). Hence, after a large impact, a large amount of heat can be buried deep below the impact site at a depth $\sim 2r_{imp}$ and contribute to the early thermal evolution of the growing moon (Kraus et al., 2011).

As already explained in the introduction, we neglect here the velocity at infinity of the impactor ($v_\infty = 0$) as we want to determine the maximal size a moon can reach without significant melting. For simulations presented here, we do not consider any transplanetary impactor with $v_{imp} \gg v_{esc}$ (Squyres et al., 1988). The impactor velocity is only determined by the gravitational acceleration of the growing target: $v_{imp} = v_{esc} = \sqrt{2gR}$ with g the gravity at the surface of a moon with radius R. The impactor velocity is therefore proportional to the satellite size. For isobaric core volume $V_{ic} = 3V_{imp}$, a balance between the kinetic energy delivered to the growing moon and the energy used to heat up the growing
<figure><img src="image_369.png" /><figcaption>**Fig. 1.** Schematic illustration of the topographical (left) and thermal (right) evolutions after large impacts. When the first large impact occurs (first line), a crater with diameter $D_f$, depth $z_f$ and rim height $\delta_0$ is formed (second line, left). Before the next large impact, the layer deposition occurs (third line, left). When a second impact occurs close enough to the first one (fourth line), the pre-existing topography is modified according to Eq. (11). When a large impact occurs (first and second line, right), heat is buried deep below the impact site following Eq. (1) while the ejecta rim temperature is the average temperature below the impact site over a volume that is $D_f$ large and $z_f$ thick. The temperature of the layer deposited before the next large impact (third line, right) obeys Eq. (13).</figcaption></figure>

moon (isobaric core and the material surrounding it) without melting leads to (Monteux et al., 2007):

$$ \Delta T_0 = \frac{4\pi \gamma_{li} \rho G \bar{R}_t^2}{9 h_m C_p} \quad (1) $$

where $\bar{R}_t$ is the mean radius of the growing moon, $\rho$ is the mean density of the moon, $h_m$ represents the volume effectively heated normalized by the volume of the isobaric core and scales with the power $m$ (see values in Table 1). $\gamma_{li}$ is the fraction of the impactor kinetic energy that is used to heat up the deep material of the impacted body. Hence, the post-impact temperature increase scales with the square of the moon radius at the time of impact (see (1)).

<table><caption>**Table 1**
Typical parameter values for numerical models.</caption><tr><td>Moon radius</td><td><i>R</i></td><td>100–2000 km</td></tr><tr><td>Impactor radius</td><td><i>r</i><sub>imp</sub></td><td>4–100 km</td></tr><tr><td>Isobaric core radius</td><td><i>r</i><sub>ic</sub></td><td></td></tr><tr><td>Average moon density</td><td><i>ρ</i></td><td>1500–2000 kg m<sup>-3</sup></td></tr><tr><td>Mean heat capacity</td><td><i>C</i><sub>p</sub></td><td>1200 J K<sup>-1</sup> kg<sup>-1</sup></td></tr><tr><td>Environment temperature</td><td><i>T</i><sub>e</sub></td><td>100 K</td></tr><tr><td>Mean heat diffusivity</td><td><i>κ</i></td><td>10<sup>-6</sup> m<sup>2</sup> s<sup>-1</sup></td></tr><tr><td>Large impact energy fraction retained</td><td>γ<sub>li</sub></td><td>0.1–0.6</td></tr><tr><td>Temperature power decrease from the isobaric core</td><td><i>m</i></td><td>3.4</td></tr><tr><td>Volume effectively heated by impact</td><td><i>h</i><sub>m</sub></td><td>5.8</td></tr><tr><td>Layer deposit energy fraction retained</td><td>γ<sub>lay</sub> ≤ γ<sub>li</sub></td><td>0.1–0.3</td></tr><tr><td>Gravitational constant</td><td><i>G</i></td><td>6.67 × 10<sup>-11</sup> m<sup>3</sup> kg<sup>-1</sup> s<sup>-2</sup></td></tr></table>

Using parameter values from Table 1 and $\gamma_{li} = 30\%$, for an impacted body with a radius ranging from 1000 km to 2500 km, $v_{imp} < 3$ km/s and $\Delta T_0$ ranges from ~10 K to 100 K. Obviously, if the velocity at infinity is non-negligible, the delivered energy and hence temperature increase would be higher. However, as we want to determine the maximum radius that a growing satellite can reach without significant melting, we consider the most favorable case where the velocity at infinity is zero.

Away from the isobaric core the peak pressure decays with the distance from the surface of the isobaric core (Pierazzo et al., 1997; Kraus et al., 2011) (see Fig. 1). This pressure decay can be faster for an ice/rock mixture than for terrestrial material because of the ice properties (Kraus et al., 2011). Just after the adiabatic pressure release, the thermal perturbation corresponds to an isothermal sphere of radius $r_{ic}$ and temperature $T_0 + \Delta T_0$ that decays when $\bar{r} > r_{ic}$ as (see Fig. 1)

$$ T(r) = T_0 + \Delta T_0 \left(\frac{r_{ic}}{\bar{r}}\right)^m \quad (2) $$

where $\bar{r}$ is the distance from the center of the isobaric core, $T_0$ is the pre-impact temperature and $m$ is the power characterizing the temperature decrease from the isobaric core (Pierazzo et al., 1997; Senshu et al., 2002). The post-impact temperature increase is a function of the pressure increase below the impact site. For small impact velocities (i.e. <3 km s<sup>-1</sup>), the pressure $P$ may increase to peak values of 8 GPa and the post-impact temperature increase scales with $P^{0.7-1}$ (Stewart and Ahrens, 2005). As the pressure typically decays from the isobaric core with $\sim(r_{ic}/r)^4$ (Kraus et al.,
2011), the post impact temperature increase decays from the isobaric core following $(r_{ic}/r)^m$ with $m$ ranging from 2.8 to 4. In this study we choose a medium value of $m = 3.4$.

## 2.2. Topographical effect

An impact leads to the formation of a transient cavity of diameter $D_s$, reaching its final size $D_f$ after some modifications. The diameter of the transient crater $D_s$ can be related to the impactor diameter $d_{imp}$ (in km) through (Zahnle et al., 2003):

$$D_s = a_0 \left( \frac{v_{imp}^2}{v_{esc}^2} \right)^{a_1} \left( \frac{\rho_{imp}}{\rho} \right)^{a_2} R^{a_3} d_{imp}^{a_4} \cos(\theta)^{a_5} \quad (3)$$

where $v_{imp}$ is the impactor velocity, $v_{esc}$ is the escape velocity of the impacted moon, $\rho_{imp}$ is the impactor density, $R$ is the radius of the moon (in km) and $\theta$ is the impact angle. For simplicity, we assume $\rho_{imp} = \rho$ and we set $\theta = 45^\circ$ (the most likely angle of impact and the average value for a uniform bombardment (Shoemaker, 1962)). $a_0, a_1, a_2, a_3, a_4$ and $a_5$ are constant values listed in Table 2. These are derived from laboratory experiments as well as numerical modeling, and are consistent with planetary surface observations.

If the transient crater diameter is smaller than a critical value $D_c$, no later significant modifications occur and its final diameter is $D_f = D_c$. Among the parameters listed in Table 2, $D_c$ is the one that exhibits the largest range of values as this parameter depends on the mechanical properties and gravity of each icy moon (McKinnon et al., 1991; Zahnle et al., 2003). $D_c$ typically ranges between 2 and 3 km for Ganymede and Callisto and up to 15 km for most of the medium-sized satellites (Schenk et al., 2004). Hence, $D_c$ is expected to vary during the growth of the icy moon. Here for simplicity we consider a single value, $D_c = 15$ km (see Table 2). In our models, the majority of the impacts leads to the formation of craters that are larger than $D_c$. Above $D_c$, the post-impact strength of the target material is insufficient to prevent collapse under gravity, crater modifications occur, resulting in a complex crater with a flat floor, a central peak or peak ring, and a terraced rim. Its final diameter thus becomes:

$$D_f = D_s \left( \frac{D_s}{D_c} \right)^{b_0} \quad (4)$$

We express the maximal depth at the center of the crater $z_f$ as a function of the transient simple crater diameter (Pike, 1977; Schenk, 1991):

$$z_f = \begin{cases} K_1 D_s^{b_1} & \text{if } D_s < D_c \\ K_2 D_s^{b_2} & \text{if } D_s > D_c \end{cases} \quad (5)$$

<table><caption>**Table 2**
Crater geometrical parameters used in our models.</caption><thead><tr><th>Parameter</th><th>Value</th><th>References</th></tr></thead><tbody><tr><td>a<sub>0</sub></td><td>1.1</td><td>Zahnle et al. (1998, 2003)</td></tr><tr><td>a<sub>1</sub></td><td>0.217</td><td>Zahnle et al. (1998, 2003)</td></tr><tr><td>a<sub>2</sub></td><td>0.333</td><td>Zahnle et al. (1998, 2003)</td></tr><tr><td>a<sub>3</sub></td><td>0.217</td><td>Zahnle et al. (1998, 2003)</td></tr><tr><td>a<sub>4</sub></td><td>0.783</td><td>Zahnle et al. (1998, 2003)</td></tr><tr><td>a<sub>5</sub></td><td>0.44</td><td>Zahnle et al. (1998, 2003)</td></tr><tr><td>D<sub>c</sub></td><td>15 km</td><td>McKinnon et al. (1991)</td></tr><tr><td>b<sub>0</sub></td><td>0.13</td><td>McKinnon et al. (1991)</td></tr><tr><td>K<sub>1</sub></td><td>0.15</td><td>McKinnon et al. (1991) and Zahnle et al. (2003)</td></tr><tr><td>b<sub>1</sub></td><td>0.88</td><td>McKinnon et al. (1991) and Zahnle et al. (2003)</td></tr><tr><td>K<sub>2</sub></td><td>0.75</td><td>McKinnon et al. (1991) and Zahnle et al. (2003)</td></tr><tr><td>b<sub>2</sub></td><td>0.3</td><td>McKinnon et al. (1991) and Zahnle et al. (2003)</td></tr><tr><td>K<sub>3</sub></td><td>0.017</td><td>Schenk (1991)</td></tr><tr><td>b<sub>3</sub></td><td>0.976</td><td>Schenk (1991)</td></tr><tr><td>p</td><td>2–3</td><td>Howard (2007)</td></tr><tr><td>n</td><td>2–3.5</td><td>Howard (2007)</td></tr></tbody></table>

We consider that the maximum ejecta thickness $\delta_0$ at the crater rim is (Schenk, 1991):

$$\delta_0 = K_3 D_f^{b_3} \quad (6)$$

$b_0, b_1, b_2$ and $b_3$ are constant values listed in Table 2. The elevation variation depends on whether we consider a position inside or outside the crater. Within the crater, the depth increases from center to the top of the ejecta rim with a power $p$. Outside the crater, elevation decreases from the top of the ejecta rim to a reference elevation with a power $-n$. We define $\Delta H(\eta, \xi)$ as the elevation variation between the post-impact topography and a reference elevation (equal to 0 far from the impact site):

$$\Delta H(\eta, \xi) = \begin{cases} z_f + (z_f + \delta_0) \left(\frac{2r}{D_f}\right)^p & \text{if } r < D_f/2 \\ \delta_0 \left(\frac{2r}{D_f}\right)^{-n} & \text{if } r > D_f/2 \end{cases} \quad (7)$$

where $\eta$ is the longitude and $\xi$ the latitude. $r$ is the distance from the crater center:

$$r = \bar{R}_t \arccos[\cos(\eta) \cos(\eta_{imp}) \cos(\xi - \xi_{imp}) + \sin(\eta) \sin(\eta_{imp})] \quad (8)$$

with $\eta_{imp}$ the impact longitude and $\xi_{imp}$ the impact latitude.

## 2.3. Ejected material and ejecta temperature

The fraction of material from the impactor and from the impacted body escaping the growing moon decreases with decreasing impact velocities (Asphaug, 2010; Korycansky and Zahnle, 2011). For impact velocities considered in our models ($v_{imp} = v_{esc} < 3$ km s$^{-1}$) and for 45° impact angle, the accretion is supposed to be efficient and this fraction should remain small (less than 10% of the impactor's mass) (Asphaug, 2010; Korycansky and Zahnle, 2011). After a large impact, part of the material beneath the impact site is excavated and redeposited within the ejecta rim (see Fig. 1). We thus set $n$ from Eq. (7) to a value typically ranging between 2 and 3 in order for the efficiency of mass accretion to be close to 100% during the whole accretion period and we consider that the whole impactor is deposited in the ejecta rim.

The temperature of this material depends on the pre-impact temperature, the temperature increase from the impact and the temperature of the impactor. The volume fraction of excavated material that is shock-heated increases with final crater size and this hot material is redeposited in the most external part of the ejecta rim (Maxwell, 1977; Barnhart and Nimmo, 2011). Hence, the thermal repartition within the ejecta rim should also depend on the interactions between the ejected material and the atmosphere during the excavation and the fallback processes (Kieffer and Simonds, 1980). For simplicity, we will consider in our models that the temperature of the ejecta rim is the average temperature below the impact site over a cylindrical volume with a diameter $D_f$ and a thickness $z_f$.

# 3. Multi-impact approach

The accretion of an icy moon is the result of material deposited from a wide range of impactor sizes (i.e. from dusts to 100 km size objects). In the following sections we describe our model of accretion from multi-impacts.

## 3.1. Impactor population

For the mass distribution of the impactor, we consider a power law distribution with an exponent equal to -2.5: $dN_c/dm \propto m^{-2.5}$, consistent with N-body simulations (Kokubo and Ida, 2000). We use Monte Carlo sampling to generate the impactor population
(Zahnle et al., 2001; Lognonné et al., 2009). By random drawing, we determine the impactor mass (or equivalently, radius) according to the above power law distribution. The time of impact is taken from a uniform probability distribution, while the latitude and longitude of the crater center are randomly drawn so that an isotropic impact flux is obtained. To limit the computation time, a lower size limit, $r_{min}$, is imposed on the impactor distribution (see Fig. 2). Below this lower limit, individual impact events are not simulated and a parameterized approach using successive deposit layers is used (see Section 3.3 for further details). We assumed a lower limit, $r_{min}$, typically between 1 and 10 km. We also prescribed an upper limit, $r_{max}$, typically 100–200 km. Above these values, the validity of the scaling laws used here becomes questionable. Accretion from such large bodies would require more complex impact simulations, which is beyond the scope of the present paper. Nevertheless, 200 km is probably a reasonable upper limit since the growing moon is likely to perturb large objects that were migrating in from the outer disk possibly leading to their breakup. Hyperion, for instance, may be considered as an example of such large satellites/imals (Mosqueira and Estrada, 2003a,b; Estrada et al., 2009). The probability of impacts with objects exceeding 200 km is thus likely low, except maybe during the very late stage of accretion (e.g., Sekine and Genda, 2012).

For simplicity, the impactor population is assumed to be infinite (meaning that the number of impactors of a given size does not decrease as a function of time) and the accretion rates of large impactors $\tau_{acc,li}$ and layer deposit $\tau_{acc,lay}$ are assumed constant during one simulation. To measure the influence of large impactors ($r_{min} < r < r_{max}$) relative to small impactors ($r < r_{min}$), we define the ratio:

$$x_{m,li} = m_{li}/m_{acc} \qquad (9)$$

where $m_{li}$ is the mass accreted from large impactors and $m_{acc}$ is the total mass accreted. We define the total accretion rate $\tau_{acc}$ as

$$\tau_{acc} = \tau_{acc,li} + \tau_{acc,lay} \qquad (10)$$

where $\tau_{acc,li}$ is the accretion rate from large impacts and $\tau_{acc,lay}$ is the accretion rate from small impactors modeled as thin layer deposits (see Section 3.3). We assume that the composition of the icy moon (and of the impactor) is a mixture of ice and rocks and that its density $\rho$ is uniform with depth.

<figure><img src="image_370.png" /><figcaption>**Fig. 2.** Schematic representation of the cumulated number of impacts as a function of the impactor mass. All the material with a mass smaller than $m_{min}$ (i.e. with $r < r_{min}$) is deposited as a thin global layer over the moon surface. The impactors with a mass ranging from $m_{min}$ to $m_{max}$ are considered here as successive impact events (selected randomly) and their effects (impact cratering and heating) are treated individually.</figcaption></figure>

## 3.2. Multi-impact-induced topography

To account for the pre-impact topography, we use the multi-cratering approach developed by Howard (2007). At the $i_{th}$ impact, the new elevation variation $\Delta E_i(\eta, \xi)$ is

$$\Delta E_i(\eta, \xi) = \begin{cases} \Delta H(\eta, \xi) + (R_{i-1}(\eta, \xi) - \overline{R_{i-1}})(1 - (2r/D_f)^2) & \text{when } r < D_f/2 \\ \Delta H(\eta, \xi) + (R_{i-1}(\eta, \xi) - \overline{R_{i-1}}) & \text{when } r > D_f/2 \end{cases} \qquad (11)$$

$\Delta E_i(\eta, \xi)$ depends on the local pre-impact topography variation $(R_{i-1}(\eta, \xi)) - \overline{R_{i-1}}$. We consider here no late deformation of the topography before the impact (the degree of inheritance is 1 inside and outside the crater (Howard, 2007)). After the $i_{th}$ impact, the local radius becomes $R_i(\eta, \xi) = R_{i-1}(\eta, \xi) + \Delta E_i(\eta, \xi)$ and the mean radius of the growing moon increases from $\overline{R_{i-1}}$ to $\overline{R_i}$.

The growth of the satellite requires that at least part of the impactor material remains on the growing satellite. Since we consider that the volume of the impactor is retained within the ejecta rim in our models, this growth requirement provides constraints on the scaling law describing the ejecta blanket distribution. For large $n$ values, the topography decreases rapidly from the crater rim and the volume of material accumulated in the ejecta rim decreases. On the contrary, for small $n$ values and for the same crater rim height, the topography decreases more linearly from the crater rim and the volume of material accumulated in the ejecta rim is large. The falloff in ejecta thickness is steep. Depending on the target properties, $n$ ranges between 2.5 and 3 (Housen et al., 1983; Moore et al., 2004). In Fig. 3, we monitor the average radius of the growing moon as a function of time for different values of $n$ and compare it with the theoretical mean radius resulting from the 100% accretion of $1.4 \times 10^6$ impactors ranging from 10 to 100 km radii. From this figure, we see that increasing $n$ decreases the mass accumulated and leads to a growth that is less than 100% accretive. For $n = 3$, the accretion is not fully efficient and about 30% of the impacted mass remains on the impacted body while for $n = 2.5$, 95% is accreted (see Fig. 3). For $n$ values smaller than 2.5, the growth is unrealistic since it is more than 100% accretive. We choose a value of 2.5 which maximizes the fraction of accreted material.

<figure><img src="image_371.png" /><figcaption>**Fig. 3.** Time evolution of the average radius of the growing icy moon after the accretion of $1.4 \times 10^6$ impactors ranging from 10 to 100 km radii with $n = 2.5$ (red solid line), $n = 2.7$ (green solid line) and $n = 3$ (blue solid line). For comparison, we also represent the time evolution of the average radius consisting in the 100% accretive accumulation of the impactor bodies (black solid line). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>
### 3.3. Layer deposits from small impactors

As explained previously, for numerical reasons, individual impact events for $r < r_{min}$ are not simulated. We consider that the accreted mass from small impactors is averaged and uniformly added on the surface. For a prescribed accretion rate, $\tau_{acc,lay} = \tau_{acc} \times (1 - x_{m,li})$, the thickness $\delta_{lay}$ of the uniform layer deposit between two individual large impacts is then:

$$ \delta_{lay} = \left( \frac{3\tau_{acc,lay}\Delta t}{4\pi\rho} + R_i^3 \right)^{1/3} - R_i \quad (12) $$

At any point at the surface, this additional layer is added uniformly. We assume that the temperature of this deposit layer is homogeneous over the entire thickness $\delta_{lay}$. The layer temperature depends on the radius of the growing moon $R_t$ and is calculated following an approach that is similar to the “classic” one from Schubert et al. (1981). In their 1D thermal evolution models, Schubert et al. (1981) considered that a fraction $h$ of the kinetic energy accumulated during accretion progressively heats up the near surface of the growing satellite (Kaula, 1979; Schubert et al., 1981; Lunine and Stevenson, 1987; Grasset and Sotin, 1996). Hence the corresponding temperature profile is:

$$ T(\bar{R}_t) = \frac{hGM(\bar{R}_t)}{C_p \bar{R}_t} \left( 1 + \frac{\bar{R}_t v_\infty^2}{2GM(\bar{R}_t)} \right) + T_e \quad (13) $$

Considering that $v_\infty^2 = 0$ (i.e. $v_{imp} = v_{esc}$), Eq. (13) becomes

$$ T(\bar{R}_t) = \frac{\gamma_{lay}}{2C_p} v_{imp}^2 + T_e \quad (14) $$

where $C_p$ is the heat capacity of the icy satellite material/mixture and $T_e$ is the temperature of the surrounding environment. The coefficient $\gamma_{lay}$ represents the fraction of energy that is retained in the layer as heat. Note that the coefficients $\gamma_{li}$ and $\gamma_{lay}$ defined here differ from the coefficient $h$ used in Eq. (13). $h$ implicitly includes the post-impact surface cooling, while $\gamma_{li}$ and $\gamma_{lay}$ only represent the fraction of kinetic energy converted as heat from the small impacts deposited as an uniform layer ($\gamma_{lay}$) or from large impacts ($\gamma_{li}$). $\gamma_{lay}$ is considered as a free parameter. It accounts for the effect of mechanical mixing in the shallow layers which has been described in Squyres et al. (1988) by a larger thermal diffusivity. Due to the heat removal by this “gardening” effect of numerous small impacts (Davies, 2009), it is reasonable to assume that $\gamma_{lay} \le \gamma_{li}$.

### 3.4. Numerical method

As the satellite grows, impactors bring material and thermal energy used to build-up and heat-up the moon. We monitor the thermal evolution of a growing icy satellite using the 3D-tool OEDIPUS (Choblet et al., 2007) to obtain a three-dimensional solution of the energy equation in a spherical shell. We use a finite-volume formulation and a mesh based on the “cubed sphere” transformation, the resulting grid consisting in six identical blocks. The computational grid in one block consists typically of 128 × 64 × 64 discrete cells. Initially, the growing satellite in our models consists of a core surrounded by a shell with a thickness leading to a $R_0$ radius body. In the numerical domain, the overlaying shell (between $R_0$ and the final moon radius) is initially empty and gradually filled by impacted material during the accretion history. As the accretion time is relatively short compared to the onset time of solid-state convection (e.g., Robuchon et al., 2010), we consider only the diffusion of heat with no advective term. Melt transport and water/rock separation are not considered here and simulations are stopped when a few percent of material exceeding the melting point of water ice is reached. The accreted material is assumed to

be an undifferentiated mixture of ice and rocks with a thermal diffusivity that does not depend on temperature, $\kappa = 10^{-6} \text{ m}^2 \text{ s}^{-1}$ (Squyres et al., 1988; Barr et al., 2010).

To maintain an accurate spatial resolution in our models during the entire accretion, we subdivide the accretion in successive stages between which the mesh grid is modified. Between two stages, the temperature field from the previous regime is interpo- lated on the mesh grid that we use in the next regime (see Fig. 4). The free accretionary parameters of our models are the ratio of material accreted from a large impacts $x_{m,li}$ and the accretion rate $\tau_{acc}$. The free energy conversion factors are $\gamma_{lay}$ (layer heating) and $\gamma_{li}$ (large impact heating). $\gamma_{lay}$ and $\gamma_{li}$ are independent parameters.

### 3.5. Post-impact surface cooling

After an impact, the efficient radiative heat transfer at the surface leads to a rapid cooling of the uppermost part of the heated zone (including the impact site and the surrounding ejecta blanket). As such a rapid post-impact cooling cannot be properly described in the framework of the relatively coarse grid mesh used by the 3D OEDIPUS tool, we have implemented a more precise description of heat transfer in this region. In the uppermost grid mesh of OEDIPUS, the conduction of heat for uniform heat conduc- tivity is solved in the radial direction using refined sublayers with a Crank–Nicholson method (similarly to Tobie et al. (2003)). The number of sub-layers varies between 50 and 150, depending on the distance between the local surface radius $R_i(\eta, \xi)$ and the first underlying OEDIPUS grid mesh. A radiative heat flux boundary condition is imposed at the surface:

$$ F = \sigma (T(R_i)^4 - T_{eq}^4) \quad (15) $$

with $\sigma$ the Stefan–Boltzman constant and $T_{eq}$ the expected equilibrium surface temperature. In the calculations presented below, $T_{eq} = 100$ K. The temperature at the base of the refined col- umn correspond to the temperature value provided in OEDIPUS. The conductive heat flux predicted in the refined column at the base of the first underlying OEDIPUS mesh interface is then imposed as heat flux boundary conditions at the top of the coarse grid domain.

## 4. Numerical results

### 4.1. Early and intermediate regimes: from 100 km to 1000 km

We first consider the accretion of a 1000 km size ice-rock body from a 100 km satellite embryo. For simplicity, the initial temperature from $R = 30$ km to $R = R_0 = 100$ km is set to a uniform value, here $T = T_e = 100$ K. To maintain a good spatial resolution, we subdivide the accretion history of the icy satellite in two stages: an early stage where the moon is growing from 100 km to 500 km, an intermediate stage where the moon is growing from 500 km to 1000 km.

Fig. 4 illustrates the temperature evolution during these two accretionary regimes. In order to test the influence of the early and intermediate regimes on the late accretive stage, we consider two accretionary different scenarios for both the early and inter- mediate stages: a “cold accretion” where $\gamma_{li} = \gamma_{lay} = 0.1$, $x_{m,li} = 10\%$ (Fig. 4, left column) and a “hot accretion” where $\gamma_{li} = \gamma_{lay} = 0.3$, $x_{m,li} = 33\%$ (Fig. 4, middle column). The accretion parameters used for the “Early regime” simulation are $r_{min} = 4$ km and $r_{max} = 10$ km, while for the “Intermediate regime”, we used $r_{min} = 8$ km and $r_{max} = 20$ km. At the end of the intermedi- ate regime, $t_{acc} = 0.5$ Myr and the impactor velocities remain small ($<1 \text{ km s}^{-1}$) which corresponds to small temperature increases deep below the impact site ($<10$ K).
<figure><img src="image_372.png" /><figcaption>**Fig. 4.** Equatorial cross sections of the temperature field (left and middle columns) and 3D topographical representations (right) of the growing icy moon as a function of time (from top to bottom). The left column represents the “cold accretion” evolution where, up to the end of the intermediate regime, $\gamma_{li} = \gamma_{lay} = 0.1$, $x_{m,li} = 10\%$ while the middle column represents the “hot accretion” evolution where $\gamma_{li} = \gamma_{lay} = 0.3$, $x_{m,li} = 33\%$ (Fig. 4, middle column). Temperature color scale is saturated in white for temperature at the melting point (>273 K). Between each regime (early, intermediate, late), the temperature field is interpolated to a larger mesh grid. In the “Late regime”, $\gamma_{li} = 0.1$, $\gamma_{lay} = 0.3$, $x_{m,li} = 33\%$, $r_{min} = 10$ km and $r_{max} = 100$ km for both the left and middle columns. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

When the moons reach a radius of 1050 km, the temperature barely exceeds 120 K in the cold accretive case, while it can reach values up to 250 K (near the melting point of water ice) for the hot accretive scenario. As we will show later in Section 4.2, although the obtained temperature fields are very different in these two cases, this has no major influence on the evolution of the temperature field in the outer part above 1000 km. Fig. 4 (third column) also represents the 3D topography at the surface of the icy moon at the end of the two stages. As we increase the $r_{min}$ and $r_{max}$ values between the two simulations, the impact craters become larger and the contrast in topography (the difference between the $R(\eta, \xi)$ and the mean radius $\bar{R}$) also increases.

## 4.2. Late accretive regime: >1000 km

To simulate the evolution for $R > 1000$ km (late accretion regime), we use the thermal state reached at the end of the intermediate regime as the initial thermal state. In Fig. 4 we show results obtained for the same accretionary parameters in the late regime ($\gamma_{li} = 0.1$, $\gamma_{lay} = 0.3$, $x_{m,li} = 33\%$, $r_{min} = 10$ km and $r_{max} = 100$ km) but for different initial temperature fields: “cold accretion” scenario (left column) and “hot accretion” scenario (middle column) obtained at the end of the corresponding intermediate regime. Fig. 4 illustrates that the temperature field obtained from the intermediate regime (hot or cold accretion
scenario) only plays a minor role on the critical radius above which melting becomes significant during the late regime. Using the intermediate thermal state obtained from the cold accretion regime leads to $R_{crit} = 1609$ km while using the intermediate thermal state obtained from the hot accretion regime leads to $R_{crit} = 1608$ km (Fig. 4, last line). For this reason, in the following, the temperature field and topography from the “hot accretion scenario” are considered as initial conditions for all simulations of the accretion of bodies larger than 1000 km.

As explained previously, we assume that the impactor velocity is only determined by the gravitational acceleration, and we specifically test the influence of (1) accretion rate $\tau_{acc}$, (2) mass fraction provided by large impactors $x_{m,li}$ and (3) energy conversion factors (i.e. $\gamma_{li}$ and $\gamma_{lay}$) on the thermal state of the growing moon. We monitor the temperature field evolution as well as the volume fraction of satellite material that reaches the melting temperature of pure water ice (i.e. with $T > 273$ K) as a function of satellite growth (see Fig. 4). As complex physical processes associated with melting and water-rock separation are beyond the scope of the present study, we interrupt the simulations when the volume fraction of the growing moon where $T > T_{melt}$ exceeds a threshold value fixed to 5% here. We define $R_{crit}$ as the satellite radius at which this threshold is reached. In this “late regime”, the accretionary parameters can be different from the values used in the previous regimes which may lead to temperature “discontinuities” within the growing moon as emphasized in Fig. 4). As indicated above, such artefacts do not influence the value of $R_{crit}$. As illustrated in Fig. 4, the regions where melting occurs (the regions where the temperature scale is saturated in white) are mainly confined in the most external parts of the growing moon.

### 4.3. Influence of the accretion rate, $\tau_{acc}$ and of the fraction of large impactors, $x_{m,li}$

For this simulation, we assume that the conversion rate of impact energy is similar for small and large impactors: $\gamma_{li} = \gamma_{lay} = 30\%$ or 10%. and we focus only on the late accretive regime. From our models, we can measure the influence of large impacts relative to layer deposition of small impactors by varying the value of $x_{m,li}$. Fig. 5 shows the evolution of $R_{crit}$ as a function of $x_{m,li}$ and for three different accretion rates. For a better comparison

<figure><img src="image_373.png" /><figcaption>**Fig. 5.** Critical radius $R_{crit}$ (above which more than 5% of the volume of the icy moon has a temperature larger than the melting temperature) as a function of the fraction of material accreted from large impacts $x_{m,li}$ for different accretion rates ranging from 0.015 $M_{Titan}$/Myr to 1.55 $M_{Titan}$/Myr. Black symbols represent $R_{crit}$ for $\gamma_{lay} = \gamma_{li} = 0.3$ while red circles represent $R_{crit}$ for $\gamma_{lay} = \gamma_{li} = 0.1$. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)</figcaption></figure>

with other studies, we express the accretion rate, $\tau_{acc}$, in terms of $M_{Titan}$/Myr, where $M_{Titan}$ is the mass of Titan ($=1.345 \times 10^{23}$ kg) and we consider values ranging between 0.015 $M_{Titan}$/Myr ($=2 \times 10^{15}$ kg yr$^{-1}$) and 1.5 $M_{Titan}$/Myr ($=2 \times 10^{17}$ kg yr$^{-1}$). $\tau_{acc} \le 1.5 M_{Titan}$/Myr corresponds to a relatively slow accretion, which is commonly assumed for the accretion of Callisto (Mosqueira and Estrada, 2003a; Barr and Canup, 2008).

Fig. 5 shows that, even for the least efficient conversion rate of impact energy ($\gamma_{li} = \gamma_{lay} = 10\%$), the satellite cannot grow above 1500 km without significant melting, if the accretion is dominated by large impactors ($x_{m,li} \sim 1$). For $\gamma_{li} = \gamma_{lay} = 30\%$, the critical radius is even below 1200 km. The critical radius can be increased only if a significant fraction of small impactors ($<10$ km) is considered. However, even if small impactors dominate, the critical radius does not exceed 1400 km if $\gamma_{li} = \gamma_{lay} = 30\%$. The critical radius can exceed 2000 km only if $\gamma_{lay} = 10\%$ and if at least 50% of the accreted mass is brought by small impactors ($x_{m,li} < 0.5$).

The accretion rate has some influence on the results only if the accretion is dominated by small impactors, as the rate at which new layers are added limits the cooling of the previously accreted layers. For simulations dominated by large impactors, as most of the energy is buried a few kilometers below the surface, the cooling is very inefficient and the progressive temperature increase only weakly depends on the accretion rate. Therefore, the size distribution of impactors is more crucial than the accretion rate in controlling the thermal evolution of growing satellites. However, as illustrated by the comparison between $\gamma_{li} = \gamma_{lay} = 10\%$ and $\gamma_{li} = \gamma_{lay} = 30\%$ in Fig. 5, the energy conversion rate remains the most crucial parameters, and we explore in more details the sensitivity of our results to $\gamma_{lay}$ and $\gamma_{li}$ in the next subsection.

### 4.4. Influence of the energy conversion parameters, $\gamma_{lay}$ and $\gamma_{li}$

As shown in Fig. 6, for $x_{m,li} = 33\%$ and $\tau_{acc} = 0.15 M_{Titan}$/Myr, $\gamma_{lay}$ and $\gamma_{li}$ must be smaller than 0.12 to allow the accretion of a body larger than 2000 km without significant melting. Conversion parameters as low as 0.1 correspond to the lowest value usually considered in previous studies (e.g., Squyres et al., 1988; Coradini et al., 1995). Such low values could be obtained for small impactors, but are probably a strong underestimation for large impactors. Fig. 6 also illustrates the relatively weak

<figure><img src="image_374.png" /><figcaption>**Fig. 6.** Critical radius $R_{crit}$ (above which more than 5% of the volume of the icy moon has a temperature larger than the melting temperature) as a function of the energy conversion coefficients ($\gamma_{lay}$ and $\gamma_{li}$) for two density values ($\rho = 1500$ kg m$^{-3}$ and $\rho = 2000$ kg m$^{-3}$). In these models, the energy conversion coefficients are set to be equal $\gamma_{lay} = \gamma_{li}$, the accretion rate is set to 0.15 $M_{Titan}$/Myr and the mass fraction of material accreted from large impacts is $x_{m,li} = 33\%$.</figcaption></figure>
more impact energy. Hence, the temperature increase would be higher and melting even more likely. Therefore, the maximal radii of the accreted satellite reached without significant melting in our simulations can be considered as upper limits.

Based on our simulations, when more than 10% of the accreted mass is brought by impactors larger than 1 km, it seems unlikely that a satellite larger than 2000 km may accrete without significant melting unless the environment is extremely cold and the conversion rate of impact energy unrealistically low (<10–15%). If the accretion is dominated by km-size impactors, impact-induced melting may occur for radii as small as 1100–1500 km. Above this critical radius, separation between liquid water and rock should initiate, thus leading to the accumulation of dense rock blocks above the undifferentiated core consisting of a mixture of rock and ice (e.g., Kirk and Stevenson, 1987). The dense layer of accumulated rock is gravitationally unstable, and in such conditions it is difficult to avoid subsequent full separation of rock and ice phases. Depending on the size of the core and thickness of the rocky layer, the differentiation may be catastrophic (Kirk and Stevenson, 1987) or more gradual (Nagel et al., 2004). Recently, O'Rourke and Stevenson (2014) showed that although rock–ice separation may be delayed by double-diffusive convection in the ice–rock interior, ice melting due to progressive radiogenic heating and subsequent differentiation cannot be prevented. Further modeling efforts are needed to better understand the processes controlling rock–ice segregation and how the internal structure inherited from the accretion process has evolved to the present-day state.

A series of arguments now questions the apparent partially differentiated state of Callisto and Titan, suggested by their elevated moment of inertia as estimated using the Radau-Darwin Approximation (e.g., Anderson et al., 2001; less et al., 2010; Gao and Stevenson, 2013). On Titan, the existence of a non-negligible degree-three in the gravity field as well as significant topography suggest that non-hydrostatic effects may significantly affect the estimation of the Moment-of-Inertia factor (less et al., 2010; Gao and Stevenson, 2013; Baland et al., in press) and that the Mol factor may be significantly smaller than the value estimated from the Radau-Darwin Approximation. On Callisto, similar non-hydrostatic contributions originating in the lithosphere may also affect the estimation of its moment of inertia (McKinnon, 1997; Gao and Stevenson, 2013). On these two moons, the hydrostatic dynamical flattening is relatively small as they orbit relatively far from their planet, and therefore the non-hydrostatic contributions need to be correctly estimated in order to accurately infer the moment of inertia and the density profile of their interior. On Callisto, future measurements performed by the ESA JUICE mission that will be launched in 2022 (Grasset, 2013) will provide constraints on the non-hydrostatic contribution by measuring independently the different quadrupole coefficients, as well as by estimating the degree three and four coefficients of the gravity field. On Titan, future measurements during Cassini flybys will also permit a better determination of the degree-four (less, 2012), which will provide pertinent tests on the topography compensation process in the outer ice shell (Hemingway et al., 2013; Lefevre et al., 2014), and consequently on the non-hydrostatic corrections required to infer more precisely the moment of inertia.

## Acknowledgments

The authors thank the anonymous reviewers for constructive comments. J. Monteux is funded by Agence Nationale de la Recherche (Accretis decision No. ANR-10-PDOC-001-01). The research leading to these results has received funding from the European Research Council under the European Community's Seventh Framework Programme (FP7/2007-2013 Grant Agreement No. 259285).

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<figure><img src="image_375.png" /></figure>

<figure><img src="image_376.png" /></figure>

# Evolution of Titan's atmosphere during the Late Heavy Bombardment

<figure><img src="image_377.png" /></figure>

CrossMark

Nadejda Marounina<sup>a,*</sup>, Gabriel Tobie<sup>a</sup>, Sabrina Carpy<sup>a</sup>, Julien Monteux<sup>a,b</sup>, Benjamin Charnay<sup>c</sup>, Olivier Grasset<sup>a</sup>

## ARTICLE INFO

**Article history:**

Received 4 November 2014

Revised 20 April 2015

Accepted 11 May 2015

Available online 19 May 2015

**Keywords:**

Titan, atmosphere

Impact processes

Atmosphere, evolution

## ABSTRACT

The mass and composition of Titan's massive atmosphere, which is dominated by $N_2$ and $CH_4$ at present, have probably varied all along its history owing to a combination of exogenous and endogenous processes. In the present study, we investigate its fate during the Late Heavy Bombardment (LHB) by modeling the competitive loss and supply of volatiles by cometary impacts and their consequences on the atmospheric balance. For surface albedos ranging between 0.1 and 0.7, we examine the emergence of an atmosphere during the LHB as well as the evolution of a primitive atmosphere with various masses and compositions prior to this event, accounting for impact-induced crustal $NH_3-N_2$ conversion and subsequent outgassing as well as impact-induced atmospheric erosion. By considering an impactor population characteristic of the LHB, we show that the generation of a $N_2$-rich atmosphere with a mass equivalent to the present-day one requires ammonia mass fraction of 2–5%, depending on surface albedos, in an icy layer of at least 50 km below the surface, implying an undifferentiated interior at the time of LHB. Except for high surface albedos ($A_s \ge 0.7$) where most of the released $N_2$ remain frozen at the surface, our calculations indicate that the high-velocity impacts led to a strong atmospheric erosion. For a differentiated Titan with a thin ammonia-enriched crust ($\le 5$ km) and $A_s < 0.6$, any atmosphere pre-existing before the LHB should be more than 5 times more massive than at present, in order to sustain an atmosphere equivalent to the present-day one. This implies that either a massive atmosphere was formed on Titan during its accretion or that the nitrogen-rich atmosphere was generated after the LHB.

© 2015 Elsevier Inc. All rights reserved.

## 1. Introduction

Saturn's largest satellite Titan is the only satellite in the Solar System possessing a dense atmosphere. Presently, it is composed predominantly of $N_2$ (~98%) and $CH_4$ (~2%) (e.g. Griffith et al., 2013). However, this atmospheric composition has probably varied through time since the accretion of the satellite due to various external and internal processes. Owing to continuous photochemical destruction and atmospheric escape, the lifetime of atmospheric methane is currently of the order of ~20 Ma (e.g. Griffith et al., 2013). Moreover, the isotopic $^{13}$C/$^{12}$C ratio in $CH_4$, measured by the mass spectrometer of the Huygens probe (GCMS) (Niemann et al., 2010), indicates that the present-day methane is not fractionated relative to Solar System standards, implying that it has been recently injected in the atmosphere (less than 1 Gyr ago (Mandt et al., 2012)).

The origin of $N_2$ is probably more ancient. The main constraint on its origin is provided by the $^{36}$Ar/$N_2$ ratio measured by the Huygens GCMS (~2.7 $\times 10^{-7}$, Niemann et al. (2010)), which is ~3 $\times 10^5$ times smaller than the solar value (Owen, 1982). As $N_2$ and $^{36}$Ar should be trapped in similar rates either by direct condensation or clathration in the solar nebula (Owen, 1982; Mousis et al., 2002), this low value indicates that the nitrogen was not originally captured as $N_2$ but as easily condensible nitrogen compounds such as $NH_3$ (Atreya et al., 2009). Moreover, Mandt et al. (2014) showed that the $^{14}$N/$^{15}$N ratio measured in Titan's $N_2$ (Niemann et al., 2010) is consistent with isotopic ratio recently inferred from $NH_2$ radicals produced by the photodissociation of $NH_3$ in comets (Rousselot et al., 2014; Shinnaka et al., 2014), providing additional evidence for ammonia as the main source of nitrogen on Titan. Several mechanisms have been proposed to explain the conversion of $NH_3$ into $N_2$ at Titan's conditions: photochemical conversion (Atreya et al., 1978), impact-induced conversion in the atmosphere (McKay et al., 1988; Ishimaru et al., 2011) or in a $NH_3$-enriched icy crust (Sekine et al., 2011), as well as endogenic processes (Glein

<footnote><sup>a</sup> Laboratoire de Planétologie et Géodynamique, Université de Nantes, CNRS, UMR-6112, 2, rue de la Houssinière, 44322 Nantes cedex 03, France</footnote>

<footnote><sup>b</sup> Laboratoire Magmas et Volcans, Université Blaise Pascal - CNRS - IRD, OPGC, 5 rue Kessler, 63038 Clermont-Ferrand, France</footnote>

<footnote><sup>c</sup> Virtual Planetary Laboratory, University of Washington, Seattle, WA 98195, USA</footnote>

<footnote>* Corresponding author.</footnote>
et al., 2009; Tobie et al., 2012). Here we focus on the conversion proposed by Sekine et al. (2011).

Delivery of volatiles by impact has likely occurred all along Titan's history (Griffith and Zahnle, 1995), with more intense flux during the accretion period and the Late Heavy Bombardment (LHB). Following the Nice model (Gomes et al., 2005; Morbidelli et al., 2005; Tsiganis et al., 2005), this intense bombardment would have affected the entire Solar System, due to the destabilization of the planetesimal disk beyond Neptune's orbit. Because of the gravitational focusing of Saturn (Zahnle et al., 2003), the cumulative mass delivered on Titan during the LHB is estimated to $3 \times 10^{20}$ kg (Barr et al., 2010). This intense bombardment, characterized by high impact velocities ($\gg v_{esc}$, the escape velocity of the planet or the satellite), may have supplied a huge amount of volatiles (e.g. CH$_4$ and NH$_3$) either by direct contribution from impactor volatilization or by impact-induced degassing of Titan's crust, as suggested by Sekine et al. (2011). High-velocity impactors are also expected to erode the atmosphere during the impact. Studies of impact-induced atmospheric erosion based either on analytic approaches or numerical simulations have been mostly focused on Mars and the Earth (Melosh and Vickery, 1989; Svetsov, 2000, 2007; Genda and Abe, 2005; Shuvalov, 2009). Parameterizations of impact-induced erosion were used by Pham et al. (2011) and de Niem et al. (2012) to investigate the atmospheric balance between erosion and volatile supply during the LHB on Mars, Earth and Venus. Here we follow a similar approach for Titan, combining a pressure-induced ammonia conversion from Sekine et al. (2011) and a parameterization of atmospheric erosion by impact from Shuvalov (2010).

To investigate the predominant mechanism that governs the fate of Titan's atmosphere during the LHB, we combine a stochastic approach for the impactor sampling and we monitor the atmospheric mass balance between the supply of N$_2$ and CH$_4$ by both impactor and crustal degassing and atmospheric erosion induced by impact. At pressures and temperatures expected on Titan, part of the supplied volatiles may condense at the surface. To estimate the partitioning between the volatiles in the atmosphere and those condensed at the surface, we implemented the atmospheric model developed by Lorenz et al. (1999) including radiative and gas-liquid equilibrium. In the particular case of a pure N$_2$ atmosphere, we use the atmospheric equilibrium constrained from 3D GCM simulations.

A detailed description of the model is provided in the following section. Simulations of the evolution of Titan's atmosphere during the LHB, considering a wide range of initial conditions, are presented in Section 3. Implications for the origin and evolution of Titan's atmosphere are discussed in Section 4 and our conclusions are summarized in Section 5.

## 2. Model description

### 2.1. Model of the atmospheric equilibrium

The composition of Titan's atmosphere may have varied throughout Titan's history. As the mass and composition of Titan's atmosphere before the LHB is uncertain, we consider various initial conditions prior to the LHB. The presence of methane in Titan's atmosphere might only be recent and/or episodic (Tobie et al., 2006; Mandt et al., 2012). It is therefore possible that the atmosphere was composed solely of nitrogen at the time of the LHB. Then we consider either pure N$_2$ or N$_2$-CH$_4$ atmospheres up to 10 times Titan's present-day atmospheric mass. To model the radiative balance as well as the thermodynamic equilibrium between atmospheric and surface volatile reservoirs, we adopt

two different modeling approaches for pure N$_2$ and mixed N$_2$-CH$_4$ atmosphere model as detailed hereafter.

#### 2.1.1. Pure N$_2$ atmosphere

In the case of a pure N$_2$ atmosphere, the greenhouse effect is limited to collision-induced absorptions of N$_2$-N$_2$, as N$_2$ has no absorption band in the infrared and visible (Lorenz et al., 1997; Charnay et al., 2014). Equilibrium temperatures and pressures have been evaluated for surface albedos varying between 0.2 and 0.7 (see Table 1), from full 3D GCM simulations, using the Generic LMDZ code employed and described in Charnay et al. (2014), accounting for Rayleigh scattering by N$_2$ and N$_2$ condensation and precipitation at the surface. These simulations showed that the greenhouse effect is counterbalanced by the increase of atmospheric albedo due to Rayleigh diffusion, so that the surface temperature is mostly determined by the surface albedo. Moreover, although the condensation of N$_2$ and the atmospheric pressure are mostly controlled by the pole temperature, the GCM simulations showed that the average surface temperature remains a good parameter to predict the average atmospheric pressure and hence the average amount of condensed N$_2$ at the surface. For a surface albedo A$_s$ lower than 0.68, the surface temperature is above the freezing point and atmospheric N$_2$ is in equilibrium with liquid N$_2$ at the surface. For A$_s$ > 0.68, the atmospheric N$_2$ is in equilibrium with solid N$_2$. Atmospheric pressures displayed in Table 1 correspond to saturation pressures of N$_2$. Any excess of N$_2$ is condensed at the surface. Although latitudinal variations of nitrogen condensation are expected based on GCM simulations (Charnay et al., 2014), we assume a uniform distribution of solids or liquids at the surface, and we consider the average temperature and pressure as representative of the surface conditions. When the atmospheric pressure drops below the saturation pressure, the surface temperature is maintained at the same value i.e. the value predicted and displayed in Table 1, for a given surface albedo. As the greenhouse effect is very small (~1–2 K), assuming a constant temperature once the pressure is below the saturation pressure remains a very good approximation.

#### 2.1.2. N$_2$-CH$_4$ atmosphere

Once CH$_4$ is incorporated, the radiative equilibrium and the thermal structure of the atmosphere is significantly affected. In addition to the N$_2$-N$_2$ collision-induced absorption, those of CH$_4$-N$_2$ and N$_2$-H$_2$ (resulting from CH$_4$ dissociation) as well as absorption of sunlight by CH$_4$ and photochemical haze in the upper atmosphere contribute to the radiative balance (McKay et al., 1999; Lorenz et al., 1999). To determine the atmospheric radiative equilibrium, we use the parameterized model of Lorenz et al. (1999), based on a semi-empirical gray radiative formulation. It requires the presence of a small amount of methane in the atmosphere and therefore cannot be applied to the pure N$_2$ case. For the thermodynamical equilibrium, the vapor phase is described by an ideal gas. The liquid phase is described following the regular solution theory (Thompson, 1985; McKay et al., 1999) using the saturation

<table><caption>**Table 1**
Average surface pressure, P$_s$, and temperature, T$_s$, as a function of surface albedo, A$_s$, evaluated using GCM simulations for a pure N$_2$ atmosphere.</caption><thead><tr><th>A<sub>s</sub></th><th>P<sub>s</sub> (bar)</th><th>T<sub>s</sub> (K)</th><th>Nitrogen condensed state</th></tr></thead><tbody><tr><td>0.2</td><td>1.14</td><td>79.9</td><td>Liquid</td></tr><tr><td>0.3</td><td>0.89</td><td>77.5</td><td>Liquid</td></tr><tr><td>0.4</td><td>0.63</td><td>75.1</td><td>Liquid</td></tr><tr><td>0.5</td><td>0.41</td><td>72.0</td><td>Liquid</td></tr><tr><td>0.6</td><td>0.21</td><td>67.2</td><td>Liquid</td></tr><tr><td>0.7</td><td>0.09</td><td>61.7</td><td>Solid</td></tr></tbody></table>
vapor pressures from NIST Chemistry WebBook (Linstrom and Mallard, 2015).

## 2.2. Volatile supply by impact

During an impact, volatile compounds can be released in the atmosphere by degassing of both impactor and target materials. The amount of released volatiles depends on their compositions as well as on the impact-induced pressure increase. In the present study, we consider only the release of N₂ and CH₄. We consider an impact-induced conversion scenario of NH₃ into N₂ following Sekine et al. (2011). NH₃ is assumed to be the only source of nitrogen, as it is the main N-bearing molecule detected in comets (e.g. Bockelée-Morvan and Crovisier, 2004). In the following, we discuss the abundances and the localization of the chemical species of interest and the parametrization of the conversion of the NH₃ into N₂ in the impactor and in the Titan's icy crust.

### 2.2.1. Volatile content of the impactor and the target

Assuming a cometary composition for the LHB impactors, NH₃ and CH₄ concentrations vary typically between 0.5% and 1%, 0.1% and 1% relative to bulk water (Bockelée-Morvan and Crovisier, 2004), respectively. In our model, the mass fractions are fixed to 1% for NH₃ and to 0% or 1% for CH₄. Other volatiles such as CO, CO₂, HCN, and H₂S, which are also present in significant abundances in comets and might be converted upon impacts are, for simplicity, not considered in our model.

The composition of the target (i.e. Titan's crust) is determined by the composition of the primordial bricks that formed Titan as well as by chemical differentiation processes subsequent to the accretion of the satellite. Abundance of ammonia in Enceladus' plume (Waite et al., 2009), volatile enrichments in Saturn's atmosphere (Hersant et al., 2008) and models of Saturn's subnebula (e.g. Alibert and Mousis, 2007) suggest that Titan's building blocks contained typically between ~1% and 5% of NH₃. If Titan was undifferentiated at the time of the LHB, NH₃ would be rather uniformly distributed in the interior and the concentration of the outer layer would be equivalent to the primordial value. If Titan was differentiated, NH₃ would be mostly contained in the subsurface ocean owing to the high solubility of ammonia into liquid water (Grasset et al., 2000; Tobie et al., 2012). However, even in this condition, some NH₃ might have been incorporated into the crust by intrusion of ammonia-enriched liquids (Kargel, 1991; Fortes et al., 2007; Choukroun and Grasset, 2010). To encompass various differentiation states for Titan's interior at the time of the LHB, either homogeneous distribution of NH₃ or NH₃-enriched icy crust of 1-to-5-km thickness must be considered, with ammonia mass fractions ranging between 1% and 5%.

### 2.2.2. Degassing by impact

For a fixed ammonia content, the amount of N₂ that is degassed upon an impact depends on the pressure increase and the volume affected by the impact (see Fig. 1). During an impact, the kinetic energy of the impactor is transferred to the satellite as a shock wave that propagates downward in the target and upward in the projectile. Two domains can be considered (Croft, 1982). The first domain is a truncated spherical zone called an isobaric core. It is a zone below the target surface where the pressure is constant. The burial depth of the isobaric core depends on the impactor velocity. Beyond the isobaric core limit, the shock wave attenuates and pressure decays following a power law, where the exponent also depends on the impactor velocity. The distribution of the impact pressure also depends on the impact angle (e.g. Pierazzo, 2000; Kraus et al., 2011). Here, we consider only vertical impacts, which maximize impact-induced degassing.

The impact-induced pressure increase in the isobaric core can be expressed as (Senshu, 2002):

$$P_{ic} = \frac{\rho_t}{2} v_i \left[ C_t + \frac{1}{2} S_t v_i \right], \quad \text{for } r < R_{ic}, \qquad (1)$$

where $\rho_t$ is the density of the icy crust, $C_t$ its bulk sound wave velocity, $S_t$ a dimensionless constant, $v_i$ the velocity of the impactor immediately above the impact site in m s⁻¹, r the distance from the center of the isobaric core and $R_{ic}$ is the radius of the isobaric core. Material constants for pure water ice are summarized in Table 2. To account for the deceleration of the impactor owing to the atmospheric drag, the impactor velocity is computed following Svetsov (2000):

$$v_i = v_0 \exp \left( - \frac{\rho_{atm}}{\rho_i} \left( \frac{H}{2r_i} + \frac{4H^2 \rho_{atm}^{1/2}}{3r_i^2 \rho_i^{1/2}} + \frac{2H^3 \rho_{atm}}{r_i^3 \rho_i} \right) \right), \qquad (2)$$

where $v_0$ is the initial impactor velocity before entering Titan's atmosphere, $\rho_{atm}$ the atmospheric density, $\rho_i$ is the density of the impactor, hereafter considered equal to the density of pure water ice (see Table 2), and H is the atmospheric scale height $H = RT_s/(Mg_T)$ where R is the ideal gas constant, T_s is the surface temperature, M is the mean molar molecular mass of the atmosphere and g_T is Titan's surface gravity (see Table 2). When the velocity computed with Eq. (2) is equal to zero, the impactor is considered to be disintegrated into the atmosphere. Otherwise, the impactor hits the surface with the velocity $v_i$.

The size of the isobaric core depends on the size of the impactor, the impactor velocity and the physical properties of the compressed material, e.g. porosity or composition (Davison et al., 2010; Barr and Citron, 2011). To characterize the isobaric core radius $R_{ic}$ and burial $z_{ic}$, we use scaling laws from Kraus et al. (2011), for pure water ice:

$$\log_{10} \left( \frac{R_{ic}}{r_i} \right) = 0.22 - 0.18 \log_{10} (v_i/1000), \qquad (3)$$

where $r_i$ is the radius of the impactor. The depth of the burial of the isobaric core $z_{ic}$ is described by:

$$\log_{10} \left( \frac{z_{ic}}{r_i} \right) = -0.17 - 0.09 \log_{10} (v_i/1000). \qquad (4)$$

Outside the isobaric core, the pressure decreases with distance (Croft, 1982; Pierazzo and Melosh, 2000):

$$P(r) = P_{ic} \left( \frac{r}{R_{ic}} \right)^{-n}, \quad \text{for } r > R_{ic}, \qquad (5)$$

with n being defined as (Kraus et al., 2011):

$$n = 2.9 + 0.47 \log_{10} (v_i/1000). \qquad (6)$$

Finally, the total amount of N₂ produced from NH₃ dissociation is determined using the N₂ production efficiency f, taken from Sekine et al. (2011):

$$f = \begin{cases} 0\%, & \text{for } 0 < P < 8 \text{ GPa}, \\ 100 \times \frac{P-8}{23-8}\%, & \text{for } 8 < P < 23 \text{ GPa}, \\ 100\%, & \text{for } P > 23 \text{ GPa}. \end{cases} \qquad (7)$$

For the impactor, we assume a uniform pressure equal to the isobaric core pressure. CH₄ contained in the impactor – if any – is assumed to be entirely released into the atmosphere.

## 2.3. Atmospheric erosion by impact

When entering the atmosphere, the impactor induces an atmospheric escape following two major processes: (i) Along its trajectory through the atmosphere, the impactor forms a rarefied hot