ORIGIN OF THE DIFFERENT ARCHITECTURES OF THE JOVIAN AND SATURNIAN SATELLITE SYSTEMS

First, we briefly summarize descriptions for evolution of the actively supplied accretion disk model with relatively small mass ∼10⁻⁴M_p(p = Jupiter, Saturn) developed by Canup & Ward (2002, 2006), which we follow in the present paper. More detailed descriptions are given in Appendix A.

Gas infall from the proto-planetary disk onto the proto-satellite disk is limited to the regions inside a critical radius r_c. Canup & Ward (2006) assumed that the infall flux inside r_c is independent of orbit radius and r_c has a fixed value of 30R_p, where R_p 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 enough integration times to constrain the radial and time dependence of the infall.) That is, the infall flux per unit area has a constant value of (F_p/πr²_c) at r < r_c and vanishes at r>r_c, where r is the orbital radius around the host planet and F_p is the total mass infall rate onto the disk. In the steady accretion state of the proto-planetary disk, the infall rate is defined with a parameter τ_G as F_p,0 = M_p/τ_G. 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)

$$\begin{equation} \Sigma _g \,{\simeq}\, \left\lbrace \begin{array}{@{}ll@{}} {\displaystyle 0.55\frac{F_p}{3\pi \nu } \,{\simeq}\, 100f_g\!\left(\!\frac{M_p}{M_J}\!\right)\!\left(\!\frac{r}{20R_J}\!\right)^{-3/4}\!\!\!\!\! \mbox{g cm$^{-2}$}} & [r < r_c], \\ 0 & [r > r_c], \end{array} \right. \end{equation} \tag{ 1 }$$

where

$$\begin{equation} f_g \equiv \left(\frac{\alpha }{5\times 10^{-3}}\right)^{-1}\left(\frac{\tau _G}{5\times 10^6\;\mbox{yr}}\right)^{-3/4}, \end{equation} \tag{ 2 }$$

where we used the turbulent viscosity of the proto-satellite disk with ν = αH²_SDΩ_K (Shakura & Sunyaev 1973), $\Omega _{\rm K} = \sqrt{GM_p/r^3}$, and temperature distribution to calculate the disk scale height (H_SD) is given by Equation (3). Since ν ∝ T_d ∝ F^1/4_p (Equation (3)), Σ_g ∝ F_p/ν ∝ F^3/4_p ∝ τ^−3/4_G. After a quasi-steady state of accretion and destruction of satellites (see Sections 2.2 and 2.3) is established, we start the exponential decay of infall as F_p = F_p,0exp(−t/τ_dep) with τ_dep = (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),

$$\begin{equation} T_{d,0} \simeq 160\left(\frac{M_p}{M_J}\right)^{1/2} \left(\frac{\tau _G}{5\times 10^6 \;\mbox{yr}}\right)^{-1/4} \left(\frac{r}{20R_J}\right)^{-3/4}\mbox{ K}. \end{equation} \tag{ 3 }$$

The disk is assumed to be vertically optically thin and isothermal (Appendix A). After the exponential decay of F_p is imposed, the disk temperature is decreased as

$$\begin{equation} T_d = T_{d,0} \exp \left(-\frac{t}{4 \tau _{\rm dep}}\right). \end{equation} \tag{ 4 }$$

The disk structure (gas surface density and temperature) is therefore regulated by F_p (=M_p/τ_G), in the actively supplied disk model. We discuss how F_p is likely to evolve with time for Jovian and Saturnian disks in the following. We will also point out that the disk inner boundary condition may also be regulated by F_p (Section 2.4).

Except for the late stages, growth of a gas giant is a runaway process of accreting gas from the disk. The mass increase is regulated by Kelvin–Helmholtz contraction with a timescale (see, e.g., Ida & Lin 2004),

$$\begin{equation} \tau _{\rm KH} = \frac{M_p}{\dot{M}_p} \simeq 10^6 \left(\frac{M_p}{10 \;M_{\oplus }}\right)^{-3}\;{\rm \mbox{yr}}, \end{equation} \tag{ 5 }$$

which can have different numerical factors depending on opacity, but the runaway nature is robust. Then, F_p ∼ 3 × 10⁻⁶(M/0.1M_J)⁴ M_J yr⁻¹ ∝ t^4/3. After the planet accretes all the gas in feeding zone with width ∼2r_H, where Hill radius r_H = (M_p/3M_*)^1/3r, at M_p ∼ 0.3M_J in the case of the solar nebula, the growth would be regulated by supply of global disk accretion, $\dot{M}_{\rm disk}$, which is ∼10⁻⁵M_J yr ⁻¹ and decays on timescales of 10⁶–10⁷ yr (see Figure 1).

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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 τ_G = 2 × 10⁶ yr (F_p = 5 × 10⁻⁷M_J yr⁻¹) for Jupiter. Although the choice of initial value of τ_G contains uncertainty, the critical satellite mass to start migration and disk temperature depend on τ_G only weakly: M_mig ∝ τ^−3/16_G (Equation (15)) and T_d ∝ τ^−1/4_G (Equation (3)). So, the mass distribution and compositions of final satellites do not sensitively depend on the initial value of τ_G. 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, F_p 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),

$$\begin{eqnarray} M_{\rm g,vis} &\simeq & \frac{40 \nu }{r_p \Omega _{\rm K}} M_* \simeq 40 \alpha \left(\frac{H_{\rm PD}}{r_p}\right)^2 M_* \nonumber\\ &\simeq & 30 \left(\frac{\alpha }{10^{-3}}\right) \left(\frac{r_p}{1\;{\rm AU}}\right)^{1/2} \left(\frac{M_*}{M_{\odot }}\right) \;M_{\oplus }, \end{eqnarray} \tag{ 6 }$$

where r_p is heliocentric orbital radius of the planet, Ω_K is Keplerian frequency at r_p, and M_* is the host star mass. For the scale height H_PD of the proto-planetary disk, we used the values of an optically thin disk (Hayashi 1981),

$$\begin{equation} H_{\rm PD} = 0.05 \left(\frac{r_p}{1\;{\rm AU}} \right)^{1/4} r_p. \end{equation} \tag{ 7 }$$

The thermal condition is determined by balance between the gravitational scattering and pressure gradient, r_H ∼ 1.5H_PD (Lin & Papaloizou 1985; Ida & Lin 2004). The critical mass is given by

$$\begin{equation} M_{\rm g,th} \simeq 400 \left(\frac{r_p}{1\;{\rm AU}}\right)^{3/4} \left(\frac{M_*}{M_{\odot }}\right) \;M_{\oplus }. \end{equation} \tag{ 8 }$$

Note that M_g,th is generally larger than M_g,vis, because typical values of the α parameter are ∼10⁻³–10⁻² for turbulence induced by magneto-rotational instability (e.g., Sano et al. 2004).

Since M_g,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 enough to halt gas accretion for M_p ≲ 5M_J (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 ($\dot{M}_{\rm disk}$) or less for M_p ∼ M_J and quickly vanishes with increasing M_p (Lubow & D'Angelo 2006; Tanigawa & Ikoma 2007). We will show below that 2 orders of magnitude reduction in the mass flux onto the proto-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 the gap opening depletes the circum-planetary proto-satellite disk on its own viscous diffusion timescale τ_diff ∼ R²_SD/ν where R_SD is the disk size and ν is disk viscosity. With the alpha model and R_SD ∼ 20R_p (where R_p is a physical radius of the planet),

$$\begin{equation} \begin{array}{ll} \tau _{\rm diff} & {\displaystyle \sim \frac{R_{\rm SD}^2}{\alpha H_{\rm SD}^2 \Omega _{\rm K}} \sim \left(\frac{R_{\rm SD}}{H_{\rm SD}} \right)^2 \frac{1}{\alpha } \frac{T_{\rm K}(R_{\rm SD})}{2 \pi } }\\ & {\displaystyle \sim 10^3 \left(\frac{\alpha }{10^{-3}}\right)^{-1} \;{\rm yr}}, \end{array} \end{equation} \tag{ 9 }$$

where H_SD is the scale height of the proto-satellite disk and T_K is an orbital period around the planet given by

$$\begin{equation} T_{\rm K}(r) = 0.032 \left(\frac{r}{20 R_p}\right)^{3/2} \;{\rm yr}. \end{equation} \tag{ 10 }$$

The gap opening itself would occur on the viscous diffusion timescale for the scale height (∼H_PD) in the proto-planetary disk, which is

$$\begin{equation} \tau _{\rm gap} \sim \frac{H_{\rm PD}^2}{\nu } \sim \left(\frac{H_{\rm PD}}{R_{\rm PD}} \right)^2 \frac{R_{\rm PD}^2}{\nu } \sim (10^{-3}\hbox{--}10^{-4}) \times (1\hbox{--}10) \;{\rm Myr}, \end{equation} \tag{ 11 }$$

where R_PD is size of the proto-planetary disk, which would be much larger than 5 AU, and t_dep ∼ R²_PD/ν is global disk evolution timescale. Both τ_diff and τ_gap are much shorter than formation and orbital evolution timescales of satellites (see Section 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 migration of satellites were "frozen" at that time.

Even if we consider the imperfect truncation of infall, this feature is not changed as long as the infall rate is reduced by a factor larger than 100. The equilibrium gas surface density of the proto-satellite disk (Σ_g) is determined by infall flux (F_p) as Σ_g ∝ F^3/4_p (Section 3.1). If the infall rate is decreased by 2 orders of magnitude, Σ_g is decreased by a factor of 30. According to the decrease in Σ_g and the associated decrease in surface density of satellitesimals, accretion and migration timescales of proto-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 with formation of a Callisto analog (Section 4.3).

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 Saturn should 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 from planetesimals 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 not undergone 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 diversity of 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 gradually depleted on the global depletion timescales of the proto-planetary disk, t_dep ∼ 1–10 Myr (Figure 1). The many orders of magnitude difference in depletion timescales of the proto-satellite disk between Jovian (τ_diff ∼ 10³ yr) and Saturnian (τ_dep ∼ 10⁶–10⁷ yr) systems would significantly affect the final configuration of satellite systems, because a typical type I migration timescale of proto-satellites (τ_mig ∼ 10⁵ yr; Tanaka et al. 2002) are in between the two timescales. Jovian satellites may retain their orbital configuration in a phase of the proto-satellite 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 less massive disk (Figure 1). The migration timescale depends on the magnitude of infall flux (F_p) through Σ_g and ν, which remain uncertain. However, since the migration timescale does not strongly depend on the flux (τ_mig ∝ F^−1/2_p), the relation of τ_diff < τ_mig < τ_dep is not changed by the uncertainty in F_p even if the uncertainty is 2–3 orders of magnitude. We start calculations from the last stage with τ_G = 5 × 10⁶ yr (F_p = 7 × 10⁻⁸M_J yr⁻¹) for Saturn, which is the same as the calculations by Canup & Ward (2002, 2006).

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 than the 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 the disk through the magnetic field. In this case, it is expected that the 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 mass accretion rate onto a planet of ∼10⁻⁶M_J yr⁻¹, if we use a standard formula of magnetospheric radius (Königl 1991). Such a 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 the magnetic coupling.

Although evolution of the inner cavity in the proto-satellite disk is not theoretically clear, the observations for proto-planetary 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 maintain the magnetic field strong enough for the magnetic coupling to transfer spin angular momentum to the disk and create a cavity. 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 truncated boundary 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 Saturnian system (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 the present paper is to demonstrate the important link between gas giant planet formation and satellite formation through evolution of the proto-satellite disk due to changes in infall rate and magnetic coupling.

In set II for Saturn, a quasi-steady state in which accretion of satellites is balanced with their removal to the host planet via type I migration is established. After the quasi-steady state is established, we start the exponential decay of infall as F_p = F_p,0exp(−t/τ_dep) with τ_dep = (3–5) × 10⁶ yr. We integrate the systems until disk gas surface density decreases so much that satellites no longer undergo orbital migration. This setting is the same 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 (∼2.25R_p). 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, the innermost 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 in the same way as in set II. However, we stop integration randomly at (0.5–0.8)τ_dep 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, which corresponds to imperfect gap opening.

While the evolution of the gas surface density of the proto-satellite 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 f_d as

$$\begin{equation} \Sigma _d = \eta _{\rm ice}f_d\left(\frac{M_p}{M_J}\right)\left(\frac{r}{20R_J}\right)^{-3/4} \mbox{g cm$^{-2}$}, \end{equation} \tag{ 12 }$$

where η_ice is an enhancement factor due to condensation of icy grains at r>r_ice where disk temperature is lower than 160 K; we adopt η_ice = 3 for r>r_ice and η_ice = 1 otherwise. In the infall flux, the mass fraction of solid components is given by η_ice/f, where f is the mass ratio of gas relative to silicate dust. For a fiducial value, we adopt f = 100. If f_d = f_g, the gas-to-dust ratio of the disk is the same as that of infall. But, we found that f_d is usually 10–100 times higher than f_g in outer regions in a steady state (Section 4.1). The method used to calculate the evolution of Σ_d is described in Appendix A.

We randomly select initial locations of 10–20 satellite seeds with M = 10²⁰ g (∼10⁻¹¹M_J, ∼10⁻¹⁰M_S) from a log-uniform distribution in the regions inside r_c = 30R_p and calculate their growth, migration, resonant trapping, and re-generation, simultaneously calculating the evolution of Σ_d, until disk gas is depleted. The log-uniform distribution for initial locations corresponds to the spacing between the seeds being proportional to orbital 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 effects of type I migration. The numbers of the initial satellite seeds (N = 10–20) do not affect the mass and orbital distributions of final satellites as well, as long as the initial mass is small enough, because τ_acc 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 $\dot{M} = M/\tau _{\rm acc}$ and the accretion timescale is (Appendix B)

$$\begin{eqnarray} \tau _{\rm acc} & \simeq & 0.5\left(\frac{r}{R_p}\right) \left(\frac{M_p}{\Sigma _d r^2}\right) \left(\frac{M}{M_p}\right)^{1/3} T_K \nonumber \\ &\simeq & 10^6 f_d^{-1}\eta _{\rm ice}^{-1} \left(\frac{M}{10^{-4}M_p}\right)^{1/3}\left(\frac{M_p}{M_J}\right)^{-5/6} \left(\frac{r}{20R_J}\right)^{5/4} \mbox{ yr},\nonumber\\ \end{eqnarray} \tag{ 13 }$$

where M is the satellite mass. The type I migration rate is $\dot{r} = r/\tau _{\rm mig}$ and the migration timescale is (Tanaka et al. 2002)

$$\begin{eqnarray} \tau _{\rm mig} & \simeq & 0.3 \left(\frac{c_s}{r\Omega _K}\right)^2\frac{M_p}{M}\frac{M_p}{r^2\Sigma _g}\Omega _K^{-1} \nonumber \\ &\simeq & 10^5 f_g^{-1}\left(\frac{M}{10^{-4}M_p}\right)^{-1}\left(\frac{M_p}{M_J}\right)^{-1}\left(\frac{r}{20R_J}\right)^{1/2}\nonumber \\ &&\times \left(\frac{\tau _G}{5\times 10^6\;\mbox{yr}}\right)^{-1/4} \mbox{ yr} \nonumber \\ &\simeq & 10^{5} \left(\frac{\alpha }{5 \times 10^{-3}}\right)\left(\frac{M}{10^{-4}M_p}\right)^{-1}\left(\frac{M_p}{M_J}\right)^{-1}\left(\frac{r}{20R_J}\right)^{1/2}\nonumber \\ &&\times \left(\frac{\tau _G}{5\times 10^6\;\mbox{yr}}\right)^{1/2} \;\mbox{yr}. \end{eqnarray} \tag{ 14 }$$

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 seed with mass M = 10²⁰ g is generated in the regions out of which preceding runaway bodies have migrated leaving many satellitesimals.

When τ_acc becomes larger than τ_mig, migration actually starts. From Equations (13) and (14), the critical satellite mass for the migration is given by

$$\begin{eqnarray} M_{\rm mig} & \simeq & 2 \times 10^{-5} \eta _{\rm ice}^{-3/4} \left(\frac{M_p}{M_J}\right)^{-1/8} \left(\frac{r}{20R_J}\right)^{-9/16}\nonumber \\ &&\times \left(\frac{\tau _G}{5\times 10^6\;\mbox{yr}}\right)^{-3/16} M_*. \end{eqnarray} \tag{ 15 }$$

When the orbits of proto-satellites approach each other due to their differential type I migration speed, angular momentum is transferred 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 is slow enough that resonant trapping occurs with very high probability. We assume that the two approaching proto-satellites are trapped at a resonance near an orbital separation of 5r_H.

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 trapped satellites that are lined up outside the inner disk edge with the innermost 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 investigated this halting mechanism in detail and found that asymmetric eccentricity damping for the body in an eccentric orbit straddling the 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 ∼(c_s/v_K)²∼ O(10⁻³) (Tanaka et al. 2002; Tanaka & Ward 2004), this angular momentum supply is great enough to compensate for the loss due to type I migration torques on all the trapped bodies (for details, see Ogihara et al. 2010). For the angular momentum supply mechanism to work, eccentricity of the innermost 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 the other bodies in the resonance. The maximum number of trapped satellites 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 mass of the trapped satellites exceeds the total disk mass, angular momentum 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 inner edge 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 state in which the total mass in trapped satellites is kept almost constant with small amplitude oscillations. We here adopt the latter condition rather than the condition derived by Ogihara et al. (2010). (The former condition is much more important for 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.

To check the validity of our semi-analytical model, we carried out the calculations with the same disk conditions as those in Figure 2 of Canup & Ward's N-body simulations (Canup & Ward 2006). Here, F_p is constant without the exponential decay. The non-cavity condition and f = 100 are adopted for a Jupiter mass planet. The total mass in satellites, M_T, in our simulation is plotted as a function of time for α = 10⁻⁴ (solid line), α = 5 × 10⁻³ (dashed line), and α = 5 × 10⁻² (dotted line) in Figure 2. After an initial buildup, an equilibrium state for M_T is attained in which disposal of proto-satellites to the host planet due to type I migration is balanced with the supply of 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 M_T in their N-body simulations are in good agreement.

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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 is complementary to N-body simulations. Because our model is faster 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.

We first show the results of set II, which adopts the same disk 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>10⁻⁵M_p 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>10⁻⁴M_p, which are analogous to the present Saturnian system.

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In about 70% of the runs, only one body remains in the outer regions of the disk. Figure 4 shows the silicate-dust-to-gas ratio in the disk in the steady state. If f_d/f_g = 1, the ratio is 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 steady state, f_g is radially constant, f_d itself is 10–100 times higher in outer regions than in inner regions, so that τ_acc is shorter in outer 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. From these two facts, large satellites are predominantly formed in the outer 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 to push them to the host planet, because type I migration is faster for heavier satellites (Tanaka et al. 2002). This process results in significant depletion of Σ_d (equivalently f_d) in the inner regions. In the outer regions, the retention of proto-satellites is efficient against disk gas accretion onto the host planet and the disk becomes metal-rich (f_d/f_g ∼ 10–100).

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This process repeats until the disk gas is so depleted that type I migration becomes ineffective. Since type I migration is slower in the outer regions and proto-satellites keep growing near the original locations until M exceeds M_mig, only one large body usually 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 three bodies, ∼1% for four bodies). The probability we found may be much lower than the results by N-body simulations by Canup & Ward (2006), although the evolution of the total satellite mass M_T 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 materials of dust grains are replaced by embryos with the isolation mass, which is relatively large. Since evolution of M_T 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 semi-analytical model and higher-resolution N-body simulations will be needed.

In Figure 5(a), the average mass (M_s) and semimajor axis (a) of the largest satellites with their standard deviations obtained from 67 runs that produced only one large satellite are plotted 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 body contains (1) more than 95% of the total satellite system mass in 31 runs out of 67 cases, (2) more than 90% in 21 runs, and (3) less than 90% in 15 runs. The actual significant mass concentration in Titan (∼95% of the Saturnian satellite system mass) is reproduced in our simulations.

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The colors of the results in Figure 5 show the average fraction of rocky (dark gray) and icy (light gray) components. Since the final satellites are the last survivors formed in the last stage with low disk temperatures due to reduced viscous heating, they are generally composed of ice. These satellites were usually formed on 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.

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Castillo-Rogez et al. (2007, 2009) have argued that Saturn's outer 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 the necessary amount of the short-lived radioisotope ²⁶Al to ensure its partial differentiation and tidal despinning to its current state of synchronous rotation. Mosqueira et al. (2010) also emphasize the important constraint that Iapetus places on any successful model of Saturnian satellite formation. In future work, we plan to address the formation of Iapetus by considering a more detailed evolution model of the outer edge of the proto-satellite disk.

For the Jovian case (set I), our numerical simulations produced 4 or 5 large bodies with M>10⁻⁵M_p 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 in all runs, while the outermost one is not trapped in a resonance in 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 the Saturnian system, this stage may be followed by the low-Σ_g phase 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), their averaged 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 and orbits 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 have migrated from the regions outside the ice line in the proto-satellite disk. In the Jovian systems, the final satellites are those formed in disks with relatively high infall rates onto the disk, the disk temperature is high enough (T_d ∝ F^1/4_p; Equation (A2)) that rocky materials are a major component of satellites formed in the inner regions (Equation (3)). Outer satellites are formed in the outer regions and migrate inward to be trapped in resonances. Our simulations are consistent with the compositional 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 correspond to Galilean satellites for both sizes and locations. Although some smaller 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 & Canup 2008). 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 the gap opening, we do not truncate the infall but reduce the infall rate by a factor of 100. In almost all runs, we found that after the gap opening one additional big body tends to accrete without migration in outer regions of the decayed disk on timescales longer than O(10⁶) yr, which comes from the fact that τ_acc ∝ f⁻¹_d. 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 is five. 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 details of final orbital configurations and formation timescale of the outermost satellite, we need to take into account the time evolution of radial dependence of gas and dust infall rates.

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 is organized 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 simulations of the rings, Robbins et al. (2010) estimate that the total mass of Saturn's rings may be twice the mass of the satellite, Mimas, or about 10⁻⁷M_p. The larger mass estimate for Saturn's main rings 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 mainly composed 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 of silicate rock are hidden under ice regoliths in the dense core of the B ring. Recent Cassini observations indicate the existence of 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 the catastrophic disruption of a massive progenitor, implying that the 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's Roche zone (RZ) and was destroyed by a passing comet. (The small satellite itself may have formed outside RZ and migrated or been scattered into RZ.) In this model, a key aspect is the survival of the satellite against tidal evolution in the planet's RZ. Since the tidal bulge of the host planet leads to inward migration if the satellite's orbital radius (r) is below the synchronous radius (R_sync) with the planet's spin. Typical inward migration timescales are ∼ O(10⁸) yr for M ∼ 10⁻⁷M_p, which is much shorter than the solar system age, so that for the satellite to survive, the condition R_sync < r < R_RZ is required (Charnoz et al. 2009), where R_RZ is the radius of RZ and is inversely proportional to cube root of the material density of the satellite (R_RZ ∝ ρ^−1/3_s). For current Jupiter, Saturn, Uranus, and Neptune, R_sync/R_p = 2.24, 1.86, 3.22, and 3.36, respectively. Even if a satellite is an icy body with material density of ρ_s = 1 g cm⁻³, R_RZ for Uranus and Neptune are 2.79R_p and 2.89R_p that are still smaller than their R_sync. 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 the seed of the ring, should be mixture of ice and rock for Saturnian system, while it may be rocky for Jovian system (Figure 5). Since rocky bodies have larger values of ρ_s, R_RZ may be smaller for Jupiter. For example, R_RZ = 1.90R_p with ρ_s = 1.5 g cm⁻³ for Saturn, which is larger than Saturn's R_sync = 1.86R_p, while R_RZ = 1.99R_p with ρ_s = 2.5 g cm⁻³ for Jupiter, which is smaller than Jupiter's R_sync = 2.24R_p. Therefore, Saturn is the 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 does not.