The ammonia–water system at high pressures: Implications for the methane of Titan

ARTICLE IN PRESS

Planetary and Space Science 53 (2005) 371–384 www.elsevier.com/locate/pss

The ammonia–water system at high pressures: Implications for the methane of Titan O. Grasset, J. Pargamin Faculte des Sciences, Lab. Plancetologie Geodynamique, UMR-CNRS 6112, Universite´ de Nantes, 2, Rue de la Houssinie´re, 44322 Nantes, Cedex 03, France Received 24 March 2004; received in revised form 7 July 2004; accepted 16 September 2004 Available online 15 December 2004

Abstract The Cassini/Huygens mission will provide an accurate description of Titan’s surface features. One important outcome of these data is that it will help for understanding the processes of methane exchange between Titan’s interior and its atmosphere. Such a correlation between surface features and internal processes involving methane will be highly simplified if the nature of methane reservoirs is understood. In this paper, the behavior of methane within Titan is investigated using both data on methane clathrate stability and data on the ammonia–water system. A mathematical description of the different liquidus of the ammonia–water system is proposed. It is shown that the low pressure and water rich domain of the system is very well constrained. On the contrary, both high pressure ices and ammonia hydrates domains are still very badly understood because of the lack of experimental data. Nonetheless, several important characteristics of both ices and hydrates stability are described. These data are used for proposing a new model which computes the thermodynamical characteristics of the liquid layer within Titan. This provides new constraints on the temperature and composition fields within the liquid layer of Titan which indicates that the dissociation of methane clathrates in the deep interior is almost impossible. In the last part, the methane clathrate behavior within the different layer of Titan’s interior is investigated. Due to the density contrasts between methane clathrates and ices, it will be shown that methane is certainly trapped within large clathrate reservoirs below the upper conductive lid of Titan. Further ascent and dissociation of clathrate into gaseous methane + ice must then be associated with tectonic and/or volcanic processes which allow rapid ascent without cooling of clathrates. Indeed, the dissociation is only possible at very shallow depth only if hot material from the ice layer can reach the surface rapidly. r 2004 Elsevier Ltd. All rights reserved. Keywords: Methane clathrates; Ammonia hydrates; Phase diagrams; Titan’s interior

1. Introduction: Titan and the methane Titan, the largest satellite of Saturn is surrounded by a thick atmosphere, mainly composed of nitrogen and methane (Sagan and Thompson, 1984; Lellouch et al., 1989; Samuelson, 2003). This methane is very unstable and is dissociated by photochemical processes in the stratosphere into ethane and hydrogen. Since the hydrogen escapes rapidly, the photodissociation is Corresponding author. Tel.: +33 2 51 12 54 69; fax: +33 2 51 12 52 68. E-mail address: [email protected] (O. Grasset).

0032-0633/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.pss.2004.09.062

basically irreversible. Assuming that no methane is added to the atmosphere, the present amount must be depleted in about 10 millions years (Strobel, 1982; Lunine, 1994). Two hypothesis can then be explored. The present amount of methane may be a remnant of a very large methane reservoir which is almost fully depleted, in which case we are witnessing the last stage of an atmosphere initially rich in methane. However, using an evolutionary turbulent model of the Saturn’s subnebula, Mousis et al. (2002) have shown that the total amount of methane which has been trapped in Titan’s interior is almost 30 times larger than the amount required for having the present atmospheric

ARTICLE IN PRESS 372

O. Grasset, J. Pargamin / Planetary and Space Science 53 (2005) 371–384

abundance after 4.5 billions years of irreversible photodissociation. It is then difficult to explain why only 3% of the methane may have come to the surface while the largest amount was trapped deep into Titan’s interior. That’s why the second hypothesis which assumes that methane is added constantly to the atmosphere either from below (internal and surface sources: ocean, lakes, volcanism,. . .), or from above (comets) is more favored. The external source of methane provided by comets was probably important early in Titan’s history (Jones and Lewis, 1987; Zahnle et al., 1992). But the present flux of these objects in the region of Saturn is too low for explaining a methane replenishment of the atmosphere (Lunine, 1994). It is then probable that the methane replenishment is due to important exchange of methane between the ground and the surface. The methane was originally trapped within ices forming clathrates (Prinn and Fegley, 1981; Lunine and Stevenson, 1985; Mousis et al., 2002). The present replenishment of methane of Titan’s atmosphere is commonly attributed to important exchanges through the icy surface. These transfers are possible only if methane is present in large quantities within the upper ice I layer of Titan. In order to explain the methane abundance in ice I, the most common scenario is the scenario proposed initially by Lunine and Stevenson (1987). The methane was initially trapped within Titan as clathrate. During the accretion, the clathrates were pressurized above their stability zone and methane was free to escape its icy cages. Following this dissociation, the ascent of methane through icy layers and ammonia–water liquid was supposed to be both rapid and easy mainly because methane has a very low density, its viscosity is weak, and it must be highly superheated during its ascent (Lunine and Stevenson, 1987). During the last two decades, questions concerning methane were mainly focused on the nature of methane reservoirs within ice I. In a precise review, Lunine (1994) indicated that, from all the scenarii available, the most probable was that methane was trapped into an underground layer within pores of the crust. This scenario was indeed the only one consistent with all the constraints (chemical, radar, near-infrared and tidal). But other methane reservoirs such as large liquid oceans at the surface or shallow liquid layers were also possible. Recently, Campbell et al. (2003) have shown from Arecibo radar observations that large liquid hydrocarbon reservoirs must be present on the surface. Nonetheless, Griffith et al. (2003) show that Titan’s leading hemisphere albedo resembles that of Ganymede’s leading hemisphere. Using infrared narrow windows for which Titan’s atmosphere is transparent, they find that Titan’s surface is mainly composed of water ice. Thus, the previous review from Lunine (1994) indicating that methane might be trapped in the interior seems to

be more valid than ever. But the question which is still not resolved is about the nature of these reservoirs. Hirai et al. (2000) suggested that a partial dissociation of methane clathrates into ice VI + fluid occurs around 1.5 GPa and the complete dissociation above 2.5 GPa. But this dissociation was later interpreted by the same authors as the result of the very high compression rate imposed in their experiments (Hirai et al., 2001). On the other hand, Loveday et al. (2001a) have shown that methane clathrates are actually very stable at high pressures. Finally, Loveday et al. (2001c) proved that the interpretations of Hirai et al. (2000, 2001) of diffraction data were erroneous, and proposed a reconciliation of the various experimental results. Despite slight discrepancies concerning the pressure of transitions, almost all recent studies favor the stability of clathrates structures with pressure. A transition from structure I to an hexagonal structure II occurs around 1 GPa, which is followed by a transition to a ‘‘filled ice’’ structure above 2–3 GPa (Chou et al., 2000; Loveday et al., 2001b; Hirai et al., 2003; Takashi and Kiyoyuki, 2003). These theoretical and experimental works indicate that dissociation of methane in the early stages of Titan’s history did not occur and the present methane reservoirs near the surface might be clathrate reservoirs. On the other hand, methane clathrates may be dissociated in Titan’s interior because ammonia is present. Ammonia hydrates are stable in the subnebulae of Jupiter and Saturn according to thermochemical models (Prinn and Fegley, 1989; Mousis et al., 2002) and it is believed that ammonia is abundant within Titan (Consolmagno and Lewis, 1978; Lunine and Stevenson, 1987; Grasset and Sotin, 1996; Mousis et al., 2002). Ammonia tremendously lowers the melting point of water and favors the existence of a deep liquid layer within the satellite (Grasset et al., 2000; Sohl et al., 2003). Its importance for the understanding of Titan’s interior motivated both experimental and theoretical works in the pressure range [0–1 GPa] during the last two decades (Johnson and Nicol, 1987; Lunine and Stevenson, 1987; Croft et al., 1988; Dalrymple et al., 1988; Cynn et al., 1989; Hogenboom et al., 1989; Hogenboom and Kargel, 1990; Boone and Nicol, 1991; Grasset et al., 1995; Sotin et al., 1998; Hogenboom et al., 1997; Leliwa-Kopystynski et al., 2002; Mousis et al., 2002). In the next section, a precise description of the ammonia–water system is proposed using both experimental and theoretical data from the last two decades. A putative Temperature—Composition—Pressure phase diagram is presented. This diagram is exploited in Section 3 for describing Titan’s interior. The main difficulty is to define thermodynamic conditions of liquid–ice interfaces since, at each depth, the temperature of the interface is a function of both composition and pressure. In a previous paper, Grasset et al. (2000)

ARTICLE IN PRESS O. Grasset, J. Pargamin / Planetary and Space Science 53 (2005) 371–384

373

range [0–1 GPa]. The three important characteristics of the diagram known at moderate pressure are: (i) the melting temperature of ices decreases strongly when ammonia compound is added; (ii) there is a peritectic curve corresponding to the reaction L+Ice 2 Dihydrate; (iii) the eutectic curve (L 2 Dihydrate + Monohydrate) is located at very low temperature (around 180 K). The first point has been extensively studied both at ambient pressure (see Kargel, 1992 for a synthesis) and high pressure (Grasset et al., 1995; Hogenboom et al., 1997; Leliwa-Kopystynski et al., 2002). Nonetheless, the experimental data available are mainly located in the [0–300] MPa pressure range and in the water-rich domain (Fig. 1). Thus, except for ice I, the influence of ammonia on the melting temperature of high pressure ice polymorphs is not well known. Both eutectic and peritectic paths in the P-T-X space are unknown. It seems nonetheless that both composition and melting temperature of the eutectic system do not vary strongly with pressure (Hogenboom et al., 1997). In this section, a mathematical description of the ammonia–water phase diagram is proposed which is based on the available experimental data. Let us assume that liquidus surfaces can be described with elliptic paraboloids such as

proposed to use a second order polynomial approximation which provided rough estimates of equilibrium temperature because it was based on the schematic P-TX diagram from Sotin et al. (1998). Furthermore, the coefficients of this polynom were a function of the initial ammonia concentration of Titan and were only provided for 5 and 15 wt% of ammonia. Recently, LeliwaKopystynski et al. (2002) have proposed a relation which was more precise for describing (P-T-X ) relations on liquidus surfaces. Unfortunately, this relation, based on their experiments, is only precise for pressures below 300 MPa and cannot be used for describing the interface between high pressure ice and liquid layer. Furthermore, application of these results to Titan is not straightforward. Thus, a more general algorithm is proposed for computing thermodynamic conditions of stability for both low and high pressure interfaces and for any ammonia initial composition. Plausible temperature and density profiles are then presented. Finally, thermodynamical conditions within Titan are confronted to the present experimental data about clathrates in order to provide new insights on methane stability within Titan.

2. The ammonia–water phase diagram—a synthesis 2.1. Experimental data and inversion model

x2 y2 z þ ¼ ; a2 b2 c

The water rich region of the ammonia water phase diagram presents at least six solid phases in the pressure

where a; b; and c are parameters to be described. Coordinates x; y; and z can be found from T; P; X using

(1)

50 Grasset et al. (1995) Jonhson et Nicol (1987) Hogenboom et al. (1997)

40 Composition (NH3 wt%)

Leliwa Kopistynski et al. (2002) Ice I

30

Ice III

Ice V

Ammonia Dihydrate NH3.2H20

Ice VI

Ammonia Monohydrate NH3.H20

20

10

0

200

400

600

800

1000

1200

Pressure (MPa) Fig. 1. Experimental data in the Pressure–Composition space used for describing the ammonia–water system. The low-pressure domain (below 300 MPa) has been well investigated by different authors and the data are consistent with both data at room pressure (Kargel, 1990; not plotted) and data for pure water (Chizhov, 1993, not plotted). Above 300 MPa, the data are rare and below 15 wt% NH3 : Data from Johnson and Nicol (1987) are not consistent with those at ambient pressure.

ARTICLE IN PRESS O. Grasset, J. Pargamin / Planetary and Space Science 53 (2005) 371–384

374

the transformation equation: 0 1 0 1 T  T0 x B C B C @ y A ¼ Rða; b; gÞ  @ P  P0 A; X  X0 z

the liquidus for ammonia concentration lower than 5% is very well constrained both by the pure water melting curve and the data from Leliwa-Kopystynski et al. (2002). A higher concentrations, we have used data from Hogenboom et al. (1997). These authors did not suggest in their work that they were working with ice III between 150 and 300 MPa, but since their data at low concentration are consistent with those of LeliwaKopystynski et al., we have interpreted their data in this range of pressure as those of the ‘‘ice III + Liquid’’ liquidus. This interpretation is consistent with the results of Fortes et al. (2003a,b) which suggest an incongruent melting curve between 250 and 350 MPa. Furthermore, data from Grasset et al. (1995) with 15 wt% of ammonia have also been used. To our knowledge, no data are available in the ammonia rich region of this domain. With the available data, the fit is rather accurate since the distribution of errors is almost Gaussian. At higher pressures (Fig. 2c–d), the available data are from Chizhov (1993) in the pure water case, from Grasset et al. (1995) for a 15 wt% NH3 rich solution, and from Johnson and Nicol (1987). For both ice V and ice VI, it is difficult to reconcile the data. It is possible to fit together data from Chizhov (1993) and Grasset et al. (1995), but the data from Johnson and Nicol (1987) suggest a surface which cannot intersect the melting curve of pure water. Thus, diagram of errors show a very large dispersion around the zero due to the data of Johnson and Nicol (1987). Parameters proposed in Table 2 are only rough estimates of the paraboloid function coefficients because experimental constraints are rare and not fully consistent. Hogenboom et al. (1997) have studied the liquidus evolution of the system with ammonia concentration larger than 32.1 wt%. These data added to the work of Kargel (1992) at ambient pressure can be used for computing an estimate of the parameters which describe the liquidus of both ammonia dihydrate I (DHI) and ammonia monohydrate I (MHI) at low pressure. The fits are almost perfect since relative errors are always lower than 0.3 K (Fig. 2e–f). This is only due to the fact that the number of data is small ðo36Þ: Once again, estimates are rather imprecise because of the lack of

(2)

where ðT 0 ; P0 ; X 0 Þ are the coordinates of the origin of the Cartesian system, R is the matrix of rotation and ða; b; gÞ the Euler angles. Combination of (1) and (2) indicates that liquidus surfaces can be described using nine parameters (see Table 1). Parameters have been computed for each liquidus surfaces using a generalized inversion method (Tarantola and Valette, 1982) and the experimental data found in the litterature (Table 2, Fig. 1). 2.2. The liquidus surfaces Numerical results are shown in Table 1 and in Fig. 2. Left part of Fig. 2 represents the distribution of the relative temperature differences between data ðT exp Þ and estimates from elliptic paraboloids ðT ep Þ: e¼

ðT exp  T ep Þ ; s

(3)

with s the standard deviation of the measured temperatures. In general, s is not given by the authors of the different experiments. Except for Grasset et al. (1995) for which data were available, a standard value of 1% of the liquidus temperature has been assumed. A Gaussian distribution centered at zero is representative of a good fit. Experimental data and elliptic surfaces in the T-P-X space are also plotted on the right part of Fig. 2. In Fig. 2a, it is shown that the ice I liquidus surface is very well constrained. The 130 experimental data from different works are consistent and the inversion is robust. The error diagram indicates that more than 60% of the measured melting temperatures differ by less than 2 K to their elliptic paraboloid estimate. Thus, we argue that the ice I liquidus surface is well described using the parameters given in Table 1. The liquidus of ice III is given in Fig. 2b. Forty four data have been used for the inversion. The description of Table 1 Parameters of the elliptic paraboloid function describing liquidus surfaces Parameters

ICE I

DH I

MH I

ICE III

ICE V

ICE VI

a b c a b g T0 P0 X0

199.17 1005.20 595.49 0.12 2.27 4.72 31.31 11.58 46.62

175.75 7.24 108.83 0.19 3.14 0.16 198.82 7.03 533.21

90.71 29.30 312.36 1.19 1.50 0.14 406.90 174.40 20.33

298.76 89.80 131.50 2.86 1.29 2.84 109.01 3.69 11.49

100.00 18.13 1.44 2.87 0.98 2.93 200.01 14.94 123.6

99.98 36.13 9.79 3.27 1.11 0.27 200.29 10.72 26.26

ARTICLE IN PRESS O. Grasset, J. Pargamin / Planetary and Space Science 53 (2005) 371–384

375

Table 2 Experimental data used for the inversion problem Liquidus

Pres. range (MPa)

Composition range ðNH3 wt%Þ

Nb of points

Authors

Liquid + ice I

0.1–220 0.1 100; 200 0.1–220 0.1–220

0 0–32 0–32 0–15 3.34; 11.1; 17.6; 22.9

25 16 34 15 40

Chizhov (1993) Kargel (1992) Hogenboom et al. (1997) Johnson and Nicol (1987) Leliwa-Kopystynski et al. (2002)

Liquid + ice III

150–300 300 100–350 150–300

0 0–32 15 3.34; 4.72

9 16 6 13

Chizhov (1993) Hogenboom et al. (1997) Grasset and Sotin (1996) Leliwa-Kopystynski et al. (2002)

Liquid + ice V

270–530 340–550 300–600

0 0–11 15

19 20 7

Chizhov (1993) Johnson and Nicol (1987) Grasset and Sotin (1996)

Liquid + ice VI

480–1200 600–1800 600–1200

0 0–11 15

27 90 6

Chizhov (1993) Johnson and Nicol (1987) Grasset and Sotin (1996)

Liquid + DH

0.1 0.1–300

32–36 32–36

6 30

Kargel (1992) Hogenboom et al. (1997)

Liquid + MH

0.1 100–200

36–60 36–52

16 16

Kargel (1992) Hogenboom et al. (1997)

data. But in the particular case of the dihydrate, it appears that the liquidus surface is almost isothermal and that a more precise description is probably not of fundamental importance for planetological applications. Finally, experimental data have been obtained at pressures lower than 300 MPa. At higher pressure, the hydrates may change into denser polymorphs (Hogenboom et al., 1997; Fortes et al., 2001; Mousis et al., 2002; Fortes et al., 2003a,b) and the liquidus proposed in this section are certainly not applicable. Unfortunately, the lack of experimental data impedes to describe the liquidus of hydrates at pressures larger than 300 MPa. 2.3. Description of the ammonia– water system Characteristics of the boundaries between the different icy polymorphs have been inferred from the P-T-X curves corresponding to liquidus intersections (Fig. 3). The first interesting result is the peritectic curve L+ice+DHI along which composition varies strongly. Located around 32% at ambient pressure ðP1 Þ; its composition decreases down to 28% at 170 MPa ðP2 Þ and increases again when DHI + Liquid are in equilibrium with high pressure ices. Along the curve, temperature increases slightly from 176 K at P1 to 180 K at P2 and decreases again from P2 to E 2 : This temperature variation is consistent with the data of Hogenboom et al. obtained for a composition of

32.1 wt%. The eutectic path indicates a slight decrease of both temperature and composition with pressure. This point needs to be confirmed since the two liquidus surfaces are not fully constrained due to the lack of data in this region (Fig. 1). The ice I–III boundary presents a slight curvature towards low pressure when ammonia concentration increases. We believe that this estimate of the boundary is rather correct because both ice I and ice III liquidus surfaces are well constrained by experimental data, at least below 10 wt% of NH3 : The boundary between ice III and V presents a surprising curvature towards low pressure at high ammonia concentration. We argue that this curvature is certainly overestimated because data at high pressures are mainly those of Johnson and Nicol (1987) which overestimate the melting temperatures. This point is confirmed by the positions of both the L-ice III-ice V and L-ice V-ice VI curves in the pure water case which are slightly shifted from their exact position. The curvature implies that ice III does not exist at high ammonia concentration and must be replaced either by ice II (as suggested by Fortes et al., 2003a,b) or ice V. There is no way to conclude about this point since no experimental data are available in this domain. Fortunately, it is not of fundamental importance for the understanding of Titan’s interior as will be seen in the next section. The ammonia dihydrate I surface covers a narrow range of the P-X diagram. It seems that DHI is no

ARTICLE IN PRESS O. Grasset, J. Pargamin / Planetary and Space Science 53 (2005) 371–384 36 32 28 24 20 16 12 8 4 0

274

Ice I

0 -6

-4

-2

0

2

4

17 NH3 (% wt)

34 300

150 Pressure (MPa)

0.0

NH3 17 (%wt)

34 400

250 Pressure (MPa)

100

400 Pressure (MPa)

200

259

10

Ice III

8 6

Temp (K)

12

Nb of points

200

126

-8

(a)

Temp (K)

Nb of points

376

4

202 144 0

2 0 -3

-2

-1

0

1

2

3

273

28 24 20 16 12 8 4 0 -14

Temp (K)

Nb of points

(b)

Ice V

230 186 0

-12

-10

-8

-6

-4

-2

0

2

NH3 17 (%w t)

4

34 600

36 32 28 24 20 16 12 8 4 0 -25

336

Ice VI

-21

-17

-13

-9

-5

1

3

Temp (K)

Nb of points

(c)

263

189 0 NH 17 3( %w 34 t)

7

1100 Pressure (MPa)

1700

Temp (K)

DH I

179 176.6 174.2 30 34

t) 38 -0.1

0

0.1

0.2

150 Pressure (MPa)

300

0.3

(f)

0

199.8

MH I

Temp (K)

Nb of points

(e) 7 6 5 4 3 2 1 0 -1.2

500

( %w

16 14 12 10 8 6 4 2 0 -0.2

NH3

Nb of points

(d)

186.4 173.1 35

NH

3 ( 48 %w t)

-0.8

-0.4

0

0.4

0.8

1.2

Relative error

61

340

170 Pressure (MPa)

0

Fig. 2. (a) Diagrams of normalized errors. (b) Liquidus surfaces in the P-T-X space. Mathematical description of the liquidus surfaces is accurate for ice I, ice III, dihydrate and monohydrate. Ice V and ice VI are not described accurately because data are not consistent and too rare. An experimental effort will be needed for describing more precisely these two liquidii.

longer stable with liquid above 300 MPa. This result is in agreement with the experimental study of Mousis et al. (2002) which predicts a transition of the ammonia dihydrate I above 400 MPa. It is also consistent with the results of Fortes et al. (2003a,b). At higher pressures, the

lack of experimental data impedes to describe the liquidus of ammonia hydrates. Furthermore, the ammonia polymorph which is stable at high pressure is not well known. Fortes et al. (2003a,b) suggest that ammonia dihydrate is stable again above 450 MPa.

ARTICLE IN PRESS O. Grasset, J. Pargamin / Planetary and Space Science 53 (2005) 371–384

50

40

L + MH I 175

30

MHI+DHI+ L

172

P1I L + DH I L +D HI+ HI+ I+D I I L 176

?

E2

P2 180

202

20

L+I

245

? 231 V+L III+

I+III+L

10

L+V

V+VI+L

Composition (NH3 wt%)

E1

L + VI

L + III 248

0

200

265

260

400

600

800

Pressure (MPa) Fig. 3. Ammonia–water phase diagram in the pressure–composition space. Boundaries between the liquidus surfaces are plotted with bold lines. Bold characters are temperatures (in K).

Mousis et al. (2002) indicate that at least two new high pressure phases must exist. Finally, Hogenboom et al. (1997) and Johnson and Nicol (1987) argue that a high pressure polymorph of dihydrate must exist. Due to these large discrepancies, no boundary limits have been plotted in the high pressure domain for ammonia concentration larger than 20 wt%. We argue that experimental results are still required for describing this domain of the P-X space.

3. Methane stability within Titan The previous section proposed an overview of the PT-X diagram of the ammonia–water system. The mathematical description of the liquidus surfaces inferred from both experimental and theoretical works can now be used for providing an accurate description of the methane stability within Titan. 3.1. Thermal profiles The main planetological interest of the mathematical description proposed in the previous section is that it provides an easy way for computing ðT; P; X Þ conditions at the boundaries of an ammonia-rich liquid layer whatever the primordial composition of ammonia is. In Fig. 4, a numerical algorithm is proposed. Assuming a

377

primordial ammonia amount of the liquid layer ðx0 Þ and an initial radius for the ice I—liquid boundary ðr0i Þ; the algorithm provides in a initialization step the temperature and pressure of the two interfaces. In a second step, it computes for any ammonia enrichment of the liquid layer (corresponding to the cooling of Titan) the new characteristics of each boundary. Results obtained are plotted in Figs. 5 and 6 for a primordial ammonia composition of 5% and 10%, respectively. We believe that 5% is realistic (see discussion below) but a richer composition has also been studied for a purpose of comparison. From top to bottom, thicknesses, temperatures, radii and pressures of the interfaces are plotted as a function of the ammonia concentration. From left to right, the ammonia concentration is enriched in the liquid layer from the primordial composition 5% (Fig. 5), or 10% (Fig. 6). It is important to notice that each position along the x-axis represents an equilibrium state and that the corresponding enrichment is related to the degree of cooling of the satellite. The curve of the ice I layer thickness presents a peculiar form which indicates that solidification process can be splitted into two separate steps. First, the temperature decrease in the liquid layer involves crystallization of ices both at the top and at the bottom of the liquid layer. This scheme is the classic one (Grasset et al., 2000; Sohl et al., 2003). But surprisingly, a second process occurs when ammonia concentration in the liquid layer reaches 15% (Fig. 5) or 18% (Fig. 6). Further ammonia enrichment is associated with a thinning of the ice I layer and the solidification process is only due to the important crystallization of ice V at the bottom of the liquid layer. In the extreme case, cooling may induce ammonia saturation in the liquid layer and solidification of ammonia hydrates leading to complete solidification of the satellite. But it is worth noting that such a case is unrealistic because the very high viscosity of ices at low temperature impedes vigorous convection and efficient cooling of ammoniarich liquids (Grasset et al., 1995, 2000; Sohl et al., 2003). Dashed lines on top of Figs. 6 and 7 correspond to the plausible values of the ice I layer thickness (Sohl et al., 2003). The corresponding gray areas indicate the most probable zone which describe the characteristics of the liquid layer boundaries. The ammonia liquid layer must be relatively thick (300–550 km) and its ammonia concentration lies between 7.5 and 17.5 wt%. The high pressure ice in contact with the liquid layer is probably ice V, and pressures at the interfaces are around 100 MPa at the top, and in the range 500–600 MPa at the bottom. Thermal profile within Titan has been drawn in a semi-logarithmic plot (Fig. 7) using the boundary characteristics described above. In the figure, black and white ellipsoids indicate the positions of the liquid boundaries for a primordial ammonia concentration of

ARTICLE IN PRESS

Temperature (K)

u

600 120 400

80

200

40

0

0 5

10

15

20

25

30

5

10

15

20

25

30

5

10

15

20

25

30

5

10

15 20 % wt NH3

25

30

Ice I layer thickness

O. Grasset, J. Pargamin / Planetary and Space Science 53 (2005) 371–384

Liquid layer thickness

378

280 240 200

Radius (km)

2600 2400 2200 2000

Pressure (MPa)

800 600 400 200 0

Fig. 5. Characteristics of the liquid layer boundaries versus ammonia concentration assuming an initial concentration of 5 wt%. In the upper diagram, the plain and dashed lines are for the ice I and liquid layers respectively. Dotted lines and gray area indicate the possible domain of the ice I layer thickness inferred from Sohl et al. (2002). In the following diagrams, plain lines are for the lower interface and dashed lines for the ice I—liquid boundary.

Fig. 4. The numerical algorithm which computes equilibrium conditions of the liquid–ices interfaces. Variables r; x; P; T are radius, composition, pressure, and temperature respectively. Indices u and l are for upper (ice I–Liquid) and lower (HP ices–Liquid) boundaries, respectively. V is the volume of the liquid layer. On the lower boundary, temperature can be estimated either from equations 1–2 ðT liq Þ or using the adiabat ðT ad Þ: Equilibrium conditions are obtained once these two temperatures are almost similar for a given liquid composition. Program COMP_PRESS computes the pressure at a given depth assuming that the density is constant for each layer. Program COMP_T.LIQ computes with a Brent’s method (Brent, 1973) the liquidus temperature for a given pressure and ammonia composition using Eqs. (1) and (2) and parameters described in Table 1. Program COMP_T.AD is used for computing T ad by extrapolating an adiabat through the liquid layer from the upper ice I–liquid interface where temperature and pressure are known.

10 and 5 wt% respectively as inferred from Figs. 5 and 6. The temperature profile corresponds to the hottest case, that is the thinnest ice I layer proposed by Sohl et al. (2003) for a primordial ammonia concentration of 5 wt%. At this stage, the composition of the liquid layer is 7.5 wt% (Fig. 5). Temperature through the ice I layer is computed assuming a conductive profile. This profile is probably not a realistic one because convective motions in ice I are not taken into account. But it provides the highest possible temperature within the ice I layer. The temperature profile in the liquid layer is assumed adiabatic. Finally, the temperature profile through the high pressure ices is composed of a thin upper thermal boundary layer and a thick layer which is at the melting temperature of pure ice. Indeed, one must not see this icy layer as a rigid one. It is heated from below (silicate core producing heat from radiogenic decay) and from within (tidal heating) and its surface is

ARTICLE IN PRESS

TTemperature (K)

80

600

60

400

40 200

20 0

0 10

15

20

25

30

10

15

20

25

30

10

15

20

25

30

10

15

20 % wt NH3

25

30

Ice I layer thickness

Liquid layer thickness

O. Grasset, J. Pargamin / Planetary and Space Science 53 (2005) 371–384

280 240 200

Radius (km)

2600 2400 2200

Pressure (MPa)

2000

800 600 400 200 0

Fig. 6. Characteristics of the liquid layer boundaries versus ammonia concentration assuming an initial concentration of 10 wt%. Legends are the same as in Fig. 5.

at its melting temperature. Since at least part of the heating is from below, convective motions create hot thermal plumes. Thus, ice rises along adiabatic paths which probably cross the melting curve of pure ice. The high pressure icy layer must be understood as a ‘‘sluggish’’ layer because melting occurs permanently along plumes. Its temperature profile must be close to the melting temperature and heat transfer through the layer is probably achieved by processes of melting/ recrystallization from the bottom to the top. The upper thermal boundary layer is due to the temperature drop between the pure ice melting curve and the melting temperature of ice in equilibrium with ammonia–water liquid. 3.2. Density profiles In order to study the behavior of methane clathrates within Titan, density contrasts in the different layers must be estimated. This point is illustrated in Fig. 8. The

379

density profiles have been computed for both 5 and 10 wt% primordial ammonia amount in the median cases corresponding to a liquid composition of 8.5% (Fig. 5) or 16.5% (Fig. 6), respectively. Density of ice I has been inferred from the data of Hobbs (1974). The density decreases with depth because temperature goes up (Fig. 7). Densities in the liquid layer are from Croft et al. (1988) at low pressure and from Sourirajan and Kennedy (1963) up to 140 MPa. These data are then extrapolated at higher pressures. The temperature effect has been taken into account using the data of Sourirajan and Kennedy (1963). Densities of high pressure ices are not plotted since there are around 1300 kg=m3 ; a much higher density than methane clathrates. In addition, the density of methane clathrates has been added using the data of Loveday et al. (2001a,b,c) at high pressure and Sloan (1998) at low pressure. In the figure, it seems that clathrates are denser than ice I above 20 MPa, but this point cannot be certain because of the inaccuracy of our thermal profile (conductive) through ice I and because of the lack of clathrate density measurements in this pressure range. Furthermore, the temperature effect has not been taken into account in spite of the large thermal expansivity of clathrates (Tanaka et al., 1997; Shpakov et al., 1998) because we cannot be sure that thermal expansivity is also large at high pressure. Nonetheless, the temperature increase in the ice I layer must probably shift the density profile of clathrates plotted in Fig. 8 to the left. In spite of these simplifications, the main point which is clearly seen in Fig. 8 is that densities of ice and methane clathrates are very similar on the whole depth of the ice I layer. 3.3. Methane clathrates within Titan After the experimental works of Loveday et al. (2001a,b,c) and Hirai et al. (2001, 2003), it is no longer possible to envisage the dissociation of clathrates during the early stages of Titan’s history. Methane clathrates are stable at very high pressures and whatever the primordial history of Titan is, methane remains trapped as clathrates. In their study, Loveday et al. (2001a,b,c) were interested by the clathrate stability and did not provide a precise scenario of the methane ascent. In the following section, we propose to investigate the process of methane clathrate ascent through icy layers and ammonia–water liquid shell using the thermodynamical properties of the layers described in the previous sections. First of all, it is necessary to investigate the stability of methane clathrates within Titan. This can be done using Fig. 7. In this figure, the dissociation curve of methane clathrates has been plotted. On the right of this curve, methane clathrate cannot exist. The curve is divided into two parts. At pressures below the critical pressure of the quadrupole point Q1 (Sloan, 1998), where ice I + liquid

ARTICLE IN PRESS O. Grasset, J. Pargamin / Planetary and Space Science 53 (2005) 371–384

380

Temperature (K) 100

200

300 2575 km

0.01

ice I Ice

liquid layer

ne

25 73 km

th a me cl

250 6k m

at

r a te L

h

Q1

n etha +m

100

i q u id

10

212 6k m

1

+

Pressure (MPa)

I+

0.1

HP ices

rock and iron core

e+

c

1000

Fig. 7. Semi-log plot of the pressure and temperature conditions within Titan. Pressure ranges from 0.01 MPa, the partial pressure of methane at the surface (Samuelson, 2003) to roughly 800 MPa at the ice–silicate interface. Black and white ellipsoids indicate the position of the liquid layer boundaries assuming a primordial ammonia concentration of 10 and 5 wt% respectively. The bold curve represents the hottest thermal profile in the particular case where the primordial ammonia amount is 5% and the ammonia concentration in the liquid layer is 7.5 wt%. The profile is conductive in the ice I layer, adiabatic in the liquid layer and along the melting curve of pure ice in the high pressure ices domain (see text for details). The dotted line is the methane clathrate dissociation curve inferred from Sloan (1998) at low pressures and Loveday et al. (2001a) at high pressure. The quadrupole point Q1 is the point where water ice, liquid water, methane clathrate and gaseous methane can coexist. Its exact position is 272.9 K and 2.56 MPa (Sloan, 1998). The melting curve of ice is also added (thin dashed line) for information. On the right, a schematic view of Titan’s interior has been plotted (from Sohl et al., 2003).

+ gaseous methane + methane clathrate coexist, the curve corresponds to the three phase domain ice I + gaseous methane + methane clathrate. At higher pressures, the three phases in equilibrium along the dissociation curve are: liquid + methane + methane clathrates. This dissociation curve is based on the numerous data provided in Sloan (1998) at low pressure and on the recent high pressure data from Loveday et al. (2001a,b,c). Comparison of this curve with the ‘‘hottest’’ thermal profile within Titan indicate that methane clathrate are stable at each depth because the temperature is colder than the critical temperature of the dissociation curve. This result is interesting because it proves that clathrate reservoirs are very stable on Titan and may not be able to produce gaseous methane as it is commonly assumed. The case of Titan is strongly different from Earth where clathrate are commonly dissociated because thermodynamic conditions of reservoirs are very close to the dissociation curve conditions. In Titan, it seems that the clathrate dissociation which is required for explaining the replenishment of methane in the atmosphere is difficult to achieve. Methane clathrates can be stored either in the high pressure icy layer, or in the liquid layer, or in the upper icy layer. In order to see what is the preferential zone of storage, it is necessary to use Fig. 8. The layer of high pressure ices is composed of a thick ice VI shell and a shallow ice V shell. In this domain of pressure and

temperature, the structure of methane clathrates is the well-known low pressure structure I (CH4 -5:75 H2 O) because Loveday et al. (2001a,b,c) found that the pressure transition from structure I to high pressure clathrate is above 800 MPa. Its density is much lower than the density of ices which means that the storage of a large amount of methane in these deep layers is almost impossible. Similarly, storage within the liquid layer cannot occur. On the contrary, ice I has similar densities than methane clathrate in the pressure range of the upper layer (Fig. 8). It is then highly possible that the two compounds are intimately mixed over the whole layer. An other possibility is that the high compressibility of clathrates (Fig. 8) implies a preferential storage of methane in the deepest part of the ice I layer. Due to the lack of density data in the pressure and temperature range of the ice I layer, there is unfortunately no way to affirm which storage is favored. From the last two sections, it appears that methane clathrates are stable at all depth but, due to density contrasts, went rapidly into the ice I layer. This point, in association with the fact that gaseous methane must go to the surface for explaining the atmosphere replenishment, indicate that internal motions within ice I must explain: (1) the ascent of clathrate through ice; (2) the dissociation of clathrates into ice + gaseous methane during or after this ascent. To our knowledge, hot plumes are the best candidates for the first step. Since convective motions are suspected in the ice I layer

ARTICLE IN PRESS O. Grasset, J. Pargamin / Planetary and Space Science 53 (2005) 371–384

900

1000

1100

1200

1300

0.01

381

(Stevenson, 1992) or a gas-driven water volcanism process (Crawford and Stevenson, 1988) may lead to the emission of methane directly into the atmosphere of Titan.

0.1

4. Discussion and conclusion

Pressure (MPa)

1

Ice I layer

Methane clathrates

10

Liquid layer High pressure ices

100

1000

10000 900

1000

1100

1200

1300

Density (kg/m3) Fig. 8. Density profiles within Titan. The density profiles have been computed for both 5 wt% (plain line) and 10 wt% (dashed line) primordial ammonia amount in the cases corresponding to a liquid composition of 8.5% or 16.5%, respectively. Methane clathrate density has also been plotted (dotted line) using the data of Sloan (1998) and Loveday et al. (white triangles). Densities of high pressure ices have not been precisely computed since they are far above the density of methane clathrates (gray area).

(Grasset et al., 2000; Sohl et al., 2003), vigorous hot plumes must exist and may drive part of the methane clathrates through ice I. These motions, added to the fact that methane clathrates are slightly less dense at low pressure than ice I (Fig. 8), must lead rapidly to the formation of a clathrate reservoir located below the thermal conductive lid of the ice I layer. Due to the small time scale of convective motions compared to Titan’s history, it is probable that a large amount of methane clathrates was stored rapidly at shallow depth forming large reservoirs. Finally, the ascent of methane clathrates through the ice I lid remains difficult to explain and it is out of the goal of this paper to provide a precise description of this process. It is possible that clathrates go up through cracks generated because of the upward constraint caused by the density contrast between ice and clathrates. Some of these cracks must be important enough for allowing a rapid ascent of clathrates up to the dissociation depth. Once dissociation has occurred, methane can be either stored in porous ice or caverns

The primordial amount of ammonia within Titan is not easy to determine. Lunine and Stevenson (1987) proposed a high value (15 wt%) for the primordial concentration of the ammonia liquid layer. In their recent evolutionary turbulent model of Saturn’ subnebula, Mousis et al. (2002) obtained different amounts depending on the initial N2 =NH3 ratio. For N2 =NH3 ¼ 0:1; ammonia concentration must be around 11.6 wt%. For a very high N2 =NH3 ratio (10), an ammonia concentration below 1% was obtained. Dartois and d’Hendecourt (2001), analysing spectroscopic ISO measurements of the ISM, conclude to an NH3 upper limit of a few percent of the total water ice content. The optimal value might be NH3 =H2 O around 4% which corresponds to N2 =NH3 ¼ 3 in the model of Mousis et al. (2002). On the other side, Ruffle and Herbst (2000), in a study of dense interstellar clouds evolution, predict that the chemistry on grain surfaces might result in a N2 =NH3 ratio much higher than 1 before the end of the first million year. It is possible that these conditions are not relevant for the presolar cloud from which the solar nebula was formed, but it suggests nonetheless that very low N2 =NH3 ratio leading to very high ammonia concentration within Titan may not be realistic. In our study, a primordial ammonia concentration of 5% has been chosen as the most plausible value for Titan. A 10 wt% concentration has also been studied for a purpose of comparison. The dissociation curve of methane clathrates which has been plotted in Fig. 7 is based on experimental data in the binary system water + methane. This curve suggests that the dissociation of clathrates within Titan is not possible except at very shallow depth if hot plumes drag clathrates during their ascent. However, one can wonder whether the dissociation of clathrates occurs in different conditions when ammonia is present. In other words, the behavior of methane clathrates during their ascent through the liquid layer must be discussed. There are, at our knowledge, no experimental data, in the ternary system NH3 2H2 O2CH4 at high pressure. Nonetheless, the effect of pollutants on the clathrate stability is a well-known problem at low pressures. Two cases exist depending on the behavior of the added compound. Either this compound is insoluble and able to enter the clathrate structure or it is soluble and included in the liquid system in equilibrium with the clathrate. In the first case, the compound commonly increases the stability of the clathrate structure

ARTICLE IN PRESS 382

O. Grasset, J. Pargamin / Planetary and Space Science 53 (2005) 371–384

(Mooijer-van der Heuvel et al., 2000). In the second case, the dissociation curve is generally shifted towards lower temperatures. The ammonia is obviously in the second category. It is actually known as a good inhibitor of clathrate formation twice as effective as methanol. But it has not been extensively studied in gas industry as methanols and glycols because it reacts with carbon dioxide and water to form ammonium carbonate, bicarbonate and carbamate (Sloan, 1998). A simple law was developed in 1939 by Hammerschmidt (1939) which related the temperature drop ðDTÞ to the molecular weight of the inhibitor and the weight percent of the inhibitor in the liquid. This equation has been used for more than 50 years in gas industry for both alcohols and ammonia inhibitors (Sloan, 1998, p. 203). It has been shown recently that this law is not very accurate for alcohols at high concentration and an improved law has been proposed at least for methanol (Nielsen and Bucklin, 1983): DT ¼ 72: lnð1  xÞ;

(4)

where x is the methanol mole fraction. Assuming that ammonia is twice as efficient as methanol (Sloan, 1998), the dissociation curve plotted in Fig. 7 might be shifted of roughly 12  C for the ammonia concentration of 7.5 wt% corresponding to the plotted thermal profile. Such a decrease is consistent with the theoretical estimate provided by Lunine and Stevenson (1985) using the statistical mechanical theory developed by van der Waals and Platteeuw (1959). This slight decrease is not sufficient for involving clathrate dissociation within the liquid layer (Fig. 7). In the above discussion, the pressure effect has been neglected. Eq. (4) is valid at low pressures up to 20 MPa but is probably not accurate at pressures involved in the liquid layer of Titan. It is possible that the ammonia effect on the dissociation is much more important at high pressure. Thus, experimental data are required in the ternary system methane–ammonia–water for describing precisely what is the methane dissociation curve in the high pressure domain. It has been shown that the dissociation of clathrates, if it occurs, must be located at shallow depth. If hot plumes drag methane clathrates through the ice I layer close to the surface, they are nonetheless unable to go through the upper conductive layer. Convective models with variable viscosity fluids predict relatively thick conductive lid (Solomatov, 1995; Grasset and Parmentier, 1998; Sohl et al., 2003; Sotin and Tobie, 2004). In their recent paper, Sotin and Tobie (2004) predict that the lid must be at least 10 km thick in the case of Titan. At this depth, the pressure is roughly 14 MPa and the temperature of a hot plume is roughly 250 K (Fig. 7). With these conditions, the clathrate is stable, even if Eq. (4) is taken into account for decreasing the temperature of clathrate dissociation. Then, hot plumes drag

clathrates at shallow depth but not shallow enough for dissociation to occur. Large reservoirs of methane clathrates at shallow depths are consistent with the previous internal structure proposed by Stevenson (1992) and the review of Lunine (1993), but the scenario leading to these reservoirs is strongly different. In the previous models, clathrates were supposed to be dissociated very early in Titan’s history at high pressure. Gaseous methane went up rapidly towards the surface and its storage into clathrate reservoirs was explained because it was supposed to react again with ice close to the surface. In our model, based on recent experimental constraints on both methane–water and ammonia–water system, it appears that clathrates have probably never been dissociated into Titan. The deep reservoirs of clathrates are actually reservoirs of primordial clathrates. One problem which still needs to be solved concerns the exchanges between clathrates buried into several kilometers of ice and the surface. Both tectonic and volcanic events must be taken into account for explaining how relatively deep clathrates can be extruded rapidly towards the surface and dissociated into gaseous methane + ice. There is no doubt that the Cassini–Huygens mission will provide valuable information about this processes since tectonic and volcanic features of Titan’ surface will be revealed. In this paper, a new description of the ammonia–water phase diagram has been proposed. Based on recent experimental and theoretical works on the ammonia water system, new insights about liquidus domains and peritectic and eutectic curves have been provided. These data have been used for characterizing the deep interior of Titan using a new algorithm which computes thermodynamical characteristics of the liquid layer boundaries. Results have been presented for both a primordial ammonia composition of 5 and 10 wt%. It appears that temperature profiles within Titan are much colder than the dissociation curve of methane clathrates. Thus, clathrates are stable at each depth and, due to density contrasts, must be located in shallow reservoirs below the ice I conductive lid. Ascent through the lid and dissociation into ice + methane must be related to tectonic processes involving cracks and/or gas-driven volcanism. It is hoped that the processes responsible for the ascent of methane through the lid will be soon understood because Titan’s surface will be revealed by the Cassini/Huygens mission.

Acknowledgements This work was supported by grants from the French Programme National de Planetologie (INSU/CNRS). We thank C. Sotin and G. Tobie for their helpful comments.

ARTICLE IN PRESS O. Grasset, J. Pargamin / Planetary and Space Science 53 (2005) 371–384

References Boone, S., Nicol, M.F., 1991. Ammonia-water mixtures at high pressures—Melting curves of ammonia dihydrate and ammonia monohydrate and a revised high-pressure phase diagram for the water-rich region. Lunar Planet. Sci. Conf 21st. Brent, R.P., 1973. Algorithms for Minimization without Derivatives. Prentice-Hall, Englewood Cliffs, NJ (Chapters 3–4). Campbell, D.B., Black, G.J., Carter, L.M., Ostro, S.J., 2003. Radar evidence for liquid surfaces on Titan. Science 300 (5644), 431–434. Chizhov, V.E., 1993. Thermodynamic properties and thermal equation of state of high-pressure ice phases. Translated Prikl. Mekh. Tekhn. Fiz. 2, 113–123. Chou, I.M., Sharma, A., Burruss, R.C., Shu, J., Mao, H.K., Hemley, R.H., Goncharov, A.F., Stern, L.A., Kirby, S.H., 2000. Transformation in methane hydrates. Proc. Natl. Acad. Sci. 97 (25), 13484–13487. Consolmagno, G.J., Lewis, J.S., 1978. The evolution of satellite interiors and surfaces. Icarus 34, 280–293. Crawford, G.D., Stevenson, D.J., 1988. Gas-driven volcanism and the resurfacing of Europa. Icarus 73, 66–79. Croft, S.K., Lunine, J.I., Kargel, J.S., 1988. Equations of state of ammonia–water liquid: derivation and planetological applications. Icarus 73, 279–293. Cynn, H.C., Boone, S., Koumvakalis, A., Nicol, M., Stevenson, D.J., 1989. Phase diagram for ammonia-water mixtures at high pressures—Implications for icy satellites. Lunar Planet. Sci. Conf 19th. Dartois, E., d’Hendecourt, L., 2001. Search for ammonia ice in cold dust envelopes around YSOs. Astron. Astrophys. 365, 144–165. Dalrymple III, W., Hogenboom, D.L., Consolmagno, G.J., 1988. The density of ammonia–water solution to 400 MPa. Lunar Planetary Science Conference XIX, pp. 241–242. Fortes, A.D., Brodholt, J.P., Wood, I.G., Vocadlo, L., Jenkins, H.D.B., 2001. Ab initio simulation of ammonia monohydrate ðNH3: H2 OÞ and ammonium hydroxide ðNH4 OHÞ: J. Chem. Phys. 115 (15), 7006–7014. Fortes, A.D., Wood, I.G., Brodholt, J.P., Vocadlo, L., 2003a. The structure, ordering and equation of state of ammonia dihydrate ðNH3 -H2 OÞ: Icarus 162, 59–73. Fortes, A.D., Wood, I.G., Brodholt, J.P., Alfredsson, M., Vocadlo, L., McGrady, G.S., Knight, K.S., 2003b. A high-resolution neutron powder diffraction study of ammonia dihydrate (ND3.2D2O) phase I. J. Chem. Phys. 119 (20), 10806–10813. Grasset, O., Parmentier, E.M., 1998. Thermal convection in a volumetrically heated fluid with strongly temperature dependent viscosity: implications for planetary thermal evolution. J. Geophys. Res. 103 (B8), 18171–18181. Grasset, O., Sotin, C., 1996. The cooling rate of a liquid shell in Titan’s interior. Icarus 123, 101–112. Grasset, O., Beauchesne, S., Sotin, C., 1995. Investigation of the NH3 H2 O phase diagram in the range 100 MPa–1.5 GPa using in situ Raman spectroscopy: application to Titan’s dynamics. Comptes Rendas Acad. Sci. 320 (IIa), 249–256. Grasset, O., Sotin, C., Deschamps, F., 2000. On the internal structure and dynamics of Titan. Planet. Space Sci. 48, 617–636. Griffith, C.A., Owen, T., Geballe, T.R., Rayner, J., Rannou, P., 2003. Evidence for the exposure of water ice on Titan’s surface. Science 300 (5619), 628–630. Hammerschmidt, E.G., 1939. Gas 15 (5). Hirai, H., Kondo, T., Hasegawa, M., Yagi, T., Yamamoto, Y., Komai, T., Nagashima, K., Sakashita, M., Fujihisa, H., Aoki, K., 2000. Methane hydrate behavior under high pressure. J. Phys. Chem. 104, 1429–1433. Hirai, H., Uchihara, Y., Fujihisa, H., Sakashita, M., Katoh, E., Aoki, K., Nagashima, K., Yamamoto, Y., Yagi, T., 2001. High-pressure

383

structures of methane hydrate observed up to 8 GPa at room temperature. J. Chem. Phys. 115 (15), 7066–7070. Hirai, H., Tanaka, T., Kawamura, T., Yamamoto, Y., Yagi, T., 2003. Retention of filled ice structure of methane hydrate up to 42 GPa. Phys. Rev. B 68, 172102. Hobbs, P.V., 1974. Ice Physics. Clarendon Press, Oxford, p. 79, p. 348. Hogenboom, D.L., Winebrake, J., Consolmagno, G.J., Dalrymple, W., III, 1989. Preliminary Densities and Phase Diagram of the Water/NH3 System at P-T Conditions Relevant to the Icy Moons of the Outer Planets. Lunar Planet. Sci. Conf 19th. Hogenboom, D.L., Kargel, J.S., Consolmagno, G., Holden, T.C., Lee, L., Buyyounouski, M., 1997. The ammonia–water system and the chemical differentiation of icy satellites. Icarus 128, 171–180. Johnson, M.L., Nicol, M., 1987. The ammonia–water phase diagram and its implications for icy satellites. J. Geoph. Res. 92, 6339–6349. Jones, T.D., Lewis, J.S., 1987. Estimated impact shock production of N2 and organic compounds on early Titan. Icarus 72, 381–393. Kargel, J.S., 1992. Ammonia–water volcanism on icy satellites: phase relations at 1 atmosphere. Icarus 100, 556–574. Leliwa-Kopystynski, J., Maruyama, M., Nakajima, T., 2002. The water–ammonia phase diagram up to 300 MPa: application to icy satellites. Icarus 159 (2), 518–528. Lellouch, E., Coustenis, A., Gautier, D., Raulin, F., Dubouloz, N., Frere, C., 1989. Titan’s atmosphere and hypothesized ocean: a reanalysis of the Voyager 1 Radio-occultation and IRIS 7:7 mm data. Icarus 79, 328–349. Loveday, J.S., Nelmes, R.J., Guthrie, M., Belmonte, S.A., Allan, D.R., Klug, D.D., Tse, J.S., Handa, Y.P., 2001a. Stable methane hydrate above 2 GPa and the source of Titan’s atmospheric methane. Nature 410, 661–663. Loveday, J.S., Nelmes, R.J., Guthrie, M., Klug, D.D., Tse, J.S., 2001b. Transition from cage clathrate to filled ice: the structure of methane hydrate III. Phys. Rev. Lett. 87 (21), 215501. Loveday, J.S., Nelmes, R.J., Guthrie, M., 2001c. High pressure transition in methane hydrate. Chem. Phys. Lett. 350 (5–6), 459–465. Lunine, J.I., 1993. Does Titan have oceans? A review of current understanding of Titan’s surface. Rev. Geophys. 31–2, 133–149. Lunine, J.I., 1994. Does Titan have oceans? Am. Sci. 82, 134–143. Lunine, J.I., Stevenson, D.J., 1985. Thermodynamics of clathrate hydrate at low and high pressures with application to the outer solar system. Astrophys. J. Suppl. Ser. 58, 493–531. Lunine, J.I., Stevenson, D.J., 1987. Clathrate and ammonia hydrates at high pressure—application to the origin of methane on Titan. Icarus 70, 61–77. Mooijer-van der Heuvel, M.M., Peters, C.J., de Swaan Arons, J., 2000. Influence of water-insoluble organic components on the gas hydrate equilibrium conditions of methane. Fluid Phase Equilibria 172, 73–91. Mousis, O., Gautier, D., Bockele´e-Morvan, D., 2002. An evolutionary turbulent model of the Saturn’s subnebula: implications on the origin of the atmosphere of Titan. Icarus 156 (1), 162–175. Nielsen, R.B., Bucklin, R.W., 1983. Hydrocarbon Processing. p. 71. Prinn, R.G., Fegley, B., 1981. Kinetic inhibition of CO and N2 reduction in circumplanetary nebulae—implications for satellite composition. Astrophys. J. 249, 308–317. Prinn, R.G., Fegley, B., 1989. Solar nebular chemistry: origin of planetary, satellite, and cometary volatiles. In: Atreya, S.K., Pollack, J.B., Matthews, M.S. (Eds.), Origin and Evolution of Planetary and Satellite Atmospheres. University of Arizona Press, Tucson, pp. 78–136. Ruffle, D.P., Herbst, E., 2000. New models of interstellar gas-grains chemistry—I. surface diffusion rates. Mon. Not. R. Astron. Soc. 319, 837–850. Sagan, C., Thompson, W.R., 1984. Production and condensation of organic gases in the atmosphere of Titan. Icarus 59, 133–161.

ARTICLE IN PRESS 384

O. Grasset, J. Pargamin / Planetary and Space Science 53 (2005) 371–384

Samuelson, R.E., 2003. Titan’s atmospheric engine: an overview. Planet. Space Sci. 51, 127–145. Shpakov, V.P., Tse, J.S., Tulk, C.A., Kvamme, B., Belosludov, V.R., 1998. Elastic moduli calculation and instability in structure I methane clathrate hydrate. Chem. Phys. Lett. 282, 107–114. Sloan, E.D., 1998. Clathrates Hydrates of Natural Gases, second ed. Marcel Dekker, Inc., New York. Sohl, F., Hussmann, H., Schwentker, B., Spohn, T., Lorenz, R.D., 2003. Interior structure models and tidal Love numbers of Titan. J. Geophys. Res. 108 (E12), 1–4. Solomatov, V.S., 1995. Scaling of temperature and stress dependent viscosity convection. Phys. Fluids 7 (2), 266–274. Sotin, C., Tobie, G., 2004. Comparison Titan-Europa: effect of tidal heating on their evolution, in preparation. Sotin, C., Grasset, O., Beauschesne, S., 1998. Thermodynamic properties of high pressure ices: implications for the dynamics and internal structure of large icy satellites. In: Schmitt, B., De Bergh, C., Festou, M. (Eds.), Solar System Ices. Kluwer, Dordrecht, pp. 79–96. Sourirajan, S., Kennedy, G.C., 1963. Specific volumes of liquid ammonia–water mixtures in the temperature range

01–251 and pressure range 100–1400 bars. J. Geophys. Res. 68, 4149–4155. Stevenson, D.J., 1992. Interior of Titan. In: Proceedings Symposium of Titan, European Space Agency Special Publication, SP-338, pp. 29–33. Strobel, D.R., 1982. Chemistry and evolution of Titan’s atmosphere. Planet. Space Sci. 30, 849–858. Takashi, I., Kiyoyuki, T., 2003. Structural transformation of methane hydrate from cage clathrate to filled ice. J. Chem. Phys. 119 (13), 6784–6788. Tanaka, H., Tamai, Y., Koga, K., 1997. Large thermal expansivity of clathrate hydrates. J. Phys. Chem. 101, 6560–6565. Tarantola, A., Valette, B., 1982. Generalized non-linear inverse problems solved using the least squares criterion. Rev. Geophys. Space Phys. 20, 219–232. van der Waals, J.H., Platteeuw, J.C., 1959. Clathrates solutions. Adv. Chem. Phys. 2, 1. Zahnle, K., Pollack, J.B., Grinspoon, D., Dones, L., 1992. Impact generated atmospheres over Titan, Ganymede, and Callisto. Icarus 95, 1–23.