formation of the galilean satellites: conditions of accretion

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FORMATION OF THE GALILEAN SATELLITES: CONDITIONS OF ACCRETION

Robin M. Canup and William R. Ward

Department of Space Studies, Southwest ResearchInstitute, 1050Walnut Street, Suite 400,Boulder, CO 80302;[email protected]

Received 2002 June 17;accepted 2002 September9

ABSTRACT

We examine formation conditions for the Galilean satellites in the context of models of late-stage giantplanet accretion and satellite-disk interactions. We first reevaluate the current standard, in which thesatellites form from a ‘‘ minimum mass subnebula ’’ disk, obtained by augmenting the mass of the currentsatellites to solar abundance and resulting in a disk mass containing about 2% of Jupiter’s mass. Conditionsin such a massive and gas-rich disk are difficult to reconcile with both the icy compositions of Ganymede andCallisto and the protracted formation time needed to explain Callisto’s apparent incomplete differentiation.In addition, we argue that disk torques in such a gas-rich disk would cause large satellites to be lost to inwarddecay onto the planet. These issues have prevented us from identifying a self-consistent scenario for theformation and survival of the Galilean satellites using the standard model. We then consider an alternative,in which the satellites form in a circumplanetary accretion disk produced during the very end stages of gasaccretion onto Jupiter. In this case, an amount of gas and solids of at least $0.02 Jovian masses must still beprocessed through the disk during the satellite formation era, but this amount need not have been present all

at once. We find that an accretion disk produced by a slow inflow of gas and solids, e.g., 2 Â 10À7 Jovianmasses per year, is most consistent with conditions needed to form the Galilean satellites, including disk tem-peratures low enough for ices and protracted satellite accretion times of !105 yr. Such a ‘‘ gas-starved ’’ diskhas an orders-of-magnitude lower gas surface density than the minimum mass subnebula (and for many casesis optically thin). Solids delivered to the disk build up over many disk viscous cycles, resulting in a greatlyreduced gas-to-solids ratio during the final stages of satellite accretion. This allows for the survival of Galilean-sized satellites against disk torques over a wide range of plausible conditions.

Key words: planets and satellites: formation — solar system

1. INTRODUCTION AND BACKGROUND

The Galilean satellites have played an influential role inplanetary science, from helping to establish the Copernicanview of the solar system, to displaying a myriad of physicalcharacteristics and intriguing dynamical and chemical his-tories. At Jupiter is a satellite system worthy of the term ‘‘amini solar system,’’ whose origin may provide us withinsights into planetary formation processes, in particularthose involving the growth of gas giant planets.

In this work, we address the formation of the Galileansatellites in the context of an improved understanding of both Jovian accretion and the nature of satellite-disk inter-actions. Previous works have assumed a gas-rich protosatel-lite disk containing about 2% of Jupiter’s mass—the mass of the current satellites augmented to solar composition, or a‘‘ minimum mass subnebula,’’ in direct analogy to the plane-tary ‘‘ minimum mass nebula ’’ (see Fig. 1, from Pollack &Consolmagno 1982). However, we show in

x2 that pre-

dicted conditions in such a high surface density disk are dif-ficult to reconcile with the characteristics of the satellitesthemselves, especially when the effects of satellite-disk inter-actions are included (e.g., Ward & Canup 2002). Indeed, weare unable to construct a simple and self-consistent scenariofor the formation and survival of the Galilean satellites insuch a ‘‘ gas-rich ’’ disk model.

We then consider an alternative view in x 3. Although thecurrent satellites imply a lower limit $0.02M J for the massprocessed through the circumjovian disk during the satelliteformation era, all need not have been present in the disk atany one time. By analogy to mineral deposits in a pipe— whose thickness is indicative of the total amount of water

slowly processed through the pipe, rather than the total vol-ume contained in the pipe at any one moment—a circum-

jovian disk supplied by a slow inflow of nebular gas couldhave had an instantaneous disk mass much less than that of

the minimum mass subnebula, but so long as the diskexisted for multiple disk supply cycles, a buildup of anappropriate total mass in solids could occur. The satelliteswould then form more slowly, and in a less gas-rich environ-ment. Preliminary calculations (Canup & Ward 2002a) sug-gested that this model held promise for simultaneouslyaccounting for several key constraints, including the varyingice/rock content of the satellites, a partially differentiatedCallisto, and satellite survival against orbital decay. Steven-son (2001) independently reasoned that such a scenario— which he referred to as the ‘‘ gas-starved ’’ disk model, aname we adopt here—might provide an attractive alterna-tive. Here we demonstrate that conditions consistent withthe formation and survival of the Galilean satellites result

for a wide range of gas-starved disks produced during theslow inflow of gas and solids into circumjovian orbit.In the remainder of this section, we review some issues

concerning late-stage Jovian accretion, modes of satelliteformation, and relevant properties of the Galilean satellites.We discuss key disk processes in x 1.4.

1.1. Growth of Jupiter

In the core accretion model (e.g., Bodenheimer & Pollack1986; Pollack et al. 1996; Bodenheimer, Hubickyj, &Lissauer 2000), a prolonged phase of solid and gas accretiongives way to the runaway accretion of gas once a proto-planet has achieved a critical core mass of $15 M È. During

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the runaway phase, a positive feedback exists as the rate of inflowing gas is regulated by the rate of contraction of theenvelope, with the latter becoming increasingly rapid as the

planet grows (Pollack et al. 1996). As long as gas accretionis unabated, the giant planet remains extended to its accre-tion radius, which for Jupiter’s final growth stages is compa-rable to its Hill radius, RH ¼ aðM p=3 M Þ1=3, where a andM p are the planet orbital radius and mass, respectively, andM is the solar mass. Since RH=RJ % 744ðM p=M JÞ1=3

(where RJ ¼ 71; 492 km and M J ¼ 1:90 Â 1030 g ¼ 318 M Èare the current radius and mass of Jupiter), the planet in itsexpanded stage is vastly larger than the current Galileansystem, for which a=RJ ¼ 5:9 26:4 (Table 1).

The planet’s radius shrinks relative to RH when the rateof gas accretion can no longer keep up with that needed tocompensate for the planet’s increasing rate of contraction(e.g., Bodenheimer & Pollack 1986). A reasonable upper

limit on the gas accretion rate is _M M max $ 10À2 M yrÀ1

(e.g., Hayashi, Nakazawa, & Nakagawa 1985; Bodenheimer& Pollack 1986; Bodenheimer et al. 2000). Calculations of the growth of Jupiter by Tajima & Nakagawa (1997) thatdid not impose any limit on the gas accretion rate foundthat for M p > 70 M È, an accretion rate in excess of _M M max $ 10À2 M È yrÀ1 would be required to maintain

R p % RH (their Fig. 7). This suggests that in the later stagesof its growth, Jupiter would likely not fill its entire Hillsphere (see also Pollack & Bodenheimer 1989). Recent mod-els predict a full-mass Jupiter would require 106 yr to con-tract to R p $ 1:9RJ, at which time it would have an effectivetemperature $800 K (e.g., Burrows et al. 1997, 2001); by 107

yr, R p $ 1:5RJ and T eff $ 500 K (e.g., Hubbard, Burrows,& Lunine 2002). Thus, the Galilean satellites presumablyformed late in Jupiter’s growth.

Jupiter will likely have opened a gap in the proto-planetary disk prior to the completion of its growth; fortypical assumed values of the solar nebular viscosity, gapopening (see x 2) is predicted for M p ! Oð102Þ M È (e.g.,Bryden et al. 1999; Ward & Hahn 2000). However, recenthydrocode simulations in both two dimensions (e.g.,Lubow, Siebert, & Artymowicz 1999; Bryden et al. 1999)and three dimensions (e.g., Kley, D’Angelo, & Henning2001; D’Angelo, Henning, & Kley 2002; Lubow, Bate, &Ogilvie 2002) find continuing accretion onto Jupiter aftergap opening, with _M M $ 10À2 to 10À4 M È yrÀ1 (or a Jovianmass in 104 –106 yr), depending on the assumed nebular vis-cosity, disk surface density, and scale height. Actual gasaccretion rates could be smaller if the nebula were dissipat-ing (simulations typically assume a full minimum massnebula), or if the solar nebula had a lower viscosity than canbe adequately treated by hydrocode models.1

Simulations suggest that post–gap-formation gas flowwithin the Hill sphere (e.g., Fig. 2, from Lubow et al. 1999)involves streams of material entering near the L1 and L2points, which then mutually collide to yield two shockedregions. A significant decrease in specific angular momen-tum of the shocked material then causes it to spiral inwardtoward the planet (Lubow et al. 1999; D’Angelo et al. 2002).Within $0.1RH, there is evidence of a circumplanetary diskwith a generally circular prograde flow.

1.2. Satellite Formation Scenarios

Several modes of formation have been proposed for theGalilean satellites (see, e.g., review by Pollack, Lunine, &Tittemore 1991). In the ‘‘ spin-out disk’’ model, it is

Fig. 1. —Minimum mass subnebulae for the Jupiter and Saturn satellitesystems (from Pollack & Consolmagno 1984). The protosatellite disksurface density profile was constructed by spreading the mass of eachsatellite, augmented to match a solar proportion of elements, over anannulus centered on the current satellite orbit. Similarly dense disks havebeen utilized in previous models of Galilean satellite formation.

1 There is numerical viscosity associated with hydrocode simulationswhich generally limits modeled nebular disk viscosities to > O

ð10À4

Þ(e.g.,see discussion in Brydenet al.1999).

TABLE 1

Properties of the Galilean Satellites

Satellite a/RJ

M S

(1025 g)

RS

(km)

S

(gcmÀ3) f Si ¼ 1 À ðice=S Þ1 À ðice=SiÞ

Io ........................... 5.9 8.93 1821 3.53 1Europa................... 9.4 4.80 1561 3.01 0.851Ganymede ............. 15.0 14.8 2631 1.94 0.430.73

Callisto .................. 26.4 10.8 2410 1.83 0.420.66

Note. —Silicate massfractions fromSchubertet al. 1986.

FORMATION OF THE GALILEAN SATELLITES 3405



assumed that the contraction of the planet to a scale smallerthan the satellite system postdates nebular dispersal. As theplanet contracts, outer layers of the envelope containingsufficient specific angular momentum to remain in boundorbit are ‘‘ shed,’’ forming a circumplanetary disk (Kory-cansky, Bodenheimer, & Pollack 1991). A requirement of this model is that sufficient rock and ice are contained in theouter envelope to account for the mass of solids in the satel-lites, perhaps accomplished through convective mixingwithin the planet (Bodenheimer & Pollack 1986).

The ‘‘ accretion disk ’’ model assumes that the contrac-tion of the planet to a scale of less than a few RJ

occurred prior to the complete removal of the solar neb-ula. Late-inflowing gas containing sufficient angularmomentum for centrifugal force balance at an orbit of $20–30 RJ (e.g., Ruskol 1982), then leads to the forma-tion of a circumplanetary disk, similar to the situationdepicted in Figure 2. A requirement for this model isthat inflowing material contain a mixture of gas and sol-

ids, or that some fraction of local ice and rock still becontained in particles small enough to be carried inwardwith the gas. The model we advocate in x 3 is a varia-tion on this theme.

Satellite formation from an ‘‘ impact-generated disk ’’ isthe leading candidate for the formation of the Moon—andan intriguing possibility for Uranus—but the case for a lategiant impact for Jupiter is weaker, given its extremely lowobliquity. The ‘‘ co-accretion ’’ model involves the creationof orbiting material through collisions of planetesimalsoccurring within Jupiter’s Hill sphere, and appears mostconsistent with the scale and orbital characteristics of thesmall irregular satellites.

1.3. Galilean Satellite Constraints

We infer from the very existence of the Galilean satellitesthat at least the last generation of large, similarly sized satel-lites formed in conditions that allowed for their survival

Fig. 2. —Density close-up of the Roche lobe of a Jovian mass planet located in a gap, shown in Cartesian coordinates scaled by the planet’s orbital radius.Crosses indicate theL1 and L2 points; samplestreamlinesare shown by dashedlines (from Lubow et al.1999, their Fig. 4).

3406 CANUP & WARD Vol. 124



against orbital decay due to density wave interaction withtheir precursor gas disk (e.g., Ward 1997) and aerodynamicgas drag (e.g., Adachi, Hayashi, & Nakazawa 1976). Thephysical and orbital characteristics of the current satellitesalso provide key constraints (e.g., Table 1). The currentorbital radii (in particular those of the inner three) havebeen altered from their primordial values as orbits expandeddue to tidal interaction with Jupiter (e.g., Yoder & Peale1981). Even after whatever tidal expansion they may haveexperienced in their history, the Galilean satellites are cur-rently located in very central orbits compared to the JovianHill radius, with the outermosthaving a $ 0:035RH.

The total mass in solids (ice þ rock) in the current satel-lites is %4 Â 1026 g, or $2 Â 10À4M J. Assuming a solar gas-to-solids ratio Oð102Þ implies a required total mass to yieldthe satellites of 4 Â 1028 g. The observed radial gradient inthe satellites’ bulk densities is associated with a decreasingsilicate mass fraction, f Si, with distance. A silicate composi-tion for Io, a hydrated silicate composition for Europa, anda Ganymede and Callisto that are roughly 50% rock and50% ice are implied (e.g., Schubert, Spohn, & Reynolds1986). The standard interpretation is that the protosatelliteenvironment had a decreasing radial temperature profile,with sufficiently low temperatures exterior to the orbit of Ganymede to allow for significant incorporation of waterice (e.g., Lunine & Stevenson 1982).

The final constraint we consider is the apparent incom-plete differentiation of Callisto. The most recent measure-ments (Anderson et al. 2001) of the moment of inertia of Callisto find I =MR2 $ 0:3549 Æ 0:0042, i.e., too small for ahomogenous Callisto ðI =MR2 $ 0:4Þ, but too large fora completely differentiated object according to two- andthree-layer interior models that assume hydrostatic equili-brium (Anderson et al. 2001).2 At face value, these resultsimply that Callisto’s formation did not involve the large-scale melting of ice. Accounting for only the release of gravi-tational binding energy during accretion, and assuming thatenergy is deposited and radiated away from the growing sat-ellite’s surface, an accretion time !105 yr is required for asurface temperature rise DT 270 K in a Callisto-sized sat-ellite (e.g., Stevenson, Harris, & Lunine 1986, Fig. 6; also B.Lane and D. Stevenson 2000, private communication). Thisis an extremely restrictive constraint, given that the orbitalperiod at Callisto is only about 2 weeks. Predicted accretiontimes for Callisto assuming a minimum mass subnebula (seebelow) are 103 yr; this time interval would implyDT $ 1000 K from release of binding energy alone.

In contrast, similarly sized and composed Ganymedeappears highly differentiated, with I =MR2 ¼ 0:3118

Æ0:0005 (W. B. Moore 2002, private communication) and

an apparently molten core (Anderson et al. 1996). Thesurface of Ganymede also shows evidence of extensiveresurfacing, which is absent on Callisto. From the stand-point of a satellite formation model, the apparent Gany-mede-Callisto dichotomy reduces to three basic

possibilities: (1) Callisto and Ganymede both formed rap-idly, but conditions during accretion were such as to pro-duce a differentiated Ganymede but only a partiallydifferentiated Callisto. This possibility was evaluated byLunine & Stevenson (1982) and Coradini et al. (1989) andgenerally found to be quite restrictive.3 (2) Callisto’s accre-tion interval was substantially prolonged relative to those of the other Galilean satellites (e.g., Mosquiera et al. 2001). (3)The accretion interval of all of the satellites was prolonged,with observed evidence of melting/heating on the innerthree satellites influenced by subsequent heating due to tidalinteractions, possibly higher silicate mass fractions,4 and/orhigher subnebula disk temperatures. We believe that themost successful and least restrictive disk models are consis-tent with scenario 3.

1.4. Disk Processes

We define a ‘‘ minimum mass subnebula ’’ (MMSN) diskto contain a mass in solids equal to the mass of the currentsatellites augmented to solar composition and spread out to30 RJ, with an average surface density of solids of

s % 3 Â 10

3

g cmÀ2

and an average gas surface density G $ f s % 3 Â 105 g cmÀ2, where f $ 102 is the solar gas-to-solids ratio.

Orbital periods range from about 2 days at Io’s orbit toabout 2 weeks at Callisto’s orbit, with correspondinglyrapid collision times for orbiting material. The timescale foraccretion of an object of radius RS at orbital distance r (e.g.,Lissauer & Stewart 1993; Ward 1996) is

A % 1

S RS

sF g$ 50 yr

RS

2500 km

S

2 g cmÀ3

10

F g

Â 3 Â 103 g cmÀ2

s

r

15RJ

3=2

; ð1Þ

where is orbital frequency, S is the bulk density of theobject accreting, and F g 1 þ ðvesc=v1Þ2 is the gravita-tional focusing factor for colliding objects with a relativevelocity at infinity of v1 and a mutual escape velocity vesc.We stress that equation (1) is appropriate for a disk with anapproximately constant total mass. Alternatively, if mass isdelivered to the disk on a timescale that is long compared toequation (1), then the overall satellite accretion time will beregulated by the slower inflow rate, as discussed in x 3(eq. [27]).

Accreting solids will interact with the gaseous componentof the disk through both aerodynamic gas drag (e.g., Adachiet al. 1976) and gravitational torques (e.g., Ward 1993,

1997; Papaloizou & Larwood 2000). The orbital decay time-scale of an object due to gas drag within a disk with sound

2 The allowable amount of differentiation in Callisto has increased withrecently improved measurements that imply decreased values for I =MR2

over those obtained earlier. This has led to some skepticism in the undiffer-entiated Callisto interpretation (Peale 1999); others (McKinnon 1997)point out that nonhydrostatic effects could mimic the appearance of apartially differentiated satellite.

3 For satellites with optically thick atmospheres, LS82 found that a pre-ferred size range of infalling icy planetesimals would be subject to gas dragbreakup and then aerodynamic breaking during their descent into asatellite’s atmosphere. Icy fragments reaching the surface could then coolCallisto,which in combination with cooler circumjovian disk temperatures,could prevent Callisto from substantially melting while still predicting afullymelted Ganymede.

4 Thedegree of postformation heating in a satellitecan be affected by thesilicate mass fraction, since the amount of long-lived radioactive isotopes isproportional to f Si (Friedson& Stevenson 1983; Coradini et al. 1989).

No. 6, 2002 FORMATION OF THE GALILEAN SATELLITES 3407



speed c is

GD % 8

3C D

1

S RS

G

r

c

3

$103 yr10

C D

RS

2500 km

S

2 g cmÀ3

Â0:1

c=r

33 Â 105 g cmÀ2

G r

15RJ

3=2

;ð2Þ

where C D ¼ C 1 þ C 2½GM S =ðRS c2Þ2 is the drag coefficient(e.g., Takeda et al. 1985; Ohtsuki, Nakagawa, & Nakazawa1988), and C 1 and C 2 are constants of order unity. For aGalilean-sized satellite and c $ Oð1Þ km sÀ1, C D $ Oð10Þ;for objects too small to gravitationally perturb the gas,C D $ Oð1Þ.

A perhaps more important mode of interaction forlarge satellites is disk tidal torques (e.g., Ward 1993,1997). A satellite excites density waves in a gas disk atLindblad resonances, whose reaction torques on the satel-lite cause a net inward migration (‘‘ type I decay ’’) with atimescale that is inversely proportional to the satellite’s

mass, M S ,

I % 1

C a

M J

M S

M J

r2 G

c

r

2

$102 yr3

C a

2500 km

RS

3 2 g cmÀ2

S

Â c=r

0:1

2 3 Â 105 g cmÀ2

G

15RJ

r

1=2

; ð3Þ

where C a is a torque asymmetry parameter that is a func-tion of the disk’s radial surface density and temperatureprofiles (e.g., Goldreich & Tremaine 1980; Ward 1986,

1989, 1997; Artymowicz 1993; Tanaka, Takeuchi, &Ward 2002). The ratio of a satellite’s lifetime against typeI decay versus gas drag is independent of G ,

I GD

$ 0:1C D

10

3

C a

2500 km

RS

4

Â 2 g cmÀ3

S

2c=r

0:1

5 15RJ

r

2

; ð4Þ

implying that the most rapid loss mechanism for the finalstages of satellite growth may well be disk torques ratherthan aerodynamic gas drag.

Type I inward migration would transition to the typi-

cally slower type II decay if a satellite became largeenough to open a gap in the disk. One criterion for gapopening (e.g., Lin & Papaloizou 1986; Ward 1986) is thatthe cumulative torque on the disk by the satellite mustexceed the rate of angular momentum transfer due to thedisk’s viscosity. Adopting an ‘‘ alpha model ’’ for the diskviscosity (Shakura & Sunyaev 1973), cH ¼ c2=(where H % c= is the disk scale height), the gap openingcriterion assuming local wave damping is (e.g., Ward &Hahn 2000)

M S

M J> c

ffiffiffiffi

p c

r

5=2$ 10À4c

10À3

1=2 c=r

0:1

5=2

; ð5Þ

where c is Oð1 10Þ, and typical values range from 0.1for a highly turbulent disk to 10À4 (e.g., Cabot et al.1987). The mass cutoff in equation (5) is just larger thanthe mass of Ganymede for c ¼ 1 and ¼ 10À3. In thelimit of an inviscid disk, the gap opening criterionasymptotes to an inertial one based upon a comparisonbetween the gap opening time and the time for a satelliteto migrate across a distance equal to the gap width dueto type I decay (Hourigan & Ward 1984; Ward & Houri-gan 1989; Ward 1997). This gap opening criterion is

M S

M J>

G r2

M J

c

r

3$ 2 Â 10À5 M d =M J

0:02

c=r

0:1

3

: ð6Þ

If a satellite opens a gap, it will still generally experiencean inward migration that is controlled by the viscousspreading of the disk (type II decay), with a timescale

II % ¼ r2d

% 103 yr

10À3

rd

30RJ

3=20:1

c=r

2

; ð7Þ

where rd is the disk outer radius.The Kelvin-Helmholtz cooling time for the MMSN

disk assuming the only energy source is the internal heatof the disk is (e.g., Lunine & Stevenson 1982)

KH % C G T c

2 SBT 4d

$ Oð103Þ yr250

T c

3

; ð8Þ

where SB ¼ 5:67 Â 10À5 erg cmÀ2 sÀ1 KÀ4 is the Ste-phan-Boltzman constant, C is specific heat, T c and T d

are the midplane and effective photospheric disk tempera-tures, respectively, and is the vertical gas optical depth, G K , where K is the disk opacity in cm2 gÀ1. Fromthe equations of thermal and radiative equilibrium (e.g.,Schwartzchild 1958; see also Lunine & Stevenson 1982),

for a constant K with height the midplane temperature isapproximately related to the effective temperaturethrough

T c ffi 1 þ 3

21 À T neb

T d

4" #

( )1=4

T d ; ð9Þ

where T neb is the ambient nebular temperature, likely$100–150 K near Jupiter (e.g., Lewis 1974).

Relevant disk energy sources include luminosity fromJupiter and viscous dissipation. For a radially opticallythick disk in vertical hydrostatic equilibrium with scaleheight, H ¼ c=, Jupiter’s luminosity will provide energy toan annulus in the disk extending from r to

ðr

þDr

Þata rate

_E E J % 2 SBT 4JRJ

r

2

2r½H ðr þ DrÞ À H ðrÞ

% 9

4 SBT 4J

RJ

r

2c

r

2rDr ; ð10Þ

where T J is Jupiter’s temperature, and we have assumed aT c / 1=r3=4 dependence. Viscous dissipation and energy lossdue to radiative cooling within an annulus yield

_E E ¼ 94 2 G 2rDr ;

_E E rad ¼ À 2 SBðT 4d À T 4nebÞ2rDr : ð11Þ




2. SATELLITE FORMATION FROM AGAS-RICH MMSN DISK

A straightforward approach to satellite formation is toassume that a protosatellite disk containing all of the massnecessary to form the satellites was created on a timescalethat is short compared to the disk evolution timescale—i.e.,the minimum mass subnebula model described above. Inthis section, we discuss a number of problems with this

approach that have led us to abandon it in favor of thestarved disk model presented in the next section. Readerswishing to advance directly to this new model can skip to x 3without loss of continuity.

2.1. Previous Work

The best candidate mechanism for producing an MMSNdisk is the spin-out model. Another alternative would beforming a $0.02M J disk during a final stage of gas inflow toJupiter (e.g., Coradini et al. 1989; Makalkin, Dorofeeva, &Ruskol 1999), also discussed below.

Korycansky et al. (1991) modeled the growth of aSaturnian-mass planet, using a one-dimensional model

including rotation. They found that as gas accretion ceased,the planet rapidly contracted and shed its outer layers toform a disk containing roughly 1% of the primary’s massbetween about 25 and 400 planetary radii.5 Depending onthe composition of the outer envelope layers, the resultingdisk gas-to-solid ratio, f , could be similar to solar composi-tion or even higher (e.g., Pollack et al. 1991).

These results suggest the deposition of spin-out materialoccurs well outside the region of the regular satellites on atimescale of only $102 yr. However, unless gas accretionwas terminated on a similarly short timescale, this contrac-tion would actually be accompanied by additional gasinflow, which could affect the spin-out and disk depositionprocess. The dynamics of the creation of a Jovian spin-out

disk might also differ from that of a Saturnian-mass planet,given the planet’s higher mass (and thus faster initial con-traction rate in the absence of accretion; Pollack & Boden-heimer 1989), and that Jupiter may not have completelyfilled its Hill sphere during all of its gas accretion phase.Given such uncertainties—the resolution of which would beof great interest but is beyond the scope of the work here— we consider a spin-out–produced MMSN disk as definedabove.

The most comprehensive model of Galilean satellite for-mation in an MMSN-style disk is the work of Lunine &Stevenson (1982, hereafter LS82). They considered a staticdisk, with no mass inflow or viscous radial transport, the

justification being that the timescale for satellite accretion

(eq. [1]) is much shorter than that of the cooling time of thedisk (eq. [8]). Several difficulties were revealed by their inves-tigation. The disk profile in LS82 was constructed by settingthe disk temperature at Ganymede’s orbit to allow for thecondensation of ice. However (as the authors point out), forthe high MMSN gas surface density, such a low temperaturecould be achieved only for a practically inviscid disk. Ignor-ing any possible contribution from Jupiter’s luminosity, an

upper limit on the disk viscosity that allows for potential icestability near Ganymede’s orbit can be established by bal-ancing viscous dissipation (eq. [10]) with radiation fromthe disk surface (eq. [11]). Assuming a low, solely gaseousdisk opacity with, e.g., K ¼ 10À4 cm2 gÀ1 gives $ Oð10Þand T c $ 2T d ; for an ambient nebular temperatureT neb ¼ 150 K, Oð10À6Þ is then required for even thephotospheric temperature T d ð15RJÞ 230 K. Thus, form-ing ice-rich Ganymede and Callisto necessitates a very lowviscosity compared to typical values associated with circum-stellar disks, e.g., $ 10À4 –10À2 (e.g., Stone et al. 2000).Given that A5 KH for an optically thick disk, it could bedifficult to delay satellite accretion until the appropriatelycold and viscously quiescent disk conditions were achievedto produce icy satellites. Instead, a first generation of rock-rich satellites might result. The satellite lifetimes againsttype I radial decay discussed in x 1 (and indeed cautionedabout in LS82) would be exceedingly short in such a massivegas disk (eq. [3]). Thus, it is not obvious how asystem of rocky satellites could be retained long enough forice condensation to ensue.

In the LS82 study, Ganymede and Callisto form on simi-larly short timescales and with optically thick atmospheres.Estimates of their resulting surface temperatures suggestedthat both satellites would have undergone significant differ-entiation. If a protracted accretion of !105 yr is required toaccount for Callisto’s incomplete differentiation, this con-straint would not be satisfied in a MMSN, in which the sat-ellite accretion times would be short and similar (eq. [1]).6

An important advance was made by Coradini et al. (1989;see also Coradini & Magni 1984 for the Saturnian satellitesystem), who incorporated viscous evolution of an accretiondisk formed via nebula mass inflow into circumjovian orbit.With a methodology similar to that which we adopt in x 3,they computed the steady-state disk conditions using theLynden-Bell & Pringle (1974) formalism. The disk was con-ceived to be highly convective with a strong turbulence vis-cosity parameter $ 10À1 in the inner satellite region. Astarting assumption was that the steady-state mass of gasand solids in the disk should equal the augmented mass of the current satellites. This constrained the rate of mass infallto _M M $ 10À1 M È yrÀ1, corresponding to J ¼ M J= _M M J $103 yr.

The midplane temperatures predicted by the Coradini etal. models were >103 K inside 30RJ (e.g., Coradini et al.1989, their Fig. 18). A two-part scenario was thus proposedwhose first stage involved the creation of a hot, highly con-vective accretion disk during the rapid mass infall to Jupiter,while the second phase would occur after inflow stopped,during which time disk cooling allowed for ices andhydrated silicates to condense and the disk to becomequiescent. During the initial phase, solid accretion would beprecluded by high temperatures or viscous turbulence in thedisk, whereas in the latter, satellite accretion would proceedalong the lines of the LS82 model, i.e., in the absence of gasinflow. The assumption that satellite accretion would occurafter mass inflow to Jupiter has ceased would require that

5 This process provides one means of removing sufficient angularmomentum from the planet to reduce its rotation rate to somethingcomparable to that of the current gas giants, although there may be others(e.g., Takata & Stevenson 1996; Quillen & Trilling 1998).

6 We note that if special circumstances are invoked to prolong the accre-tion time of just Callisto(Mosquieraet al. 2001 LPSC), then theinner threemore rapidly forming satellites would still be subject to loss as discussedbelow in x 2.2—due toeither typeI orII decay orgas drag—so longas theirprecursor gas disk persists.




the inflow cut off extremely rapidly, specifically on a time-scale that is short compared to disk evolution processes.The initially hot and convective disk considered in Coradiniet al. would viscously spread on a timescale of only $ r2

d = , or Oð10Þ yr for the assumed values rd ¼ 102RJ

and $ 1015 cm2 sÀ1, a timescale much shorter than KH

from equation (8). Thus, it seems likely that much of thedisk mass would viscously evolve onto the central objectbefore temperatures cooled enough to allow for ices tocondense.

Makalkin et al. (1999) considered an accretion disk modelwith a much slower inflow rate _M M $ 10À6 M È yrÀ1, or J ¼ M J= _M M J $ 108 yr. They emphasized the impossibility of simultaneously satisfying the temperature and total massrequirements for the Galilean satellites. Two possible alterna-tive disk models were suggested: a highly inviscid MMSN diskwhich satisfied the total mass requirement (similar to thatwhich we consider in this section), or a low-mass, viscous andappropriately low-temperature disk (similar to that consideredin the next section). Makalkin et al. (1999) concluded that it isuncertain which disk model is more promising. In this sectionand the next, we extend this and earlier works by considering awide range of inflow rates and possible disk viscosities, as wellas the effects of satellite-disk interactions; the latter providesubstantial additional constraints, as we discuss next for thecase of the MMSN disk.

2.2. Satellite Survival

It was argued in x 1.4 that both accretion and type I decaytimescales are extremely short in a MMSN disk with

A I

% 0:5C a

3

RS

2500 km

4S

2 gm cmÀ3

2

Â 0:1

c=r

210

F g

f

100

r

15RJ

2

: ð12Þ

Given the closeness of this ratio to unity, explaining how aGalilean-sized satellite could accrete without being lost tothe primary is a key issue. Since A= I / f , one way toincrease the likelihood of satellite survival is to decrease theeffective f in the accreting region. A disk composed at theoutset of material with f 5 f would alleviate this situation,but a primary concern that has been raised with the spin-outmodel is whether it would be able to provide even a solarproportion of solids, given that solids in the planet will havelikely been primarily concentrated in the core and wouldneed to be well mixed in the outer envelope to be present inthe spin-out disk at all (e.g., Pollack et al. 1991). Similarconcerns exist for material inflowing from the solar nebulaafter its partial depletion of solids by the formation of the

giant planet cores. Alternatively, one could postulate thatthe gaseous portion of the disk viscously spreads duringaccretion. Recall, however, the temperature argumentabove, which implies that a nearly inviscid disk is needed, sothat 4 A; I .

Assuming large satellites were able to accrete, they mustcontinue to survive for the lifetime of their precursor disk.Here the relevant timescale comparison is

I

$ 1

C a

M J

M S

M J

r2 G

r2d

c

r

2

$Oð10À4Þ

10À6

ð13Þ

for the values given in equations (3) and (7), so that thesatellites are lost inward on timescales much shorter thanthat needed to viscously remove the disk for a cool, nearlyinviscid disk.

So far, the above expressions assume full-strength type Itorques. Could such torques and the resulting migration havebeen mitigated by wave reflection from disk boundaries (e.g.,Ward & Canup 2002; Tanaka et al. 2002)? If waves reflect off the outer and inner disk edge, they can return a portion of theirangular momentum flux to the resonance site. A standingwave pattern then develops on each side of the satellite as aresult of the superposition of outgoing and incoming wavetrains. In the case of total reflection, the returning wave modi-fies the launch conditions so that outgoing and incomingwaves are symmetrical about the primary/secondary line andthe net torque of the satellite vanishes. However, total reflec-tion can occur only if the waves do not dissipate during theirentire round trip; if they suffer significant dissipation, littletorque reduction will occur.

From the dispersion relationship for density waves, thewavenumber k $ D=c2ð Þ1=2

, where D m2ðS À Þ2 À 2

is the frequency distance from resonance, where D ¼ 0and is the epicyclic frequency (e.g., Goldreich &Tremaine 1979). The wavenumber increases as the wavespropagate radially, causing them to wrap up in thefamiliar spiral pattern with m number of spiral arms.Far external to the resonance, D m22

S andk mS =c; the most important resonances have wave-numbers of the order of m $ rS =c, for which k approaches r=H 2 and the wavelength is only a few per-cent of the scale height H ¼ c=. On the other hand,the surface density at the outer edge of the disk cannotfall off more sharply than a scale height, since steepergradients would be Rayleigh unstable (e.g., Lin et al.2000; see also Agnor & Ward 2002). Consequently,should the outbound waves make it as far as theMMSN disk outer edge, they would penetrate this wingof the disk and shock dissipate. On the inward side of the satellite, some wave reflection may be possible if theinner disk extends to the planet, but this would exacer-bate the tendency for satellite loss by reducing the out-ward torque on the satellite.

Even if the type I torques were somehow mitigated, satel-lites in a MMSN disk would still be vulnerable to orbitaldecay via aerodynamic drag. As noted by Lunine & Steven-son (1982), the gas drag timescale for the satellites is quiteshort in such a disk, $O(103) yr (eq. [2]); thus, they could belost on times comparable to or shorter than that needed toviscously remove the disk for 10À3.

From equation (6), Galilean-sized satellites could poten-tially open gaps in a low-viscosity disk, reducing the effec-tiveness of aerodynamic drag and causing a transition totype II decay. However, since the type II decay timescale isessentially the disk’s viscous lifetime, satellites would still belost as long as the disk is removed by diffusion. This fatecould be avoided if the disk were removed by some nonvis-cous process on a timescale shorter than % OðÀ1Þ yr.Photoevaporation is sometimes invoked as a nonviscousdisk removal mechanism, although the timescale for this isso long ($107 yr, e.g., Hollenbach, Yorke, & Johnstone2000) that the disk must be even more inviscid than our pre-vious estimate with < 10À7. Survival of the Galilean satel-lites via gap opening and transitioning to type II decaywould then require that (1) the disk was extremely inviscid




and long-lived, (2) all four satellites were able to open gaps,and (3) the protosatellite disk is eventually removed viaphotoevaporation.

One inconsistency in this picture is the requirement of such a low disk viscosity. The interaction of the satellitesthemselves with the disk provides a source of angularmomentum transport (e.g., Goodman & Rafikov 2001). Inthis regard, one must remain cognizant that gap openingdoes not actually shut off disk torques, but merely providesa mechanism whereby a satellite can seek a quasi-equili-brium where the inner positive and outer negative torquesbalance. Angular momentum continues to flow outwardacross the satellite’s orbit, and this transport can be charac-terized as an effective viscosity. For instance, setting thetime, tDL $ DL=À, necessary to remove all of the angularmomentum, DL % Oð G r4Þ, from the interior portionof the disk by the one-sided disk torque,À % l

2S G r2ðrÞ2ðr=wÞ3, equal to an equivalent diffusion

timescale $r2/ eff , allows one to estimate an effectiveeff $ O½ l2

S ðr=H Þ5ðH =wÞ3 % 10À3ðH =wÞ3, where w is thegap half-width, H is the disk scale height, and lS M S =M J.For w $ H , eff is orders of magnitude larger than that nec-essary to allow for a photodissociative removal of thecircumplanetary disk.

An extreme case would be to assume that a Galilean-sizedsatellite could open a very broad gap, so that disk materialwas shepherded out beyond the 2 : 1 and 1 : 2 resonances(although this may not be possible given the close spacing of the satellites). This would act to reduce bothC and eff , withðH =wÞ3 $ a few Â 10À3 and eff $ 10À6. The zero-orderdisk torqueswould then be largely shut off, but the more dis-tant first-order resonances (i.e., the 3 : 1 and 1 : 3) wouldstill remain in the disk. These are m ¼ 2 Lindblad resonan-ces with pattern speeds of ps $ 3S =2, S =2, respectively;both act to excite the satellite’s eccentricity (Goldreich &Tremaine 1980; Goldreich & Sari 2002). Concentrating onthe inner Lindblad resonance (ILR) as an example, the first-order torque is (e.g., Ward & Hahn 2000; Ward & Canup2000)

À1 $ 2

6e2

l2S ð G r

2ÞðrS Þ2 S

res

2

Â 2d 2b

ð2Þ1=2

d 2þ 10

dbð2Þ1=2

d þ 20b

ð2Þ1=2

0@

1A2

%Oð10Þ e2 l

2S ð G r

2ÞðrS Þ2 ; ð14Þwhere the bracketed quantity is evaluated at the 3 : 1 reso-nance, ¼ rres=r ¼ 1=32=3 ¼ 0:481. The satellite eccentricity

would then be excited at the rate

7

1

e

de

dt% ÀÀ1

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 À e2

p

M S e2r2S

1 À ps

S

ffiffiffiffiffiffiffiffiffiffiffiffiffi1 À e2

p % 3 lS ld S ; ð15Þ

where ld M d =M J. For r ¼ 15RJ, this corresponds to acharacteristic e-growth time of only $O(103) yr. To shut off this torque, the satellite would have to further shepherd thedisk material DM d 2 G r2ð1 À Þ beyond the 3 : 1, i.e.,

an additional distance r=r ¼ 2À2=3 to 3À2=3 ¼ 0:15, butwould have only the first-order torque C1 available to dothis. This requires

R À1dt ¼ DL $ DM d r2ð r=2rÞ % a few

Â10À2 ld M Jr2S more angular momentum be removedfrom the inner disk. Combining this with the eccentricityequation then implies a total e-growth during a hypotheticalwide-gap phase of De2 $ a few Â 10À2ð ld = lS Þ $ Oð1Þ. Atlarge eccentricities, other higher order resonances wouldbecome important as well. Thus, even assuming a highlyinviscid disk and the formation of very broad gaps aroundthe satellites, it is dubious that the Galilean satellites couldhave occupied a MMSN disk for $107 yr (i.e., $104 e-fold-ing times) without developing instabilities due to forcedeccentricity growth.

2.3. Jovian Obliquity

We close this discussion by mentioning some completelyindependent evidence against a long-lived $107 yr diskbased on Jupiter’s small $3 obliquity. The current spin axisprecession period of Jupiter is $4:5 Â 105 yr, due mostly tothe solar torque exerted on the Galilean satellites (e.g.,

Ward 1975; Harris & Ward 1982; Tremaine 1991). How-ever, this would have been up to $O(102) shorter if aMMSN were present. If this disk were subsequently photo-evaporated after the solar nebula itself was dissipated,Jupiter’s precession frequency would have drifted throughone of the mutual orbital precession frequencies of Jupiterand Saturn, i.e., the so-called 16 that describes the preces-sion of their orbital nodes with a period of P 16 $ 5 Â 104 yr.An adiabatic passage could generate an obliquity of (Henrard & Murigande 1987; Ward & Hamilton 2002)

cos ¼ 2

ð1 þ tan2=3 I Þ3=2À 1 ; ð16Þ

where I $ 0=36 is the amplitude of Saturn’s perturbation of the Jovian orbital inclination (e.g., Applegate et al. 1986).

This gives a limiting obliquity of ¼ 25=6. If passage is fastenough to be nonadiabatic, the final obliquity is ratedependent, i.e.,

% 2I

P 16

2

_s

1=2

ð17Þ

(Ward, Columbo, & Franklin 1976), where s denotes thespin axis precession parameter, which is a function of the circumplanetary disk and satellite masses in addition tothe Jovian oblateness (e.g., Ward 1975). We define pole s= _s and calculate its value when equation (17) iscomparable to Jupiter’s current obliquity, viz., pole $ P 16ð=2I Þ2 % Oð105Þ yr. But since the change in s

would be due primarily to dissipation of the protosatellitedisk, we conclude that a disk life much longer than this isnot consistent with Jupiter’s low-obliquity spin state.

Thus, multiple difficulties emerge from the standard‘‘ minimum mass subnebula ’’ model, and a single, self-consistent scenario does not simultaneously satisfy the sys-tem constraints. The simplest means of removing these diffi-culties would be to have the final stages of satelliteformation occur in an environment that was enhanced insolids relative to the solar gas-to-solids ratio. By reducing f ,the lifetimes of the satellites against various processes asso-ciated with the gas disk are increased; in addition, a lower

7 The innermost corotation resonance that would damp e would beinsidethe gap.




mass gas disk eases the constraints on the disk’s viscosityneeded to yield sufficiently low temperatures.

3. SATELLITE FORMATION FROM A GAS-STARVEDACCRETION DISK

Given the extremely rapid circumplanetary disk processtimescales, it is difficult to imagine that satellite accretioncould ‘‘ wait ’’ until gas inflow to Jupiter had completelystopped. Instead, we consider it more reasonable that gasaccretion onto Jupiter ended on a timescale that is longcompared to relevant timescales in a circumjovian disk, sothat satellite growth occurred in the presence of the final gasinflow. Assuming that the cutoff in gas accretion was suffi-ciently prolonged that Jupiter had time to contract to a scalesmaller than the current satellite orbits, the gas inflow ratesduring satellite accretion would presumably be lower thanthe peak rates experienced during Jovian runaway gasaccretion; otherwise, the planet would have likely still beenquite distended. The creation of a circumplanetary accre-tion disk during such a slow-inflow period would yield a lessmassive steady-state disk than the MMSN considered aboveand by earlier works. Here we explore this alternative gas-starved disk model.

3.1. Origin of an Accretion Disk

Recent hydrocode simulations describing planet-nebulainteractions in the waning stages of Jupiter’s growth revealdynamical behavior within its Hill sphere and approachingthe region of the regular satellites. Lubow et al. (1999) usedtwo-dimensional simulations in which the planet’s orbit wasfixed to consider the behavior of a Jovian mass planetembedded in a disk with an assumed viscosity parameter of ¼ 4 Â 10À3. The pattern of flow shown in Figure 2 is simi-lar to that obtained by recent three-dimensional simulationsby D’Angelo et al. (2002) that additionally consider the

back reaction of the perturbed disk on the planet. Flow intothe Hill sphere enters through the L1 and L2 Lagrangepoints via two streams of material. After traveling abouthalfway around the Hill sphere, the streams mutually col-lide, producing a shocked region at a distance from theplanet of about 100–200 RJ. The shocked material thenflows toward the planet, forming a more quiescent progradeinterior disk. D’Angelo et al. (2002) find a high-density‘‘ core ’’ surrounding the central planet’s position with anorbital radius $0:04RH $ 30RJ for a Jovian mass planet(their Fig. 17).

Unfortunately, with current resolutions the inner satelliteregion is typically described by only a few numerical‘‘ cells.’’ Mass is removed from the innermost region in the

course of a simulation to mimic accretion onto the planet atsome assumed rate, which could influence inner disk surfacedensities. The hydrocode simulations do not self-consistently calculate the size of the central planet;8 it is thusnot clear what physical planet radius would be appropriatefor the implied mass accretion rates, and the question of therelative timing of planetary contraction versus nebular dis-persal remains an open and important one.

Below we formulate a generic disk model motivated bythe overall properties of the above results, in which several

key (but uncertain) quantities are treated as freeparameters.

3.2. Gas Disk Model

Consider a circumjovian accretion disk (see Fig. 3) sup-plied by an inflow of material with some characteristic maxi-mum specific angular momentum, j (e.g., Cassen &Summers 1983; Coradini et al. 1989). Inflowing material,

containing some mass ratio f of gas-to-solids, is assumed forsimplicity to be deposited into centrifugally supportedorbits with a uniform flux per area, F in, in the region extend-ing to the radial distance rc j 2GM J. The total rate of massinflow is F Ã ¼ F inr2

c . The disk is assumed to have someouter edge, rd , at which disk material is stripped from thedisk, perhaps by solar torques or through collision with thehighly shocked regions that appear to be associated withmass flow into the Hill sphere. The steady-state gas surfacedensity is then (Lynden-Bell & Pringle 1974; see Appendix)

G ðrÞ % 4F Ã15

5

4À

ffiffiffiffi ffirc

rd

r À 1

4

r

rc

2

for r < rc ;

ffiffiffiffirc

rr À ffiffiffiffiffir

crd

r for r ! rc ;

8>>><>>>:

ð18Þ

where again the disk viscosity is cH ¼ c2=, and c isthe midplane sound speed.

A simple model estimates disk temperature by consider-ing a balance between energy produced from mass infall,viscous dissipation, heating due to Jupiter’s luminosity, andradiative cooling from the disk surface (eqs. [10] and [11];e.g., Coradini et al. 1989; Makalkin et al. 1999). With theeffective and midplane sound speeds given byc2

d ðrÞ RT d ðrÞ= lmol and c2ðrÞ RT cðrÞ= lmol, whereR ¼ 8:31 Â 107 ergs moleculeÀ1 KÀ1 is the gas constant,and lmol is the molecular weight in g moleculeÀ1, with

lmol ¼ 2 for molecular hydrogen, the energy balance at anorbital radius r is

9

4 SBT 4J

RJ

r

2c

rþ c2

G

" #¼ 2 SB

2c2d

R

4

ÀT 4neb

" #;

ð19Þwhere 1 þ 3

2½rc=r À 1

5À1, with the second term arising

from the mass infall. At each disk radius, extending from

8 An exception are recent simulations by A. Coradini presented at the2002June Eurojove conference in Lisbon, Portugal.

Fig. 3. —Schematic of accretion disk model described in x 3. Solids andgasare assumed to be deliveredto circumplanetary orbit with some charac-teristic specific angular momentum such that they achieve orbit about theplanet in a region extending from the surface of the planet out to somedistance, rc. Once in orbit, solids rapidly accumulate into objects largeenough to decouple from the gas, while the gas component of the diskspreads viscously both onto the planet, and out to some assumed outeredge, rd .




the surface of the planet to outer edge rd , we use an iterativemethod to determine T d ðrÞ, T cðrÞ, ðrÞ, and G ðrÞ self-con-sistently. Given an input value for , F Ã, and K , an initialguess for G ðrÞ is used to calculate T cðrÞ from equation (18).These values, together with c ¼ cd

È1 þ 3=2

Â1 À ðT neb=

T d Þ4ÃÉ1=8

, are then used to recalculate G ðrÞ from equation(19), and the process is repeated until convergence isachieved.

Table 2 contains our nominal disk parameters. The rele-vant disk opacity is quite uncertain. For a dust-free disk, themean Rosseland gaseous opacity is K $ 10À4 cm2 gÀ1 (e.g.,LS82; Ikoma, Nakazawa, & Emori 2000) for T $ 200 Kand a density G $ 10À5 g cmÀ3 (Mizuno 1980; cf. Steven-

son et al. 1986), while absorption by micron and submicronsized grains could provide opacities as high as K $ 10À1 to 1cm 2 gÀ1 for relevant disk temperatures (e.g., Bodenheimeret al. 1980; Chiang et al. 2001). Given this, we consider caseswith K ¼ 10À4, 10À2, and 1 to bracket a range of plausibleconditions.

3.3. Disk Temperature Profile

The inflow rate is constrained by the requirement thattemperatures be low enough for ice in the general regionnear Ganymede and Callisto’s orbits. Assuming that vis-cous dissipation is the dominant energy source (as it is in theregular satellite region for all of the cases considered here),and T 4

d

4T 4

neb

,

T 4d %92

8 SB G ¼ 32F Ã

8 SB1 À 4

5

ffiffiffiffiffirc

rd

r À 1

5

r

rc

2" #

ð20Þ

for r < rc, and T d / 1=r3=4 approximately. The effective disk temperature is a function of the rate of inflow, F Ã, but inde-

pendent of both the gas surface density and the disk viscosity .Thus, for optically thin disks, F Ã and the disk scale parame-ters (rc and rd ) determine the disk temperature profile. Foroptically thick cases, the disk central temperature is depend-ent on the choice of and K , as these affect the disk gassurface density and therefore (from eq. [9]). Makalkin etal. (1999) considered a restricted range of inflow rates andconcluded that

!10

À3 would be needed to produce an

appropriately cool disk. However, we show below that forany (and K ), there exist values of F Ã that yield therequired disk temperatures. Whether they are realistic ornot must be determined by gas accretion models.

Figures 4 and 5 display predicted disk temperature pro-files for two values of the inflow timescale, G M J= _M M J, forthe nominal model with K ¼ 10À4 cm2 gÀ1. For G ¼ 104 yr(Fig. 4), temperatures are !O(103) K throughout the regu-lar satellite region even with K ¼ 10À4. A slower inflow rateyields a more appropriate disk temperature profile; Figure5c shows T cðrÞ and T d ðrÞ for G ¼ 5 Â 106 yr and ¼ 5 Â 10À3. A disk pressure is estimated byP ¼ RG T c= lmol; Figure 5d shows the implied temperature

versus pressure profile. For low opacities and G ! 3 Â 106

yr, disk conditions are consistent with ice stability exteriorto Ganymede’s orbit. The water-ice stability line shown inFigure 5d (from Prinn & Fegley 1989) should not be viewedas a sharp boundary, since the inflowing solids will be a mix-ture of ice and rock, and their survival against evaporationwill be size-dependent. In addition, some inward migrationof the satellites themselves likely occurred as a result of typeI interaction with the disk (see

x3.5 and Figs. 7–9 below), so

it is possible that they formed at somewhat greater orbitaldistances.

Figure 5a shows G ðrÞ for G ¼ 5 Â 106 yr with ¼ 5 Â 10À3. For a given inflow rate, G / 1=, and evenfor a relatively low-viscosity disk ½ $ Oð10À4Þ, predictedgas surface densities for F Ã slow enough for ice stability areorders of magnitude below the MMSN. In steady state, thetotal mass of gas with r rc in Figure 5a is only $afew Â 1025 g; for r < rc, the radial surface density profile isapproximately G / 1=r3=4. Figure 5b plots ðc=rÞ as afunction of radius; in the inner disk, the disk aspect ratioincreases gradually with radius, with ðc=rÞ / r1=8. In theouter disk T d T neb and the disk profile ‘‘ flares ’’ morerapidly with radius, with

ðc=r

Þ /r1=2.

In the case of a very high opacity disk ðK ¼ 1Þ, evenslower inflow rates are required to yield appropriately lowdisk temperatures. Figure 6 shows the resulting disk for G ¼ 108 yr and K ¼ 1. Midplane temperatures are some-what high in this case for significant incorporation of icesnear $15RJ (although inward migration of the satellitescould ease this constraint; see Fig. 9). A somewhat broaderand perhaps more likely range of inflow rates is possibleassuming lower opacities and a less dust rich disk, which hasbeen argued for previously (LS82).

A growth time of G ¼ 5 Â 106 yr is orders of magnitudelonger than that characteristic of runaway gas accretion,implying that the Galilean satellites formed as gas accretiononto Jupiter was abating, either as the nebula itself was dis-sipating or as inflow slowed as a result of Jupiter opening agap. In the latter case, G ¼ 5 Â 106 yr corresponds closelyto that obtained by Bryden et al. (1999) for accretionthrough a gap onto Jupiter for a nebular viscosity alphavalue of $a few Â 10À4. This slow rate of gas accretionwould need to persist for at least $0:02 G $ 105 yr in orderfor a mass equal to the reconstituted mass of satellites to beprocessed through an appropriately low-temperature disk.

The inner disk properties are relatively insensitive tochanges in the assumed outer edge of the disk. The valuerd ¼ 150RJ was motivated by the shocked regions seen inLubow et al. (1999) and D’Angelo et al. (2002); a moregeneric choice would be the radius at which a disk of orbit-ing gas would begin to extend outside Jupiter’s Hill sphereand thus be subject to removal via solar tides, orr > 0:7RH $ 500RJ. For the case shown in Figure 5, settingrd ¼ 500RJ leads to an increase in G by a few tens of per-cent for r < rc over that obtained with rd ¼ 150RJ. Thesteady-state gas surface densities are also not very sensitiveto the choice of rc.

3.4. Evolution of Disk Solids

In order to yield satellites from an accretion disk, solidmaterial ðrock þ iceÞ must be supplied either by direct trans-port of small material into the disk with the gas inflow, orby capture of heliocentrically orbiting solids as they pass

TABLE 2

Nominal DiskModel

T J 500 K

rc ............... 30RJ

rd ............... 150RJ

f ................ 100




through the disk. A Sun-orbiting particle of radius Rs wouldhave its motion dominated by the motion of the gas if itsstopping time due to gas drag, te, is shorter than its orbitalperiod (e.g., Weidenschilling & Cuzzi 1993), withte Rss=ðcG Þ5 1=, or

Rs5Oð1Þ m2 g cmÀ3

s

G

4 Â 10À11 g cmÀ3

Â c

0:8 km sÀ1

r

5 AU

3=2; ð21Þ

where the nominal nebular values shown are for a minimum

mass solar nebula at 5 AU with 150 K and ðH =rÞ ¼ 0:07.Thus, particles much smaller than $1 m in radius would beexpected be delivered to the accretion disk with the hydro-dynamic gas inflow.

Once a circumplanetary disk has formed, solids couldalso be captured as a result of gas drag as they passedthrough the disk. This process would require that a tra-versing particle encounter a mass much greater than itsown during passage through the disk, or

Rs5Oð1Þ m G

500 g cmÀ2

2 g cmÀ3

s

: ð22Þ

For disk surface densities appropriate to a slow inflow

rate and ! 10À3, particles small enough to be capturedas they pass through the disk would also be small enoughto have had their motion dominated by the nebular gasmotion (eq. [21]), and so would have likely instead beendelivered to the disk with the inflowing gas. It is possiblethat both processes played some role in supplying disksolids, although if capture had been significant it is diffi-cult to explain why a more extended regular satellite sys-tem would not have resulted.

It is uncertain what the appropriate gas-to-solid massratio of inflowing material would be; a nominal guess wouldbe a solar ratio of f $ 102, but this fraction could be largerif most of the mass in nebular solids at the end of Jupiter’sgrowth is contained in larger sized planetesimals. However,some fraction of material at small sizes would be expectedbecause of ongoing production during fragmenting planet-esimal-planetesimal collisions, particularly given the poten-tial collisional energetics at the edge of a Jovian-openedgap, from which most of the Hill entering material appearsto originate (Lubow et al. 1999).

Solids delivered to the disk with the inflowing gas wouldachieve orbit around Jupiter at a similar orbital distancethan that at which the gas achieved centrifugal balance, i.e.,with r < rc. The timescale for collisional accretion assumingsome mass fraction of solids 1= f can then be compared tothe disk’s characteristic viscous time, to determine to what

Fig. 4. —Disk steady-state surface densities and temperature profiles for a fast-inflow circumjovian accretion disk, with the nominal model and G M J= _M M J ¼ 104 yr and K ¼ 10À4 cm2 gÀ1. The disk midplane, T c, and effective, T d , temperatures are shown by the solid and dashed lines, respectively.(a – b) A low-viscosity disk, with ¼ 10À4, and (c – d ) a high-viscosity disk with ¼ 0:01. In both cases, resulting temperatures are too high for the incorpora-tion of ices in theregion of theregularsatellites.




extent particles could grow before being viscously removedfrom the disk, provided that turbulence is insufficient to pre-vent accretion. Setting A ¼ , with G from equation (18)for r < rc, and s ¼ G = f gives

Rs $ Oð102Þ mF g

10

100

f

5 Â 10À3

22 g cmÀ3

s

Â F ÃM J=5 Â 106 yr

15RJ

r

1=20:1

c=r

4

: ð23Þ

This estimate implies that once in circumplanetary orbit,

objects large enough to decouple from the gas could colli-sionally aggregate on timescales much shorter than those onwhich they would be carried inward or outward with the gasdisk as it viscously spread, over a wide range of f values.

Once having grown large enough to decouple from thegas disk, solids orbiting Jupiter would still interact with thegas through aerodynamic gas drag. The orbiting gas will bepressure-supported, so that the difference in its orbitalvelocity from Keplerian velocity is (e.g., Whipple 1972; Wei-denschilling & Cuzzi 1993)

Dv vK À vG ¼ Àð2G ÞÀ1 dP

dr$ c

c

r

: ð24Þ

For a disk with minimum temperature of $150 K andðc=rÞ $ 0:1, Dv ! 1 km sÀ1.9

The ratio of the accretion time to the lifetime against gasdrag for a Jupiter-orbiting particle too small to gravitation-ally focus the gas (e.g., Ward 1993; Makalkin et al. 1999) is

A GD

$ 2 Â 10À3 10

F g

f

100

c=r

0:1

3

: ð25Þ

Thus, objects would accumulate on timescales much shorterthan their orbital decay time as a result of gas drag for awide range of gas-to-solids ratios. We expect that solids

delivered to the disk will rapidly accrete near to the radialdistance at which they achieve centrifugal force balance, sothat a region of accreting solids will exist for RJ < r < rc.This implies that in order to account for the current locationof the Galilean satellites, rc $ 20–30 RJ; this range could besomewhat larger if some inward type I migration isaccounted for (see x 3.5), perhaps rc $ 35–40 RJ.

Fig. 5. —Predicted steady-state (a) surface density, (b) scale height, (c) temperature, and (d ) temperature vs. midplane pressure profiles for a slow-inflow,low-opacity circumjovian accretion disk, with ¼ 5 Â 10À3, G M J= _M M J ¼ 5 Â 106 yr, K ¼ 10À4 cm2 gÀ1, and the nominal disk model (see Table 2). Diskmidplane and effective temperatures are shown by the solid and dashed lines, respectively; positions of the current Galilean satellites are starred. The diskbecomes isothermal (with T d ¼ T neb ¼ 150 K)for r ! 40RJ. Shown in(d ) is thewater-ice stabilitycurve(dot-dashedline) from Prinn & Fegley(1989),whichisclose to thepredicted disk temperaturesat the current locations of Ganymede and Callisto.For a fixed value of G , the gas surface density is inversely propor-tional to , asis the pressure. For the choiceof K and G shownhere,the disk is opticallythin (with 0:01 < < 0:1 in theregion of theregularsatellites), so that

T d % T c, anda similar temperature profileresults forany choiceof . Lower (higher) temperatures result for slower (faster) inflow rates.

9 In comparison, the radial inward/outward velocity of the circumjoviangas due to its viscous evolution is orders of magnitudes less, $0.1–1 m sÀ1

fordisk values 10À3 to10À2.




As gas and solids are delivered to the disk, the gas thensustains a quasi–steady state, while the surface density of solids will build up over time. While the gas component of the disk viscously spreads outward and onto the planet, thesolids rapidly accumulate in the region where they are ini-tially delivered, providing a mechanism for accreting largesatellites in a limited region extending from the surface of the planet to a characteristic distance, rc, with the latterdetermined by the specific angular momentum of the inflow-ing material. This effective spatial fractionation of solidsprovides a means to explain why large satellites are notfound at distances more commensurate with tidal stability,i.e.,

$RH, as might be expected, e.g., in the spin-out model.

In the slow-inflow accretion disk, the overall formationtime of the satellites is regulated by the time necessary todeliver an amount of solids M S to the disk,

acc;in ¼ G f M S =M Jð Þ ; ð26Þrather than by equation (1), with acc;in ! 105 yr for G ! 5 Â 106 yr and f ¼ f . For this accretion timescale,and in the limit that an accreting satellite forms from smallmaterial whose energy is deposited near to the satellite’s sur-face, the balance of accreted versus radiated energy gives asurface temperature rise DT 300 K for a 2500 km radiussatellite (Stevenson et al. 1986, Fig. 6), potentially consistentwith an initial ‘‘ cold start ’’ for Callisto.

For the disk shown in Figure 5, the ratio of gas tosolids in the region of the regular satellites decreasesfrom f ¼ 102 to O(1) in the time required to buildup the satellites’ combined mass in solids, implying thatthe final stages of satellite accretion occur in relativelygas-free conditions. This appears consistent with theresults of Coradini et al. (1989), who calculated pre-dicted satellite masses and spacing as a function of theprotosatellite disk gas content using a feeding zoneapproximation; the best agreement with the Galileansatellites was obtained for a low-mass gas disk case.

3.5. Satellite Migration and Survival

From equation (3), the timescale for orbital decay of a large satellite due to type I interaction with the gasdisk can be very rapid. In the cases favored here, thesteady-state mass of the gas disk, M d , is significantlyless than 0:02M J, which acts to lengthen the migrationtimescale I. However, the time necessary to accrete thesatellites is also increased (eq. [26]), and so the conse-quences of potential satellite migration must still beaddressed.

We first consider the dependence of the migration rate onsatellite orbital radius. From equations (1), (18), and (20),the timescale for orbital decay due to type I interaction with

Fig. 6. —Predicted steady-state(a) surface density, (b) scale height, (c) temperature,and (d ) temperature vs. midplanepressureprofilesfor an extremely slowinflow, high-opacity disk, with ¼ 5 Â 10À3, G M J= _M M J ¼ 108 yr, K ¼ 1 cm2 gÀ1, and the nominal disk model. The disk is isothermal for r ! 25RJ. Posi-tions of the current Galilean satellites are starred. Predicted effective disk temperatures at Ganymede and Callisto’s orbits are close to the water-ice stabilitycurve; however, disk midplane temperatures at Ganymede are somewhat higher. For this choice of K , the resulting disk is optically thick, with 3 < < 30 inthe region of the regular satellites.




the disk increases with orbital radius, with10

I / r1=2

M S

1 À 4

5

ffiffiffiffi ffirc

rd

r À 1

5

r

rc

2" #À1

; ð27Þ

where M S is the satellite mass, r < rc, and a viscosity-domi-nated energy production is assumed.11

The predicted rate of type I migration then increases as

the orbital radius decreases, so that once a satellite began tomigrate, its inward speed would continue to increase as itapproached Jupiter. A dispersal of the gas or some othertorque-modifying action would then be needed to halt a sat-

ellite’s migration prior to its demise. A simple expectationfor a satellite system that had undergone significant butincomplete migration might be to find a single exterior satel-lite—whose type I decay timescale had, e.g., exceeded thedisk lifetime—with all massive interior satellites having beenlost to inward decay. While this configuration is reminiscentof the Titan-dominated Saturnian system (Canup & Ward2002b), it is quite distinct from the Galilean multiplesatellite system.

In fact, the predicted type I decay timescales for the Gali-lean satellites are all quite similar (e.g., Figs. 7b, 8b, and 9b)given their current positions and masses. If these rates weremore dissimilar, e.g., if IðCallistoÞ= IðIoÞ41, then a forma-tion history involving significant migration could be arguedagainst by the very existence of the inner satellites. Instead,their similar type I timescales suggests that (1) they allmigrated somewhat and were saved from destruction by thecessation of gas supply to the disk on a timescale theirmigration time, or (2) migration was insignificant for thedisk conditions in which they formed. We consider bothpossibilities below.

A basic requirement for forming a satellite in the presenceof type I torques is that I > acc;in, which places an upperlimit on the gas surface density at the end of satellite forma-

11 The expression above is for full-strength interior and exterior disktorques acting on the satellites. Significant reflection off an inner boundaryedge (e.g.,the planetarysurface) could reducethe interior torques forclose-in satellites and actto increase their inwardmigration rate; similarly, reflec-tion off an outer boundary edge could mitigate the exterior torques andlessen the tendency to migrate inward for outer satellites, although in x 2.2

we concluded that such a situation is unlikely for an expected outer diskedge profile.

10 In equation (3) we considered an arbitrary surface density, while inequation (27) the radial surface density and temperature profiles areaccounted for.

Fig. 7. —Slow-inflow, low-opacity steady-state disk, with ¼ 5 Â 10À3, G M J= _M M J ¼ 5 Â 106 yr, K ¼ 10À4 cm2 gÀ1, and the nominal disk model (samecase as in Fig. 5). The temperature profile (with the solid line indicating midplane temperature, and the dashed line effective temperature) and the water-icestability curve (dot-dashed line) are shown in (a). In (b) are plotted the type I and gas drag orbital decay timescales for the Galilean satellites with their currentpositions and masses (stars and diamonds, respectively), as well as the time required to deliver a mass in solids equal to the mass of the satellites to the disk, acc;in (solid line) for this value of G . An estimated migration history for the satellites to end at their current locations is shown in ( c), including effects due togasdrag and type I torques (assuming C a ¼ 3:5 and C D ¼ 10). Theeffectof a linearsatellite growthin mass over time acc;in hasbeen included.The type I time-scales and migration historiesassume full-strengthtorques, and that the satellites migrate independently.




tion, and therefore a lower limit on the disk -value. From(4), (19), and (27), for r

rc,

I acc;in

% 3

fC a

M J

M S

2c

r

4

% Oð1Þ 100

f

c=r

0:1

41026 g

M S

2

5 Â 10À3

: ð28Þ

Assuming an inflow with f ¼ f , type I migration would nothave been significant over the satellite formation intervalfor a quite viscous protosatellite disk, i.e., with 410À2.For low disk viscosities, i.e., 5Oð10À3Þ, full-strength typeI torques would preclude the formation and survival of Gal-ilean-sized satellites.

Intermediate disk viscosities, with Oð10

À3

Þ< <

Oð10À2Þ, would yield some degree of inward migrationduring the satellite formation era. For example, with G ¼ 5 Â 106 yr; K ¼ 10À4 cm2 gÀ1, and ¼ 5 Â 10À3 (thecase shown in Fig. 5), Figure 7 shows the disk temperatureprofile and the water-ice stability line (Fig. 7a), and acc;in, I, and GD for the Galilean satellites (Fig. 7b). Figure 7cshows a satellite migration history, including effects due toboth gas drag and type I torques, that would leave the satel-lites at their current positions after t ¼ acc;in. Here we haveincluded the effect of a linear growth in satellite mass overthis time period on the calculated ðda=dtÞI and ðda=dtÞGDrates and have assumed that each satellite’s migration isindependent. The latter would not be valid if the satellites

were locked in mutual mean-motion resonances. Figure 8shows the same for G

¼107 yr,

¼10

À3, and

K ¼ 0:1 cm2 gÀ1, and Figure 9 the same for the very slowinflow high-opacity case shown in Figure 6.

The inward differential type I migration of the satellitesshown in Figures 7–9c could provide an opportunity for theestablishment of the Laplace resonance from ‘‘ outside-in ’’as Ganymede’s larger mass causes its orbit to converge onthat of both Europa and Io, providing an opportunity forresonance capture. Callisto is naturally omitted from thisconfiguration because of its slower decay rate. Interestingly,Peale & Lee (2002) have recently demonstrated that the pri-mordial origin of the Laplace resonance in this manner,combined with continuing tidal interaction with the planetafter the subnebula has dissipated, can reproduce all of thefeatures of the current configuration.

What general limits can be placed on the extent to whichthe satellites could have migrated inward, and thus on theirformation in a somewhat more exterior (and thus typicallycolder for a given inflow rate) region of the disk? WhileGanymede could have captured Europa and Io into mean-motion resonances as it migrated inward, the orbits of Ganymede and Callisto diverge, and thus these satelliteswould have been closer together in the past. A crudeestimate for the upper limit on the degree of migrationcan then be determined by back-tracking their orbits untilthey were marginally stable, with ðaC À aG Þ ¼ 3:5RH ¼3:5aG ½ðM G þ M C Þ=3M J1=3, or aG =aC % 0:9. Assuming amass ratio between the two satellites similar to the current

Fig. 8. —Sameas Fig.7, but herewith ¼ 10À3, G M J= _M M J ¼ 107 yr, K ¼ 0:1 cm2 gÀ1, andthe nominal disk model




one, this implies that Callisto could not have migrated infrom greater than about 35R

J(assuming that this region of

the disk would have been isothermal), while Ganymedewould have needed to form within about 30RJ.

In a moderate-migration scenario, somewhat higher ratesof inflow for a given disk opacity could then be consistentwith an ice-rich Ganymede, since the satellite could form ina somewhat more exterior portion of the disk. For example,ice stability exterior to 30RJ in the low-opacity disk shownin Figure 5 requires G ! 106 yr, while for the high-capacitycase shown in Figure 6, G ! 5 Â 107 yr is needed.

We note that for satellite formation from a ‘‘ gas-starved ’’ disk, the build-up of the appropriate mass in solidsoccurs over multiple disk viscous cycles, implying that satel-lites massive enough to open gaps would be lost due to typeII decay, which would occur on a viscous timescale with 5 acc;in. It may not be coincidental that the Galilean sat-ellites fall quite near in mass to that required to open a gapfrom equation (5), as more massive satellites might not sur-vive. It is also plausible that earlier generations of satellitesaccreted during periods of more rapid gas inflow ontoJupiter and were lost to either type I or II decay to theplanet, so that the Galileans were simply the last generationof satellites to form and survive. Satellites forming duringperiods of higher gas inflow would have been more rock-rich than the Galileans for r rc, given the higher predicteddisk temperatures. Alternatively, it may be that theGalileans were the only substantial satellite system to form;e.g., it is possible that only during the very last phase of its

growth did Jupiter contract sufficiently to allow for the for-mation of a circumplanetary accretion disk.

4. SUMMARY AND DISCUSSION

The standard model holds that the Galilean satellitesformed from a disk containing their total mass in solids,augmented in gaseous components to match the bulk solarcomposition. Formation from the implied gas-rich, 0:02M J‘‘ minimum-mass subnebula’’ (MMSN) presents multipledifficulties for Galilean satellite formation and survival,including

1. Rapid accretion times for all four satellites 103 yr, inapparent contradiction with the long, >105 yr formationtimes implied by Callisto’s partially differentiated state.

2. Disk temperatures too high for ices in the region of Ganymede and Callisto unless the disk is extremely inviscid(requiring 10À6 using a Shakura and Sunyaev viscos-ity parameterization), with a correspondingly long disk vis-cous lifetime of e106 yr.

3. Extremely short lifetimes of Galilean-sized satellitesagainst type I orbital decay due to disk torques ($102 yr)and aerodynamic gas drag ($103 yr), in contradiction to sat-ellite survival given the disk lifetimes implied by point 2.

4. Loss of satellites due to type II decay on a timescale of $O(1/) yr if the satellites are able to open gaps in the disk.Survival of satellites in gaps would require the disk to beremoved by a nonviscous means, e.g., via photoevapora-

Fig. 9. —Sameas Figs. 7 and 8,with ¼ 5 Â 10À3, G M J= _M M J ¼ 108 yr, K ¼ 1 cm2 gÀ1, and the nominal diskmodel(same caseas inFig.6)




tion, which would then necessitate an even longer disk life-time of $107 yr and a lower viscosity, 10À7.

5. The effective viscosity imposed by a Galilean-sized sat-ellite as a result of its density wave interaction with the diskimplies $ 10À4 –10À3, in contradiction to both points 2and 4.

6. If the satellite-induced viscosity from point 5 could beavoided through the opening of wide gaps, Galilean-sizedsatellites occupying such gaps would have their eccentric-ities excited by first-order Lindblad resonances on time-scales of only $103 yr, making the orbital stability for 107 yrrequired by point 4 difficult to reconcile.

7. A key independent trait—Jupiter’s current very lowobliquity—provides strong circumstantial evidence againstthe existence of a long-lived MMSN circumjovian disk. If such a massive disk had been present after removal of thesolar nebula, Jupiter would likely have passed through a res-onance involving its spin precession rate and the 16 orbitalprecession frequency as the circumplanetary disk dissipated(Ward & Hamilton 2002). Such a passage would have gener-ated an obliquity significantly larger than % 3.

We instead find that a self-consistent, and much less

restrictive, scenario results by considering satellite forma-tion in a circumjovian accretion disk produced during theslowing of gas inflow onto Jupiter. The creation of such adisk is supported by results of recent hydrocode simulationsof gas accretion onto Jupiter at the end stages of the planet’sgrowth and as the planet has opened a gap in the nebulardisk (e.g., Lubow et al. 1999; D’Angelo et al. 2002). In sucha model:

1. Gas inflow rates of a Jovian mass in ! a few Â 106 yryield disk temperatures sufficiently low to allow for icestability in orbits exterior to that of Ganymede, and forhydrated silicates in the region of Europa. Varying theassumed values of disk opacity and assumed viscosity alphaparameters affects the required inflow time for a given case,but successful models can be found over a wide range of these parameters.

2. Given the temperature requirements in point 1, theresulting steady-state circumplanetary gas disk is orders of magnitude less massive than the minimum-mass subnebulamodel. In many cases, this ‘‘ gas-starved ’’ disk is opticallythin.

3. Over a wide range of assumed gas-to-solid ratios in theinflowing material, disk solids will rapidly accumulate oncethey are placed into orbit, decoupling from the gas disk andgrowing faster than they can be removed via gas drag. Themass of solids and the solid-to-gas ratio in the disk thusincrease with time, so that the final stages of satellite growthoccur in a relatively gas-free and low-pressure environment.

4. The overall formation time of the satellites is con-trolled by the inflow timescale of the solar nebula material,rather than the satellites’ short orbital timescales. Thus,long satellite accretion times of !105 yr—such as that appa-rently needed to account for Callisto—result for the inflowrates implied by point 1.

5. Satellite survival against type I decay for the formationinterval in point 4 is predicted if the disk viscosity alphaparameter is ! 10À3 for inflowing material with a solargas-to-solids ratio.

6. Disk viscosities with Oð10À3Þ < < Oð10À2Þ yieldinward migration of Galilean-sized satellites during thesatellite formation era as a result of type I torques. The dif-

ferential migration rates provide an alternative means of establishing the Laplace resonance among the inner threesatellites (Peale & Lee 2002).

7. Satellite formation occurs during Jupiter’s accretionof the final few percent of its mass. The circumplanetarydisk is removed as gas inflow to Jupiter stops, presumablyas the nebula itself dissipates. Once inflow ceases, the gasdisk is viscously removed on a timescale of $O(1/ ) yr, or

103 yr from point 5.

Thus, given current understanding of giant planetgrowth and satellite-disk interactions, we believe that theformation of the Galilean satellites from a ‘‘ gas-starved ’’accretion disk is strongly favored over formation from agas-rich MMSN disk. This implies that satellite forma-tion occurred within a much lower gas density environ-ment than considered by previous models for satellitegrowth (e.g., Lunine & Stevenson 1982; Coradini et al.1989) and Jovian subnebular chemistry (Prinn & Fegley1981, 1989).

Several key aspects of the accretion disk model areuncertain and merit further investigation. In order toaccount for the radial scale of the Jovian regular satel-lite system, the inflowing material should be character-ized by some maximum prograde specific angularmomentum, j GM Jrcð Þ1=2, so that it achieves centrifu-gal force balance orbiting Jupiter interior to25RJ rc 35RJ, with this value depending on theamount of inward migration of the satellites via point 6above. This corresponds to j ð0:3–0.4)JR2

H, expressedin terms of Jupiter’s orbital frequency and Hill radius.It appears likely that the next generation of hydrocodesimulations will have sufficient resolution to test thisrequirement. In addition, models of the late growth of Jupiter that self-consistently account for the planet’scontraction could address whether the inflow ratesimplied in point 1 are consistent with a Jovian radiussmaller than the current satellite system.

Recent N -body accretion simulations suggest that asatellite system similar in mass and number to that of theJovian system could result from a gas-free disk with amass and angular momentum similar to that contained inthe satellites (Canup & Ward 2000). However, in themodel favored here, satellite growth will be regulated bya slow inflow of material. It is possible that in an inflow-regulated disk, the protosatellite system at a given timewould appear as a distribution of satellites, sufficientlyseparated in orbital distance to maintain short-termstability (e.g., 5–10 mutual Hill radii). Inflowing materialaccreting onto the satellites would gradually increasetheir mass until the point of system destabilization, colli-sions, and ultimately a decrease in the number of satel-lites. The potential for such an episodic growth patternhas not been explored previously, and will be a focus of our future work. Indeed, the distribution of resulting sat-ellites could provide constraints on the required radialdependence of the inflow flux, which was assumed to beuniform in this work.

In the absence of such detailed accretion studies, wenote that the gas-starved disk model predicts long accre-tion times for all of the Galilean satellites. There areother factors that could have contributed to the greaterobserved degree of resurfacing and differentiation onGanymede than that seen on Callisto. First, Ganymede




may have been tidally heated as a result of its involve-ment in orbital resonances subsequent to its formation(e.g., Showman et al. 1997; cf. Peale 1999). Second,Ganymede is somewhat more massive, and would havelikely formed in a higher subnebula temperature environ-ment. Assuming a gas-free accretion scenario and a lowvalue for the fraction of heat deposited at depth in thesatellite per accretional impact (which seems plausible foraccretion of predominantly small, inflowing solids),Ganymede and Callisto can fall on opposite sides of thewater-melting curve (e.g., Stevenson et al. 1986). Animportant test of the gas-starved disk model will be howwell it can fit into a more detailed scenario for the originof the Ganymede-Callisto dichotomy. Another issue con-cerns the accretion of rock versus ice during the finalstages of growth of these two large satellites, as accretingices may upon initial impact be sublimed or vaporized,since the gravitational potential energy per unit mass atthis point is roughly the latent heat of sublimation of water. Assuming that the ambient subnebula temperatureand pressure still fall below the water-ice stability curve,it seems likely that this material would recondense whilestill in Jovian orbit and reaccrete onto the satellites, sothat the net accreting material would have a similar rock-to-ice ratio as that in the inflowing solids. However, thispoint deserves a closer examination as well.

The similarities in the believed growth processes of Jupiter and Saturn, as well as in the features of their satellitesystems, suggest that a common mode of origin producedthe regular satellites of both planets. We are beginning toexplore the ramifications of a slow-inflow–producedaccretion disk model and type I migration for the Saturniansystem (Canup & Ward 2002b).12 We hope that continuedstudy of satellite origin in both the Jupiter and Saturn sys-tems can provide valuable constraints on the late-stagegrowth of gas giant planets and related nebular conditions,as well as the specific formation histories that set the condi-tions for satellite chemical inventories, thermal history, andlater dynamical evolution.

We wish to acknowledge helpful discussion with D.Stevenson at the 2001 Jupiter Conference in Boulder, CO;his talk at that meeting included some concepts similar tothose we presented, and which are explored further here.We thank S. Peale for sharing some unpublished resultswith us on the establishment of the Laplace resonance viainward orbital migration. The manuscript benefited fromconstructive comments from an anonymous reviewer, D.Hamilton, S. Peale, A. Stern, and J. Wisdom. This researchwas supported by NASA’s Planetary Geology and Geo-physics Program.

Correspondence and requests for materials should beaddressed to R. M. C. ([email protected]).

APPENDIX

After Lynden-Bell & Pringle (1974), we consider a gasprotosatellite disk with viscosity and surface density G , inwhich shearing stress causes a viscous couple

gðrÞ ¼ 3 G h ðA1Þ

on material in the disk orbiting at radius r with specificangular momentum h ¼ r2. For a fixed central gravityfield,

F dh

dr¼ À @ g

@ r; ðA2Þ

where F ¼ 2 G ru is the flux of material moving outwardwith velocity u through radius r.

We desire a quasi–steady-state solution for F ðrÞ and G ðrÞ as a function of the inflow supply to the disk. Weassume that inflowing material to the disk is character-ized by a specific angular momentum, j , in the range of

GM JRJð Þ1=2 j in GM Jrcð Þ1=2 (e.g., Cassen & Summers1983; Coradini & Magni 1984; Makalkin et al. 1999), so

that material is deposited into centrifugally supportedorbits with a uniform flux per area F in in the regionextending to the radial distance rc. The total infalling fluxis then just F Ã ¼ F inr2

c . It is further assumed that thereis a disk outer edge, rd , at which point material isremoved from the circumplanetary system and gðrd Þ ¼ 0;and that the viscous couple at the planet’s surface, r p,also vanishes, gðr pÞ ¼ 0.

From the continuity equation for a disk in steady-state,

@ G

@ t¼ À1

2r

dF

drþ F in ¼ 0 ;

dF

dr¼

2rF in :

ðA3

ÞSince F in ¼ 0 for r > rc, the flux of material moving outwardin the disk exterior to rc is F ðr > rcÞ ¼ F 0 ¼ constant, andfrom equation (A2),

F 0½hðr > rcÞ À hd ¼ À ½gðr > rcÞ À gd ¼ Àgðr > rcÞ ; ðA4Þ

where the subscript d corresponds to values at the disk outeredge, rd . Interior to rc, F ¼ F ðrÞ because material is beingadded to the disk across a range of radii. SubstitutingF dh=dr ¼ d =drðFhÞ À h dF =dr into equation (A2) and inte-grating gives

hF

ðr

Þ þg

ðr

Þ ¼45 r2hF in

þA ;

ðA5

Þwhere A is a constant, defined by the boundary conditiongðR pÞ ¼ 0tobe A ¼ F ph p À 4

5 F inr2 ph p.

The condition that F must be continuous at r ¼ rc

requires that F ðrcÞ ¼ F 0, which when combined with equa-tions (A4) and (A5) gives

F 0hd À F ph p ¼ 45 F inðr2

c hc À r2 ph pÞ : ðA6Þ

From conservation of mass, the sum of the magnitudes of the flux onto the planet and that flowing outward past rd

must equal the total inflowing flux to the disk; combiningjF pj þ jF 0j ¼ F Ã À F inR2

J with equation (A6) gives the

12 Titan is intermediate in mass between Ganymede and Callisto and itsdegree of differentiation is uncertain, but differentiation exceeding that of Callisto is not necessarily inconsistent with long accretion times for Titansimilar to those found here. It hasbeen suggested(e.g., Schubert et al.1986;Stevenson et al. 1986) that lower ambient nebular temperatures at Saturnlikely allowed for the incorporation of the more volatile ammonia-water iceand clathrate hydrate, which could act to reduce the freezing temperatureandincrease thedegree of expected melting in that environment(alsoCora-dini et al.1989).




steady-state flux exterior to rc,

F ðr > rcÞ ¼ F 0

¼ F inr2c

4

5

rc

r p

1=2

þ 1

5

r p

rc

2

À1

" #

Â rd

r p 1=2

À1

" #À1

: ðA7Þ

Interior to rc, the inward flux past each radius r must equalthe difference between the total supply flux exterior to rminus the outward flux F 0; since we have defined F < 0 tobe an inflowing flux, this translates to the condition that

ÀF ðrÞ ¼Z rc

r

2rF in dr À F 0 ; ðA8Þ

which in combination with equation (A7) gives the steady-state flux interior to rc,

F ðr < rcÞ ¼ ÀF inr2c

Â1

À

r

rc

2

À

4=5ð Þ rc=r pÀ Á1=2þ 1=5ð Þ r p=rcÀ Á

2À1

rd =r pÀ Á1=2

À1" # :

ðA9Þ

For the cases of interest here, ðr p=rcÞ251 and

ðrd =r pÞ1=241, the steady-state fluxes then reduce to

F ðr ! rcÞ ¼ F 0 % F Ã 4

5

rc

rd

1=2" #

;

F ðr < rcÞ % À F Ã 1 À r

rc

2

À 4

5

rc

rd

1=2" #

; ðA10Þ

and the flux onto the planet is

F p % ÀF Ã 1 À 4

5

rc

rd

1=2" #

: ðA11Þ

The fluxes in equation (A10), together with equa-tions (A4) and (A5), yield the steady-state disk gassurface density:

G

ðr

Þ %

4F Ã15

5

4À

ffiffiffiffiffirc

rd

r À 1

4

r

rc

2

for r < rc ;

ffiffiffiffirc

r

r À ffiffiffiffi ffirc

rd

r for r ! rc :

8>>><>>>: ð

A12

Þ

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