Thermal evolution of Kuiper belt objects, with implications for cryovolcanism

Icarus 202 (2009) 694–714

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Thermal evolution of Kuiper belt objects, with implications for cryovolcanism

Steven J. Desch a,∗, Jason C. Cook b, T.C. Doggett a, Simon B. Porter a

a School of Earth and Space Exploration, Arizona State University, P.O. Box 871404, Tempe, AZ 85287-1404, USAb Southwest Research Institute, 1050 Walnut St., Suite 300, Boulder, CO 80302, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 25 July 2008Revised 4 February 2009Accepted 8 March 2009Available online 14 March 2009

Keywords:CharonIcesSpectroscopyThermal historiesTrans-neptunian objectsVolcanism

We investigate the internal thermal evolution of Kuiper belt objects (KBOs), small (radii <1000 km), icy(mean densities <2500 kg m−3) bodies orbiting beyond Neptune, focusing on Pluto’s moon Charon inparticular. Our calculations are time-dependent and account for differentiation. We review evidence forammonia hydrates in the ices of KBOs, and include their effects on the thermal evolution. A key findingis that the production of the first melt, at the melting point of ammonia dihydrate, ≈176 K, triggersdifferentiation of rock and ice. The resulting structure comprises a rocky core surrounded by liquids andice, enclosed within a >100-km thick undifferentiated crust of rock and ice. This structure is especiallyconducive to the retention of subsurface liquid, and bodies the size of Charon or larger (radii >600 km)are predicted to retain some subsurface liquid to the present day. We discuss the possibility that thisliquid can be brought to the surface rapidly via self-propagating cracks. We conclude that cryovolcanismis a viable process expected to affect the surfaces of large KBOs including Charon.

© 2009 Elsevier Inc. All rights reserved.

1. Introduction

Volcanism on Earth, where silicates are the dominant rock-forming mineral, plays an important role in reshaping the surfaceof this planet. Cryovolcanism is the process by which subsurfacewater, which as ice is a major “rock”-forming “mineral” in KBOs,is brought to the surface. Strong evidence for cryovolcanism comesfrom spacecraft images of the icy satellites of the giant planets.Moving outward in the Solar System, Europa has been recognizedsince the Voyager mission to have an extremely young surface(Smith et al., 1979), ∼50 Myr (Zahnle et al., 2003). Support forresurfacing by cryovolcanism exists (Figueredo and Greeley, 2004),although its role on large scales, and the extent of diapirism inresurfacing Europa, are currently debated (Fagents, 2003). Mea-surements by the Galileo magnetometer strongly indicate the ex-istence of a subsurface ocean on Europa (Kivelson et al., 2000),supporting a role for cryovolcanism. Evidence for heat fluxes in ex-cess of what could be produced by radioactive decay suggest tidaldissipation drives this surface activity (Ruiz and Tejero, 2000). Thebright terrain of Ganymede has been interpreted as cryovolcanic inorigin, with liquid filling low-lying grabens (Showman et al., 2004).High inferred heat fluxes in the past suggest tidal dissipation asthe driver of this activity as well (Nimmo et al., 2002; Nimmoand Pappalardo, 2001; Dombard and McKinnon, 2001). Some ofthe surface of Enceladus appears extremely uncratered and young,<108 yr, and shows tectonic features highly indicative of cryovol-

* Corresponding author. Fax: +1 (480) 965 8102.E-mail address: [email protected] (S.J. Desch).

0019-1035/$ – see front matter © 2009 Elsevier Inc. All rights reserved.doi:10.1016/j.icarus.2009.03.009

canism (Kargel and Pozio, 1996). The Cassini mission has revealedthe south polar region is especially young and active, with a plumeof icy material escaping the moon from this region (Porco et al.,2006; Hansen et al., 2006). Whether the plume is attributable tothe boiling of liquid water near the surface (Porco et al., 2006) orto decomposition of clathrates (Kieffer et al., 2006) is unknownat this time. The heat fluxes inferred to exist within Enceladus(Giese et al., 2008) also exceed the fluxes driven by radioactive de-cay alone. The Cassini Titan Radar Mapper observed a large domeor shield volcano, three calderas and, several extensive flows, con-sistent with cryovolcanism (Lopes et al., 2007). The inferred rheo-logical properties of the flows are consistent with water–ammoniaslurries (Lopes et al., 2007). Titan is large enough to maintainsubsurface liquid by radiogenic heating alone, although tidal heat-ing probably also contributed (Grasset et al., 2000; Tobie et al.,2005). Miranda and Ariel have relatively young surfaces in places,no older than 0.1–1 Gyr (Plescia, 1987, 1988; Zahnle et al., 2003),as well as grabens and flow features that have been interpreted ascryovolcanic (Kargel, 1994). There are no obvious sources of tidalheating for Miranda and Ariel, although orbital resonances in thepast cannot be excluded (Peale, 1988). Images of Triton (itself likelya captured KBO: Agnor and Hamilton, 2006) show large uncrateredterrains that are among the youngest in the Solar System, <10 Myr(Schenk and Zahnle, 2007). These regions contain large depres-sions that have apparently been extensively modified by flooding,melting, faulting and collapse, and several episodes of partial fill-ing and removal of material, features that have been interpretedas cryovolcanic in origin (Croft, 1990; Kargel and Strom, 1990;Kargel, 1994). Triton could have been substantially tidally heated

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Cryovolcanism on Kuiper belt objects 695

as a consequence of its capture (McKinnon, 1984), and there is ev-idence for high heat fluxes due to tidal dissipation in Triton’s past(Ruiz, 2003). In short, there is abundant evidence for cryovolcan-ism on the surfaces of icy satellites. It is likely that in each of thesecases, tidal dissipation enhanced the heating in these satellites. Itis therefore not known whether KBOs of similar size should havealso experienced cryovolcanism or not.

Cryovolcanism on KBOs has also been argued for, based on ob-servations of crystalline water ice or ammonia/ammonia hydrates.Crystalline water ice is converted to amorphous ice by doses ofGalactic cosmic rays (GCRs) ∼3 eV per molecule (Mastrapa andBrown, 2006; Cook et al., 2007, and references therein), and thespectral signatures of ammonia hydrates are likewise removed bydoses ∼100 eV per molecule (Strazzulla and Palumbo, 1998). Usingthe GCR fluxes of Cooper et al. (2003), Cook et al. (2007) calculatedthat the NIR spectral signatures of crystalline water ice and ammo-nia hydrates (which probe to depths ∼350 μm) should be erasedin times 1.5 and 50 Myr, respectively. When the ability of solarultraviolet photons to amorphize ice was considered, these timeswere reduced to 0.03 and 1 Myr (Cook et al., 2007). The presenceof these ices on the surface of a KBO therefore demands a physicalexplanation, possibly cryovolcanism. At the same time, ammoniaor ammonia hydrates are especially suggestive of cryovolcanismbecause ammonia depresses the melting point of ice, from 273to 176 K, enabling ice to be melted and mobilized (Hogenboomet al., 1997). Presumably any liquid brought to the surface wouldbe ammonia-rich. Jewitt and Luu (2004) observed crystalline waterice on the KBO 50000 Quaoar, based on absorption at 1.65 μm, aswell as absorption at 2.21 μm consistent with ammonia hydrates.More recently, Schaller and Brown (2007) have argued that the ab-sorption at 2.21 μm is predominantly due to adsorbed methane,rendering it unclear whether ammonia hydrates exist on Quaoar’ssurface or not. Cook et al. (2007) also observed crystalline waterice and ammonia hydrates on Pluto’s moon, Charon, and arguedfor cryovolcanism as a resurfacing process on this KBO. More re-cently, Barucci et al. (2008) have claimed detection of crystallinewater ice and ammonia ices on the KBO 90482 Orcus. These KBOsmay be experiencing cryovolcanism.

The use of crystalline water ice as a diagnostic of surface re-newal is strengthened by the apparent lack of physical mechanismscapable of converting amorphous ice back to crystalline form atthe cold temperatures of KBO surfaces. Cook et al. (2007) quanti-tatively analyzed many possible annealing mechanisms and foundnone of them to be competitive against amorphization. The mosteffective mechanism they considered was thermal annealing by de-position of kinetic energy by micrometeorite impacts, which theycalculated could anneal ice on Charon over timescales ∼5 Myr.This is to be compared to the amorphization by GCRs, which takes∼1.5 Myr, and by solar ultraviolet (UV) photons, which Cook et al.(2007) calculate could take as little as 3×104 yr. It is possible thatthe micrometeorite dust fluxes within the Kuiper Belt are largerthan those measured between 1 and 18 AU by Pioneer 10 (Humes,1980) by orders of magnitude. These fluxes will be measured bythe Student Dust Counter instrument on the New Horizons missionwhen it reaches the Pluto–Charon system in 2015 (Horányi et al.,2008). If they are very large, then localized heating by micromete-orite impacts may be why water ice exists in the crystalline state.Otherwise, according to the analysis of Cook et al. (2007), cryovol-canism may be the only explanation for the crystalline water iceand ammonia hydrates on the surfaces of KBOs.

It is worth clarifying that while Cook et al. (2007) interpretedthe spectral signature of crystalline water ice to be diagnostic ofcryovolcanism on KBOs, their work should not be construed tomean that icy satellites with crystalline water ice necessarily ex-perience cryovolcanism. Crystalline water ice is observed on essen-tially all icy satellites with water ice on their surfaces (Grundy et

al., 1999), including some with heavily cratered, ancient surfaces(e.g., Oberon), and others too small to have experienced internalheating (e.g., Nereid: Brown, 2000). However, due to the enhancedorbital velocities of satellites around their planets, as well as theincreased dust fluxes around the giant planets, it is likely that mi-crometeorites could anneal ice on satellite surfaces, even if theyare not quite able to anneal the ice on KBOs (Porter et al., 2009, inpreparation).

Observational diagnostics such as crystalline water ice beinginconclusive, then, the question remains, do KBOs experience cry-ovolcanism? For now, the question must be answered approx-imately, with theoretical models. Direct imaging of Pluto andCharon will be possible in 2015 with the arrival of the New Hori-zons probe (Stern, 2008), and should largely settle the question ofwhether Pluto or Charon experience cryovolcanism. But even then,proper interpretation of the data will depend on a theoretical un-derstanding of cryovolcanism on KBOs. We are strongly motivated,therefore, to study the possibility.

This article represents a theoretical modeling effort to assesswhether KBOs can experience cryovolcanism. A mechanism for de-livering liquid to the surface is described in Section 6, and thefocus in this paper is on whether subsurface liquid can persist onsmall, icy bodies, and Charon in particular. A prerequisite for main-tenance of liquid is ammonia, and we begin in Section 2 by re-viewing ammonia abundances on icy satellites, comets, and KBOs.In Section 3 we describe a numerical code written to calculate thethermal evolution of a KBO, including the effects of ammonia, dif-ferentiation, and heating by decay of long-lived radionuclides. Re-sults for a Charon-like KBO are described in detail in Section 4, andsensitivity of the results to key parameters is explored in Section 5.We then draw conclusions in Section 6 concerning the persistenceof subsurface liquid in KBOs, and the likelihood of cryovolcanismon them. We explore the implications of subsurface liquid in KBOsin Section 7.

2. Ammonia in the early Solar System

The potential role of ammonia in depressing the melting pointof water and affecting the rheology of ice has long been appre-ciated (Stevenson, 1982; Consolmagno, 1985; Croft et al., 1988;Kargel et al., 1991; Kargel, 1992; Durham et al., 1993; Hogenboomet al., 1997). A key parameter in this discussion is X , the weightfraction of ammonia relative to ammonia plus water. Based on thecosmochemical abundances of elements in the solar nebula, thisprimordial ratio could have been as high as X = 14.5%, assum-ing all N condensed into NH3 (Lodders, 2003). In fact, a significantfraction of the N must have condensed into organics, and an un-known fraction could have existed in the form of N2, which wasnot obviously accreted. It is therefore difficult to predict from firstprinciples the value of X in icy bodies in the early Solar System.Nonetheless, a variety of lines of evidence suggest that X is at leasta few percent.

Theoretical models (Charnley and Rodgers, 2002) predict thatN2 is efficiently dissociated into atomic form in molecular clouds,and that the pathways to return the atomic N back to N2 are inhib-ited by a lack of oxygen radicals in the gas phase. Instead, a sub-stantial amount of N is reduced to NH3 which then freezes ontoicy grain mantles. This NH3 may then undergo photolysis by ultra-violet radiation in the icy mantles, which converts it to N-bearingorganics. Once in the Solar System, atomic N may be converted toN2 or NH3. This scenario is also consistent with the fractionationof N isotopes in various Solar System nitrogen reservoirs, as well.Observations of molecular clouds confirm a lack of gas-phase N2(Maret et al., 2006), as predicted by the chemistry models. Thesesame models predict that as the Solar System forms, roughly 25%of the N is in the form of NH3 ice (Charnley and Rodgers, 2002).

696 S.J. Desch et al. / Icarus 202 (2009) 694–714

Likewise, Maret et al. (2006) model the chemistry in the Barnard68 core and infer X ≈ 7%. So, even if no NH3 formed within theSolar System from the remaining atomic N, a large value of X(≈ 4–7%) might be expected.

On the other hand, observations of Solar System objects do notsupport such high values of X . Perhaps the best observational con-straints come from observations of comets. The abundance of am-monia can be fixed by observing it directly in the radio spectrum,or by observing UV and optical lines of its photodestruction prod-ucts, NH2 and NH. From NH2 observations, a value X ≈ 0.5% wasfound to be typical among a large database of comets (Kawakitaand Watanabe, 2002). Note, though, that these same authors de-rived a range of values, from 0.25 to 1.0%. Many comets aredepleted in ammonia. For example, 21P/Giacobini–Zinner is con-sidered depleted (Beaver et al., 1990) and a value X < 0.1% is in-ferred from the NH2 observations (Kawakita and Watanabe, 2002).C/1996 B2 (Hyakutake) is inferred to have X ≈ 0.3% from radiolines of NH3 (Palmer et al., 1996) and X = 0.35% from lines ofNH (Meier et al., 1998). Comet 1P/Halley has slightly higher am-monia abundance. From lines of NH a value X = 0.44–0.94% isinferred (Feldman et al., 1993), and from optical lines of NH2,X = 0.75±0.18% is inferred (Kawakita and Watanabe, 2002). Directmeasurements from mass spectrometers on the Giotto probe yieldvalues closer to 1.5%: 0.8–2.0% using the neutral mass spectrom-eter (Meier et al., 1994), and X = 1–2% using the ion mass spec-trometer (Allen et al., 1987). Finally, radio observations of NH3 inC/1995 O1 (Hale–Bopp) yield X = 1.0–1.8% (Bird et al., 1997). Thecombined evidence seems to point towards short-period cometsthat have made repeated passages around the Sun being more de-pleted in ammonia. This implies that the primordial abundance ofammonia in comets must exceed the highest abundances observedin comets, X ≈ 2%.

Observational searches for ammonia or ammonia hydrates onKBOs and icy satellites are less conclusive. Strong evidence existsfor the presence of ammonia hydrates on the surface of Charon(Brown and Calvin, 2000; Dumas et al., 2001; Cook et al., 2007),based on absorption at 2.21 μm. Ammonia hydrates also appearpresent on Quaoar (Jewitt and Luu, 2004), although this absorptionhas also been interpreted as absorption by methane (Schaller andBrown, 2007). Tentative hints of ammonia hydrate absorption havealso been reported for Miranda (Bauer et al., 2002), and evidencefor ammonia on Enceladus is accumulating (Emery et al., 2005;Verbiscer et al., 2006). Intriguingly, observations of cryovolcanicplains on Miranda and Ariel yield estimates of the viscosity of thecryolavas; these are most consistent with a water–ammonia slurry(Schenk, 1991), with X large enough to reduce the viscosity. Thisimplies X on the order of a few percent or more.

Spectral measurements unfortunately probe only the top mil-limeter of each body’s surface, and do not provide a reliable esti-mate of the global values of X . Radiation damage at the surface, orcryovolcanism itself, may drastically alter the inferred values of X .Nonetheless, the spectral fits of Cook et al. (2007) (their model 2)do suggest that the surface ices are roughly 20–30% “ammonia hy-drate”. The “ammonia hydrate” component is modeled based onoptical constants obtained from experimental data by Brown etal. (1988), who froze a 3% molar solution of NH4OH in water, orX ≈ 3%. The (globally averaged) value of X on the Charon’s surfaceis therefore probably ∼1% within the observed ices.

To summarize, theoretical models of solar nebula chemistryindicate that X ≈ 7% or more are possible. Direct evidence forammonia in the outer Solar System comes from observations ofcomets, although at levels closer to X ≈ 1–2%. Morphological fea-tures on icy satellites and the presence of ammonia hydrates onCharon support X ≈ a few percent on these bodies as well. In whatfollows, we consider X = 1% and X = 5% to bracket the reasonablerange of ammonia abundances with which KBOs form.

3. Internal thermal evolution model

3.1. Background

We have written a numerical code to calculate the thermal evo-lution of KBO interiors. This code, like many others, essentiallysolves the 1-D (spherical) heat conduction equation. As discussedbelow, a time-dependent solution is absolutely necessary. The roleof differentiation is also of critical importance. Commonly solu-tions consider a completely homogeneous body or a completelydifferentiated body. As we show below, however, the maintenanceof subsurface liquid is optimized when a rocky core is in contactwith an icy layer, both of them surrounded by a crust of an undif-ferentiated rock/ice mixture. Also, as is well known, the effects ofammonia are important in lowering the thermal conductivity andviscosity of water ice, and especially in suppressing the meltingpoint. To study the internal thermal evolution of KBOs, it is there-fore essential to construct a time-dependent code, including therole of ammonia, and allowing for partial differentiation. In thissection we describe such a code we have written.

To our knowledge, ours is the first code including all the rel-evant effects (time-dependence, ammonia, partial differentiation)to be applied to KBOs. Previous codes used to study KBOs andicy satellites have been incomplete in one or more areas. Ruiz(2003) used a steady-state model to estimate the depth to the bot-tom of the ice layer on Triton’s surface. Similarly, Hussmann etal. (2006) considered steady-state, homogeneous or fully differen-tiated models of icy satellites and KBOs. De Sanctis et al. (2001)considered the thermal evolution of porous KBOs, and Shchukoet al. (2006) modeled the thermal evolution of Varuna, a porous,relatively small KBO. These models did not include differentiation(appropriate considering the small size of the objects they consid-ered). Thermal evolution models of icy satellites were developedby Ellsworth and Schubert (1983), but did not apparently followthe change in structure due to differentiation. Prialnik and Bar-Nun (1990) studied the thermal evolution of icy satellites subjectto heating by short-lived and long-lived radionuclides but did notinclude differentiation or ammonia. McKinnon (2002) consideredthe thermal evolution of KBOs, but also did not include differen-tiation. The code used by Schubert et al. (2007), used to studyEnceladus, incorporates time evolution and differentiation but doesnot include ammonia. To our knowledge, only one code describedin the literature includes all of the relevant effects. The code usedby Castillo-Rogez et al. (2007a) is a time-evolution model that in-cludes ammonia and differentiation. This code has been appliedto Iapetus (Castillo-Rogez et al., 2007a) and Rhea (Castillo-Rogez,2006) and Ceres (Castillo-Rogez et al., 2007b), but has not beenapplied to KBOs.

3.2. Description of model

Our code assumes 1-D spherical geometry and divides the KBOinto N radial zones (typically N ≈ 200). The zones are evenly dis-tributed in r so that the outer radius of each zone obeys ri =Rp(i/N), where Rp is the KBO radius. The code tracks the time-varying internal energy Ei(t) within each zone. These are set att = 0 to the energies corresponding to the temperature Tsurf; thatis, we assume the KBOs accreted “cold” (did not retain accretionalheat). Given the energy at some time t , the energy at a later timet + �t is updated, including contributions from radiogenic heatingand the fluxes of heat energy into and out of each zone:

Ei(t + �t) − Ei(t)

�t= Q i(t) + 4πr2

i−1 Fi−1 − 4πr2i F i . (1)

Here Fi refers to the heat flux out of zone i, into zone i + 1. Atthe innermost zone, Fi−1 vanishes, since by symmetry F = 0 at


Table 1Adopted long-lived radionuclide heating parameters.

Isotope t1/2

(Gyr)Initial# per 106 Si

wt%(ppm)

�E(MeV)

H0

(mW/kg elem)H0

(pW/kg rock)40K 1.265 5.244 1.389 0.6087 0.0254 35.2232Th 14.0 0.04399 0.0676 39.0 0.0253 1.70235U 0.704 0.00592 0.0092 42.7 0.5444 5.02238U 4.47 0.01871 0.0295 46.1 0.0913 2.69

r = 0. An “equation of state” is used to convert the energy Ei ineach zone to a temperature Ti . At the outermost zone, the energyin that zone is not calculated; rather, the temperature is fixed:T N = Tsurf. This equation evolves the temperature in a physicallyreasonable way so long as the timestep satisfies a Courant stabilitycriterion: �t < min[(�r)2/(2κ)], where κ is the thermal diffusiv-ity. In practice we find that rather than readjusting the timestep itis simpler to fix the timestep at �t = 1000 yr, as this satisfies theCourant condition for all of the cases we consider and still allowsthe full thermal evolution to be computed in a few hours on adesktop computer (Apple G5, 2 GHz processor). The thermal evo-lution is then completely determined once the rate of radiogenicheating and the heat fluxes are determined, and once an equationof state for the matter, T (E), can be established. These quantitiesare in turn affected by the amounts of rock, ices and liquids ineach zone, which are affected by differentiation. We now discusseach of these effects.

3.3. Radiogenic heating

We consider radiogenic heating by four long-lived radionuclidesonly: 40K, 232Th, 238U and 235U. On the premise that formation ofseveral-100-km diameter KBOs will take longer than ∼10 Myr, wedo not include decay of 26Al or 60Fe here, although we intend toconsider their effects in future work. The rate of radiogenic heatingdue to an isotope x with half-life t1/2, per mass of that isotope, iseasily found to be (�E)x(ln 2/t1/2)/mX , where mX is the mass ofan atom of x, and �E X represents the heat energy per decay. These�E X include heating due to emission of α and β particles andγ rays, but not the emission of neutrinos, which escape planetsaltogether.

Because of the loss of neutrino energies, the radiogenic heatingrate cannot be determined from the mass deficit between the par-ent and daughter nuclei alone. The uncertainties in the neutrinoenergy during radioactive decay are often on the order of 10%,and lead to uncertainties almost as large in �E X . We list our as-sumed values of t1/2 and �E X for each radionuclide in Table 1. Foreach radionuclide, we also list in Table 1 each one’s assumed initialmass. The masses of each isotope decrease exponentially with timeaccording to their half lives. Details on how we estimated neu-trino energy losses and computed the initial abundances of eachradionuclide are provided in Appendix A.

We calculate an initial radiogenic heating rate of S0 = 44.6pW/kg of rock, dominated by decay of 40K. For comparison,Ellsworth and Schubert (1983) used S = 52 pW/kg, and Multhaupand Spohn (2007) used S0 = 45 pW/kg. For the present day,Ellsworth and Schubert (1983) used S = 6 pW/kg while Multhaupand Spohn (2007) find S = 5.2 pW/kg; we find, in comparison,S = 5.6 pW/kg. The variation of heating rate with time is depictedin Fig. 1.

3.4. Fluxes

We generally assume that the heat fluxes are carried by con-duction, so that F = −k∂T /∂r. We finite-difference this as

Fi = −ki + ki+1

2

Ti+1 − Ti

(r − r )/2. (2)

i+1 i−1

Fig. 1. Adopted heating rate per mass of rock due to the radioactive decay of 40K,232Th, 235U, and 238U, as a function of time relative to the present day (t = 0 onthis graph only).

As the energy in the outermost zone (EN ) is fixed, F N is not cal-culated. By symmetry F0 = 0. Below (Section 3.6) we describe howwe calculate the total thermal conductivity of a mixture of solids.

There will arise in our calculations many instances where theheat fluxes are not carried by conduction alone but by convectionas well, especially in the layer of pure water ice that we predictwill develop during the thermal evolution of a KBO. A commonprescription for determining whether a fluid heated from belowwill convect is to determine the Rayleigh number, comparing it toa critical value. In this context, which assumes a uniform viscosityη across the fluid layer, the fluid will convect if the Rayleigh num-ber between the top of the layer (here, at r = Rdiff , the boundarybetween the ice layer and the undifferentiated crust, discussed be-low) and the base of the ice layer (at r = Rbase) exceeds the criticalvalue, ≈1100 (Spohn and Schubert, 2003). The Rayleigh number iscalculated as

Ra = αρice g(�T )(�r)3

κη, (3)

where �r = Rdiff − Rbase and �T = T (Rbase) − T (Rdiff). We cal-culate the other quantities—the expansivity α, the gravitationalacceleration g , the thermal diffusivity κ and especially the vis-cosity η—at the midpoint of the ice layer. The thermal diffusivityκ = k/ρice/cP, where k is the thermal conductivity and cP is theheat capacity. Evaluation of these quantities is discussed below(Sections 3.6 and 3.7).

This simple model for the onset of convection may not applyto icy satellites, however, where the viscosity contrasts across thefluid layer are great. Instead, satellites are understood to convectin the “stagnant lid” regime, where much of the ice layer will behigh-temperature and low-viscosity, and will participate in vigor-ous convection; this is capped on top (and in some treatments, atthe bottom as well) by a stagnant lid of ice that transports heat byconducting fluxes alone (Solomatov, 1995; McKinnon, 1998, 1999;Multhaup and Spohn, 2007). The onset of convection in the stag-nant lid regime may require even higher Rayleigh numbers, ∼106

(McKinnon, 1999), so the onset of convection is not ensured inour simulations without modeling stagnant lid convection. Unfor-tunately, a true simulation of stagnant lid convection lies outsidethe scope of this paper. Inventing an algorithm for parameterizingfluxes in the stagnant lid regime is one obstacle; numerically re-solving the temperature structure in the relatively thin stagnant lidis another.


Despite the difficulties in modeling the stagnant lid regime, wecan still approximate the thermal structure within a KBO by thefollowing prescription. First, we are not particularly interested inthe thermal structure within the ice layer per se, just the totaltemperature drop across it. Two regimes are relevant. The first isthe case where convection is presumed to be inhibited, and fluxesare carried by thermal conduction alone. This case is easily ac-commodated in our models. The second case is where stagnant lidconvection is initiated. In that case, we assume there is a nearlyisothermal profile within the ice layer (at T = Tc), capped at thetop and bottom by stagnant boundary layers across which theflux is carried by thermal conduction (e.g., Ellsworth and Schubert,1983; Multhaup and Spohn, 2007). The thickness of the bound-ary layers is δ = (�r)/Nu, where Nu is the Nusselt number of theconvecting layer. The temperature drop across the ice layer is es-sentially the temperature drop across the stagnant lids, which is�T = F δ/k = F (�r)/(k × Nu), where k is the thermal conductivity.Here it is understood that k represents the average of the thermalconductivities in the two stagnant lids on top and bottom, whichwill be equivalent to the thermal conductivity in the convectinglayer. Our code will correctly compute the same temperature dropacross the ice layer if the effective thermal conductivity keff is cho-sen appropriately; i.e., if �T = F (�r)/keff = F (�r)/(k × Nu), or ifkeff = k × Nu. This is how we implement convection in our code:within the convecting ice layer we increase the thermal conduc-tivity by a factor of the Nusselt number. To calculate the Nus-selt number, we adopt the parameterization Nu = (Ra/Rac)

0.25 forRa > Rac (Spohn and Schubert, 2003), where the Rayleigh numberis calculated at the midpoint of the ice layer. While this treatmentdoes not accurately reflect the likely temperature profile within theice layer (nearly isothermal, except in the stagnant lids), it shouldwell approximate the temperature drop across the ice layer.

Our treatment of convection is crude, but does provide a meansfor calculating the effects of convection on the thermal evolutionwithin a KBO, assuming convection is initiated. We have also per-formed runs where convection was shut off. As we will discussbelow (Section 5.4), the mass of liquid within a KBO, and the timefor which it can resist freezing, are not very sensitive to how con-vection is treated. Typically the mass of liquid varies by severaltens of percent, and the time for which it stays liquid varies by afew ×108 years, depending on whether or not convection is shutoff. So despite the crudeness of our treatment of convection, weexpect it to describe the temperature drops across the ice layeradequately enough for our purposes.

Within layers containing liquid, the Rayleigh number almost al-ways greatly exceeds Rac because the viscosity of water is so low,yielding typical values of Nu ∼ 104. The actual temperature gradi-ent in the convecting liquid layer could be found by using a corre-spondingly high effective thermal conductivity ∼104 W m−1 K−1;however, this approach is inadvisable in the liquid layer, as thecode is limited by a Courant condition from taking timesteps�t < (�r)2/(2κ), and high thermal conductivities would lead toproportionally small timesteps (�103 yr). On the other hand, thephysical significance of the large effective thermal conductivity issimply that the temperature drop across the liquid layer is small.It is likely, in fact, to be <10−2K, but none of the main conclu-sions of this paper is altered if the temperature drop is allowedto be as large as 1 K. Within any layer with at least 2% liquid(by weight), we therefore impose an effective thermal conductiv-ity keff = 400 W m−1 K−1. (Low values of the liquid mass fractionpresumably refer to heat transport by fluid flow through the rock,rather than liquid convection per se.) For a maximum heat flux<10 mW m−2, the temperature gradient therefore will not exceed0.025 K km−1, and across a liquid layer 40 km thick the temper-ature drop will not exceed 1 K. This is a small discrepancy com-pared to the temperatures typically achieved in the core, which are

at least several ×102 K, and typically the heat fluxes and the tem-perature discrepancies are even smaller. We therefore assume thatk = keff in any layers containing at least 2% liquid. We note thateven the small temperature gradient this imposes is basically elim-inated in the next step of our algorithm, which homogenizes theinternal energy of the slush layer after each timestep (Section 3.9).

3.5. Ice viscosity

In assessing the extent of differentiation or the effectiveness ofconvection, we must calculate the viscosity, especially of the wa-ter ice layer. At the low stress levels to be found in KBO interiorsit is expected that the dominant deformation mechanism for iceis dislocation-accommodated grain boundary sliding (McKinnon,1999; Durham and Stern, 2001). The effective viscosity is ηeff =σ/(3ε), where ε is the strain rate, σ is the stress, and d the grainsize of the ice, and where the other constants are parameters de-pendent on the phase and temperature of the ice (Durham andStern, 2001). We adopt the strain rate associated with Nabarro–Herring creep, in which case the strain rate is given by

ε = 9.3σb3 Dν

d2kT, (4)

where b = 4.52 × 10−10 m is the Burgers vector, Dν =D0ν exp(−Q /RT ) is the lattice diffusion frequency, D0ν = 9 ×10−4 m2 s−1, Q = 59.4 kJ mol−1 is the activation energy for self-diffusion, and R is the gas constant (Song et al., 2006). Based onthese parameters, we calculate

η ≈ 3.8 × 1014(

d

1 mm

)2

exp

[26.2

Tm − T

T

]Pa s, (5)

where Tm = 273 K. For grain sizes slightly smaller than a millime-ter, this formula matches well the commonly used viscosity law ofThomas et al. (1987):

η(T ) = η0

[25

(273 K

T− 1

)], (6)

where η0 = 1.0×1014 Pa s. We choose to adopt the viscosity law ofThomas et al. (1987) in our runs. Many of the parameters enteringinto the more exact viscosity law are uncertain, and this viscositylaw is commonly used and simple to implement. The differencewill be slight in any case, because the viscosity is only ever usedto change the effective thermal conductivity in the convecting icelayer. The conductivity is multiplied by the Nusselt number, whichscales as the fourth root of the viscosity, so different treatmentswill not change the viscosity much. Moreover, relevant quantitiessuch as the mass of liquid and the time at which it all freezes arefairly insensitive to whether the ice layer is convecting or is not, sominor changes in the viscosity law will not be expected to mattermuch anyway.

Using the viscosity law of Thomas et al. (1987), we estimatethat at temperatures just below 176 K, the viscosity in pure wa-ter ice is ≈1 × 1020 Pa s. If this ice contained some proportionof ADH, the viscosities would be lowered slightly. In an ice withX = 4.7%, just below 176 K the viscosity ≈1018 Pa s, while in anice with X = 9.5%, the viscosity ≈ 1017 Pa s (Durham et al., 1993;Arakawa and Maeno, 1994). At temperatures just above 176 K, theviscosity of pure water ice is still ≈1 × 1020 Pa s. In an ice bear-ing ADH or ammonia, however, the melting of the ADH just above176 K dramatically lowers the viscosity, to <1012 Pa s for a mix-ture with X = 4.0%, and to <1010 Pa s for a mixture with X = 8.4%(Arakawa and Maeno, 1994). It is not clear what the viscosityshould be for an ice mixtures with abundance X ≈ 1%, as is in-ferred from observations of comets. Clearly there is some thresholdbetween 0% and 4%, below which ammonia does not significantly


alter the viscosity of the ice; probably it is on the order of 1–2%,but this threshold is unknown.

3.6. Thermal conductivities

Besides the heat fluxes, no quantity is as important for deter-mining the temperature gradient within an icy body as its thermalconductivity. We are somewhat surprised, then, that most of themodels cited above have paid so little attention to this funda-mental input. The conductivity of crystalline ice is well knownto increase with decreasing temperature (e.g., Klinger, 1980), andthe conductivities of rock can be quite variable. Nevertheless,with little or no justification in many cases, particular valuesof the thermal conductivities are merely imposed. For example,Hussmann et al. (2006) assume a temperature-independent kice =3.3 W m−1 K−1 in their models of KBOs. Schubert et al. (2007) intheir modeling of Enceladus assume a temperature-independentkice = 2.1 W m−1 K−1 and krock = 3 W m−1 K−1. In their model-ing of the KBO Varuna, Shchuko et al. (2006) assume kice =5.67(T /100 K) W m−1 K−1 (Klinger, 1980), but a temperature-independent value krock = 2.4 W m−1 K−1. Castillo-Rogez et al.(2007a) in their modeling of Iapetus took kice = [0.47 + 4.88(T /

100 K)] W m−1 K−1 and krock = 4.2 W m−1 K−1. Finally, McKinnon(2002) similarly adopted krock = 2 W m−1 K−1, but recognized thatthe conductivities of amorphous quartz and feldspar near 100 Kcan be much lower, ≈1 W m−1 K−1 (Cahill et al., 1992). To sum-marize, modelers sometimes adopt a temperature-varying formulafor the conductivity of (crystalline water) ice based on laboratorymeasurements, but usually adopt an arbitrary value for the con-ductivity of rock.

The thermal conductivity of rock is an essential input, and lab-oratory measurements of suitable analogs are demanded. The typ-ical values adopted in the literature, krock ≈ 2–4 W m−1 K−1, areappropriate for terrestrial rocks at room temperature, but are notobviously applicable to the type of rock from which KBOs formed,at the typical temperatures of KBOs. Instead, we adopt the thermalconductivities measured for ordinary chondrites by Yomogida andMatsui (1983). Perhaps because of the aggregate nature of chon-drites, these conductivities typically do not exceed 1 W m−1 K−1;also, they exhibit no variation with temperature from 100 K toover 500 K. We therefore adopt krock = 1.0 W m−1 K−1, with novariation with temperature. Note that this means that rock in ourmodel is effectively an insulator in comparison with ice.

At higher temperatures, we do not expect the conductivities toincrease significantly. We note that practically all terrestrial rockstend to exhibit decreased thermal conductivities at higher tem-peratures (Clauser and Huenges, 1995). In fact, a variety of sedi-mentary, volcanic, plutonic and metamorphic terrestrial rocks havethermal conductivities that are nearly independent of temperatureabove about 600 K, generally not exceeding ≈2 W m−1 K−1. Sedi-mentary rocks (limestone, dolomite, shale and quartz sandstone),which may be the closest terrestrial analogs to the rocks in KBOs,generally are lower in conductivity, ≈1–1.5 W m−1 K−1 (Clauserand Huenges, 1995). We accordingly consider the likely range ofthermal conductivities to be krock = 1–2 W m−1 K−1, and we favorthe lower end of this range.

For the thermal conductivity of water ice we adopt kH2O(s) =5.67(T /100 K)−1 W m−1 K−1 (Klinger, 1980). For the thermal con-ductivity of ammonia dihydrate, we use the data of Lorenzand Shandera (2001) and Kargel (1992) to infer a temperature-independent conductivity kADH = 1.2 W m−1 K−1. (See also thediscussion in Ruiz, 2003, who came to the same conclusion.) Forthe sake of simplicity, we combine the conductivities in a waterice–ADH mixture by taking their geometric mean, using gH2Os, thefraction of the H2O(s)–ADH volume that is H2O(s):

Fig. 2. Thermal conductivity of an ice–rock mixture as a function of temperature.The rock conductivity krock (lower solid line) conforms to measurements of ordinarychondrites (Yomogida and Matsui, 1983). The ice conductivity kice(T ) (upper solidcurve) assumes an ice that is mostly H2O(s), with 1% NH3 by weight, or 3.1% ADH.Assuming a rock mass fraction of 0.63, we plot the total thermal conductivity (thicksolid curve), calculated using the formulation of Sirono and Yamamoto (1997). Thisis well approximated as 3.49(T /100 K)−0.816 W m−1 K−1 (dash–dot curve).

kice = kgH2OsH2O(s)k

1−gH2OsADH . (7)

With the thermal conductivities of the ices and rock found sep-arately, we combine them to find the conductivity of an intimatemixture of ice and rock, using the formulation (motivated by per-colation theory) of Sirono and Yamamoto (1997). Specifically, iff Vrock is the volume fraction of rock in the combined ice–rock mix-

ture, then ktot is found from:

2k2tot − [

krock(3 f V

rock − 1) + kice

(2 − 3 f V

rock

)]ktot − krockkice = 0. (8)

Fig. 2 shows the variation of the total conductivity of a mix-ture of 63% rock and 37% ice (by mass), with the ice being 1%NH3, or 3.1% ADH by weight (which is potentially representativeof Charon). As it happens, this conductivity is well described bya power law, ktot ≈ 3.49(T /100 K)−0.816 W m−1 K−1. We calculatea thermal conductivity at 100 K (for a Charon-like compositionwith frock = 0.63 and X = 0.01) of only 3.49 W m−1 K−1 in itssurface layers. For a heat flux of 1 mW m−2, this corresponds toa temperature change over 200 km of 57 K. Using the param-eters and formulas of Castillo-Rogez et al. (2007a) for the samemixture of rock and ice, one would calculate a conductivity of4.97 W m−1 K−1; over 200 km, the temperature change would beonly 40 K. Finally, assuming a fully differentiated body with apure water ice shell, the thermal conductivity in the outer layerswould be 5.67 W m−1 K−1, leading to a temperature jump of only35 K over the same distance. Significant differences in temperaturegradient can therefore arise depending on how the thermal con-ductivity is modeled. In particular, as this exercise illustrates, theinclusion of rock in an undifferentiated crust is insulating.

3.7. Other physical properties

Besides thermal conductivities, there are other physical proper-ties of materials to which attention should be paid. First are thedensities. It is common when calculating the fraction of an icybody that is rock to adopt an ice density of 1000 kg m−3 (Gulbiset al., 2006; Hussmann et al., 2006; Schubert et al., 2007). If theice contains contaminants such as CO2, as Schubert et al. (2007)assume, the density may indeed be this high, but if not assum-ing this, a more appropriate ice density would be 918 kg m−3, atleast at the freezing point of ice. The density increases slightly with


colder temperatures, and at 100 K, appropriate for KBO interiors,the density is 935 kg m−3 (Croft et al., 1988), which is the valuewe adopt here. For ammonia dihydrate (ADH), we adopt a densityof 965 kg m−3 (Croft et al., 1988). For the density of rock, we con-sider chondrites to be a reasonable analog. Consolmagno and Britt(1998) and Wilkison and Robinson (2000) measured the densitiesof ordinary chondrites and found them to vary with porosity, rang-ing from ≈3600 kg m−3 for chondrites with near zero porosity,to ≈3200 kg m−3 for chondrites with ≈ 10% porosity. Yomogidaand Matsui (1983) find the same trends; from their tabulationsof the properties of L chondrites, we adopt a typical porosity of10%, an intrinsic density 3600 kg m−3 and an average bulk densityρrock = 3250 kg m−3. The latter value is more consistent with thethermal conductivities used above, since these are based on theaverage conductivities measured by Yomogida and Matsui (1983).

If the ammonia content X and average density ρ of a KBO areknown, it is straightforward to use that information and the den-sities above to determine its rock (mass) fraction:

frock = ρrock

ρ

ρ − ρice

ρrock − ρice. (9)

For example, taking Charon’s mean density, ρ = 1710 ± 80 kg m−3

(Sicardy et al., 2006), and densities of rock ρrock = 3250 kg m−3

and ice ρice = 936 kg m−3 (appropriate for X = 0.01), we esti-mate Charon’s rock fraction to be frock = 0.636 ± 0.036 (ignoringself-compression). For comparison, Person et al. (2006) find a rockfraction 0.58 ± 0.04 assuming ice and rock densities of 1000 and3000 kg m−3.

For the heat capacity of H2O(s), we use the approximation

cH2O(s)(T ) = 773.0

(T

100 K

)J kg−1 K−1. (10)

Over the range 40 < T < 273 K, this approximation matches theexact expression of Shulman (2004) to within a maximum devia-tion of 15% (at 75 K), and to within <1% above 200 K. For the heatcapacity of ADH, we use the approximation

cADH(s)(T ) = 1120

(T

100 K

)J kg−1 K−1. (11)

This matches the exact results of Fortes et al. (2003) to within 4%throughout the temperature range 40 < T < 176 K. For liquid waterwe set cP = 4188.5 J kg−1 K−1, and adopt cP = 4700 J kg−1 K−1 forammonia. Finally, the heat capacity of rock is a significant input,since the core will store significant amounts of heat over geolog-ical history. These heat capacities can be inferred from the datapresented by Yomogida and Matsui (1983) for ordinary chondrites.We approximate the heat capacity of rock as

crock(T ) =⎧⎨⎩

770( T

275 K

)J kg−1 K−1, 0 � T < 275 K,

607 + 163( T

275 K

)J kg−1 K−1, 275 � T < 1000 K,

1200 J kg−1 K−1, 1000 K � T .

(12)

We note that the implied heat capacity is of the same or-der as common minerals; for example, our assumed value at1000 K, 1200 J kg−1 K−1, is comparable to the values c(1000 K) =1250 J kg−1 K−1 for forsterite and 1210 J kg−1 K−1 for enstatite(Navrotsky, 1995). Heat capacities in mixtures combine arithmeti-cally, weighted by the masses of the components.

The expansivity of ice, α, is important for determining whetherit can convect. It is common to assume a temperature-independentvalue for α that is appropriate to 273 K, e.g., α = 1.56 × 10−4 K−1

(Hussmann et al., 2006). In fact, α drops nearly proportionally withtemperature, falling below 10−5 K−1 below 100 K, and actuallychanging sign below 60 K (Röttger et al., 1994). This means thatat the frigid surface temperatures <60 K of many KBOs, ice cannotundergo solid-state convection, regardless of its viscosity, because

Fig. 3. Our assumed approximation to the phase diagram of the ammonia–watersystem. Region 1 corresponds to solid H2O ice and ammonia dihydrate (ADH). ADHis assumed to melt in region 2 over a temperature range 174–178 K, producing H2Oand NH3 liquids. In region 3, solid H2O ice gradually melts, in equilibrium withliquid H2O and NH3. At low values of the mass fraction of NH3 (X < 0.0466), H2Oice is assumed to melt in region 4 over a temperature range 271–275 K. In region 5,only liquid H2O and NH3 are stable. Our assumed liquidus temperature T liq(X) isdisplayed in bold, and compares favorably with an exact function from Croft et al.(1988), displayed as a thin curve.

ice at these temperatures actually contracts upon heating. We ap-proximate this behavior as α/(10−5 K−1) = 1.5(T /50 K) − 2.0 forT > 50 K, and = −0.5(T /50 K) for T < 50 K. We also adopt theexpansivity of ADH from Fortes et al. (2003), which we approxi-mate as α = 2.81 × 10−5(T /176 K)K−1. We combine expansivitiesarithmetically, weighted by the volume of each component.

3.8. Equation of state

Within each zone, we track 5 phases: rock; water ice, H2O(s);solid ammonia dihydrate, ADH(s); liquid water, H2O(l); and liquidammonia, NH3(l). At the beginning of each timestep, the mass ofrock, the total mass of H2O (in water ice, liquid water and boundin ADH), and the total mass of NH3 (in liquid ammonia and boundin ADH) are specified in each zone, as well as the total combinedinternal energy Ei of all phases. The fraction of the total mass thatis rock is denoted frock,i . Also specified in each cell is the frac-tion of the non-rock mass that is in ammonia [either as NH3(l) orthe ammonia portion of ADH(s)], Xi . Regardless of phase transi-tions (e.g., ADH melting) within each zone, the internal energy Eiand total mass Mi are conserved, as are Xi and frock. Given Mi ,Xi , frock,i and Ei at a particular timestep, we define the temper-ature Ti in that zone to be that temperature such that Ei is theenergy required to raise the mix of rock and ices from 0 K to thetemperature Ti , including phase transitions. Computing the energyneeded to raise the mixture to a given temperature is done usingan equation of state.

To make the problem tractable, we have adopted a simpli-fied version of the phase diagram of the ammonia–water system(Fig. 3). We do not consider ammonia monohydrate and so weignore the distinction between the peritectic and eutectic points.We only consider 0 � X � Xc , where Xc = 0.3210 (the fraction ofthe weight of ADH that is bound NH3). We also ignore the effectsof pressure on the melting point of water ice. While the meltingpoint of pure H2O ice can be reduced to only 251 K at pressuresof 200 MPa (Hogenboom et al., 1997), the peak pressures inside aKBO the size of Charon are only about (3/8π)GM2/R ≈ 135 MPa(assuming an homogeneous body); at the base of the ice shell ina partially differentiated Charon we predict even lower pressures,≈80 MPa. In the bodies we consider, the melting point of H2O


ice is depressed by less than 10 K, and we neglect this effect.Up to pressures of 300 MPa, there appears to be no variation inthe melting point of ammonia dihydrate (Hogenboom et al., 1997;Leliwa-Kopystynski et al., 2002). The phase transition from Ice Ito Ice II likewise requires pressures of about 220 MPa (Leliwa-Kopystynski et al., 2002), that are unlikely to be achieved on thebodies in consideration here. A description of the phase diagramcan be found in Appendix B.

This phase diagram is used to determine the fractions of non-rock mass that is in each phase [H2O(s), ADH(s), H2O(l) andNH3(l)], and the total specific energy e(T ) required to raise awater–ammonia mixture (characterized by Xi) from 0 K to a tem-perature T. Once this is determined, the energy required to raisethe rock and ices mixture in zone i from 0 K to a temperature Tis readily found:

E(T ) = frock,iMi

T∫0 K

crock(T ′)dT ′ + (1 − frock,i)Mie(T ). (13)

In practice, however, we are faced with the inverse problem: givenfrock, X and E , we must find the temperature Ti in shell, by findingthe temperature at which E(T ) = Ei . We accomplish this rapidlyusing simple bracketing techniques (Press et al., 1992). Thus, fora given internal heat energy E , we find the temperature T (E),as well as the apportionment of non-rock material into ices andliquids, that are consistent with that internal energy and tempera-ture. Volume changes are explicitly neglected in this treatment.

3.9. Differentiation

In our code we allow for differentiation of rock, liquids and ices.At the start of each simulation, we assume that each icy body is anhomogeneous mix of rock and ice; but we allow that these phasesmay separate as the temperature is increased. If the conditions fordifferentiation are met, our code allows the process to proceedpractically instantaneously (i.e., within the 1000 year timestep).Two types of differentiation must be considered. The first is theseparation of the rock from the other components, forming a rockycore. The second is the separation of liquid water/liquid ammo-nia/ammonia dihydrate from water ice. Each depends critically butin different ways on the viscosity.

We assume that separation of the rock from the ice occurs viasolid-state creep of ice around the denser rock, and only via thismechanism. We consider the rock to exhibit Stokes flow, with avelocity

V = 1

18

(ρrock − ρice)g D2rock

η, (14)

where Drock and ρrock are the diameter and density of the rock,and g (∼0.30 m s−2 inside a typical KBO) is the gravitational accel-eration. We find that differentiation usually takes several ×107 yr,so we demand that rocks slip one grid zone (≈2 km) on timescales<106 yr to be consistent with our numerical code. For rocks withdiameters of 1 m, differentiation over 1 Myr therefore demandsη < 6 × 1011 Pa s, and differentiation over 1 Gyr demands η <

6 × 1014 Pa s, both thresholds increasing as the square of the rockdiameter.

The viscosities required for differentiation are lower than canbe typically achieved in pure water ice, which are ∼1014 Pa s justbelow the melting point, and significantly higher at colder temper-atures. In a KBO made of pure water ice, differentiation probablywould not occur until the ice melted. But, if the ice contains justa few percent NH3 by weight, differentiation can be expected tooccur when the ADH within it melts, at about 176 K. Just belowthis melting point, the measured viscosities are about 1017 Pa s

for X ≈ 5–10% (Durham et al., 1993), and differentiation wouldtake a time greatly exceeding the age of the Solar System. But assoon as ADH begins to melt, producing some H2O and NH3 liquid,the viscosity drops to <1012 Pa s for X = 4.0% or more (Arakawaand Maeno, 1994). It is not known what the viscosity of water–ice mixtures is for lower concentrations of ammonia (X ), but acasual glance at the data of Arakawa and Maeno (1994) suggeststhat η ≈ 1013 Pa s for X ∼ 1%. Thus differentiation would proceedeven with X ∼ 1%, provided the rock component is predominantlyin objects at least 10 m in diameter. We assume within our codethat differentiation occurs instantaneously wherever the tempera-ture exceeds 176 K (actually, 174 K, the temperature at which meltis first produced in our approximate phase diagram).

The maximum radius within a KBO at which the temperatureever has exceeded 176 K we denote Rdiff . (In general, Rdiff startssmall and grows with time as the KBO heats up, to an asymp-totic value.) All the rock inside Rdiff is assumed to have differ-entiated into a rocky core. Of the remaining phases, water ice isless dense (≈935 kg m−3 at 100 K; Croft et al., 1988) than ADH(≈965 kg m−3; Croft et al., 1988) and the liquid water–ammoniamixture (>950 kg m−3 at the same temperatures; Croft et al.,1988). At temperatures above 176 K, the slushy ADH and liquidwater–ammonia should fall relative to the less dense water ice,and we assume that all the ADH and liquid phases end up sand-wiched between a rocky core and a pure water ice layer insideRdiff . Beyond Rdiff , material is undifferentiated.

After the rock separates from other components, it is expectedthat ice and liquid in the remaining material can quickly separatevia convection. The Rayleigh criterion for convection is that theRayleigh parameter (see Eq. (3)) exceed the critical value Rac ∼103. Assuming typical parameters of α ∼ 3 × 10−5 K−1, ρice ∼1000 kg m−3, g ∼ 0.10 m s−2, κ ∼ 10−6 m2 s−1, �T ∼ 100 K, and�r ∼ 100 km, we find Ra ∼ 3 × 1020 Pa s/η. Since the viscosity ofthe remaining ice is by assumption <1012 Pa s at the time of sepa-ration, vigorous convection is easily initiated in the remaining ice.This convection is expected to overturn the remaining phases ina time ∼D2/κ/Nu, where D ∼ 100 km is the thickness of the icelayer, κ ∼ 2 × 10−6 m2 s−1 is the thermal diffusivity, and Nu ∼ 4is the initial Nusselt number. Thus, within 40 Myr, the remainingphases should separate according to density. In particular, waterice will rise to the top and the ammonia will be concentrated inthe liquid layer at the base.

Although ADH and the liquid phases also have different densi-ties, we do not assume that they differentiate. ADH is denser thanits melt, and could be expected to sink to the bottom of the liq-uid layer upon freezing, but then it would warm up, melt, and riseagain. This keeps the ADH and liquids well mixed, so we retainthem within a single system, a “slush” layer that is well mixedchemically and energetically. This also may lead to effective heattransport, which we account for by assuming a high effective ther-mal conductivity (see Section 3.6).

In our code, differentiation is implemented as follows. At eachtimestep, Rdiff is determined. The code searches for zones insideRdiff containing rock, starting at the innermost zone. The mass andinternal energy of rock from each zone are then used to fill upa new grid, starting with the innermost zones. After all the rockinside Rdiff has been moved, the code searches for all slush com-ponents (liquids and ADH) inside Rdiff and moves their mass andinternal energy to the new grid, building on the rocky core there.Last, the code searches for pure water ice inside Rdiff and moves itto the new grid, filling up the new grid up to a radius of Rdiff . Thedensities of each component are chosen so that the total volume ofthe system does not depend on chemical phases; in this way thetotal volume of the system is conserved during differentiation, andthe new grid is filled to exactly Rdiff . Mass outside Rdiff is movedto the new grid without change. As a last step, the chemical com-


position and internal energies in the slush layer are homogenized,to account for the vigorous mixing expected in these layers. Morespecifically, we assign each cell associated with the slush layers thesame ammonia concentration X equal to the average value in theslush layers, and the same energy per mass. This last step stronglyimposes isothermality within the slush layers.

Finally, we also calculate the gravitational potential energy Ugbefore and after each timestep:

Ug = −G

Rp∫0

4πξ2ρ(ξ)M(ξ)

ξdξ, (15)

where M(ξ) is the mass enclosed within a sphere of radius ξ . Ifdifferentiation has redistributed mass and changed Ug during atimestep, we deposit that energy difference uniformly throughoutall shells interior to Rdiff . We find that this additional heating isgenerally unimportant in KBOs, amounting to an increase of only afew K within the core during differentiation.

Differentiation is a potentially complex phenomenon. For ex-ample, if rock is much smaller than the size we have assumed(meter-sized), or if liquid can somehow be lost from the ice be-fore rock can differentiate, then differentiation would not proceedas we have described. We assume differentiation occurs in themanner described above; it remains to be seen how robust ourassumptions are.

4. Thermal evolution of Charon

As a canonical case, we now apply this code to the thermalevolution of a body comparable to Pluto’s largest moon, Charon.Charon’s radius has been reported as Rc = 606 ± 8 km (Gulbiset al., 2006) and Rc = 603.6 ± 1.4 km (Sicardy et al., 2006);combined observations yield Rc = 606.0 ± 1.5 km (Person et al.,2006). Charon’s mean density has been reported as ρ = 1720 ±150 kg m−3 (Gulbis et al., 2006) assuming its mass is 1.60±0.12×1021 kg (Olkin et al., 2003), or ρ = 1710 ± 80 kg m−3, assuminga mass 1.58 ± 0.07 × 1021 kg (Sicardy et al., 2006). Person et al.(2006) revised this downward to ρ = 1630 ± 70 kg m−3 by assum-ing a mass 1.52 ± 0.064 × 1021 kg (Buie et al., 2006). Rather thanfix particular values of mass and radius, here we consider a bodywith radius of 600 km and a mean density 1700 kg m−3, yield-ing a mass 1.538 × 1021 kg. To be conservative, we consider a lowammonia fraction, X = 0.01. From these choices we derive a rockmass fraction frock = 0.63. The mass of rock inside our Charon ana-log is therefore inferred to be 0.97 × 1021 kg. Assuming a valueS ≈ 5.66 pW kg−1 in rock, this yields a steady-state surface heatflux due to the decay of long-lived radionuclides of 1.216 mW m−2

at the present day.The surface temperature of Charon is another input that is

not well known. Fits to the shape of the absorption feature at1.65 μm due to crystalline water ice suggest temperatures of 43and 53 K on the sub-Pluto and anti-Pluto hemispheres of Charon(Cook et al., 2007). We regard these as lower limits to the sur-face temperature, because the absorption occurs in ice that is ofhigher albedo (and therefore lower temperature) than the rest ofthe surface. The presence of a poorly conducting regolith can alsolead to a steep temperature gradient just below the surface. Thethermal inertias of icy satellite regoliths have been determinedto be typically (kρc)1/2 ≈ 30–70 J m−2 s−1/2 K−1 (Spencer, 1989;Spencer and Moore, 1992; Spencer et al., 1999). From this we infera thermal conductivity k ≈ 10−3 W m−1 K−1 within the regolith.Coupled to a heat flux ∼1 mW m−2, this implies that a temper-ature jump of 1 K can occur across a regolith only 1 m deep.We therefore adopt a surface temperature that is a few degreeshigher than the 53 K observed in ice on the anti-Pluto hemisphereof Charon; specifically, we adopt 60 K.

Fig. 4. Temperatures within a Charon analog, at times t = 0 (dotted line), t = 1 Gyr(dotted curve), and (solid curves, from top to bottom) t = 2, 3, 4 Gyr and the presentday, t = 4.56 Gyr.

4.1. Numerical results

The evolution of our Charon analog proceeds as follows. Afteraccreting cold, the planet warms steadily as long-lived radionu-clides decay. After roughly 75 Myr of evolution, temperatures nearthe planet center increase to the critical value of 176 K at whichthe ADH within the ice begins to melt, dramatically lowering theviscosity and allowing for differentiation. A rocky core quickly sep-arates out; by t = 80 Myr it incorporates all rock within a ra-dius Rdiff = 420 km and has a radius of about 292 km (radii arerounded off to the nearest grid point; here the grid resolution is2 km). Roughly half of the core’s eventual mass differentiates dur-ing this rapid initial stage. For the next roughly 700 Myr, the dif-ferentiated region and the rocky core continue to grow, but muchmore slowly. By t = 0.75 Gyr the core reaches its maximum size of356 km, and Rdiff reaches its maximum value of about 516 km. Atthis time a slush layer extends from about 356 km to the base ofthe water ice layer at about 368 km, and the pure water ice layerextends from about 368 to 516 km.

The temperatures within the body are illustrated in Fig. 4 forseveral different times. Key features to note from this figure areas follows. First, temperatures in the core rise steadily for the first2 Gyr; energy from the decay of long-lived radionuclides drivesa heat flux outward, but at a rate reduced from the steady-statevalue, as energy also is consumed in heating the rocky core. Themaximum temperature in the core is 1418 K, reached at 2.07 Gyr.It is worth noting that the solidus temperature of chondritic com-positions is 1375 K (Agee et al., 1995), so a small fraction ofthe core should actually melt. After 2.07 Gyr, the temperaturesin the core drop and the heat flux is enhanced by the releaseof stored heat. The heat flux emanating from the core is carriedefficiently through the “slush” layer between 356 and 396 km be-cause this layer is assumed to be partially liquid and convective,and is assigned a very large effective thermal conductivity. Con-vection also carries heat effectively through the ice layer, but isstill associated with a non-negligible temperature gradient. TheNusselt number in the ice layer exceeds unity, but not by much,rising to its maximum of 4.4 at t = 1 Gyr, and dropping to 2.9 att = 2 Gyr, 1.7 at t = 3 Gyr, and 1.0 at t = 4 Gyr. Thereafter convec-tion shuts off.

The distribution of phases within each layer, at t = 2 Gyr, isseen in Fig. 5. The distributions at other times look essentially thesame as soon as differentiation is initiated, except that the valuesof Rcore and Rdiff change slightly, and the extent of the slushy layervs. the ice layer will change. At the present day, Rcore and Rdiff are


Fig. 5. Distribution of phases within a Charon analog at time t = 2 Gyr. The linesrepresent rock (solid line), solid or liquid water (dashed lines), and ammonia dihy-drate or liquid ammonia (dash–dot lines). A core of pure rock exists within 355 km.Between 355 and 362 km there is a layer of water/ammonia liquid, and from 362to 515 km there is a layer of pure water ice. From 515 km to the surface at 600 kmthere is an undifferentiated crust of rock, water ice and ADH.

Fig. 6. Total mass of liquids within a Charon analog (solid curve), and that mass ofliquids derived from melting of ADH (dashed curve), as a function of time. The firstliquid arises from ADH suddenly melted at t ≈ 75 Myr during initial differentiation.Continued heating melts water, but it eventually refreezes. By t = +4.5 Gyr, theremaining liquid refreezes as ADH, until by t = 4.61 Gyr it is completely frozen.

the same as in Fig. 5, except that much of the water liquid in theslush layer has frozen to water ice. At the present day, the liquidlayer is predicted to extend from 356 to about 362 km.

In Fig. 6 we display the total mass of liquids as a function oftime. The total mass of ice on this Charon analog is 0.57 × 1021 kg,and the total mass of ADH is 0.0177 × 1021 kg. After about 80 Myrhas elapsed, the temperatures rise to 176 K and the ADH withinabout 420 km all melts, producing 0.0061 × 1021 kg of liquid.(Eventually, as Rdiff increases, the mass of ADH that will be meltedis 0.011 × 1021 kg.) At first, this liquid has an ammonia concentra-tion near the eutectic, as seen in Fig. 7. Then, over the next several100 Myr, water ice melts, diluting the ammonia. The liquid con-tent peaks at about 1 Gyr, at a mass 1.8 × 1019 kg (about 1% themass of the KBO), and an ammonia fraction X ≈ 0.2. Thereafterthe core continues to heat but the heat fluxes escaping the coredecrease. The water liquid begins to refreeze, a process that con-tinues for several Gyr. The ammonia concentration increases as thewater freezes onto the bottom of the ice mantle. As it happens,in these simulations the ammonia concentration reaches the eu-

Fig. 7. Mass fraction X of ammonia in the liquid inside a Charon analog, as a func-tion of time. The eutectic concentration, 0.321, is plotted for comparison.

Fig. 8. Pressures inside a Charon analog, as a function of radius within the body,at a time t = +2 Gyr, when differentiation is complete. The vertical dotted linesrepresent (from left to right) the boundaries between the rocky core, the liquidlayer, the water ice layer, and the undifferentiated ice/rock crust.

tectic and the last of the liquid begins to refreeze right at aroundt = 4.5 Gyr, which is the present day. The last of the liquid takesabout 200 Myr to refreeze. This duration partially reflects the tem-perature range of 4 K over which we presumed refreezing to takeplace, but is probably of the right magnitude. The latent heat as-sociated with the freezing of 1.1 × 1019 kg of ADH over 200 Myris equivalent to a surface heat flux of 0.05 mW m−2, or 3% of thetotal heat flux. If the refreezing took place over 10 Myr, it wouldhave an appreciable impact on the internal thermal structure.

Finally, we plot in Fig. 8 the pressure as a function of depthwithin the KBO at t = 4.56 Gyr. The pressure was calculated sim-ply by integrating

dP

dr= − GM(r)ρ(r)

r2(16)

from the surface, where P = 0. Here M(r) is the mass enclosedwithin a shell at radius r. For this modeled KBO, the pressure atthe center of the body is 269 MPa, falling to 78 MPa at the edgeof the rocky core. The central pressure is nearly doubled over itsvalue before the differentiation, 135 MPa, due to the concentra-tion of mass at the center. The pressures in the liquid layer areroughly 78 MPa, and those in the ice layer vary from 78 to about38 MPa. The pressures in the undifferentiated crust are lower still,<38 MPa. These low pressures justify the neglect of higher-order


phases of ice and other pressure effects. These effects could be-come important, however, on larger icy bodies. Taking 200 MPa asthe limit for which our water–ammonia phase diagram can be con-sidered important, we estimate these pressures would be reachedin the liquid layer for bodies larger than about 900 km in radius.Our models therefore cannot capture the complex physics in thelarge satellites (Europa, Ganymede, Callisto, Titan), but might beapplied with some care to large KBOs (Pluto, Triton, Eris). All otherKBOs and icy satellites, with radii <800 km, should be describeddecently by these models.

4.2. Analysis

The smooth variation of temperature with radius seen in ournumerical results suggests that an analytical solution may be ob-tained. We are motivated to find such a solution, both as a checkon the numerical results, and also because it will provide physi-cal insight into the behavior of the system. To do so, we assumethe same compositional stratification as found by the numericalsimulations, then integrate the heat conduction equation. We as-sume a heat flux that is the sum of two components: a flux drivenby the decay of long-lived radionuclides, F rad(Rp); and an addi-tional flux arising from the release of stored heat from the rockycore, ≡ε Frad(Rp). Because stored heat could be released from otherlayers besides the core, this approach is necessarily approximate;nonetheless, it does provide useful insights.

We first examine the undifferentiated crust layer. Although thislayer contains radioactive rock that generates heat, we assume noflux arises from release of heat stored within this layer, so ε refersto heat released from the rocky core and the ice and liquid layers.The flux at radius r < Rp is straightforwardly found to be

F (r) = F rad(Rp)

[r

Rp+ ε

(Rp

r

)2]. (17)

Assuming a power-law thermal conductivity in this rock–ice mix-ture, k(T ) = k0(T /T0)

−p , the heat conduction equation is easilyintegrated to find the temperature at r = Rdiff:(

T (Rdiff)

T0

)1−p

=(

T (Rp)

T0

)1−p

+ (1 − p)F rad(Rp)Rp

k0T0

×[

1

2

(1 − R2

diff

R2p

)+ ε

(Rp

Rdiff− 1

)]. (18)

We assume T (Rp) = 60 K, a thermal conductivity in the undif-ferentiated crust of k(T ) = 3.49(T /100 K)−0.816 W m−1 K−1 (seeFig. 2 and Section 3.6), and a present-day radiogenic heat fluxat the surface F rad(Rp) = 1.216 mW m−2. These quantities arebasically parameters of the problem; the last two are takenfrom the numerical simulations, from which we find Rdiff =515.3 km = 0.859Rp, and observe a heat flux at the surface of1.724 mW m−2 at t = 4.56 Gyr, implying ε = 0.418. With theseparameters in hand, our analytical solution predicts T (Rdiff) =100(0.961 + 0.0632ε)5.435 K = 93.2, which compares extremelywell with the observed value, T (Rdiff) = 93.3 K. We would alsopredict a heat flux at r = 517 km (near the bottom of the undiffer-entiated crust) equal to F = 1.732 mW m−2, which is within 1% ofthe value we observe in the simulations, F = 1.747 mW m−2.

We next integrate inward from Rdiff to the base of the ice layerat Rbase. The flux in the ice layer is

F (r) = F rad(Rp)

[(Rdiff

Rp

)3

+ ε

](Rp

r

)2

. (19)

Combining this with the conductivity of pure water ice, k(T ) =k0T0/T , where k0 = 5.67 W m−1 K−1 and T0 = 100 K (Section 3.6),we derive the temperature at the base of the ice layer:

T (Rbase) = T (Rdiff)

× exp

[F rad(Rp)Rp

k0T0

(Rp

Rbase− Rp

Rdiff

)(R3

diff

R3p

+ ε

)]. (20)

Here it is assumed that the temperatures are low enough inthe ice layer to render it non-convective. Our numerical simula-tions show that at t = 4.56 Gyr, Rbase = 362.2 km, so this an-alytical solution would predict T (Rbase) = T (Rdiff) exp(0.401 +0.633ε) = 181.5 K, which is within 2.5% of the value observedin the simulations, T (Rbase) = 177.1 K. Likewise, we would pre-dict a heat flux at r = 367 km (near the base of the ice layer) ofF = 3.417 mW m−2, which matches to within 2% of the observedflux, F = 3.362 mW m−2. Our simulations show that the heat fluxthrough the undifferentiated crust is augmented by 42% over thevalue arising from instantaneous radioactive decay alone.

The additional heat flux is completely consistent with that ex-pected from the release of stored heat from the rocky core, aswe now demonstrate. Within the rocky core, assuming a spatiallyuniform rate of radiogenic heating and a constant thermal conduc-tivity, the temperature in the core is easily shown to conform to

T (r) = T (0) − [T (0) − T (Rcore)

]x2, (21)

where Rcore is the outer boundary of the core and x = r/Rcore. Thetemperature at the outer boundary ≈200 K, but the central tem-perature is much higher. The heat energy of the core is then easilyfound by integrating the internal energy (Eq. (13)) over the volumeof the core:

Ecore =Rcore∫0

4πr2ρrockerock[T (r)

]dr = 3Mcore

1∫0

x2erock[T (x)

]dx.

(22)

To proceed we must integrate the heat capacity of rock (Sec-tion 3.7), and use temperatures computed in the simulations. Att = 4 Gyr, T (0) = 1077 K and T (Rcore) = 197 K; at t = 4.56 Gyr,T (0) = 932 K and T (Rcore) = 176 K. Between these two times, it istedious but straightforward to integrate the heat capacity of rockover the core to show that the core loses an amount of heat energy≈6.85 × 104 J kg−1. The core mass is 6.15 × 1020 kg, so the aver-age heat loss over the last 0.56 Gyr of the simulation is equivalentto ε ≈ 0.43 times the present-day radiogenic heating.

It is possible to quantify other sources of stored heat. Fromabout t = 2.0Gyr to 4.5 Gyr, about 4 × 1018 kg of water freezes,releasing latent heat of freezing equivalent to ε = 0.003. The heatflux from freezing of water is therefore negligible. More significantis the freezing of the last remaining liquid from t = 4.51Gyr to 4.68Gyr, which releases a burst of latent heat equivalent to ε = 0.049.The other sources of stored heat, cooling of the ice, liquid, andouter layers, are even less important. So it is the release of storedheat from essentially just the rocky core that is increasing the heatflux through the KBO.

The heat stored in the core was generated during the earliestphases of the KBO’s evolution. While it was being stored, the heatescaping to the surface necessarily was lower than it would oth-erwise be; later, when the heat from the core was released, theheat flux is enhanced over the amount due to instantaneous ra-dioactive decay alone. This is illustrated in Fig. 9, which showsthat the point at which the core begins to release heat it has al-ready stored is reached at about t = 1.6 Gyr. At the present day(t = +4.56 Gyr), the heat flux at the surface is enhanced over theinstantaneous radiogenic heat flux by a factor of 1.42, or ε = 0.42.This is a significant increase in the heat flux, and demonstratesthe importance of the KBO’s past history in determining its cur-rent thermal structure. In fact, without these stores of heat (setting


Fig. 9. Heat flux at the surface of a Charon analog, as a function of time (solid curve).For comparison we plot the flux that would arise from the instantaneous rate ofradioactive decay (dashed line). During the first 1.6 Gyr, the rocky core absorbs moreheat than it emits, but thereafter the core releases heat and enhances the heat flux.

ε = 0), the temperature at Rdiff would be only 81 K, and the tem-perature at Rbase would be only 120 K, and no liquid layer wouldbe possible. The value of ε is a function of the prior heating of thecore. Thus, it is impossible to judge whether or not subsurface liq-uid can persist within KBOs without performing a time-dependentcalculation.

5. Sensitivity to various parameters

5.1. Ammonia

Ammonia abundance has a large bearing on the thermal evolu-tion of a KBO, so we consider two departures from our canonicalabundance X = 1%. The first case involves an increase in ammonia,to X = 5%, near the upper limit of what we deemed plausible inthe ices accreted by KBOs (Section 2). Surprisingly, there is verylittle difference between this case and the canonical case. Theadditional ADH in the undifferentiated crust reduces the thermalconductivity somewhat, and allows differentiation to proceed outto roughly 528 km instead of 516 km, but this is a relatively minoreffect. The amount of liquid that exists within Rdiff is, at all times,larger than in the canonical case, because there is more ammoniaavailable as antifreeze; but the profile of Mliq versus time looksidentical to Fig. 6, merely scaled upward by a factor of exactly 5.Significantly, the time at which the liquid freezes is not changedmuch. The extra latent heat released upon freezing of the liquidbuffers the freezing slightly more effectively than in the canonicalcase. Total freezing does not occur until 4.57 Gyr, instead of 4.54Gyr, an insignificant difference (both simulations using 100 zones).The amount of liquid varies proportionally to X , but the behaviorof the system is very insensitive to the value of X , provided it islarge enough (a few percent) to allow differentiation at 176 K.

It is especially instructive to compare the canonical results ob-tained above to the case where ammonia is absent. The presenceof ammonia with X exceeding a few percent was found sufficientto trigger a change in viscosity at the melting point of ammo-nia dihydrate, 176 K. Here we consider a KBO with Rp = 600 km,ρ = 1700 kg m−3, but with X = 0. In this scenario without ammo-nia, the viscosity of the ice remains high at all temperatures. Inice just below the melting point, the assumed viscosity, 1014 Pa s,is high enough to prevent the Stokes flow of rocks <1 m in diam-eter from flowing faster than ∼10 km Gyr−1, an inconsequentialamount. We therefore consider differentiation to take place only

Fig. 10. Same as Fig. 6, but for the case X = 0. By t = 2.81 Gyr all the liquid iscompletely frozen.

when temperatures exceed 271 K, the point at which ice begins tomelt in our code.

Because of the lack of ammonia, no melt is produced untilabout 200 Myr, much later than the time taken to produce meltin our canonical case, roughly 70 Myr. When melt is produced anddifferentiation occurs, it ultimately is limited to Rdiff = 455 km,compared to 515.3 km in the canonical case, and Rcore = 313.2 km,compared to 354.5 km in the canonical case. The mass of the rockycore that forms in the absence of ammonia is only 70% as large asthe core in the canonical case. The smaller size of the core leadsto smaller temperatures within it. Temperatures at r = 0 peak att = 1.94 Gyr at 1210.3 K, rather than at 1418.0 K at t = 2.06 Gyr.Instead of storing an average 4.90×105 J kg−1 in the core, the rockcore in the X = 0 case stores only about 4.52 × 105 J kg−1. Therocky core in the X = 0 case stores only 2/3 the heat of the X = 1%case (at t = +2 Gyr), about 1.9 × 1026 J instead of 3.0 × 1026 J.

The lack of a substantial heat reservoir allows considerablymore heat to be released into the outer layers early on, but consid-erably less heat later. In contrast to the X = 1% case, the rocky coreabsorbs more heat than it emits only until t ≈ 0.5 Gyr (as opposedto 1.6 Gyr). The difference in heat energy that is released early on,an additional 1.1×1026 J, is sufficient to melt roughly 3.3×1020 kgof water ice. producing an order of magnitude more liquid thanin the X = 1% case. As illustrated in Fig. 10, the mass of liquid(all water) peaks between 1.0 and 1.5 Gyr, at about 2.3 × 1020 kg.Thereafter the liquid gradually refreezes. The transition to a com-pletely frozen state is buffered by the immense amount of liquidthat must refreeze, and the latent heat of fusion released as it does.The heat released by the refreezing of 2 × 1020 kg of water over1 Gyr is equivalent to a surface heat flux of 0.5 mW m−2, whichalone is a significant fraction (≈10%) of the total heat flux escap-ing the body at this time. Essentially, the liquid freezes only asfast as heat can be released; this is in contrast to the X = 1% case,where the freezing of the liquid had less bearing on the overallheat flux. The liquid is completely frozen by t = +2.81 Gyr, andthereafter the heat flux drops considerably.

5.2. Thermal conductivity

We have performed an additional simulation of our canonicalcase, with Rp = 600 km, ρ = 1700 kg m−3 and X = 1%, but withthe rock conductivity doubled to krock = 2 W m−1 K−1. This slightlyincreases the conductivity in the undifferentiated crust, and greatlyincreases it in the core. The increased conductivity thus has two ef-fects, both of which decrease the internal temperatures. For a fixed


flux, the increased conductivity reduces the temperature gradientthrough the ice crust shell, and lowers the temperatures through-out the body. But an increased rock thermal conductivity reducesthe fluxes, too, because less heat can be stored in the core whileits temperature is increasing in the first 2 Gyr. This manifests it-self at the present day via a reduced ε . Both these effects have thepotential to depress the temperatures and limit the time for whichliquid can exist.

A quick comparison with the k = 1 W m−1 K−1 serves to illus-trate the differences. In the increased conductivity case, at t =+4.56 Gyr, Rdiff = 515.0 km, and Rcore = 354.5 km, so these quan-tities are unchanged. On the other hand, the last liquid on this KBOfreezes after 4.05 Gyr, instead of after 4.67 Gyr, as in the canonicalcase. The temperatures are generally lower: at the base of the un-differentiated crust, T (Rdiff) = 84.1 K, instead of 93.3 K, and at thebase of the ice layer T (Rbase) ≈ 143 K, instead of 177 K. The rea-son is the lower heat stored in the core. At t = +4.56 Gyr, again,the temperature at r = 0 is 434 K, rather than 942 K. Less heatenergy in the core translates into less heat flux escaping the core.Our results are consistent with a flux enhanced over the steady-state radiogenic heating rate, but with ε = 0.32 rather than 0.42.That is, the heat being released from the core at t = +4.56 Gyr isreduced below the canonical case by about 25%.

The case described above shows that increasing the conductiv-ity of rock by a factor of 2 has a significant quantitative effect,accelerating the freezing of the body by about 0.6 Gyr. We havealso run a simulation with the rock conductivity reduced by afactor of 2, to 0.5 W m−1 K−1. As might be expected, the lowerconductivity leads to higher internal temperatures and delays thefreezing of the body, by about 0.3 Gyr. The thermal conductivity ofrock is not well known and variations of a factor of 2 may be con-sidered reasonable; while we favor the measured value for coldordinary chondrites, krock ≈ 1 W m−1 K−1 (Yomogida and Matsui,1983), we do note that terrestrial rocks tend to have higher con-ductivity, krock ≈ 2 W m−1 K−1. The uncertainty in the conductivityleads to an uncertainty of about 0.4 Gyr in the time at which theliquid in the body completely freezes. The evolution of a KBO withdifferent thermal conductivity is not, however, qualitatively differ-ent from the canonical case we have considered.

5.3. Size and density

Increasing the size or density of a KBO allows it to retain sub-surface liquid longer. A higher mean density implies a higher rock-to-ice fraction, which enhances the radiogenic heating at all times,and also provides a larger core that can store heat; both effectsprolong the time for which some ice can remain liquid. Likewise,a larger object will contain a greater mass of rock. By inspectionof Eqs. (18) and (20), the planet radius enters into the temper-ature most strongly as the product F rad(Rp)Rp. All other thingsbeing equal, increasing the planet radius by a factor of 2 shouldallow liquid to be retained with only half the surface flux, whichis achieved on this larger body after an additional 4 Gyr of decay.Equivalently, an increase in radius by 10 km might be expected toextend the duration for which liquid exists by about 100 Myr, notaccounting for changes in Rdiff , Rbase and ε due to heat stored inthe rocky core.

In Fig. 11, we display the results of 40 simulations we haveperformed, with ρ = 1300, 1500, 1700, 1900 and 2100 kg m−3, andRp = 300, 400, 500, 600, 700, 800, 900 and 1000 km. An ammoniaabundance of X = 1% has been assumed. Plotted are contours ofthe time at which all liquid in the KBO freezes. We note that thesesimulations were carried out at a lower numerical resolution (100zones) than our canonical simulation (300 zones). Direct compar-ison between the cases with Rp = 600 km and ρ = 1700 kg m−3

shows that the poorer numerical resolution accelerates the freez-

Fig. 11. Times after formation at which all liquid freezes within a KBO with ini-tial ammonia content X = 1%. KBOs with mean densities and radii that place themto the right of the rightmost curve potentially can retain subsurface liquid to thepresent day.

ing by 0.13 Gyr (from t = 4.67 Gyr to t = 4.54 Gyr). The freezeouttimes are probably underestimated uniformly by ∼0.1 Gyr in thisplot.

The numerical simulations largely confirm that liquid can per-sist for many Gyr, to the present day in fact, in objects the sizeand density of Charon. The boundary between objects having liq-uid at the present day and those that do not is given roughly byρRp ≈ 1.0 × 109 kg m−2. This suggests that the heat flux at thesurface (∝ frockMp/R2

p ∝ frockρRp) is the most important quan-tity controlling liquid content. The simulations demonstrate thatfor bodies with smaller density or radius, the time at which com-plete freezing occurred is roughly proportional to ρRp:

tfreeze ≈ 4.6

(ρ

1700 kg m−3

)(Rp

600 km

)Gyr. (23)

The simulations also verify that every 10 km increase (decrease)in radius can prolong (shorten) the duration for which liquid canpersist by about 100 Myr. We caution that this time of completefreezing is uncertain by many ×108 yr, as described below.

These conclusions do not change substantially when the am-monia abundance is increased to X = 5%. We have run additionalsimulations with identical radii and densities as in Fig. 11, but withX = 5%. There is a slight tendency for the more ammonia-rich bod-ies to retain liquid for a longer time, but the difference amounts toonly about 0.1 Gyr. The later freezeout time is due in part to aslight increase in Rdiff , by about 10 km, a consequence of a slightlylower thermal conductivity in the ice, now richer in ADH. An in-crease to X = 5% will increase the total amount of water–ammonialiquid there is in a KBO at any time, roughly as X ; it will alsoslightly increase the amount of water liquid. The later freezeouttime is also due in part to the greater release of latent heat offreezing of this greater mass of liquid.

Finally, we note that Charon’s position in these plots (R =603.6 km, ρ = 1710 kg m−3) implies that it is just now reachingthe point where its last liquid is about to freeze; however, it is dif-ficult to assign too much certainty to this result. Several sourcesof systematic errors (e.g., in how convection is treated) and otheruncertainties (e.g., in thermal conductivities) are all seen to shiftthe freeze-out times in Fig. 11 by several ×108 yr (even if the KBOradius is known accurately). Also, to generate results rapidly, themodels used to create Fig. 11 used only 100 radial zones; however,convergence is found to require a numerical resolution of at least300 zones, which would increase the freeze-out times by roughly0.1 Gyr. The crudeness of the calculations preclude us from saying


Fig. 12. Times after formation at which all liquid freezes within a KBO with initialammonia content X = 5%, in the cases where convection is treated as describedin the text (solid curves) and where convection is shut off (dashed curves). Thedifferences in freezeout times are minor, and are comparable to the variations infreezeout time due to uncertainties in the inputs.

when the fluid within Charon should freeze entirely, other thansometime between 4 and 5 Gyr after formation (today, plus or mi-nus 0.5 Gyr).

5.4. Treatment of convection

Our treatment of convection is crude: we do not treat convec-tion in the stagnant lid regime, and there are uncertainties in icerheology. To quantify the effects of uncertainties due to how con-vection is treated, we have also performed a series of simulationswith X = 5% (and 100 grid zones), but with convection turned off.Turning convection off effectively decreases the conductivity earlyon, allowing the heat to be stored more effectively in the core.KBOs that do not convect should therefore retain liquid at latertimes than the KBOs considered here. The results of our simula-tions with and without convection are shown in Fig. 12, and con-firm our expectations. For example, a KBO with radius 600 km anddensity 1700 kg m−3 is seen to freeze at t ≈ 4.5 Gyr in the casewith convection, and at a later time, t ≈ 4.7 Gyr, when convec-tion is shut off. In all cases, however, the differences in freeze-outtime are small, ≈0.1 Gyr, and are comparable to the variations infreeze-out time due to uncertainties in the inputs to the calcu-lation. This result indicates that the details of how convection istreated, and the exact viscosity used, are relatively unimportantfor KBOs.

6. Discussion

6.1. Summary of findings

We have presented models of how KBOs thermally evolve overtime due to the heating by long-lived radionuclides. A key fea-ture of our models is the inclusion of ammonia. Ammonia playsthree important roles, all of which aid in the retention of liquid.Obviously, ammonia is an antifreeze, depressing the melting pointof water (at least at the eutectic) by 97 K. Ammonia is the rea-son liquid can exist even when the temperatures are near 176 K.More subtly, ammonia also has the effect of lowering the ther-mal conductivity of its host ice to a small degree. In the modelspresented here, however, the major role of ammonia is its effecton the viscosity. The presence of a few percent ammonia in wa-ter ice drastically lowers the viscosity by orders of magnitude once

the first melt is produced at 176 K. The drastically lowered vis-cosity enables differentiation of rock and ice, allowing the KBO toevolve to a new density configuration. This configuration allowsliquid to exist just outside a rocky core but under a thermally insu-lating crust, and allows ammonia to be concentrated as the liquidrefreezes. As demonstrated by the simulations without ammonia(Section 5.1), wherein differentiation takes place only at the melt-ing point of water, retention of liquid on KBOs today is essentiallynot possible without at least ∼1% ammonia (by weight, relative towater).

Another key feature of our simulations is the allowance for par-tial differentiation. Because differentiation in our model requires theproduction of some water–ammonia melt, at 176 K, differentia-tion does not reach the surface of the KBO, which is always muchcolder (≈30–60 K). For the typical temperature gradients in thecrusts of KBOs (≈1 K km−1), the maximum radius of differentia-tion is likely to lie under about 100 km of rock/ice crust. The roleof this crust is significant, because a rock/ice mixture is much lessconducting than a pure ice layer (see Section 3.6). At depths of afew 100 km, the insulating properties of this crust raise the tem-perature by many tens of K typically.

The existence of a high-density (∼1700 kg m−3) rock/ice crustoverlying an ice layer of density <1000 kg m−3 is, of course, gravi-tationally unstable. To the extent that the crust can flow as a fluid,it seeks to sink under the ocean. The key to the stability of thisconfiguration is that the crust is rigid and viscous and unable toflow. Throughout the undifferentiated crust, temperatures do notever exceed 176 K (by definition), and the viscosity of ice is noless than 1017–1018 Pa s. Stokes flow of rock through this ice oc-curs at velocities great enough to be relevant (e.g., a few km perMyr) only if the rock diameters exceed at least 300 m (cf. Sec-tion 3.9), Especially in light of the fact that ice and rock are likelyto co-accrete into KBOs, it is difficult to see how kilometer-sizedregions of pure rock would arise. Likewise, following the logic dis-cussed in Section 3.9, this layer cannot initiate convection. Despitethe technically unstable nature of the 100-km-thick rock–ice crustsat the surfaces of our model KBOs, we accept that they might re-main in this state because we see no way for the layers to overturnthemselves at the cold temperatures of KBOs.

Finally, the third key feature of our simulations is that they aretime-dependent, and can account for the storage of heat at earlytimes when radiogenic heating is greater, and release of that heatlater as radiogenic heating wanes. This effect is often neglected,but we find that for Charon-sized KBOs the release of stored heatis a major factor. The surface heat flux is increased by 42% in ourcanonical model, mostly by heat stored in the core and released att > 2 Gyr, after the core has reached its maximum temperature.

The combination of these effects is that KBOs and icy satel-lites can retain liquid water to the present day if they are as smallas 600 km in radius or roughly 1.5 × 1021 kg in mass, providedthey accreted with some ammonia (X ∼ 1% or greater). The ex-act time for which liquid can be retained depends most stronglyon the treatment of convection and on the thermal conductivity ofrock, and is uncertain to within a few ×108 yr.

6.2. Implications for cryovolcanism

A significant implication of our findings on the thermal evolu-tion of KBOs is that they offer the possibility of bringing subsur-face liquid to the surfaces of KBOs. On bodies the size of Charonand larger, subsurface liquid can exist, provided it is mixed withammonia. We hypothesize that this liquid can be brought to thesurface via self-propagating cracks that open in the brittle ice ly-ing between the surface and the subsurface ocean. Crawford andStevenson (1988) developed this model in the context of Europa.Using linear elastic fracture mechanics, they showed that for cracks


of length l propagating from the bottom of the ice layer (from thetop of the ocean), the stress at the tip of the crack is

K = (π l)1/2[T − 2g(�ρ)l/π], (24)

where T is the tidal stress that they considered to initiate thecrack, and �ρ is the density of the liquid, minus the densityof the ice or medium through which it is propagating. The liq-uid in the crack transmits pressure to the tip of the crack, whichtends to widen it. If the stress exceeds the critical value Kc = 6 ×106 N m−3/2 for water ice, Crawford and Stevenson (1988) showed,the crack will continue to propagate. On Europa, the appropriatedensities are 920 kg m−3 for the ice and 1000 kg m−3 for the liq-uid (essentially just water), leading to �ρ = +80 kg m−3. PositiveK therefore requires T to be large and positive; tidal stresses arerequired for cracks to propagate in the europan ice shell.

On KBOs the situation is very different. The density of the ice is≈930 kg m−3, and the density of the liquid, being ammonia-rich, islikely to be much lower than the density of water alone. The den-sity depends on temperature: a liquid with X = 0.32 has a density880 kg m−3 at T = 288 K, rising with decreasing temperature tothe density of ice at 230 K, and about 945 kg m−3 at T = 176 K(Croft et al., 1988). The density of the liquid is therefore less thanthe density of the ice if the liquid is warmer than about 230 K,and is never more than 15 kg m−3 denser. In our Charon analog,the liquid reaches temperatures ≈230 K at about t = +2 Gyr, dur-ing which time �ρ < 0. Thus, even without tidal stresses (T = 0),K > Kc and cracks would self-propagate, provided l starts witha large enough value. For the typical parameters being discussedhere, that minimum length is l ∼ 1 km. If a crack is initiated in theice at the bottom of the ice layer, and the length is at least 1 km, itpotentially will propagate all the way through the ice layer to theundifferentiated crust. There the density contrast is even greaterand the crack potentially can reach the surface. An additional dif-ference between KBOs and Europa is that the base of the ice layerin a KBO, being at roughly 200 K, is far less ductile than the baseof an ice layer on Europa.

Initiation of a crack is likely due to stresses associated withthe expansion of ice upon freezing. It is worth noting that despitethe low density of an ammonia/water liquid, the liquid must al-ways expand as the water within it freezes into water ice (this nolonger holds once the eutectic composition is reached and the liq-uid freezes directly to ADH). A trapped pocket of liquid surroundedby ice will, upon freezing, generate enormous overpressures as itfreezes, because of the incompressibility of the liquid; as Fagents(2003) have demonstrated, these overpressures are alone sufficientto drive liquid to the surface given a conduit. We view such pock-ets of trapped liquid likely locales for cracks to be initiated, if theyexist. The volumes of liquid involved, and the needed size of theinitial crack, demand that pockets of trapped liquid would need tobe many kilometers in size. This mode of initiating cracks wouldserve to counteract the slight overdensity of the liquid that occursfor t > 2 Gyr, when the liquid is colder than 230 K.

Once a crack is initiated, it will propagate to the surface atseveral meters per second (Crawford and Stevenson, 1988). Crackscould therefore traverse the 240 km to the surface, in our canon-ical case, in about 105 s, or about 1 day. It is worth noting thatduring this time, relatively little cooling of the liquid is expected.Taking a temperature of 200 K, we estimate the thermal diffusivityof a water–ammonia mixture to be no higher than 10−6 m2 s−1,so that the timescale for losing heat out of a crack with radiusR ∼ 1 m would be ∼106 s, about 10 times longer than the timetaken to travel along the crack to the surface. While some ofthe water may freeze out of the water–ammonia liquid during itspassage (raising the ammonia content of the liquid), for reason-able parameters this effect would seem to be limited. In fact, it

seems more probable that the fluid would entrain xenoliths of na-tive “rock”, mostly water ice crystals. Such a possibility has beenconsidered in the case of cryovolcanism on Titan and Enceladus,and xenolith weight fractions of several tens of percent are con-sidered plausible, by analogy to terrestrial volcanos (Fortes, 2007;Fortes et al., 2007). This would add water ice to the surface. In anycase, it seems likely that the liquid reaching the surface would beammonia-rich, but with some water ice entrained in it. This ma-terial will then spill over the surface; or, if it contains dissolvedgases, might more violently explode upon reaching the surface.

The globally averaged rate at which liquid is brought to the sur-face is likely to be the rate at which the volume available to theliquid decreases, due to the expansion of the ice as it freezes. Asdiscussed by Cook et al. (2007), the volume of ammonia–watermixtures with X < 0.321 must increase as they freeze, by anamount comparable to 7%, with little dependence on X . In ourcanonical simulation, between about t = 1 Gyr and the presentday, about 7 × 1018 kg of liquid freezes, implying the displacementof 7% of that liquid, or 5 × 1017 kg, over 3.5 Gyr. Provided self-propagating cracks provide a conduit for this liquid to reach thesurface, it is energetically favored to do so and will resurface ourCharon analog to a depth of 120 m, at a fairly uniform (global) rateof 0.033 km Gyr−1. Freezing of ADH does not involve an expansion,and it seems unlikely that cryovolcanism could occur during thisstage, nor, of course, once all the liquid is frozen.

The rate at which liquid is predicted to be brought to the sur-face is consistent with both the presence of crystalline water iceand trace amounts of ammonia hydrates observed on Charon’ssurface. First we consider the ammonia hydrates. Since the spec-tral signatures of ammonia hydrates on the surface are destroyedby GCRs (Section 1), ammonia hydrates are destroyed to depths≈350 μm on 20 Myr timescales. For ammonia hydrates to beseen on the surface, they must be emplaced at a globally aver-aged rate of at least 2 × 10−5 km Gyr−1. As the liquid brought tothe surface is near the eutectic, the rate at which ammonia dihy-drate melt is brought to the surface is less than but comparableto 0.033 km Gyr−1. Ammonia hydrates can be emplaced on thesurface about three orders of magnitude faster than they can bedestroyed by GCRs, while cryovolcanism is occurring.

Cook et al. (2007) likewise estimated that to preserve the spec-tral signatures of crystalline water ice, new crystalline water icewould have to be emplaced at a globally averaged rate equalto the penetration depth of the near-infrared (NIR) observations,≈350 μm, divided by the amorphization timescale of 3 × 104 yr(if due to UV amorphization) to 1.5 Myr (by GCRs). Emplace-ment of crystallization water ice would outpace amorphization,then, if the globally averaged deposition rate of water exceeded0.012 km Gyr−1 (UV) or 2.3 × 10−4 km Gyr−1 (GCRs). Just beforecryovolcanism ends, the fraction of liquid brought to the surfacethat will freeze as water ice is small. For example, in our canon-ical run, 0.5 Gyr before cryovolcanism ends, X ≈ 0.28 in the liq-uid. Thus the fraction of liquid that can freeze as water ice is1 − X/0.321 = 13%. To this several tens of percent more of theliquid mass may be present as water ice crystal xenoliths. At thepresent day on Charon, the rate at which water ice is emplacedon the surface is therefore on the order of several tens of percentof the total mass of cryomagma. We therefore estimate the rate ofdeposition of water ice on the surface to be about ∼0.01 km Gyr−1

near the time cryovolcanism ends, if xenoliths are significantly en-trained in the flows. These rates are just adequate to compete withthe rate of UV amorphization estimated by Cook et al. (2007), andare more than adequate to compete with GCR amorphization. Therates of emplacement of ADH and crystalline water ice on the sur-face will scale upward if X exceeds the conservatively low valueof 1% assumed in the canonical case. Thus sufficient quantities ofcrystalline water ice and ammonia hydrates can be emplaced on


the surface to account for the spectroscopic observations, even inthe face of destruction by GCRs and UV.

The cryomagma brought to the surface is necessarily ammonia-rich. It cannot be at the eutectic composition or else it wouldnot expand upon freezing and be extruded; but it does haveX > 0.2 throughout our canonical simulation. Indeed, if the liquidbecomes too water-rich it will lack the buoyancy possibly neededto drive a self-propagating crack through the ice layer. If all ofthe water ice and all of the ammonia dihydrate on the surfaceof Charon were emplaced by cryovolcanism, we should predicta high abundance of ADH in the spectrum. Using the estimatesabove, if water ice is emplaced at ∼0.01 km Gyr−1 and ADH ata rate ∼0.03 km Gyr−1, then we would predict X ≈ 0.24, signifi-cantly higher than the value X ≈ 1% inferred from spectroscopicmeasurements (Cook et al., 2007). This discrepancy implies thatwater ice is more widespread than ADH on the surface. One ex-planation is that the spectroscopic measurements are includingsignificant amounts of ancient, amorphized water ice as crystalline.Laboratory analysis of the spectra of ices shows that ice that is amixture of 20% crystalline water ice and 80% amorphous water iceis difficult to distinguish from 100% crystalline water ice (R. Mas-trapa, 2008, personal communication,). So if cryovolcanism onlyaccounted for 20% of the ice observed on the surface, with ancient,amorphous water ice accounting for the other 80%. This would re-duce our prediction of X on the surface by a factor of 5, to X ≈ 5%.Another factor which might ameliorate the discrepancy is if waterice simply spreads out more than ammonia-rich liquid upon reach-ing the surface. If the cryovolcanism is explosive, entrained waterice xenoliths may be launched in a plume around a vent, whileADH might ooze out. Of course, a third interpretation is that crys-talline water ice is not exclusively produced by cryovolcanism (e.g.,it is due to micrometeorite impacts), in which case only ADH onthe surface would need to be explained by cryovolcanism. Cryovol-canism covering only a few percent of Charon’s surface would thenbe sufficient to produce the spectral signature of ADH on Charon.

Finally, as a consistency check, we note that the rates at whichfluid is transferred to the surface are not sufficient to violate theimplicit assumption made here, that they do not transport signif-icant heat to the surface. The fluids move from a region of about176 K to the surface, at about 60 K, a drop of ∼102 K; usingthe heat capacities of water and ammonia, we estimate the ex-cess heat stored to be ∼5 × 105 J kg−1, plus the latent heats offusion. No more than 106 J kg−1 can be stored in the liquid. Atthe globally averaged rate at which it is emplaced, the maximumheat flux associated with cryovolcanism is <10−3 mW m−2, whichis small compared to the large-scale heat flux. Likewise, the lossof liquid is necessarily limited to about 7% of the water fractionof the liquid, The largest effect this loss of liquid may have is inaccelerating the moment at which the liquid reaches the eutecticand cryovolcanism stops. We did not account for this effect in ourcode, but we estimate it would accelerate that moment by about0.07 × 3.5 = 0.2 Gyr, an amount within the range of other uncer-tainties.

7. Final speculations

Our thermal evolution models have great significance towardunderstanding the evolution of KBOs generally, but Charon in par-ticular. It may seem at first that the models cannot apply in thecontext of the hypothesized formation of the Pluto–Charon systemvia giant impact on Pluto (Canup, 2005). However, these modelstend to favor scenarios in which Charon strikes Pluto and escapesmore or less intact, but gravitationally bound. During these colli-sions, Charon is predicted to experience only minor (∼30 K) tem-perature increases; later tidal dissipation would only heat Charonan additional 20 K. This pulse of heat would then dissipate on the

thermal diffusion timescale R2pρcp/k which, for T = 100 K, is ap-

proximately 1.5 Gyr. There is hope, then, that after 4.5 Gyr thetemperature difference between the real Charon and our mod-els would approach 50 exp(−4.5/1.5) K, or only a few degrees.Our models are therefore probably not completely inapplicable.The largest discrepancy probably arises from the assumption thatCharon accreted cold, thereby underestimating the effects of heat-ing early in Charon’s evolution, and the extent of differentiation,A Charon that collided with Pluto would have a rocky core largerthan we predict. This would increase the temperatures today, dueto increased stored heat. Charon in particular, then, is somewhatfavored over an identical but isolated KBO to experience cryovol-canism, but could still be cryovolcanic if it was isolated and heatedonly by long-lived radionuclides.

Our models point out an interesting, relevant fact: Charon islikely to have partially differentiated at the time of the Pluto im-pact. The models of Canup (2005) favor the impact of an undif-ferentiated Charon in order to more likely produce a planet–Moonsystem. If the impact did indeed take place before Charon differen-tiated, our models imply that the Pluto–Charon binary was formedin the first 70 Myr of the Solar System.

As for observational evidence of cryovolcanism today on Charon,we find it intriguing to note that the albedo maps of Charon byBuie et al. (1992) include not just apparent polar caps of high-albedo (>0.7) material, presumably water ice; they also reveal aregion of high-albedo material centered on the sub-Pluto point.We estimate this region occupies about 10% of Charon’s surfacearea, in contrast to the two polar caps, which we estimate com-prise about 15% of Charon’s total surface area. If cryovolcanism isrestricted to one location on Charon, we speculate that it is thisequatorial region.

An additional complication to the above analysis is the pos-sible presence of methanol in the ice. Methanol is also an ef-fective antifreeze, and a eutectic mixture of water and methanolfreezes at about the same temperature as a water–ammonia mix-ture, 180 K (Kargel, 1994). A ternary composition of water, ammo-nia and methanol (in the ratio 47:23:30) freezes at an even lowertemperature, 153 K (Kargel, 1994). Methanol is routinely detectedin comets, at levels relative to water that range from <0.15 to 6%;in most comets the ratio is ≈2% (Bockelee-Morovan et al., 2004).Methanol is in all likelihood at least as important as ammonia inreducing the melting point of ice in KBOs, and their effects willprobably reinforce each other. This would manifest itself in KBOsin two ways. First, the lower melting point would probably trig-ger differentiation at a larger radius Rdiff , which would lead to theformation of a larger rocky core that could retain heat for longer.Second, the last liquid, being driven towards the ternary eutectic(just as ammonia was concentrated in the liquid in our models),the last freezing would not occur until temperatures of 153 Kwere reached. In our canonical model, the liquid layer cooled ata rate 12 K Gyr−1 until right before it froze (i.e., at t < 4 Gyr), thencooled at a rate 38 K Gyr−1 during the 0.5 Gyr before it froze. Atthese rates, it would take and additional 0.6–2 Gyr to cool from176 K to 153 K. Inclusion of methanol is likely to reduce the mini-mum radius of a KBO with liquid today, probably to somewhere inthe 400–500 km range. Clearly, future studies should include theeffects of methanol as well as ammonia.

Unfortunately, determination of whether other specific KBOsbeside Charon could be experiencing cryovolcanism must awaitmore precise determinations of their radii and densities. Basedon tentative observations of ammonia hydrates on their surfaces,Quaoar (Jewitt and Luu, 2004) and Orcus (Barucci et al., 2008)are obvious candidates for cryovolcanic KBOs. Both KBOs are nowknown to possess satellites (Brown and Suer, 2007), so a deter-mination of their masses should not be long coming. Unfortu-nately, radius determinations are notoriously difficult. For example,


Stansberry et al. (2008) estimate Quaoar’s radius to be 422±99 kmbased on its thermal emission, whereas Brown and Trujillo (2004),using the Hubble Space Telescope to image it, estimate its radius tobe 615 ± 95 km. In the first case, Quaoar would have frozen solid2 Gyr ago, but in the latter case, it would still be experiencing cry-ovolcanism. Likewise, Stansberry et al. (2008) estimate the radiusof Orcus to be 473 ± 36 km, based on its thermal emission, but ifthermal emission modeling underestimates the radius to the samedegree implied by the data for Quaoar, then Orcus, too, may retainsubsurface liquid and be experiencing cryovolcanism. Alternatively,these bodies are large enough that they may possibly experiencemethanol–ammonia–water cryovolcanism.

We close with the provocative thought that KBOs may be astro-biologically significant. The proximity of subsurface oceans to a hotrocky core implies the possibility of hydrothermal circulation. Thehighly ammonia-rich nature of the liquid (X > 0.2, typically), cou-pled with the extreme cold temperatures (≈200 K) certainly makethese subsurface oceans a challenge for life, but not impossible inany obvious way. Certainly, there is abundant liquid in the outerSolar System. Assuming X = 2%, each KBO the size of Charon orlarger would contain ≈2 × 1019 kg of liquid at the present time;KBOs the size of Pluto would contain ≈2 × 1020 kg of liquid. Thereare three KBOs (Triton, Pluto, Eris) with radii larger than 1200 km,and together these might contain about 6×1020 kg of liquid today.Adopting a conservative power law size distribution dN/dD ∝ D−3

(e.g., Donnison, 2006), we would estimate about 3 times as manyKBOs in the 600–1200 km size range, or about 9. The total amountof liquid on objects in this size range would be about 2 × 1020 kg.Altogether, we estimate a total mass of liquid in the Kuiper Belt, to-day, ∼1021kg, which is comparable to the mass of Earth’s oceans,1.6 × 1021 kg (Mottl et al., 2007). It is probable that there ex-ists as much liquid in the outer Solar System, now, as exists onEarth.

Acknowledgments

We thank Southwest Research Institute for hosting SJD duringthe final writing of this paper. We thank Wendy Hawley for assis-tance with running the calculations. We thank Bill McKinnon andMichael Brown for stimulating discussions.

Appendix A. Radiogenic heating rates

As described in the text, the radioactive heat released per kilo-gram of rock is only approximately known, because of uncertain-ties in the energy carried off by the neutrinos. Few descriptions ofthis calculation exist in the literature. For completeness, we pro-vide details here.

Consider the case of 40K decay. In 89.14% of decays, this iso-tope emits an electron and a neutrino, decaying to 40Ca. Only thekinetic energy of the electron, which is 0.5083 MeV, can heat thematerial. In 10.66% of decays, 40K captures an electron and emits aneutrino, decaying to an excited state of 40Ar. The 40Ar then decaysto the ground state with the emission of a 1.4609 MeV gammaray, which can heat the material. The 0.20% of remaining decaysare by electron capture directly to the ground state of 40Ar, cre-ating a neutrino but not heating the material. The average heatenergy generated per decay of 40K is therefore �E = 0.6088 MeV,which is considerably smaller than the average mass deficit of thedecays, 1.332 MeV. Put another way, the neutrinos carry off an av-erage of 0.72 MeV per 40K decay, which is consistent with theirpredicted spectrum of energies (Araki et al., 2005). We note thatthe predicted heat energy is most sensitive to the uncertainty inthe kinetic energy of the emitted β particles, and is estimated atabout 10%.

The decay chains of U and Th also involve beta decays and therewill be energy losses by neutrinos as well, although these are notas severe. The decay chains of 232Th, 238U and 235U involve theemission of 4, 6 and 4 neutrinos, as they decay to 208Pb, 206Pb and207Pb, respectively. The energies of these neutrinos do not ever ex-ceed 3.3 MeV, and their average energies are comparable to thoseemitted during beta decay of 40K, about 1 MeV. For the sake ofsimplicity, we calculate the heat energy released in each decaychain as the mass deficit between the radionuclide and their stabledaughters, minus 1 MeV per emitted neutrino. Our assumed valuesof the heat energies per decay of each isotope, and their half-lives,are given in Table 1.

To converting the energy per decay to a heating rate per kilo-gram of rock, we use the initial abundances of each isotope ascomputed by Lodders (2003), extrapolated back to the origin ofthe Solar System. Their abundances (relative to 106 Si atoms) areprovided in Table 1. As these elements should condense easily inthe Kuiper Belt they are assumed to be present in KBO rock inthe same proportions. To convert these abundances into weightsper kilogram of rock, we assume that the rock is formed by com-plete condensation of the (Lodders, 2003) solar abundances of Ca,Al, Fe, Mg, Si and S (plus O), into gehlenite, spinel, olivine, pyrox-ene, and troilite. By appropriately averaging the densities of thesecomponents we calculate an average density of this mixture is3590 kg m−3, corresponding very closely to the average density oflow-porosity chondrites (Wilkison and Robinson, 2000). Each gramof this combination of rock contains 1/151 moles of Si atoms. Wecalculate initial weight percents in rock of each radionuclide, listedin Table 1. The masses of each isotope decrease exponentially withtime according to their half lives.

Appendix B. Equation of state of water–ammonia system

Our thermal evolution model requires knowledge of e(T ), theenergy per mass needed to raise the temperature of an ammonia–water mixture from 0 K to a temperature T, given the mass fractionthat is ammonia, X . This appendix describes how we compute thisquantity.

We first require knowledge of which part of the phase diagramwe are in, which is a function of X and T . Our approximate phasediagram is broken up into 5 regimes, as depicted in Fig. 3. Regime1 refers to temperatures below the solidus, which we take to be176 K, the melting point of ADH. Actually, we model the meltingof ADH over a small range in temperatures from 174 to 178 K,so regime 1 applies to any X and T < 174 K. In this regime, thefractions of the total mass of water plus ammonia in each phaseare given by

gH2O(s) = 1 − X/Xc,

gADH(s) = X/Xc,

gH2O(l) = 0,

gNH3(l) = 0. (B.1)

Regime 2 refers to the small range of temperatures over whichADH is modeled to melt, 174 to 178 K. In this regime, the massfractions are

gH2O(s) = 1 − X/Xc,

gADH(s) = (X/Xc)(178 K − T )/(4 K),

gH2O(l) = X[1/Xc − 1](T − 174 K)/(4 K),

gNH3(l) = X(T − 174 K)/(4 K). (B.2)


Regime 3 refers to temperatures between the solidus (actually,above 178 K) and the liquidus. To simplify matters, we use a crudeapproximation to the liquidus temperature:

T liq(X) ={

273 − 95(X/Xc)2 K, Xb � X � Xc,

271 K, 0 � X < Xb,(B.3)

where Xb = 0.0466. The advantage of this approximation is that itcan be algebraically inverted to find Xliq(T ), the ammonia concen-tration of the liquid at a temperature T :

Xliq(T ) = Xc

[273 K − T

95 K

]1/2

, 178 K � T � 271 K. (B.4)

This quantity is used via the “lever rule” to find the mass fractionsof the various phases in regime 3:

gH2O(s) = 1 − X/Xliq(T ),

gADH(s) = 0,

gH2O(l) = X/Xliq(T ) − X,

gNH3(l) = X . (B.5)

Regime 4 applies only if X < Xb , so that some solid water ice re-mains unmelted even at 271 K. The melting of the remaining waterice is assumed to take place over the temperature range 271 to 275K, and the mass fractions in this regime are given by

gH2O(s) = [1 − X/Xb](275 K − T )/(4 K),

gADH(s) = 0,

gH2O(l) = [1 − X/Xb](T − 271 K)/(4 K) + X/Xb − X,

gNH3(l) = X . (B.6)

Finally, regime 5 refers to temperatures above the liquidus; that is,T > T liq(X) if X > Xb , or T > 275 K for X � Xb . In this regime, themass fractions are simply

gH2O(s) = 0,

gADH(s) = 0,

gH2O(l) = 1 − X,

gNH3(l) = X . (B.7)

The next problem is determining the energy required to raise amass of water and ammonia (with given X ) to a temperature T .We will find it convenient to define

eH2O(s)(T ) =T∫

0

cH2O(s)(T ′)dT ′ = 3.87 × 104(

T

100 K

)2

J kg−1 (B.8)

and

eADH(s)(T ) =T∫

0

cH2O(s)(T ′)dT ′ = 56.0 × 104(

T

100 K

)2

J kg−1. (B.9)

Within regime 1 (T < 174 K), the specific energy e (per mass ofwater and ammonia) required to reach a temperature T is simple:

e1(T ) = gH2O(s)

T∫0

cH2O(s)(T ′)dT ′ + gADH(s)

T∫0

cADH(s)(T ′)dT ′, (B.10)

or

e1(T ) = eH2O(s)(T ) + X [eADH(s)(T ) − eH2O(s)(T )

]. (B.11)

Xc

As the temperature is increased into regime 2 (174 K < T <

178 K), one must account for the fact that the abundances of prod-ucts change as the ADH melts, and one must include the latentheat of melting of ADH, lADH = 1.319 × 105 J kg−1. Thus,

e2(T ) = e1(174 K) +T∫

174 K

gH2O(l)(T ′)cH2O(l) dT ′

+T∫

174 K

gNH3(l)(T ′)cNH3(l) dT ′

+ gH2O(s)

T∫174 K

cH2O(s)(T ′)dT ′

+T∫

174 K

gADH(s)(T ′)cADH(s)(T ′)dT ′

+T∫

174 K

(−∂ gADH(s)

∂T ′

)lADH dT ′. (B.12)

Only because we have simplified the form of Xliq(T ) does this per-mit an analytic solution:

e2(T ) = e1(174 K) +(

1 − X

Xc

)[eH2O(s)(T ) − eH2O(s)(174 K)

]

+ X

Xc

(T − 174 K

4 K

){lADH +

(182 K − T

2 K

)cADH(s)(174 K)

+(

T − 174 K

2 K

)XccNH3(l) +

(T − 174 K

2 K

)(1 − Xc)cH2O(l)

}.

(B.13)

Within regime 3 (178 K < T < T liq(X)), one must again accountfor melting, this time of water ice, including its latent heat of melt-ing, lH2O = 3.335 × 105 J kg−1, so

e3(T ) = e2(178 K) +T∫

178 K


+T∫

178 K

gNH3(l)cNH3(l) dT ′

+T∫

178 K

gH2O(s)(T ′)cH2O(s)(T ′)dT ′

+T∫

178 K

(−∂ gH2O(s)

∂T ′

)lH2O dT ′. (B.14)

After much tedious algebra we derive

e3(T ) = e2(178 K) + [eH2O(s)(T ) − eH2O(s)(178 K)

]+ X

[cNH3(l) − cH2O(l)

](T − 178 K) + X

Xc(1 − r)

×{

lH2O

r+ 2(95 K)

[cH2O(l) − cH2O(s)

]

+ 2(95 K)2

3(273 K)cH2O(s)

(1 + r + r2)}, (B.15)

where r = [(273 K − T )/(95 K)]1/2, and cH2O(s) is evaluated at T =273 K.


If X > Xb, then further heating takes the system into regime 5(T > T liq(X)), so

e5(T ) = e4(T liq(X)

) + gH2O(l)

T∫T liq(X)

cH2O(l) dT ′

+ gNH3(l)

T∫T liq(X)

cNH3(l) dT ′, (B.16)

or

e5(T ) = e3(T liq(X)

) + [XcNH3(l) + (1 − X)cH2O(l)

][T − T liq(X)

](B.17)

Otherwise, if X < Xb, the system must first pass through regime 4(271 K < T < 275 K), in which the last of the solid water ice melts,so

e4(T ) = e3(271 K) +T∫

271 K


+ gNH3(l)

T∫271 K

cNH3(l) dT ′

+T∫

271 K

gH2O(s)(T ′)cH2O(s)(T ′)dT ′

+T∫

271 K

(−∂ gH2O(s)

∂T ′

)lH2O dT ′, (B.18)

or

e4(T ) = e3(271 K) +[

XcNH3(l) +(

X

Xb− X

)cH2O(l)

](T − 271 K)

+(

1 − X

Xb

)(T − 271 K

4 K

)

×{

lH2O + 1

2cH2O(l)(T − 271 K)

+ 1

2cH2O(s)(279 K − T )

}, (B.19)

where cH2O(s) is evaluated at 273 K.Finally, in regime 5 (T > 275 K), if X � Xb,

e5(T ) = e4(275 K) + gH2O(l)

T∫275 K

cH2O(l) dT ′

+ gNH3(l)

T∫275 K

cNH3(l) dT ′, (B.20)

or

e5(T ) = e4(275 K) + [XcNH3(l) + (1 − X)cH2O(l)

](T − 275 K). (B.21)

For a given temperature and X , these formulas allow us to de-termine what fraction of the water–ammonia mixture is in eachphase, and the total specific energy required to heat that mixtureto a temperature T .

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Thermal evolution of Kuiper belt objects, with implications for cryovolcanism