The Astrophysical Journal, 714:1052–1064, 2010 May 10

doi:10.1088/0004-637X/714/2/1052


C 2010. The American Astronomical Society. All rights reserved. Printed in the U.S.A.

ORIGIN OF THE DIFFERENT ARCHITECTURES OF THE JOVIAN AND SATURNIAN SATELLITE SYSTEMS
T. Sasaki1 , G. R. Stewart2 , and S. Ida1
1 Earth and Planetary Sciences, Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8551, Japan; takanori@geo.titech.ac.jp,

ida@geo.titech.ac.jp
2 Laboratory for Atmospheric and Space Physics, University of Colorado, Campus Box 392, Boulder, CO 80309-0392, USA; gstewart@lasp.colorado.edu

Received 2009 October 12; accepted 2010 March 22; published 2010 April 15

ABSTRACTThe Jovian regular satellite system mainly consists of four Galilean satellites that have similar masses andare trapped in mutual mean-motion resonances except for the outer satellite, Callisto. On the other hand, theSaturnian regular satellite system has only one big icy body, Titan, and a population of much smaller icy
moons. We have investigated the origin of these major differences between the Jovian and Saturnian satellite
systems by semi-analytically simulating the growth and orbital migration of proto-satellites in an accreting protosatellite disk. We set up two different disk evolution/structure models that correspond to Jovian and Saturniansystems, by building upon previously developed models of an actively supplied proto-satellite disk, the formationof gas giants, and observations of young stars. Our simulations extend previous models by including the (1)different termination timescales of gas infall onto the proto-satellite disk and (2) different evolution of a cavity
in the disk, between the Jovian and Saturnian systems. We have performed Monte Carlo simulations and haveshown that in the case of the Jovian systems, four to five similar-mass satellites are likely to remain trapped
in mean-motion resonances. This orbital configuration is formed by type I migration, temporal stopping of
the migration near the disk inner edge, and quick truncation of gas infall caused by Jupiter opening a gap inthe solar nebula. The Saturnian systems tend to end up with one dominant body in the outer regions caused by theslower decay of gas infall associated with global depletion of the solar nebula. The total mass and compositional
zoning of the predicted Jovian and Saturnian satellite systems are consistent with the observed satellite systems.
Key words: planets and satellites: formation – planets and satellites: individual (Galilean satellites, Titan) – planetsand satellites: rings

disk onto the circum-planetary proto-satellite disk. The disk
temperature profile in the CW model is dominated by viscousheating of the evolving disk and the luminosity of the giant
planet plays a lesser role. The “satellitesimals” in this model are
assumed to form immediately from dust grains that are suppliedby gas infall (Canup & Ward 2006). Since the motions of the“satellitesimals” are decoupled from gas accretion, they areretained while the disk gas is actively replenished. As a result,a high dust-to-gas ratio is realized in the disk. The satellites
that have grown massive enough are disposed through type I
migration into the planet due to satellite–disk interaction (e.g.,
Ward 1986; Tanaka et al. 2002) and new generations of satellites
are repeatedly accreted from the supplied dust grains. The finallysurviving satellites are those formed at the very end of the hostplanet’s accretion. In the present paper, we will focus on thelatter model.
Canup & Ward (2006) performed N-body simulations ofaccretion of satellites from small bodies and their type I
migration in decaying accretion disks. They showed that theequilibrium total mass of satellites resulting from a balance
between disposal by type I migration and repeated satellitesimalaccretion is universally ∼10−4 Mp , where Mp is the host planet
mass. The proto-satellite disk gas is eventually depleted due tothe decline of infall gas due to global depletion of the protoplanetary disk, then type I migration stops and satellites withtotal mass ∼10−4 Mp survive. The result is consistent withactual Jovian and Saturnian systems. They suggested that thedifference between Jovian and Saturnian systems is caused bystochastic timing between the repeated formation/migration anddepletion of disk gas.
The disk evolution in their model is regulated by the mass
infall rate from the proto-planetary disk onto the proto-satellite

1. INTRODUCTION
The four Galilean satellites around Jupiter have similar
masses and the inner three bodies are trapped in mutual meanmotion resonances. They exhibit a trend of decreasing meanbulk density with increasing orbital radius, and a diversity ofaxial moments of inertia (Table 1). These properties are likely
caused by a progressive increase in ice-to-rock ratio with anorbital radius and different states of differentiation. The inferredundifferentiated interior of the outermost satellite, Callisto,
would require a relatively long accretion timescale, 5 × 105
yr or more (Stevenson et al. 1986; Schubert et al. 2004; Barr
& Canup 2008). On the other hand, Saturn has only one big
icy satellite, Titan, which is located relatively far from Saturn.
Recent Cassini data indicates that Titan also has an incompletely
differentiated interior (Iess et al. 2010; Table 1). The origin of
the pronounced difference between the Jovian and Saturniansystems is an intriguing question.
Two important models for circum-planetary proto-satellite
disks have been proposed. One is the “solids enhanced minimummass” (SEMM) model (Mosqueira & Estrada 2003a, 2003b;
Estrada et al. 2009) and the other is an actively supplied gaseous
accretion disk (CW) model (Canup & Ward 2002, 2006, 2009).
The SEMM model postulates a massive quiescent disk with apeak surface density near 105 g cm−2 and a temperature profile
determined by the luminosity of the gas giant planet. The Jovian/Saturnian satellites in this model form from solid materials
supplied by ablation and capture of planetesimal fragments
passing through the massive disk (Mosqueira et al. 2010). The
CW model postulates a low mass, viscously evolving disk witha peak surface density near 100 g cm−2 , that is continuously
supplied by mass infall from the circum-stellar proto-planetary1052

No. 2, 2010

FORMATION OF JOVIAN AND SATURNIAN SATELLITESTable 1
Satellite Properties

Satellite

a/Rp

M/Mp (10−5 )ρ (g cm−3 )C/MR 2

Io
EuropaGanymede
Callisto
Titan

5.9
9.4
15.0
26.4
20.3

4.70
2.53
7.80
5.69
23.7

3.53
2.99
1.94
1.83
1.88

0.378
0.346
0.312
0.355
0.34

Note. Source: Schubert et al. (2004); Iess et al. (2010).

disk (Section 2.1) and the infall is responsible for the final stage
gas giant planet formation. Although Canup & Ward (2006)
assumed the same proto-satellite disk evolution for both Jovian
and Saturnian systems, recent theories of gas giant formationand observational data of extrasolar planets raise the possibility
that the disk evolution can be different between Jovian and
Saturnian systems as follows.
Unless the growth of gas giants is truncated or at leastsignificantly slowed down at some critical mass, only a single
gas giant should exist in each planetary system, because gas
accretion onto a planet is a runaway process (e.g., Bodenheimer& Pollack 1986; Pollack et al. 1996; Ikoma et al. 2000). Coresfor gas giants do not generally form at the same time, and the rateof gas mass accretion in the proto-planetary disk that can supply
gas to giants is observationally ∼ O(10−5 )MJ yr−1 (where MJ
is a Jupiter mass). However, the observed data of extrasolar
planets show that there are many systems with multiple gas
giants. Our solar system also has two gas giants. Furthermore,
the mass distribution of extrasolar gas giants is centered at
Mp,ave ∼ a few MJ with an upper cut-off of Mp,max ∼ 10MJ .
The observationally inferred proto-planetary disk masses have amean value of ∼10MJ and a maximum value of ∼100MJ (e.g.,
Beckwith & Sargent 1996). These mean and maximum values
are one order greater than Mp,ave and Mp,max , respectively. These
facts suggest that the giants did not accrete most of disk gas,so they are consistent with existence of the rapid truncation(e.g., Lin & Papaloizou 1985) or severe decline (e.g., Lubow &D’Angelo 2006; Tanigawa & Ikoma 2007) of gas infall onto theplanets due to gap opening in the proto-planetary disks.
Since the proposed gap opening conditions show that the critical mass for gap opening is larger in outer regions (Section 2.2),
outer gas giant planets, in principle, tend to be larger than inner ones. However, the outer one can be smaller, if the disk
gas is depleted before the outer planet can complete its formation. Since Saturn is three times less massive than Jupiter, it ismost likely that Jupiter opened up a gap to halt its growth while
Saturn did not and its growth was terminated by global depletionof the proto-planetary disk (the solar nebula). Although reduced
gas infall can still continue after the gap opening (Lubow &D’Angelo 2006; Tanigawa & Ikoma 2007), the severe reduction in infall rate makes the Jovian disk evolution significantly
different from the Saturnian one (see Section 2).
Here, we explore a possible path to produce the pronounced
different architectures of the Jovian and Saturnian satellite
systems, by introducing the different evolution of proto-satellite
disks due to the gap opening to the actively supplied gaseous
accretion proto-satellite disk model (Canup & Ward 2002,
2006). The purpose of the present paper is to demonstrate how
this diversity in gas giant formation can profoundly affect theregular satellite formation process.
Since we survey a wide range of parameters for initial andboundary conditions, we adopt a semi-analytical model to sim-

1053

ulate accretion and migration of satellites from satellitesimals
that has been developed by Ida & Lin (2004, 2008) for modelingsequential planet formation. While the semi-analytical calculations inevitably introduce approximations, N-body simulations
have to employ unrealistic initial and/or boundary conditions
because of the computational limitations (e.g., Kokubo & Ida
1998 started from planetesimals of ∼(10−4 –10−3 ) M⊕ ; Canup& Ward 2006 replaced infalling dust grains by embryos with theisolation mass of an accreting satellite). We show in Section 4.1
that our calculation produces results consistent with Canup &Ward’s (2006) N-body simulation, although inconsistency may
exist in detailed features (we need more careful comparison toreconcile the inconsistency).
In Section 2, we explain our disk evolution models for Jovian
and Saturnian systems in details, highlighting their differences.
In Section 3, semi-analytical treatments to simulate accretionof satellites from satellitesimals, orbital migration, and resonant
trapping are described. The simulation results are presented in
Section 4. We successfully explain the pronounced difference
between Jovian and Saturnian systems in the framework of
our model. We also discuss implications for the formation of
Saturn’s rings. Section 5 is devoted to a summary of the major
results.
2. SATELLITE FORMATION CIRCUMSTANCES
We modify the actively supplied disk model developed by
Canup & Ward (2002, 2006; see Section 2.1), by introducing
the effects of the gap formation around a host planet’s orbit.
If a gap is opened, the proto-satellite disk rapidly dissipates(Section 2.2), so that the mass distribution and orbital configuration of satellites are “frozen” in the state of a relatively massive
and hot disk. The hot disk condition allows retention of rocky
satellites corresponding to Io and Europa. The massive disk
condition could also result in formation of an inner cavity ofthe disk (Takata & Stevenson 1996; also see Section 2.4) thatplays a crucial role in formation of resonantly trapped multiple
satellites like the Galilean satellites.
Observations of Classical T-Tauri Stars (CTTSs) with relatively strong mass accretion and Weak line T-Tauri Stars(WTTSs) with relatively weak mass accretion suggest that magnetic coupling between disks and host stars becomes weak from
CTTSs to WTTSs. It may imply that the inner cavity would
vanish as disk surface density and disk accretion rate decay,
corresponding to evolution from CTTS stage to WTTS one.
We adopt the same picture for evolution of proto-satellite disks(Section 2.4).
The Saturnian disk would also have experienced the massive
disk stage with an inner cavity to have resonantly trapped
multiple satellites. However, if Saturn did not open up a gap in
the protoplanetary disk, then as the cavity vanishes, the massive
satellites are released to migrate onto the host planet. The final
surviving satellites are therefore generally formed in depleted
and cold disk conditions, in which only ice-rich bodies exist(Section 2.3).
2.1. Actively supplied Disk ModelFirst, we briefly summarize descriptions for evolution of theactively supplied accretion disk model with relatively small
mass ∼10−4 Mp (p = Jupiter, Saturn) developed by Canup &Ward (2002, 2006), which we follow in the present paper. Moredetailed descriptions are given in Appendix A.

1054

SASAKI, STEWART, & IDA

Gas infall from the proto-planetary disk onto the protosatellite disk is limited to the regions inside a critical radius
rc . Canup & Ward (2006) assumed that the infall flux inside rc
is independent of orbit radius and rc has a fixed value of 30Rp ,where Rp is the physical radius of the host planet. (Realistically,
the infall mass flux can have a radial dependence and should
evolve with time, but current hydrodynamic simulations of gas
giant planet formation lack sufficient resolution and long enoughintegration times to constrain the radial and time dependence of
the infall.) That is, the infall flux per unit area has a constantvalue of (Fp /π rc2 ) at r < rc and vanishes at r > rc , where r is
the orbital radius around the host planet and Fp is the total mass
infall rate onto the disk. In the steady accretion state of the protoplanetary disk, the infall rate is defined with a parameter τG as
Fp,0 = Mp /τG . Considering non-uniform inflow distributions
would be an important area of future investigation.
Neglecting a diffused-out extended faint disk to make clear
dynamical process of accretion and migration of proto-satellites,
surface density of disk gas is given by (Appendix A)
⎧
 
−3/4
Mp
Fp
r
⎨
0.55
g cm−2 [r < rc ],

100f
g
Σg 
3π ν
MJ 20RJ⎩
0[r > rc ],
(1)
where
−1 
−3/4

τG
α
fg ≡
,
(2)
5 × 10−3
5 × 106 yr

Vol. 714

Fp [MJ/yr]
10-5

Mdisk

Jupiterformation of
final satellites

transition to
supply limit

Saturn10-6

10-7

formation of
final satellites
residualflow?
gap opening
massive, hot disk
with an inner cavity

globaldepletion

faint, cool disk
without a cavitytFigure 1. Schematic illustration for evolution of the infall rate (Fp ) onto theproto-satellite disks and timing of satellite formation in Jovian and Saturnian
3/4
1/4
systems. Since Σg ∝ Fp and Td ∝ Fp , high Fp means a massive and hot
disk.

regulated by Kelvin–Helmholtz contraction with a timescale(see, e.g., Ida & Lin 2004),

−3
Mp
Mp
6
 10
yr,
(5)
τKH =
10 M⊕
Ṁp

2.2. Jovian System with Gap Opening

which can have different numerical factors depending on opacity, but the runaway nature is robust. Then, Fp ∼ 3 ×
10−6 (M/0.1MJ )4 MJ yr−1 ∝ t 4/3 . After the planet accretes
all the gas in feeding zone with width ∼2rH , where Hill radiusrH = (Mp /3M∗ )1/3 r, at Mp ∼ 0.3MJ in the case of the solar nebula, the growth would be regulated by supply of global
disk accretion, Ṁdisk , which is ∼10−5 MJ yr −1 and decays ontimescales of 106 –107 yr (see Figure 1).
As we will discuss in Section 2.4, satellite formation would
attain a quasi-steady state between accretion of satellites from
infalled materials and their migration onto the host planet. Since
we are concerned with the architecture of satellite systems
formed by the last survivors, we start calculations from relatively
late stage with τG = 2 × 106 yr (Fp = 5 × 10−7 MJ yr−1 )for Jupiter. Although the choice of initial value of τG contains
uncertainty, the critical satellite mass to start migration and−3/16
disk temperature depend on τG only weakly: Mmig ∝ τG
−1/4
(Equation (15)) and Td ∝ τG
(Equation (3)). So, the mass
distribution and compositions of final satellites do not sensitively
depend on the initial value of τG . Furthermore, since Canup &Ward’s (2006) N-body simulations adopted similar values, we
can directly compare our results with theirs.
As discussed in Section 1, we assume that Jupiter opened up a
gap in the solar nebula. Then, Fp quickly decays (Equation (9)).
It is proposed that a gap is opened if both viscous and thermal
conditions are satisfied (Lin & Papaloizou 1985; Crida et al.
2006). The former condition is derived by comparison between
the viscous diffusion of the gas and gravitational scattering by
the planet and the critical planetary mass is given by, Lin &Papaloizou (1985); Ida & Lin (2004),


40ν
HPD 2
Mg,vis M∗  40α
M∗
rp ΩK
rp


 r
 α
1/2
M∗
p
M⊕ ,
 30
(6)
10−3
1 AU
M

Except for the late stages, growth of a gas giant is a runaway
process of accreting gas from the disk. The mass increase iswhere rp is heliocentric orbital radius of the planet, ΩK is
Keplerian frequency at rp , and M∗ is the host star mass. For

where we used the turbulent viscosity of the proto-satellite2disk with ν = αHSD ΩK (Shakura & Sunyaev 1973), ΩK =3
GMp /r , and temperature distribution to calculate the disk
scale height (HSD ) is given by Equation (3). Since ν ∝
1/4
3/4
−3/4
Td ∝ Fp (Equation (3)), Σg ∝ Fp /ν ∝ Fp ∝ τG .
After a quasi-steady state of accretion and destruction ofsatellites (see Sections 2.2 and 2.3) is established, we start theexponential decay of infall as Fp = Fp,0 exp(−t/τdep ) with
τdep = (3–5) × 106 yr, corresponding to global depletion of the
proto-planetary disk.
We use the disk temperature in the steady state that is derived
by a balance between viscous heating and blackbody radiation(Appendix A),
 
−3/4
−1/4 
Mp 1/2
τGr
Td,0  160
K.
MJ
5 × 106 yr
20RJ(3)
The disk is assumed to be vertically optically thin and isothermal(Appendix A). After the exponential decay of Fp is imposed,
the disk temperature is decreased ast
Td = Td,0 exp −
.
(4)
4τdep
The disk structure (gas surface density and temperature) is
therefore regulated by Fp (= Mp /τG ), in the actively supplied
disk model. We discuss how Fp is likely to evolve with time for
Jovian and Saturnian disks in the following. We will also pointout that the disk inner boundary condition may also be regulated
by Fp (Section 2.4).

No. 2, 2010

FORMATION OF JOVIAN AND SATURNIAN SATELLITES

the scale height HPD of the proto-planetary disk, we used thevalues of an optically thin disk (Hayashi 1981),
 r
1/4
p
HPD = 0.05
rp .
(7)
1 AU
The thermal condition is determined by balance between thegravitational scattering and pressure gradient, rH ∼ 1.5HPD(Lin & Papaloizou 1985; Ida & Lin 2004). The critical mass isgiven by


 r
3/4
M∗
p
M⊕ .
(8)
Mg,th  400
1 AU
MNote that Mg,th is generally larger than Mg,vis , because typicalvalues of the α parameter are ∼10−3 –10−2 for turbulence
induced by magneto-rotational instability (e.g., Sano et al.
2004).
Since Mg,th is comparable to Jupiter mass, it has been
proposed that Jupiter’s final mass is determined by the gap
opening (Lin & Papaloizou 1985; Ida & Lin 2004). Recent fluid
dynamical simulations suggest that the gap is not clear enoughto halt gas accretion for Mp  5MJ (e.g., Lubow et al. 1999;
D’Angelo et al. 2003). However, the gas flow across the gap is
not necessarily accreted by the planet (Dobbs-Dixon et al. 2007)
and even if all the passing flow is accreted by the planet, it is
only 10% of exterior disk mass accretion flow (Ṁdisk ) or less for
Mp ∼ MJ and quickly vanishes with increasing Mp (Lubow &D’Angelo 2006; Tanigawa & Ikoma 2007). We will show belowthat 2 orders of magnitude reduction in the mass flux onto theproto-satellite disk is enough to discriminate proto-satellite disk
evolution with the gap from that without the gap.
The truncation of gas infall or its severe decline due to thegap opening depletes the circum-planetary proto-satellite disk2
on its own viscous diffusion timescale τdiff ∼ RSD/ν where RSD
is the disk size and ν is disk viscosity. With the alpha model and
RSD ∼ 20Rp (where Rp is a physical radius of the planet),
τdiff



2
RSD
RSD 2 1 TK (RSD )
∼
∼
2
H
α 2π
αHSD
Ω
 αK −1 SD
∼ 103
yr,
10−3

(9)

where HSD is the scale height of the proto-satellite disk and TK
is an orbital period around the planet given by3/2
r
TK (r) = 0.032
yr.
(10)
20Rp
The gap opening itself would occur on the viscous diffusion
timescale for the scale height (∼HPD ) in the proto-planetarydisk, which is 22
HPD 2 RPD
HPD
τgap ∼∼∼ (10−3 –10−4 ) × (1–10) Myr,
ν
RPD
ν
(11)
where RPD is size of the proto-planetary disk, which would
2
be much larger than 5 AU, and tdep ∼ RPD/ν is global disk
evolution timescale. Both τdiff and τgap are much shorter thanformation and orbital evolution timescales of satellites (seeSection 3.1). This separation of timescales allows us to assume
that the proto-satellite disk around Jupiter was abruptly depleted
when the gap opened, such that the growth and orbital migrationof satellites were “frozen” at that time.

1055

Even if we consider the imperfect truncation of infall, thisfeature is not changed as long as the infall rate is reduced bya factor larger than 100. The equilibrium gas surface densityof the proto-satellite disk (Σg ) is determined by infall flux (Fp )3/4
as Σg ∝ Fp (Section 3.1). If the infall rate is decreased by 2
orders of magnitude, Σg is decreased by a factor of 30. According
to the decrease in Σg and the associated decrease in surface
density of satellitesimals, accretion and migration timescales ofproto-satellites become comparable to or longer than the global
depletion timescale of the proto-planetary disk (∼1–10 Myr;
Meyer et al. 2007). Therefore, although the residual infall after
the gap opening is considered, the “frozen feature” is mostly
preserved. We also carried out runs with the residual continuous
infall with a reduction factor of 100 and confirmed the above
argument. We will show that the introduction of the imperfect
truncation produces results that may be rather consistent withformation of a Callisto analog (Section 4.3).
2.3. Saturnian System without Gap Opening
Both thermal (Equation (8)) and viscous (Equation (6))
conditions show that the critical mass for gap opening is larger
in outer regions of the solar system. So, the final mass of Saturnshould be larger than Jupiter’s mass, unless significant disk mass
has already been accreted by the Sun and Jupiter such that the
total residual disk mass is reduced to the level of Saturnian mass
when the Saturn’s core is formed. Since the core accretion fromplanetesimals is generally slower at larger orbital radius in the
proto-planetary disk (e.g., Ida & Lin 2004), it is reasonable that
Saturn’s core accreted after Jupiter has fully formed (Figure 1).
Since it is most likely that Jupiter has not undergone significant
type II migration, Jupiter may have formed in a partially depleted
proto-planetary disk as well. Therefore, it is not unreasonable
to assume that Saturn formed in a significantly depleted disk
and did not open up a gap in the disk (Figure 1). Actually,
planets in further outer regions, Uranus and Neptune, have notundergone runaway gas accretion although their cores have large
enough mass for runaway gas accretion. This timing between
core accretion and disk depletion would also produce diversityof extrasolar planets (e.g., Ida & Lin 2008).
On the assumption that Saturn did not open up a gap,
the Saturnian proto-satellite disk should have been graduallydepleted on the global depletion timescales of the protoplanetary disk, tdep ∼ 1–10 Myr (Figure 1). The many ordersof magnitude difference in depletion timescales of the protosatellite disk between Jovian (τdiff ∼ 103 yr) and Saturnian(τdep ∼ 106 –107 yr) systems would significantly affect the
final configuration of satellite systems, because a typical type I
migration timescale of proto-satellites (τmig ∼ 105 yr; Tanaka
et al. 2002) are in between the two timescales. Jovian satellites
may retain their orbital configuration in a phase of the protosatellite disk with relatively high mass that was frozen at
the time of abrupt disk depletion, while Saturnian satellites
must be survivors against type I migration in the final lessmassive disk (Figure 1). The migration timescale depends onthe magnitude of infall flux (Fp ) through Σg and ν, which
remain uncertain. However, since the migration timescale does
−1/2
not strongly depend on the flux (τmig ∝ Fp ), the relation ofτdiff < τmig < τdep is not changed by the uncertainty in Fp even if
the uncertainty is 2–3 orders of magnitude. We start calculations
from the last stage with τG = 5×106 yr (Fp = 7×10−8 MJ yr−1 )for Saturn, which is the same as the calculations by Canup &Ward (2002, 2006).

1056

SASAKI, STEWART, & IDA
2.4. Difference in a Disk Inner Cavity

Takata & Stevenson (1996) studied magnetic coupling between the proto-satellite disk and proto-Jupiter, in order to address why Jupiter’s spin rotation is substantially slower thanthe break-up spin rate. If Jupiter accretes disk gas rotating at
Keplerian velocity near its surface, Jupiter’s spin rate should
be close to the break-up one. If the coupling is strong enough,angular momentum is transferred from the Jupiter’s spin to thedisk through the magnetic field. In this case, it is expected thatthe disk is truncated at the corotation radius where the disk gas
corotates with the planetary spin. The critical magnetic field
for the magnetic coupling is a few hundred Gauss for massaccretion rate onto a planet of ∼10−6 MJ yr−1 , if we use a standard formula of magnetospheric radius (Königl 1991). Sucha strong magnetic field for proto-Jupiter could be consistent
with a scaling argument (Stevenson 1974). Actually, Takata &Stevenson (1996) found through numerical calculation that the
proto-Jovian disk may have had an inner cavity created by themagnetic coupling.
Although evolution of the inner cavity in the proto-satellite
disk is not theoretically clear, the observations for protoplanetary disks around T Tauri stars suggest that the inner cavity
exists in the early active stage and vanishes as the disk accretion
onto a central body becomes weak. The observed spin period
distribution of young stars is bimodal and CTTSs tend to rotate
slower than WTTSs (Hartman 2002; Herbst & Mundt 2005).
A possible idea to account for the bimodality is as follows.
The relatively high disk accretion onto CTTSs may maintainthe magnetic field strong enough for the magnetic coupling
to transfer spin angular momentum to the disk and create acavity. On the other hand, with the lower disk accretion rate in
the following WTTS stage, the coupling and hence the cavity
vanishes and the host star accretes high angular momentum gas
from the disk, becoming a rapid rotator (Herbst & Mundt 2005).
Recent Spitzer observations also support this idea (Rebull et al.
2006; Cieza & Baliber 2007).
Based on the analogy of the evolution from CTTSs to
WTTSs, we here assume simulation conditions of (1) a truncatedboundary with an inner cavity and abrupt termination of infall
for the Jovian system (set I) and (2) a non-truncated boundary
without a cavity and gradual depletion of infall for the Saturniansystem (set II). The detailed settings are described in Section 3.1.
The assumption may include large uncertainty and needs
to be studied in greater detail. However, the purpose of thepresent paper is to demonstrate the important link between gas
giant planet formation and satellite formation through evolutionof the proto-satellite disk due to changes in infall rate andmagnetic coupling.
3. METHODS FOR SIMULATIONS FOR SATELLITE
FORMATION
Given a disk evolution model, we apply the population
synthesis planet formation model (Ida & Lin 2004, 2008) tosimulate growth of proto-satellites through accretion of small
bodies (“satellitesimals”) and their inward orbital migration
caused by tidal interactions with the gas disk.
3.1. Evolution of Infall
In set II for Saturn, a quasi-steady state in which accretionof satellites is balanced with their removal to the host planet
via type I migration is established. After the quasi-steadystate is established, we start the exponential decay of infall as

Vol. 714

Fp = Fp,0 exp(−t/τdep ) with τdep = (3–5)×106 yr. We integrate
the systems until disk gas surface density decreases so much thatsatellites no longer undergo orbital migration. This setting is thesame as that adopted by Canup & Ward (2002, 2006).
In set I for Jupiter, we set the disk inner edge at a corotation
radius of Jupiter (∼2.25Rp ). In this case, satellites are piled
up outside the disk edge. As we will argue in Section 3.3, if
the amount of trapped satellites exceeds a critical value, theinnermost one is released. Thus, a quasi-steady state different
from set II is also established in set I. After the establishment of
the quasi-steady state, we start the exponential decay of infall inthe same way as in set II. However, we stop integration randomlyat (0.5–0.8)τdep after the exponential decay starts, which reflects
the very rapid depletion of the disk due to the gap opening. We
also did calculations (set I’) with 100 times reduced infall flux
after the gap opening, instead of the complete termination, whichcorresponds to imperfect gap opening.
While the evolution of the gas surface density of the protosatellite disk is analytically given by Equation (1), the surface
density of satellitesimals is consistently calculated with removal
due to accretion by proto-satellites and supply from solid
components in the infalling gas. We scale the solid surface
density with a scaling factor fd as


−3/4
Mp
r
g cm−2 ,(12)Σd = ηice fd
MJ
20RJ
where ηice is an enhancement factor due to condensation of icy
grains at r > rice where disk temperature is lower than 160 K;
we adopt ηice = 3 for r > rice and ηice = 1 otherwise. In theinfall flux, the mass fraction of solid components is given byηice /f , where f is the mass ratio of gas relative to silicate dust.
For a fiducial value, we adopt f = 100. If fd = fg , the gas-todust ratio of the disk is the same as that of infall. But, we foundthat fd is usually 10–100 times higher than fg in outer regions ina steady state (Section 4.1). The method used to calculate theevolution of Σd is described in Appendix A.
3.2. Accretion and Migration of Proto-satellites
We randomly select initial locations of 10–20 satellite seeds
with M = 1020 g (∼10−11 MJ , ∼10−10 MS ) from a log-uniform
distribution in the regions inside rc = 30Rp and calculate their
growth, migration, resonant trapping, and re-generation, simultaneously calculating the evolution of Σd , until disk gas is depleted. The log-uniform distribution for initial locations corresponds to the spacing between the seeds being proportional toorbital radius, which is the simplest and natural choice (Ida &Lin 2004). Although the choice of the initial locations is arbitrary, it would not change the overall results thanks to the effectsof type I migration. The numbers of the initial satellite seeds(N = 10–20) do not affect the mass and orbital distributionsof final satellites as well, as long as the initial mass is smallenough, because τacc is shorter for smaller M (Equation (13))
and only a few bodies undergo runaway growth from the seeds.
The accretion rate of a proto-satellite from satellitesimals
is given by Ṁ = M/τacc and the accretion timescale is(Appendix B)


 
Mp
M 1/3
r
τacc  0.5
TKRpΣd r 2Mp1/3 

5/4
Mp −5/6
Mr−1
 106 fd−1 ηiceyr,
10−4 Mp
MJ
20RJ
(13)

No. 2, 2010

FORMATION OF JOVIAN AND SATURNIAN SATELLITES

1057

Table 2
Simulation Settings for Jovian (set I) and Saturnian (set II) Systems
Simulation Setting
Jovian systems (set I)
Jovian systems (set I’)
Saturnian systems (set II)

Fp,0/(5 × 106 yr)

Mp
Mp /(5 × 106 yr)
Mp /(2 × 106 yr)

Inner Cavity
Yes
Yes
No

Disk Depletiona

α

1
2
3

10−3 to 10−2 b
10−3 to 10−2
10−3 to 10−2

Notes.
a (1) The evolution of F is the same as (3) until it is truncated at t = (0.5–0.8)τ
p
dep , which is corresponding to the gap
formation; (2) the same as (1) except that Fp is reduced by a factor of 100 instead of complete vanishment at the gap
formation; (3) after establishment of steady state, the infall is decayed as Fp = Fp,0 exp(−t/τdep ) with τdep = (3–5) ×
106 yr (we assume a log-uniform distribution).
b A log-uniform distribution.

where M is the satellite mass. The type I migration rate isṙ = r/τmig and the migration timescale is (Tanaka et al. 2002)
cs 2 Mp Mp −1Ωτmig  0.3
rΩK
M r 2 Σg K

−1 
 
1/2
Mp −1
M
r
 105 fg−1
10−4 Mp
MJ
20RJ
−1/4

τG
×
yr
5 × 106 yr


−1 
 
1/2
Mp −1
α
M
r
5
 10
5 × 10−3
10−4 Mp
MJ
20RJ
1/2

τG
×
yr.
(14)
5 × 106 yr


We include neither reversed torque due to a cavity (Masset
et al. 2006) nor entropy gradient (Paardekooper et al. 2009)
for simplicity. If a proto-satellite migrates to the host planet
or collides with another proto-satellite, a next-generation seedwith mass M = 1020 g is generated in the regions out of
which preceding runaway bodies have migrated leaving manysatellitesimals.
When τacc becomes larger than τmig , migration actually starts.
From Equations (13) and (14), the critical satellite mass for the
migration is given by−1/8 −9/16r−3/4 Mp
Mmig  2 × 10−5 ηice
MJ
20RJ
−3/16

τG
×
M∗ .
(15)
5 × 106 yr
3.3. Trapping by Resonances and Disk Edge
When the orbits of proto-satellites approach each other due totheir differential type I migration speed, angular momentum istransferred between them through mutual gravitational perturbations and they are often trapped in mean-motion resonances.
As explained in Appendix C, migration in proto-satellite disks isslow enough that resonant trapping occurs with very high probability. We assume that the two approaching proto-satellites aretrapped at a resonance near an orbital separation of 5rH .
For the Saturn case, we will show that large proto-satellites
generally form in the outer regions and they sweep up smaller
proto-satellites in the inner regions with the resonant trapping,because larger bodies migrate faster.
For the Jupiter case, migration is halted at the disk inner edge
and subsequently migrating satellites are trapped in resonances

one after another. Individual satellites, except the innermost
one, should keep losing angular momentum through type I
migration even after the trapping. However, Ogihara & Ida(2009) used N-body simulations that included the damping
due to disk–planet interactions to produce resonantly trappedsatellites that are lined up outside the inner disk edge with theinnermost one pinned up at the edge even in the case without
the effect of reversed type I migration torque near the edge
(Masset et al. 2006). Ogihara et al. (2010) has investigatedthis halting mechanism in detail and found that asymmetriceccentricity damping for the body in an eccentric orbit straddlingthe edge transfers angular momentum from the disk to the body.
Since the timescale of the eccentricity damping is much shorter
than type I migration timescale by a factor of ∼(cs /vK )2 ∼O(10−3 ) (Tanaka et al. 2002; Tanaka & Ward 2004), thisangular momentum supply is great enough to compensate forthe loss due to type I migration torques on all the trappedbodies (for details, see Ogihara et al. 2010). For the angularmomentum supply mechanism to work, eccentricity of theinnermost body must be continuously excited. In the case of
the resonantly trapped bodies, the eccentricity of the innermost
body is continually excited by resonant interactions with theother bodies in the resonance. The maximum number of trappedsatellites may be several, if their formula is applied.
However, there is another limitation for the trapping, which
is not considered in Ogihara et al. (2010). When the total massof the trapped satellites exceeds the total disk mass, angularmomentum loss due to type I migration is absorbed by outward
diffusion of the disk to outer regions in which the specific orbital
angular momentum is higher. As a counter-reaction, the inneredge is no longer able to halt the satellites’ inward migration.
The innermost satellite is therefore released to the planetary
surface. Thus, Jupiter systems also attain a quasi-steady statein which the total mass in trapped satellites is kept almost
constant with small amplitude oscillations. We here adopt thelatter condition rather than the condition derived by Ogihara
et al. (2010). (The former condition is much more importantfor studying formation of close-in super-Earths in extrasolar
planetary systems; Ida & Lin 2010.)
We summarize the simulation settings for Jovian (set I) and
Saturnian (set II) systems in Table 2.
3.4. Comparison with N-body Simulations
To check the validity of our semi-analytical model, we carriedout the calculations with the same disk conditions as thosein Figure 2 of Canup & Ward’s N-body simulations (Canup& Ward 2006). Here, Fp is constant without the exponential
decay. The non-cavity condition and f = 100 are adopted fora Jupiter mass planet. The total mass in satellites, MT , in our1058

SASAKI, STEWART, & IDA

Vol. 714

different random numbers for generating the initial locations ofproto-satellite seeds and values for the parameter α for the disk
viscosity and the timescale τdep for exponential decay of mass
infall onto the disk. In addition to the initial locations of thesatellite seeds, we assume log-uniform distributions for α in
10−2 –10−3 and τdep in (3–5) × 106 yr. The dependence on thevalue of α is also very weak as long as α is in this range. The
inner boundary and infall truncation conditions are the mostimportant factors that can change the results.
4.1. Saturnian Satellite System

Figure 2. Results of calculations with time-constant inflows to be comparedwith Canup & Ward’s (2006) N-body simulation (Figure 2 in their paper). The
non-cavity condition is adopted for a Jupiter mass planet. The total mass insurviving satellites, MT , scaled by the planet’s mass (Mp ) is shown vs. timescaled to τG . The solid, dashed, and dotted lines correspond, respectively, tocalculations with α = 10−4 , 5 × 10−3 , and 5 × 10−2 . Dotted horizontal lines
are predicted values of MT /Mp from Equation (3) of Canup & Ward (2006).

simulation is plotted as a function of time for α = 10−4 (solidline), α = 5 × 10−3 (dashed line), and α = 5 × 10−2 (dottedline) in Figure 2. After an initial buildup, an equilibrium statefor MT is attained in which disposal of proto-satellites to thehost planet due to type I migration is balanced with the supplyof satellitesimals from the infall. The dotted horizontal lines
are Canup & Ward’s (2006) analytical estimates for individual
values of α with which the asymptotic values of MT in theirN-body simulations are in good agreement.
Our semi-analytical model also reproduces evolution curves
that asymptote to the dashed lines. In addition to the asymptotic
values, the evolution of the initial buildup stage and oscillation
frequencies in the steady state are also very similar to Canup &Ward’s (2006) N-body simulations, although our results show
more regular oscillations.
As mentioned in Section 1, our semi-analytical model iscomplementary to N-body simulations. Because our model isfaster than N-body simulations by many orders of magnitude,
we can perform enough number of runs for statistical arguments
and survey different boundary and initial conditions.
4. RESULTS OF MONTE CARLO SIMULATIONS
For each set of simulations for Jovian and Saturnian systems(set I and set II; for details, see Table 2), we repeat 100 runs with(a)

We first show the results of set II, which adopts the samedisk setting as Canup & Ward’s (2006) N-body simulations. We
apply this setting to simulations of Saturnian satellite systems. In
this case, we found that satellites formed in outer regions of the
proto-satellite disk repeatedly sweep up inner small satellites,in a steady state after an initial buildup and before significant
depletion of disk gas. Figure 3(a) shows the distributions of
the number of final surviving satellites with M > 10−5 Mp
produced from 100 simulations for set II. Dark gray parts show
the runs that produced satellite systems in which the largest
satellite is mostly icy and M > 10−4 Mp , which are analogous
to the present Saturnian system.
In about 70% of the runs, only one body remains in theouter regions of the disk. Figure 4 shows the silicate-dust-togas ratio in the disk in the steady state. If fd /fg = 1, the ratiois the same as that of infalling gas, 1/100. This figure shows
the disk is highly metal-rich in outer regions. (We are using
the astronomical term “metal-rich” to denote all condensable
solids, including ices, rocks, and metals.) Since in the steadystate, fg is radially constant, fd itself is 10–100 times higher inouter regions than in inner regions, so that τacc is shorter inouter regions (Equation (13)). The total mass flux to satellite
feeding zones is larger in the outer regions of the disk because
the infall mass flux per unit area is assumed to be constant. Fromthese two facts, large satellites are predominantly formed in theouter regions. As large satellites migrate from the outer regions,the satellites accrete satellitesimals in the inner regions during
migration and trap smaller inner satellites into resonances topush them to the host planet, because type I migration is fasterfor heavier satellites (Tanaka et al. 2002). This process results insignificant depletion of Σd (equivalently fd ) in the inner regions.
In the outer regions, the retention of proto-satellites is efficientagainst disk gas accretion onto the host planet and the disk
becomes metal-rich (fd /fg ∼ 10–100).
This process repeats until the disk gas is so depleted that typeI migration becomes ineffective. Since type I migration is slower(b)

Figure 3. Distribution of the number of final surviving satellites with M > 10−5 Mp produced from 100 simulations for each system, (a) Saturnian system and (b)Jovian system. Dark gray parts show the runs that the produced satellite systems are analogous to the real one: for the Saturnian system, the largest satellite is icy and
M > 10−4 Mp , while for the Jovian system, the inner two bodies are rocky and the outer two are icy satellites.

No. 2, 2010

FORMATION OF JOVIAN AND SATURNIAN SATELLITES

Figure 4. Dust-to-gas ratio in the steady state disk for three typical Saturniancases. The solid, dashed, and dotted lines correspond, respectively, to calculations with N = 15, 18, and 20. If fd /fg = 1, the ratio is the same as that ofinfalling gas, 1/100.

in the outer regions and proto-satellites keep growing near theoriginal locations until M exceeds Mmig , only one large bodyusually exists in outer regions after the disk gas is depleted.
Several large bodies like the Galilean satellites are found under the same simulation setting in the previous work (Canup &Ward 2006). Our simulations also reproduce the Galilean satellites’ analogs but with a very low probability (∼5% for threebodies, ∼1% for four bodies). The probability we found may bemuch lower than the results by N-body simulations by Canup &Ward (2006), although the evolution of the total satellite mass
MT is very similar (Figure 2). As mentioned in Section 1, while
the semi-analytical calculations inevitably introduce approximations, N-body simulations have to employ unrealistic initial
and/or boundary conditions to save computational time. In the
Canup & Ward’s (2006) N-body simulations, infalling materialsof dust grains are replaced by embryos with the isolation mass,
which is relatively large. Since evolution of MT is very similar,
our semi-analytical calculations and Canup & Ward’s (2006)
N-body simulations are consistent with each other to some level.
To reconcile the inconsistent details, a better calibrated semianalytical model and higher-resolution N-body simulations will
be needed.
In Figure 5(a), the average mass (Ms ) and semimajor axis (a)of the largest satellites with their standard deviations obtained
from 67 runs that produced only one large satellite are plotted
(a)

1059

and compared with Titan (open circle). The mass and semimajor
axis of the largest bodies agree with those of Titan within 1σ
standard deviations of 67 runs. Furthermore, the largest bodycontains (1) more than 95% of the total satellite system massin 31 runs out of 67 cases, (2) more than 90% in 21 runs,and (3) less than 90% in 15 runs. The actual significant massconcentration in Titan (∼95% of the Saturnian satellite systemmass) is reproduced in our simulations.
The colors of the results in Figure 5 show the average fractionof rocky (dark gray) and icy (light gray) components. Since thefinal satellites are the last survivors formed in the last stage withlow disk temperatures due to reduced viscous heating, they are
generally composed of ice. These satellites were usually formedon long timescales ∼ O(106 ) yr, which suggests that Titan is not
fully differentiated. In fact, recent Cassini data indicated that
Titan has an inertia factor of 0.34 and is therefore incompletely
differentiated just like Callisto (Iess et al. 2010).
Therefore, our calculations are able to produce many features(orbits, masses, and compositions) of Saturnian satellite system.
The results of typical successful runs are shown in Figure 6(a).
In these cases, we produced only one large satellite at Titan’s
location and some smaller satellites at inner satellites’ region.
Castillo-Rogez et al. (2007, 2009) have argued that Saturn’souter regular satellite, Iapetus, must have formed between 3.4
and 5.4 Myr after the production of the calcium–aluminum
inclusions (CAIs) found in meteorites in order to accrete thenecessary amount of the short-lived radioisotope 26 Al to ensure
its partial differentiation and tidal despinning to its current stateof synchronous rotation. Mosqueira et al. (2010) also emphasize
the important constraint that Iapetus places on any successfulmodel of Saturnian satellite formation. In future work, we plan toaddress the formation of Iapetus by considering a more detailed
evolution model of the outer edge of the proto-satellite disk.
4.2. Jovian Satellite System
For the Jovian case (set I), our numerical simulations produced 4 or 5 large bodies with M > 10−5 Mp in about 80% of
the runs (Figure 3(b)). Dark gray parts show the runs that the
produced satellite systems in which the inner two bodies are
rocky and the outer two are icy satellites. In the four large satellite cases, the inner three are trapped in mutual resonances inall runs, while the outermost one is not trapped in a resonancein about half of the runs. Such resonant trapping of inwardly
migrating satellites would reproduce the detailed orbital properties of the Galilean satellites (Peale & Lee 2002). Note that in(b)

Figure 5. Theoretically predicted satellite systems in (a) set II corresponding to Saturnian systems with the data of Titan and (b) set I for Jovian systems with thedata of Galilean satellites. For the Saturnian case, the average mass (Ms ) and semimajor axis (a) of the largest satellites with their standard deviations are plotted withfilled circles with bars for the highest probability cases (67 runs among 100 runs) in which only one large satellite (Ms > 10−5 Mp ) is produced. The open circle
represents Titan. For the Jovian case, the averaged mass–semimajor axis and their standard deviations from the highest probability runs (39 runs) that produced fourlarge satellites are plotted. Galilean satellites are represented by open circles. The colors of each plot show the average fraction of rocky (dark gray) and icy (lightgray) components of the formed satellites.

1060

SASAKI, STEWART, & IDA(a)(b) 0.001

0.001

0.00010.0001

1e-05

1e-05

1e-06

1e-06

1e-07

Vol. 714

0

5

10

15

20

25

30

1e-07

0

5

10

15

20

25

30

Figure 6. Final satellite systems for successful Saturnian and Jovian-like cases in (a) set II corresponding to Saturnian systems with the data of large satellites (Tethys,
Dione, Dione, and Titan) and (b) set I for Jovian systems with the data of Galilean satellites. The mass (Ms ) and semimajor axis (a) of the produced satellites areplotted with crosses, pluses, and stars for each successful case, while the open squares represent observed satellite systems.

the Saturnian system, this stage may be followed by the low-Σgphase without a cavity, in which all the trapped satellites fall
onto the host planet and only one dominant icy body formed in
the last stage remains, as explained in Section 4.1.
Figure 3(b) shows that for set I, formation of four large
satellites is the most likely outcome. In Figure 5(b), theiraveraged mass and semimajor axis and their standard deviations
from 39 runs that produced four large satellites are plotted with
Galilean satellites (open circles). The simulated masses andorbits are consistent with those of Galilean satellites.
In Figure 5(b), the compositions of the simulated satellites
are also indicated. In more than half of the runs, the inner
two bodies are composed mostly of rocky materials and while
the outer two are formed mostly of icy materials that havemigrated from the regions outside the ice line in the protosatellite disk. In the Jovian systems, the final satellites arethose formed in disks with relatively high infall rates onto
1/4
the disk, the disk temperature is high enough (Td ∝ Fp ;Equation (A2)) that rocky materials are a major componentof satellites formed in the inner regions (Equation (3)). Outersatellites are formed in the outer regions and migrate inward to
be trapped in resonances. Our simulations are consistent with thecompositional gradient observed in the Galilean satellites. The
results of typical successful runs are shown in Figure 6(b). In
these cases, we produced four large satellites which correspondto Galilean satellites for both sizes and locations. Although somesmaller satellites were also formed near Io’s location, they could
be merged with Io afterward.
The moment of inertia of Callisto suggests that its interior
is differentiated only partially. It implies that Callisto must be
formed slowly such as accretion timescale 5 × 105 yr or more
(Stevenson et al. 1986; Schubert et al. 2004; Barr & Canup2008). In the case of Jovian system, the outermost large satellite
corresponding to Callisto is formed on timescales of O(105 )yr in some cases, but generally the timescales would be much
shorter than the above estimate.
We also performed additional set (set I’) of calculations
with the effects of imperfect gap opening, that is, after thegap opening, we do not truncate the infall but reduce theinfall rate by a factor of 100. In almost all runs, we foundthat after the gap opening one additional big body tends toaccrete without migration in outer regions of the decayed disk
on timescales longer than O(106 ) yr, which comes from the factthat τacc ∝ fd−1 . Because the outermost body hardly migrates,
it is not captured by a resonance. The pronounced difference
between Jovian and Saturnian satellite systems is preserved,
although the most probable number of remaining satellites isfive. Thus, if we take into account the effects of imperfect gap
opening, we found that the formation timescale and orbital
configuration of the outermost satellite in our simulations
becomes much more consistent with Callisto. To discuss detailsof final orbital configurations and formation timescale of theoutermost satellite, we need to take into account the time
evolution of radial dependence of gas and dust infall rates.
4.3. Formation of Saturn’s Rings
We will comment on another pronounced difference between
Jovian and Saturnian systems, planetary rings. The discovery
by the Cassini mission that most of the mass in Saturn’s rings isorganized into opaque, elongated clumps of ring particles (Colwell et al. 2007, 2009; Hedman et al. 2007) has led to revised
estimates of the total mass contained in the rings. By comparing the amount of light transmitted by the rings during stellar
occultations with the transparency of local N-body simulationsof the rings, Robbins et al. (2010) estimate that the total massof Saturn’s rings may be twice the mass of the satellite, Mimas,
or about 10−7 Mp . The larger mass estimate for Saturn’s mainrings makes a ring formation scenario that captures a passing
large Kuiper belt object less probable and also causes problems
for ring formation by the collisional breakup of a former Saturnian satellite unless the impact occurred within the first 750
Myr after Saturn’s formation when the impactor flux at Saturn’s
orbit was much higher than today (Charnoz et al. 2009). Spectroscopic observations indicate that Saturn’s rings are mainlycomposed of water ice with less than a few percent by mass of a
non-icy component (Cuzzi et al. 2009). This is a much smaller
fraction of rock than is typical for the icy regular satellites in the
Saturn system, which presents a problem for ring formation by
the breakup of a former satellite unless substantial amounts ofsilicate rock are hidden under ice regoliths in the dense core of
the B ring. Recent Cassini observations indicate the existenceof macroscopic bodies that produce “propeller”-shaped structures, in the A ring. The existence of these macroscopic bodies
gives new support to a scenario of the ring origin through thecatastrophic disruption of a massive progenitor, implying thatthe ring is old (Tiscareno et al. 2008). The origin of Saturn’s
main ring system is therefore still an unsolved question.
Charnoz et al. (2009) showed that the ring may be formed
from an ancient satellite which was originally in Saturn’sRoche zone (RZ) and was destroyed by a passing comet.
(The small satellite itself may have formed outside RZ andmigrated or been scattered into RZ.) In this model, a key aspectis the survival of the satellite against tidal evolution in theplanet’s RZ. Since the tidal bulge of the host planet leads toNo. 2, 2010

FORMATION OF JOVIAN AND SATURNIAN SATELLITES

inward migration if the satellite’s orbital radius (r) is below
the synchronous radius (Rsync ) with the planet’s spin. Typicalinward migration timescales are ∼ O(108 ) yr for M ∼ 10−7 Mp ,which is much shorter than the solar system age, so that for thesatellite to survive, the condition Rsync < r < RRZ is required(Charnoz et al. 2009), where RRZ is the radius of RZ and isinversely proportional to cube root of the material density of the−1/3
satellite (RRZ ∝ ρs ). For current Jupiter, Saturn, Uranus, and
Neptune, Rsync /Rp = 2.24, 1.86, 3.22, and 3.36, respectively.
Even if a satellite is an icy body with material density ofρs = 1 g cm−3 , RRZ for Uranus and Neptune are 2.79Rp and
2.89Rp that are still smaller than their Rsync . Thus, Uranus and
Neptune may be unable to maintain satellites inside their RZs(Charnoz et al. 2009).
Our results indicate that the inner satellite, which can be theseed of the ring, should be mixture of ice and rock for Saturniansystem, while it may be rocky for Jovian system (Figure 5).
Since rocky bodies have larger values of ρs , RRZ may be smaller
for Jupiter. For example, RRZ = 1.90Rp with ρs = 1.5 g cm−3
for Saturn, which is larger than Saturn’s Rsync = 1.86Rp , while
RRZ = 1.99Rp with ρs = 2.5 g cm−3 for Jupiter, which issmaller than Jupiter’s Rsync = 2.24Rp . Therefore, Saturn isthe most plausible planet to maintain a satellite inside the RZ
against the tidal evolution. If the disruption scenario is correct,
our result that finally surviving satellites are formed in a warmer
disk for Jovian system than for Saturnian system is consistent
with the fact that Saturn has a massive ring while Jupiter doesnot.
5. CONCLUSION
We simulated growth and orbital migration of proto-satellites
in an accreting circum-planetary disk that is modified from thedisk model by Canup & Ward (2002, 2006), in order to address
the different architectures between Jovian and Saturnian satellite
systems.
1. Jovian system has four similar-sized satellites (Galilean
satellites) locked in mean-motion resonances except theoutermost one (Callisto), while Saturnian system has only
one big satellite, Titan, in outer region.
2. Inner two satellites are rocky and outer two are mostly icy
for Jovian system, while Titan is mostly icy.
We modified the semi-analytical model for the population
synthesis model for planet formation developed by Ida &Lin (2004, 2008) to apply the model for satellite formation,
introducing the effect of resonant trapping that characterizes
Galilean satellites. The semi-analytical model reproduces theresults consistent with N-body simulations by Canup & Ward
(2006), although details such as mass distribution of surviving
satellites are not necessarily consistent (it is a future work toreconcile the inconsistency).
We considered a coupled system of gas giant planet formationand satellite formation around the planet, although planet
formation is not simultaneously simulated. The fact that multiple
gas giant systems are found in extrasolar systems in addition to
our solar system suggests that gas accretion onto gas giants is
truncated or at least significantly reduced by gap formation in
the proto-planetary disk. Since Saturn is smaller than Jupiterand it is theoretically expected that Saturnian core is formed
later than that of Jovian core, we consider a working hypothesisthat Jupiter opened up a gap while Saturn did not. Accordingly,
we assume the following.

1061

1. Gas infall onto a proto-satellite disk is truncated or significantly reduced at the time of gap opening for Jupiterwhile that onto Saturnian disk slowly decays as the protoplanetary disk (the solar nebula) globally decayed on atimescale of τdep ∼ 1–10 Myr. The Jovian disk decays on
its viscous diffusion timescale of τdiff ∼ 103 yr after thegap opening.
2. Based on inference from observations of different spin
periods between CTTSs and WTTSs, the proto-satellite
disk has an inner cavity in the early, high disk accretion
stage and the cavity vanishes in the subsequent, low disk
accretion stage.
Since typical type I migration timescales are much longer
than τdiff and much shorter than τdep , the Jovian satellite system
may have been “frozen” at the time of gap opening in which disk
mass and temperature were relatively high while the Saturnian
satellites may be final survivors in the last stage in which disk
mass and temperature were relatively low. Thus, we carriedout simulations with a cavity and abrupt decay of disk gas for
Jupiter and those without a cavity and with slow disk depletionon timescales of 1–10 Myr for Saturn. Then, we have found thefollowing results.
1. In Jupiter-condition systems, four or five similar-sizedsatellites are formed in 80% of the runs and the inner ones
tend to be locked in mean-motion resonances due to type I
migration and its stoppage at the disk inner edge. In Saturncondition systems, predominant bodies repeatedly accretein outer regions and migrate onto the surface of the hostplanet, sweeping up inner small satellites. In 70% of runs,
only one big body remains. The big bodies have more than
95% of total mass of remaining bodies in the system in
about half of the runs.
2. Because Jupiter-condition systems are formed in relatively
hot disks, the inner two satellites tend to be rocky and outerones are ice-rich. On the other hand, in Saturn-condition
systems, the finally surviving bodies form in the longer
lasting cold disks and are therefore ice-rich. Moreover,the surviving massive bodies corresponding to Titan wereusually formed on long timescales (∼megayears), whichis consistent with the incomplete differentiation of Titan’s
interior.
If we consider a small amount of residual mass infall onto
the circum-Jovian disk after the gap opening, the formation
timescale of the outermost satellite is longer than megayears,
which is consistent with incomplete differentiation of Callisto’s
interior. Thus, our simulations produced the Galilean satellites’
analogs and the Titan’s analogs in each setting with highprobability.
We have demonstrated that our assumption on gap formation
and associated existence of a disk cavity naturally explain the
different architectures between Jovian and Saturnian satellite
systems, although more detailed fluid dynamical simulations areneeded for the radial dependence and its time evolution of infall
onto proto-satellite disks and gap formation in proto-planetarydisks, and more detailed MHD simulations are necessary forevolution of a cavity. The different architectures may be fossil
evidence that Jupiter opened up a clear gap in the proto-planetary
disk to terminate its growth. This result gives deep insights into
the mass distribution and multiplicity of extrasolar gas giant
planets (Ida & Lin 2008) as well as into the observed bimodal
distributions of spin periods of young stars (Herbst & Mundt
2005).

1062

SASAKI, STEWART, & IDA

Notice that the Jovian/Saturnian satellite systems are analogous to extrasolar super-Earth systems, because superEarths accrete only solid materials and have masses of∼(10−5 –10−4 ) M . A high fraction of solar-type stars may haveclose-in super-Earths (e.g., Mayor et al. 2009), while many systems including the solar system do not have any close-in superEarths. Our results should also give deep insights into the originof the diversity of super-Earth systems found around other stars.
We gratefully acknowledge the referee’s helpful comments.
This work was supported by a JSPS Research Fellowship. G.R.S.
was supported by NASA’s Outer Planet Research Program andthanks the Tokyo Institute of Technology for hosting his visit
in 2007. S.I. thanks the University of Colorado at Boulder for
hosting his visit in 2008.
APPENDIX ACIRCUM-PLANETARY DISK MODEL
The growth rate of a proto-satellite depends on the surface
density of solid materials of the proto-satellite disk (Ida & Lin
2004), and its orbital migration rate depends on the surface
density of gas (Tanaka et al. 2002). We adopt the model of anactively supplied accretion disk by Canup & Ward (2002, 2006).
The asymptotic disk gas surface density distribution at r < rc ,in which the gas from the circum-stellar proto-planetary disk
inflows, is given by
Fp
4
Σg =
1−
3π ν
5

rc
1
−
rd
5



r
rc

2

Fp
,
 0.553π ν

9 20.55 × 3 2
ΩK νΣg 
ΩK Fp ,
88π

where ΩK is Keplerian frequency. That is, in the steady statewith a constant Fp,0 ,
Td,0  160

Mp
MJ1/2 

−1/4 

r
20RJ

−3/4
K.

Note that Td depends solely on Fp (equivalently, τG ) but is
independent of the value of the viscosity ν (equivalently, α)which has a large uncertainty. After establishment of a steadystate, Td is decreased from this value according to the decay of
Fp , so proto-satellites are mostly composed of ice in the final
state.
The midplane temperature, Tc , is given by Tc  (1 +3τ/8)1/4 Td , where τ is optical depth. Because we are concernedwith disk temperature mainly for evolution of the ice line atTd ∼ 160 K that divides compositions of satellitesimals and islocated in relatively outer regions, we assume that Tc  Td to
avoid uncertainty in opacity.
Adopting the alpha prescription for viscosity (Shakura &Sunyaev 1973), ν  αcs2 /ΩK , the disk gas surface density
distribution becomesΣg  100fg

(A1)

(A2)

τG
5 × 106 yr(A3)
After the exponential decay of Fp is imposed, the disk temperature decreases as


t
.
(A4)
Td = Td,0 exp −
4τdep



where Fp is the total infall rate to whole disk regions at r < rc ,
ν is disk gas viscosity, and rd is the diffused-out disk outeredge, which is ∼150Rp . The critical radius for the infall, rc ,is defined as the orbital radius at which the specific angularmomentum of a circular orbit equals that of the inflow average.
We adopt rc = 30Rp that is consistent with three-dimensionalhydrodynamic simulations with high spatial resolution within
the planet’s Hill sphere (Machida et al. 2008; Machida 2009).
The total infall rate is defined with a parameter τG as
Fp,0 = Mp /τG for the steady accretion state. According toCanup & Ward (2006), we adopt τG = 5 × 106 yr for Saturnwhile 2 × 106 yr for Jupiter. The infall rate decays exponentiallywith timescale τdep = 3 × 106 –5 × 106 yr in the final state of
the Saturnian case and abruptly vanishes due to the gap opening
in the proto-planetary disk for the Jovian case. The second andthird terms in the bracket of the above equation are correction
terms due to outward diffusion. If they are neglected, Σg is noother than the distribution of an ordinary steady accretion disk.
The proto-satellite disk is heated by luminosity from the central planet, viscous dissipation, and energy dissipation associated with the difference between the free-fall energy of theincoming gas and that of a Keplerian orbit. We make the simplification that the viscous dissipation is the dominant energy
source, as is typically the case in most of the regular satellite
regions (Canup & Ward 2002). Then, the photosurface temperature of the proto-satellite disk (Td ) is determined by a balance between viscous heating and blackbody radiation from thephotosurface,
σSB Td4 Vol. 714

Mp
MJ



r
20RJ

−3/4

g cm−2 ,

(A5)

where

fg ≡

α
5 × 10−3

−1 

τG
5 × 106 yr

1/4

−3/4
.

(A6)

−3/4

3/4

Since ν ∝ Td ∝ Fp , Σg ∝ Fp /ν ∝ Fp ∝ τG .
For simplicity, it is assumed (Canup & Ward 2002, 2006) thatthe infall flux per unit area is constant (Fp /π rc2 ) at r < rc andvanishes at r > rc . The total disk mass inside rd isrdMdisk =

Σg 2π rdr  1×10−4 fg



0

Mp
MJ



rd
150RJ

5/4
MJ .

(A7)
We scale the solid surface density with a scaling factor fd asΣd = ηice fd

Mp
MJ



r
20RJ

−3/4

g cm−2 ,(A8)

where ηice is an enhancement factor due to condensation of icy
grains at r > rice where disk temperature is lower than 160K;
we adopt ηice = 3 for r > rice and ηice = 1 otherwise. Since
the satellitesimals’ motions are decoupled from gas accretion/diffusion, they stay on-site and Σd remains zero for r > rc .
The infall rate of solid components per unit area of the disk isgiven by (Fp /π rc2 )/(f/ηice ), where f is the ratio of gas to rocky
dust grains in the Solar nebula. In our calculations, the radial
distribution of solid materials (Σd or fd ) in the proto-satellite disk
quickly relaxes to a steady state in which the supply of solid
components in the infalling gas is balanced with the removal
due to accretion by proto-satellites and their migration. In theNo. 2, 2010

FORMATION OF JOVIAN AND SATURNIAN SATELLITES

steady state, the rate of increase of fd due to infall is(ηice /f ) Fp /π rc2dΣd /dt
dfd==
dtΣd (fd = 1)
Σd (fd = 1)


 
−1

Mp −2/3 f −1
τG
= 0.029
MJ
100
5 × 106 yr
3/4

r
×
yr−1 .
(A9)
20RJ
The increase in the masses of the proto-satellites is subtracted
from their feeding zones of width 10rH . fd is assumed to
be locally uniform in the feeding zones. The proto-satellites
migrate inward and eventually fall onto the host planets in thepresence of disk gas. Thereby, the total amount of solid materials
is decreased by the growth and migration of proto-satellites. Our
calculation shows that asymptotic values of fd are ∼1 in inner
regions and (10–100)fg in outer regions (Figure 4), which meansthat the outer disks are generally metal-rich due to the higher
retention rate of solid materials compared to the gas.
APPENDIX B
ACCRETION AND MIGRATION RATES OF
PROTO-SATELLITES
The growth and migration of satellites in circum-planetary
proto-satellite disks proceed similarly to those of solid planetsin circum-stellar proto-planetary disks that have been studied
in detail (Ida & Lin 2004). Given a disk model, the accretionand migration rates of proto-satellites are analytically evaluated.
For values of velocity dispersion (σ ) larger than the Hill velocity(rH ΩK ) but smaller than surface escape velocity from satellites,
the accretion rate of a proto-satellite with mass M at an orbital
radius r is (Ida & Lin 2004)




2GM
GM
2
2
Ṁ  Cπ R ρdσ  2Cπ R Σd ΩK
Rσ 2
Rσ 2



 
2
rΩK
Σd r 2RMΩK ,(B1) 2π Cr
Mp
σwhere ρd is the spatial mass density of solid components(ρd  Σd ΩK /σ ), R is a physical radius of the satellite, and C
is a numerical factor of 2–3 (Stewart & Ida 2000; Ohtsuki et al.
2002). In principle, the velocity dispersion σ of satellitesimals
would be determined by a balance between gas drag damping
and stirring by the satellite. However, because the typical massof satellitesimals, which determines gas drag damping, accretedby the satellite is not clear, we simply set


σ
M 1/3
β
,
(B2)
rΩK
3Mp

1063

where we used C = 2.5 and ρ and ρp are bulk densities ofthe satellite and the host planet (we neglect the dependence on
ρ/ρp ).
The type I migration timescale (Tanaka et al. 2002) is
cs 2 Mp Mp −11rτmig = =Ωṙ
2.7 + 1.1qg rΩK
M r 2 Σg K

−1 
 
1/2
Mp −1
M
r
1
 105
fg 10−4 Mp
MJ
20RJ
−1/4

τG
×
yr,
(B4)
5 × 106 yr
where qg is defined by qg = − ln Σg / ln r.
APPENDIX CRESONANT TRAPPINGFor two satellites on nearly circular orbits, the expansion of
the difference in their semimajor axes (b) during an encounteris given by linear calculations (Goldreich & Tremaine 1982;
Hasegawa & Nakazawa 1990) as δb  30(b/rH )−5 rH . The
Hill radius (rH ) is defined by rH = (M/3Mp )1/3 r, where M,
Mp , and r are masses of the proto-satellite and the central
planet, and the orbital radius of the proto-satellite, respectively.
Since the scattering occurs at every synodic period [Tsyn =2π r/((dΩ/dr)b)  (4π r/3bΩK )], −4 
rH
b
db
δb
7

dt
Tsyn
rH
r

2

vK .

(C1)

The embryos may be trapped in a resonance close to theequilibrium value of b that satisfies db/dt = vmig , where v mig is
relative radial migration speed for the non-interacting case thatis given by Equation (14). Then, the equilibrium value is givenby1/6 
vmig −1/4Mbtrap  0.29
rH .
(C2)
10−4 MJvK√
If the estimated value of btrap is smaller than 2 3rH , the trapping
does not actually occur, because the separation is within theirfeeding zone and δb saturates due to the non-linear effect (e.g.,
Ida 1990). Substituting Equation (14) into Equation (C2), we
find thatwith β  10, following the case of planetesimals in the protoplanetary disks (Ida & Lin 2004). The accretion timescale is−1/12 
1/8
Mp −1/4Mr
10−4 MJ
MJ
20RJ
−1/16

τG
×
rH .
(C3)
5 × 106 yr


  
  −1/3
Mpρ
M 1/3 β 2Mrτacc =
TK
 0.5
Rp
ρp
Σd r 2Mp
10Ṁ −1/3 1/3 

Mp −5/6ρM
6 −1 −1 10 fd ηiceρp10−4 Mp
MJ
5/4
 2 
r
β
×
yr,
(B3)
10
20RJ

√
Since the estimated value of btrap is much greater than 2 3rH , weassume perfect trapping in our simulations. In our simulations,several satellites are usually trapped, so the final trapping
location tends to be deeper than the above value due to collective
interactions, which is also shown in N-body simulations (e.g.,
Ogihara & Ida 2009). We therefore assume that the two
approaching proto-satellites are trapped at a resonance nearb = 5rH .

btrap  16fg−1/4



1064

SASAKI, STEWART, & IDA
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