Download - Three-layered models of Ganymede and Callisto - Compositions, structures, and aspects of evolution

,!CARUS 76, 437-464 (1988)

Three-Layered Models of Ganymede and Callisto Compositions, Structures, and Aspects of Evolution

S T E V E M U E L L E R 1 AND W I L L I A M B. M c K I N N O N

Department o f Earth and Planeta~ Sciences and MeDonnell Center for the Space Sciences, Washington University, Saint Louis, Missouri 63130

Received March 27, 1987; revised March 9, 1988

Three-layered structural models are determined for Ganymede and Callisto. Each consists of a rock core, a mixed ice-rock lower mantle, and a pure ice upper mantle. This structure results from differentiation subsequent to accretional melting. Attention is given to evaluating various candidates for the rock component and three alternatives, representing various degrees of silicate hydration and oxidation, are modeled and incorporated. Structures are calculated on the basis of a 250°K isotherm, which is a reasonable approximation to the gentle adiabats expected to occur in icy satellites. Differentiation of an ice-rock satellite generally involves an increase in radius, and the three-layered approach allows this process to be examined in some detail. It is determined that satellite expansion is most significant early in the process and much less so as differentiation proceeds to completion. If tectonics are due to global expansion, distinguishing on this basis between a completely differentiated satellite and one that is only partially differentiated is difficult. The postaccretionai global expansion of Ganymede, which may have left a tectonic record, was probably limited to 1% in radius, in agreement with observed limits. Useful quantities such as silicate mass and volume fraction, uncompressed density, J2, C22, binding energy, and surface heat flow are also determined. Nonhydrostatic contributions to 3"2 and Cz2 are estimated and shown to be nonnegligible. Encounters with Jupiter-orbiting spacecraft are unlikely to determine Callisto's degree of central condensation. We conclude by calculating the relative likelihood of postaccretional melting caused by radiogenic heating. Three-layered satellites have generally hotter interiors, because additional thermal boundary layers divide the separately convecting upper and lower mantles, inhibiting heat transport. Ganymedes and Callistos that are less than about i differentiated (by mass) should experience a second episode of melting, as these boundary layers are either above the level of the water-ice minimum-melting temperature or intersect the melting curve at deeper levels. Runaway differentiation to at least a depth corresponding to a pressure in the ice V stability field is likely. The main point here is that if satellite tectonics are tied to differentiation by melting or its aftermath (as in the instability following ocean closure of Kirk and Stevenson), moderate or small amounts of accretional differentiation are unlikely to explain an absence of tectonics (as on Callisto), because extensive differentiation ultimately occurs. © 1988 Academic Press, Inc.

INTRODUCTION

A major obs tac le to u n d e r s t a n d i n g the Gal i l ean satell i te sys t em is the absence of an agreed u p o n m e c h a n i s m to accoun t for

Now at the Department of Geological Sciences, Southern Methodist University, Dallas, TX 75275.

437

the geological d i f ferences b e t w e e n Gany- mede and Callisto. The younge r terra ins on G a n y m e d e are resur faced uni ts formed of re la t ively pure ice and p robab ly emplaced dur ing episodes of l i thospher ic ex tens ion . There is, however , little c onse nsus on the cause and na tu re of the ex tens ion or the reason that s imilar te r ra ins are not ob-

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438 MUELLER AND MCKINNON

served on Callisto (for recent reviews see McKinnon and Parmentier 1986, Schubert et al. 1986, Squyres and Croft 1986).

Ganymede and Callisto are approximately half rock and half ice by mass, but the present-day distribution of these is uncertain. It is generally considered that the satellites formed by homogeneous accretion of planetesimals that were themselves intimate ice-rock mixtures (e.g., Steven- son et al. 1986). If the ice component does not melt during accretion, an initially homogeneous ice-rock satellite is produced. If some melting of the ice component ac- companies the later stages of accretion, the liberated silicate inclusions sink to the bottom of an outer liquid region. The resulting intermediate rock layer is considerably denser than the underlying ice-rock core and is unstable to overturn on a geologic time scale (Kirk and Stevenson 1987). The stable configuration that is expected to eventually evolve is what we formally des- ignate a "three-layered ice-rock satellite," consisting of a silicate core, a lower mantle region of mixed rock and ice, and an outer shell of pure ice formed by the refreezing of the liquid-water region (Mueller and Mc- Kinnon 1984). The purpose of this work is to characterize the compositions, structures, and aspects of the evolution of such satellites, with an eye toward the Gany- mede/Callisto dichotomy.

Interior models of satellites (and planets) are in themselves prosaic accomplish- ments. Their true value lies in applications and the inferences that can be drawn, however generic. They are an essential founda- tion from which other results follow. The only precise interior models of the icy Gall- lean satellites currently available are those of Lupo and Lewis (1979) and Lupo (1982). These authors considered their principal sources of error to be the composition and density of the rock component, which they modeled as ordinary chondrite. In contrast, we regard CI carbonaceous chondrite, a rather "wet" mineral assemblage, as the best analog to primordial planetary rock in

the Jupiter region (cf. Lewis and Prinn 1984). Anhydrous ordinary chondrite is an extreme choice, and is best viewed as an end-member of a suite of possible mineralogies. We note the degree of silicate hydration is far from academic; it is critical in evaluating the probability of nonaccretional melting (Friedson and Stevenson 1983). Our structural modeling incorporates the following improvements: (1) alternative silicate or, better, rock mineralogies representing various degrees of hydration (and oxidation); and (2) structural determinations for models that are partially differentiated (i.e., three-layered satellites). Geo- physically useful quantities such as silicate mass and volume fraction, uncompressed density, gravitational J2 (and C22), binding energy, and surface heat flow are calculated based on the models.

A three-layered ice-rock satellite should have two convecting regions (ignoring the silicate core): the ice-rock lower mantle and the pure ice upper mantle. Density con- trasts between these regions act as barriers to convection; these inhibit the efficiency of heat transport and result in a hotter satellite interior. An important conclusion of this work is that melting due to radiogenic heating alone is likely in this situation. Ice-rock satellites subjected to a small-to-moderate degree of melting of the ice component during accretion (-2-35% by mass) are potentially unstable to a second episode of melting (and differentiation) Ibllowing postaccretional refi-eezing. In contrast, large degrees of melting during accretion pro- duce satellites that convectively "run hotter" only at depths where the melting temperature of ice is relatively high. In this case, a second episode of melting is not expected, and these satellites do not possess what we term an "accretional trigger." (If overturn of the intermediate rock layer originally created during accretion is sufficiently delayed, melting and differentiation may begin first, triggering the overturn [Kirk and Stevenson 1987]. In this situation, our three-layer models are proxies for

THREE-LAYER GANYMEDES AND CALLISTOS 439

structures that we will argue are more susceptible to melting and differentiation.)

Schubert et al. (1981) estimated that a large, homogeneous ice-rock satellite should not melt due to radiogenic heating alone. Friedson and Stevenson (1983) concluded that, were the silicate volume fraction large enough, the suspended silicate particles would increase the bulk viscosity so as to significantly inhibit heat transport and melting might occur, and possibly only in Ganymede. Friedson and Stevenson also introduced the concept of "runaway differentiation." If the interior of Ganymede or Callisto were convecting, the gravitational energy released by melting and differentiation (plus continued radioactive heating) is capable of melting more ice and driving further differentiation. It is possible that, once melting is initiated, differentiation necessarily proceeds to completion. Because ice is more compressible than rock, differentiation results in satellite expansion as more ice is displaced upward; it had been argued that formation of tectonic terrains on Gan- ymede is associated with expansion (Squyres 1980). Both Schubert et al.

(1981) and Friedson and Stevenson (1983) postulated that the different surface char- acteristics of Ganymede and Callisto are explainable on the premise that Ganymede is largely differentiated whereas Callisto has remained essentially undifferentiated (see also Schubert et al. 1986).

Lunine and Stevenson (1982) studied the formation of the Galilean satellites in the presence of the proto-Jovian nebula. Their initial calculations suggested that both Ganymede and Cailisto experience large degrees of accretional melting and differentiation (>50% by mass), but modifying as- sumptions produced a model in which Cal- listo is subjected to a smaller amount of melting (~23% by mass). Kirk and Steven- son (1987) modeled the freezing of a deeply accretionally melted Ganymede, and found that the final freezing of the "ocean" at the water-ice minimum-melting temperature results in a vigorous convective overturn as

the adiabats readjust. They argue that grooved and smooth terrains form as a result, and their preferred hypothesis is that Callisto was not deeply melted enough to suffer the consequences. Thus, Lunine and Stevenson (1982) and Kirk and Stevenson (1987) concluded that differences in the amount of accretional melting may account for the divergent evolution of Ganymede and Callisto.

We show there are complications to these scenarios. Primary among them is the second episode of melting followed by (possibly runaway) differentiation described above. Even if differentiation does not penetrate much below the ice III to ice V transition isobar, because of the endothermic nature of the ice II-to-V transition and the positive Clayperon slope of the ice V melting curve, its occurrence suggests that both Ganymede and Callisto are at least ~ differentiated. One could argue Ganymede is nearly completely differentiated and Cal- listo barely ½ so, and seek an explanation for the dichotomy on this basis. We show, however, that most of the expansion ac- companying differentiation happens in the early stages, with 75-90% occurring during the first half. Thus even if Callisto is only differentiated, it should have experienced a major amount of expansion, and might be expected to exhibit the same tectonic and volcanic manifestations as Ganymede. The lack of grooved and smooth terrain on Cal- listo indicates that either differentiation alone is not responsible for the formation of these features (not unreasonable), or that Callisto differentiated earlier than Gany- mede (if at all), prior to the formation of its observed crater population. On the other hand, if grooved and smooth terrain are caused by the "heat pulse" phenomenon of Kirk and Stevenson (1987), then an accretionally triggered second differentiation for Callisto is probably not reconcilable with Callisto's geologic record. (It is reconcilable only if the second differentiation is slow, orders of magnitude slower than the runaway time scale of Friedson and Steven-


son [1983]). To avoid the accretional trigger, a satellite must be deeply accretionally melted from the beginning, or hardly melted at all. The latter remains difficult to justify a priori.

ROCK MINERALOGY

Geologically, the interior of an icy satellite is not closely analogous to any terrestrial environment. This and the fact that most reasonable alternatives for '°Gany- mede-rock" consist of unfamiliar mineral assemblages account for the lack of an accurate model for the rock component . While admitting that carbonaceous chondrite might be more appropriate, Lupo and Lewis (1979) chose ordinary chondrite to represent the rock. Two reasons were cited: first, an equation of state was available for ordinary but not carbonaceous chondrite; and second, their satellite models were completely differentiated, and heating subsequent to differentiation could dehydrate the silicate core.

We take a more general approach, draw- ing on various observed and theoretical mineral suites to model rock types representative of varying degrees of hydration and, simultaneously, oxidation (carbon- ation, as noted below, is not explicitly considered). This is done to account for both the plausible variation in starting material condensing from the proto-Jovian nebula and the possible petrological evolution of the rock condensate after incorporation into a satellite, particularly into a satellite core. We then examine the stability of each rock type in the lower mantles and cores of three-layer ice- rock satellites, mostly from the point of view of pressure-induced dehydration. This allows the selection of models that are geophysically reasonable.

C a r b o n a c e o u s Chondr i te

Our work is concerned with satellites that are not completely differentiated, so it is necessary to consider wet-rock alternatives. The obvious candidate is CI carbonaceous chondrite. CI chondrites represent

virtually unfractionated samples of "non- volatile" Solar System material; CI elemental abundances compare well with solar abundances (certainly better than any other meteorite class), and there are no significant irregularities in the abundance curve (Anders and Ebihara 1982, Ebihara et al. 1982). There is also no petrological evidence for major nonisochemical alteration.

Evidence for mineralogical changes is substantial, however. Although the formation of carbonaceous chondrites is far from understood, most CI minerals did not origi- nate as nebular condensates; rather, they are the products of (isochemical) aqueous alteration on parent bodies (e.g., DeFresne and Anders 1962, Kerridge and Bunch 1979, McSween 1979, Bunch and Chang 1980, Clayton and Mayeda 1984). In the case of the Orgueil (CI) meteorite, this alteration may have occurred episodically, with the last episode as recent as 10 myr ago (possibly coinciding with a low-energy impact event responsible for the breakup of its parent body) (MacDougall et al. 1984). In addition, as much as one-half of the water reported in CIs may be terrestrial in origin (Lewis and Prinn 1984). This may be simply the result of exchange, affecting only the isotopic signature, or it may represent additional water. J. S. Lewis (personal communication, 1987) favors a CI origin in which something akin to ordinary chondrite or amorphous interstellar grains, mixed in ice, causes radiolysis of the ice (or water) via U, Th, and K decay, releasing free oxygen to then alter the grains.

CI chondrite is a good analog to the rock in the interior of an icy satellite despite these complications. Processes indigenous to CI parent bodies could occur on (1) planetesimals formed in the proto-Jovian nebula, (2) planetesimals formed in the solar nebula that enter the proto-Jovian nebula, and (3) larger, planetary-sized bodies. If aqueous alteration on meteorite parent bodies results from mild brecciation and heating, the same is expected during the accretion of outer planet satellites. If CIs ac-


quire additional water upon entering the terrestrial environment, they might do the same in an H20-rich icy satellite. CI containing terrestrial water may thus be a better representation of "Ganymede-rock" than unaltered CI precursor. Radiolysis and low-temperature oxidation could occur in all the above environments. Ultimately, though, CI chondrites exist (and are estimated to be abundant in the asteroid belt and possibly beyond), and this is not necessarily true for other, theoretical rock types.

There is no whole-rock equation of state for CI carbonaceous chondrite, but densities, compressibilities, and volume thermal expansion coefficients are available for most major CI minerals (see Table I). Thus an approximate equation of state can be

constructed for the entire assemblage given mass fraction estimates. The few CI mineral abundance determinations differ significantly (see, e.g., Nagy 1975), but the same major minerals are generally reported, with about 100 mg g-1 listed as "residue." This "residue" consists chiefly of organic com- pounds (several 10's of mg g-l), plus smaller amounts of free sulfur and other poorly characterized materials. We opt to constrain the mineral abundances by tying them to CI elemental abundances (derived from Dodd 1981), which are reasonably consistent. Our "CI rock" is assumed to consist of the five major CI minerals: serpentine, epsomite, magnetite, "troilite," and gypsum. We are thus able to constrain four elements, Mg, Fe, S, and Ca, to their

T A B L E I

MINERAL COMPONENTS OF THE MODEL ROCK TYPES (IN m g g ~), WITH ASSOCIATED S T P

DENSITIES, VOLUME THERMAL EXPANSION COEFFICIENTS, AND COMPRESSIBILITIES

CI r o c k P / F r o c k P T C r o c k P0" a " /3" (g c m 3) (10 5 K l) (10 11 p a - t )

A n o r t h i t e - - 16 - - 2 .760 1.42 1.15

D i o p s i d e - - - - 78.1 3 .277 2 .4 1.07

E n s t a t i t e - - - - 208 .8 3 .190 3 .00 1.01 E p s o m i t e 128.1 - - - - 1.677 7.25 b 2 .29

F e l d s p a U - - 31 - - 2 .623 4.791 2 .05

F o r s t e r i t e - - - - 225 .7 3 .213 2 .4 0 .79

G y p s u m 52 .0 - - - - 2 .305 7.25 2.5

J a d e i t e - - - - 90 .0 3.35 2 .0 0 .75 M a g n e t i t e 75.1 111 165.2 5 .20 4 .394 0 .859

Mi l le r i t e - - 26 - - 5 .374 6 .6 a 0 .730 e

S e r p e n t i n e 692 .4 457 - - (2.918) f 7.0~ 1.8 h

T r e m o l i t e - - 134 - - 2 .977 3.131 1.3 Tro i l i t e 52 .4 225 232.2 4.83 6 .6 J 0 .730

a D a t a f r o m B i r c h (1966), R o b i e et al. (1978), S k i n n e r (1966), C o a t e s a n d A s l a m (1968), E v a n s

(1979), H a z e n a n d F i n g e r (1978), MaD et al. (1981), S u e n o et al. (1973). b G y p s u m v a l u e .

~ P r i n n a n d F e g l e y (1981) r e f e r to th is as f e l d s p a r p lus n e p h e l i n e ; w e a s s u m e d it to be o r t h o c l a s e .

a P y r r h o t i t e v a l u e . e Tro i l i t e v a l u e .

I D e n s i t y f o r CI r o c k s e r p e n t i n e ; a d j u s t e d v a l u e f o r P / F r o c k is 2 .67 g c m 3 ( M g : F e =

2 .74 : 0 .26) .

E s t i m a t e . h C h l o r i t e v a l u e .

i A c t i n o l i t e v a l u e .


Si-normalized CI abundances. Minor carbonate is not modeled, but all carbonate cations are accounted for. The organic portion of the residue possesses a low density similar to that of ice, and rather than model its behavior, we simply treat it as part of the ice component . This should not be a serious source of error.

The exact identity of the CI chondrite layered silicate has been hard to determine because of its extremely fine-grained habit; it has been reported as both serpentine and chlorite (see Nagy 1975, Dodd 1981, and for a recent analysis, McSween 1987). Our primary concern is its thermomechanical properties, however, so because those of serpentine and chlorite are not dissimilar, we assume the layered silicate is serpentine, Mg~Fe3 ~Si205(OH)4. Chlorite is the host mineral for most of the aluminum in CI chondrites (<1% by mass), but since this minor element is not constrained in our model, serpentine is the logical choice. Kerridge (1976) determined x to be 1.9 for the Orgueil phyllosilicate, which we adopt.

"Troi l i te" deserves some comment. Its abundance is overest imated if the sulfur in the residue is not accounted for. Therefore, we constrain the actual troilite content instead of the S/Si ratio. We chose as a mean abundance one representative of the larger model values reported in Nagy (1975), - 5 2 mg g ~. As such, troilite accounts for only 26% of the CI sulfur, with 37% residing in the residue and the rest in sulfates. The troilite abundance or, more generally, sulfide abundance, as Kerridge et al. (1979) show that the phase is dominantly the Fe- deficient variety, pyrrhotite, may well be smaller. The model CI equation-of-state is rather insensitive to this, however, as changes in the amount of sulfide cause com- pensating changes in the amount of magnetite.

The resulting mineralogical abundances are given in Table I. The serpentine density is approximated by using the molar volume of antigorite, the Mg end-member. The other model CI minerals contain no signifi-

cant solid solution, and their densities are obtained from Robie et al. (1978). CI densities are believed to range from 2.2 to 2.3 g cm 3 although measurements are quite scarce (Mason 1962, Wasson 1974). Our model CI rock has an STP density of 2.77 g cm -3. Part of the difference in these values is due to the exclusion of the residue; up to half of the discrepancy is eliminated if the residue has a density of 1.5 g cm 3. Al- though the fraction of residue could be larger or its density lower, there is no doubt some contribution by porosity. No porosity measurements are available for CI chondrites, but a value of 24% was obtained for a CM carbonaceous chondrite (Wasson 1974). These chondrite types are similar in many respects, and a porosity of this size would, by itself, more than reconcile the two density values. With the residue, a perhaps more realistic porosity for a thor- oughly aqueously altered rock of ~<10% would suffice. Nevertheless, porosity would not be expected to persist at depth in a large icy satellite, and the model density is more realistic for the rock fraction.

Prinn-Fegley Rock

The second alternative for Ganymede- rock is an assemblage predicted to have condensed from the proto-Jovian nebula (Prinn and Fegley 1981). This "Pr inn-Feg- l ey ," or P/F, rock possesses a water content intermediate to that of CI carbonaceous chondrite and completely anhydrous rock (CI contains adsorbed and bound water, P/F rock only the latter). P/F rock may be a more appropriate model for primordial Ganymede-rock than CI rock, i f nebular condensation arguments are strictly applied. (Hydration reactions do have a better chance of going to completion in the proto- Jovian nebula, where pressures are at least 104 times greater at a given temperature than in the solar nebula [Prinn and Fegley 1981].) P/F rock is derived from solar composition gas, though, so it is plausible that partial dehydrat ion of CI chondrite (as well as reduction of sulfates, decarbonation,


etc.) in a silicate core results in something mineralogically similar to P/F rock. It also frees us from the necessity of calculating the assemblage that would result from such a partial dehydration, reduction, etc.

The mineral percentages in Prinn and Fegley (1981) were revised to account for the approximately 8% decrease in the estimated solar abundance of iron (cf. Cameron 1973, 1982), and are given in Ta- ble I.

Pretremoli te Condensate

The third alternative for Ganymede-rock is an anhydrous mineral assemblage. We prefer not to use the ordinary chondrite described by Lupo and Lewis (1979), because its calculated elemental abundances are nonsolar. P/F rock was derived from equilibrium condensation calculations, and we appeal to the condensation sequence for an anhydrous mineralogy. The first hydroxyl- bearing mineral to appear is tremolite, so we restrict ourselves to the pretremolite condensate, or PTC. Conditions in the outer proto-Jovian nebula proceeded to at least water-ice condensation, so it is unlikely that anhydrous silicates were the sole rocky constituents of Ganymede and Cal- listo (it would be difficult to prevent reactions between anhydrous silicates and nebular water vapor or icy condensate). On the other hand, PTC rock may result from the complete dehydration of CI or P/F rock in a silicate core. We thus use PTC rock solely as a core component .

PTC rock consists of six minerals, so it is possible to constrain five elements, Mg, Fe, S, Ca, and AI, to their Si-normalized CI abundances. We modify PTC rock to include magnetite rather than metallic iron. This is done because metallic iron is not found in CI or P/F rock, and it is not reasonable that heating and dehydration in a silicate core will be accompanied by the reduction of magnetite due to the large oxygen fugacity expected (although the presence of organic matter in CI chondrites makes this less certain). Pretremolite con-

densate contains the major minerals found in the ordinary chondrite of Lupo and Lewis (1979) (except for magnetite), but in different proportions. The quantity of magnetite is probably overestimated because Fe substitution in olivine and pyroxene is neglected. This neglect also decreases the divalent cation to Si ratio and stoichiometrically favors the production of pyroxene at the expense of olivine.

Mineral Assemblage Stabilities

Given the three alternative mineral assemblages, which may be loosely described as wet, damp, and dry Ganymede-rock, the next step is to ask whether they are stable in the pressure-temperature environment of a large icy satellite and whether mineralogical changes are indicated.

Some stability considerations are easily accommodated. Albite, a component of pretremolite condensate, cannot exist sta- bly in the high-pressure environment of a silicate core of Ganymede or Callisto. It should break down to jadeite plus quartz, and the quartz will react with the forsterite to form enstatite. Thus, under core conditions, PTC rock takes the form given in Ta- ble I.

Other stability considerations, notably dehydration reactions, are of such importance that in their presence the applicability of the hydrated rock types is called into question. Although we do not attempt to fully characterize the mineralogical transformations the hydrated assemblages may undergo, we consider dehydration to be the most important indicator of whether the full suite of dehydration, reduction, and solid- solid reactions that could plausibly trans- form CI rock to P/F rock and P/F rock to PTC rock might occur. Coupled with some temperature considerations, constraints are placed on the use of the hydrated rock types in the models.

The satellite cores were chosen to be relatively cool, as discussed in the next section. We thus focus on possible pressure- induced dehydration reactions. The higher


polymorphs of ice are denser than water, and dehydrat ion reactions may proceed at lower pressures in an ice-rich environment (vs a water-rich one), notwithstanding the stabilizing effect of lower temperatures. We calculate the dehydrat ion pressures of key minerals as a function of temperature. If the Gibbs free energy of a mineral is known at a given state, its variation as a function of pressure can be calculated from (OG/OP)T =

V(P), where V is the molar volume, if the compressibility is also known. As long as the free energy of the hydrous phase is less than that of the dehydrated assemblage, the hydrous phase is stable.

For simplicity, we examine reactions in- volving complete dehydrat ion only (see, e.g., Turner 1981, p. 169):

(Mg,Fe)3Si2Os(OH)4 ~- (Mg,Fe)2SiO4 (serpentine) (olivine)

+ (Mg,Fe)SiO3 + 2H20 (la) (opx)

CazMgsSi8Ozz(OH)2 ~ 2CaMgSi206 (tremolite) (cpx)

+ 3MgSiO3 + SiO2 + H20 (lb) (opx)

MgSO4 • 7H20 ~- MgSO4 + 7H20 (epsomite)

(lc)

CaSO4 • 2H20 ~ CaSO4 + 2H20 (gypsum)

(ld)

As discussed in the next section, the interior models were also chosen to be isothermal. Stability calculations were carried out at 250 and 298°K; magnesium end-members were assumed for Eq. (la). Thermody- namic data were taken from Robie et al.

(1978), except for Mg3SizOs(OH)4 (Helge- son et al. 1978), MgSO4 (Dickerson 1969), and H20 (Eisenberg and Kauzmann 1969). The free energies in these references were corrected from STP to 250°K and 105 Pa by use of the constant pressure relation (OG/ OT)e = - S , where S is entropy. The equation-of-state for water ice is from Lupo and Lewis (1979).

The calculated dehydrat ion pressures at 250°K are given in Table II; the 298°K calculations do not yield significantly different results. The highest pressures in our structural models occur in the cores of completely differentiated satellites and are about 8 GPa; we therefore assume tremolite and serpentine do not experience pressure- induced dehydrat ion there. We interpret this to mean that P / F rock is a stable lower mant le cons t i tuent and a stable core con-

s t i tuent under the cooler conditions that will prevail immediately after core formation. (We recognize that tremolite breaks down to diopside and talc at low pressures, but as no water is released, we do not consider it further.) Whether rising temperatures force further dehydrat ion (and other) reactions is a question we prefer to leave open for two reasons. First, the temperatures may be self-regulated by solid-state convect ion (though perhaps only in the outer cores) to remain below the up to -900-1200°K necessary to drive these reactions at core pressures. Second, even if the core fully dehydrates, then, simply, PTC rock is the appropriate choice for the core material. We consider PTC rock to be stable throughout the full range of satellite interior conditions modeled here.

The maximum pressure experienced in the i ce - rock lower mantles are about 4 GPa, and this value is only attained in ho- mogenous satellites. We therefore assume epsomite can exist in the lower mantles but not in the cores. The pressures in the outer cores of completely differentiated satellites are about 2 GPa. Although epsomite is technically stable there if cool, we assume

TABLE 11

COMPLETE DEHYDRATION PRESSURES AT 250°K

(GPa)

Serpentine >8.0 Tremolite 7.6 Epsomite 4.2


for ease of calculation that the cores are epsomite free (one could argue that they become sufficiently warm that this is certainly so).

The dehydrat ion pressure of gypsum cannot be stated for certain, because the difference between the STP free energy of gypsum and that of the corresponding dehydrated assemblage is small compared to the errors in the published free energy determinations. Gypsum accounts for 52 mg g-l of CI rock, and the density difference between gypsum and anhydrite is about 28%, so dehydrat ion results in a whole-rock density change of 1.5% at STP. Because gypsum is more compressible than anhydrite, the density change is much less at depth where the reaction is most likely to occur. For this reason the uncertainty in the dehydrat ion pressure introduces an in- significant error, and we simply assume that the reaction does not occur in the lower mantles. Gypsum is certainly unstable as a core mineral, though.

On this basis, we a c c e p t CI rock as a p laus ib le lower m a n t l e cons t i tuen t , but re- j e c t it as a core c o m p o n e n t in favor of P/F and PTC rock.

S u m m a r y

Three alternatives for Ganymede-rock have been presented: CI rock (high water content, 157 mg g-Z), P/F rock (intermediate water content , 61 mg g-l) , and PTC rock (anhydrous). Whole-rock equations- of-state determined from the component mineral propert ies (Table I) are given in Ta- ble III according to the form

p(T, P) = p0[1 - c~(T - To) + /3(P - P0)], (2)

where To and P0 refer to STP. This simple, linear equation-of-state is adequate for our purposes, and is accurate to within a fraction of a percent for the range of pressures and temperatures we explore.

Because P/F and PTC rock are based on solar abundances, and CI chondrite and solar abundances correlate so well (if not

"FABLE 11I

EQUA'IION-OF-STATE PARAMETERS FOR THE

MODEL ROCK TYPES

P0 ~ /3 ( g c m 3) (10 5o K i) (10 l i p a i)

Cl rock 2.766 6.953 1.878 P/F rock 3.262 5.997 1.479 PTC rock 3.756 3.507 0.862

agree by definition), it may be possible to create all three model rock types from one another by the addition or removal of water and oxygen. Condensation considerations exclude PTC rock from the i ce - rock lower mantles, and stability considerations exclude CI rock from the cores. We construct internal models for the four remaining com- positional alternatives: (1) P/F rock in the core and lower mantle (PF.PF); (2) P/F rock in the core and CI rock in the lower mantle (PF.CI); (3) PTC rock in the core and CI rock in the lower mantle (PTC.CI); and (4) PTC rock in the core P/F rock in the lower mantle (PTC.PF).

I S O T H E R M A L S T R U C T U R E S

A vigorously convecting i ce - rock satellite should possess adiabatic thermal gradients throughout, with conductive gradients restricted to thin boundary layers. In the case of ice the adiabat can be realistically approximated with an isotherm; for the relatively high heat flow associated with 4- gyr-old chondritic heating rates, 250°K is reasonable (e.g., Zuber and Parmentier 1984), which we adopt. Isothermal models have the advantage of eliminating the effects of thermal variation and isolating the effects of structural and composit ional variation. (The effects of thermal variation through time are discussed in Zuber and Parmentier [1984].)

Figures 1 and 2 depict the present-day allowable hydrostat ic structures of Gany- mede and Callisto, based on the mass and radius values in Morrison (1982) and our model rock mineralogies (the small differ-


C O R E = P / F & M A N T L E = CI C O R E = P / F & M A N T L E = P / F

~3

t~

0.555 0.580 0.605 0.830 0.655 0,490 0.515 0.540 0.565 0.590

C O R E -- PTC & M A N T L E -- Cl C O R E = PTC & M A N T L E = P / F

ROCK

MIX

ICE

g

0.575 0.600 0.625 0.650 0.480 0.505 0.530 0.555

CI S I L I C A T E F R A C T I O N PF S I L I C A T E F R A C T I O N

ROCK

MIX

[ ~ ICE

0 . 5 8 0

FIG. 1. Structural diagrams for Ganymede as a function of global silicate mass fraction. All models are isothermal at 250°K. Global silicate fraction is calculated as if the rock mineralogy is that of the lower mantle; the lower mantle and global average so determined are identical.

C O R E = P / F & M A N T L E = CI C O R E = P / F & M A N T L E = P / F

~0

t~

0.530 0.555 0.580 0.605 0.630

C O R E = PTC & M A N T L E = CI

0,465 0.490 0.515 0.540 0.565

C O R E = PTC & M A N T L E = P / F

ROCK

MIX

r--'l ,CE

0.530 0.558 0.580 0.605 0.630 0,460 0.485 0.510 0.535 0.560

CI S I L I C A T E F R A C T I O N PF S I L I C A T E F R A C T I O N

m ROCK

MIX

r ~ ICE

FIG. 2. Structural diagrams for Callisto as a function of global silicate mass fraction. The remarks of Fig. I apply.

T H R E E - L A Y E R G A N Y M E D E S A N D C A L L I S T O S 447

ences be tween Morrison [1982] and the up- dated values of Campbell and Synnott [1985] are easily subsumed into the uncertainty in the tempera ture profile). We iter- ate the equations of hydrostat ic equilibrium for a given mineralogy and silicate mass fraction, the latter falling within certain bounds. For each combinat ion there is only one structure, meaning one value for the fraction of rock or silicates in the core (hereon referred to as the core fraction), that can account for the observed mass and radius. The core radius and the outer radius of the i c e - r o c k lower mantle are plotted as a function of silicate mass fraction in Figs. 1 and 2. The tempera tures of the cores in the models are also fixed at 250°K; raising the core t empera ture to 1000°K increases the modeled internal radii by 0.3 to 0.6%. Fig- ure 3 shows the allowable combinat ions of silicate and core fraction for each satellite.

For models with core silicates that are dehydra ted with respect to silicates in the i c e - r o c k lower mantle, the overall silicate fraction is depicted in Figs. 1 and 2 as if " w e t . " In reality and in the models presented here, core dehydrat ion involves water leaving the rock componen t to join the ice component . Thus the " t r u e " silicate fraction does not remain constant during the course of evolution, but we choose to por t ray results in a manner uncomplicated by this apparent change. We count the original water of hydrat ion contained in the core silicates as part of the rock fraction instead of part of the ice fraction. The mass fraction of ice in the diagrams is thus reduced f rom its actual value. Never theless , the wet silicate approach is convenient if undifferentiated i c e - r o c k satellites are en- visioned as possessing hydrous mineral assemblages, with dehydrat ion taking place only during (or following) subsequent differentiation. The " w e t " silicate fraction is then equivalent to the mass fraction of the lower mantle mixture and the primordial silicate fraction.

Limiting values for the silicate mass fractions of both satellites are given in Table IV

GANYMEDE 0 . 6 7 5

0 . 6 2 5

I , I . 0 . 5 7 5

I I J I - -

U m , . ,J

0 . 5 2 5

' ' ' ' I i , , , I ' ' ' ' I , , , , .

. . . . . . . . . .

j . j , ~ " ~ P F . C I

~ . . . . . . P F . P F

= ' . . . . P T C . C I

B - - B P T C . P F

0 . 4 7 5 , , , I , , , , I , , J , I L i J

0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0

CALLISTO 0 . 6 3

0 . 5 9

0 . 5 5

, , , , i . . . . [ , , , , i , , , ,

. . . - '

o.51 ",~:;'~ , #

/ /

0 . 4 7 . . . . i , , i i I I i i i I i i I i

0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0

C O R E F R A C T I O N

FIG. 3. Global silicate mass fraction of Ganymede and Callisto as a function of the fraction of silicates in the core. Silicate mass fractions for the various models (Figs. 1 and 2) are calculated as if the rock mineralogy is that of the lower mantle.

for the various models . Silicate mass fractions are lower for the undifferentiated models, of course, due to the relatively high compressibi l i ty of ice. Perhaps more interesting are the volume fractions implied. These can be calculated f rom

Vs = (ms/ps)[(ms/pO + (mi/pi)] -I, (3)

where V~, m~, and ps are the volume fraction, mass fraction, and density of the rock component , and mi and p~ are the mass fraction and density of the ice component . At zero pressure , the volume fraction of rock


TABLE IV

LIMITS ON ROCK MASS FRACTION

Ganymede Callisto

Undifferentiated models CI rock 0.561 0.536 P/F rock 0.491 0.474

Differentiated models P/F rock core 0.586 0.559

Expressed as CI (0.652) (0.622) PTC rock core 0.541 0.516

Expressed as P/F (0.576) (0.549) Expressed as CI (0.641) (0.612)

in Ganymede ranges f rom a minimum of 0.21 for an undifferentiated PF-rock model to a "p r imord ia l " max imum of 0.38 for a completely differentiated PF-rock core model, i f the primordial silicate in the latter case was CI rock. Volume fractions for Cal- listo are similar but slightly smaller. Impli- cations of these values are taken up in the discussion.

The silicate mass fractions can also be used to calculate uncompressed densities, (P), f rom

(p) = [(mJps) + (mi/pi)] J. (4)

The range for Ganymede , corresponding to the volume fraction limits above, is 1.42 to 1.63 g cm -3. The minimum uncompressed density for Callisto is 1.39 g c m 3, corresponding to an undifferentiated PF-rock model. This low value approaches those of the intermediate-sized satellites of Saturn.

Derived Quantities

The Galileo Orbiter and future spacecraft may determine the actual internal structures of Ganymede and Callisto by measuring the strength of the second-degree harmonics of their gravitational potentials. The unnormalized coefficient of the zonal harmonic, J2, can be calculated for a synchro- nously rotating satellite in hydrostat ic equilibrium from its moment of inertia by means of the relationships (Hubbard and Ander-

son 1978, Hubbard 1984; see also Kaula 1968)

q = w2R3/GM

A = ~{[(15C/4MR 2) - _~]2 + 1} i _ _

5 J2 = ~Aq,

(5)

(6)

(7)

where q is a dimensionless measure of the strength of the centrifugal potential, oJ is the angular velocity of the satellites, R and M are the satellite radius and mass, A is a dimensionless response coefficient, C is the moment of inertia, and J2 contains contributions f rom both rotationally induced and tidally induced flattening along the spin axis. Values of J2 for the various structural models are given as a function of core fraction in Fig. 4. The unnormalized coefficient of the sectoral harmonic, C22, in this in- stance equals 43Aq or ~J2. Although Eqs. (6) and (7) are accurate to first order in q ( - 1 0 4 for Ganymede and Callisto), and second-order terms should be entirely negligible, nonhydrostat ic contributions may be important . These are evaluated in the Discussion section. Satellite shapes are briefly mentioned as well.

Gravitational binding energies are calculated for the various structural models from f ~ rg(r)dM(r) (Fig. 5). Differentiation could have significantly affected the thermal and tectonic evolution of Ganymede and Cal- listo. If either satellite were to experience instantaneous solid-state differentiation, the release of gravitational energy ( - 2 × 105 J kg ~) would be sufficient to melt all the ice (Friedson and Stevenson 1983). The heat of fusion of water ice per kilogram of the rock - i ce mixture is - 1 . 0 - 1 . 5 × 10 5 J kg ~, dependent on rock mass fraction (Ta- ble IV).

Figure 6 depicts s teady-state (or "equi- l ibr ium") surface heat flow 4 gyr ago predicted by the structural models; present- day values are reduced by ~. Rock heat production rates are determined using solar abundances of the major long-lived radioac-


0 1 6 . 2 5

GANYMEDE 2 1 . 2 5 . . . . I . . . . I . . . . I . . . .

k P F . C l

. . . . . . PF. PF 1 8 . 7 5 ' ~ . , ~ . . . . P T C . C I

1 3 . 7 5 ~ ' ~ ~

1 1 . 2 5 i i , , i , , , , J . . . . i , , , , 0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0

f- 0

4 . 2 5

3 . 7 5

3 . 2 5

CALLISTO

2 . 7 5

2 . 2 5 0.OO

f , ' ' T . . . . i . . . . i . . . .

. . . . i , . , , i . . . . L , ~ i

0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0


FIG. 4. Second-degree gravitational harmonic coefficients ./2 for Ganymede and Callisto. Values are plotted as a function of core silicate fraction for the structural models in Figs. 1 and 2.

T A B L E V

HEAT PRODUCTION RATES FOR MODEL ROCK TYPES (W kg t)

Today 4 gyr ago

CI r o c k 4.242 × 10 ~2 2.545 × 10 -It

P /F r o c k 4.723 × 10 -12 2.834 × l0 11

model of the differentiation process, of course, because differentiation here implies localized (nonaccretional) melting and mi- gration of liquid water. Thus one of our evolutionary tracks is not a simply con-

0

>-

,.n'- U.I Z U.I

(9 Z

Z

- -3 .5

- -3 .6

- -3 .7

- - 3 .8

- -3 .9 0 . 0 0

GANYMEDE . . . . i . . . . i . . . . i . . . .

P F . C I

. . . . . . . . PF. PF

. . . . PTCmCI

- - - - - P T C . P F

~ ' ~ , ' . ~ . . .

. . . . i . . . . i , , , • i , , . 1 ~

0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0

CALLISTO - 1 . 9 , ' , , ~ . . . . I ' ' ' ' ~ . . . .

tive elements, U, Th, and K, relative to Si (Basaltic Volcanism Study Project 1981), and are given in Table V. The most straight- forward calculation of the heat flow uses the "wet" silicate fraction, so there is no need to calculate the heating rate of pretremolite condensate.

Other aspects of differentiation can be studied with these models. "Evolutionary" tracks can be set up, linking specific three- layer Ganymedes and Callistos with others possessing the same mass and "wet" silicate fraction, but with greater or lesser degrees of differentiation. This is not an exact

CJ

>-

¢1: tU

Z Ud

O Z

Z

03

- 2 . 0

- 2 . 1

- 2 2 " ~ < ' ~ ~

- 2 . 3 i J , , I r i i i I i , , , I . . . .

0 , 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0


FIG. 5. Gravitational binding energies of Ganymede and Callisto as a function of core silicate fraction (cf. Figs. 1-3).

450 M U E L L E R A N D M C K I N N O N

&.-.

E

q EL

<C U. I '1"

LU U

EL (Z:

U~

GANYMEDE 4 B.Y. AGO

2 9 . 0

2 7 . 5

2 6 . 0

2 4 . 5

. . . . , . . . . r . . . . i . . . . .

/ . . . . P T C . C I

m ' m P T C . P F

2 3 . c . . . . i , m , = i , , . . i . . . .

o . o o 0 . 2 5 0 . 5 0 0 . 7 5 1 . o o

E

E

S

w

w

CALLISTO 4 B.Y. AGO

2 3 . 7 . . . . i . . . . i . . . . r . . . .

22.7 S # , ~ ' " . /

~'.'/

2,.7 /':;/ 2 0 . 7

1 9 . 7 i , , , i . . . . L . . . . J , , , ,

0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0


F~6. 6. Steady-state surface heat flows for Gany- mede and Callisto 4 gyr ago (cf. Figs. 1-3). Long-lived radioactive element abundances are based on solar ra- tios to Si for each model rock mineralogy. Present-day steady-state heat flows are a factor of 6 less.

nected sequence of structures, but rather, the locus of potential beginning and end states. We note, however, that refreezing of a liquid-water region generally involves very little change in satellite radius (Squyres 1980), so these evolutions provide a suitable approximation to reality, at least as far as the discussion below is concerned.

Radius changes due to satellite differentiation are of potentially great consequence in satellite tectonics (see Squyres and Croft 1986). Figure 7 displays radius changes

along "evolutionary" tracks for Ganymede and Callisto in which the fully differentiated cases conform to the present-day radius (and mass). Assuming Ganymede and Cal- listo are at other stages of differentiation results in similar curves, as radius change is not as sensitive to the structure as total radius. The overall expansion is, of course, due to the much greater effective compres- sion of ice (caused by phase changes) compared to rock (Squyres 1980). What may

0 . 0 4

0 . 0 3

0 . 0 2

0 . O 1

O.OO

0.00

GANYMEDE

0 , 2 5 0 . 5 0 0 . 7 5 1 . 0 0

CALLISTO 0 . 0 4 . . . . r . . . . I . . . . I . . . .

0 . 0 3

0.02

0 . 0 1

o.oc .... i .... i .... l , , , ,

0 . 0 0 0 . 2 5 0 . 5 0 0 . 7 5 1 . 0 0


FIG. 7. Radius change as a function of core silicate fraction or "degree of differentiation." The core silicate fraction here is not that of Figs. 3-6. Rather, only fully differentiated cases are constrained to the present-day radius, and the curves represent the possible evolution of global volume change. Note the level- ing off of expansion during the later stages of differentiation.


not be appreciated is that as the rock mi- grates downward and the average pressure the ice is subjected to decreases, the difference in average pressure between any two neighboring differentiation states also decreases. This translates to rapid satellite expansion early in differentiation followed by a period of less vigorous expansion. This "plateau effect" is apparent in most of the curves in Fig. 7, and is discussed more fully in the last section.

In the final stages of differentiation, some of the models actually contract. This occurs for some of the structures that involve core dehydration. Under normal circumstances pre s sure - i nduced dehydration alone always results in a net volume loss, and in the final stages of differentiation it may be the most important contributor to volume change.

Finally, Fig. 8 indicates how the binding energy varies during differentiation along the evolutionary tracks of Fig. 7 (compare with Fig. 5). As differentiation proceeds, gravitational energy is converted to heat, and surface heat flow may be higher than steady-state values resulting from radiogenic heating alone.

THERMAL CALCULATIONS

The presence of relatively clean ice on the surface of Ganymede suggests that some fraction of its primordial ice-rock mixture was at some time subjected to melting. Although the lack of "clean" ice on Callisto is not necessarily an indication that Callisto never experienced melting at depth, careful evaluation of the likelihood of melting in both satellites should provide considerable insight into their evolution.

Neglecting tidal dissipation, there exist three major heat sources that may induce melting on a planetary scale: accretion, differentiation, and radioactive element decay. In this section we evaluate the likelihood of melting in a three-layer Ganymede or Callisto caused by the latter. We determine whether the postaccretional temperature profile, which generally follows a convective adiabat, either intersects the

GANYMEDE

~ 3.5

~ -3.6

~ -3.7

- 3 . 8

--3.9 0.00 0.25 0.50

. . . . r . . . . I . . . . I . . . .

PF.CI ~ ........ PF.PF

. . . . PTC.Cl ---~ PTC. PF

\

0.75 1 .~

CALLISTO -1.95 . . . . [ . . . . I . . . . [ . . . .

~ -2.01 ~

-2.07

Z~ -2,13

-2.19 \

-2.25 . . . . J , , , , I 0,00 0.25 0.50 0.75 1.00

CORE FRACTION

FIG. 8. Gravitational binding energies of Ganymede and Callisto corresponding to the "evolutionary" tracks of Fig. 7 (cf. Fig. 5).

well-known minimum melting temperature for water ice (the ice 1-ice III-liquid water triple point at 251°K and 209 MPa; Kell and Whalley 1968) or, depending on the place- ment of boundary layers, intersects the melting curve at greater depths.

To determine the temperature profiles (horizontally averaged), we appeal to estab- lished "recipes" of parameterized convection (e.g., Reynolds and Cassen 1979, Schubert et al. 1986, and references therein). The Rayleigh number, Ra, the dimensionless measure of convective vigor, is related to the Nusselt number, Nu, a di-


mensionless measure of the global efficiency of heat transport, by

Nu = b(Ra/Racr) ~ (8)

where Ra~r is the critical Rayleigh number (of order 103), and b is a constant of order unity. The exponent/3 has been determined to be approximately 1 for vigorously convecting systems (e.g., Elder 1978; but also see, for example, Christensen 1985). For a fluid shell heated from below, the Rayleigh and Nusselt numbers are given by

Ra = go~pD3AT/i.tK (9)

Nu =- FD/kAT, (10)

where g is the gravitational acceleration, D is the shell thickness, AT is the temperature difference across the shell in excess of the adiabatic drop, F is the heat flux through the shell, and ~ , /z , K and k are the volume coefficient of thermal expansion, viscosity, thermal diffusivity, and thermal conductiv- ity of the fluid, respectively. Intrinsic quali- ties such as viscosity are usually averaged over the convect ive cell. Appropriate modi- fications to Eqs. (9)-(10) occur for a fluid shell or sphere heated from within or a shell heated by both mechanisms.

We expect subsolidus convect ion to be vigorous within Ganymede and Callisto. A not atypical value of Ra, evaluated from Eq. (9) for D = I00 km, AT = 100°K, g = 1 . 4 m s e c 2 , ~ = 10 4 K - I , p = 1 .0gcm 3, K = 10 6 m 2 sec ~, and tz determined for Newtonian creep (discussed below) at T = 220°K, would exceed 106. In boundary layer theory, which should apply at very high Ra, convect ive heat transport is governed solely by conductive heat loss through boundary layers, so the rate of heat transport is independent of the depth of the convective cell. In this case /3 = ~, and Eqs. (8)-(10) can be combined to solve for the convect ive heat loss

F = k(gap/I.zKRacr)l/3(AT) 4/3, (11)

where b has been set equal to one. Thermal boundary layers have finite tern-

perature differences associated with them. Thus, as three-layer i ce - rock satellites possess at least two separately convecting icy shells, they possess at least four boundary layers: one below the surface lithosphere of the the pure ice upper mantle, two adjacent at the contact of the pure ice upper mantle and mixed i ce - rock lower mantle, and one at the base of the lower mantle. The convective properties of the silicate core are not considered here. With at least twice as many boundary layers as a completely differentiated or undifferentiated satellite, three-layer satellites have an enhanced potential for melting. This is the principal point we explore in this section.

The application of boundary layer theory is complicated by the enormous temperature-dependent viscosity variations that occur for both silicates and ices. Variable viscosity can apparently be accounted for by evaluating the viscosity at the average of the temperatures bounding the convective cell (Booker 1976), which owing to the thin- ness of the low-viscosity bot tom boundary layer (Richter 1978; and see Stevenson et al. 1983) or to the nonexistence of the bottom boundary layer in a convecting sphere (Friedson and Stevenson 1983), amounts to the mean temperature of the upper boundary layer. Kirk and Stevenson (1987) assume that all boundary layers are main- tained at the critical Rayleigh number, and if true, the thermal profile in a convecting planet or satellite can be simply determined. It is not clear that all the boundary layers adhere to this condition of criticality, though, so we simply require that all the boundary layers in a three-layer Ganymede have the same Rayleigh number, whatever that may be. (The Rayleigh number applied to a boundary layer is distinct from that applied to the entire convecting cell, as the latter is convect ive and the former is conductive or marginally convective. The boundary layer Rayleigh number, Rabl, should be critical or subcritical. As an example, an isoviscous shell heated from below can be shown to possess two boundary


layers, each with Rau equal to Rcr/16.) Be- cause increased Rayleigh numbers imply larger temperature drops, we determine for a given internal structure and heat flow the minimum boundary layer Rayleigh number for the thermal profile to intersect the ice melting curve. We thus determine the rela- t ive likelihood of melting.

For simplicity, we evaluate the likelihood of melting 4 gyr ago, a time considered characteristic of the ages of the surfaces of Ganymede and Callisto (Shoemaker and Wolfe 1982) and sufficiently late so that the intermediate silicate layer originally created during accretion should have largely formed a core. The upper three boundary layers are required to carry the steady-state heat flux (Fig. 6); the bottom boundary layer carries only the heat flux from the core. Calculations were based structurally on PF.PF models, but assumed the mini- m u m radiogenic heating possible (i.e., that of the undifferentiated state). Secular cool- ing, stored accretional heat, and the heat of differentiation are ignored. Fine silicate particles suspended in the water-ice upper mantle or potassium-containing brines in a residual liquid layer at the ice I-ice III interface (Reynolds and Cassen 1979, Kirk and Stevenson 1987), both of which might contribute to heating within the upper mantle, are also ignored; this is justified post priori. The explicit connection of this work to the calculations of Kirk and Stevenson (1987) is deferred until the Discussion section.

The Rayleigh number for the boundary layers is easily shown, using Fourier's law of heat conduction and constant k, to be

RaN -- g a p k 3 ( T b - Tt)4/t.6blKF 3, (12)

where Tt and Tb are the temperatures at the top and bottom of the boundary layer. Al- though Eq. (12) can be derived from Eq. (11), we stress that it is independently derived for the boundary layers. The formal- ism is thus independent of the value of fi and is in the spirit of Stevenson et al. (1983) and Kirk and Stevenson (1987). The quan-

tity ak3/K is taken to be 2 x 10 3 W 3 sec m -5 ° K - 4 for all the ice phases; g is evaluated at the radius of each boundary layer, but the surface value of g is used for the uppermost boundary layer. The viscosity/zu is evaluated at the central temperature of each boundary layer and is assumed to have an activated form and a stress-dependent preexponential term

/Zbl = i&o0.| n exp[2B/(Tt + Tb)], (13)

where/x0 is the preexponential coefficient, tr is the differential stress, and B is the activation enthalpy divided by the gas constant. If Tt, o-, and the viscosity (and other) parameters are known, and a Rau is chosen, Eqs. (12)-(13) can be solved for Tb, allowing the thermal profile to be determined by working downward into the satellite.

The convective regions are assumed to be adiabatic and include the effects of ice phase changes where appropriate, with the thermal effects associated with such transformations being reduced by an amount corresponding to the mass fraction of any silicates present. Heats of transformation are from Eisenberg and Kauzmann (1969); if variable numbers were given, minimum values for exothermic transformations (with depth) were chosen and maximum values for endothermic ones. We also assume phase changes do not act as convective barriers or otherwise reduce the efficiency of convective heat loss. This is a conservative assumption in that it minimizes the number of thermal boundary layers and thus the likelihood of melting. Lin- ear stability calculations indicate that some phase changes may act as barriers to convection (Thurber et al. 1980, Bercovici et al. 1986), but the application of these calculations to vigorously convecting icy satellites is debated (see Friedson and Steven- son 1983, Kirk and Stevenson 1987).

In general, when confronted with alternatives, that least likely to result in melting is adopted. The boundary layer approach is itself conservative, as it represents the

454 MUELLER AND MCK1NNON

most efficient means of extracting heat from a satellite and hence minimizes the likelihood of melting. Embedded in this approach is the use of horizontally averaged "p lane to therms ." This is again conservative as the ascending limbs of convection cells are appreciably hotter and hence more susceptible to melting (and differentiation).

For all but the outermost boundary layer, Tt is determined by the temperature profile above it. We determine the initial Tt by min- imizing Tb with respect to it. Differentiating Eq. (12) with respect to Tt leads to an expression for OTb/OTt, which is zero (for 02Tb/OT~ positive) if

Tb = {(B/4) - [(B/4) 2 - BTt] ~/2} - Tt. (14)

This expression and Eq. (13), when substi- tuted in Eq. (12), determine Tt for a chosen Rabl such that the likelihood of melting is minimized. This procedure is essentially equivalent to those in Friedson and Steven- son (1983) and Kirk and Stevenson (1987).

R heo logy

The viscosity parameters chosen are those which minimize the viscosity in a given situation. Non-Newtonian, stress-dependent (n ~ 1) flow laws are compared to Newtonian ones once the m a x i m u m stress level is determined. It is easily shown, using relationships in Turcotte and Schubert (1982), that this stress is

O-max ~ 2gapDA T/3Ra~ 3. (15)

For the values of AT, g, ~, and 0 given above, and Racr = 1 0 3, O'max - - D kPa km -~. This estimate is used in Eq. (13) to minimize the viscosity.

The empirical ice Ih flow laws of Kirby et al. (1985) are used to model the power-law behavior of ice Ih, and for simplicity, that of ice I1. The lowest temperature ice I law of Kirby et al. (1985) is unnecessary, because the midtemperature of even the uppermost and coldest boundary layer never falls into the temperature range of its applicability (<195°K). The non-Newtonian ice II1 flow law is taken from Durham et al.

(1987); it has a moderate effect on the results. The ice V! flow law of Sotin et al. (1985) is used for the power-law behavior of ices VI-VII I . The recently determined ice V flow law of Sotin and Poirier (1987) is also incorporated, and its great stiffness compared with ice VI proves important. The Newtonian behavior of all the ice phases is modeled with the flow law fa- vored by Friedson and Stevenson (1983), but with a (zero-pressure) melting point viscosity of 10 ~3 Pa sec (1014 P). This reflects our belief that little could be gained by an elaborate and ultimately uncertain charac- terization of the diffusive flow of ice, given its dependence on grain size. Furthermore, the uncertainty in grain size may be grouped with the uncertainty in the value actually taken by Rabl; the result is still a valid calculation of relative likelihood of melting. We also do not include a hydrostatic pressure dependence for ice I, because the activation volume determined by Kirby et al. (1985), - - 1 0 cm 3 mole 1, is less than a third of that needed to permit a truly homologous definition of the viscosity. Parameters for the flow laws used are given in Table VI.

A non-Newtonian flow law for ice II is available (Durham et al. 1985, 1987). Ice II appears to be stiffer than ice I at the same temperature and stress level (see also

T A B L E V I

VISCOSITY PARAMETERS FOR W A T E R |CE a

/~0 n B (MPa ~ ~ Pa sec) (°K)

Ih, T > 243°K 5.28 × l0 7 4.0 10,950 Ih, T < 243°K 2.65 4.0 7,340 l lI 1.33 × 10 2~ 4.3 18,160 V 3.09 x 106 2.67 3,897 + 1,214P ~' VI 1.57 × 104 1.93 3,430 + 974P ~' Newton ian , all 1.39 × 102 1 6,830

po lymorphs

" Cf. Eq. (13); pa rame te r s ave der ived f rom Kirby et al . (1985), D u r h a m et al . (1987), Sotin and Poir ier (1987), Sotin et al . (1985), and Fr iedson and S tevenson (1983) us ing ~ = o-[j/2~j, where e~j and o-[j are respec t ive ly the strain rate and devia tor ic s t ress tensors .

h P res su re in GPa.


Echelmeyer and Kamb 1986), but ignoring this minimizes the non-Newtonian ice II viscosity. We ignore the stiffening effect of silicate inclusions (McKinnon 1982, Fried- son and Stevenson 1983) in the lower mantle boundary layers for the same reason.

Ice III is less stiff than ice I under conditions probed in experiments to date, but its large power-law exponent and activation enthalpy (which remain uncertain [Durham er al. 19871) complicate its behavior here. In general, it proves to be softer than even the Newtonian law, but this is not the case if the older version of the ice III law (Dur- ham et al. 1985) is used.

We note that the two data points for steady-state creep of ice V in Durham et al.

(1985) are poorly fit by the ice V flow law (Sotin and Poirier 1987), but are well fit by the ice VI law (Sotin et al. 1985) when cor- rections for experimental geometry are applied. An alternate ice V law (Durham ef al.

1987) is in sharp disagreement with that of Sotin and Poirier, but we choose to not use the former because of its extreme stress dependence (n = 6.9), given that our calculations assume much lower stresses than in the experiments of Durham et al.

Results

Figure 9 plots the boundary-layer Ray- leigh numbers necessary for melting as a function of core rock fraction (closely related to the structures in Figs. 1 and 2) for Ganymede and Callisto. The curves are similar for both worlds. The key datum in any of the calculations is the position of the upper/lower mantle interface; this deter- mines the position of two of the thermal boundary layers (the adjacent “second” and “third” boundary layers). For small degrees of differentiation (5 15% for Gany- mede and ~20% for Callisto), this interface remains near the surface and above the minimum melting isobar. There are then three thermal boundary layers to reach the minimum melting temperature (Fig. lOa), and the minimum Rabl required are small, -30-50. All three of the boundary layers

0 Ganymede

$ 103 r

i

2 102

r. .9

9. 2 10’ Accretional Trigger

Pure Ice Melting

,“O ’ 1 1 ’ ’ - 0.0 0.2 0.4 0.6 08 1 0

‘04E a E Callisto

2 103 .-f/

2 E

t 102

.c

.o, z IO'

$

Accretional Trigger

Pure Ice Melting

loo w. 0.0

Core Rock Fraction

FIG. 9. Critical boundary layer Rayleigh numbers

for melting as a function of core silicate (rock) frac-

tion. Results are similar for Ganymede and Callisto.

Solid lines (“accretional trigger”) indicate melting be-

low the boundary layer at the top of the ice-rock lower

mantle. The three continuous segments correspond to

upper/lower mantle interfaces in the ice 1, III, and V

stability fields, respectively. Variation in Ra,,, from one

segment to another is continuous, but omitted for clar-

ity. Squared line indicates melting within the pure ice

upper mantle, and corresponds to upper/lower mantle

interfaces in the stability field of ice VI or higher. The

calculations are for models with P/F rock in the core

and lower mantle.

are in the ice I stability field, but as the stress levels in the pure ice upper mantle are low (of order 0.1 MPa), the Newtonian law governs the rheology of the upper two boundary layers. Only in the deep ice-rock lower mantle are stresses large enough to make the non-Newtonian ice I law weaker and preferrable. Because the third boundary layer, the one that reaches the minimum-melting temperature of 251”K, is non- Newtonian and relatively hot, it is relatively thin. Temperature drops across it are a few Kelvins, and it is generally true


Thermal Lithosphere Boundary

~Pu re l c e ~ @ Lay> c e

Thermal Lithosphere . . . . . . . Boundary

@ Layer I

. . . . . . . . . . .

® \ (a) T

(b) T

Thermal Lithosphere i Boundary

. . . . . . - - - - ~ . ~ " '~\1 nt e..rfd ace

f ® \

Thermal Lithosphere -- Boundary

~| Layer I

@ \'::;

T ~ No melting T

(c) (d)

FIG. 10. Schematic temperature profiles illustrating the concept of accretionally triggered melting, The upper/lower mantle interface is a barrier to convection, resulting in two additional thermal boundary layers and a hotter lower mantle. (a) Three boundary layers occur above the ice minimum-melting temperature and melting is easily initiated; (b) the second and third boundary layers occur in the ice II1 pressure field and melting is more difficult to initiate; (c) the lower boundary layers are somewhat deeper than the ice lII field and melting is much more likely, at least until the adiabat moves to cooler temperatures along the.ice ll-ice V phase boundary; (d) in structures similar to (c) but with the upper/ lower mantle interface in the ice VI field, melting first occurs at the minimum-melting temperature, is achieved by means of a single boundary layer only (compare to dashed alternative), and is thus relatively difficult to initiate compared to (a) and (b). See text for discussion.

that the third b o u n d a r y layer plays a minor role in ach iev ing mel t ing (it is only un t rue when the third b o u n d a r y layer is ice V, as no ted below). The four th and deepes t b o u n d a r y layer is also n o n - N e w t o n i a n , but as the mel t ing t empera tu re is so high at the base of the mant le , mel t ing is neve r initiated there in any of the model calcula t ions .

As the in ter face drops into the region of ice II and III s tabil i ty (at - 2 1 0 MPa), the

lower v iscos i ty of ice III more than coun- teracts the exo the rmic na tu re of the ice l- to-II t r ans i t ion (Fig. 10b), and mel t ing is more difficult to ini t iate (in the sense of higher Rab~ values requi red; Fig. 9). The second and third b o u n d a r y layers are wholly con ta ined in the ice III field, and both obey the ice III power law. The critical Rayleigh n u m b e r at first increases a factor of 2 or less, but fur ther increases as the


interface moves to higher pressures are moderated because the temperature range for ice III stability shrinks.

The transition from control by ice III to control by ice V, as the upper/lower mantle interface penetrates the ice III-to-V transition (at -345 MPa), is dramatic. Ice V is much stiffer than ice IlI, and as a consequence the second boundary layer again be- comes Newtonian. The critical Rabl drops so much that the second boundary layer is initially in the ice II field (all phases have the same Newtonian behavior, of course). Critical Rabl values increase rapidly with increasing interface depth, because the ice V melting temperature increases with pressure a n d the ice II-to-V transition is strongly endothermic, causing adiabats to rapidly move rapidly away from the melting curve (Figs. 9, 10c). The third boundary layer is no longer negligible, and temperature drops reach 15°K; the rheology switches from power-law ice V to Newto- nian also. The rise in Rabl (into the 100's) eventually slows as the adiabat drops off the ice II-V phase boundary into the ice V field. These Rabl values are also high enough that adiabats intersect the ice I-III phase boundary rather than the I-II (Fig. 10d).

If the interface occurs below the ice V-to- VI transition (at -625 MPa), effects are also dramatic. Ice VI is sufficiently less vis- cous than ice V that melting more readily occurs at the level of the minimum-melting isobar, between the first and second boundary layers (Figs. 9, 10d). For all greater degrees of differentiation, melting within the pure ice upper mantle is the preferred mode. Rabl values are relatively constant until the increasing stress level in the upper mantle throws the uppermost boundary layer into non-Newtonian behavior.

Purely Newtonian versions of Fig. 9 are quite similar. Most thermal boundary layers in Fig. 9 are Newtonian. The notable changes would occur for interface depths in the ice III and VI range. In the first case, Rao~ values would decline smoothly from

values appropriate to interfaces in the ice I range to the low values appropriate to interfaces at depths in the ice V range. In the second, Rabl values should continue to smoothly increase as the interface penetrates the ice V-to-VI transition. The increase will not be much, however, before melting below the first boundary layer is preferred.

Use of the older version of the ice III law (Durham et al. 1985) lowers the critical Rabl values by as much as a factor of several. I f the ice V law of Durham et al. (1987) is used, the appropriate Rab~ increase by about a factor of 2 at the low Rabl end, but there is virtually no change at the high Rabl end. Over time, we have tested quite a few rheologicai laws. While details vary, overall results have remained qualitatively similar.

In summary, the calculations yield a roughly bimodal distribution of the minimum boundary-layer Rayleigh numbers required for the initiation of melting. Large ice-rock satellites the size of Ganymede and Callisto that are only moderately differentiated (<~30% of the silicate mass residing in the core for Ganymede and ~<38% for Callisto) require Rabl in the -10-100 range; more differentiated structures require much larger values, -500-1000 (Fig. 9). The values depend on the position of the adjacent second and third thermal boundary layers, determined by the position of the upper/lower mantle interface. For interface depths that are above the ice V-to-VI transition, melting most readily occurs below the third boundary layer, within or at the top of the ice-rock lower mantle. For deeper interface levels, melting more readily occurs at the position of the minimum-melting isobar, in the pure-ice upper mantle.

We note that the boundary layer Ray- leigh numbers given above should not be so strictly interpreted as to imply melting in nearly all cases (most values are "subcritical" in the usual sense); the uncertainties in the calculations preclude such sweeping


generalizations. Rather, it is the likelihood of melting that is addressed. Still, minimum Rab~ values under - 5 0 are strong indicators that melting will occur.

In highly differentiated structures melting can be initiated between the two uppermost thermal boundary layers, but because pure ice is melted, further differentiation does not follow. Less differentiated structures can experience renewed differentiation, because melting first occurs below the third thermal boundary layer in the primordial i ce - rock mixture (Figs. 10a-c). The gravitational energy released may then lead to a runaway differentiation similar to that described by Friedson and Stevenson (1983). This second differentiation occurs only for those satellites that have an opti- mal three-layer configuration, one ultimately due to primary accretional melting. Such renewed differentiation can be said to be "accret ional ly t r iggered."

The presence or absence of an "accretional tr igger" is determined by the position of the upper/lower mantle interface relative to the bottom of the ice V stability field. This occurs at a depth of approximately 400 km for Ganymede and 450 km for Callisto, if the regions above are pure ice, and corresponds to a degree of differentiation of about 40% for Ganymede and 50% for Cal- listo, with some variation dependent upon rock mineralogy. These amounts of differentiation, while far from trivial, are not as great as might occur during accretion (Lunine and Stevenson 1982).

Even if an "accret ional trigger" exists, it may not get pulled. The calculations determine the relative likelihood of melting, and the Rabl necessary may not be achieved. Therefore melting may not occur, or differentiation may not penetrate, below some level in the ice V field. Possibly, melting may not occur or penetrate below the base of the ice III field.

Melting calculations were carried out only for those models that included P/F rock in both the core and lower mantle. Results for other structural models cannot

be significantly different. Any variations are tied to heat flow, and most of the variation in heat flow depends on absolute silicate fraction rather than specific silicate type. As noted above, the minimum rock fraction was chosen. The amount of heating within the water-ice upper mantle possible, due to effects mentioned in the previous section, is unlikely to be more than some fraction of that caused by the silicate mass separated from it. The neglect of this internal heating causes smaller temperature drops for all but the upper boundary layer, but the effect is slight for satellites possessing the accretional trigger, and easily ab- sorbed into the uncertainty in the radioactive element abundance. It may not be trivial for satellites that are deeply differentiated, though.

DISCUSSION

Our calculated interior structures, along with the geophysical quantities and thermal profiles derived from them, lead to a number of interesting conclusions. We discuss each in turn.

Rock Volume Fraction

The range in rock volume fractions estimated for Ganymede and Callisto from Eq. (3) and Table IV are all generally under the "cr i t ica l" silicate volume fractions of Friedson and Stevenson (1983), at which convect ive self-regulation of internal temperature in an undifferentiated Ganymede or Callisto breaks down as melting is initiated at the water-ice minimum-melting temperature. Greater values can occur only for CI-rock containing models in which organic matter is counted as silicate (though, theo- logically, the organics may be softer than water ice). Even these values for the silicate volume fraction are less than the approximately 60% that could result in a rigid silicate framework, which would undoubt- edly lead to internal melting (Friedson and Stevenson 1983, Schubert et al. 1986). Con- clusions in these works as to the relative ease of melting in an undifferentiated Gany-


mede versus difficulty of the same in Cal- listo should be viewed with caution.

Refinement of these rock fraction estimates requires that the thermal and petrological evolution of the possible cores of Ganymede and Callisto be more completely modeled. This is, of course, justifiable on general grounds. A better treatment of the metamorphic and rheological behavior of the organic component is especially important. Such a study would have broad application to other satellites such as Europa and to "carbonaceous" asteroids.

Gravi ta t ional H a r m o n i c s

Nonhydrostatic contributions to JR and C22 may be important. These can be roughly estimated by scaling them to the J2 or C22 of the Moon, which are considered to be supported by finite strength (Phillips and Lam- beck 1980). In this case

J2 ~- 0.5(O-max/O- ~max)(g~/g)2(r/R)5/2JC2 , (16)

where J2 ~ is the lunar value (2.02 x 10 -4, Ferrari et al. 1980), r is the effective radius within Ganymede or Callisto at which nonhydrostatic density anomalies are supported, g and g c are the gravitational accel- erations at the effective radius of density anomaly support for Ganymede (or Cal- listo) and the Moon (lunar anomalies are lithospheric in origin, so gC is essentially the surface value, 1.62 m sec-2), and O'ma x and O-m~ax are the maximum stress differences maintainable on each body. The factor of 0.5 arises from a portion of J ( being due to a fossil tidal bulge, and not to density anomalies (Lambeck and Pullan 1980).

If Ganymede or Callisto are undifferentiated, then density anomalies should be supported in their outer ice-rich lithospheres. In this case r ~ R and (O-max/O" Cmax) ~ 0.I is appropriate. Estimates of J2 for Ganymede and Callisto are then approximately 1.3 × 10 -5 and 1.7 × 10 5, respectively. For fully differentiated models, density anomalies should also be supported in the relatively cool outer regions of the silicate cores (essentially internal lithospheres). In this case

(Ormax/O-~max) ~ 1, and information in Figs. 1 and 2 and Table IV can be used to estimate maximum values of J2 of approximately 3.8 × 10 -5 and 4.3 x 10 -5 for Ganymede and Callisto, respectively, for core anomal ies alone. Comparison of these values to Fig. 4 suggests the Galileo experiment at Callisto will be severely compromised. A differentiated Callisto may have larger second-degree harmonics than an undifferentiated one!

We stress that these estimates are approximate. If J2 and C~2 contribute equally to the normalized power spectrum of the potential, we expect C22 = J e / ' x / ~ . For the Moon Cf2 is weaker than this estimate by another factor of 2.5. The actual values for the "intrinsic" J2's of Ganymede and Cal- listo may range between the values given above and those a factor of a few less. The importance of making measurements of both J2 and C22 to test for hydrostatic equilibrium (Hubbard and Anderson 1978) is clear. As of 1984, the plans for Galileo make this seem feasible at Ganymede, where the 1-o- errors on J2 and C22 are 1.5 × 10 -5 and 2 x 10 -6, respectively (Campbell 1984); the experiment at Callisto appears dubious, with corresponding 1-o- errors of 9.6 × 10 5 and 2 × 10 -6. The satellite tour ultimately flown will likely give similar errors. Perhaps the only way to constrain Callisto's internal structure will be to correct for the nonhydrostatic component of the second-degree response by measuring the power in the third and fourth (or more degrees. This requires a satellite orbiter.

The prospects for measuring satellite shapes from Galileo images are poor. The maximum hydrostatic triaxial radius variation is (12J2/5 + 2q)R, and does not exceed 2.3 km for Ganymede and 410 m for Callisto (cf. Zharkov et al. 1985). Even if the metic- ulous care is taken (e.g., Dermott and Thomas 1988), it is unlikely that measure- ment precision can exceed -0 .5 pixel, which for Galileo at Ganymede and Callisto means uncertainties o f - 1 . 5 km in radius.


Global Expansion

Most of the expansion caused by the differentiation of a Ganymede or Callisto occurs early in the process; 75-90% takes place before differentiation is one-half complete (Fig. 7). There are several ways to view this. First, if satellite tectonics and volcanism are directly correlated with global expansion, then it is difficult on this basis to distinguish between a partially differentiated satellite and one that is completely differentiated. More to the point, Callisto could be significantly less differentiated than Ganymede, and still be expected to manifest the same geologic vitality. Cal- listo's apparent lack of geologic activity would then be evidence for a virtually undifferentiated interior. This is, however, a naive interpretation of icy satellite geology. Ganymede most likely underwent major differentiation during accretion (Schubert et al. 1981, Coradini et al. 1982, Lunine and Stevenson 1982); any tectonic record from this period of very high heat flow and impact flux is irrevocably lost. It is only differentiation at later epochs that might be expected to leave a tectonic imprint via global expansion. Thus, Fig. (7) could imply that later differentiation in Ganymede results in a modest amount of expansion, an amount consistent with the upper limit of - 1 % radius change determined by McKinnon (1981). Callisto's dead appearance then only implies that later differentiation-expansion was insufficient for tectonic expression.

As discussed in McKinnon and Parmen- tier (1986), the limit of McKinnon (1981) (and that of Golombek [1982]) applies to the elastic stress and strain that lead to brittle failure. Grooved terrain tectonics are generally regarded as a manifestation of brittle failure, but if expansion is not rapid enough, strain is accommodated viscously. McKinnon and Parmentier (1986) estimated the Maxwell time of the lithosphere to be -108 years from previous crater relaxation studies, although Kirk and Stevenson

(1987) argue for a viscoelastic strain time of less than 104 years from the theological data of Durham et al. (19841. If the smaller amounts of expansion derivable from later differentiation (Fig. 7) are to leave their mark, they must accumulate relatively rapidly. So we may alternatively conclude that differentiation and expansion alone are not responsible for tectonics and volcanism on Ganymede or their absence on Callisto.

Second Differentiation

The formation of grooved and smooth terrain on Ganymede was a distinct episode, so a continually active mechanism such as gradual global expansion is a poor choice for its cause. The implication is clear, though, that a new episode of differentiation or other specific phenomenon (such as the "hea t pulse" of Kirk and Stevenson [1987]) could be responsible. This new episode might be represented by the differentiation-driven thermal runaway of Friedson and Stevenson (1983) (which is in this case the first differentiation!), in that the process they describe accelerates asymptotically and may be delayed sufficiently so as to leave a visible record. If Ganymede partially differentiated during accretion as seems likely, then the new episode could be "accret ional ly tr iggered" by the optimum configuration of internal thermal boundary layers we have described. Runaway differentiation into at least the top of the ice V stability field is likely.

Accretionally triggered second differentiation does not conflict with the model of Kirk and Stevenson (1987). Their model calls on a Ganymede so deeply melted during accretion that the accretional trigger simply does not exist. A Ganymede not so deeply melted initially is susceptible to second differentiation. This is true whether the water upper mantle is thin (all in the ice I pressure range), so it rapidly refreezes postaccret ion, or whether it is thicker, extending into the ice III or V pressure field. In the latter case freezing is slower and the heat pulse occurs (perhaps early enough


that it leaves no permanent surface record), but after the closing of the ocean, normal convect ion should operate between the ice I and II or III regions. Residual liquid due to the freezing point depression caused by dissolved salts, other solutes, and ammonia will " p o n d " in regions between ascending and descending limbs of the convect ion cells (Kirk and Stevenson 1987).

The consequences of melting caused by accretionally triggered second differentiation are, most likely, initial conditions similar to that in Kirk and Stevenson (1987). The runaway differentiation envisaged by Friedson and Stevenson (1983), and appli- cable to differentiation in a three-layer satellite, accelerates rapidly once started; 90% may occur in as little as l03 years. Because the maximum gravitational energy releas- able is more than sufficient to melt all the ice in Ganymede or Callisto and cannot be removed by solid-state convect ion over such short time scales, an internal ocean must open up. Only if the runaway is slowed down by several orders of magnitude will second differentiation occur with- out creation of an ocean (the gravitational energy released is equivalent to -108 years of radiogenic heat production). This could happen if melt extract ion from the partially molten region of the i ce - rock layer is ineffi- cient. Although we doubt this, the issue is worthy of future study.

There is another time scale issue. We choose to evaluate the likelihood of melting 4 gyr ago partially in order to be instructive. Obviously, if melting can occur at 4 gyr, it could occur earlier when the heat flow is higher (although this may be mitigated or counteracted by lower conduct ive core heat flow). All that is necessary is for the convect ive adiabat to be set up. This should happen no later than -108 years after accretion, but could occur before overturn of the original postaccretional rock outer core is complete. If a three-layer model is unstable to melting, however , inserting a conductive rock layer above the i ce - rock should make it more unstable. Kirk and Stevenson

(1987), in fact, invoke widespread inner- core ice melting at a late time (t - 108 years) to initiate core overturn. A Friedson-and- Stevenson-like unmixing of the inner core may proceed concurrent ly with overturn. Despite these possibilities, the longevity of the rock outer core in the model of Kirk and Stevenson is self-admitted to be somewhat extreme, and the whole issue of core formation and whether melting is necessary for overturn, especially for the thinner rock cores appropriate to models possessing the accretional trigger, merits further study as well. Our point is that the calculations of the likelihood of melting and differentiation can be applied to all these circumstances.

In summary then, if the expansion associated with second differentiation is inade- quate to form grooved terrain, it is most likely followed by something much more vi- olent. This is true whether the second differentiation is major (corresponding to a thin upper mantle with all three boundary layers in the ice I stability field, Fig. 10a) or much less so (which would occur when the upper mant le - lower mantle interface is in the pressure range of ice III or V stability, Figs. 10b,c). The latter would supply additional liquid water to the residual ocean or oceans between the ice I and II or 111 layers. Warm-ice diapirism could then be reinitiated, perhaps solving the problem of extending the lifetime of the heat pulse phenomenon to cover the range in ages of grooved terrain.

The Ganymede/Callisto Dichotomy

Whether Ganymede is deeply differentiated during accretion or less so such that, as a convect ive adiabat is set up, a second and potentially runaway differentiation en- sues (perhaps concurrent with core overturn as above), the results are likely similar: Ganymede is deeply melted and the potential exists for convect ive and tectonic vio- lence during closure of the internal ocean. To escape this fate Callisto must be essentially undifferentiated. Although it might be suggested that Callisto was simply not dif-

462 M U E L L E R A N D M C K I N N O N

ferentiated deeply enough for an internal ocean to be created and the heat pulse phenomenon to occur, the presence of the accretional trigger makes this implausible (un- fortunately). To avoid the trigger (and not be deeply melted in the first place), the water-ice upper mantle must be very thin. An upper limit on the thickness can be set by requiring it to conductively carry the 4-gyr- old heat flow between bounding temperatures of -130°K (the surface) and 251°K (melting). This works out to -15 km or about 2% differentiation. This is marginal differentiation, and considering the depth of mixing caused by the ancient impact flux, is of marginal significance.

If Bercovici et al. (1986) are correct and the ice I to II or III transformations act as barriers to convection, then Callisto is doomed regardless. Kirk and Stevenson (1987) suggest some possible ways Callisto could be differentiated but avoid the heat pulse (having to do with ice viscosity and ammonia incorporation). Perhaps Callisto cannot avoid a Ganymede-like evolution, but went through it early enough that its surface expression has been erased by cra- tering. Or perhaps Callisto is truly undifferentiated, notwithstanding the present in- ability of accretion models to create such a world. The divergent evolution of the two satellites remains a profound enigma. Even if the instability proposed by Kirk and Stevenson (1987) is the Ganymedean equivalent of the Grail, it does not serve Callisto.

ACKNOWLEDGMENTS

We thank Roger Phillips for helpful discussions of boundary layer theory and for providing the opportu- nity to present this work at the Kona Coast, Hawaii, DPS meeting. We also appreciate perceptive com- ments from John Lewis and reviewer Paul Thomas, rheologic reprints from Bill Durham and Christopbe Sotin, and the detailed review of Randolph Kirk. Steve Mueller is supported at Southern Methodist Univer- sity by NASA Grant NAGW-459. This research was drawn from SM's Master's Thesis, and was supported by Grant NAGW-432 from the NASA Planetary Geol- ogy and Geophysics Program to Washington Univer- sity and completed at SMU and Washington.

REFERENCES

ANDERS, E., AND M. EBIHARA 1982. Solar System abundances of the elements. Geochim. Cosmochim. Acta 46, 2363-2380.

Basaltic Volcanism Study Project 1981. Basaltic Vol- canism on the Terrestrial Planets, Chap. 4. Perga- mon, New York.

BERCOVICI, D., G. SCHUBERT, AND R. T. REYNOLDS 1986. Phase transitions and convection in icy satellites. Geophys. Res. Lett. 13, 448-451.

BIRCH, F. 1966. Compressibility: Elastic constants. In Handbook o[" Physical Constants (S. P. Clark, Jr., Ed.), pp. 97-173. GSA Memoir 97.

BOOKER, J. R. 1976. Thermal convection with strongly temperature-dependent viscosity. J. Fluid Mech. 76, 741-754.

BUNCH, T. E., AND S. CHANG 1980. Carbonaceous chondrites. II. Carbonaceous phyllosilicates and light element geochemistry as indicators of parent body processes and surface conditions. Geochim. Cosmochim. Acta 44, 1543-1577.

CAMERON, A. G. W. 1973. Abundances of the elements in the Solar System. Space Sci. Rev. 15, 121- 146.

CAMERON, A. G. W. 1982. Elementary and nuclidic abundances in the Solar System. In Essays in Nu- clear Astrophysics (C. A. Barnes, D. N. Schramm, and D. D. Clayton, Eds.), pp. 23-43. Cambridge Univ. Press, New York.

CAMPBELL, J. K. 1984. Determination of satellite gravity harmonics from Galileo radio tracking data. Bull. Amer. Astron. Soc. 16, 705.

CAMPBELL, J. K. , AND S. SYNNOTT 1985. Gravity field of the Jovian system from Pioneer and Voyager tracking data. Astron. J. 90, 364-372.

CHRISTENSEN, U. R. 1985. Thermal evolution models for the Earth. J. Geophys. Res. 90, 2995-3007.

CLAYTON, R. N. , AND T. K. MAYEDA 1984. The oxygen isotope record in Murchison and other carbonaceous chondrites. Earth Planet Sci. Lett. 67, 151- 161.

COATES, D. F., AND M. ASLAM 1968. The equations of state up to 250 kb of magnetite and a quartzite. Int. J. Rock Mech. Min. Sci. 5, 495-500.

CORADINI, A., C. FEDERICO, AND P. LANCIANO 1982. Ganymede and Callisto: Accumulation heat content. In Comparative Stu~v of the Planets (A. Coradini and M. Fulchignoni, Eds.), pp. 61-71. Reidel, Dordrecht.

DERMOTT, S. F., AND P. C. THOMAS 1988. The shape and internal structure of Mimas. Icarus 73, 25-65.

DICKERSON, R. E. 1969. Molecular Thermodynamics. Benjamin, Menlo Park, CA.

DODD, R. T. 1981. Meteorites: A Petrologic-Chemical Synthesis. Cambridge Univ. Press, New York.

DUFRESNE, E. R., AND E. ANDERS 1962. On the chemical evolution of carbonaceous chondrites. Geochim. Cosmochim. Acta 26, 1085-1114.

T H R E E - L A Y E R G A N Y M E D E S A N D C A L L I S T O S 463

DURHAM, W. B., S. H. KIRBY, AND H. C. HEARD 1984. Flow and fracture of H20 ices Ih, II, and III: Latest experimental results. Lunar Planet. Sci. XV, 234-235.

DURHAM, W. B., S. H. KIRBY, AND H. C. HEARD 1985. Rheology of the high pressure H20 ices II, III, and V. Lunar Planet Sci. XVI. 198-199.

DURHAM, W. B., S. H. KIRBY, H. C. HEARD, AND L. A, STERN 1987. Inelastic properties of several high pressure crystalline phases of H20: Ices II, III and V. J. Phys. 48, C-221-C-226.

EBIHARA, M., R. WOLF, AND E. ANDERS 1982. Are CI chondrites chemically fractionated? A trace element study. Geochim. Cosmochim. Acta 46, 1849- 1861.

ECHELMEYER, K., AND B. KAMB 1986. Rheology of ice II and ice III from high-pressure extrusion. Geophys. Res. Lett. 13, 693-696.

EISENBERG, D., AND W. KAUZMANN 1969. The Struc- ture and Properties of Water. Oxford Univ. Press, London/New York.

ELDER, J. 1978. The Bowels of the Earth. Oxford Univ. Press, London/New York.

EVANS, H. T., JR. 1979. The thermal expansion of anhydrite to 1000°C. Phys. Chem. Minerals 4, 77- 82.

FERRARI, A. J., W. S. SINCLAIR, W. L. SJOGREN, J. G. WILLIAMS, AND C. F. YODER 1980. Geophysical parameters of the Earth-Moon system. J. Geophys. Res. 85, 3939-3951.

FRIEDSON, A. J., AND D. J. STEVENSON 1983. Viscos- ity of rock-ice mixtures and applications to the evolution of the icy satellites. Icarus 56, 1-14.

GOLOMBEK, M. P. 1982. Constraints on the expansion of Ganymede and the thickness of the lithosphere. Proc. Lunar Planet. Sci. Conf. 13th in J. Geophys. Res. Suppl. 87, A77-A83.

HAZEN, R. M., AND L. W. FINGER 1978. The crystal structures and compressibilities of layer minerals at high pressure. II. Phlogopite and chlorite. Amer. Mineral. 63, 293-296.

HEEGESON, H. C., J. M. DELANY, H. W. NESBITT, AND D. K. BIRD 1978. Summary and critique of the thermodynamic properties of rock-forming minerals. Amer. J. Sci. 278-A, 1-229.

HUBBARD, W. B. 1984. Planetary Interiors. Van Nos- trand-Reinhold, New York.

HUBBARD, W. B., AND J. D. ANDERSON 1978. Possi- ble flyby measurements of Galilean satellite interior structure. Icarus 33, 336-341.

KAULA, W. M. 1968. An Introduction to Planetary Physics: The Terrestrial Planets. Wiley, New York.

KEEL, G. S., AND E. WHALEEY 1968. Equilibrium line between ice I and ice III. J. Chem. Phys. 48, 2359- 2361.

KERRIDGE, J. F. 1976. Major element composition of phyllosilicates in the Orgueil carbonaceous meteorites. Earth Planet. Sci. Lett. 29, 194-200.

KERRIDGE, J. F., AND Z. E. BUNCH 1979. Aqueous

activity on asteroids: Evidence from carbonaceous meteorites. In Asteroids (T. Gehrels, Ed.), pp. 745- 764. Univ. of Arizona Press, Tucson.

KERRIDGE, J. F., J. D. MACDOUGALL, AND K. MARTI 1979. Clues to the origin of sulfide materials in Cl chondrites. Earth Planet. Sei. Lett. 43, 359- 367.

KIRBY, S. H., W. B. DURHAM, AND H. C. HEARD 1985. Rheologies of H20 ices lh, II, and III at high pressures: A progress report. In Proc. NATO Work- shop Ices in the Solar System (J. Klinger, D. BeHest, A. Dollfus, and R. Smoluchowski, Eds.), pp. 89-107. Reidel, Dordrecht.

KIRK, R. L., AND D. J. STEVENSON 1987. Thermal evolution of a differentiated Ganymede and implications for surface features. Icarus 69, 91-134.

LAMBECK, K., AND S. PULLAN 1980. The lunar fossil bulge hypothesis revisited. Phys. Earth Planet. In- ter. 22, 29-35.

LEWIS, J. S., AND R. G. PRINN 1984. Planets and Their Atmospheres: Origin and Evolution. Aca- demic Press, Orlando.

LUNINE, J. I., AND D. J. STEVENSON 1982. Formation of the Galilean satellites in a gaseous nebula. Icarus 52, 14-39.

LuPo, M. J. 1982, Mass-radius relationships in icy satellites after Voyager. Icarus 52, 40-53.

LuPO, M. J., AND J. S. LEWIS 1979. Mass-radius relationships in icy satellites. Icarus 40, 157-170.

MAcDOUGAEE, J. D., G. W. LUGMAIR, AND J. F. KERRIDGE 1984. Early Solar System aqueous activity: Sr isotope evidence from the Orgueil CI meteorite. Nature 307, 249-251.

MAD, H. K., G. Zou, AND P. M. BELL 1981. High- pressure experiments on FeS with bearing on the composition of the Earth's core. Carnegie Inst. Washington Yearbook 81, 267-272.

MASON, B. 1962. Meteorites. Wiley, New York. McKINNON, W. B. 1981. Tectonic deformation of

Galileo Regio and limits to the planetary expansion of Ganymede. Proc. Lunar Planet. Sci. Conf. 12B, 1585-1597.

McKINNON, W. B. 1982. Problems pertaining to the internal structures of Ganymede and Callisto. Lunar Planet. Sci. XllI, 499-500.

McKINNON, W. B., AND E. M. PARMENT1ER 1986. Ganymede and Callisto. In Satellites (J. A. Burns and M. S. Matthews, Eds.), pp. 718-763. Univ. of Arizona Press, Tucson.

McSWEEN, H. Y., JR. 1979. Are carbonaceous chondrites primitive or processed? A review. Rev. Geophys. Space Phys. 17, 1059-1078.

McSWEEN, H. Y., JR. 1987. Aqueous alteration in carbonaceous chondrites: Mass balance constraints on matrix mineralogy. Geochim. Cosmochim. Acta 51, 2469-2477.

MORRISON, D. 1982. Introduction to the satellites of Jupiter. In Satellites of Jupiter (D. Morrison, Ed.), pp. 3-43. Univ. of Arizona Press, Tucson.

4 6 4 M U E L L E R A N D M C K I N N O N

MUELLER, S. W., AND W. B. McKINNON 1984. Three- layer generic Ganymedes . Bull. Amer. As- tron. Soc. 16, 685-686.

NAGY, B. 1975. Carbonaceous Chondrites. Elsevier, Amste rdam.

PHILLIPS, R. J., AND K. LAMBECK 1980. Gravity fields of the terrestrial planets: Long-wavelength anomalies and tectonics. Rev. Geophys. Space Phys. 18, 27-76.

PRINN, R. G., AND B. FEGLEY 1981. Kinetic inhibition of CO and N2 reduction in c i rcumplanetary nebulae: Implications for satellite composit ion. Astrophys. J. 249, 308-317.

REYNOLDS, R. T., AND P. M. CASSEN 1979. On the internal s tructure of the major satellites of the outer planets. Geophys. Res. Lett. 6, 121-124.

RICHTER, F. M. 1978. Exper iments on the stability of convect ion rolls in fluids whose viscosity depends on temperature . J. Fluid Mech. 89, 553-560.

ROBIE. R. A., B. S. HEMINGWAY, AND J. R. FISHER 1978. The rmodynamic properties of minerals and related subs tances at 298.15 K and I bar (105 Pascals) pressure and at higher temperatures . U.S. Geol. Survey Bull. 1452.

SCHUBERT, G., T. SPOHN, AND R. T. REYNOLDS 1986. Thermal histories, composi t ions and thermal structures of the moons of the Solar System. In Satellites (J. A. Burns and M. S. Mat thews, Eds.), pp. 224- 292. Univ. of Arizona Press, Tucson.

SCHUBERT, G., D. J. STEVENSON, AND K. ELLSWORTH 1981. Internal s t ructures of the Gali- lean satellites. Icarus 47, 46-59.

SHOEMAKER, E. M., AND R. F. WOLFE 1982. Crater- ing t imescales for the Galilean satellites. In Satel- lites t?f Jupiter (D. Morrison, Ed.), pp. 277-339. Univ. of Arizona, Press, Tucson.

SKINNER, B. J. 1966. Thermal expansion. In Hand- book t~fPhysical Constants (S. P. Clark, Jr., Ed.), pp. 75-96. GSA Memoir 97.

SOTIN, C., P. G1LLET, AND J. P. POIRIER 1985. Creep of high-pressure ice. VI. In Proe. NATO Workshop Ices in the Solar System (J. Klinger, D. Benest , A. Dollfus, and R. Smoluchowski , Eds.), pp. 109-118. Reidel, Dordrecht.

SOTIN, C., AND J. P. POIRIER 1987. Viscosity of ice V. J. Phys. 48, C1-233-C1-238.

SQUYRES, S. W. 1980. Volume changes in Ganymede and Callisto and the origin of grooved terrain. Geophys. Res. Lett. 7, 593-596.

SQUYRES, S. W., AND S. K. CROFT 1986. The tectonics of icy satellites. In Satellites (J. A. Burns and M. S. Mat thews, Eds.), pp. 293-341. Univ. of Arizona Press, Tucson.

STEVENSON, D. J., A. W. HARRIS, AND J. i. LUN1NE 1986. Origins of satellites. In Satellites (J. A. Burns and M. S. Mat thews, Eds.), pp. 39-88. Univ. of Arizona Press, Tucson.

STEVENSON, D. J.. T. SPOHN, AND G. SCHUBERT 1983. Magnet i sm and thermal evolution of the terrestrial planets. Icarus 54, 466-489.

SUENO, S., M. CAMERON, J. J. PAPIKE, AND C. T. PREWITT 1973. The high temperature crystal chem- istry of tremolite. Amer. Mineral. 58, 649-664.

THURBER, C. H., A. T. HsuI , AND M. N. TOKSOZ 1980. Thermal evolution of Ganymede and Callisto: Effects of solid-state convect ion and constraints from Voyager imagery. Proc. Lunar Planet. Sci. Cortf. llth, 1957-1977.

TURCOTTE, D. L., AND G. SCHUBERT 1982. Geody- namics: Applications of Continuum Physics to Geo- logical Problems. Wiley, New York.

TURNER, F. J. 1981. Metamorphic Petrology: Mineral- ogic, Field, and Tectonic Aspects, 2nd ed. Hemi- sphere Publishing Corp. , Washington.

WASSON, J. T. 1974. Meteorites: Classification and Properties. Springer-Verlag, New York.

ZHARKOV, V. N., V. V. LEONTJEV, AND A. V. KO- ZENKO 1985. Models , figures, and gravitational mo- ments of the Galilean satellites of Jupiter and icy satellites of Saturn. Icarus 61, 92-100.

ZUBER, M. T., AND E. M. PARMENTIER 1984. Litho- spheric s t resses due to radiogenic heating of an ice- silicate planetary body: implicat ions for Gany- mede ' s tectonic evolution. Proc. Lunar Planet. Sci. Conf. 14th in J. Geophys. Res. Suppl. 89, B429- B437.