Geophysical evolution of Saturn’s satellite Phoebe, a large planetesimal in the outer Solar System

Icarus 219 (2012) 86–109

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Geophysical evolution of Saturn’s satellite Phoebe, a large planetesimal in the outer Solar System Julie C. Castillo-Rogez a,⇑, T.V. Johnson a, P.C. Thomas b, M. Choukroun a, D.L. Matson a, J.I. Lunine b a b

Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, United States Center for Radiophysics and Space Research, Space Sciences Building, Cornell University, Ithaca, NY 14853, United States

a r t i c l e

i n f o

Article history: Received 10 January 2011 Revised 29 January 2012 Accepted 5 February 2012 Available online 15 February 2012 Keywords: Saturn, Satellites Geophysics Planetesimals

a b s t r a c t Saturn’s satellite Phoebe is the best-characterized representative of large outer Solar System planetesimals, thanks to the close flyby by the Cassini spacecraft in June 2004. We explore the information contained in Phoebe’s physical properties, density and shape, which are significantly different from those of other icy objects in its size range. Phoebe’s higher density has been interpreted as evidence that it was captured, probably from the proto-Kuiper-Belt. First, we demonstrate that Phoebe’s shape is globally relaxed and consistent with a spheroid in hydrostatic equilibrium with its rotation period. This distinguishes the satellite from ‘rubble-piles’ that are thought to result from the disruption of larger proto-satellites. We numerically model the geophysical evolution of Phoebe, accounting for the feedback between porosity and thermal state. We compare thermal evolution models for different assumptions on the formation of Phoebe, in particular the state of its water, amorphous or crystalline. We track the evolution of porosity and thermal conductivity as well as the destabilization of amorphous ice or clathrate hydrates. While rubble-piles may never reach temperatures suitable for porous ice to creep and relax, we argue that Phoebe’s shape could have relaxed due to heat from the decay of 26Al, provided that this object formed less than 3 Myr after the production of the calcium–aluminum inclusions. This is consistent with the idea that Phoebe could be an exemplar of planetesimals that formed in the transneptunian region and later accreted onto outer planet satellites, either during the satellite’s formation stage, or still later, during the late heavy bombardment. Ó 2012 Elsevier Inc. All rights reserved.

1. Introduction Despite its small size and lack of obvious, recent, surface activity, Saturn’s irregular satellite Phoebe may be a key to better understanding the early history of the outer Solar System. Phoebe differs from other small icy objects in important ways, as we will demonstrate with the Cassini–Huygens observations of June 11, 2004. Indeed, this mission has provided the most extensive body of information yet on a mid-sized, water-rich planetesimal. Survey of small bodies across the Solar System indicates a peak in the size range bin 100–200 km (radius) (Cuzzi et al., 2010), an observation supported by the theoretical consideration that objects of this size formed early and within 1 Myr of the formation of the CalciumAluminum Inclusions (CAI) (e.g., Kenyon et al., 2008). Thus Phoebe may be a representative of an early generation of large planetesimals. Imaging by the Cassini–Huygens remote sensing instruments shows that Phoebe has a complex surface composition. The surface is composed of water ice, a variety of organic material, possibly ⇑ Corresponding author. Address: M/S 79-24, JPL/Caltech, 4800 Oak Grove Drive, Pasadena, CA 91109, United States. E-mail address: [email protected] (J.C. Castillo-Rogez). 0019-1035/$ - see front matter Ó 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.icarus.2012.02.002

hydrated minerals, carbon dioxide ice (Clark et al., 2005), and has overall spectral properties similar to C-type asteroids. Clark et al. (2005) have suggested that Phoebe’s surface may be composed of a variety of primitive materials present in the early, outer Solar System. Also, it is possible that the dark material that gives Phoebe its C-type appearance may postdate, or possibly be a consequence of, the bombardment that formed its heavily cratered surface. Most importantly Phoebe appears to be the best-observed C-type object to date, and as such, it may help in understand the origin of that population of small bodies. Phoebe’s density (1630 kg/m3) is significantly higher than the average density for the regular saturnian satellites, about 1240 kg/m3. This property led Johnson and Lunine (2005) to suggest that Phoebe’s density is closer to the density for Kuiper-Belt objects, but lighter by 15–20% due to its bulk porosity. This is evidence that Phoebe was captured by Saturn rather than formed from the same source material as the regular satellites. Capture is also suggested by its peculiar dynamical properties: a retrograde orbit with an inclination of 176° with respect to the ecliptic and an eccentricity of 0.164. An origin in the Kuiper Belt, i.e., a common genetic pool with comets, centaurs, trojans (e.g., Gomes et al., 2005) and perhaps small asteroids (Levison et al., 2009), is also

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consistent with the organic compounds and possibly hydrated silicates identified on Phoebe’s surface. Such a large density is unexpected for an object of this size formed at a temperature as low as 50 K. In absence of creep compaction, the porosity expected in the object should range from at least 35% in the deep interior to up to 50% close to the surface (Durham et al., 2005a). The relatively bulk porosity inferred by Johnson and Lunine (2005) implies that compaction could be accommodated by the creep of certain species, water ice if warmed above 170 K, or thanks to the presence of compounds that promote low-temperature creep (e.g., ammonia hydrates, methane ice, etc.). This study is also motivated by the observation that Phoebe’s global shape appears mostly relaxed (see Appendix in Porco et al. (2005)), which is a surprising characteristic for an object of this temperature, and is another piece of evidence that Phoebe’s material became mobile enough to drive the evolution of the object toward a hydrostatic state. We explore the question of Phoebe’s relaxed shape in further detail in Section 2. Then we explore the parametric space of possible scenarios for Phoebe’s initial conditions and material properties (Section 3) in order to identify in which context the satellite could compact and relax toward an equilibrium shape. Models differ in the assumptions made about the origin and early history of Phoebe as well as for late scenarios of disruption and reaccretion and possibly a dynamical history that took it closer to the Sun, at some point (Section 4). Implications of the results are discussed in Section 5. 2. Phoebe’s origin, composition, and physical properties We review the physical and spectral observations obtained during the Cassini–Huygens flyby performed in June 2004. We discuss the information contained in these observations in terms of composition, origin, and geophysical evolution. 2.1. Dynamical properties and origin Phoebe has a retrograde orbit, with an inclination of 176° (with respect to Saturn’s equator), and an eccentricity of 0.16. Turrini et al. (2009) showed that the secular variations of Phoebe’s dynamical properties are regular and limited, leading to the long-term preservation of its dynamical properties after its capture by Saturn. Several studies have assessed whether or not Phoebe could have been disrupted by the intense bombardment that has sculpted its surface. This idea was first suggested by Cuk et al. (2003) who noted the lack of evident family members in the vicinity of the satellite. These authors, as well as Nesvorny´ et al. (2003) and Turrini et al. (2008) demonstrated that the small irregular and retrograde satellites detected by Gladman et al. (2001) are not dynamically consistent with being derived from Phoebe, i.e., it is not possible to backtrack their dynamical evolution to Phoebe. Cuk et al. (2003) pointed out that the absence of a Phoebe dynamical family may indicate that the satellite was never disrupted. It may have avoided most of the late heavy bombardment impacts if it was captured at Saturn late during that event. This was confirmed by Turrini et al. (2009) who computed a low probability for the collision of Phoebe with heliocentric comets, estimating about one impact every Gy. On the other hand, Nesvorny´ et al. (2003) predicted that Phoebe’s surface would be very cratered as a result of impacts by prograde, irregular satellites in its vicinity. This scenario was confirmed by Turrini et al. (2008) who demonstrated that Phoebe has a high probability of collision with objects in these orbits. This ‘‘sweeping’’ effect is probably responsible for the gap in the orbital (semi-major axis) distribution of prograde and retrograde irregular satellites with low inclinations in Phoebe’s vicinity.

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Ground-based observations by Owen et al. (1999) indicated the presence of water ice on Phoebe’s surface. These authors inferred that the satellite originated in the outer Solar System. Although evidence of hydration is common at the surface of asteroids (e.g., Ghosh et al., 2006; Rivkin and Emery, 2010), and some of these objects may have significant ice content (e.g., Ceres), dislodging an asteroid of Phoebe’s mass from the asteroid belt requires significant energy and there is no obvious scenario for moving such an object from the asteroid belt out to Saturn. It is much easier, from a celestial mechanics point of view, to conceive that Phoebe was ejected from the vicinity of the current Kuiper Belt and captured at Saturn, perhaps after spending some time as a centaur. Numerical simulations by Di Sisto and Brunini (2007) suggested that Phoebe could have been a component of a binary centaur before being captured at Saturn. Turrini et al. (2009) studied the capture of Phoebe in the context of the Nice model introduced by Gomes et al. (2005) and Morbidelli et al. (2005). They were not able to determine the exact dynamical origin of Phoebe, but in the context of the Nice model Phoebe, along with other irregular satellites, may have been captured as a result of the high flux of planetesimals during the late heavy bombardment. Both comets and centaurs provide a good match for Phoebe’s observed inclination and eccentricity. Turrini et al. (2009) proposed a mechanism of capture as a result of the change in momentum of the satellite induced by one or several oblique collisions with impactors 50–100 km in diameter, which resulted in the formation of some of Phoebe’s large craters, especially Jason. Sufficiently oblique collisions would not destroy Phoebe. However they also demonstrated that even if the massive satellite had been the object of catastrophic disruption, it would have reaccreted its fragments in the form of a rubble-pile, explaining the absence of a ‘‘Phoebe’’ family. In February 2009 Verbiscer et al. (2009) used the Spitzer Space Telescope’s Multiband Imaging Photometer (MIPS) to scan regions near Phoebe’s orbit to search for the broad debris ring which they in fact discovered. From their analysis they estimated that the mass of the ring was 1011 kg. This is factor of 107 the mass of Phoebe. Or, as Verbiscer et al. (2009) noted, sufficient to fill only a 1 km diameter crater on Phoebe’s surface. While the ring have some effect on the texture and chemistry of the very upper part of Phoebe’s surface layer, it is unlikely to constrain Phoebe’s early geophysical history, which is the subject of this paper. 2.2. Global shape Tracking of the spacecraft and imaging of Phoebe during the flyby by the Cassini–Huygens spacecraft in June 2004 yielded Phoebe’s mass and shape (Porco et al., 2005; Thomas et al., 2006, 2007; Jacobson et al., 2007). We have updated fits for Phoebe’s principal axes: the mean radius is 106.5 ± 0.7 km, with ellipsoidal radii of: a = 109.3 ± 0.6 km, b = 108.4 ± 1.4 km, and c = 101.8 ± 0.3 km. The mean density is 1638 ± 33 kg m3. Phoebe’s shape is thus close to an oblate spheroid (Fig. 1), with a = b to within the uncertainties of the data. Phoebe’s shape is affected by large, unrelaxed impact craters that provide over 15 km of relief relative to equipotentials calculated with or without a differentiated interior structure (e.g., Giese et al., 2007). The satellite has at least seven craters that are 50–100 km in diameter, which testify to a severe collisional history, consistent with the above-mentioned dynamical studies (Nesvorny´ et al., 2003; Turrini et al., 2008). Richardson and Thomas (2007) noted that Hyperion and Phoebe display similar cratering records with saturation of craters larger than 10 km in radius but not for smaller craters. Phoebe’s limb, roughened by much cratering, is distinctly different from the obviously more relaxed surfaces of objects such as Mimas and Enceladus (Thomas et al., 2007). Its suggestive oblate form invites more detailed comparisons to the smaller, irregularly-shaped objects to

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Fig. 1. Phoebe’s shape (Porco et al., 2005, supplemental material).

test for any geophysical significance of the shape. Phoebe can be compared to other objects by the range of topography compared to the object’s mean radius. Here we take the topographic measure as the heights relative to an equipotential, or more particularly its proxy, the dynamical heights (Thomas, 1993). These quantities are calculated for the shapes and mean densities (usually assumed to be homogeneous, but having little influence on this measure) allowing for spin and tidal effects where relevant. Table 1 shows values of the ranges (in units of standard deviation) of the topography on these objects, and, for reference, on Enceladus, as a fraction of mean radius. Enceladus is clearly different. Phoebe, on the other hand, is the ‘‘smoothest’’ of the irregular, small objects, though it is not grossly distinct from Phobos or even Janus by this measure. We have tested whether heavy cratering with its consequent roughness would inhibit the retention and recognition of a useful geophysical shape. That is, how much should cratering alter a relaxed shape once it has become rigid? Does the current shape of Phoebe reflect its relaxed form? How uncertain do the craters make an interpretation of ‘‘original’’ shape? The results of this simulation are summarized in Table 2. For the first step in this modelTable 1 Range of topography on small objects. Topography values are root mean square of topography relative to fit ellipsoids expressed as a percentage of the object’s mean radius. Object

Mean radius (km)

Topography (% radius)

Enceladus Phoebe Phobos Janus Epimetheus Eros Ida Amalthea Deimos Hyperion Pan Atlas

252.1 ± 0.1 106.5 ± 0.7 11.1 ± 0.1 89.5 ± 1.4 58.1 ± 1.8 8.46 ± 0.02 15.7 ± 0.6 83.5 ± 2 6.2 ± 0.2 135 ± 4.0 14.1 ± 1.3 15.1 ± 0.9

0.2 4.7 5.4 6.3 6.9 8.6 8.6 10.4 11.0 12.4 15.1 17.6

ing we started with an oblate spheroid model of mean radius 106.5 km, and an (a  c) of 7.0 km (Fig. 2). The model is a 2° by 2° latitude–longitude-radius table (Simonelli et al., 1993). A crater population was generated with a largest crater of 130 km diameter, and a power-law distribution with a slope of 2.26 and we used it to produce a few more craters of diameters greater than 45 km than the number observed (i.e., 14 vs. 8). This is the size range that most affects shape estimation. The areas of these craters were then randomly placed on the shape model, with a maximum crater depth of 14.4 km, and all excavated material retained on the surface as ejecta (Thomas, 1998). This is simply a geometric simulation to evaluate the effect of large craters on the shape estimate and no dynamical calculations are involved. The orientations of the moments of inertia were then calculated assuming a homogeneous interior, and the dimensions of the ellipsoids rotated to give a c-axis coincident with maximum moment calculated. (Assuming a core model would have only a small effect on changing orientation of moments.) We ran 100 examples of such random crater placement, and obtained an average resulting (a  c) (actually, (a + b)/2  c) of 7.9 ± 2.1 km, and an average (a  b) of 2.8 ± 1.5 km. Secondly, the same exercise was carried out on a spherical model and this resulted in an average (a  c) of 3.9 ± 1.1 km. For the oblate model, the average change in the spin orientation was 13.1°. Starting with an oblate spheroid of (a  c) = 10 km, 100 trials yielded an average (a  c) of 10.5 ± 1.8 km, and (a  b) of 2.9 ± 1.4 km. These results suggest that even an impact history more severe than that recorded by the currently visible craters would retain a shape ‘‘signal’’ to within 2 km in the critical measure of (a  c). As long as there are few craters larger than the object’s mean radius, cratering has a relatively modest effect upon the global shape, tending, on average, to make spherical and near-spherical objects slightly more elongate, but leaving the axes, on average, within 2 km of their ‘‘initial’’ values. The results also indicate that it would be unlikely for any amount of impact shaping to result in a truly spherical body. This Monte Carlo result becomes intuitively obvious when one thinks of sphericity as the limiting case that real

J.C. Castillo-Rogez et al. / Icarus 219 (2012) 86–109 Table 2 Results of the shape cratering models presented in Fig. 2. Model – 100 trials cratering trials

Initial shape hri = 106.5 km

Final shape

‘Phoebe’ oblate spheroid

(a  c) = 7.0 km (a  b) = 0

(a  c) = 7.9 ± 2.1 km (a  b) = 2.8 ± 1.5 km

‘Super-oblate’ Phoebe spheroid

(a  c) = 10.0 km (a  b) = 0

(a  c) = 10.5 ± 1.8 km (a  b) = 2.9 ± 1.4 km

Sphere

(a  c) = 0 km (a  b) = 0

(a  c) = 3.9 ± 1.1 km (a  b) = 1.6 ± 0.9 km

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Phoebe’s unusual near-spherical shape compared to similar size saturnian satellites Hyperion and Janus was noted by Thomas et al. (1986) based on Voyager data for their a/c values. Cassini images show a/c to be about 1.06 for Phoebe, vs. 1.6 for Hyperion and 1.2 for Janus (Thomas et al., 2007). Phoebe’s oblate shape is close to equilibrium with its spin state. Phoebe’s period is 9.27 h (Bauer et al., 2004). If the asteroid were homogenous and hydrostatically relaxed, the difference between the equatorial and polar radii (a  c) would be about 10.7 km. However, the data are (a  c) = 7.1 ± 1.1 km (formal error for limb fitting), or as noted, 7.1 ± 2.1 km if considering the modification of shape by post-relaxation cratering. This difference between the values of (a  c) could be due to some degree of concentration of mass toward the center. This could be due either to reduced porosity at depth, or to stratification due to a more rock-rich core below an icy shell resulting from internal melting and differentiation. Possible geophysical evolution scenarios for Phoebe are addressed in more detail below. However, it is also possible that Phoebe’s spin rate could have changed as a result of impacts and the present shape is an unrelaxed fossil from the earlier dynamical history. If Phoebe achieved its present state while it was spinning one hour slower than the measured value, then its shape actually shows a deviation of the difference (a  c) from the hydrostatic value by about 1 km. If the observed figure relaxed with the current rotation period, then it is more consistent with a differentiated and low-porosity internal structure (Fig. 3). On the other hand, if the shape is frozen from an earlier state when the rotation period was 12 h, then it is consistent with an undifferentiated, very porous interior. These are the two structural possibilities that we address later. 2.3. Surface composition

Fig. 2. Comparison between the distribution of large craters observed at Phoebe’s surface and the distribution of craters resulting from cratering simulation.

bodies can only approach. Additionally, equal values for the a and b axes of Phoebe are also unlikely (but not impossible) by cumulative impacts. While Phoebe shows no near-catastrophic crater such as that shaping the south pole of Deimos, or the 90 + km crater on Epimetheus (mean radius 55 km), could it have been completely disrupted and reaccreted with near-spherical shape? So far, obvious rubble piles such as Itokawa, do not resemble spheres, nor do some asteroids hypothesized to be rubble piles such as Eros and Mathilde. Some bodies thought to show effects of major mass movement, such as 199KW4 (Scheeres et al., 2006) require a fractionally large amount of small particles to reach an equilibrium shape by mass wasting (as opposed to viscous relaxation). While we cannot rule out a reaccreted, near-spherical Phoebe, we believe that this is an unlikely scenario. Whether or not the figure of Phoebe is geophysically meaningful or the result of random collisions is important for constraining the satellite’s origin and internal structure. In this context, it is interesting to note that Hyperion, thought to be a collisional fragment of a larger precursor (e.g., Farinella et al., 1983), has a very irregular, elongated shape despite its mean radius of 135 km. As another example, Janus and its co-orbital Epimetheus are thought to result from either the disruption of a larger object, or the aggregation of small fragments of ring material around a core (Porco et al., 2007). The deviation from sphericity can be expressed through the ratio of the equatorial radius to the polar radius a/c,

Phoebe has been observed with ground-based instruments (e.g., Simonelli et al., 1999; Owen et al., 1999) and high-resolution observations have been made by the Cassini Visual and Infra-red Mapping Spectrometer (VIMS) and the Ultra-Violet Imaging Spectrograph (UVIS). Analysis of the infrared data indicates the presence of a variety of compounds, especially water ice, trapped carbon dioxide, amorphous carbon, ferrous-iron-bearing minerals, and a variety of organics (Clark et al., 2005; Buratti et al., 2008; Coradini et al., 2008; Cruikshank et al., 2008). Buratti et al. have suggested a compositional link between Phoebe and Triton based on the similarity of the spectra of Triton’s tholins with VIMS spectra of Phoebe, with additional support provided by the interpretation of UVIS measurements (Hendrix and Hansen, 2008). Coradini et al. (2008) found a correlation of carbon dioxide and aromatic hydrocarbons with exposed water ice. Buratti et al. (2008) mapped the reflectance of Phoebe and identified two different units with very different bolometric Bond albedo. The disk-averaged bolometric Bond albedo is 0.023 ± 0.007. The low albedo is possibly caused by amorphous carbon and organic molecules (Clark et al., 2005). Clark et al. (2005) noticed a possible decreasing gradient in water content from the rims to the center of Phoebe’s craters, which was also confirmed by Hendrix and Hansen (2008). Both teams found a high amount of ice in the south polar region, and an enrichment in dark material, such as organic species. Buratti et al. (2008) quantified the variations in ice content visible at the surface from 5% to 20%. Clark et al. (2005) suggested that this gradient may indicate that the observed water is a thin coating layer possibly of cometary origin. This is debatable because the corresponding amount of cometary material would have to have been huge. The difficulty of modeling the dynamical interaction between Phoebe and heliocentric comets has prevented the testing of that scenario. Still, it seems more likely that this material is either indigenous to Phoebe or was provided to Phoebe as a

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postdate the formation of the very cratered surface. It could be a lag deposit resulting from intense impacting that led to separation of the rock phase and vaporization of the volatile component. Separation could have also occurred if Phoebe traveled closer to the Sun before being captured by Saturn and the volatile component was subjected to preferential sublimation. However, it is interesting to note that the small bright areas visible on the edge of some of the craters, thanks to localized landsliding of the dark coating, are probably richer in water ice. Their photometric properties indicate that these regions also contain a fraction of dark contaminants, whose abundance remains to be constrained. 2.4. Cosmochemical constraints on composition

Fig. 3. Theoretical (a  c) values as a function of the ratio of core radius to mean radius for Phoebe models. Models are compared with the measured (a  c) assuming hydrostatic equilibrium. Details about the computation of this chart are provided in Appendix E. To the upper right are undifferentiated models with a solid density higher than the measured mean density and a porous icy shell with porosity ranging between 20 and 40%. At the lower left are multilayered models composed of a rocky core, icy shell and a porous outer layer with porosity ranging between 25 and 40%. The core may be composed of anhydrous silicate with a density between 3200 and 3500 kg/m3 and/or hydrated silicate with a density between 2500 and 3000 kg/m3. Note that we find no realistic models with an (a  c) as small as the lowest values allowed by the uncertainties in the data.

result of interactions with its neighbors when it was still residing in the trans-neptunian region. Specifically, no silicate has been identified in the ice seen at the surface of Phoebe although the dark hydrated material may include silicates and the ferrous-iron bearing minerals may be silicates as well. Differentiation of the silicate (i.e., rock phase) from ice, could be the consequence of volatile melting as a consequence of temperature and/or impact. Cruikshank et al. (2008) pointed out the possible presence of phyllosilicates on Phoebe, which would indicate either the occurrence of hydrothermal activity in Phoebe or supply from the material that impacted on Phoebe. Phyllosilicates have also been inferred from observations with the Spitzer Space Telescope at the comet Wild 2, although the robustness of this inference remains to be confirmed by observations obtained over a wide range of wavelengths. Both observations suggest that hydrated silicates may be common in some outer Solar System planetesimals. Their origin is debated though. In the case of inner Solar System icy planetesimals, hydrated silicates are thought to be the result of early aqueous alteration resulting from 26Al heating possibly starting in the planetesimals (e.g., Grimm and McSween, 1989; Wilson et al., 1999; Castillo-Rogez and McCord, 2010). It has recently been suggested that anhydrous silicates could have been hydrated in the solar nebula as a result of bow shock processes (e.g., Ciesla et al., 2003), although it is not clear whether or not the conditions in the nebula were favorable to efficient kinetic processing of anhydrous silicate (e.g., Fegley, 2000). It is also possible that silicates could have been hydrated by interfacial water at relatively low temperature (>190 K), (e.g., Rietmeijer, 1985; Möhlmann, 2008). An important piece to the puzzle is the peculiar spectra of organics analyzed by Cruikshank et al. (2008) that may be evidence of some mild thermal processing of the material. In that regard, these authors suggest that Phoebe’s material is intermediate between the CI/CM and CO/CV categories of chondrites. The dark coating covering most of the surface gives Phoebe the spectral characteristics of a C-type object. This coating seems to

2.4.1. Genetic link with other Solar System objects based on color Comparison of remote sensing observations of Phoebe suggests a genetic link between Phoebe and outer Solar System primitive objects such as comets and KBOs. Based on its composition and reflectance properties, Phoebe has also been compared to low-albedo asteroids, and especially C-type (carbonaceous) asteroids (e.g., Buratti et al., 2008). The satellite’s surface color has been compared to Neptune’s satellite Nereid (Nicholson et al., 2008). The origin of the latter satellite (regular, disrupted from a larger protosatellite, or captured as suggested by its mean eccentricity of about 0.75) is debated. However, Brown et al. (1999) pointed out that Nereid’s surface color, especially for the dark material, differs from the spectroscopic properties of Proteus, one of Neptune’s inner satellites, and is closer to the spectral properties of the Centaur 1997 CU26 and KBO 1996 TO66. 2.4.2. Carbon composition vs. density (Fig. 4 and Appendix A) Johnson and Lunine (2005) noted that Phoebe’s density is close to the average uncompressed value measured for objects observed past the orbit of Uranus, such as Triton, Pluto, Charon, centaurs, and some KBOs, which is about 1.9 g/cm3. If Phoebe had 15–20% porosity its density would be compatible with a similar composition. The relatively high density of this class of object, e.g., with respect to satellite systems formed in a subnebula, can be interpreted in terms of the availability of carbon, which controls the amount of oxygen available for the satellite’s water and hence the density through the rock-to-ice ratio. ‘Rock’ in this context includes refractory silicates and metals. Equilibrium condensates from a gas of solar nebular composition should have a density of about 1.45 gm/cm3, consisting of 47% rock and 53% water ice, with most of the carbon in the gas phase as CH4. Lewis and Prinn (1980) pointed out however that in the colder regions of the outer solar nebula chemical equilibrium might not be achieved in the age of the Solar System, with CO being the primary carbon-bearing gas in this case. Condensates from CO rich gas have a higher density of about 2.1 gm/cm3 with 74% rock and 26% water ice, the difference being due to the oxygen in the CO gas not being available to form water ice. Equilibrium conditions may be still obtained, however, in the warmer gas and dust disks surrounding forming giant planets such as Saturn and Jupiter (Prinn and Fegley, 1981; Prinn et al., 1989). Carbon in the solar nebula may also exist as refractory organic compounds. Based on observations of cometary particles, interplanetary dust particles (IDPs), and the interstellar medium, Pollack et al. (1994) concluded that about 60% (±15%) of the carbon in molecular clouds was in the form of refractory organic grains. In cold, CO rich nebular conditions the fraction of carbon in solid form reduces the amount of oxygen tied up as CO and thus increases the water ice fraction in the resulting condensates, lowering their density. If all the carbon is in solid form having a density of 1.7 gm/cm3 (for our example, we use a slightly higher value than the 1.5 gm/cm3 assumed by Pollack et al. – however, the precise properties of meteoritic com-

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plex solid carbon compounds are not well constrained), the condensate density is 1.5 gm/cm3, with 38% rock, 43% water ice and 28% solid carbon. The relationship of material density of condensates to the redox state of gaseous carbon and the fraction of carbon in heavy organics has been explored by Johnson and Lunine (2005) and Wong et al. (2008) using solar composition values from Grevesse et al. (2005). Fig. 4, taken from Johnson and Estrada (2009), illustrates these relationships. Appendix A contains tables of condensate densities for sample points within the solar composition region displayed in Fig. 4. Johnson and Lunine (2005) and Wong et al. (2008) have considered a range of possible composition models for Phoebe depending on the carbon redox state. Johnson and Lunine (2005) pointed out that Phoebe’s density is not consistent with a Saturn subnebula chemistry rich in CH4 – which would produce a low ratio of CO/Cto3 tal value and hence a bulk density of 1.45 g/cm for CO/Ctotal equal to zero. Assuming that Phoebe’s material is characteristic of the outer solar nebula, where the CO/Ctotal should be greater than in a hot Saturn subnebula case, there is a range of possible densities, porosities and rock fractions that could apply to Phoebe. For the case of no solid carbon compounds: If Phoebe is not porous, its density of 1.6 g/cm3 implies 55% rock and 45% ice and a CO/ Ctotal ratio of 0.4. If on the other hand Phoebe has some porosity, as suggested by Johnson and Lunine, and has the same uncompressed density as Pluto and Triton (1.9 g/cm3), this corresponds to a silicate mass fraction equal to 66%, 34% ice, with a bulk porosity of 14%. This is consistent with CO/Ctotal  0.8. The latter situation is plausible for cold planetesimal-disk bodies formed beyond Saturn where kinetics prevents reaching equilibrium as noted above. If the carbon is all in the form of gaseous carbon monoxide in the outer Solar System, the material density is about 2.1 g/cm3, which corresponds to 74% rock, 26% ice, and would require a bulk porosity of 22% in order to match Phoebe’s density. For the case of solid carbon compounds in the solar nebular: If Phoebe is not porous and CO/Ctotal  1, then the solid carbon fraction is 0.6, with 47% rock 38% ice and 15% solid carbon. If Phoebe’s material density is 1.9 g/cm3 like Pluto and Triton this implies a solid carbon fraction of 0.2 and a composition of 62% rock, 31% ice with 7% solid carbon in the condensate, and 14% porosity to explain the current density. In summary, depending on plausible values of initial outer nebular conditions the silicate mass fraction for Phoebe can range from about 47% to 74%, the corresponding total porosity from 0 to 22%, and the solid density from 1.6 to 2.1 g/cm3. However, it is doubtful that Phoebe’s global porosity could be lower than 5% con-

g m-3 x 103) Material Density (kg

2.2 Gas-phase Carbon CO CH4

2.0

1.8

16 1.6

1.4 Rock+Metal Water Ice

1.2 Solid-phase Carbon

1.0 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fraction of carbon in heavy organics ( = 1700 kg m-3) Fig. 4. Possible compositions for Phoebe as a function of assumed density and the state of carbon, which depends on the origin of the satellite (after Wong et al., 2008). Details about the computation of this figure can be found in Appendix A.

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sidering that its surface morphology (e.g., crater shape, Giese et al., 2007) seems to be built of low-strength material with a porosity that could be at least 45%. For the mid- to upper-range of porosities the solid density of Phoebe strongly resembles that of Triton (Smith et al., 1989), Pluto (Buie et al., 2006) and Eris (Brown and Schaller, 2007)—all thought to have originated in the Kuiper Belt. There are few constraints on the state of the silicate phase accreted into Phoebe. Astrophysical observations of circumstellar disks have found silicates in both amorphous and crystalline form (e.g., Keller and Messenger, 2007). 2.4.3. Composition of volatiles In the context of the Nice model, we assume that Phoebe formed in the transneptunian region where temperatures were likely lower than 50 K (Lunine et al., 1991). Two scenarios are generally considered for the nature of water accreted in these conditions: accretion in the form of gas-laden amorphous ice (Notesco and Bar-Nun, 1997; Notesco et al., 2003) or condensation of polycrystalline ice, hydrated salts, and clathrates hydrates (Lunine and Stevenson, 1985; Gautier and Hersant, 2005; Mousis et al., 2009). These two scenarios continue to be debated because astrophysical observations and laboratory data are not sufficient to confirm either scenario. Despite the common assumption that amorphous ice is favored at lower temperatures in the interstellar medium, Kouchi et al. (1994) pointed out that conditions in the solar nebula were actually more suitable for the crystallization of water. Gautier and Hersant (2005), Hersant et al. (2008), and Mousis et al. (2009) also demonstrated that the pressures and temperatures in the outer solar nebula were suitable for the condensation of hydrates and the trapping of volatiles in the form of clathrate hydrates. Clathrate hydrate compounds form by reaction of nebular gas with predeposited crystalline ice. The formation of clathrate hydrates is a possible mechanism that explains enrichments in volatiles observed in the atmospheres of Jupiter and Saturn (Hersant et al., 2004, 2008; Mousis et al., 2009). The main issue for constraining the amounts and relative fractions of volatiles trapped in planetesimals is the poor knowledge of the kinetics of clathrate formation at the relevant conditions. Without such data, the degree of conversion of ice into clathrates is another parameter to be accounted for in cosmochemical models (Mousis et al., 2009). On the other hand, experimental work by Notesco and Bar-Nun (2000) support the idea that the main gas species observed at comets could be trapped in the form of amorphous ice. Prialnik and Bar-Nun (1992) and Prialnik et al. (2008) have succeeded in explaining some aspects of cometary activity with models based on amorphous ice. However, models of comets involving clathrates have also been suggested (e.g., Delsemme and Swings, 1952; Smoluchowski, 1988; Dartois and Deboffle, 2008). The lack of a detection of amorphous ice at the surface of Phoebe is not relevant because the spectral signature of clathrates is not easily distinguishable from water ice. Thus it is not possible to disprove the presence of such compounds at the surface of icy bodies with available space-borne and ground-based observations techniques. Also, clathrate hydrates are unlikely to be thermodynamically stable at the surface of icy satellites, because of the lack of a sufficient partial pressure by their guest gas(es), which then need to escape from the structure to generate such partial pressure (e.g., Choukroun et al., 2012). The dissociation rates of clathrates under high vacuum and very low temperatures are poorly constrained, but they are expectedly low given the low diffusion rate of water in the hydrate lattice (Ripmeester and Davidson, 1981). Furthermore, irradiation by solar UV and electrons (Hand et al., 2006), potential endogenic activity, impact-induced heating, and/or pre-capture orbit closer to the Sun are further possible explanations for the absence of detection of clathrates hydrates at the surface of Phoebe, and icy satellites in general. On the other hand, as discussed in

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Section 4, clathrate hydrates would likely be present in Phoebe’s subsurface and interior at least for part of its history. Our geophysical models are very much dependent on the thermophysical properties of the assumed volatile species. Thus we will distinguish between different initial conditions: accretion of amorphous ice and volatile condensates (e.g. Notesco and BarNun, 1997; Notesco et al., 2003), and accretion of crystalline water ice and clathrate hydrates of volatile phases, along with a fraction of condensates of these phases (if clathration was not complete) (e.g. Iro et al., 2003; Hersant et al., 2008; Mousis et al., 2009). Similarly we will assume that at the time of accretion the rocky phase could be either in amorphous or crystalline form. Molster (2000) identified the two populations of silicate grains in protoplanetary disks. The thermophysical properties of amorphous and crystalline silicates are very close, thus assumptions on the initial state of silicate material has little affect on our models. In either case, other highly-volatile species are expected, such as nitrogen, methane ices, as well as ammonia and methanol hydrates. Methanol has been suggested to be in pristine material in planetesimals (e.g., Notesco and Bar-Nun, 1997) and has been observed in at least one centaur object (Cruikshank et al., 1998) and on KBOs (e.g., Barucci et al., 2008). However the methanol observed at the surface of KBOs could also be the product of exogenic processes such as UV photolysis (e.g., Hodyss et al., 2008). It is possible that highly-volatile material was lost if Phoebe journeyed closer to the Sun and that any such material that remains may now be buried at depth as has been suggested to be the case for comets (e.g., Prialnik and Bar-Nun, 1992) and Nereid (Brown et al., 1999). The potential occurrence of certain volatile compounds may bear important implications for the geophysical evolution of these objects, provided that they are abundant. They act in depressing the melting temperature of an icy mixture, e.g., down to 176 K in the case of ammonia–water systems (Kargel et al., 1991), and promote compaction at temperatures as low as 100 K (e.g., Leliwa-Kopystyn´ski and Kossacki, 1995). Desch et al. (2009) developed a geophysical model of mid-sized KBOs in part driven by the impact of ammonia hydrates on the ice thermophysical properties. That modeling was motivated by the detection of ammonia-hydrates and crystalline water ice at the surface of Charon (Cook et al., 2007), which they imputed to cryovolcanic activity. The fraction of ammonia hydrates expected in KBOs is however debated, as summarized in Desch et al. (2009). These authors reviewed the observational evidence as well as cosmochemical models and concluded that ammonia hydrates should represent at least 2 wt.% of the total volatile content accreted in KBOs, but an upper bound is more difficult to evaluate. The most recent condensation models, such as Mousis et al. (2009), suggest a fraction of ammonia of 5.5–7.5 wt.%. The geophysical impact of ammonia is of little significance at concentrations of a few percent (e.g., Grasset et al., 2000). However, a small fraction of ammonia implies that the nitrogen is preferentially stored in the form of N2, commonly observed at the surface of KBOs (e.g., Pluto). Of course, the condensation of the latter compound would, in theory, have an even greater impact on the geophysical evolution of KBOs, although internal heating would drive off this very volatile compound. 2.4.4. Nature of silicates The nature of silicates in outer Solar System planetesimals is a matter of debate. Based on the meteoritic record, it is commonly believed that inner Solar System planetesimals were subject to hydrothermal activity driven by heating from 26Al decay, resulting in hydration of most or all the silicate phase (e.g., Grimm and McSween, 1989; Wilson et al., 1999). Castillo-Rogez and Lunine (2010) showed in the case of Titan that if heating from 26Al was a driver of melting and differentiation, then conditions would have

been suitable for serpentinization of most of the silicate phase, leading to an interior model consistent with gravity data obtained by the Cassini orbiter at that satellite. This situation requires that the temperature of water in contact with rocks be greater than 273 K (Allen and Seyfried, 2004). Desch et al. (2009) assumed that sedimentary rocks are the closest analogs to the rock accreted in KBOs. However the rationale for this choice is not obvious. First, the contrast in heat budget and chemistry between inner and outer Solar System planetesimals prevents a direct analogy between the two categories of objects. Even if the planetesimals that formed Phoebe and other medium-sized KBOs formed while live 26Al was abundant, the ambient temperature of 50 K or so limited the volume of material affected by aqueous alteration to less than 50% (Johnson et al., 2009). Then, the extent of hydration during melting and differentiation of KBOs is a function of the temperature of the liquid phase. If internal melting was driven by loweutectic species, then the kinetics of serpentinization should have been extremely slow (e.g., Allen and Seyfried, 2004). If melting occurred rapidly, at the water ice melting temperature as a consequence of 26Al decay heating, then the conditions were more suitable for extensive serpentinization, as suggested by CastilloRogez and Lunine (2010) for Titan. Based on accretion models (e.g., Kenyon et al., 2008) Phoebe could have formed while 26Al was an important heat source, in which case it is possible that aqueous alteration affected a large fraction of its rocky material. If Phoebe formed on a longer timescale, then its content of hydrated silicate would have been limited to those provided by planetesimals. Larger KBOs, such as those considered by Desch et al. (2009) probably accreted on a timescale of tens of Myr, in which case 26Al was a minor player in their geophysical evolution, affecting a 100-km thick, innermost, layer. 3. Thermal evolution models 3.1. General modeling approach We model Phoebe after the approach developed by CastilloRogez et al. (2007a) and Castillo-Rogez and McCord (2010) but add to it the possible presence of amorphous ice and clathrate hydrates. Special care has been devoted to modeling the thermophysical properties of amorphous ice. We have noted in the literature discrepancies between some of the thermophysical properties of amorphous ice depending on experimental procedure (Appendix B). We account for the evolution of material thermophysical properties of the materials as a function of temperature, and as a consequence of the aqueous alteration of the rock phase and the destabilization of trapped volatiles. 3.2. Initial conditions We carry out a systematic investigation of possible models with a range of assumed initial properties and formation environments. Initial conditions are summarized in Table 3. We now discuss these assumptions. 3.2.1. Accretion timescale If Phoebe formed in the trans-neptunian region, the current state of understanding of early Solar chronology suggests that it could have accreated within less than 4 Myr after the production of calcium–aluminum inclusions (CAIs). Arguments in favor of early and rapid accretion of large KBOs in the transneptunian region come from the growing observational evidence that protoplanetary disks clear in less than 10 Myr, and typically in about 3 Myr (e.g., Haisch et al., 2001). Using Callisto’s and Rhea’s possibly undifferentiated interiors as constraints, Barr and Canup (2008) in-

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J.C. Castillo-Rogez et al. / Icarus 219 (2012) 86–109 Table 3 Initial conditions used in models of Phoebe. Parameter

Value or range

References

Time of formation after CAIs t0_CAIs (My) Accretional timescale Grain density (kg/m3) Initial temperature (K) Fraction of accretional energy used for interal heating Ha Nature of the ice

3–10 Instantaneous 1900–2300 50 0–0.5 Amorphous vs. crystalline and clathrate hydrates

Kenyon et al. (2008) Assumption Johnson and Lunine (2005) Jewitt and Luu (2004) Ranged Sarid and Prialnik (2009) Hersant and Gautier (2005)

ferred a lower bound on the time at which the solar nebula cleared of about 4 Myr after the formation of CAIs in the context of the giant planet satellite formation model of Canup and Ward (2002). Also, based on geophysical and dynamical observations, Castillo-Rogez et al. (2007a, 2009) suggested that Iapetus formed within 3.4–5.4 Myr after CAIs. Phoebe, as a possible planetesimal, is likely to have formed before the giant planet satellites. Gladman et al. (2001) and Kenyon et al. (2008) suggest that in the transneptunian region, large objects up to 100 km radius could form in less than 1 Myr. Farinella et al. (2001) and Bottke et al. (2005) have demonstrated that objects of that size are more likely to escape collisional grinding and could retain their integrity since formation. Cuzzi et al. (2010) have also identified a large population of transneptunian and asteroid objects 100–200 km in diameter, which they suggest are primordial. It is important to point out that large KBOs (e.g., dwarf planets like Pluto) may have longer formation time scales, over tens and up to a hundred Myr (e.g., Kenyon et al., 2008). Phoebe’s 100-km radius appears to be just at the threshold below which relatively rapid accretion is expected. Thus we consider both short (live 26Al) and long times of accretion (not involving a geophysically significant amount of 26Al). 3.2.2. Initial temperature profile An initial temperature of 50 K, at most, was expected in the transneptunian region (e.g., Lunine et al., 1991; Jewitt and Luu, 2004). Since Phoebe is assumed to have formed in a planetesimal belt, it was probably subject to small, low-velocity collisions, additive collisions leading to the retention of relatively little accretional heat. Even if the usual model of accretional heating for icy satellites by Squyres et al. (1988) (based on Safronov’s (1979) model for heliocentric impacts) were applicable, the maximum temperature increase would be only a few tens of K. We do not model incremental accretion but start the models at the end of accretion, which may introduce some error in the initial temperature profile for the models formed earlier than 4 Myr after CAIs, as 26Al decay heat could contribute to internal warming during accretion. However, proper incremental accretion modeling requires accounting for the thermal state of the planetesimals at the time of accretion, which is work in progress (Johnson et al., 2009). For this study, we assume that Phoebe accreted from planetesimals smaller than 10 km across that were little affected by 26Al heating. 3.2.3. Initial composition We assume that the ice is either amorphous or crystalline. If the planetesimals that formed Phoebe were affected by heat from 26Al decay, it is even possible that Phoebe accreted mixtures of crystalline and amorphous ice. Accretional energy may also destabilize amorphous ice, but in absence of constraints, we consider endmember compositions. Models with crystalline material also include ammonia as the main impurity of the volatile phase, at a concentration no greater than 8 wt.%. On the basis of cosmochemical models (Section 2.3, Fig. 4), we consider the lower and upper bound on the silicate mass fraction in Phoebe to be between 45 wt.% and 75 wt.%. Although amorphous and crystalline differ in their thermophysical properties,

the state of the silicate phase has relatively little consequences for the models. 3.2.4. Initial porosity profile The initial porosity profile is based on experimental measurements by Durham et al. (2005a) and Yasui and Arakawa (2009). From tests on pure water ice, Durham et al. (2005a) determined the amount of porosity that could be sustained at temperatures of 77–120 K (at which water ice is known to deform in the brittle regime). For the maximum pressure inside Phoebe, about 4.5 MPa, compaction as a result of pressure sintering is predicted to reduce porosity to about 30–35% if the internal temperature remained cold enough to prevent significant ice creep. A change in porosity is expected for a pressure of about 1 MPa, or a depth of about 15 km. Above that limit, the porosity may be as large as 45–50% while it may decrease to 30% at the center of the body. From this porosity distribution we have inferred an initial radius for Phoebe of about 125 km. 3.3. Input parameters All the input parameters for the models are gathered in Appendices C and D. The parameters account for the effects of volatile composition, porosity, silicate, ice crystallinity, and temperature on the thermophysical parameters. Properties are recomputed or updated for each computational time step. For crystalline ice and clathrate hydrates, the data are those compiled by Ross and Kargel (1998) and the compaction measurements performed by Leliwa-Kopystyn´ski and Kossacki (1995) and Durham et al. (2005a, 2009). Thermophysical properties of amorphous ice are inferred from Jenniskens and Blake (1994, 1996), Kouchi et al. (1992), Andersson and Inaba (2005), and Hessinger et al. (1996) (Appendix B). 3.3.1. Amorphous ice and silicate Amorphous water ice is a complex material whose thermophysical properties and phase diagram are not fully constrained. The different types of laboratory-produced low-density amorphous ice are summarized in Andersson and Inaba (2005): hyperquenched liquid water, vapor deposited on cold substrate, and isothermal depressurization of high-density amorphous ice produced at a pressure of 1.13 GPa and a temperature less than 77 K. It has been demonstrated that depending on the mode of production, amorphous ice exhibits large variations in its microstructure, which affects its thermal and mechanical properties (e.g., Hessinger et al., 1996). Vapor-deposited ice is probably the most appropriate analog available for Solar System amorphous ice, which is thought to have been deposited from vapor phase onto dust grains (e.g., Lunine et al., 1991; Jenniskens et al., 1995; Wang et al., 2005). It has even been suggested, based on astrophysical observations by Jenniskens et al. (1995), that amorphous ice deposited on silicate grains at temperatures less than 50 K is in the form of high-density amorphous ice (hereafter labeled HDA). Transition to low-density ice occurs at about 50 K over geological timescales (Jenniskens and Blake, 1994). Similar to previous studies, we consider the presence

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of gas species, such as CO, CO2, N2, and CH4 that are thought to have been trapped in the porous ice (e.g., Notesco and Bar-Nun, 1997; Notesco et al., 2003; Choi et al., 2002). Amorphous ice is typically believed to crystallize into cubic ice at about 137 K, although the transition temperature may be affected by kinetic effects. This study distinguishes itself from other geophysical models of amorphous-ice-based models by its tracking the evolution of nanoporosity and its impact on the material thermophysical properties. Other studies on the topic have neglected that property, and some of these amorphous-ice-based studies also assume some thermophysical data obtained on crystalline ice instead of basing their approach on data available for amorphous ice (e.g., Sarid and Prialnik, 2009 about the application of macroporosity evolution data obtained on crystalline ice). Our modeling contrasts with previous studies by accounting for the dependence of amorphous ice thermophysical properties on nanoporosity. That property is a function of temperature with a reorganization of amorphous ice structure due to annealing observed at about 113 K (e.g., Mayer and Pletzer, 1987). Hence, thermal conductivity ranges from 104 for fresh, low-temperature ice, and increases by three orders of magnitude as a result of annealing. Although mechanical measurements are scarce for that material, available data reveal a temperature-dependence of amorphous ice viscoelastic properties that contrast with those observed for crystalline ice. Hessinger and Pohl (1996) showed that fresh amorphous ice exhibits a shear modulus that is one order of magnitude less than that observed for crystalline ice. Also, amorphous ice reaches its lowest viscosity at low temperature (50 K in the Hessinger and Pohl’s experiments) and increases with increasing temperature. To the best of our knowledge, this is the first study considering the impact of amorphous ice mechanical properties on the structural evolution of planetesimals. As to silicates, their glass transition temperatures are very much above the maximum temperatures obtained in any of our models, i.e., above 1000 K for most compositions (e.g., Avramov et al., 2005). 3.3.2. Crystalline ice Low-density amorphous ice crystallization leads to cubic ice, stable up to a temperature of about 160 K, at which it crystallizes into hexagonal ice. There are relatively few thermophysical data on cubic ice. However, several studies were able to record the properties of cubic ice directly formed form the destabilization of the low density form of amorphous ice (LDA) (e.g., Kouchi et al., 1992). Interestingly, discrepancies in the cubic ice thermophysical data quoted in the literature seem to be correlated with the method used for amorphous ice sample preparation (e.g., Sack and Baragiola, 1992). Hexagonal ice properties are well known and most frequently used in icy satellite models, so that we summarize briefly the information relevant to the present study. Detailed information on the thermophysical properties of crystalline ice used in this work can be found in Ross and Kargel (1998), Durham and Stern (2001), and Fortes and Choukroun (2010). 3.3.3. Clathrate hydrates and other hydrates Clathrate hydrates are crystalline compounds with a hydrogenbonded H2O skeleton that forms cages within which gas molecules (size within 3.5–7.5 Å) may be trapped individually by weak interactions. Their physical properties, and especially their spectroscopic properties, exhibit strong similarities with the H2O ice phase Ih (e.g., Sloan and Koh, 2007, and references therein). Also, their stability is largely pressure-dependent, and these species are unstable on most planetary surfaces – although their presence may be thermodynamically expected at depth (see review by Choukroun et al. (2012)). These two factors make the detection

of these compounds on planetary surfaces extremely difficult. However, decades of experimental data acquisition on their stability (e.g., Deaton and Frost, 1946; Miller and Smythe, 1970; Delsemme and Wenger, 1970; Kuhs et al., 2000; Choukroun et al., 2010) and associated thermodynamic modeling (e.g., Lunine and Stevenson, 1985; Yoon et al., 2002; Iro et al., 2003; Mousis et al., 2009) provides a reasonable framework for addressing the stability of these species in planetesimals. Clathrate hydrates have a very low thermal conductivity compared to water ice, about 4–5 times lower at a temperature around 260 K (e.g. Ross and Kargel, 1998). Also, clathrates are generally stronger than ice with a viscosity greater by at least one order of magnitude (Durham et al., 2003, 2005b). Being poor conductors and strong materials, clathrate hydrates are likely to affect the thermal response of Phoebe. Their high viscosity may impede the mechanical relaxation, or at least affect the timescales of cometary materials relaxation predicted by Cheng and Dombard (2006). However, ice–clathrates mixtures may exhibit similar creep properties as water ice, even at low ice fractions (Lenferink et al., 2009). Ammonia hydrates and methanol hydrates, which are other crystalline hydrated compounds that likely formed upon condensation of the presolar nebula (e.g. Lewis, 1972; Mousis et al., 2009), may affect both the stability of icy compounds and the mechanical properties of the planetesimals. Indeed, these materials melt incongruently at 176 K (ammonia dihydrate) and 172 K (methanol hydrate) to form aqueous solutions (e.g. Kargel et al., 1991, and references therein). These icy impurities, if present in amounts above a few percent, could generate enough volume of aqueous solutions to drive the separation of the rock from the ice matrix well below the melting point of pure water ice. According to Leliwa-Kopystynski and Kossacki (2000), a weight fraction of 10% of ammonia is required to have a significant impact on the viscous response of the icy shell. This would weaken significantly the icy shell, thus facilitating the mobility of soluble compounds and affecting the creep and dissipation properties. Furthermore, both methanol and ammonia are strong clathrate hydrate inhibitors (e.g. Sloan and Koh, 2007, and therein; Choukroun et al., 2010), and as such may play a role in outgassing from the interior of icy moons (Tobie et al., 2006; Kieffer et al., 2006; Fortes, 2007; Choukroun et al., 2010), including, possibly, Phoebe.

3.4. Processes 3.4.1. Differentiation Differentiation in a low-gravity field is not well understood. To the best of our knowledge the separation of silicate from water in very small objects bodies has not been studied, neither theoretically nor experimentally. Fast spinning, hydrothermal circulation, and Brownian motion are likely to counteract the settling of fine particles subject to a weak gravity field. In the conditions of pressure expected in Phoebe, less than 5 MPa, we can take terrestrial permafrost as an analog to an undifferentiated Phoebe. Thus, we expect that Phoebe, if heated, probably would not undergo complete differentiation into a rock core and that a pure ice mantle, preservation of an undifferentiated region is likely under these conditions. If early melting of the ice is triggered by the decay of 26Al, then the heat pulse would be large enough to promote rapid and extensive melting of the interior (Castillo-Rogez et al., 2007b). Differentiation at cold temperature, i.e., at the ammonia-hydrate peritectic, has been suggested for large icy satellites, e.g., Titan (Grasset et al., 2000), assuming significant amounts of ammonia are present. As melting starts at temperatures below the water ice creep temperature, the melt separates and the rock phase remains trapped in the very cold ice matrix.

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3.4.2. Gas escape A main aspect of transneptunian object (TNO) models based on amorphous ice is about quantifying the desorption of various gas species from the ice matrix. In these models vapor from the ice and other volatiles migrate toward the surface where it can freeze when the conditions of temperature and pressure are appropriate, or escape if the temperature is high. This process has been modeled for meteorite parent bodies (e.g., Young et al., 2003) and comets (e.g., Haruyama et al., 1993; Choi et al., 2002; Prialnik and Merk, 2008). The amount of material sublimated and refrozen at lower temperatures, or lost, is a function of the porosity and permeability of the material. The gas desorption/sublimation temperature under vacuum is 20 K for CO, 30 K for methane, 70 K for CO2, 120 K for water ice and increases under increasing partial pressure (Choi et al., 2002 – Table V). For the conditions considered in these models described in Section 4, N2, CH4, and CO, if initially present, will probably escape. If the material is fully impermeable to gas flow, then equilibrium may be reached with the formation of a partial pressure of gases in the pores. The permeability decrease resulting from porosity collapse prevents migration of volatiles throughout the body. That situation is moderated by the amount of early heat available considered in the model. Accretional heating is no more than 10 K and does not affect that picture. Intense heating from 26Al decay rapidly leads to crystallization so that the long-term evolution of the model is controlled by crystalline ice thermophysical properties. Two different situations are encountered in the models described below. If the interior melts, the gas species will be released, and the most volatile species will be lost. This is expected if melting is close to the surface and that event is accompanied by volume changes resulting in surface cracking. As described in further detail below, it is unlikely that these species present in the transient liquid layer will be trapped in the form of clathrate hydrates. If the interior temperature slowly increases, then the mobility of the species depends on the form in which they are trapped in the interior as well as on the permeability of the material. In the case of amorphous ice, the gas species will be desorbed at the temperatures mentioned above. Stability of the clathrated forms of these species is presented in Fig. 5. 3.4.3. Hydrothermal activity Ice melting can be accompanied by hydrothermal circulation and consequent alteration of the material chemistry and physical properties, e.g., silicate serpentinization. The extent of hydrothermal alteration is a function of the melting temperature that will determine the reaction kinetics. At temperatures above the water ice melting point, serpentinization is pervasive (see Castillo-Rogez and Lunine, 2010). For the present study, we assume that hydrothermal activity could be associated with rapid melting resulting from intense heating due to the decay of 26Al. However, we lack constraints on the extent of hydration and consider endmember models of zero and full silicate hydration. The impact of hydration on silicate thermal properties is detailed in Castillo-Rogez and McCord (2010) and Castillo-Rogez and Lunine (2010). The presence of liquid water most likely supported some hydrogeochemistry as has been studied for meteorite parent-body models in the same size range (e.g., Young et al., 2003). The link between the very altered matrix in carbonaceous chondrites with up to 80% of hydrated minerals, and 26Al induced melting has been considered by a number of studies (e.g., Grimm and McSween, 1989; Cohen and Coker, 2000; Young et al., 2003). Quantifying the extent of the hydrogeochemical activity is beyond the scope of the present study. The maximum temperature achieved in the model is a function of the material permeability. This has been studied by Young et al. (2003) and Travis and Schubert (2005) who showed that in a object of the size of Phoebe, water was likely

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Fig. 5. Pressure–temperature plot comparing the phase diagram of H2O (black lines) with the stability curves of some major, single-guest, clathrate hydrate species. Q1: quadruple point clathrate – H2O ice – H2O liquid – guest gas. Q2 (CO2, H2S): quadruple point clathrate – H2O liquid – guest gas – guest liquid. The shaded area indicates the envelop of dissociation curves. Average surface conditions are shown for Titan and Mars. Modified from Choukroun et al. (2012, and references therein). The thick curves (near the ordinate) show the temperature profiles in the models of Phoebe presented in Fig. 6a (late time of formation, thick solid curve) and Fig. 6b (early time of formation, thick dashed curve) for different times after accretion indicated in Myr next to the curves (1, 10, 100, and 1000 Myr after formation).

to convect, which could mitigate the temperature rise due to the significant heating by chemical reactions. 3.4.4. Evolution of upper boundary condition Significant changes in Phoebe’s surface temperature could have resulted from its orbital evolution. We do not know exactly the minimum perihelion that Phoebe might have reached or how much surface modification could have resulted from traveling closer to the Sun. Our model considers a simple scenario where Phoebe escaped from the transneptunian region and was captured by Saturn during the ‘‘terminal cataclysm’’ at 3.9 Ga. This corresponds to a possible change of surface temperature from about 50 K to 70 K at about 700 Myr after formation, and is of little significance for our models. In this paper we assume that significantly higher surface temperature did not occur. This assumption has no impact on the global evolution of Phoebe, because all our models are characterized by rapid freezing within a few hundred Myr after formation in the transneptunian region. 4. Numerical results Model results are illustrated in Fig. 6 and summarized in Table 4. 4.1. Late formation models In this section we explore models where Phoebe formed after Al ceased to be an effective heat source, i.e., later than 4 Myr after the production of CAIs, so that it had little influence on Phoebe’s geophysical evolution.

26

4.1.1. Accretion of amorphous ice We consider the case where Phoebe accretes dust and amorphous water ice rich in gases. In amorphous ice the viscosity following accretion At a temperature of about 50 K, macroporosity

as a mixture of the presence of was very small. evolves rapidly

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for the first 10 Myr after formation. As a result, despite the thick insulating layer assumed in the modeling and amorphous ice’s low thermal conductivity, the temperature remains under 80 K, and the interior comes into equilibrium with the upper boundary condition within about 50 Myr after formation. In these conditions, amorphous ice is expected to preserve its high nanoporosity (Angell, 2002, 2004) and low density. The overall evolution is eventless, and in particular we do not expect significant desorption of volatiles and their migration to the surface. This is consistent with models previously obtained by Sirono and Yamamoto (1999) who compared the geophysical consequences of using an endothermic latent heat against an exothermic value. The latter case leads to runaway heating and gas desorption. In the endothermic case the

crystallization of amorphous ice is limited in extent and entirely dependent on available heat sources. This is contrary to the predictions of other models (e.g., Sarid and Prialnik, 2009). Also, neither the ice nor the silicates crystallize. This implies that the phyllosilicates observed in Phoebe’s dark material must be of external origin or come from some sort of a lag deposit on the surface that resulted from exogenic processes (e. g., impact induced hydrothermal alteration). This scenario produces a completely undifferentiated model with a porosity gradient from the interior to the surface. It is expected that such a model would relax early to an oblate shape and remain globally in hydrostatic equilibrium, due to amorphous ice low viscosity at low temperature. As a corollary, Phoebe’s deep

a

Fig. 6. Thermal evolution models for Phoebe after the method of Castillo-Rogez et al. (2007a, 2009). The different panels show, from left to right, starting from the top: sketch of the initial internal structure, thermal evolution, internal structure expected at present, porosity evolution, and density profile expected at present. In the thermal evolution plot, isotherms are represented every 50 K. The two models assume accretion of polycrystalline materials (as described in Section 4.1.2). They differ in the time of formation assumed for Phoebe. (a) Formation 5 Myr after the production of calcium–aluminum inclusions (CAIs). Porosity starts collapsing after 30 Myr and the radius decreases, as seen at the top of the plot. We have assumed enough ammonia–water dihydrate for that component to control the porosity decrease. (b) Formation about 3 Myr after CAIs. Shortlived radioisotope decay leads to extensive melting of the volatile phase leading to separation of the rock phase and hydrostatic relaxation of the body. The extent of differentiation in the low-gravity conditions relevant to Phoebe is not well constrained. In this example we have assumed that it is advanced and results in the separation of a volatile-rich shell overlaying a rocky core. For the temperature achieved in Phoebe’s core, organics and hydrated salts may separate from the rock and form a layer at the interface between the rock and ice-rich shell (presented here for illustration purposes but needs to be studied in detail in the future).

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b

Fig. 6 (continued)

interior would be very volatile-rich as the amorphous ice has never achieved conditions suitable for crystallization. 4.1.2. Accretion of crystalline ice and clathrates In this case we assume that Phoebe’s ice accreted in crystalline form and gases were trapped in clathrate hydrates. In this framework the maximum temperature achievable in the model is dependent upon the evolution of porosity as temperature increases. In the following discussion we evaluate the amount of compaction in Phoebe, if that satellite formed in the absence of heat from short-lived radioisotopes, and its internal evolution was governed chiefly by the rheology of ammonia-hydrate and water ice. First, we assume a silicate mass fraction of 46%, i.e., the upper bound for the amount of long-lived radioactive heating in a cold end-member model. In this case the maximum temperature achievable in Phoebe is 150 K, if there were no compaction (i.e., highest insulation). The maximum temperature becomes 180 K if the silicate mass fraction is 75%. At a temperature of 150 K, ice deforms at a strain rate of 1022– 1020 s1, and that means the relaxation of the object cannot occur in the age of the Solar System. The presence of ammonia hydrates

and possibly methanol hydrates may promote compaction at temperatures as low as 100 K (e.g., Leliwa-Kopystyn´ski and Kossacki, 1995). Additionally, Leliwa-Kopystynski and Kossacki (2000) showed that the gravitational self-compression of objects of the size of Phoebe is so slow that they most likely freeze before full compaction is achieved, unless other heat sources, especially short-lived radioisotopes, are present. These authors indicate that if the ice temperature is only a few tens deg. above the temperature for the onset of creep, a residual porosity of at least 5% will always remain for pressures of only a few MPa. Another important aspect of the problem concerns the effect of the silicate phase on the rheology of the mixture. The presence of up to 75 wt.% of silicate, or 45vol.%, assuming a silicate density of 2.7 g/cm3 can increase the viscosity by one order of magnitude, and thus affect creep-driven compaction (Friedson and Stevenson, 1983; Durham et al., 2009). As a result, this category of models (e.g., represented in Fig. 6a) includes a 20-km thick outer porous layer with a porosity of up to 50%, plus at least 5% porosity for the remainder of the satellite, assuming that ammonia-creep-driven compaction occurred. The corresponding grain in density by this process must be at least 2.3 g/cm3 in

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Table 4 Summary of the models investigated in this study and their respective outcomes. Early formation (within 4 Myr of CAI))

Late formation (26AI does not contribute)

 Outer layer is mostly pristine  Gas freed from clathrate may ascend and retrap at d < 1 km, either as pure ices or in clathrates  Internal temperatures high enough for relaxation to hydrostatic shape

 Does not differentiate  Deep interior is volatile-rich

Amorphous

Crystalline

Amorphous

Crystalline

Accretion from a mixture of dust and amorphous water ice rich in gases

Accretion of crystalline ice. Gases trapped in clathrate hydrates and in ice condensates

Accretion from a mixture of dust and amorphous water ice rich in gases

 Rapid crystallization of at least 60% of Phoebe’s volume, possibly associated with melting for times of formation shorter than 3 Myr

 Extensive melting and shape relaxation

 The overall evolution is eventless; no significant desorption of volatiles and migration to the surface

Accretion of crystalline ice. Gases trapped in clathrate hydrates and in ice condensates  Little interior compaction unless ammonia hydrates are abundant, porous outer layer

 Interior compaction on a timescale of the order of 1 Myr

 Possible hydrothermal and geochemical activity similar to that suggested for some meteorite parent bodies  Destabilization of clathrate hydrates in most of the interior  Volatile composition (at depth) is mostly waters at present

order to explain Phoebe’s observed density. Since the outer layer is porous and its strength is reduced by at least one order of magnitude with respect to pure water ice (e.g., Keller et al., 1999) it is possible that it could relax somewhat, although this remains to be modeled. The porosity gradient between the surface and the interior could result in some density contrasts corresponding to a hydrostatic (a  c) greater than 9 km, which is less inconsistent with the observed flattening, assuming it is characteristic of Phoebe’s current spin properties. In the extreme case, if there were no or little creep of the ice when the temperature was above 100 K, then the expected total porosity in most of Phoebe’s interior could be as high as 35%. This result would imply that Phoebe’s grain density1 is 2.5 g/cm3, placing it at the edge of the range of values suggested by Johnson and Lunine (2005). Also, in this context, it is doubtful that Phoebe could have relaxed its global shape. The thermal profile corresponding to the maximum temperature achieved in this model is represented on the phase diagram of clathrate hydrates (Fig. 5). This figure illustrates the fact that the temperature never becomes warm enough to allow the destabilization of clathrate hydrates, so that Phoebe should have preserved the integrity of its volatile content throughout its history if accreted as clathrates. Furthermore, at depths corresponding to pressures larger than 104–105 Pa, the most abundant volatiles (CH4, Ar, N2, CO, etc.) are above their triple point, which implies that they will melt upon heating in the deep interior, before turning into gases. This limits their ability to escape as soon as they are destabilized, and it suggests looking in greater detail into their potential influence on the relaxation and volatile budget and evolution of Phoebe, and other icy objects where such pressure conditions are attained. 4.2. Early formation times 4.2.1. Crystalline ice (Fig. 6b) If Phoebe formed within 4 Myr after the production of CAIs, then it must have been affected by rapid melting of its volatile 1 The grain density is the true density of the material making up the body once porosity has been removed.

 Relaxation may not be warranted

 Would relax early to oblate shape and remain globally in hydrostatic equilibrium

 Volatile inventory is maintained, clathrates not destabilized

 Phyllosilicates in Phoebe’s dark material must be of external origin or come from a lag deposit on the surface that resulted from exogenic processes (e. g., impact induced hydrotherma alteration)

component. For times of formation between 2 and 3 Myr, the internal temperature can reach the water ice melting temperature in less than 10 Myr. For formation between 3 and 4 Myr, the temperature can reach the ammonia water peritectic during the same timeframe. In all cases, melting affects most of the interior, except for the upper 5–10 km that remains little affected by internal warming. The thickness of that layer is a function of the thermal conductivity assumed for the porous material. Note that macroporosity from cratering may extend to greater depth and has not been accounted for in our models. Ice melting resulting from 26Al decay heat could have driven hydrothermal and geochemical activity similar to that suggested for some meteorite parent bodies. While the low thermal conductivity of hydrated silicate will help keep the rocky core warm for a few tens of Myr, the entire body is expected to freeze within 100 Myr of formation. In the present case, the destabilization of clathrate hydrates affects the deepest parts of the model interior. The total volume of clathrates affected by destabilization is a function of the time of formation with respect to CAIs. It should be rapid, compared to geological timescales, particularly at depth where the melting point of water ice has been reached. It is even possible that dissociation of clathrate hydrate took place during the first 1 Myr after accretion, especially if the clathrates do not contain large quantities of CO2 or another very stable hydrate-former (Fig. 5). The final state of these volatiles would mostly depend on two factors: condensation temperature as a function of depth, and time available for interaction of gas with water ice to form new clathrate hydrates. Permeability of the icy shell and potential fracturation due to pressurization by released gases are the limiting/facilitating factors that would control the rate of gas ascent through the icy shell. Nevertheless, the very low subsurface temperatures on Phoebe (see Fig. 5), and its low escape velocity, do not seem to allow for contact between porous ice and the gases over long periods of time. Therefore, it is unlikely that re-clathration of released volatiles occured extensively, and we favor a scenario in which gases released by clathrate hydrate dissociation would, for the most part, simply escape, although a fraction of them could re-condense as

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pure ices at shallow depths. The rock may not cleanly separate from the ice and form a core because of the low central pressure. As a point of reference, a pressure of 4.5 MPa corresponds to a depth of about 150 m in terrestrial permafrost, which can be considered as a relevant (but not exact) analog for the center of Phoebe. As in Phoebe gravity gets weaker with depth, the density contrast becomes even less effective for separating rock from ice. The amount of ice in terrestrial permafrost varies depending on the geographical context, the region history, and the nature of the rock (Phillips et al., 2003). The ice volume fraction can range from 3 to 50%: alluvial sediments (analogs to some hydrated silicates) have ice contents measured between 10 and 50% while sand terrains contain on average 20% ice. For 20 vol.% ice mixed with rock, Phoebe’s core density ranges from 2.8 g/cm3 (assuming a silicate density qsil = 3.3 g/cm3) to 2.2 g/cm3 (assuming qsil = 2.5 g/cm3). For this volume fraction of ice in Phoebe, the core radius ranges from 76 km to 87 km. For the latter value, the core is mostly undifferentiated except for a thin layer of ice (5 km) and a 15-km thick outer layer of porous material. A gradient in the relative amount of ice from the surface to the interior is also possible. These models all achieve internal temperatures high enough that relaxation to a hydrostatic shape is possible (see discussion in Appendix E). Assuming hydrostatic equilibrium, differentiated models (with cores ranging from pure rock to a mixture of silicate and ice) have (a  c) between 6.8 and 8 km. This range is within the error bars of the measured oblateness, (Fig. 3). Models that are more differentiated, i.e., with an icy layer above a rock–ice core, are more consistent with the mean values obtained for the shape measurements. However, within the error bars on these measurements and on the amount of material that we believe was removed from the equatorial region (see Section 2.2), further interpretation is precluded. Assuming that there is no significant macroporosity (e.g., impact produced), the corresponding solid density (porosity removed) ranges from 1.9 g/cm3 (for a 5 km thick outer layer) to 2.2 g/cm3 (for a 10 km thick outer layer). 4.2.2. Amorphous ice The general evolution scenario presented in the previous section is similar if water has condensed in the form of amorphous ice. Heating from 26Al decay can lead to rapid crystallization of at least 60% of Phoebe’s volume, possibly associated with melting for for formation times after CAIs shorter than 3 Myr. This results in a stratified object whose outer layer is mostly pristine, as previously suggested by Prialnik et al. (2008) for large KBOs. This crystallization event results in a drastic increase of the thermal conductivity below 50 km depth. Since most of the volatile species were trapped in the amorphous ice and may have been released upon crystallization, we expect the remaining volatile composition (at depth) to be mostly water ice. Similar to the situation described for clathrates hydrates, part of the volatile phase should have recondensed closer to the surface. At the time of crystallization, temperatures around 135 K, the amorphous ice viscosity is estimated at about 1012 Pa s, which allows compaction over a timescale of the order of 1 Myr. The thermal conductivity increase resulting from compaction leads to rapidly freezing the interior at about 100 Myr after accretion. Compaction may actually start at lower temperatures, in such case further limiting the maximum temperature reached in the model. The lack of constraints on amorphous ice viscosity prevents a more detailed assessment of this scenario. 5. Discussion 5.1. Thermal evolution of a disrupted phoebe We consider the possibility that Phoebe could have reaccreted from debris produced by a catastrophic collision that destroyed

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the original ‘‘proto-Phoebe’’. In this scenario Phoebe reaccreted its own fragments after collision with an interloper in Saturn’s orbit of up to 200 km in size (Turrini et al., 2008). If we assume that conditions in the proto-Phoebe were such that it was mostly compacted at the time of disruption (cf. Fig. 6b), then the reaccreted body had some macroporosity but little microporosity. If we assume a time of reaccretion a few hundred Myr after the formation of the proto-Phoebe the object could not benefit from the heat from 235U, a major heat source that is responsible for the isotherm ‘‘peak’’ achieved at about 90– 100 Myr in Fig. 6a. Assuming 15% porosity for a solid density of 1.9 g/cm3, the maximum temperature achieved in the satellite barely reaches 108 K, i.e., slightly above the ammonia-hydrate creep temperature. While such a scenario could explain the observed density, the temperature reached in this model is too low to allow relaxation to a nearly-spherical shape. Other irregular satellites objects share similar properties with Phoebe and comparisons between these objects can help to better constrain the geophysical evolution of large water-rich planetesimals. For example, Jupiter’s irregular satellite Himalia presents some resemblance to Phoebe. This object is the largest representative of the Himalia family. Its mean radius is about 85 km and its mean density appears to be between 1.6 and 2.6 g/cm3 (Emelyanov et al. (2005); JPL Solar System Dynamics database http://ssd.jpl.nasa.gov), i.e., about the same as the properties inferred for Phoebe’s core (considering that we cannot accurately evaluate its density). The seismic waves generated by an impact onto a differentiated icy body are reflected at the core–shell interface due to the large contrast in mechanical properties between the rocky core and icy shell, and the latter is mostly stripped off. A very good illustration for this is 24 Themis, which mainly represents the rocky core of the Themis family progenitor, the rest of the family members being enriched in volatiles (Castillo-Rogez and Schmidt, 2010). Similarly, one could imagine that Himalia represents the rock-rich core of its progenitor, while the other family members are water-rich, a hypothesis that future missions to the Jovian system may be able to test. 5.2. Time of formation of Phoebe Our models in which significant porosity collapse is achieved are consistent with the suggestion by Johnson and Lunine (2005) that Phoebe’s grain density is representative of a solar nebula composition. From the thermal models we can narrow down the range of possible material densities to between 1.9 and 2.2 g/cm3, assuming that Phoebe is not affected by significant macroporosity. Porosity not accounted for in the model would result in increasing the upper bound of the possible solid density range. We have identified two very different types of models promoting porosity collapse and shape relaxation. One assumes accretion of amorphous ice and formation after 26Al has largely decayed, so that ice nanoporosity determines the mechanical properties of Phoebe’s material. On the other hand, accretion of crystalline ice and clathrates requires an early time of formation, in less than 3.5 Myr after CAIs, so that the heat resulting from 26Al decay is the main driver for early compaction and shape evolution. This model is also applicable if Phoebe accreted amorphous ice. Thus these geophysical models alone cannot provide constraints on the nature of the accreted volatile phase and on the time of formation. As discussed earlier, Phoebe exhibits surface properties similar to those inferred for the parents of carbonaceous asteroids. If some of the carbonaceous asteroids in the outer main belt are captured KBOs (Levison et al., 2009) and/or if some of the carbonaceous chondrites are samples from KBOs (Gounelle et al., 2008; Zolensky et al., 2009), then this would be evidence that 26Al-driven processes also took place in Kuiper Belt objects.

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If Phoebe accreted a significant amount of 26Al, rapid melting of the interior probably resulted in hydrothermal activity and in intense degassing with some of the volatiles refreezing near or at the surface. Future studies should aim toward assessing how much degassing and aqueous alteration could take place in planetesimals that accreted into larger ice–rock objects (icy satellites, large KBOs). Hydrothermal activity creates a context for the formation of complex organic compounds and leads to questions as to whether Phoebe’s surface composition is pristine, the result of material provided from later impactors, the signature of internal processes, or the product of all of these scenarios. The occurrence of similar processes in KBOs has been suggested by Busarev et al. (2003) who pointed out that they could lead to significant alteration of the initial composition of these building blocks with respect to cosmochemical models. This alteration however may have been less extensive than suggested by Busarev et al. since models by Prialnik et al. (2008) actually show that planetesimals should be stratified with a deep layer processed by 26Al heating and an outer layer that could remain mostly pristine except for potential exogenic processing depending on the long-term dynamical history of these objects. 5.3. Stability of a undifferentiated and porous outer shell One aspect of the modeling not addressed in detail is the stability of the outer layer in the models affected by internal melting. In the case of Ceres, Castillo-Rogez and McCord (2010) suggested that it was very likely for that outer layer to founder during the period when a large part of Ceres’ interior was molten. In the case of small transneptunian objects the layer may remain stable because of the following conditions: (a) a surface temperature lower than the temperature of creep onset for any of the volatiles expected in Phoebe; (b) significant remnant porosity decreasing the contrast in density between that outer layer and the interior. In the present case, the outer layer may contain up to 50% of porosity and thus have a density of about 1.0–1.1 g/cm3. 5.4. Formation of Phoebe’s surface Various exogenic processes could alter the properties of the surface over the long term, resulting in the formation of the dark, dusty material observed by Cassini–Huygens. Volatile migration may be triggered by internal or surficial temperature increase, impacting and gardening. Phoebe’s surface could also have matured as a consequence of interaction with the large ring in which the satellite is embedded, resulting in continuous micro-impacting and implantation or gardening of surface dust. 6. Conclusions We demonstrate that Phoebe’s global shape is actually close to that for a hydrostatic-equilibrium spheroid rotating with Phoebe’s spin period (Section 2). Our models suggest that Phoebe formed within a few Myr after the production of CAIs. Such a scenario is consistent with the increasing number of detected protoplanetary disks that show planet accretion within less than 5 Myr after formation (e.g., Currie et al., 2008). If Phoebe formed early enough to have had significant heating from 26Al (i.e., before 3 Myr after CAIs), a plausible and consistent (but not unique) scenario is that it is a body that formed in the outer planetesimal disk contemporaneously with carbonaceous chondrite parent bodies, with a density of 2000 kg/m3. Whether Phoebe started with amorphous or crystalline material, our models suggest that today it should have a layered internal structure with concentration of rock at depth, possibly in hydrated form, and a very porous outer most icy shell.

Phoebe’s low bulk porosity and near-spherical shape suggest that the satellite was not disrupted and reaccreted. If it were, then the disruption and reaccretion conditions would have to have been very exceptional for Phoebe not to end up as a rubble-pile. The fact that Phoebe could remain whole is consistent with the Bottke et al. (2005) suggestion that most 100-km size bodies in the Solar System did not lose their integrity as a result of collisions and represent a primordial population. This is also consistent with dynamical models (cf. Section 2.1 and Farinella et al., 2001; Weidenschilling, 2004). However, due to 26Al heating, only part of the interior of these objects would actually remain primordial, unprocessed material. If so Phoebe may be typical of many objects in the outer Solar System including the present KBOs and TNOs. It may be a member of a remnant class of large planetesimals in the 100–200 km diameter range that predominanted throughout the Solar System (Cuzzi et al., 2010), and are believed to have form on a 1 Myr timescale in the transneptunian region (Kenyon et al., 2008; Cuzzi et al., 2010). The fraction of primordial material in the transneptunian belt organized into larger objects early on is unknown. However, the rapid emergence of a class of planetesimals large enough to undergo some internal evolution and especially hydrothermal activity bears important implications. First, the larger the object, the longer the period during which hydrothermal processing of organics could take place. This implies that if these planetesimals, or the products of their disruption, were involved in the formation of outer planet satellites, then they could supply these objects with hydrated minerals and processed organics. This idea has already been suggested by Scott et al. (2002) and Castillo-Rogez and Lunine (2010) in the case of Ganymede and Titan, respectively. This type of framework leads to a class of satellite models that contrasts with the classical models based on pure water ice and anhydrous rock, and is a matter that needs to be studied further. More generally, the observations obtained by the Cassini–Huygens mission provide us with a good baseline for the types of observations that should be obtained for large, primordial planetesimals. Phoebe is the only roughly equidimensional and lowporosity object in the 100-km size range visited by a spacecraft. Our interpretation of these observations allows us to address a key question highlighted in the recently released Decadal Survey Report, ‘‘Vision and Voyages’’ (National Research Council, 2011): ‘‘were KBOs and comets formed too late to have included significant amounts of live 26Al as a heat source?’’ The present study suggests that a primordial population of large planetesimals formed rapidly and early enough for short-lived radioisotopes to be effective and produce partial to full melting and differentiation of their interiors. Our modeling does not set a lower bound for Phoebe’s time of formation. This is constrained mostly by the time needed to accrete objects of the size of Phoebe in current models (e.g., Kenyon et al., 2008). However the earlier it formed, the more heat would have been available for its physical and chemical evolution. Also, while we speculated on the origin of Phoebe, the material developed here does not directly bring new constraints on its origin. The capture of a large object migrating along a heliocentric orbit is difficult and remains to be demonstrated. This study also provided the opportunity to introduce a new framework for modeling objects containing amorphous materials. We recalled that amorphous ice is a very porous material, whose nanoporosity significantly affects its thermophysical properties and thus drives thermal evolution. Further experimental data on the mechanical properties of cubic ice, clathrate hydrates, amorphous ice, and highly volatile condensates are needed to place new and accurate constraints on the mechanical properties of these species in order to quantify the timescales for topographic relaxation. Also, experimental data on the kinetics and stability of clathrate hydrate formation at solar nebula conditions, or at

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Table A1 (continued) CO/RC

CO/RC

Fraction of condensate Rock + metal

Density

Solid carbon

Ice 3

103 kg m3

3

Fraction of carbon in solid phase = 0 (q = 1.7  10 kg m ) 1.0 0.74 0.00 0.26 0.9 0.70 0.00 0.30 0.8 0.66 0.00 0.34 0.7 0.63 0.00 0.37 0.6 0.60 0.00 0.40 0.5 0.57 0.00 0.43 0.4 0.55 0.00 0.45 0.3 0.53 0.00 0.47 0.2 0.50 0.00 0.50 0.1 0.49 0.00 0.51 0.0 0.47 0.00 0.53

2.13 1.99 1.88 1.79 1.72 1.66 1.60 1.56 1.52 1.48 1.45

Fraction of carbon in solid phase = 0.1 (q = 1.7  103 kg m3) 1.0 0.67 0.04 0.29 0.9 0.64 0.03 0.32 0.8 0.62 0.03 0.35 0.7 0.59 0.03 0.38 0.6 0.57 0.03 0.40 0.5 0.54 0.03 0.43 0.4 0.52 0.03 0.45 0.3 0.51 0.03 0.47 0.2 0.49 0.03 0.49 0.1 0.47 0.02 0.50 0.0 0.46 0.02 0.52

1.98 1.89 1.81 1.74 1.68 1.63 1.59 1.55 1.52 1.49 1.46

Fraction of carbon in solid phase = 0.2 (q = 1.7  103 kg m3) 1.0 0.62 0.07 0.31 0.9 0.60 0.06 0.34 0.8 0.57 0.06 0.36 0.7 0.55 0.06 0.39 0.6 0.54 0.06 0.41 0.5 0.52 0.05 0.43 0.4 0.50 0.05 0.45 0.3 0.49 0.05 0.46 0.2 0.47 0.05 0.48 0.1 0.46 0.05 0.49 0.0 0.45 0.05 0.51

1.87 1.80 1.75 1.69 1.65 1.61 1.57 1.54 1.51 1.49 1.46

Fraction of carbon in solid phase = 0.3 (q = 1.7  103 kg m3) 1.0 0.57 0.09 0.34 0.9 0.56 0.09 0.36 0.8 0.54 0.09 0.38 0.7 0.52 0.08 0.39 0.6 0.51 0.08 0.41 0.5 0.49 0.08 0.43 0.4 0.48 0.08 0.44 0.3 0.47 0.07 0.46 0.2 0.46 0.07 0.47 0.1 0.45 0.07 0.48 0.0 0.44 0.07 0.50

1.79 1.74 1.69 1.66 1.62 1.59 1.56 1.53 1.51 1.49 1.47

Fraction of carbon in solid phase = 0.4 (q = 1.7  103 kg m3) 1.0 0.53 0.11 0.35 0.9 0.52 0.11 0.37 0.8 0.51 0.11 0.39 0.7 0.50 0.10 0.40 0.6 0.48 0.10 0.41 0.5 0.47 0.10 0.43 0.4 0.46 0.10 0.44 0.3 0.45 0.10 0.45 0.2 0.44 0.09 0.46 0.1 0.43 0.09 0.47 0.0 0.43 0.09 0.48

1.72 1.68 1.65 1.62 1.60 1.57 1.55 1.53 1.51 1.49 1.47

Fraction of carbon in solid phase = 0.5 (q = 1.7  103 kg m3) 1.0 0.50 0.13 0.37 0.9 0.49 0.13 0.38 0.8 0.48 0.13 0.39 0.7 0.47 0.12 0.41 0.6 0.46 0.12 0.42 0.5 0.45 0.12 0.43 0.4 0.45 0.12 0.44 0.3 0.44 0.12 0.45 0.2 0.43 0.11 0.46

1.66 1.64 1.62 1.59 1.57 1.56 1.54 1.52 1.51

0.1 0.0

Fraction of condensate

Density

Rock + metal

Solid carbon

Ice

103 kg m3

0.42 0.42

0.11 0.11

0.47 0.47

1.49 1.48

Fraction of carbon in solid phase = 0.6 (q = 1.7  103 kg m3) 1.0 0.47 0.15 0.39 0.9 0.46 0.15 0.39 0.8 0.45 0.14 0.40 0.7 0.45 0.14 0.42 0.6 0.44 0.14 0.42 0.5 0.44 0.14 0.43 0.4 0.43 0.14 0.44 0.3 0.42 0.13 0.44 0.2 0.42 0.13 0.45 0.1 0.41 0.13 0.46 0.0 0.41 0.13 0.46

1.62 1.60 1.59 1.57 1.56 1.54 1.53 1.52 1.50 1.49 1.48

Fraction of carbon in solid phase = 0.7 (q = 1.7  103 kg m3) 1.0 0.44 0.16 0.40 0.9 0.44 0.16 0.40 0.8 0.43 0.16 0.41 0.7 0.43 0.16 0.42 0.6 0.42 0.16 0.42 0.5 0.42 0.15 0.43 0.4 0.41 0.15 0.43 0.3 0.41 0.15 0.44 0.2 0.41 0.15 0.44 0.1 0.40 0.15 0.45 0.0 0.40 0.15 0.45

1.58 1.57 1.56 1.55 1.54 1.53 1.52 1.51 1.50 1.49 1.48

Fraction of carbon in solid phase = 0.8 (q = 1.7  103 kg m3) 1.0 0.42 0.18 0.41 0.9 0.41 0.17 0.41 0.8 0.41 0.17 0.42 0.7 0.41 0.17 0.42 0.6 0.41 0.17 0.42 0.5 0.40 0.17 0.43 0.4 0.40 0.17 0.43 0.3 0.40 0.17 0.43 0.2 0.40 0.17 0.44 0.1 0.39 0.17 0.44 0.0 0.39 0.16 0.45

1.55 1.54 1.54 1.53 1.52 1.52 1.51 1.51 1.50 1.49 1.49

Fraction of carbon in solid phase = 0.9 (q = 1.7  103 kg m3) 1.0 0.39 0.19 0.42 0.9 0.39 0.19 0.42 0.8 0.39 0.19 0.42 0.7 0.39 0.19 0.42 0.6 0.39 0.18 0.43 0.5 0.39 0.18 0.43 0.4 0.39 0.18 0.43 0.3 0.39 0.18 0.43 0.2 0.38 0.18 0.43 0.1 0.38 0.18 0.43 0.0 0.38 0.18 0.44

1.52 1.52 1.51 1.51 1.51 1.51 1.50 1.50 1.50 1.50 1.49

Fraction of carbon in solid phase = 1.0 (q = 1.7  103 kg m3) 1.0 0.38 0.20 0.43 0.9 0.38 0.20 0.43 0.8 0.38 0.20 0.43 0.7 0.38 0.20 0.43 0.6 0.38 0.20 0.43 0.5 0.38 0.20 0.43 0.4 0.38 0.20 0.43 0.3 0.38 0.20 0.43 0.2 0.38 0.20 0.43 0.1 0.38 0.20 0.43 0.0 0.38 0.20 0.43

1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50 1.50

least at conditions that may be extrapolated reliably to those in the solar nebula, would be essential to quantify the extent and composition of clathrate hydrate formation upon nebula condensation and planetary accretion. Furthermore, in-depth studies are needed to truly assess the effect of melting of highly volatile condensates

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at pressures above 105 Pa on the thermomechanical behavior of icy planetesimals and satellites. Further study of the Cassini remote sensing data may help to constrain the origin of Phoebe’s surface composition and the impact of space weathering, e.g., photolysis, radiolysis, volatile migration, escape, if these observations are also supported by complementary experimental research. Acknowledgements Part of this work has been carried out at the Jet Propulsion Laboratory, California Institute of Technology. Copyright 2011. All rights reserved. Government sponsorship acknowledged. Appendix A. Density and composition of solar composition condensates The density and composition of solid condensates formed in a solar composition protoplanetary nebula beyond the ‘snow line’ is determined primarily by the distribution of the available oxygen among non-condensable gas (CO), refractory silicates (MgSiO3) and metal (FeS,FeO), and condensed volatiles (H2O). The resultant mass fraction of rock and metal fr–m, the fraction of water ice, fH2O, and the fraction of any solid carbon, fCsolid, determines the uncompressed material density of the condensate. The method of calculation is described in (Johnson and Lunine, 2005). The calculation was later extended to include updated solar abundance values and to include the effect of carbon in the solid phase in Wong et al. (2008) and Johnson and Estrada (2009). Table A1 shows results for the mass fractions of rock + metal, solid carbon and water ice in the condensates as a function of the redox state of the nebular gas (CO/total C) and the fraction of the solar C in solid form in the nebula. In these calculations ‘rock’ is assumed to be anhydrous enstatite (MgSiO3) with density 3360 kg/m3, water ice density 940 kg/m3, ‘metal’ is Fe in the form of FeS with Ni added with a density of 4880 kg/m3 and solid carbon in the form of refractory organics with density 1700 kg/m3. The solid carbon component is defined to be carbon-bearing molecules other than CO, CH4 or CO2, that is, refractory organics. Because we cannot determine a specific molecular composition for such material, we assign it a single density of 1700 kg m3, which is that of amorphous carbon. The input solar composition abundance values relative to H for the relevant elements are from (Asplund et al., 2009). These values are slightly modified from Grevesse et al. (Grevesse et al., 2005; Grevesse et al., 2007), used in Wong et al. (2008) and Johnson and Estrada (2009), but have little significant effect on densities reported earlier. Following the astrophysical convention, values are expressed in dex, where A(H) = 12 (by definition): so that A(i) = log[n(i)/n(H)] + 12, where n(i) is the number of ‘‘i’’ atoms, the important solid forming elemental abundances are: A(O) = 8.73 A(C) = 8.47 A(Si) = 7.54 A(Fe) = 7.54 A(Ni) = 6.26 A(S) = 7.16

fi ¼ li X i =Mtotal ;

where M total

¼ lice X ice þ lrock X Si þ lmetal X metal þ fCsolid lCsolid X C

ðA2Þ

where lice = 18; lrock = 100; lmetal = 88; and lCsolid = 12 and Xmetal = XFe + XNi. In the table, rock and metal are combined as rock plus metal, where fr–m = frock + fmetal. The density, q, of the condensate is then given by:

q ¼ ðfrock =qrock þ fmetal =qmetal þ fice Þ

ðA3Þ

Appendix B. Thermophysical properties of amorphous ice B.1. Phase diagram Phase stability of amorphous ice as a function of temperature is matter of debate. A review of the state of knowledge can be found in Martonak et al. (2005) and Cyriac and Pradeep (2008). Transition from high-density amorphous ice (HAD) to the low-density form (LDA) occurs rapidly in the conditions in the transneptunian region at about 50 K. The kinetics of transitions as a function of temperature can be found in Jenniskens et al. (1994). However, recent studies show that the transition of amorphous to crystalline ice is not a straightforward process, especially when the initial amorphous ice is gas-rich. Most models of comets assume that the transition from amorphous to cubic ice could occurs very rapidly when the temperature reaches 137 K. However, the transitions from Ia to Ic and Ic to Ih are not complete. Jenniskens et al. (1994) reported the formation of another form of amorphous ice at 131 K that can coexist metastably from 148 K to 188 K. That phase has been identified as responsible for anomalous retention of gas species at temperatures beyond their normal desorption temperatures so some gas species can remain trapped at temperatures up to 150 K. Jenniskens et al. (1996) found that even at 140 K amorphous ice crystallization is a limited phenomenon and that the remaining amorphous ice is metastable in the form of ‘‘restrained amorphous ice’’ (see Section B.5 about amorphous ice mechanical properties below). We compute the amount of crystallized ice as a function of temperature based on Prialnik and Merk (2008) and Jenniskens et al. (1996). B.2. Nanoporosity and density

We express the relative abundance of oxygen atoms available to form water ice compared with refractory condensates as:

X ice ¼ X O  3X Si  X C ð1  fCsolid ÞðCO=Ctotal Þ  X Fe ð1  X S =X Fe Þ

number of Si atoms); the third term accounts for the effect of C in the form of non-condensable CO removing O from the pool available for ice condensation, where (1  fCsolid) is the fraction of C in gaseous form to start with and CO/Ctotal is the fraction gaseous C in the form of CO, relative to CH4, depending on the kinetics and redox state of nebular models; the final term accounts for O in the form of metal oxide (FeO), after most of the Fe has been taken up by S as FeS. The mass fractions, fi, of rock, metal, solid carbon, and water ice are computed from the relative abundances and the molecular or atomic weight of each component:

ðA1Þ

where Xi = 10A(i). In Eq. (A1), XO is the total number of solar O atoms before condensation; the second term represents the number of O atoms removed by condensation of MgSiO3 (i.e., three times the

Pugh et al. (2002) showed that amorphous solid water ice is nanoporous when grown below 120 K. Nanoporosity can be as large as 0.6 for vapor-deposited ice (Hessinger et al., 1996) although Baragiola et al. have found a density of 0.82 g/cm3 and a porosity of 0.13 with this method. Microporosity is negligible for low-density amorphous ice formed by depressurization of a high-density amorphous form. This is the reason we believe it is important to check the origin of the thermophysical parameters used in geophysical models. Previous models have pointed out discrepancies of up to three orders of magnitude in ice thermal con-

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ductivity measurements. These discrepancies appear actually to be due to different preparation procedures for the tested ice. It has been shown that microporosity anneals with temperature with a dramatic reorganization of the ice structure at about 113 K that leads to a densification of the material (Mayer and Pletzer 1987; Cyriac and Pradeep, 2008). Mayer and Pletzer (1987) measured a corresponding decrease in microporosity from 0.21 to 0.12. That threshold is not clearly established and some experiments have demonstrated that the reorganization could take place between 100 and 150 K (Hessinger et al., 1996; Zondlo et al., 1997). The actual value does not change the general result of our geophysical model. On the other hand Westley et al. (1998) found a porosity of about 0.13 at 110 K and no dependence of the density on temperature. Baragiola et al. (2002) linked the densification to stress relaxation associated with annealing. In summary, amorphous ice density depends on pressure and temperature that drive the amount of nanoporosity (Angell, 2002, 2004). As a function of temperature, the density can evolve from 0.4 g/cm3 at temperatures less than 100 K, to 0.94 g/cm3 after significant annealing and gas desorption has taken place. In the case of Phoebe, the small internal pressure is of lesser affect, resulting in no more than 10% density increase in the center of Phoebe (at 4 MPa). B.3. Thermal conductivity The thermal conductivity of vapor-deposited water ice ranges from 104 W/m/K (Kouchi et al., 1992; Sack and Baragiola, 1992) to 0.6 W/m/K (Klinger, 1980; Andersson and Suga, 1994; see also

Ross and Kargel (1998) for a review) Andersson and Suga (1994) pointed out that that parameter is a function of the ice preparation method, i.e., the amount of microporosity allowed in the structure, while Kouchi et al. (1992) demonstrated that thick deposition layers of amorphous ice include a fraction of cubic ice that increases the overall thermal conductivity of the sample. Sack and Baragiola (1992) also demonstrated the dependence of the thermal conductivity measurement on the thickness of the sample layers. A low thermal conductivity of fresh amorphous ice is consistent with the large fraction of nanoporosity that decreases heat transfer efficiency. Our calculation of thermal conductivity takes into account nanoporosity annealing at a temperature of 113 K. For the lowerend conductivity we take the Kouchi et al. (1994) value of about 104 W/m/K, and for the high-end we adopt a value of about 1.4 W/m/K measured by Andersson and Inaba (2005) on low-density amorphous ice formed by transition from high-density amorphous ice, hence devoid of porosity. B.4. Specific heat capacity and latent heat of crystallization Specific heat data are scarce. Leliwa-Kopystynski and Kossacki (2000) used a specific heat linearly dependent on temperature, ca = 9 T. Angell (2002) mentions that the specific heat is similar to that measured for crystalline ice, that can be found in Table D1. While models of comets (e.g., Prialnik and Bar-Nun, 1992) and some models of satellites (e.g., Leliwa-Kopystynski and Kossacki, 2000) use very exothermic latent heat of crystallization for the pure amorphous water ice, calorimetric measurements obtained on gas-laden amorphous ice show that that the crystallization is

Table C1 Input parameters for the geophysical model, some of which is presented as a function of temperature T. Parameters

Value

Source

0.4–0.92 104 to 1 1425

Multiple references (see text) Multiple references (see text) Kouchi and Sirono (2001)

1012 Pa s s at 50 K 9T

Hessinger and Pohl (1996) After Leliwa-Kopystynski and Kossacki (2000)

Amorphous silicate Density (g/cm3) Thermal conductivity (W/m/K)

3.4 1–2

Assumed Cahill et al. (1992)

Crystalline ice Ic Density (g/cm3) Thermal conductivity (W/m/K) Latent heat of crystallization to ice Ih

0.92 1–2 35.5

Andersson and Inaba (2005) Handa et al. (1986)

Amorphous ice Density (g/cm3) Thermal conductivity (W/m/K) Latent heat of crystallization to ice Ic (J/kg) Viscosity (Pa s) Specific heat

Crystalline ice Ih Density Thermal conductivity (W/m/K) Viscosity (Pa s) Specific heat

0.92 0.4685 + 488.12/T Temperature-dependent, see Durham and Stern (2001) and Fig. C2 185 + 7.037 T

Crystalline silicate Density (g/cm3)

3.4

Thermal conductivity (W/m/K) Viscosity (Pa s) Specific heat

4.2 >1021 920

Clathrate hydrates Density (g/cm3) Thermal conductivity (W/m/K) Viscosity (Pa s) Specific heat kJ kg1 K1

1.0 0.5–2 Equal or greater than for water 2.1

Choukroun et al. (2010) Ross and Kargel (1998)

Hydrated silicate Density (g/cm3) Thermal conductivity (W/m/K) Latent heat of hydration (kJ/kg) Specific heat (kJ/kg)

2.5 1.5–2.4 233 1300–2000

Chondritic assemblages (Britt et al., 2002) Clauser and Huenges (1995) Allen and Seyfried (2004) Waples and Waples (2004)

Ross and Kargel (1998)

Temperature-dependent, see Castillo-Rogez et al. (2007a) Mean ordinary chondrite value, Britt and Consolmagno (2003) Clauser and Huenges (1995) 1021 Pa s taken as a lower bound Waples and Waples (2004)

Handa (1986), at 250 K

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present study, there is no reason to discard the scenario that the amorphous ice trapped a significant amount of gas. Thus we assume an endothermic latent heat following Kouchi and Sirono (2001 – Fig. 3). However, as the amorphous ice loses its gas content, the latent heat of crystallization might evolve toward exothermic behavior.

Thermal Conductivity (W/m/K)

12

10

8

6

B.5. Mechanical properties 4

2

0

50

70

90

110

130

150

170

190

210

230

250

Temperature (K) Fig. C1. Thermal conductivity of pure water ice (black, solid curve), a mixture of 50:50 water:anhydrous silicate (gray), and water with 16wt.% ammonia hydrates (dashed) as a function of temperature.

10 12

η (Pa s)

10 8

10 4

1

10 -4 50

70

90

110

130

150

170

Temperature (K) 10 58 10 54 10 50 10 46

η (Pa s)

10 42 10 38 10 34 10 30 10 26 10 22 10 18

Amorphous ice mechanical properties have been little studied because large samples are difficult to produce in the laboratory. We summarize the state of knowledge but will have to make extrapolations and even assumptions about its mechanical properties. Hessinger et al. (1996) and Hessinger and Pohl (1996) have carried out internal friction measurements at high frequency. They measured the evolution of ice mechanical properties as a function of temperature, i.e., the amount of annealing of the structural microporosity. The shear modulus is found to increase from about 0.1 GPa at 48 K to 1.5 GPa at 121 K. The dependence of viscosity on temperature is not available except for a few data found in Angell (2002 – Fig. 10). However, the friction coefficient of LDA has been measured as a function of temperature by Hessinger et al. (1996) and Hessinger and Pohl (1996). These authors found a decrease in the friction coefficient by about two orders of magnitude over the temperature range 50–120 K. The friction coefficient being an inverse function of material viscosity, the observed trend suggests that amorphous ice is very soft at low temperature and its viscosity increases as temperature increases. This may be explained by the correlation between porosity content and friction coefficient. At low temperature, high porosity makes it easier for the material to deform in response to cyclic stressing. Collapse of nanoporosity as a consequence of temperature increase alters the creep properties by strengthening the material. There are few direct measurements of amorphous ice viscosity. Jenniskens et al. (1996) found that at the glass transition, amorphous ice is very mobile and is characterized by the viscosity of a ‘‘strong liquid’’. That ‘‘restrained amorphous ice,’’ which is an intermediate form between all amorphous and all crystallized, has a viscosity between 1010 and 1012 Pa s. At temperatures lower than the glass transition temperature, the ice viscosity should rapidly become large, although measurements are not available. Extrapolation from measurements obtained at higher temperatures and general understanding of the mechanical behavior of strong liquids (e.g., Angell, 2002) indicates that the ice viscosity could be of the order of 1015 Pa s at 113 K, which is about 10 orders of magnitude less than the viscosity of crystalline water ice at that temperature.

10 14 50

70

90

110

130

150

170

190

210

230

250

Temperature (K) Fig. C2. Viscosity of amorphous (top) and crystalline ice (bottom) as a function of temperature. The amorphous ice data comes from Angell (2004). Crystalline ice is assumed to flow in the diffusion creep regime (Newtonian and grain-size independent).

actually endothermic when the gas content becomes greater than 2% (Kouchi and Sirono, 2001; M. Gudipati, personal communication), because the crystallization heat triggers the desorption of volatiles. Models of comets indicate that the initial content in gas in the amorphous phase is at least 2% gas (e.g., Prialnik and barNun, 1992; Lunine et al., 1991). This experimental observation has very important implications for modeling the degassing evolution of comets and KBOs crossing the inner Solar System. For the

Appendix C. Thermophysical properties Thermophysical parameters used in the modeling are summarized in Table C1. We also present the thermal dependence of thermal conductivity (Fig. C1) and viscosity (Fig. C1), as a function of temperature, for some of the materials assumed in the modeling (crystalline and amorphous ice, water and ammonia hydrates) (see Figs. C1 and C2). For solid mixtures, thermal conductivity is computed after the volume fraction fl and thermal conductivity kl of each component:

kmix ¼

X

fl kl

ðC1Þ

l

with kice and ksil the thermal conductivities of ice and silicates, respectively (see example in Fig. C1).

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J.C. Castillo-Rogez et al. / Icarus 219 (2012) 86–109 Table D1 Decay information for the radioisotopes used in this study, after Van Schmus (1995) and Castillo-Rogez et al. (2009). Element

Aluminum

Potassium

Thorium

Uranium

Isotope Isotopic abundance (wt.%) Decay constant (y1) Half-life k (Myr) Specific heat production (W/kg of elements)

26

40

232

235

Al 5  105

0.716 0.350

K 0.01176 5.54  1010 1277 29.17  106

In the case of a porous mixture, Shoshany et al. (2002) have established a correction factor u to the thermal conductivity of a medium with high porosity p0:

/ðp0 Þ ¼ ð1  p0 =pc Þaðp0 Þ

ðC2Þ

where pc the percolation limit of the solid through the porous medium (Shoshany et al. suggest pc = 0.7). Shoshany et al. (2002) infer from their calculations that

aðpÞ ¼ 4:1p þ 0:22

ðC3Þ

in practice pore size and geometry plays a role, but we have used Eq. (C2) as first approximation. For amorphous ice, we have used a thermal conductivity of 1e5 W/m/K over the temperature range 40–113 K and 104 W/ m/K over 113–136 K (based after Kouchi et al. (1992) and references in Appendix A). The specific heat capacity of a mixture is computed as a function of the mass fractions xl and specific heat capacities Cl of each phase in the mixture (found in Table C1):

C mix ¼

X

xl C l

ðC4Þ

l

Appendix D. Numerical modeling We perform one-dimensional finite-element heat transfer modeling and compute the evolution of porosity and thermophysical parameters as a function of time. The time step is 104 yrs. Conductive thermal heat transfer is computed after the following equation solved by a Crank–Nicholson approach:

  @ðkðTÞ:@TðrÞ=@rÞ 2 @TðrÞ þ kðTÞ @r r @r   dTðrÞ ¼ qðrÞC p ðTÞ  HðrÞ dt

ðD1Þ

where T is temperature (in Kelvin), r is local radius (in meters), k is thermal conductivity, q is material density, Cp is specific heat, t is time, and H describes the internal radiogenic heating (Table D1). The thermal parameters Cp and k are recomputed at each time step, as a function of temperature and porosity. The initial porosity profile is determined by the strength of the material compared against pressure. In the models based on crystalline material, the porosity profile is based on the experimental study of brittle compaction by Durham et al. (2005a). In the case of an amorphous-material based model, we have assumed that the object is mostly compacted following accretion as amorphous ice is expected to have a low viscosity at 50 K (Angell, 2002). Long-term porosity evolution is determined by the temperature, as a consequence of ‘‘creep compaction,’’ i.e., compaction resulting from the deformation of the material when the temperature becomes warm enough to decrease its viscosity. Creep compaction is computed following the method presented in Leliwa-Kopystynski and Kossacki (2000) using empirical creep functions derived for mixtures of rock, water ice, and ammonia hydrates. This states that porosity w evolves as a function of time t as:

Th 100.00 4.95  1011 14,010–14,050 26.38  106

U 0.71 9.85  1010 703.81 568.7  106

238

U 99.28 1.551  1010 4468 94.65  106

dw ¼ wð106 PÞa1 w dt   a5  a6 lnð1  fs Þ  a7 x ½s1   exp a2 þ a3 w þ a4 wT  T ðD2Þ where P is pressure (Pa), T temperature (K), x is the ammonia to water mole fraction, fs is the rock volume fraction in the solid phase. The empirical coefficients are a1 = 9.8 a2 = 8.727 a3 = 54.11 a4 = 0.2625 K1 a5 = 4663 K a6 = 6000 K a7 = 12,644 K The relationship (D2) is valid for 0 6 x 6 28. It neglects the compressibility of the material itself, which is acceptable considering the low pressures (<5 MPa) relevant to this study in comparison to the bulk moduli (>5 GPa) of the material under consideration. There is no empirical relationship available yet that also includes a fraction of organics, but their presence is expected to increase the compaction rate (Castillo-Rogez, 2011). Appendix E. Calculation of the theoretical, hydrostatic shape The potential of a Phoebe-sized object to relax to a spherical shape is determined by two parameters: the intrinsic strength of the material and the temperature, which drives viscous relaxation (Johnson and McGetchin, 1973). These authors showed that small objects (<100 km in radius) are expected to preserve large nonhydrostatic topography features as the pressure achieved in these objects is too small to compensate for the strength of a mixture of ice and rock. While that result was inferred for solid bodies, the presence of porosity is expected to significantly weaken the material and decrease its strength by up to one order of magnitude (Keller et al., 1999). While we lack measurements on the strength of amorphous ice, it is likely that the high nanoporosity intrinsic to that material should significantly weaken its capacity to withstand non-hydrostatic loads. Compressive strength is also a strong function of temperature. The compressive strength of water ice and clays at 150 °C can be as large as 20 MPa and 40 MPa, respectively (Zelinin et al., 1958) decreases by one order of magnitude at a temperature of 10 °C (Ladanyi, 2003). Also, the compressive strength of permafrost material has been found to decrease from about 5 MPa to 1 MPa when the temperature increases from 20 °C to 2 °C (Li et al., 2003) and for a strain rate of 106 s1. Hence, the relaxation of Phoebe’s shape to an ellipsoid in spite of the low internal pressure (<5 MPa) involved weakening of the material, either due to porosity or amorphous ice, or as a consequence of internal warming and possibly melting. Another possible source of weakening, although not well constrained, may be due to the presence of low-strength material, such as organics. Organics tend to flow at very low temperature (Akbulut et al., 2006) and

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those expected in outer Solar System planetesimals may even be liquid at temperatures less than 150 K (e.g., methane and ethane). The impact of organics on the overall thermophysical properties of icy materials remains to be quantified, but we suspect it could be significant. The presence of ammonia would also likely decrease the overall material strength (Shandera and Lorenz, 2001), although data on the extent of that effect as a function of the amount of ammonia needs to be evaluated experimentally. It is important to note that the compressive strength of the solid (and crystalline) material expected in Phoebe is still at least 1 MPa. This means that near-surface material (<15 km depth) is less likely to relax and could create large non-hydrostatic topographic features. Full relaxation to hydrostatic equilibrium requires the temperature to become warm enough near the surface. In the conductive heat transfer regime typical of such a small object, we have found that if long-lived radioisotope decay is the only heat source, this situation cannot be achieved. This is due to the fact that the time it takes for heat produced at depth to reach the surface is of the order of 100 Myr. In these conditions the objects freezes before its interior can achieve significantly warm temperatures. This is illustrated in Fig. 6a. On the other hand, short-lived radioisotope decay results in warming up the entire object on a very short timescale, as presented in Fig. 6b. The capacity of the object to relax is also constrained by the relaxation timescale compared against the period of time during which the material strength was decreases as a consequence of temperature. The relaxation time s of a topographic feature of the same wavelength as the satellite’s radius (i.e., global scale relaxation) can be computed from (Johnson and McGetchin, 1973):

19g 2qgR

s¼

ðE1Þ

where g is the material viscosity, q the bulk density (we will take the average density), g is the surface gravity, and R is Phoebe’s mean radius. Eq. (E1) indicates that Phoebe’s shape cannot relax to a nearspherical during its lifetime if its viscosity remains greater than about 1020 Pa s. In the present situation, the temperatures needed for the pressure to overcome the material strength are such that they also imply very low viscosities (see Appendix C). In summary, the low internal pressure in Phoebe compared against material strength suggests that Phoebe’s interior had to achieve very warm temperatures, or be composed of amorphous ice, in order for this satellite’s shape to relax to an ellipsoid. The theoretical shape for the thermal models presented in Section 4, and in Fig. 3 was computed from the Clairaut–Radau–Darwin equation (hereafter labeled as CRD). There is some confusion in the literature about the applicability of the CRD equation, which is summarized in Denis et al. (1997). In particular, the form of the Radau-Darwin equation that is a function of the secular Love number ks (e.g., Zharkov, 1985) is less accurate in the case of rapid rotators such as Phoebe. Denis et al. (1997) recommend that the form of the CRD read:

I MR2

¼

 1=2 2 4 5m  1 3 15 2f

ðE2Þ

where I, R, and M correspond to the satellite’s mean moment of inertia, mean radius, and mass, respectively. The geometric flattening parameter, f, is equal to (a-c)/c. The mean moment of inertia is computed from the following integration, as a function of the radius r and density q(r)

I¼

8p 3

Z

R

qðrÞr4 dr

ðE3Þ

The parameter m is the ratio of centrifugal to gravity acceleration at the equator:

m¼

x2 R3 GM

ðE4Þ

From Eqs. (E2) and (E4) we inferred the geometrical flattening f. Under the assumption of hydrostatic equilibrium, the radii a and c are linked by the following relationship (e.g., Zharkov et al., 1985):

a ¼ R þ Da c ¼ R þ Dc

ðE5Þ

where Da and Dc are the deviations of a and c from the mean radius, R leading to

Da ¼

fR 3f

ðE6Þ

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