chapter18 thethermalevolutionandinternalstructureofsaturn...

Chapter 18The Thermal Evolution and Internal Structure of Saturn’s Mid-SizedIcy Satellites

Dennis L. Matson, Julie C. Castillo-Rogez, Gerald Schubert, Christophe Sotin, and William B. McKinnon

Abstract The Cassini-Huygens mission is returning newgeophysical data for the midsize, icy satellites of Saturn (i.e.,satellites with radii between 100 and 1,000 km). These datahave enabled a new generation of geophysical model studiesfor Phoebe, Iapetus, Rhea, Mimas, Tethys, Dione, as well asEnceladus (which is addressed in a separate chapter in thisbook). In the present chapter we consider the new modelstudies that have reported significant results elucidating theevolutionary histories and internal structures of these satel-lites. Those results have included their age, the developmentof their internal structures and mineralogies, which for great-est fidelity must be done concomitantly with coupled dynam-ical evolutions. Surface areas, volumes, bulk densities, spinrates, orbit inclinations, eccentricities, and distance from Sat-urn have changed as the satellites have aged. Heat is requiredto power the satellites’ evolution, but is not overly abundantfor the midsized satellites. All sources of heat must be eval-uated and taken into account. This includes their intensitiesand when they occur and are available to facilitate evolution,both internal and dynamical. The mechanisms of heat trans-port must also be included. However, to model these to highfidelity the material properties of the satellite interiors mustbe accurately known. This is not the case. Thus, much ofthe chapter is devoted to discussion of what is known aboutthese properties and how the uncertainties affect the estima-tion of heat sources, transport processes, and the consequen-tial changes in composition and evolution. Phoebe has anoblate shape that may be in equilibrium with its spin periodof �9:3 h. Its orbital properties suggest that it is not one ofthe regular satellites, but is a captured body. Its density ishigher than that of the other satellites, consistent with for-mation in the solar nebula rather than from material around

D.L. Matson, J.C. Castillo-Rogez, and C. Sotin

Jet Propulsion Laboratory, California Institute of Technology,Pasadena, CA, 91109, USA

G. SchubertUniversity of California, Los Angeles, Los Angeles, CA, 90095, USA

W.B. McKinnonWashington University in St. Louis, St. Louis, MO, 63139, USA

Saturn. Oblate shape and high density are unusual for objectsin this size range, and may indicate that Phoebe was heatedby 26Al decay soon after its formation, which is consistentwith some models of the origin of Kuiper-Belt objects. Iape-tus has the shape of a hydrostatic body with a rotation periodof 16 h. It subsequently despun to its current synchronousrotation state, �79 day period. These observations are suffi-cient to constrain the required heating in Iapetus’ early his-tory, suggesting that it formed several My after CAI conden-sation. Since Saturn had to be present for Iapetus to form,this date also constrains the age of Saturn and how long ittook to form. Both shape and gravitational data are avail-able for Rhea. Gravity data were obtained from the singleCassini flyby during the prime mission and within the uncer-tainties cannot distinguish between hydrostatic and non-hy-drostatic gravitational fields. Both Dione and Tethys displayevidence of smooth terrains, with Dione’s appearing consid-erably younger. Both are conceivably linked to tidal heatingin the past, but the low rock abundance within Tethys and thelack of eccentricity excitation of Tethys’ orbit today makeexplaining this satellite’s geology challenging.

18.1 Introduction

In this chapter we consider the midsize satellites of Saturnfor which the Cassini-Huygens mission has returned newgeophysical data. These data have provided much neededconstraints for models. Existing models were not adequatefor the interpretation of these data and the development ofsome new approaches has been required. Chief among these isthe realization that thermophysical modeling and dynamicalmodelingcannotbecarriedout in theabsenceoftheother.Theymust be done simultaneously. It appears that these satellitesaccretedearly (i.e., less than�10Myrafter theformationof theSolar System). Thus, depending upon the date a simulation isstarted, the correct amount of heat from short-lived radioactiveisotopes must be considered in the models. Our review isfocused on these new developments that are described in the

M.K. Dougherty et al. (eds.), Saturn from Cassini-Huygens,DOI 10.1007/978-1-4020-9217-6_18, c� Springer Science+Business Media B.V. 2009

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578 D.L. Matson et al.

recent literature, which has focused on Iapetus, Phoebe, Rhea,and Enceladus. Thus, this is not a complete review of thescientific literature on the midsize satellites. The reader isreferred to Johnson (1998) and McKinnon(1998) and relevantchapters in Satellites (Burns and Matthews 1986) for earlierreviews that more explicitly deal with Mimas, Tethys, andDione. New information on Enceladus is found in the chapterby Spencer et al. in this book.

In the following we discuss some geological considera-tions and then address why the midsize satellites are so im-portant for understanding the more general questions of thehistories of the satellites, and the Saturnian system.Geology: Satellite surfaces are shaped by both exogenic andendogenic processes. The most obvious of the exogenic pro-cesses is impact cratering. Interactions with the magneto-sphere and its trapped particles are less obvious but may drivechemical reactions on the surface. Also, through sputtering,the satellites contribute material, e.g., ions and atoms, to themagnetosphere.

Endogenic geologic processes are driven by heat and arethe main shapers of the geologic features observed. Changesin the volume of a satellite can arise from internal evolution:melting and differentiation, silicate serpentinization or otherrock hydration reactions, or transformation of ice I to/fromice II (the midsize satellites are too small to stabilize higherpressure ice polymorphs). Some of these processes have beendescribed, for example, by Ellsworth and Schubert (1983)and Squyres and Croft (1986). The volume change resultingfrom pore and void space compaction could also create large-scale compressive features.

More generally, if initial temperatures were as low as70 K, and the main heat source is from long-lived radioiso-tope decay, then the time needed for the temperature toreach the peritectic ammonia-water melting temperature, i.e.,176 K, is so long that it might occur only beneath a verythick lithosphere (given solid-ice thermal conductivity). Un-der these conditions it is difficult for the interior to influencethe geology of the surface. While the satellites, by and large,show ancient surfaces (i.e., as inferred from the crateringrecord; see Dones et al., this book), there is also plenty ofevidence of activity during their geological histories.Importance of the midsize icy satellites: The midsize icysatellites of Saturn preserve evidence of their formation andevolution, and thus, also, evidence of the history of the wholeSaturnian system, including Saturn itself. By studying thesatellites we can learn more about their present properties,particularly about their interiors, and how they came to thisstate. The desire to decipher and understand the evidence ismotivating present studies and the development of increas-ingly sophisticated models for all elements of the Saturniansystem. These better models include higher computationalaccuracy and more of the relevant physical and chemical pro-cesses than were previously available or needed to interpret

available data. If the early results from these efforts are cor-rect, then we will be learning much about the early solar sys-tem and how bodies evolve.

The recovery of evolutionary evidence started with theVoyager images that showed variations in the impact craterdensities across the surfaces of some of the satellites (seeChapman and McKinnon 1986 for a review). Clearly, therehad been geologic periods when resurfacing occurred. Theresurfacing erased existing impact craters and this allowedcrater counting to measure impacts referenced to a newstarting date. The Cassini-Huygens mission obtained muchhigher precision geophysical data for important propertiessuch as satellite shapes, motion (e.g., librations) and grav-itational moments. These facts have turned out to be veryimportant for constraining the evolutionary history of theSaturnian satellites.

There is a reason why the midsize icy satellites can makea unique contribution. They have the “right” size. Smallersatellites lose heat by thermal conductivity very fast, and arelikely to have been inactive throughout their evolution. At theother extreme, large satellites, such as Titan, have evolvedmore and, as a result, key information about their formationand evolution has been destroyed as a result of geologicalactivity. Thus, it is the midsize, icy satellites that were largeenough to have evolved in response to environmental pro-cesses (such as tidal heating) but were small enough that theirevolution reached its end before key evidence was destroyed.What was preserved depends on the evolution of each of thesatellites.

Today most of the midsize icy satellites are cold andinert. One, Enceladus, is active and may have relativelyhigh internal temperatures. Some have possibly differenti-ated and undergone endogenic activity in the not too distantgeologic past, as expressed in their geology, such as Dione.Several of them, e.g., Iapetus, Mimas, Rhea, show primitive,heavily cratered surfaces that let us wonder whether theseobjects have undergone any endogenic activity. The Cassini-Huygens mission has provided important geophysical con-straints on satellite evolution, for example gravity data forRhea, high-resolution shape and topography measurements,as well as high-resolution imaging that allows the geologicalevolution of the satellites to be studied. These data are alsocrucial for assessing the hydrostatic (compensation) state ofthe satellites, thus their internal evolution, and, in some cases,their dynamical evolution. Over the past two decades our un-derstanding of processes involved in the geophysical evolu-tion of icy satellites has grown. Work on theory has enabledmore accurate modeling, especially in areas such as convec-tion, tidal dissipation, and the understanding of the thermo-mechanical properties of ice. As examples of advances inmodeling, one can point to the most recent generation of icysatellite models that combines simultaneous dynamical andgeophysical modeling.

18 The Thermal Evolution and Internal Structure of Saturn’s Mid-Sized Icy Satellites 579

This chapter builds on these new observational, theo-retical, and laboratory developments to convey the presentstate of knowledge about the midsized icy satellites. Firstwe will present the Cassini-Huygens and other observationsthat can be interpreted as geophysical constraints on the evo-lution of icy satellites (Section 18.2). Then we present anddiscuss the key factors involved in the evolution of theseobjects – sources of heat (Section 18.3) and heat trans-fer (Section 18.4). In particular, these involve assumptionsabout the initial state of these satellites, i.e., initial composi-tion, temperature, etc. (Section 18.5). This is followed by adiscussion of the main aspects of satellite modeling whichleads to a comparison of model results with the observa-tions (Sections 18.6 and 18.7). A roadmap for future studies(Section 18.9) and a summary (Section 18.10) close out thechapter.

18.2 Satellite Properties

The midsize satellites of Saturn are a diverse group. An in-spection and comparison of the entries in Table 18.1 will re-veal the extent of this diversity. Pay particular attention to theradii, the densities, and the silicate fractions, xs. The differ-ences in these parameters are significant. Later we will arguethat these values are responsible for the model results show-ing that the satellites have different evolutionary histories,internal structures, and levels of activity today.

18.2.1 Size and Shape

Size is important. The internal pressures are relatively lowfor bodies with radii of less than about a hundred kilome-ters. As a result they retain their initial porosities because theinternal pressure is not sufficient to compact their material.They also tend to have irregular shapes. They may even be“rubble piles” especially as they have likely been subject toheavy if not catastrophic bombardment early in Solar Systemhistory (see Chapter 19). Larger satellites with radii greaterthan several hundred kilometers have higher internal pres-sures and tend to be spherical in shape.

Shapes are an indication of radial mass distribution, pro-vided that the bodies are in hydrostatic equilibrium with thetidal and rotation forces that act on them (e.g., Schubert et al.2004). The shapes measured in Cassini images (Thomaset al. 2007b) indicate that most satellites deviate by lessthan a few kilometers from uniform density bodies inhydrostatic equilibrium. The shape data are compared inFig. 18.1.

The trend is for the satellites to cluster about an .a � c/=.a�c/hydrostatic value of 1.00, indicating that they are either inhydrostatic equilibrium or close to that condition. Iapetus andthe Moon are outliers. In fact until Cassini-Huygens returneddata for Iapetus, the Moon was the most non-hydrostaticsatellite known. Iapetus is now the extreme case. It sup-ports a 33-km shape anomaly. This is huge compared to a10 m bulge that is the amplitude expected for such body inhydrostatic equilibrium with the present rotation period of79.3 days.

Why is there such a huge anomaly? Studies of satelliterotation suggest that the present periods are highly evolvedfrom their initial values that were much shorter (see discus-sion by Peale, 1977). Since the shape of Iapetus correspondsto that of a hydrostatic body with a rotation period of about16 h, this suggests a faster spin in the past, but there are fewconstraints on the initial rotation period. The present day spinrates of other satellites cannot be used for guidance becausethey also experienced despinning. For guidance we can lookto the distribution of rotation periods among the asteroids(e.g., Dermott and Murray 1982) and the transneptunian ob-jects. These distributions suggest that periods ranging from 5to 10 h should be considered.

Unless a body is spinning particularly rapidly, the the-oretical shape as a function of spin period for a uniformdensity, hydrostatic body is a Maclaurin spheroid. Its rota-tional oblateness can be computed using Chandrasekhar’s(1969) formulation. The clear implication is that all satelliteshad faster spin rates in the past. Depending upon their de-grees of hydrostaticity, their shapes also changed with timeas they despun. Castillo-Rogez et al. (2007) have suggestedthat this was the evolution for Iapetus. Over time it becameless hydrostatic as it cooled. By the time it slowed down to aspin period of about 16 h its lithosphere had become strongenough to maintain the nonhydrostatic figure. If this is thecase, then the formation and long-term preservation of the16-h figure strongly constrains models for Iapetus’ geophys-ical evolution.

18.2.2 Density

With the exception of Hyperion, the apparent densities ofall of the satellites are consistent with their being mixturesof solid rock and ice. The Cassini data produced a sig-nificant change in the density of Enceladus which is now1;608 kg=m3, an increase of �60% over that previously usedin geophysical modeling. It is now the most rock rich of themidsized satellites.

The Cassini spacecraft’s close flyby of Iapetus’ dark, lead-ing, hemisphere on December 31, 2004, collected the data for


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Fig. 18.1 Satellite shapes. This logarithmic plot shows the devia-tion from hydrostatic equilibrium (using the shape determinations byThomas et al. (2007b)). The ordinate is the difference between the polar

and an equatorial axis, a–c, scaled by the corresponding difference fora hydrostatic, uniform density body. The abscissa is a–c scaled by themean radius of the satellite. The Moon is shown for reference

determining that satellite’s density and shape (Thomas et al.2007b). Iapetus’ density of 1;090 ˙ 32 kg=m3 is the thirdlowest density, after Hyperion and Tethys (see Table 18.1).This density is �20% lower than the weighted mean den-sity of the midsized Saturnian satellites. This mean density iscomputed from the densities of Mimas, Enceladus, Tethys,Dione, Rhea, Iapetus weighted by their actual mass (seeJohnson and Lunine (2005) for details of the calculation).The lower density implies that Iapetus’ material is ice richand suggests that it accreted from solids condensed directlyfrom the circum-Saturnian-nebula rather than accreting fromrelatively unprocessed heliocentric solids (as in recent satel-lite formation models (Canup and Ward 2006)), as the lat-ter would have given a density some fifty percent higher(Prinn and Fegley 1989). It has been suggested that Iapetus iscomposed of a mixture of water ice and carbonaceous chon-dritic material with an enrichment in volatiles and, possibly,light hydrocarbons (Cruikshank et al. 2008). It is also sig-nificant that there is no simple, monotonic relation betweensatellite density and distance from Saturn (i.e., no parallel tothe situation for the Galilean satellites of Jupiter). (Also seechapter on the Origin of the Saturn System).

18.2.3 Porosity

Hyperion’s density is 540 kg=m3, so even pure, solid waterice is too dense to account for the bulk of this satellite. Thusthe presence of a significant amount of porosity must be con-sidered in order to account for Hyperion’s density. Porosity isresponsible for the difference between a satellite’s apparent

density and the density of the material of which it is com-posed. Thus the porosity must be taken into account beforethe relative amounts of rock and ice in a satellite can be in-ferred. At the time of accretion the satellites were presum-ably relatively porous. The presumption that porosity wassignificant in the early history of small ice-rock bodies is sup-ported by several studies (see McKinnon 2008 for a review).Even today a significant fraction of their volume may still bepore space.

A number of processes play a role in compacting the satel-lites. When hydrostatic pressure exceeds the strength of iceit produces compaction through brittle fracture and reorga-nization of the material (e.g., Durham et al. 2005). Severallaboratory studies (Leliwa-Kopystynski and Maeno 1993;Leliwa-Kopystynski and Kossacki 1995); Durham et al.(2005) and models (Leliwa-Kopystynski and Maeno 1993)show or imply that porosity evolution is not linear with depthbut is characterized by a substantial change in porosity dueto brittle reorganization of the material at pressures between1 and 10 MPa. Laboratory measurements by Durham et al.(2005) indicate that in pure water up to 20% porosity can besustained up to pressures as large as 150 MPa when the tem-perature is less than 120 K. Leliwa-Kopystynski and Maeno(1993) have shown that if ammonia is a substantial fractionof the volatile component present in the ice, the porosity de-creases for temperatures greater than 100 K. For pressuresless than 1 MPa, porosity can be as large as 40%. Volatile mi-gration can also play a role in decreasing porosity or modify-ing the local structure (e.g., McKinnon 2008). When condi-tions, especially temperature, become suitable, ice creep andsintering results in further structural evolution and porosityreduction.


Leliwa-Kopystynski et al. (1995) have pointed out the im-portance of composition for these different mechanisms. Forexample, the decrease of porosity is impeded by the pres-ence of rocky material in the ice. Such an aggregation of fine-grained rock particles, well mixed with the ice, can producea substantially stronger material.

The present day porosity of a satellite cannot be measureddirectly but must be inferred via an iterative procedure usingthermal models to estimate how the porosity evolved. Thisprocedure has important consequences for interpreting theobserved or apparent densities. The satellite interiors can bethought of as having three components: ice, rock, and voidspace. In order to match the apparent density, the fractionalabundance of rock must be increased to compensate for thedecrease in density due to porosity. Fortunately, it is possibleto model the evolution of porosity with time and iterativelyobtain estimates for the amount of void space in a satellite.The procedure will be illustrated later in the chapter.

18.2.4 Initial Composition

The temperature of the accreting planetesimals is a functionof the characteristics of the Saturnian subnebula that pre-sumably existed around Saturn when the giant planet wasforming. In such a subnebula, it is the temperature and pres-sure that determines whether or not some ambient chemicalspecies such as CH4 and NH3 will be present in solid formand be available to accrete (Prinn and Fegley 1981; Mousiset al. 2002; Hersant et al. 2008). In treating the satellites’composition we use the terms “ice”, “rock”, and “porosity”.We follow the definitions of Johnson and Lunine (2005). By“ice” we mean the volatile phase that contains all of theices and gases. The “rock” phase contains the silicate andoxide minerals and any metals and sulfides. “Porosity”, ofcourse, refers to void space. The presently available con-straints on the compositions of ice and rock phases are nowdiscussed.

18.2.4.1 Volatile Composition

Conditions in the solar nebula near Saturn’s position or in theSaturnian subnebula were such that the ice likely accreted incrystalline form (Hersant et al. 2004; Gautier and Hersant2005; cf. Garaud and Lin 2007). Given the weighted-averagedensity of the regular midsized satellites above, Johnson andLunine (2005) have inferred that the Saturnian subnebulawas enriched in volatiles over the composition of the Solarnebula. While the subnebula models predict the materialsthat should be present in the satellite interiors, the exactcomposition of the ice that accreted is not so obvious. For

the most part, the best information we have on the inter-nal volatile composition of the Saturnian satellites comesfrom in situ measurements, such as those made in Ence-ladus’ plumes, in the E-ring, and in Titan’s atmosphere. Asdetected by the Cassini Ion and Neutral Mass Spectrome-ter (INMS) (Waite et al. 2006), Enceladus’ plumes containsubstantial CO2, either N2 or CO, methane, some ammo-nia, and traces of propane and acetylene, Greatly improvedsignal-to-noise ratios for later encounters allow for refine-ment of these results, noted below. Enceladus is the secondSaturnian satellite, after Titan, for which methane has beendetected. It remains to be determined whether this compo-nent is primordial, i.e., trapped as clathrates (e.g., Hersantet al. 2008) and/or the result of internal thermochemical pro-cesses (e.g., Fischer-Tropsch reaction, Matson et al. 2007).However, the possible presence of N2 Is a real puzzle. It waseither trapped in clathrates (or adsorbed on grain surfaces)at very low temperatures .�27K/ in the Saturnian subneb-ula, or it was produced by the decomposition of NH3, thusindicating temperatures of at least a few hundred degrees K(Matson et al. 2007; Glein et al. 2008). Such conditions mayhave also been favorable for Fischer-Tropsch and many otherorganic synthesis reactions. If this was the case, then it wouldfavor the argument that the methane we see today was syn-thesized in Enceladus. Recent, high signal-to-noise measure-ments of plume composition by the INMS (Waite et al. 2009)have failed to detect primordial 36Ar, which supports the syn-thesis interpretation.

Spectrometric measurements of the satellites’ surface re-flectances indicate that their ices are predominantly (if notoverwhelmingly) composed of water. Also, the presence ofCO2 and HCN, as well as simple organics (Cruikshank et al.2005) have also been detected at the surface of most midsizeSaturnian satellites, Mimas and Tethys being exceptions. NoCO has been detected so far, even though it is predictedto have condensed in the Saturnian subnebula models ofHersant and Gautier e.g., (Hersant et al. 2004). Under thepresent conditions, however, CO is not stable on the satellitesurfaces, so its absence is not a surprise.

Recent Saturnian subnebula evolution models indicatethat ammonia should constitute 0.5 to 11wt% of the to-tal mass of water included in the satellites (Mousis et al.2002; Alibert and Mousis 2007). Conclusive observationalevidence of NH3 hydrates, however, has not been re-ported. Ground-based observations are also contradictory(Cruikshank et al. 2005; Emery et al. 2005; Verbisceret al. 2008). The Cassini Visual and Infra-red Spectrome-ter (VIMS) team has detected NH3 itself and sets its con-centration to be less than 2% at Enceladus’ surface (Brownet al. 2006). The Cassini Ion and Neutral Mass Spectrometer(INMS) has conclusively detected NH3 in Enceladus’ ventedplume gas at a concentration of nearly 1 mole% (Waite et al.2006, 2009). RADAR measurements, while not diagnostic,


suggest the absence of ammonia at the surface of the satel-lites, except for Iapetus (Ostro et al. 2006). For the latter,the presence of a small amount of ammonia appears as thebest possibility for explaining the electrical properties of thereturned radar signal. Ostro et al. (2006) refer to Lanzerottiet al. (1984) for an explanation of the non-detection of am-monia on the inner Saturnian satellites. They proposed thatammonia is destroyed as a result of the interactions betweenthe surfaces and Saturn’s magnetospheric plasma, which istoo weak to affect Iapetus at its relatively great distance fromSaturn.

The reasons for the difficulty in detecting ammonia inthe Saturnian system may be due to the low concentrationof ammonia in the satellites when they formed, as well assubsequent thermal and sputtering loss from their surfaces.As noted, ammonia hydrates are a crucial parameter in geo-physical evolution and modeling because they depress the icemelting temperature to as low as 176 K for a multicomponentmixture (Kargel 1992), and thus can influence thermal evolu-tion and heat transfer. However, the effects of ammonia maybe mitigated if it is present at levels below 1 or 2 wt%, suchas suggested for Iapetus (Ostro et al. 2006).

18.2.4.2 Rock Composition

Many geophysical models for the outer planet satellites as-sume or argue that anhydrous rock accreted or is represen-tative of the rock fraction today (e.g., Kuskov and Kronrod2001; Sohl et al. 2002, 2003). However, in the case of me-teorite parent bodies, it has been suggested that some finecomponents of the silicate phase could have been hydratedin the Solar nebula as a result of shock waves (Ciesla et al.2003). Moreover, bulk aqueous alteration was widespreadon carbonaceous asteroids (e.g., Kargel 1991), and similarinfalling planetesimals could have contributed to the accret-ing satellites (Canup and Ward 2006). Leliwa-Kopystynsckiand Kossacki (2000) have also suggested that Mimas couldhave accreted amorphous silicate. Cosmochemical modelsfor the Saturnian satellites, however, tend to favor a sce-nario in which the minerals were crystalline (e.g., Gautierand Hersant 2005), and evidence from the Stardust missionimplies substantial mixing of crystalline silicates through-out the solar nebula (Brownlee et al. 2006). Thus the rockphase was likely composed of silicate and metallic minerals.

The constraints on the composition of the rock phase arepoor. Ordinary (L) chondritic and mean CI (i.e., C1) carbona-ceous chondritic compositions have been used as analogsfor the rock phase. These also provide a basis for estimat-ing the radionuclide content and thus the amount of heatthat will become available from radioactive decay. Note alsothat the average ordinary chondrite density, �3;500 kg=m3

(Consolmagno et al. 1998), is very close to the density ofIo .�3;528 kg=m3/ often used for modeling the rock phasein the Galilean satellites (Schubert et al. 2004) (though it isimportant to remember that Io is very hot by comparison tometeorite samples!).

Alteration of the rock by water results in the production ofthe serpentine minerals, montmorillonite, as well as oxides(e.g., goethite) and silicate hydroxides (e.g., talc and brucite)(e.g., Scott et al. 2002). The mean density for this mixture ofminerals is between 2,300 and 2;700 kg=m3. This means thathydrated rock can sequester up to 15% water by volume.

18.2.4.3 Rhea’s Gravitational Field

Rhea is especially important among the medium size icysatellites of Saturn because it is the only moon in this familyfor which we have data on the quadrupole gravitational field(Table 18.2). The data are limited, consisting of only 1 near-equatorial flyby by the Cassini spacecraft. Moreover, the in-ference of J2 and C22 from the radio Doppler data has beencontroversial; Table 18.2 lists 3 separate sets of values forthese gravitational coefficients. The mass of Rhea, expressedby the GM values in Table 18.2, is not in contention. Themean radius of Rhea .764:3 ˙ 2:2 km/ and its mass yielda density of 1;233 ˙ 11 kg=m3 (Thomas et al., 2007b). IfRhea is composed of ice with density 1;000 kg=m3 and rockwith density 3;527:5 kg=m3 .2;500 kg=m3/ (the larger valueof rock density is the mean density of Io) then its silicatemass fraction is 0.26 (0.31). Rhea has a larger density andsilicate mass fraction than Iapetus, but a smaller density androck mass fraction than Dione. The rock mass fraction de-termines the quantity of long-term radiogenic heat availableto a satellite. Rhea is intermediate between Dione and Iape-tus in terms of the magnitude of the satellite radiogenic heatsource.

The different values of the gravitational coefficients ofRhea listed in Table 18.2 derive mainly from different

Table 18.2 Published values forRhea’s gravitational coefficients

Anderson and Schubert (2007) Iess et al. (2007) Mackenzie et al. (2008)

GM .km3 s�2/ 153:9372 ˙ 0:0013 153:9395 ˙ 0:0018 153:9398 ˙ 0:0008

J2.10�6/ 889:0 ˙ 25:0 794:7 ˙ 89:2 931:0 ˙ 12:0

C22.10�6/ 266:6 ˙ 7:5 235:3 ˙ 4:8 237:2 ˙ 4:5

J2=C22 10/3 (assumed) 3.377 3.925C=MR2 0:3911 ˙ 0:0045 0:3721 ˙ 0:0036 –


approaches to the interpretation of the radio Doppler data.The gravitational coefficients J2 and C22 are strongly cou-pled for a near equatorial flyby as can be seen by writing theterms contributing to the second-degree equatorial, gravita-tional potential, Veq

Veq D��GM

r

�"1C 1=2J2

�R

r

�2C 3C22

�R

r

�2cos 2�

#

(18.1)

where G is the universal gravitational constant, M is the massof the satellite, œ is longitude and r is the distance from thecenter of the satellite of radius R and S22 C22. Gener-ally, as a practical matter, only a linear combination of J2 andC22 can be inverted from a near equatorial flyby of a satel-lite (Schubert et al. 2004) although Mackenzie et al. (2008)have reported the values of J2 and C22 listed in Table 18.2.Anderson and Schubert (2007) assumed that Rhea is in hy-drostatic equilibrium, an assumption that connects J2 and C22by J2=C22 D 10=3; they derive the values of J2 and C22 givenin Table 18.2. Iess et al. (2007) did not make the a priori as-sumption of hydrostatic equilibrium but derived values of J2and C22 consistent with it (Table 18.2). If Rhea is in hydro-static equilibrium then the value of C22 can be used to infer

the satellite’s moment of inertia factor C=MR2, where C isthe axial moment of inertia. Anderson and Schubert (2007)find C=MR2 D 0:391 (Table 18.2) consistent with an un-differentiated Rhea in which the ice component of Rhea’sice/rock interior undergoes a phase transition from ice I toice II at depth. Iess et al. (2007) obtain a slightly smallervalue of C=MR2 (Table 18.2) implying a partial separationof ice and rock inside Rhea. Since the values of J2 and C22according to Mackenzie et al. (2008) are not consistent withhydrostatic equilibrium, nothing about Rhea’s interior canbe inferred from them (other than Rhea is not in hydrostaticequilibrium).

To better understand the differences in the reported val-ues of Rhea’s quadrupole gravitational coefficients, Ander-son and Schubert (2009) calculated the Doppler residuals forthe Rhea flyby for the three inferred gravitational fields. Theresults are shown in Fig. 18.2 where it is seen that the resid-uals are essentially indistinguishable from one another. Inother words, the three Rhea gravitational fields listed in Ta-ble 18.2 fit the Doppler data from the Rhea flyby equallywell. The reason for this agreement is that the fits to theDoppler flyby data are mainly determined by the values ofC22 that are similar to about the ten percent level for all threegravitational fields. The different values of J2 are all a pri-ori possible. The only way to distinguish among the possible

0.6

0.4

0.2

0

–0.2

–0.4

0.6

0.4

0.2

–0.2

0

–0.4

0.6

0.4

0.2

–0.2

0

Dop

pler

Res

idua

ls (

mm

s–1

)D

oppl

er R

esid

uals

(m

m s

–1)

–0.4

0.6

0.4

0.2

–0.2

0

–0.4–4000 –2000 0 2000

Time from Closest Approach (s)

Anderson & Schubert

Anderson & Schubert Iess et al.

Mackenzie et al.

4000 –4000 –2000 0 2000Time from Closest Approach (s)

4000

–4000 –2000 0 2000Time from Closest Approach (s)

4000–4000 –2000 0 2000Time from Closest Approach (s)

Dop

pler

Res

idua

ls (

mm

s–1

)D

oppl

er R

esid

uals

(m

m s

–1)

4000

Fig. 18.2 Calculated Doppler residuals for the Rhea flyby for the three inferred gravitational fields of Table 18.2


0.02

0.01

0

–0.02

–0.01

–0.03

–4500 –4000

C22=(267.6±4.9)×10–6



C22

Dop

pler

Sig

nal (

mm

s–1

)


0.4

0.3

0.2

0.1

0

–0.1

2.5

3

0.04

0.02

0

–0.02

1.5

0.5

0

1

2

–0.2

–0.3–2000 –1000 0 1000 2000 –2000 –1000 0 1000

2000 2500 3000Time from Closest Approach (s)

Dop

pler

Res

idua

ls (

mm

s–1

)

Dop

pler

Res

idua

ls (

mm

s–1

)D

oppl

er R

esid

uals

(m

m s

–1)

3500

2000

–3500 –3000 –2500

pre

Encounter

post

Fig. 18.3 Calculated Doppler residuals (with RTG heat radiation effects removed) compared with Doppler data for pre-encounter, close-encounter,and post-encounter spacecraft trajectory segments

J2 values is on the basis of some physical argument. Theassumption of hydrostatic equilibrium provides such a ba-sis although hydrostaticity cannot be proven. There is nophysical basis on which to select a non-hydrostatic J2. Thebottom line is that the single near equatorial flyby of Rheacannot distinguish between hydrostatic and non-hydrostaticgravitational fields. A hydrostatic Rhea is consistent withthe Doppler flyby data despite the claim of Mackenzie et al.(2008) to the contrary (Anderson and Schubert 2009). Andas this discussion makes clear, until we can determine withsome degree of certainty whether Rhea is truly hydrostatic,inferences about Rhea’s MOI based on C22 alone should betreated with caution.

The Doppler residuals in Fig. 18.2 contain an unmodeledsignal. Anderson and Schubert (2009) attribute this signal tothe radiation of RTG heat from the Cassini spacecraft. Thesignal can be removed and a better estimate of Rhea’s C22 ob-tained by dividing the analysis of the Doppler data into threesegments, a pre-encounter segment, a close-encounter seg-ment, and a post-encounter segment (Anderson and Schubert2009). Figure 18.3 shows the results of separate fits to theDoppler data in each of the individual segments. The sep-arate fits reveal no further residual signal. The fit to theclose-encounter segment yields the improved determination

of C22 D .267:6˙4:9/�10�6 (Anderson and Schubert 2009),in essential agreement with the value of C22 determined byAnderson and Schubert (2007) (see Table 18.2). Figure 18.3also shows that the gravitational signal from Rhea’s C22 isfully contained within the time span of the close-encountersegment (Anderson and Schubert 2009).

An undifferentiated or partially differentiated Rhea is con-sistent with its heavily cratered surface and its lunar-like,inert interaction with the Saturnian plasma (Khurana et al.2008). Rhea’s shape is also consistent with hydrostatic equi-librium although uncertainties in the shape data leave openthe question of the satellite’s homogeneity (Thomas et al.2007b).

18.3 Sources of Heat

The relative significance of the different sources of heatdepends upon the time when they occur and how longthey continue to supply heat. Heating of the satellites startsoff with accretion. Then other sources of heat becomeimportant. The decay of short-lived radioactive isotopes(hereafter, SLRI) provides a heat pulse during the first 10


My following accretion. By contrast, the decay of long-livedradiogenic isotopes (hereafter, LLRI) provides heat overthe long-term. Saturn’s luminosity plays a role, perhaps upto the first hundred million years following the subnebuladissipation (Lissauer et al. 2009). Other heat sources areprovided by transient (and sometimes dramatic) events,such as the release of gravitational energy associated withinternal structure evolution (porosity collapse, differenti-ation), and despinning. Tidal dissipation associated withorbital evolution is a more complex process, which is highlydependent on the initial dynamical situation and on theinternal temperature evolution. Let us consider these heatsources in some more detail.

18.3.1 Heating by Radioactivity

During accretion, the satellites incorporate radionuclides inproportion to their rock content. The concentration of ra-dionuclides in the rock is a function of time and is usuallyreferenced to the time when the Calcium–Aluminum Inclu-sions (CAIs) accreted in the inner solar nebula.. This, in turn,was approximately the time when the Solar system formed(Wasserburg and Papanastassiou 1982). The volumetric ra-diogenic heating rate is given by:

HR D xS

nXiD1

C0H0;i e��i t0�CAIs (18.2)

where is the density of the mixture of ice and rock, xs isthe mass fraction of silicates,C0 is the initial concentration ofradiogenic elements, n is the number of radiogenic elementsincluded in the sum, H0;i is the initial power produced byradiogenic decay per unit mass of radiogenic element i withdecay constant �i . Time, t , is time since CAIs formation,labeled as t0-CAIs .

The SLRI have half-lives of less than 10 My (see Cohenand Coker 2000, for a review of SLRI nuclides). The impor-tant radionuclides, in terms of abundance and heat produc-tion are 26Al and 60Fe. Collectively they are often referred toas “Aluminium-26” because it produces the largest amountof heat. The origin of these elements is not well constrained.Two models have been proposed: (1) the “X-wind” modelby Shu et al. (1993), since then highly explored, for exampleby Gounelle and Russell (2005); (2) the supernova injectionmodel proposed by Vanhala and Boss (2002).

SLRI are extensively referred to in the literature about as-teroids and the formation of the inner Solar system and theyare widely used in models for meteorite parent bodies. How-ever their presence in the outer Solar system has been rarelyaddressed, except in the works by Prialnik and Bar-Nun

(1990), Leliwa-Kopystynski and Kossacki (2000), Prialnikand Merk (2008) and related, see the review by McKinnonet al. (2008). While these works considered SLRI as a poten-tial component of thermal models, it is only recently that ithas been suggested that these nuclides are useful in explain-ing the Iapetus and Enceladus observations by the Cassinispacecraft (Castillo-Rogez et al. 2007; Matson et al. 2007;Schubert et al. 2007). The initial concentration of 26Al=27Alhas been established as 5 � 10�5 (Wasserburg and Papanas-tassiou 1982). This is referred to as the “canonical” value.Recently a “supercanonical” value equal to 6:5 � 10�5 hasbeen proposed by Young et al. (2005), but it remains to beconfirmed by further studies.

With respect to 60Fe, it is important to note that there hasbeen significant progress in the last decade in instrumenta-tion for measuring the 60Fe daughter product .60Ni/ in mete-orites. The initial concentration of 60Fe=56Fe is now reportedto be between 0:5 � 10�6 and 1 � 10�6 (Mostefaoui et al.2005; Tachibana et al. 2006). This is an increase of a fac-tor of �1;000with respect to the discovery measurements byShukolyukov and Lugmair (1993).

The concentrations of radionuclides for the ordinary andmean CI chondritic compositions are presented in Table 18.3.These data are based on the elemental compositions fromWasson and Kalleymen (1988) convolved with isotopicabundances from Van Schmus (1995), Kita et al. (2005) andTachibana et al. (2006). The data for LLRI are found inTable 18.4 and those for SLRI in Table 18.5. These are ourpreferred values and they are presented here because a vari-ety of values have been used for some of the parameters in therecent literature, creating a confusing situation (see Castillo-Rogez et al. 2009, for a listing of the values that have beenused). Note that the ordinary chondritic composition pro-vides about 10% more specific radiogenic heat in the longterm than the mean CI chondritic composition (though CIcompositions are of course hydrated, oxidized, and carbona-ceous by comparison).

18.3.2 Tidal Heating

Tidal heating is due to changing tides. The gravitationalfields of other bodies raise tides. The tides convert powerinto heat as they distort the bulk of the satellite and as theywork against friction when they cause motion along faultsand fractures near the surface. Concurrent with tidal heatingis dynamical evolution. The heat energy must have a source,such as the energy stored in the spin of the planet and theorbital energy and spin of the satellite. We now consider twoof the most common situations, namely the heat producedby the despinning of the satellite and by the satellite’s orbitalevolution.


Table 18.3 Major radioactiveisotopes: Initial compositions forthe CI and ordinary chondrites(Wasson and Kalleymen 1988)

CI chondrite Ordinary chondrite

Density .kg=m3/ 2,756 3,51026Al (ppb) 525 60060Fe (ppb) 111–222 107.5–21553Mn (ppb) 23.2 25.740K (ppb) 943 1,104232Th (ppb) 44.3 53.8235U (ppb) 6.27 8.2238U (ppb) 20.2 26.2

Table 18.4 Decay informationfor the long-lived radioisotopes.Adapted from Van Schmus(1995)

Element Potassium Thorium Uranium

Isotope 40K 232Th 235U 238UIsotopic abundance (%wt) 0.01176 100.00 0.71 99.28Decay constant (per year) 5:54 � 10�10 4:95 � 10�11 9:85 � 10�10 1:551 � 10�10

Half-life œ (My) 1,277 14,010–14,050 703.81 4,468Specific heat production

(W/kg of elements) today 29:17 � 10�6 26:38 � 10�6 568:7 � 10�6 94:65 � 10�6

Table 18.5 Decay data for 26Aland 60Fe

Parent nuclide 26Al 60Fe

Daughter nuclide 26Mg 60NiInitial isotopic abundance 26Al=27Al 60Fe=56Fe

5� 10�5 0:1–1� 10�6

Half-life œ (My) 0.717 1.5Specific heat production (W/kg) at 4.5 Ga .to D CAI/ 0.357 0.063

18.3.2.1 Despinning

Despinning is a rapid event for most satellites, as was recog-nized by Peale (1977). If taking place instantaneously, de-spinning increases the internal temperature by the amount(Burns 1976):

�T D 1

2��2

o � �2� R2CP

(18.3)

where �o and � are the initial and final angular rates,and � is the dimensionless moment of inertia. Cp is thetemperature-dependent specific heat of the material. For aCp � 0:5 kJ=kg=K, appropriate to a 60/40 ice-rock mixtureat 100 K (McKinnon 2002), and an initial rotation period of5 h, despinning can contribute to a globally averaged increasein temperature of up to 20 K in satellites as large as Iapetusand Rhea.

The despinning rate (in rad/s) as a function of time, t , isgiven by

d�=dt D �Œ3k2 .t/ GMp2 Req

5 .t/=ŒC .t/ D6 .t/ Q .t/

(18.4)

where Mp is Saturn’s mass, Req is equatorial radius of thesatellite, C is the polar moment of inertia of the satellite,and D is the semi-major axis of the orbit. The dissipationfactor Q and the tidal Love number k2 are functions of

the frequency- and temperature-dependent viscoelastic (andother) properties of the satellite and thus vary as a function oftime. The values of k2 andQ can be computed by numericalintegration (Takeuchi and Saito 1972) (and see Tobie et al.2005a; Wahr et al. 2009). Usually despinning occurs rapidlyandD can be taken as constant. But, if the despinning takes asufficiently long time, and the orbit evolves substantially onthat time scale, thenD must be updated as a function of time.

Peale (1977) estimated the times required for tidal dis-sipation to reduce the satellites’ spins to values equal totheir planetary orbital periods. The times are from Peale’sTable 6.1 and are plotted in Fig. 18.4. The initial rotation pe-riod was 2.3 h. The units are in “years/Q” where Q is thespecific dissipation or “quality” factor. Despinning time isproportional to Q, so a despin time/Q of 107 years means adespin time of 109 years for Q D 100. The red horizontalline is drawn for the age of the Solar System using the com-mon assumption that 100 is a reasonable quality factor foricy satellites (Goldreich and Soter 1966).

Most of the satellites despin rapidly, as expected. Thesatellites above the red line require a time longer than theage of the solar system to despin. Hyperion is rotating chaot-ically and thus this analysis does not apply to it. However,this analysis does apply to Iapetus and it should not be rotat-ing synchronously. Iapetus is relatively far from Saturn andthe difficulty in despinning a distant satellite results directly


Fig. 18.4 Despinning times for satellites of the outer solar system

from the strong (inverse sixth power) dependence on distanceof the despinning torque. However, the internal evolution ofa satellite is also a complicating factor that was beyond thescope of the assumptions used for Peale’s estimates. Castillo-Rogez et al. (2007) have shown that despinning for Iapetusproceeds non-linearly. Most of the despinning takes placewhen internal conditions are favorable for substantial tidaldissipation. Thus, in some cases, the heat from despinningis not available immediately but is supplied on a timescalethat can stretch over several hundred My. For Saturn’s innersatellites, despinning occurs in less than a few My and theheat produced by this process generally raises their internaltemperatures by less than 10K.

18.3.2.2 Orbital Eccentricity

The average heating rate due to eccentricity tides is expressedas (Peale 1999):

dE

dtD 21

2

k2

Q

�nReq

�5G

e2 (18.5)

where n is the orbital mean motion, and e is the orbitaleccentricity. A major uncertainty in the modeling of tidaldissipation is the absence of laboratory data for frequencyand temperature dependent, rheological parameters. Severalmodels for rheology have been considered in the literature.Those more frequently used are Maxwell, Cole, and Burgers.The Maxwell model is a theoretical one (see Zschau 1978)while the Cole and Burgers models are based on terrestrialanalogues. The latter are especially valid for ice for tem-peratures greater than 230 K. Moreover, for these tempera-tures, the Maxwell, Cole, and Burgers models are essentiallyin agreement for the dissipation factor as a function of fre-quency (see Sotin et al. 2009 for a detailed discussion).

At lower temperatures, there is a wide range of possiblevalues for the dissipation factor, and the discrepancy amongmodels is further exacerbated at lower frequencies (cf. Tobie

et al. 2005b). In general, the interiors of the midsize icysatellites remain cold during the larger part of their histories.Thus, better knowledge of the responses of planetary materi-als at orbital frequencies and low temperatures is necessaryto make the modeling more accurate. For example, measure-ments on ice at frequencies of 10 Hz (Nakamura and Abe1977), yielded a dissipation factor of 300 at 100 K, whereason a theoretical basis, Showman and Malhotra (1997) haveproposed that Q for icy satellites should not be greater than104. For a more detailed discussion (from a planetary per-spective) of dissipation in ice/rock, see Sotin et al. (2009).

18.3.3 Heat from the Gravitational Field

Heat is obtained from a gravitational field when a mass fallsand potential energy is converted into kinetic energy thatis subsequently dissipated as heat. This is the case duringaccretion as incoming material impacts the satellite’s sur-face. Later, when the interior of the satellite experiences aphase change or differentiation and as a result mass movesdownward, its potential energy is converted to kinetic en-ergy that is viscously dissipated, producing heat. Accretionand internal evolution will now be discussed as processesthat produce heat by extracting energy from the gravitationalpotential field.

18.3.3.1 Accretion

Accretional heating occurs at the time the satellite is formed.The temperature increase due to accretion is relatively smallfor the midsize satellites. It can be as high as 90 K for thelargest such as Rhea and Iapetus, but is only a few tens ofdegrees for the smaller satellites (Ellsworth and Schubert1983). The temperature profiles resulting from accretioncan be computed using the following formula (Squyreset al. 1988)

T .r/ D ha

Cp.T /

�4�

3Gr2 C <�>2

2

�C Ti (18.6)

where ha is the fraction of mechanical energy turned into ma-terial heat and varies between 0 and 1,Cp is the temperature-dependent specific heat of the material, Ti is the temperatureof the planetesimals, r is the instantaneous radius, and � isthe mean encounter velocity of a planetesimal with the grow-ing satellite, but outside the satellite’s gravitational sphereof influence (i.e., �1. For our application, <�>2 is usually


some small fraction (typically 1/5 to 1/3) of the square of asatellites’ instantaneous circular orbital speed, 4 Gr2=3.

The value of ha depends on the characteristics of ac-cretional processes, especially the duration of accretion(Stevenson et al. 1986), and the size-frequency distributionof the accreting planetesimals. Midsized satellite accretion isconsidered to be a rapid process (e.g., Mosqueira and Estrada2005; Canup and Ward 2006) that takes place in less in a few105 years. Coradini et al. (1989) have proposed that ha rangesbetween 0.1 and 0.5 for giant planet satellites. Recently, Barrand Canup (2008) have assumed ha D 0, characteristic of avery slow accretion process where all of the accretionary heatis thermally radiated to space as it is produced. However, theprecise link between ha and the actual accretion conditionsis not clear. It is also at least conceivable that a greenhouseeffect developed due to the formation of an atmosphere (ina very rapid accretion case for the largest midsize satellites)and this would tend to retard the loss of heat and increase thevalue of ha (see Lunine and Stevenson, 1982).

According to modeling by Squyres et al. (1988), the maxi-mum temperature is reached at a depth of about 20 km depth.They assumed that eighty percent of the accretional energywas retained as heat. If accretion occurred very slowly, muchless heat would be retained. If enough ammonia accreted tocontrol the rheology, then early compaction and early melt-ing could occur. Otherwise, the corresponding processes forwater-ice rheology will control the evolution. In most of the

satellites, the maximum temperature increase resulting fromaccretional energy is between 20 and 50 K. Assuming thatthe initial temperature is about 75 K, the ammonia creeptemperature (to the extent this is known at low tempera-tures and stresses; Durham et al. 1993) is barely reached,and this only in a relatively small region just below thesurface.

A major difference between midsized and larger icy satel-lites is the temperature profile immediately after accretion. Acomparative plot of these temperatures is shown in Fig. 18.5.Models for Europa, Ganymede, and Titan indicate that theremay have been substantial melting, if not full melting, oftheir interiors during accretion, for high enough ha (e.g.,Schubert et al. 1986; Deschamps and Sotin 2001; Grassetet al. 2001). In those cases the lithosphere grows by freez-ing of ocean water as heat is conducted to the surface andradiated to space. This cooling continues until the conditionsbecome favorable for convection to start. On the other hand,if the ammonia concentration is significant then the midsizedicy satellite interiors will reach the water-ammonia peritec-tic/eutectic melting point within a few hundred My after ac-cretion, but at temperatures too low (and thus too early) forsolid state convection to get started in water ice.

It is generally assumed that the satellites accreted homo-geneously. This is mainly for lack of information about theaccretion process. Depending on the characteristics of the ac-creting planetesimal (size, velocity, composition), chemical

Fig. 18.5 Maximum temperatures reached by the end of accretion. The creep temperature is the lowest temperature at which crystals of materialcan glide when in contact with each other, yielding substantial deformation on a geologic time scale


and structural heterogeneities could have been present in theoriginal satellite. Pressures generated by impacts during ac-cretion could also have contributed to uneven compaction(e.g., Leliwa-Kopystynski and Kossacki 1994; Blum 1995).However, at the present, there is insufficient information fortreating heterogeneous properties acquired during accretionin the models for midsize satellites.

18.3.3.2 Internal Conversion of Gravitational Potentialinto Heat

The shrinking and differentiation of a satellite releases grav-itational energy in the form of heat. This phenomenonhas been studied and modeled by Leliwa-Kopystynscki andKossacki (2000).

The specific gravitational energy of self-compaction re-lated to the closing of pores is defined by (e.g., Leliwa-Kopystynski and Kossacki 2000):

Eg D R2h1� .1 �‰/

1=3i

� 0:8�G N .in J=kg/ (18.7)

where N is the satellite’s mean density, ‰ is porosity. Formedium-sized satellites, the increase in temperature resultingfrom full compaction is a few degrees.

For phase changes and differentiation, the difference inthe gravitational potential of the mass distribution before andafter the event characterizes the heat produced. A quantitativeassessment of how much heat is produced requires a model ofthe event. In general, however, this is a small source of heatcompared, for example, to radioactivity and tidal dissipation.

18.4 Thermal Transfer

Modeling temperature as a function of time is important be-cause temperature, through the temperature dependence ofchemical and physical properties of ices, controls processessuch as chemical reactions, orbital dynamics, tectonism,and differentiation. Physical parameters like viscosity varygreatly with temperature. The stability of chemical species isalso temperature dependent.

The basic thermal transfer processes relevant to the interi-ors of midsize satellites are conduction and convection. Thepossibility of convection has obvious geological implicationsbecause convection is an important regulator of the thermalstate of the interior. Determining whether or not it can occuris a key issue for understanding a satellite’s evolution. Solid-state convection may also play a role in providing thermaland rheological anomalies that can change the distribution oftidal dissipation in a satellite’s icy shell.

18.4.1 Heat Transfer by Conduction

Because the timescales related to heat transfer are longerthan those related to accretion, one can start the modelswith a temperature profile described by Eq. 18.6. Assumingthat lateral variations are negligible, one can use the one-dimensional, radial, conservation of energy if there is noadvection of heat. Thus, in the early history, just followingaccretion, heat is transferred by conduction following:

@�k.T /:@T .r/=dr

@rC 2

r

�k.T /

@T .r/

@r

�C

.r/CP .T /

�dT .r/

dt

��H.r/ D 0 (18.8)

where k is the thermal conductivity, T is temperature, andHis internal heating (radiogenic, tidal dissipation) as a functionof radius. Radiogenic heating is computed from the parame-ters presented in Tables 18.2 to 18.4.

Other things being equal, the maximum temperature thatcan be reached inside a satellite depends on the relativeamount of the rock phase and the size of the satellite(Table 18.6). First, one can note that the maximum tempera-ture that can be reached is equal to several hundred degreesexcept for Tethys whose density is so close to that of ice thatvery little radiogenic heating is expected. Such temperaturevariations suggest that ice would melt and that chemical re-actions and differentiation could have happened. The secondobservation is that the heating rate is small (Table 18.6). Formore of these satellites, it varies from 0.1 to 0.3 K/My. It im-plies that it takes several hundreds of million years to 1 Gybefore the melting temperature of ice is reached. This cal-culation does not include the effect of short-live radiogenicelements, which may allow for a much faster heating rate ifthe accretion was completed within a few half-life times of26Al and 60Fe (cf. Section 18.3.1). The melting of water icecan take place if convection does not start before the meltingtemperature is reached. The equations describing convectiveheat transfer are now described for the specific case of themid-sized icy satellites. Then the question of the onset ofconvection will be addressed.

18.4.2 Heat Transfer by Convection

Considering the likelihood of convection occurring is impor-tant because both the thermal evolution and internal dynam-ics of the satellite depend strongly on whether or not con-vection occurs. For convection to operate, the viscosity ofthe interior must be low enough. Because these bodies are


Table 18.6 Thermal characteristics of mid-sized Saturnian satellitesMimas Enceladus Tethys Dione Rhea Iapetus

Radiogenic power (GW) (a) 0:06 0:37 0:13 3:28 4:62 2:13

Radiogenic power (GW) at t D 0 (b) 0:40 2:48 0:86 21:85 30:80 14:17

Cp at 250 K (c) 1;671:17 1;348:87 1;907:96 1;425:82 1;597:45 1;740:25

Cp at 273 K 1;789:87 1;416:64 2;064:08 1;505:75 1;704:49 1;869:87

Total temperature increase (d) 476:24 1;286:38 55:43 1;059:67 632:72 341:63

�T in first 10 My (e) 1:06 2:86 0:12 2:35 1:41 0:76

equiv heat flux .mW=m2/ (f) 0:12 0:46 0:04 0:82 0:63 0:31

TconvD

250K

TBL thickness (km) (g) 7:02 5:20 5:72 4:23 4:29 4:73

heat flux .mW=m2/ (h) 9:21 12:44 11:31 15:30 15:08 13:69

Power GW (i) 4:57 9:95 40:87 60:85 110:62 92:81

Equivalent �T (j) 22:70 21:48 10:92 12:27 9:45 9:30

variations in 10 My

TconvD

273

K

TBL thickness (km) (g) 3:06 2:26 2:49 1:84 1:87 2:06

heat flux .mW=m2/ (h) 25:22 34:05 30:96 41:89 41:28 37:48

Power GW (i) 12:53 27:24 111:90 166:57 302:80 254:08

Equivalent �T (j) 58:05 56:00 27:65 31:80 24:24 23:70

variations in 10 My

Rayleigh number at 200 K (k) 1:74 31:7 57:9 2;970 1:51 3;280

(a) and (b) are the radiogenic heating (calculated using the values in Tables 18.3 through 18.5 and the mass fractions in Table 18.1). The values ofspecific heat at two temperatures are given in row (c). These values are calculated using the mass fraction of silicates and ices with the specific heatof ice being temperature dependent (McCord and Sotin 2005). The temperature increase assuming that all the radioactive heat becomes sensibleheat (neglecting heat transfer) is given in row (d). The temperature increase during the first 10 My is provided in row (e) and the equivalent heatflux at present time (radiogenic heating in line “a” divided by the surface) is given in row (f). Rows (g) to (j) provide some similar informationobtained with models of heat transfer by parameterized convection (cf. text). (k) gives values of the Rayleigh number at T D 200 K for the differentsatellites.

mainly composed of ice (Table 18.1), the viscosity of ice isthe main parameter that controls the possibility of convec-tion. Because viscosity depends strongly on temperature, theinterior must become warm enough and thus the viscositylow enough before convection can start. The interior warmsdue to the decay of radiogenic elements contained in the rockfraction (Tables 18.3 through 18.5). At the same time the in-terior of the satellite is being cooled from its surface by theradiation of heat to space. There is a competition betweenthe heating of the interior and cooling from the surface. Thismanifests itself in the downward propagation of a cold front.

As will be described in the following, the interiors of someof the mid-sized moons of Saturn become warm enough thatconvection is likely to start, eventually. The key issue is as-sessing when it might start and what the effect would be onthe evolution of the satellite (Section 18.2.4.3). Before ad-dressing this question, however, the use of scaling laws mustbe discussed. The geometry for convection within a fluid thatis heated from within and cooled from above is driven bydescending cold plumes that form at the cold boundary layerand by global upwelling (Parmentier et al. 1994). The ge-ometry is fully three-dimensional. In addition, the curvaturedue to the size of these moons must be taken into accountas well as the dependence of the gravitational accelerationwith depth. In order to investigate a large range of param-eters, scaling laws can be used to quickly assess whether

convection can occur under a given set of conditions (e.g.,Ellsworth and Schubert 1983; Multhaup and Spohn 2007).

Following the study of Multhaup and Spohn (2007), it isassumed that the mid-sized moons of Saturn are undifferen-tiated although recent studies suggest that Enceladus may befully differentiated (Schubert et al. 2007) and that the othermoons may be partly differentiated (e.g., Dione, as suggestedby its surface morphology).

Because the viscosity depends very strongly on temper-ature, convection in these icy moons is likely to be of the“conductive lid” type (Solomatov and Moresi 2000) in whicha conductive layer overlays the convective shell in which theconvection takes place. In this regime, cold instabilities growat the bottom of the conductive lid where a thermal bound-ary layer builds up between the lid and the convecting shellbelow.

After the initial onset phase when convection begins,the convection reaches a steady-state phase during whichthe convection process can be described by simple scalinglaws. The temperature difference across the thermal bound-ary layer .�TTBL/ is proportional to a viscous temperaturescale .�T˜/ which describes how the viscosity .˜/ varieswith temperature (T) (Davaille and Jaupart 1993; Moresi andSolomatov 1995; Grasset and Parmentier 1998). It must benoted that viscosity depends on a series of parameters in-cluding the fractional amount of rock, the deviatoric stress,


and the grain size (Durham and Stern 2001). But becausetemperature is the parameter which most influences the vis-cosity, results for Newtonian fluids (i.e., strain rate propor-tional to deviatoric stresses) are emphasized here, and whichcan be justified on theoretical grounds (very low convectivestresses; (Kirk and Stevenson 1987; McKinnon 2006; Barrand McKinnon 2007). In this case the viscosity follows anArrhenius type of law:

� D A exp�Qact=RgasT

�(18.9)

whereA is a constant,Qact is the activation energy (typically�60 kJ=mol for ice) and Rgas is the gas constant. The con-stant A can be adjusted empirically to take into account thegrain size and the presence of rock. If the viscosities of terres-trial glaciers and polar ices are assumed as good analogues,the constant A is chosen such that Eq. 18.9 yields a viscosityof 1014 Pa s at the melting point. In practice, this value mayvary by a factor 10 or more and the effects of different valuescan be investigated. The viscous temperature scale, �T�, isthen defined by

�T� Dˇ̌ˇ̌ˇ

1

@ln.�/=@T

ˇ̌ˇ̌ˇTDTm

D RgasT2m

Qact(18.10)

where Tm is the temperature of the convective interior. Fol-lowing the experimental work by Davaille and Jaupart (1993)and the three-dimensional, numerical study by Grasset andParmentier (1998), the temperature variations across the ther-mal boundary layer,�TTBL, can be expressed by

�TTBL D Tm � Tc D 2:23RgasT

2m

Qact(18.11)

where Tc is the temperature at the base of the conductive lid.The constant has been empirically determined by the inver-sion of laboratory and numerical experiments (e.g., Grassetand Parmentier 1998; Solomatov and Moresi 2000).

It is then possible to determine the heat flux that can betransported by convection, because the thickness of the ther-mal boundary layer .ı/ is controlled by the growth of thermalinstabilities through the thermal boundary layer Rayleighnumber .RaTBL/, which is defined by

RaTBL D ˛g�TTBLı3

��(18.12)

where ’; ¡, g, and › are the volume thermal expansioncoefficient, density, gravitational acceleration, and thermaldiffusivity, respectively. For a volumetrically heated fluid,the value of the thermal boundary layer Rayleigh numberis about 20 (e.g., Sotin and Labrosse 1999, and references

therein). This value depends slightly on the global Rayleighnumber itself (Ra), which is defined by

Ra D ˛gH 0R5

k��(18.13)

Where H0 is the volumetric heating rate in W=m3, which con-tains both the radiogenic heating, tidal heating if any, and thecooling or heating rate,

H0 D Hrad C Htid � Cp .@T=@t/ (18.14)

Using Eqs. 18.11 through 18.13, one can determine the con-vective heat flux .qconv/:

qconv D k�TTBL

ıD k

�TTBL

R

�Ra�TTBL

RaTBLH 0R2=k

�1=3

D H 0R�

Ra

RaTBL

�1=3 ��TTBL

H 0R2=k

�4=3(18.15)

With this approach, one can determine how much heat canbe removed by convection. If the heat that can be removedby convection is smaller than internal heat sources, thenthe internal temperature increases, the viscosity decreases,the Rayleigh number increases and the convective heat fluxincreases. Table 18.6 gives two examples, one for a tem-perature close to melting temperature (viscosity close to1014 Pa s), and the other for a temperature 20 K lower whichcorresponds to a viscosity almost one order of magnitudelarger. The tabulated results show that the amount of theconvective heat that can be transported is much higher thanthe production by radiogenic heating. The temperature in-crease due to radiogenic heating (row) can also be comparedwith the temperature decrease due to convection (rows j) toshow the effect (Table. 18.6). It is obvious that for viscositiessmaller than 1016 Pa s, convection processes are sufficient toremove the radioactive heat as it is produced.

Table 18.6 shows that convection is very efficient at re-moving heat. In the case of Enceladus, row (i) shows thatconvection can support fluxes of 9–20 GW of heat to the sur-face. Barr and McKinnon (2007) derived somewhat lowerheat flows, but they specifically considered a differentiatedEnceladus. In any event, these heat flows are global values,and an important issue for Enceladus is being able to con-centrate the heat flux within a relatively small geographicarea at the south pole (see Chapter 21). In the case of Iape-tus, such an efficient process would have quickly removedso much heat that Iapetus would not have evolved to itspresent day properties. The conclusion of studies that haveused scaling laws to describe convection (e.g., Ellsworth andSchubert 1983; Multhaup and Spohn 2007) is that convec-tion is very efficient at viscosities occurring at temperatures


lower than the water-ice melting point. They predict that tem-peratures would not become high enough for ice to melt andthat convection would prevent the differentiation of the satel-lite. A legitimate question is whether the amount of rockin each satellite, if undifferentiated, sufficiently stiffens theice so that ice melting occurs before convection can startas internal temperatures increase (as in the classic paper byFriedson and Stevenson 1983). This question was not specifi-cally addressed by Multhaup and Spohn (2007), although therock volume fractions in question (Table 18.1) are generallynot very high, so this is not likely to be a significant effectfor most of the midsized satellites. Another way to look atthis is to ask whether convection can start at all. This ques-tion has been addressed by Barr and Pappalardo (2005) andBarr and McKinnon (2007) and is discussed in the followingsection.

18.4.3 Onset of Convection

Linear stability analysis (Turcotte and Schubert 1982;McCord and Sotin 2005) shows that thermal anomalies growand lead to convection once a critical value of the Rayleighnumber has been reached. This critical value depends onthe wavelength of the anomaly, on the heating mode (basalversus internal) and on the amplitude of the anomaly (Barrand Pappalardo 2005). These linear stability analyses sug-gest that typical values of the critical Rayleigh number forconvection to start are around 103. In order to comparethe different mid-sized Saturnian satellites, Rayleigh num-bers at a given temperature have been calculated for eachsatellite using Eq. (18.13) where the radius has been re-placed by an effective thickness equal to one third of theradius. This allows us to compare with analytical solutionsthat are obtained in Cartesian coordinates instead of spher-ical coordinates. The different parameters have been calcu-lated using volumetric average of ice and silicates. Valuesof thermal expansion, thermal conductivity and thermal dif-fusivity for ice and silicates were taken from McCord andSotin (2005). Equation 18.9 is used for viscosity with con-stant A being chosen such that the 1014 Pa s is obtainedat melting temperature .T D 273K/. With these data, theRayleigh number at T D 200K varies from 1.5 for Rhea to3,280 for Iapetus. Actually, Dione and Iapetus have Rayleighnumbers two orders of magnitude larger than those of Ence-ladus and Tethys and three orders of magnitude larger thanthose of Mimas and Rhea. It suggests that convection wouldstart at much lower temperature (higher viscosity) within Ia-petus and Dione. However the mid-sized Saturnian satelliteshave many complexities including the spherical shape, therate of conductive cooling versus the heating rate of the in-terior, the strong temperature dependence of viscosity, and

the depth dependence of the gravitational acceleration. Al-though all of these complexities have not yet been exten-sively explored, the following paragraphs describe some oftheir effects.

A density perturbation must occur before buoyant forcescan start parcels of “fluid” in motion. In conductive lidconvection the density perturbation might be due to atemperature variation (in the sense of being cooler) causedby irregularities in surface topography. If the size of the per-turbation is sufficiently large, both in terms of its densitycontrast and its physical size (and depending on rheology),the onset phase of convection will be initiated. Viscosity andgravity control the velocity with which the parcels can move.The three dimensional size of the volume where convectionwill occur sets the distances that the parcels must traverse.These factors govern the time it takes for convection to start.

The time required to reach a temperature large enough forthe Rayleigh number to exceed the critical value depends onthe amount of internal heating available. Table 18.6 (line e)provides some numbers for each satellite. As discussed previ-ously, the temperature must be large enough for the viscosityto reach a value that yields a Rayleigh number larger than thecritical value. This table suggests that it takes from some tensto hundreds of millions of years up to a billion years if thelong-lived radiogenic elements are the only source of internalheating. During this time, the satellite is cooled from above.Determining the onset of convection for a fluid cooled fromabove and heated from within has not yet been performed formaterials having a strongly temperature dependent viscosity.On the other hand, the scaling rules that govern the onset ofconvection have been studied in the case of a hot fluid cooledfrom above (Davaille and Jaupart 1994; Choblet and Sotin2000; Korenaga and Jordan 2003; Zaranek and Parmentier2004; Solomatov and Barr 2006). These rules have been ap-plied to convective instabilities in large icy satellites (Barret al. 2004), instabilities of the oceanic lithosphere (Kore-naga and Jordan 2003), and the onset of convection for Mars(Choblet and Sotin 2000). Although these models can be ap-plied to instabilities at shallow depth, they are not the de-signed to describe what happens much deeper in a satellitewhere curvature and gravity variations must be taken intoaccount.

The onset of convection is initiated once the internal vis-cosity, which is strongly temperature-dependent, reaches thecritical value for the colder fluid above to become unstable.Work by Barr and McKinnon (2007) provides a detailed con-sideration of this for Enceladus and their results are qual-itatively applicable to similar midsize satellites. Assumingthat the upper layer is pure water ice Ih, they found that thedriving stresses due to buoyancy were quite low, and (in theabsence of strong tidal stresses) would probably be accom-modated by volume diffusion creep of the ice crystals. Underthese conditions they found that convection could start within


Enceladus only if the ice grain size was very small, �0:3mm.They noted that for this to be the case, the grain sizes have tobe kept small, suggesting the presence of other processes ormaterials that are rigid compared to water ice, such as smallrock particles, salts, hydrated sulfates, and/or clathrates, topin the grain boundaries and limit grain growth. With regardto the larger midsize satellites, Ra is such a strong function ofR (see Eq. 18.13) that the grain size (really, viscosity) limitis swiftly relaxed for satellites the size of Tethys and Dioneand larger (McKinnon 1998).

18.4.4 Convective Evolution and an Example

There are two time periods during which convection mightstart in the medium-sized icy satellites: (a) in their early his-tory (<10 My after formation), as a result of heat producedby the decay of SLRI, (b) later, as the decay of LLRI warmsthe deep interior up to temperatures at which the viscosityof ice becomes low enough for subsolidus convection to oc-cur. As previously discussed, the Rayleigh number has to ex-ceed some critical value. Application to the Saturnian mid-

sized satellites requires including the effects of two processeswhich are the cooling from above and heating from within.Robuchon et al. (2009) provides some examples. Their studyuses a viscous law, described by Eq. 18.9, with an activationenergy of 60 kJ/mol and a viscosity of 1014 Pa s at meltingtemperature. With these conditions, the convective instabil-ities start (see also discussion above). For the viscosity lawused by Robuchon et al. (2009), illustrated in Fig. 18.6, thiscondition occurs when the temperature is above 240 K. TheRayleigh number at the onset time is larger than 106, whichis much higher than the critical value.

Now, down-welling plumes may be prevented from ex-tending all the way to the center of an undifferentiated satel-lite. First, consider the spherical geometry of such a smallsatellite as being like the layers of an onion. With succes-sively smaller areas (and volumes) for each lower layer,the relative amount of ascending hot material would veryquickly be overwhelmed by the relatively larger amount ofcold plume material coming down from above. Second, (fora homogeneous satellite) the gravity decreases with depth to-ward a value of zero at the center. Between radii of 400 and200 km, for example, the gravity acceleration decreases bya factor of two, resulting in a severe decrease in the buoy-

Fig. 18.6 Illustration of the evolution of porosity using a model forIapetus. The rock/ice ratio is initially uniform, and only accretion andlong-lived radiogenic heating is considered. The ordinate is equatorialradius; the abscissa is time (on a log scale). The time at the extremeleft is the start of the model (time of accretion); at the extreme right is

the present. The lower panels are magnified views of the upper 350 km.Left-hand panels show temperature; right-hand panels show the corre-sponding porosity. Temperature contour interval is 25 K. The left-handcolor scheme highlights geophysically significant temperature regions.Adapted from Castillo-Rogez et al. (2007)


ancy force. These two effects may constrain any convectionto a shell between about 200 and 400 km from the center (ifa satellite is large enough), leaving the center of the satellitefree of convection.

The temperature-dependence of viscosity is critical to ini-tiating the onset of convection. As a fluid cools from above,a relatively denser, colder layer is formed at the surface, butits viscosity is too large for convective instabilities to start.Thus, a conductive “lid” forms, with the temperature insidethe lid increasing with depth. When the temperature is highenough, a thermal boundary layer is formed, defining thebase of the lid. Here convective instabilities can start andgrow (Multhaup and Spohn 2007). A convective instabilitybegins once the thickness of the thermal boundary layer andits Rayleigh number (Eq. 18.12) reach critical values and theviscosity contrast across the boundary layer is less than oneorder of magnitude (Solomatov and Barr 2006).

There are at least two additional complexities that need tobe discussed. First, if dissipation is important for the satellitein question, the viscosity used for convection is not neces-sarily the same as that used for tidal dissipation, which isfrequency dependent. Internal temperatures can be favorablefor both despinning (for example) and convection. In such acase, because despinning is a very rapid process with respectto the time needed for the onset of convection, despinningfinishes before convection reaches steady-state. Second, theonset of convection is further delayed by the accretional tem-perature profile, which is coldest at the center and warmestnear the surface. This is not conducive to convection andmust be overcome before convection can start. This takes10–300 My (Ellsworth and Schubert 1983; Multhaup andSpohn 2007).

Depending on the temperature at which convection starts,it takes from 2 My (at 250 K) to 20–200 My (at 220 K) toreach steady-state (Sotin et al. 2006; Multhaup and Spohn2007). However, the initial pulse in the heating rate due toSLRI decay, if any, lasts less than 10 My and then can nolonger contribute to sustaining the development of convec-tion. Numerical simulations suggest that convection wouldexist for a few million years in this case. As soon as con-vection starts, the amount of heat that can be removed ex-ceeds the radiogenic heating. Convection cools the interiordown to a temperature at which the viscosity becomes toohigh for convection to continue. With the viscosity law usedin Robuchon et al. (2009) convection stops when the temper-ature is equal to 220 K. At that time the LLRI are the only re-maining, significant heat source. The heating rate is such thathigh temperature can be attained only at large depths. For Ia-petus, it is then possible that convection may have existedvery early in its history, preventing high enough temperature(i.e., low enough viscosity) from being achieved and thus notpermitting despinning during these first several millions ofyears. However, we note again our earlier caveat that at large

depths in these satellites the effects of increasing curvaturewith depth and waning gravity sap the vigor of convection,making it less effective.

The time needed for LLRI to heat the deep interior up to240 K (the approximate temperature for the onset of convec-tion), can be several hundred million years. The argumentsdeveloped above on the effect of sphericity and the lack ofnumerical work on convection in an infinite Prandtl numberregime, volumetrically heated, self-gravitating sphere, leadus to assume that in any case an inner core of about 200 kmin radius could be free of convection and would provide alow viscosity volume where tidal dissipation could effect de-spinning.

We close this section with a final caveat reminding thereader that much depends upon accurate data for the rheo-logical properties of the materials of which the satellites areactually composed. Presently much of this data does not ex-ist, a topic that will be revisited in Section 18.9.2.

18.5 Constraints on Thermal Parameters

A major difference between the present models and “classi-cal” models of the Saturnian satellites (e.g., Ellsworth andSchubert 1983; Schubert et al. 1986) comes from the use oftemperature-dependent thermal parameters, especially ther-mal conductivities and specific heats. In this section we focuson the effect of the material’s structural and chemical proper-ties on the thermal conductivity. The reader seeking more in-formation is invited to consult the review by Ross and Kargel(1998).

18.5.1 Ice Thermal Conductivity

The thermal conductivity of pure water ice ranges from�6:2W=m=K at 100 K to 2.3 W/m/K at 270 K (Petrenko andWhitworth 1999). A review of ice thermal properties hasbeen published by Ross and Kargel (1998, Fig. 1). Data showthat the thermal conductivity of some clathrates can be oneorder of magnitude less compared to that of water ice. Thethermal conductivity of ammonia-water has been crudelymeasured by Lorenz and Shandera (2001). They show thatthe thermal conductivity of ice can be decreased by a factorof two to three by increasing its ammonia content.

18.5.2 Rock Thermal Conductivity

The thermal conductivity of the rock phase is not that wellconstrained, or at least not that well appreciated, for the


full range of temperatures and compositions expected inicy satellite interiors. Values of 3–4 W/m/K are often used(e.g., McCord and Sotin 2005; Multhaup and Spohn 2007),which are typical STP values for igneous and metamor-phic rock, although the higher values are more applica-ble to ultramafic (i.e., mantle) rocks (Turcotte and Schubert2002 Appendix 2, Table E). Crystalline rocks are consid-erably more conductive at low temperatures, and less con-ductive at higher temperatures, whereas amorphous minerals(glasses) do not show pronounced temperature dependence(Hofmeister 1999). Hydrated silicate conductivities at STPrange between 0.5 and 3 W/m/K (Clauser and Huenges 1995)depending on composition.

18.5.3 Effect of Porosity

Porosity can have a significant effect on thermal conductiv-ity (see Ross and Kargel 1998, for a review). The affect ofporosity is difficult to assess accurately because it is a func-tion of the microstructure of the material. Indeed, depend-ing on porosity the thermal contacts between the grains willbe more or less efficient in transferring heat. Results pre-sented below from Castillo-Rogez et al. (2007) follow theapproach of McKinnon (2002) who defined two bounds forthermal conductivity. The upper bound is a function of theeffect of void volume fraction on the equation for the ther-mal conductivity of a mixture. The lower bound is set bymaking a rough assessment of the effect of the structural ar-rangement and the resulting contact between grains. Basedon laboratory and other constraints (e.g., Ross and Kargel1998), McKinnon (2002) suggested (for modeling) purposesthat imperfect thermal contacts in a rubble or otherwise un-consolidated layer can decreases the thermal conductivityof such a rock-ice-void mixture by up to an additional or-der of magnitude. Based on geological analogues, we inferthat the upper layers (and regolith) of a midsize icy satel-lite could be as much as 40% porous (also see discussionin McKinnon et al. 2008). We estimate that the thermal con-ductivity should be approximately less by a factor of 10 com-pared to solid. This value also roughly corresponds to labo-ratory measurements performed on lunar regolith (Langsethet al. 1976) and the conductivity of Europa’s surface in-ferred from telescope observations of thermal emission dur-ing eclipses (Matson and Brown 1989), and detailed (or atleast complex) theoretical models of cometary conductiv-ity (Shoshany et al. 2002). How deep a given regolith ex-tends depends on the satellite in question and its respec-tive thermal and bombardment history (e.g., Eluszkiewiczet al. 1998), aspects which are very poorly understood atpresent.

18.6 Structural Evolution

Structural evolution needs to be considered over a range ofthree different size scales: micro- and meso-scales (multi-scale porosity decrease), and global scale (melting, differ-entiation, ice phase change). These processes are driven bythe material rheology as a function of temperature, pressure,composition, and structure. Volume changes and conversionof some gravitational energy into heat occur when mass is re-distributed. We will now discuss changes due to the evolutionof porosity, melting, differentiation, and chemical alteration.

18.6.1 Porosity Evolution

As the internal temperature increases, ice creeps and poros-ity decreases as a function of pressure and composition ofthe material. We apply the results and empirical relation-ships developed by Leliwa-Kopystynski and Maeno (1993)and Leliwa-Kopystynski and Kossacki (1995) noted earlier.Although these have only been defined for a limited range ofcompositions and ice properties compared with the range ofconditions expected in the midsize Saturnian satellites, theydo give some insight into the compaction kinetics over a fewtens of millions of years when the ice is at what we termthe creep temperature and for a few million years when theice is close to the melting point (or solidus). We define thecreep temperature as the lowest temperature for which icecan viscously deform and release elastic stress (and so re-duce porosity). It is related to the well-known Maxwell time(e.g., Turcotte and Schubert, 2002), and is equivalent to theelastic blocking temperature mentioned in Section 18.4, orthe brittle-ductile transition temperature. Now, because solid-state creep is a thermally activated process, there is no one,fixed creep temperature; rather it depends on the time scaleinvolved, stress level, and composition. It is nonetheless auseful benchmark. For satellite modeling purposes, the creeptemperature can be taken �176K for pure water ice (Durhamet al. 1983). If sufficient ammonia is present in the ice thenthe creep temperature may be depressed to as low as 100 K(Leliwa-Kopystynski and Kossacki 1995).

To give a specific example, we show how porosity evolvesin a model for Iapetus (Fig. 18.6). The initial porosity profileis based on Durham et al. (2005) and Kossacki and Leliwa-Kopystynski (1993). It has a porosity of 45% at the surface,decreasing approximately exponentially with depth to about10% at 120 MPa, following the laboratory values of Durhamet al. (2005). The mechanisms for the collapse of porosityare as discussed in Section 18.2. In the model, the creeptemperature is reached between 200 and 300 K; as internaltemperatures increase the porosity collapses. The thermal


conductivity becomes close to that for solid ice and is aboutan order of magnitude more conductive than the porous ma-terial. As porosity decreases, the radius of the satellite de-creases from about 850 km to about 800 km. This occurs be-tween �200–300My. The gradual radius reduction is dueto the creep temperature being reached, and densificationachieved, at different times at different depths. A secondepisode of radius change occurs much later, at �1Gy, whenthe satellite has developed conditions (solidus temperatures)favorable for tidal dissipation. As a result, despinning occursrapidly, the model becomes more spherical, and the equa-torial radius changes from about 800 km to the present day734.5 km, but there is no global volume change. The evolu-tion of the porosity profile is best seen in Fig. 18.6d.

We note that the timescales for porosity collapse are muchshorter than the timescale for heating by LLRI decay. How-ever, the internal heating sets the time when compactioncan occur.

18.6.2 Melting and Differentiation

The mechanisms by which the rock phase separates from theice have not been studied in detail. Clearly, if the rocky phaseis agglomerated in relatively large chunks in an icy matrix,it will be easier for differentiation to occur (Friedson andStevenson 1983; Nagel et al. 2004). De La Chapelle et al.(1999) show that when a few percent of pure water ice melts,separation of the liquid (water) from ice (by gravity) occursquasi-instantaneously (as might be expected). Obviously, forsufficient ice melting in an icy satellite, rock particles can bereleased and differentiation can occur.

At temperatures below the melting point of pure ice, theseparation of rock from ice is more complicated. For exam-ple, if ammonia is present in the ice, when the ammonia-water peritectic/eutectic temperature is reached .�176K/,the remaining water-ice-and-rock matrix would be too rigidto convect, but it may be able to deform because the waterice creep temperature has been reached or because residual,interstitial melt weakens the ice. Ammonia-rich melt shouldbe able to percolate upward under the action of gravity, but itis not clear whether downward rock separation is feasible atall (the depleted ice-rock residuum would be even stiffer thanbefore). This is especially true if no more than a few percentammonia is present, which may be the case for the icy satel-lites. Furthermore, the time scale of rock separation, by eitherice melting or Stokes flow, is poorly constrained. Thus, in afundamental way, we do not know how rapidly subsolidusdifferentiation (for example) progresses compared to otherprocesses taking place in the satellite (i.e., ice-rock mixtureheating from LLRI decay versus cooling).

An understanding of the above processes is crucial, be-cause differentiation affects the long-term thermal evolution

when and if the radiogenic heat sources are concentrated ina rocky core. Depending on the temperature at which differ-entiation occurs, conditions may or may not be favorable fortidal dissipation. In the long term, the warming of the corecan heat the ice phase from below and trigger the develop-ment of hot upwelling plumes in the ice. However the con-ditions under which this develops in very cold ice and thetimescale for it starting (once conditions are favorable) havenot been studied in detail. Differentiation and its aftermathare deserving of much continued study.

Two scenarios can be envisioned for the process of dif-ferentiation when ammonia is mixed with ice: (1) the am-monia concentration is significant and the rock chunks largeenough for differentiation to take place; (2) the ammoniais present in a concentration of only a few percent and therock cannot separate from the ice when the eutectic tempera-ture is reached. At present there is no definitive constraint onthe amount of ammonia needed for favoring one process orthe other.

18.6.3 Long-Term Evolution of a Rock Core

At present, Enceladus is the only midsized satellite that isgenerally thought to have formed a rock core (e.g., Barr andMcKinnon 2007; Schubert et al. 2007; Roberts and Nimmo2008; Tobie et al. 2008). The reader is referred to the Ence-ladus chapter in this book for a detailed discussion of Ence-ladus. Nevertheless, given the possible presence of SLRI, ifearly conditions differ from the present estimates, other satel-lites might have formed rocky cores as well. If that proves tobe the case, and differentiation occurred, then depending onthe temperature at which differentiation occurred, hydrationof the silicate phase could have taken place. Silicate hy-dration kinetics are temperature dependent. Thermodynam-ically optimum conditions for hydration start at 300–350 Kand such temperatures have been observed in hydrothermalsites on the Earth (e.g., Kelley et al. 2005). Under these con-ditions, the serpentinization (for example) of the bulk of therock phase can be completed in less than 100 years, assumingthat the rock phase is relatively finely divided and in contactwith water. Thus, in an icy satellite it is possible that hydra-tion of anhydrous rock (and metal) could go to completionwhile differentiation is still taking place, and thus before coreformation has been completed. On the other hand, hydrationreactions will be slow at low temperatures. For example, ifdifferentiation could occur via ammonia-water eutectic melt-ing, hydration kinetics might be sufficiently inhibited, andlead to the formation of an anhydrous core. The reactivityof cold ammonia-water solutions with anhydrous mineralswould, however, make an interesting laboratory study.

During serpentinization, olivine and pyroxene are alteredinto the serpentine minerals, brucite and oxides, such as


magnetite (Scott et al. 2002). These reactions are also accom-panied by an abundant release of H2. An example of the re-action (assuming 20% Fe and 80% Mg by mole) is (McCordand Sotin 2005):

2 .Fe0:2;Mg0:8/2 SiO4 C 3H2O D Mg3Si2O5 .OH/4

C .Mg0:2;Fe0:8/ .OH/2

or

2 olivine C 7:4=3 H2O D 1 chrysotile C 0:2 brucite

C0:8=3 magnetite C 0:8=3 H2

(18.16)

Such serpentinization can produce a rock volume increaseof up to 60% (e.g., McCord and Sotin 2005) and is accom-panied by the release of 233 kJ per kg of rock (Grimm andMcSween 1989). Also, other geochemical activity is likely totake place with reactions such as the production of methanefrom the Fischer-Tropsch reaction (e.g., Giggenbach 1980;Atreya et al. 2006) or amino acid synthesis (Shock andMcKinnon 1993). Matson et al. (2007) have proposed thatsuch an environment inside Enceladus was favorable to thedecomposition of ammonia into the molecular nitrogen .N2/

that is observed in Enceladus’ plumes.This context also favors hydrothermal circulation, as has

been described for terrestrial analogues (e.g., Kelley et al.2005). Hydrothermal circulation is expected to play a signif-icant role in cooling Enceladus’ core, if the core was heatedby SLRI (Matson et al. 2007; Glein et al. 2008). Many pa-rameters are involved in modeling hydrothermal circulation,however. A major one is the permeability of the material,which governs the depth of penetration of the water flow,and thus, its capacity to cool down, and chemically interactwith, the core (e.g., Travis and Schubert 2005). McKinnonand Zolensky (2003) discuss the fact that sulfur compoundsreleased during the interaction of rocks and warm water,and bulking due to alteration (the volume change referred toabove), could lead to deposits that plug and/or collapse theporosity near the surface of the core (and below) and slowdown hydrothermal circulation.

18.7 Model Studies of Thermaland Dynamical Evolution

Modeling the midsize satellites requires simultaneous sim-ulation of a satellite’s thermophysical and dynamical evo-lution. This is necessary because each affects the other.For example, as the interior of a satellite warms up, itstemperature and frequency-dependent rheology properties

change and it responds differently to the external gravita-tional field. Consequently, its tidal response changes and this,in turn, directly affects its despinning (if not already despun)and the evolution of inclination, eccentricity, and semima-jor axis. The changing dynamical parameters, in turn, af-fect the interior of the satellite by presenting it with a dif-ferent, time-variable (i.e., spin and orbit), gravitational field.The tides and the amount of tidal dissipation will now bedifferent and this will be reflected by a change in the in-ternal temperature distribution that affects the interior’s rhe-ology, bringing us full cycle, back to where this examplestarted.

At the practical level, this means that all of a simula-tion’s parameters must be updated at appropriate computa-tional time steps. For example, in the models reported byCastillo-Rogez et al. (2007) for Iapetus, a one-dimensionaltemperature field is used to calculate heat transfer and thenew temperature field. All of the known sources and sinks ofenergy are included (e.g., Section 18.3) as well as applicableheat transfer processes (e.g., Section 18.4). The new temper-atures are used to update parameters related to tides and tidaldissipation and these, in turn, feed into calculations of thechanges in the rotation rate and the orbital elements. Thus,the material properties are updated at each time step basedon the current temperature and dynamical state. Part of thisalso involves keeping track of size, shape (e.g., changes involume and surface area) and structure (e.g., phase changes,degree of differentiation, etc.).

In the following we shall see what such models have tosay about the initial composition of the satellites, and theirglobal evolution, with Iapetus as the prime example. The lat-ter topic encompasses issues such as differentiation, litho-sphere formation, and global shape and age.

18.7.1 Effect of Initial Composition

An important uncertainly in initial composition is how muchammonia was present. Depending upon the amount, it canaffect the evolutionary path of a satellite. To investigate thisand other effects model studies were carried out by Castillo-Rogez et al. (2007) for three general cases: (1) the volatilephase was only pure water; (2) a small amount of ammoniawas present such that it plays a role in ice creep but is ratherineffective in causing differentiation; (3) a larger amount ofammonia is present such that it plays a significant role, en-abling internal differentiation (though we stress that the latteris an assumption, and generally requires large rock “parti-cles” or batch melting).

The main result is that depending on the composition,conditions favorable for ice creep compaction occur rela-tively late in the history of a satellite. The presence of any


volatiles that depress the water ice melting point helps inter-nal differentiation (at least to some degree) take place.

The heat from SLRI decay appears as a heat pulse duringthe first few million years following accretion. This results ina decrease in porosity early in the histories of the satellites(see Section 18.6). The ammonia-water eutectic temperatureis reached just below the surface, but the ice that is ammoniafree remains solid.

18.7.1.1 Rock-Rich Models

Some satellites, such as Enceladus or Dione, have more rockthan ice by mass. Models for these bodies that include 26Alshow quick melting that enables differentiation that then pro-ceeds rapidly, with the rock phase descending to form a core.This occurs whether or not ammonia is present, because thewater ice melting point can be reached in a few millionyears, and in this case convection cannot transfer heat as ef-ficiently as it is supplied by 26Al decay (see Section 18.4).Furthermore, conditions may become favorable for maficsilicate alteration (e.g., serpentinization) under optimum ki-netics conditions (see Section 18.6). In these cases, serpen-tinization of the whole silicate phase can be achieved on arapid timescale (Section 18.6.3).

The decay of 60Fe provides a second, smaller but broader,pulse of heat during the five million years following accre-tion. As a result the core temperature rapidly reaches sev-eral hundred K, depending on when the model started withrespect to the formation time of CAIs. For extreme modelswith intervals as short as 2 My or less, the boiling point ofwater and even critical point can be reached inside cores witha large rock fraction. In the long-term an outer icy shell de-velops, made of relatively pure water with a potentially salt-and/or-sulfate water solution at its base (e.g., Kargel et al.2000; McCarthy et al. 2006). The shell has little porosity ex-cept for that caused by impact bombardment and large-scalefaulting and tectonic activity. Soluble volatiles and salts re-leased during hydrothermal circulation concentrate in a deep,ocean-like layer, with volatiles such as H2 probably escaping(Zolotov and Shock 2003; Zolotov 2007).

18.7.1.2 Rock-Poor Models

The effect of SLRI decay is less dramatic in the case ofrock-poor bodies such as Tethys. However, the heat pulsecan be significant enough to trigger early compaction of thedeep interior. This increases the thermal conductivity, accel-erates cooling, and leads to a colder interior. We will return tothis point below when we consider some models for Iapetus(Section 18.7.2, Fig. 18.7).

18.7.2 Global Evolution

18.7.2.1 Assessing the State of Differentiationof an Icy Satellite

The long-term evolution of a satellite’s core temperature isa function of: (1) the nature of the rock phase, hydrated ornot, and its thermal properties, especially conductivity; (2)the initial amount of SLRI that can increase the temperatureby a few hundred degrees. Depending on the core pressureand temperature profiles, different processes can take place.Most probably, if the largest satellites are differentiated, theircores could be stratified into an internal dehydrated centralcore and a hydrated outer core.

The final thermal state and internal structure depend onwhether major temperature thresholds are achieved: (1) thetemperature at which the viscous creep of ice becomes im-portant which is a function of composition; (2) the meltingtemperature, also a function of composition; (3) the onsetof convection. The depth at which these temperatures areachieved strongly influences the geological evolution of thesatellites.

We identify two chief evolutionary paths and they dependon the initial conditions, especially the time of formationwith respect to CAIs. Short-lived radiogenic isotopes alwaysbring about an early porosity decrease. In the long term thisresults in the development of a lithosphere that is more elas-tically rigid than for models without SLRI. Other conse-quences include earlier differentiation and the formation ofa rocky core, and related processes, such as hydrothermalactivity, that can give rise to interesting geochemical envi-ronments. On the other hand, models without SLRI will becooler and are likely to become, at most, only partially dif-ferentiated in the long term.

Now we will compare some models with and withoutSLRI and times of formation with respect to CAIs of 2 to5 My (i.e., time when the model was started) (Fig. 18.7). Thechoice of formation time sets the amount of SLRIs present.In all cases the composition of the model is pure water iceand rock (as defined in Section 18.2). The heat sources in-clude accretion (assuming several heat retention values, ha)and the decay of long-lived radioactive isotopes (LLRI), anddifferent amounts of SLRI as indicated by the time after CAIat which the model was started (t0-CAIs). The lower boundon the initial 60Fe=56Fe ratio is 0:5 � 10�6 and the upperbound is 1:0�10�6 (see Section 18.3.1). The amount of 60Fe,even at the upper bound, does not influence the models verymuch.

The model “(a)” is the hottest of the four models shownin Fig. 18.7. It started at 2.0 My and has the highest con-centration of SLRIs. The accretion heat-retention coefficientis moderate, 0.5. It has a minimal amount of 60Fe, with


t0-CAIs = 2.0 My

Temperature (K)

100 200 300 400 500 600 700 800

ha = 0.5 C0(60Fe/56Fe) Lower bound

t0-CAIs = 5.0 My ha = 0.8 C0(60Fe/56Fe) Upper bound ha = 0.5 C0(60Fe/56Fe) Upper bound

t0-CAIs = 2.5 My ha = 0.15 C0(60Fe/56Fe) Lower bound

t0-CAIs = 5.0 My

a b

c d

Rad

ius

(km

)

Time (My) Time (My)0.1 1 10 100 1000 0.1 1 10 100 1000

0.1 1 10 100 1000 0.1 1 10 100 1000

Rad

ius

(km

)

Fig. 18.7 Sensitivity of thermal evolution to parameter values in sev-eral models for Iapetus. Temperature plots are as in Fig. 18.6. Pure waterice is assumed for the ice phase. (t0-CAIs) is the time when the modelstarted in My after CAIs condensed. ha is the accretion-heat-retention

coefficient, and C0(60Fe/56Fe) is the initial abundance ratio. Their typ-ical radial grid size was 5 km and the time step was �100 years, for athree billion year evolution. Adapted from Castillo-Rogez et al. (2007)

C060Fe=56Fe at the lower bound. Within 10 My a signifi-

cant collapse of porosity has started and some melting oc-curs near the surface. The big change in radius at �250 Myis due to change in shape as the model despins. By �300My much of the interior ice has melted (orange tone) andthe rock has sunk to the center. By �1:5 Gy temperatures inexcess of 600 K have been reached at the center. After that,the model cools down. (The method of handling tidal dissi-pation is the same in all of these models and is discussed inSection 18.3.2).

Model “(b)” has significantly less initial SLRIs, havingstarted at 2.5 My. Its value of ha D 0:15 means that 85 per-cent of the heat of accretion escapes. This corresponds to aslower rate of accretion than model “(a)”. There is ampletime for the heat deposited by infalling material to be con-ducted to the surface and radiated to space. From the center

outward �200 km, ice melting occurs at �1 Gy, and then themodel cools.

Models “(c)” and “(d)” start at 5.0 My. This is late enoughthat most of the SLRIs have decayed and there is no signif-icant heating from this source. These two models show theeffect of differing levels of retention of accretion heat. Eventhough “(d)” is the model that has the least amount of totalheat available to it, it has the longest lasting major region ofmelted ice, because its internal porosity was maintained formore than 100 Myr, promoting thermal insulation.

18.7.2.2 Evolution of the Lithosphere

For icy satellites the bottom of the lithosphere, or more pre-cisely, the mechanical lithosphere is conventionally definedby the isotherm at which the ice responds to stress by viscous


creep rather than by elastic deformation (e.g., Deschampsand Sotin 2001). For the midsize satellites the evolution ofthe lithosphere is primarily a function of the initial condi-tions that determine the evolution of the internal temperaturegradient and mechanical properties. The mechanical proper-ties are dependent upon temperature, viscosity, elastic mod-uli, and the rock and ice composition. In models withoutSLRIs (such as Figs. 18.7c and 18.7d) the warming of theinterior is primarily by the decay of the long-lived radionu-clides and the time scale for this is longer than the thermalconduction time scale for the satellite as a whole. As a con-sequence the satellites have a relatively thick, porous, outershell during the first hundred million years or so followingaccretion. If sufficient ammonia is present, conditions willbecome favorable for dihydrate-ice creep sooner. Tempera-tures near the surface never get very high and the lithosphereremains structurally weak, in the sense of being porous andpossessing a low ridigity, although it might also be thoughtof as strong in the sense of being thick. In models with sig-nificant SLRI heating, porosity collapse and compaction takeplace relatively early in the history, especially if ammonia ispresent. Full melting of the ice can also take place relativelyearly. The outer, icy shell that grows afterwards will be rela-tively stronger due to the absence of porosity in most of thelithosphere. At the surface there will still be some porositythat will be maintained by impact gardening, fracturing andother processes.

18.7.2.3 Shape Evolution

The data for the shapes of the midsize, icy satellites werepresented in Section 18.2. There was a short discussion aboutthe satellites spinning faster in the past and that perhaps someshapes reflect back to a time when they were in hydrostaticequilibrium with the earlier spin rate. Now we will considerthe evolution of the satellite shape from the time of accretionuntil the present when all but chaotically rotating Hyperionare spinning synchronously with their orbital periods aboutSaturn. While our discussion is perfectly general and appliesto all satellites, we will focus as before on Iapetus becausethe details of the models we need to consider have been pub-lished (Castillo-Rogez et al. 2007).

Cassini ISS images show that Iapetus is an oblate spheroidwith a 33 km difference between the equatorial and polarradii (Thomas et al. 2007b). This difference does not in-clude the prominent equatorial ridge. The shape is inconsis-tent with the present slow 79.33 day period. If the moon werein hydrostatic equilibrium with this period, it would have anaxial difference of only 10 m. The measured shape requires aspin period of 16 ˙ 1 h to attain such an equilibrium hydro-static shape.

Among the satellites in the Solar System with syn-chronously locked rotation, Iapetus is by far the most distantfrom its planet. Given the very strong inverse dependence onthe orbital semi-major axis of the tidal despinning torque, itis somewhat surprising that Iapetus is in a synchronous stateat all (see Fig. 18.4). The time scale and dissipation levels re-quired for despinning Iapetus have been the subjects of sev-eral studies. In a general study of the despinning of giantplanet satellites, Peale (1977) showed that due to its orbit’slarge semi-major axis, despinning Iapetus requires a dissipa-tion factor of Q < 50, corresponding to a moderately dissi-pative interior. If we model Iapetus with a Maxwell rheology(Zschau 1978) and the best values for the temperature de-pendent material properties, this constraint on Q correspondsto an internal viscosity lower than 5 � 1015 Pa s. Models ofIapetus’ thermal evolution that assume radioactive heatingonly by long-lived isotopes lack a significant, early, warm-ing phase that is needed to match the data. Furthermore, tidaldissipation in such relatively cold models is not a significantsource of heat for a body in Iapetus’ distant orbit. For ex-ample, Ellsworth and Schubert’s (1983) model never reachesviscosities lower than 1017 Pa s throughout Iapetus’ historywhereas a viscosity of less than 1015 Pa s is needed for sig-nificant tidal dissipation by Maxwell dissipation. Thus, usingthe current best knowledge of rheology and a low density,rock poor composition, we find that satellites such as Iapetuswould not have despun to a synchronous spin state in the ageof the Solar System. One solution to this problem, summa-rized by Fig. 18.8, invokes a source of heat that temporar-ily made the interior relatively warm, and thus dissipativeenough for despinning to occur. However, the preservationof the 16-h shape requires that the lithosphere become thickand strong enough to retain this shape before despinning hasbeen completed.

Heating by short-lived radionuclides provides a temporarysource of heat that makes the interior relatively warm anddissipative enough for despinning to occur. When they areadded to the model, it can produce the required evolution.

The idea of SLRI in satellites and small bodies is notnew. For example, McCord and Sotin (2005) invoked themin comparing possible models for Ceres’ interior as a func-tion of its formation time. However, Iapetus is the first casewhere other sources of heat appear not to be sufficient to ex-plain its evolution, and heat from SLRIs may be required.

18.7.2.4 Age of Iapetus

Castillo-Rogez et al. (2007, 2009), found models that ac-counted for Iapetus’ shape and its despinning if the mod-els were started between �3:4 and 5.4 My after the CAIswere formed. If we accept the Pb–Pb age of CAIs mea-sured by Amelin et al. (2002) of 4;567:2 ˙ 0:6 My, for


Fig. 18.8 Iapetus’ enigmatic geophysical properties. The nearly circu-lar, dashed red curve shows the shape Iapetus should have if it were inequilibrium with its spin period. The solid green curve traces Iapetus’actual shape which corresponds to a hydrostatic body rotating with a

period of 16 h. In addition to its broad, non-hydrostatic bulge of 33 km,it has a relatively narrow equatorial ridge that reaches heights of up to20-km in some places

example, then the age of Iapetus is between 4.5638 and4:5618 Gy ˙ 0:0006 Gy.

The situation can be succinctly summarized with the aidof Fig. 18.9. The figure plots the amount of heat that canbe obtained from SLRI versus the time after CAIs when themodel started. The horizontal band represents the additionalamount of heat that must be added to the models for themto produce the shape of a 16-h period of rotation, hydrostaticbody and to despin to the present synchronous value. Theneeded heat is available in the time interval of �3:4–5:4 My.

A different model for Iapetus has recently been reportedby Robuchon et al. (2009). They find a suggested time forIapetus’ formation of �4 My after CAI condensation.

Since tidal despinning of Iapetus requires the existence ofSaturn, and since Iapetus is almost certainly a regular satelliteof Saturn, rather than a captured object (Canup and Ward2006), the 5.4 My upper bound on its age also bounds thetime available for the formation of Saturn itself. This sort oftimescale for giant planet formation is supported by recentastronomical observations of dust clearing in circum-stellardisks suggest that giant planets form on timescales of two tofive million years (Najita and Williams 2005).

18.7.2.5 Iapetus’ Equatorial Ridge

Another surprising feature of Iapetus is its prominent equato-rial ridge. It extends along most of the equator and is centered

(60Fe/56Fe)0=10–7

(60Fe/56Fe)0=10–6

0

107

106

105

104

103

107

106

105

104

103

1 2 3

Time of Formation after CAIS (My)

Cu

mu

lati

ve H

eat

(J/k

g)

Cu

mu

lati

ve H

eat

(J/k

g)

4 5 6

Fig. 18.9 Time and heat constraints for Iapetus models (after Castillo-Rogez et al. 2007)


upon it. It is well developed over a length of �1;600 km. Thisfeature also sits right on top of the equatorial bulge. The ridgeappears to be cratered as densely as the surrounding terrain,indicating that it is comparable in age. The ridge is clearlyolder than the large basins that overlap it.

There is no consensus among researchers as to themechanism that created the ridge. Porco et al. (2005) andCastillo-Rogez et al. (2007) have suggested that it is relatedto despinning. Denk et al. (2005) proposed that it resultedfrom volcanic activity. Giese et al. (2005, 2008) suggest thatthe morphology of the ridge indicates upwarping of the sur-face due to a tectonic (rather than a volcanic) event. Ip (2006)suggested that it was due to the collapse of a ring in orbitabout Iapetus. Czechowski and Leliwa-Kopystynski (2008)and Roberts and Nimmo (2009) argue for it being a result ofconvection within Iapetus. Melosh and Nimmo (2009) sug-gested that dykes may be involved in raising the ridge.

18.8 Other Satellites

18.8.1 Phoebe

Despite its small size and obvious lack of activity at itssurface, Phoebe is a fascinating object that may hold thekey to better understanding the early history of the outerSolar system. Phoebe differs from other small icy objects inmany ways, as demonstrated by the analysis of data from theCassini-Huygens June 11, 2004 flyby.

Imaging by the Cassini-Huygens remote sensing instru-ments found that Phoebe has one of the most complex sur-face compositions in the outer Solar system. At the surface ithas patches of water ice. CO2 ice, a variety of organic com-pounds, and signs of hydrated minerals (Clark et al. 2005).Overall, its global spectral properties (albedo, spectral slope)appear similar to C-type asteroids. Clark et al. (2005) havesuggested that Phoebe’s surface may sample a variety ofprimitive materials in the outer Solar system. They also raisethe possibility that, similarly to Iapetus, Phoebe is coatedwith dark outer Solar System materials, and that its interiormay be much less ice-rich. If Phoebe is a C-type asteroid, itwould be the best-observed object of its class, and as suchit can help better understand the origin of that population ofsmall bodies (Hartmann 1987). C-type asteroids are charac-teristic of the main asteroid belt, however, so a dynamicaland chemical link between Phoebe and this asteroid class re-mains to be demonstrated. Regardless, Phoebe could havebeen part of the population of outer Solar System planetes-imals, the building blocks that led to the formation of largeicy objects in the outer Solar System.

Phoebe’s density of �1;630 kg=m3 (cf. Table 18.1) issignificantly higher than the average density for the regu-lar, midsize Saturnian satellites, which is about 1200 kg=m3.This led Johnson and Lunine (2005) to suggest that Phoebe’sdensity could be closer to the density for Kuiper-Belt Ob-jects, offset by 15–20% due to bulk porosity (see also dis-cussion in Wong et al. 2008). This is further evidence thatPhoebe was captured at Saturn, as also indicated by its pecu-liar dynamical properties: a retrograde orbit with an inclina-tion of 176 deg. and eccentricity of 0.164. An origin in theKuiper Belt (i.e., sharing a common genetic pool with eclip-tic comets, Centaurs, and possibly D-type asteroids (e.g.,Gomes et al. 2005; McKinnon 2008), is also consistent withthe variety of organic compounds and possibly hydrated sili-cates identified on Phoebe’s surface. Most interestingly, Bu-ratti et al. (2008) drew a comparison between Phoebe andthe nucleus of Comet 19P/Borrelly, which both show varia-tions in albedo that Buratti et al. (2004) linked to areas of sig-nificant degassing. Whether or not Phoebe could have spentsome time of its life as a comet or Centaur is difficult to con-strain with the available observations and dynamical model-ing (Turrini et al. 2008).

Another outstanding characteristic of Phoebe is its sub-spherical shape, which makes it the only roughly equidi-mensional and low-porosity object in the 100-km size rangevisited by a spacecraft (Johnson et al. 2009). Other objectsthat are sub-spherical and low-porosity are, for example, theasteroids of the binary 90-Antiope. However, most icy bod-ies in that size category, e.g., Amalthea, Janus, and Himalia,have very different physical properties: irregular shapes anal-ogous to rubble-piles, and (where determined) densities lessthan that of water ice, between 0.4 and 900 kg=m3 (e.g.,Anderson et al. 2005). These data indicate a porosity of about5–30%, assuming these objects are composed of pure water,and up to “rubble pile” values of 30–50% if we assume thattheir bulk densities match the average density for Saturn’smidsize regular satellites, equal to 1240 kg=m3 (Johnson andLunine 2005). The most notable object in that category isprobably Hyperion, whose density is 544˙50 kg=m3 despiteits 135 km mean radius (Thomas et al. 2007a). In compari-son, the density of the 6-km diameter Comet 9P/Tempel 1 isabout 400 kg=m3 (Richardson et al. 2007). This indicates thatthese objects (in their present physical form, i.e., subsequentto any catastrophic impact) never had internal temperatureshigh enough for them to relax to a more spherical shape.

From compaction tests on pure water ice at 77–120 K(Durham et al. 2005) demonstrated that compaction is apressure-limited process for temperatures below which ei-ther solid state creep or vapor-induced sintering is important.For the maximum pressure inside Phoebe, about 4.5 MPa,and assuming the same porosity for the whole ice and rock


assemblage, compaction as a result of ice crushing is limitedand the total remnant porosity expected in the center couldat least be 25% and possibly up to 35% assuming that theinternal temperature always remained low. Impact-inducedcompaction may decrease micro-porosity, but also can createporosity on larger size scales. Durham et al. (2004) suggestedthat, in the absence of creep-induced densification, Phoebe’sintermediate porosity could have resulted from accretion ofa “wide-size spectrum of ice-rock fragments,” i.e., a well-graded sample with efficient filling of voids. That concept,however, does not address how such an assemblage of frag-ments would have relaxed to a more-or-less spherical shape.

Crater shape analysis hints, from the number of relativelydeep, conical craters, that Phoebe’s outer “layer” is substan-tially porous (Giese et al. 2006); from numerous landslideson crater walls, Phoebe’s surface certainly appears unconsol-idated. As modeled by Johnson et al. (2009), a 10- to 15-kmthick outer porous layer (45% porosity) is sufficient to ac-count for most of the porosity in the framework of Johnsonand Lunine (2005) and is a natural result from some ther-mal models that assume that the satellite formed within a fewMy after the beginning of the Solar system (i.e., a few mil-lion years after CAI formation) (Johnson et al. 2009). Suchlow-porosity layers are thermally insulating, and Phoebe mayhave warmed up quite a bit in that case, and even LLRImay prove important for such a relatively rock-rich bodyif thermal conductivity remains low enough (Fig. 18.10).Future models should explore the evolution of Phoebe fordifferent assumed formation times and compositions (espe-cially considering suggestions of the accretion of primor-dial amorphous ice, as has been suggested to be the case forKBOs and comets (e.g., Prialnik and Bar-Nun 1992; Prialnikand Merk 2008).

18.8.2 Mimas, Tethys and Dione

Only one targeted flyby was dedicated to each of Mimas,Dione and Tethys during Cassini’s primary mission. High-resolution imaging led to better mapping the cratering recordfor these satellites (Chapter 19). However, the geophysicalconstraints on their internal structures are few. From theirphysical and geological properties, it is envisioned thatDione and Tethys have undergone some endogenic activity(e.g., Schenk and Moore 1998; Schenk and Moore 2009).Signs have been reported that Dione might be slightlyactive based on its interaction with Saturn’s magnetosphere(Khurana et al. 2007), but the interpretation of the CassiniMagnetometer data in terms of geophysical and geologicalconstraints is not definitive at this time and remains to beinvestigated further.

In comparison, Tethys’ very low density indicates a smallsilicate mass fraction and possibly some non-negligible in-ternal porosity. The most striking feature observed at Tethysis Ithaca Chasma, a rift that runs over 2,000 km in lengthand up to 100-km across (Giese et al. 2007). Interpretingthe uplifted flanks of the chasma in terms of elastic flex-ure, Giese et al. (2007) inferred constraints on the thicknessof the lithosphere and thus the corresponding heat flux atthe time the ridge formed. They inferred a significant heatflux value of 18–30mW=m2, a value dependent upon theirassumed thermodynamic properties of the surface materialand its porosity. This value is surprisingly large for sucha small satellite with a small silicate mass fraction. Aboutan order of magnitude less would be expected, even 4 bil-lion years ago. Nevertheless, this flux led Chen and Nimmo(2008) to infer that at the time Ithaca Chasma formed Tethysmust have been subject to significant tidal heating. These

Fig. 18.10 Simple thermal evolution models for Phoebe, based on theKBO evolution models in (McKinnon 2002). For Phoebe, a rock/icemass ratio of 70/30 is assumed, with an initial porosity of 18%. Onlylong-lived radiogenic heating is modeled. If the thermal conductivity isassumed to be reduced by a factor of 10 due to poor thermal contacts be-

tween (crystalline) ice grains, Phoebe’s interior may have gotten warmenough for creep densification, sintering, and perhaps shape change, al-though only the inner 50% of the satellite (by volume) may have beenaffected (from Durham et al. 2004)


authors suggested the presence of a warm interior, possiblyassociated with a deep ocean, as a likely context for substan-tial tidal heating during Tethys’ passage through an orbitalresonance with Dione (the present inclination resonance be-tween Mimas and Tethys does not excite the eccentricity ofeither body). The conditions leading to internal melting andthe conjectured creation of a deep ocean in Tethys remain tobe investigated.

Mimas was much studied after Voyager flyby byEluszkiewicz et al. (1998) and Leliwa-Kopystynscki andKossacki (2000). The latter authors inferred that the satellitehas preserved some porosity although the exact amount ismodel dependent. Recent modeling by Charnoz et al. (2008)indicates the high probability that Mimas could have beendestroyed during the last period of intense bombardment inthe outer Solar system. A better understanding of this satel-lite is important because it theoretically undergoes at least 35times as much tidal stress as Enceladus, due to its proximityto Saturn, and assuming similar internal structures. However,Mimas does not show obvious signs of recent or even past ac-tivity, and even its eccentricity is not fully tidally evolved asillustrated by its relatively high, relict eccentricity of 0.0196.This demonstrates that the satellite is barely affected by tidalheating, no doubt due to a cold, non-dissipative interior,which would be quite different compared with Enceladus(McKinnon and Barr 2007).

18.9 At the Frontiers: Space, Laboratory,Processes, and Modeling

In this section we discuss ways to advance our understandingof the midsize satellites of Saturn. The areas prime for ad-vancement are further exploration by spacecraft, laboratorydata for the properties of candidate materials for the com-position of the midsize satellites, research on the operativeprocesses like convection, and better modeling and computa-tional techniques.

18.9.1 Space

With the Cassini-Huygens mission expected to be activeuntil about 2017, there will be more data becoming availableon the properties of midsize icy satellites. The shapesand masses of the satellites are now accurately known.Observations that could potentially greatly enhance ourknowledge of satellite structure and evolution are geophys-ical measurements, especially higher-order gravity fields(presently only available for Rhea). Such provide constraintson the density structure and, if hydrostaticity prevails, the

moment-of-inertia. However, most of the satellites’ shapeshave been identified as potentially non-hydrostatic to somedegree (Thomas et al. 2007b). In these cases more flybysare desirable in order to constrain (1) non-hydrostaticanomalies due to large scale heterogeneities in porositydistributions (e.g., in the case of Mimas between the sub-and anti-Saturnian hemispheres), (2) non-hydrostatic rockcores, (3) other types of mass concentrations (e.g., bodiesof subsurface water). The last are particularly important forEnceladus as potential ways to explain its non-hydrostaticshape (Collins and Goodman 2007; Thomas et al. 2007b;Schenk and McKinnon 2009).

From image analysis it is possible to infer constraintson the geological evolution of the satellites, especially thecounting of craters and the analysis of their sizes, distribu-tions, and morphologies. The identification of surface com-position and tectonic features can also provide additionalconstraints for models of the interior processes and struc-ture (e.g., Moore et al. 2004). Furthermore, there will beimportant geological consequences for models of internalevolution when good evidence can be developed for cryovol-canism, resurfacing, and recent tectonic activity. Other typesof remote sensing of potentially great utility are heat flowmeasurements (as now done at Enceladus) and, on a futuremission to the Saturn system, radar sounding. But perhapsthe greatest advance, in a comparative sense, would comefrom a more detailed reconnaissance of the Uranian midsizeicy satellites. While a flyby with modern instruments woulditself be of great value, we would argue that what is mostneeded is an orbital mission, with its opportunities for multi-ple encounters with the bodies most analogous to those dis-cussed in this chapter.

18.9.2 Laboratory Data Needed

Data on the properties of the materials from which thesatellites are made are uneven – plentiful for water ice, forexample, but sparse for all other ice formers – and do not pro-vide adequate support for the modeling of satellite evolution.Also in the same category are mixtures such as ice combinedwith rock particles of various sizes. Impacted are the model-ing of tidal dissipation, differentiation, convection, as wellas geological and orbital evolution and the understandingof their timescales. Such measurements are not necessarilyeasy to achieve. Many of the properties are known to be tem-perature and frequency dependent. Data are needed down totemperatures as low as �80K (lower if one is to model con-ditions at even greater heliocentric distance). This must alsobe done for frequencies characteristic of satellite spin ratesand orbital periods over the whole course of their evolution.Particularly urgent are the measurement of low-temperature


properties and the characterization of icy materials near thevisco-elastic transition. Such data are essential for accuratesimulations of how a satellite evolves (geophysically and dy-namically) from the time of accretion until the present.

18.9.3 Processes

The processes of accretion and convection in satellites arenot well understood. This is especially true of accretion, forwhich laboratory simulation or field observation are not yetpossible. Improving our knowledge of these, and other, pro-cesses would allow their inclusion in more detail in modelsfor the satellites. At the present we must make many assump-tions about some of these processes and as a result it is verydifficult to assess the accuracy of the results that we obtain.Improved knowledge will make the process and evolutionsimulations more accurate.

18.9.4 Modeling

Dynamical evolution of the satellites can no longer be treatedin isolation. Similarly, the thermophysical histories cannot becalculated in isolation of the ongoing dynamical situation.Each affects the other. Satellite modeling must simultane-ously treat thermophysical and dynamical evolution. Like-wise, models with early starting dates must take into accountheat from short-lived radioisotopes.

Since the medium-sized satellites are sensitive to initialand early heating and have preserved evidence regarding thathistory, it is important to better model the processes expectedto take place in their interiors. The proper interpretation ofthe preserved evidence is one way that these satellites can beused to provide a window into conditions in the early Satur-nian system.

Because the midsize satellites are sensitive to heat, lati-tudinal variations in properties can create lateral variationsin temperature and porosity. In some cases two- or three-dimensional modeling is necessary to better characterize heattransfer and the evolution of the satellite. Another process forfuture models is the evolution of a hydrated-rock core at lowpressure and high temperature and the resulting hydrother-mal transfer of heat.

18.10 Concluding Remarks

The return of data by Cassini-Huygens has reinvigorated icysatellite research. While the previous section emphasized that

there is much work to do to learn more about these satel-lites, we should not lose sight of the major advances thathave been made. While researchers are by in large exploringdifferent assumption and parameter spaces, there are emerg-ing some common threads in their results. As a result weare learning much about the early solar system. For exam-ple, a strong case is being made that these satellites are veryold objects. The studies of Iapetus by Castillo-Rogez et al.(2007, 2009) suggest that Iapetus formed some 3.4–5.4 Myafter the nominal beginning of the Solar System. Robuchonet al. (2009), using a somewhat different approach, find 4 Myor earlier. Meanwhile, the study by Barr and Canup (2008)suggests that Rhea formed at a time no earlier than 4 Myafter the CAI condensation, assuming Rhea is indeed un-differentiated. Taking this last date as a point of reference,that means that these satellites formed some 4.563 Gy ago,with the uncertainties being discussed by investigators be-ing variations in the last decimal point, the several millionyear level!

Since Saturn had to be present before the satellites, thesedates also constrain the age of Saturn and how long it took toform. Interestingly, a formation time of a few million yearsfor a giant planet is supported by recent astronomical obser-vations of dust clearing in circum-stellar disks that suggestgiant planets can form on timescales of two to five millionyears (e.g., Najita and Williams 2005).

With these and other results, we have demonstrated ourinitial thesis. At the beginning of the chapter it was statedthat the midsized icy satellites are unique in that they havepreserved important geophysical evidence of their early his-tory and evolution. Thus, the future of icy satellite studiesis bright. There will be more data returned from spacecraft.Laboratory programs are obtaining data at the proper tem-peratures, pressures, and frequencies needed for assessingthe behavior of icy materials inside the satellites over theirentire histories. Improvements in modeling techniques areoccurring and many more are on the horizon. Numerical,computational techniques as well as computer speeds are allimproving at impressive rates. And, for the first time, we cantreat the icy satellites as a system. What we learn from onesatellite, like Iapetus, we can apply to the others to unlock ad-ditional information. Thus, we can expect that in the not toodistant future there will continue to be impressive advancesin understanding the evolution and history of the midsize icysatellites and indeed, Saturn itself, and, for that matter, thewhole outer solar system.

Acknowledgements This work has been conducted at the Jet Propul-sion Laboratory, California Institute of Technology, Under a contractwith the National Aeronautics and Space Administration. Copyright2008 California Institute of Technology. Government sponsorship ac-knowledged. W.B.M. thanks the Cassini Data Analysis Program.


Appendix: Glossary of Symbols

A D constant (Eq. 18.9)C D polar moment of inertia of the satellite (Eq. 18.4)C0 D initial concentration of radiogenic elements.(Eq. 18.2)Cp D temperature-dependent specific heat (Eqs. 18.3, 18.6,18.8)C22 D second degree gravitational harmonic (i.e., ellipticityof the equator) (Eq. 18.1)D D the semi-major axis of the orbit (Eq. 18.4)e D eccentricity (Eq. 18.5)dE=dt D average tidal heat produced during one orbit(Eq. 18.5)g D gravity (Eq. 18.12)G D universal gravitational constant (Eqs. 18.1, 18.4through 18.7)ha D fraction of mechanical energy retained as heat(Eq. 18.6)HR D volumetric radiogenic heating rate (Eq. 18.2)H D internal heating rate (radiogenic, tidal dissipation)(Eq. 18.8)H0;i D initial power produced by radiogenic decay per unitmass of element i (Eq. 18.2)J2 D second degree gravitational harmonic .D �C20/ (i.e.,oblateness) (Eq. 18.1)k D thermal conductivity (Eqs. 18.8 and 18.13)k2 D the periodic, potential, tidal Love number (Eqs. 18.4and 18.5)M D satellite mass (Eq. 18.1)Mp D Saturn’s mass (Eq. 18.4)n D number of radiogenic elements included in the sum(Eq. 18.2)n D mean orbital motion (Eq. 18.5)qconv D convective heat flux (Eq. 18.13)Q D the dissipation factor (Eqs. 18.4, 18.5, 18.10, 18.11)Qact D the activation energy (typically 60 kJ/mol for ice)(Eq. 18.9)r D distance from the center of the satellite (Eqs. 18.1, 18.6,and 18.8)R D satellite radius (Eqs. 18.1, 18.3, and 18.7)Req D equatorial radius of the satellite (Eqs. 18.4 and 18.5)Rgas D the perfect gas constant (Eqs. 18.9 and 18.11)Ra D Rayleigh number (Eq. 18.13)RaTBL D thermal boundary layer Rayleigh number(Eqs. 18.12 and 18.13)t D time (Eq. 18.8)t0-CAIs D time since CAIs formation (Eq. 18.2)T D temperature (Eq. 18.8)Ti D temperature of the planetesimals (Eq. 18.6)�T D increase in the internal temperature (Eq. 18.3)�T� D viscous temperature scale is then defined by(Eq. 18.10)

�TTBL D temperature variation across thermal boundarylayer (Eqs. 18.11 and 18.13)Tm D temperature of convective interior (Eq. 18.11)Tc D temperature at the base of the conductive lid(Eq. 18.11)Tm D temperature of the convective interior (Eq. 18.10)T .r/ D temperature profile resulting from accretion(Eq. 18.6)Veq D second degree equatorial gravitational potential(Eq. 18.1)xs D mass fraction of silicates (Eq. 18.2)’ D thermal expansion coefficient (Eq. 18.12)ı D thickness of thermal boundary layer (Eqs. 18.12and 18.13)� D moment of inertia (Eq. 18.3)� D viscosity (Eqs. 18.9 and 18.12)› D thermal diffusivity (Eq. 18.12)œ D longitude (Eq. 18.1)�i D decay constant of radiogenic element i (Eq. 18.2)<�> D mean satellitesimal encounter velocity� D the initial angular rate (Eq. 18.3) D the density (Eqs. 18.2, 18.6, 18.8, 18.12)N D the satellite’s mean density (Eq. 18.7)‰ D porosity (Eq. 18.7)

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