WATER ICE LINES AND THE FORMATION OF GIANT MOONS AROUND SUPER-JOVIAN PLANETS

The disk is assumed to be axially symmetric and in hydrostatic equilibrium. We adopt a standard viscous accretion disk model (Canup & Ward 2002, 2006), parameterized by a viscosity parameter α (10⁻³ in our simulations; Shakura & Sunyaev 1973), that is modified to include additional sources of disk heating (Makalkin & Dorofeeva 2014). We consider dusty gas disks around young ($\approx {10}^{6}$ years old), accreting giant planets with final masses beyond that of Jupiter (${M}_{\mathrm{Jup}}$). These planets accrete gas and dust from the circumstellar disk. Their accretion becomes increasingly efficient, culminating in the so-called runaway accretion phase when their masses become similar to that of Saturn (Lissauer et al. 2009; Mordasini 2013). Once they reach about a Jovian mass (depending on their distance to the star, among others), they eventually open up a gap in the circumstellar disk and their accretion rates drop rapidly. Hence, the formation of moons, which grow from the accumulation of solids in the CPD, effectively stops at this point or soon thereafter. A critical link between planet and moon formation is the combined effect of various energy sources (see the four heating terms described above) on the temperature distribution in the CPD and the radial position of the H₂O ice line.

In our disk model, the radial extent of the inner, optically thick part of the CPD, where moon formation is suspected to occur, is set by the disk's centrifugal radius (${r}_{\mathrm{cf}}$). At that distance to the planet, centrifugal forces on an object with specific angular momentum j are balanced by the planet's gravitational force. Using 3D hydrodynamical simulations, Machida et al. (2008) calculated the circumplanetary distribution of the angular orbital momentum in the disk and demonstrated the formation of an optically thick disk within about $30{R}_{\mathrm{Jup}}$ around the planet. An analytical fit to their simulations yields (Machida et al. 2008)

$$\begin{eqnarray}&&j(t)=\left\{\begin{array}{c}7.8\times {10}^{11}\left({\displaystyle \frac{{M}_{\rm{p}}(t)}{{M}_{\mathrm{Jup}}}}\right){\left({\displaystyle \frac{{a}_{\star \rm{p}}}{1\;\mathrm{AU}}}\right)}^{7/4}{\rm{m}}^{2}\;{\rm{s}}^{-1}\\ \qquad \qquad \mathrm{for}\ {M}_{\rm{p}}\lt {M}_{\mathrm{Jup}}\\ \ 9.0\times {10}^{11}{\left({\displaystyle \frac{{M}_{\rm{p}}(t)}{{M}_{\mathrm{Jup}}}}\right)}^{2/3}{\left({\displaystyle \frac{{a}_{\star \rm{p}}}{1\;\mathrm{AU}}}\right)}^{7/4}{\rm{m}}^{2}\;{\rm{s}}^{-1}\\ \qquad \qquad \mathrm{for}\ {M}_{\rm{p}}\geqslant {M}_{\mathrm{Jup}},\end{array}\right.\end{eqnarray} \tag{ 1 }$$

where we introduced the variable t to indicate that the planetary mass (${M}_{\rm{p}}$) evolves in time. The centrifugal radius is then given by

$$\begin{eqnarray}&&{r}_{\mathrm{cf}}={\displaystyle \frac{{j}^{2}}{{{GM}}_{\rm{p}}}},\end{eqnarray} \tag{ 2 }$$

with G as Newton's gravitational constant. For Jupiter, this yields a centrifugal radius of about $22\;{R}_{\mathrm{Jup}}$, which is slightly less than the orbital radius of the outermost Galilean satellite, Callisto, at roughly $27\;{R}_{\mathrm{Jup}}$. Part of this discrepancy is due to thermal effects that are neglected in the Machida et al. (2008) disk model. Machida (2009) investigated these thermal effects on the centrifugal disk size by comparing isothermal with adiabatic disk models. They found that adiabatic models typically yield larger specific angular momentum at a given planetary distance, which then translate into larger centrifugal disk radii that nicely match the width of Callisto's orbit around Jupiter.⁵ We thus introduce a thermal correction factor of 27/22 to the right-hand side of Equation (2) following Machida (2009), and therefore include Callisto at the outer edge of the optically thick part of our disk model. We note, though, that this slight rescaling hardly affects the general results of our simulations.

Recent 3D global magnetohydrodynamical (MHD) simulations by Gressel et al. (2013, see their Section 6.3) produce circumplanetary surface gas densities that agree much better with the "gas-starved" model of Canup & Ward (2006), which our model is derived from, than with the "minimum mass" model of Mosqueira & Estrada (2003a).⁶ The latter authors argue that Callisto formed in the low-density regions of an extended CPD with high specific angular momentum after Jupiter opened up a gap in the circumstellar disk. In their picture, the young Callisto accreted material from orbital radii as wide as $150\;{R}_{\mathrm{Jup}}$. In our model, however, Callisto forms in the dense, optically thick disk, where we suspect most of the solid material pile up. Simulations by Canup & Ward (2006) and Sasaki et al. (2010) show that our assumption can well reproduce the masses and orbits of the Galilean moons.

The disk is assumed to be mostly gaseous with an initial dust-to-mass fraction X (set to 0.006 in our simulations; Hasegawa & Pudritz 2013). Although we do not simulate moon formation in detail, we assume that the dust would gradually build planetesimals, either through streaming instabilities in the turbulent disk (Johansen et al. 2014, p. 547) or through accumulation within vortices (Klahr & Bodenheimer 2003), to name just two possible formation mechanisms. The disk is parameterized with a fixed Planck opacity (${\kappa }_{\rm{P}}$) in any of our simulations, but we tested various values. The fraction of the planetary light that contributes to the heating of the disk surface is parameterized by a coefficient ${k}_{\rm{s}}$, typically between 0.1 and 0.5 (Makalkin & Dorofeeva 2014). This quantity must not be confused with the disk albedo, which can take values between almost 0 and 0.9, depending on the wavelength and the grain properties (D'Alessio et al. 2001). The sound velocity in the hydrogen (H) and helium (He) disk gas usually depends on the mean molecular weight (μ) and the temperature of the gas, but in the disk midplane it can be approximated (Keith & Wardle 2014) as ${c}_{\rm{s}}\;=\;1.9\;\mathrm{km}\;{\rm{s}}^{-1}\;\sqrt{{T}_{\rm{m}}(r)/1000\;\rm{K}}$ for midplane temperatures ${T}_{\rm{m}}\lesssim 1000\;\rm{K}$. At these temperatures, ionization can be neglected and $\mu =2.34\;\mathrm{kg}\;{\mathrm{mol}}^{-1}$. Further, the disk viscosity is given by $\nu \;=\;\alpha {c}_{\rm{s}}^{2}/{\rm{\Omega }}_{\rm{K}}(r)$, with ${\rm{\Omega }}_{\rm{K}}(r)=\sqrt{{{GM}}_{\rm{p}}/{r}^{3}}$ as the Keplerian orbital frequency.

The steady-state gas surface density (${\rm{\Sigma }}_{\rm{g}}$) in the optically thick part of the disk can be obtained by solving the continuity equation for the infalling gas at the disk's centrifugal radius (Canup & Ward 2006), which yields

$$\begin{eqnarray}&&{\rm{\Sigma }}_{\rm{g}}(r)={\displaystyle \frac{\stackrel{\dot{}}{M}}{3\pi \nu }}\;\times \;{\displaystyle \frac{\rm{\Lambda }(r)}{l}}\end{eqnarray} \tag{ 3 }$$

where

$$\begin{eqnarray}\begin{array}{ccc}\rm{\Lambda }(r) & = & 1-{\displaystyle \frac{4}{5}}\;\sqrt{{\displaystyle \frac{{r}_{\mathrm{cf}}}{{r}_{\rm{d}}}}}-{\displaystyle \frac{1}{5}}{\left({\displaystyle \frac{r}{{r}_{c}}}\right)}^{2}\\ l & = & 1-\sqrt{{\displaystyle \frac{{R}_{\rm{p}}}{{r}_{\rm{d}}}}}\end{array}\end{eqnarray} \tag{ 4 }$$

is derived from a continuity equation for the infalling material and based on the angular momentum delivered to the disk (Canup & Ward 2006; Makalkin & Dorofeeva 2014), and $\stackrel{\dot{}}{M}$ is the mass accretion rate through the CPD, assumed to be equal to the mass dictated by the pre-computed planet evolution models (Mordasini 2013). We set ${r}_{\rm{d}}={R}_{\rm{H}}/5$, which yields ${r}_{\rm{d}}\approx 154\;{R}_{\mathrm{Jup}}$ for Jupiter (Sasaki et al. 2010; Makalkin & Dorofeeva 2014).

The effective half-thickness of the homogeneous flared disk, or its scale height, is derived from the solution of the vertical hydrostatic balance equation as

$$\begin{eqnarray}&&h(r)={\displaystyle \frac{{c}_{\rm{s}}({T}_{\rm{m}}(r)){r}^{3/2}}{\sqrt{{{GM}}_{\rm{p}}}}}.\end{eqnarray} \tag{ 5 }$$

We adopt the standard assumption of vertical hydrostatic balance in the disk and assume that the gas density (${\rho }_{\rm{g}}$) in the disk decreases exponentially with distance from the midplane as per

$$\begin{eqnarray}&&{\rho }_{\rm{g}}(r)={\rho }_{0}\;{e}^{\frac{-{z}_{\rm{s}}^{2}}{2h{(r)}^{2}}}\end{eqnarray} \tag{ 6 }$$

where z is the vertical coordinate and ${\rho }_{0}$ the gas density in the disk midplane. The gas surface density is given by vertical integration over $\rho (r,z)$, that is,

$$\begin{eqnarray}&&{\rm{\Sigma }}_{\rm{g}}(r)={{\displaystyle \int }}_{-\mathrm{inf}}^{+\mathrm{inf}}{dz}\ \rho (r,z).\end{eqnarray} \tag{ 7 }$$

Inserting Equation (6) into Equation (7), the latter can be solved for ${\rho }_{0}$ and we obtain

$$\begin{eqnarray}&&{\rho }_{0}(r)=\sqrt{{\displaystyle \frac{2}{\pi }}}{\displaystyle \frac{{\rm{\Sigma }}_{\rm{g}}(r)}{2h(r)}},\end{eqnarray} \tag{ 8 }$$

which only depends on the distance r to the planet. At the radiative surface level of the disk, or photospheric height (${z}_{\rm{s}}$), the gas density equals

$$\begin{eqnarray}&&{\rho }_{\rm{s}}(r)={\rho }_{0}\;{e}^{\frac{-{z}_{\rm{s}}^{2}}{2h{(r)}^{2}}}\end{eqnarray} \tag{ 9 }$$

where we calculate ${z}_{\rm{s}}$ as

$$\begin{eqnarray}&&{z}_{\rm{s}}(r)={\mathrm{erf}}^{-1}(1-{\displaystyle \frac{2}{3}}{\displaystyle \frac{2}{{\rm{\Sigma }}_{\rm{g}}(r){\kappa }_{\rm{P}}}})\sqrt{2}h(r).\end{eqnarray} \tag{ 10 }$$

The latter formula is derived using the definition of the disk's optical depth

$$\begin{eqnarray}&&\tau ={{\displaystyle \int }}_{{z}_{\rm{s}}}^{+\mathrm{inf}}{dz}\ {\kappa }_{\rm{P}}\rho (r,z)={\kappa }_{\rm{P}}{{\displaystyle \int }}_{{z}_{\rm{s}}}^{+\mathrm{inf}}{dz}\ \rho (r,z)\end{eqnarray} \tag{ 11 }$$

and our knowledge of $\tau =2/3$ at the radiating surface level of the disk, which gives

$$\begin{eqnarray}\begin{array}{ccc}{\displaystyle \frac{2}{3}} & = & \ {\kappa }_{\rm{P}}{{\displaystyle \int }}_{{z}_{\rm{s}}}^{+\mathrm{inf}}{dz}\ {\rho }_{\rm{g}}(r,z)\\ \iff {z}_{\rm{s}}(r) & = & \ {\mathrm{erf}}^{-1}(1-{\displaystyle \frac{2}{3}}\sqrt{{\displaystyle \frac{2}{\pi }}}{\displaystyle \frac{1}{h(r){\rho }_{0}(r){\kappa }_{\rm{P}}}})\sqrt{2}h(r).\end{array}\end{eqnarray} \tag{ 12 }$$

Using Equation (8) for ${\rho }_{0}(r)$ in Equation (12), we obtain Equation (10).

Following the semi-analytical disk model of Makalkin & Dorofeeva (2014), the disk surface temperature is given by the energy inputs of various processes as per

$$\begin{eqnarray}\begin{array}{ccc}{T}_{\rm{s}}(r) & = & \left({\displaystyle \frac{1+{(2{\kappa }_{\rm{P}}{\rm{\Sigma }}_{\rm{g}}(r))}^{-1}}{{\sigma }_{\mathrm{SB}}}}\left({F}_{\mathrm{vis}}(r)+{F}_{\mathrm{acc}}(r)\right.\right.\\ & & +{\left.\left.{k}_{\rm{s}}{F}_{\rm{p}}(r)\right)+{T}_{\mathrm{neb}}^{4}\right)}^{1/4},\end{array}\end{eqnarray} \tag{ 13 }$$

where

$$\begin{eqnarray}\begin{array}{ccc}{F}_{\mathrm{vis}}(r) & = & {\displaystyle \frac{3}{8\pi }}{\displaystyle \frac{\rm{\Lambda }(r)}{l}}\stackrel{\dot{}}{M}{\rm{\Omega }}_{\rm{K}}{(r)}^{2}\\ {F}_{\mathrm{acc}}(r) & = & {\displaystyle \frac{{X}_{d}\chi {{GM}}_{\rm{p}}\stackrel{\dot{}}{M}}{4\pi {r}_{\mathrm{cf}}^{2}r}}\ {e}^{-{(r/{r}_{\mathrm{cf}})}^{2}}\\ {F}_{\rm{p}}(r) & = & {L}_{\rm{p}}{\displaystyle \frac{\mathrm{sin}(\zeta (r)r+\eta (r))}{8\pi ({r}^{2}+{z}_{\rm{s}}^{2})}}\end{array}\end{eqnarray} \tag{ 14 }$$

are the energy fluxes from viscous heating, accretion onto the disk, and the planetary illumination, and ${T}_{\mathrm{neb}}$ denotes the background temperature of the circumstellar nebula (100 K in our simulations). The geometry of the flaring disk is expressed by the angles

$$\begin{eqnarray}\begin{array}{ccc}\zeta (r) & = & \mathrm{arctan}\left({\displaystyle \frac{4}{3\pi }}{\displaystyle \frac{{R}_{\rm{p}}}{\sqrt{{r}^{2}+{z}_{\rm{s}}^{2}}}}\right)\\ \eta (r) & = & \mathrm{arctan}\left({\displaystyle \frac{{{dz}}_{\rm{s}}}{{dr}}}\right)-\mathrm{arctan}\left({\displaystyle \frac{{z}_{\rm{s}}}{r}}\right).\end{array}\end{eqnarray} \tag{ 15 }$$

Taking into account the radiative transfer within the optically thick disk with Planck opacity ${\kappa }_{\rm{P}}$, the midplane temperature can be estimated as (Makalkin & Dorofeeva 2014)

$$\begin{eqnarray}\begin{array}{ccc}{T}_{\rm{m}}{(r)}^{5}-{T}_{\rm{s}}{(r)}^{4}\;{T}_{\rm{m}}(r) & = & \ {\displaystyle \frac{12\mu \chi {\kappa }_{\rm{P}}}{{2}^{9}{\pi }^{2}{\sigma }_{\mathrm{SB}}{R}_{\rm{g}}\gamma }}\times {\displaystyle \frac{{\stackrel{\dot{}}{M}}^{2}}{\alpha }}{\rm{\Omega }}_{\rm{K}}{(r)}^{3}\\ & & \times {\left({\displaystyle \frac{\rm{\Lambda }(r)}{l}}\right)}^{2}{q}_{\rm{s}}{(r)}^{2},\end{array}\end{eqnarray} \tag{ 16 }$$

where ${\sigma }_{\mathrm{SB}}$ is the Stefan–Boltzmann constant, ${R}_{\mathrm{Gas}}$ the ideal gas constant, $\gamma =1.45$ the adiabatic exponent (or ratio of the heat capacities), and

$$\begin{eqnarray*}&&{q}_{\rm{s}}(r)=1-{\displaystyle \frac{4}{3{\kappa }_{\rm{P}}{\rm{\Sigma }}_{\rm{g}}(r)}}\end{eqnarray*}$$

is the vertical mass coordinate at ${z}_{\rm{s}}$.

We use a pre-computed set of planet formation models by Mordasini (2013) to feed our planet disk model with the fundamental planetary properties such as the planet's evolving radius (${R}_{\rm{p}}$), its mass, mass accretion rate (${\stackrel{\dot{}}{M}}_{\rm{p}}$), and luminosity (${L}_{\rm{p}}$). Figure 1 shows the evolution of these quantities with black solid lines indicating an accreting gas giant that ends up with one Jupiter mass or about 318 Earth masses (${M}_{\oplus }$). In total, we have seven models at our disposal, where the planets have final masses of 1, 2, 3, 5, 7, 10, and $12\;{M}_{\mathrm{Jup}}$. These tracks are sensitive to the planet's core mass, which we assume to be $33\;{M}_{\oplus }$ for all planets. Jupiter's core mass is actually much lower, probably around $10\;{M}_{\oplus }$ (Guillot et al. 1997). Lower final core masses in these models translate into lower planetary luminosities at any given accretion rate. In other words, our results for the H₂O ice lines around the Jupiter-mass test planet are actually upper, or outer limits, and a more realistic evaluation of the conditions around Jupiter would shift the ice lines closer to the planet. Over the whole range of available planet tracks with core masses between 22 and $130\;{M}_{\oplus }$, we note that the planetary luminosities at shutdown are ${10}^{-4.11}$ and ${10}^{-3.82}$ solar luminosities, respectively. As the distance of the H₂O ice line in a radiation-dominated disk scales with ${L}_{\rm{p}}^{1/2}$, different planetary core masses would thus affect our results by less than ten percent.

Zoom In Zoom Out Reset image size

The pre-computed planetary models cover the first few 10⁶ years after the onset of accretion onto the planet. We interpolate all quantities on a discrete time line with a step size of 5000 years. At any given time, we feed Equation (14) with the planetary model and solve the coupled Equations (1)–(10) in an iterative framework. With ${T}_{\rm{s}}$ provided by Equation (13), we finally solve the fifth-order polynomial in Equation (16) numerically.

Once the planetary evolution models indicate that ${\stackrel{\dot{}}{M}}_{\rm{p}}$ has dropped below a critical shutdown accretion rate (${\stackrel{\dot{}}{M}}_{\mathrm{shut}}$), we assume that the formation of satellites has effectively stopped. As an example, no Ganymede-sized moon can form once ${\stackrel{\dot{}}{M}}_{\mathrm{shut}}\;\lt \;{M}_{\mathrm{Gan}}\;{\mathrm{Myr}}^{-1}$ (${M}_{\mathrm{Gan}}$ being the mass of Ganymede) and if the disk's remaining life time is $\lt {10}^{6}$ years (see Figure 1(c)). As ${\stackrel{\dot{}}{M}}_{\rm{p}}$ determines the gas surface density through Equation (3), different values for ${\stackrel{\dot{}}{M}}_{\mathrm{shut}}$ mean different distributions of ${\rm{\Sigma }}_{\rm{g}}(r)$. In particular, ${\rm{\Sigma }}_{\rm{g}}(r\;=\;10\;{R}_{\mathrm{Jup}})$ equals $7.4\;\times \;{10}^{2}$, $9.7\;\times \;{10}^{1}$, and $7.8\;\times \;{10}^{0}\;\mathrm{kg}\;{\rm{m}}^{-2}$ once ${\stackrel{\dot{}}{M}}_{\mathrm{shut}}$ reaches 100, 10, and 1 ${M}_{\mathrm{Gan}}\;{\mathrm{Myr}}^{-1}$, respectively, for the planet that ends up with one Jupiter mass (see Figure 4 in Heller & Pudritz 2015).

In any single simulation run, ${\kappa }_{\rm{P}}$ is assumed to be constant throughout the disk, and simulations of the planetary H₂O ice lines are terminated once the planet accretes less than a given ${\stackrel{\dot{}}{M}}_{\mathrm{shut}}$. To obtain a realistic picture of a broad range of hypothetical exoplanetary disk properties, we ran a suite of randomized simulations, where ${\kappa }_{\rm{P}}$ and ${\stackrel{\dot{}}{M}}_{\mathrm{shut}}$ were drawn from a lognormal probability density distribution. For ${\mathrm{log}}_{10}({\kappa }_{\rm{P}}/[{\rm{m}}^{2}\;{\mathrm{kg}}^{-1}])$ we assumed a mean value of −2 and a standard variation of 1, and concerning the shutdown accretion rate we assumed a mean value of 1 for ${\mathrm{log}}_{10}({\stackrel{\dot{}}{M}}_{\mathrm{shut}}/[{M}_{\mathrm{Gan}}\;{\mathrm{Myr}}^{-1}])$ and a standard variation of 1.

To get a handle on the plausible surface absorptivities of various disk, we tested different values of ${k}_{\rm{s}}$ between 0.1 and 0.5. For each of the seven test planets, we performed 120 randomized simulations of the disk evolution and then calculated the arithmetic mean distance of the final water ice line. The resulting distributions are skewed and non-Gaussian. Hence, we compute both the downside and the upside semi-standard deviation (corresponding to $\sigma /2$), that is, the distance ranges that comprise $+68.3\%/2\;=\;+34.15\%$ and $-34.15\%$ of the simulations around the mean. We also calculate the $2\sigma$ semi-deviation, corresponding to $+95.5\%/2\;=\;+47.75\%$ and $-47.75\%$ around the mean. Downside and upside semi-deviations combined deliver an impression of the asymmetric deviations from the mean, and their sum equals that of the Gaussian standard deviations.

The total, instantaneous mass of solids in the disk at the time of moon formation shutdown is given as

$$\begin{eqnarray}&&{M}_{\rm{s}}=2\pi X({{\displaystyle \int }}_{{r}_{\mathrm{in}}}^{{r}_{\mathrm{ice}}}{dr}\ r{\rm{\Sigma }}_{\rm{g}}(r)+3{{\displaystyle \int }}_{{r}_{\mathrm{ice}}}^{{r}_{\mathrm{cf}}}{dr}\ r{\rm{\Sigma }}_{\rm{g}}(r)),\end{eqnarray} \tag{ 17 }$$

where ${r}_{\mathrm{in}}$ is the inner truncation radius of the disk (assumed at Jupiter's corotation radius of 2.25 ${R}_{\mathrm{Jup}}$, Canup & Ward 2002), ${r}_{\mathrm{ice}}$ is the distance of the H₂O ice line, and ${r}_{\mathrm{cf}}$ is the outer, centrifugal radius of the disk (Machida et al. 2008). Depending on the density of the disk gas and the size of the solid grains, the water sublimation temperature can vary by several degrees Kelvin (Lewis 1972; Lecar et al. 2006), but we adopt 170 K as our fiducial value (Hasegawa & Pudritz 2013).

We simulate the evolution of the H₂O ice lines in the disks around young super-Jovian planets at 5.2 astronomical units (AU, the distance between the Sun and the Earth) from a Sun-like star, facilitating comparison of our results to the Jovian moon system. These planets belong to the observed population of super-Jovian planets at ≈1 AU around Sun-like stars (Hasegawa & Pudritz 2011, 2012; Howard 2013), and it has been shown that their satellite systems may remain intact during planet migration (Namouni 2010).

While Canup & Ward (2002) stated that accretion rates of $2\times {10}^{-7}\;{M}_{\mathrm{Jup}}\;{\mathrm{yr}}^{-1}$ (about $2.6\times {10}^{4}\;{M}_{\mathrm{Gan}}\;{\mathrm{Myr}}^{-1}$) best reproduced the disk conditions in which the Galilean satellites formed, our calculations predict a shutdown accretion rate that is considerably lower, closer to $10\;{M}_{\mathrm{Gan}}\;{\mathrm{Myr}}^{-1}$. The difference in these results is mainly owed to two facts. First, Canup & Ward (2002) only considered viscous heating.⁷ Our additional heating terms (illumination from the planet, accretion onto the disk and stellar illumination) contribute additional heat, which imply smaller accretion rates to let the H₂O ice lines move close enough to the Jupiter-like planet. Second, the parameterization of planetary illumination in the Canup & Ward (2002) model is different from ours. While they assume an ${r}^{-3/4}$ dependence of the midplane temperature from the planet (r being the planetary radial distance), we do not apply any pre-described r-dependence. In particular, ${T}_{\rm{m}}(r)$ cannot be described properly by a simple polynomial due to the different slopes of the various heat sources as a function of planetary distance.

Previous models assume that type I migration of the forming moons leads to a continuous rapid loss of proto-satellites into the planet (Canup & Ward 2002; Mosqueira & Estrada 2003a, 2003b; Alibert et al. 2005; Sasaki et al. 2010). (Alibert et al. 2005) considered Jupiter's accretion disk as a closed system after the circumstellar accretion disk had been photo-evaporated, whereas Sasaki et al. (2010) described accretion onto Jupiter with an analytical model. In the Mosqueira & Estrada (2003a, 2003b; ME) model, satellites migrate via type I but perturb the gas as they migrate and eventually stall and open a gap, ensuring their survival. In opposition to the Canup & Ward (CW) theory, their model does not postulate "generations" of satellites, which are subsequently lost into the planet, because satellite formation does not start until the accretion inflow onto the planet wanes.

There are two difficulties with the CW picture. First, type I migration can be drastically slowed down as growing giant moons get trapped by the ice lines or at the inner truncation radius of the disk.⁸ Thus it is not obvious that a conveyor belt of moons into their host planets is ever established. Second, our Figure 5 also contradicts this scenario, because the instantaneous mass of solids in the disk during the end stages of moon formation (or planetary accretion) is not sufficient to form the last generation of moons. In other words, whenever the instantaneous mass of solids contained in the circumjovian disk was similar to the total mass of the Galilean moons, the correspondingly high accretion rates caused the H₂O ice line to be far beyond the orbits of Europa and Ganymede.

We infer, therefore, that the final moon population around Jupiter and other Jovian or super-Jovian exoplanets must, at least to a large extent, have been built during the ongoing, final accretion process of the planet, when it was still fed from the circumstellar disk.⁹ In order to counteract the inwards flow due to type I migration, we suggest a new picture in which the circumplanetary H₂O ice line and the inner cavity of Jupiter's accretion disk have acted as migration traps. This important hypothesis needs to be tested in future studies. The effect of an inner cavity will also need to be addressed, as it might have been essential to prevent Io and Europa from plunging into Jupiter.

In our picture, Io should have formed dry and its migration might have been stopped at the inner truncation radius of Jupiter's accretion disk, at a few Jupiter radii (Takata & Stevenson 1996). It did not form wet and then lose its water through tidal heating. Ganymede may have formed at the water ice line in the circumjovian disk, where it has forced Io and Europa in the 1:2:4 orbital mean motion resonance (Laplace et al. 1829). From a formation point of view, we suggest that Io and Europa be regarded as moon analogs of the terrestrial planets, whereas Ganymede and Callisto resemble the precursors of giant planets.

Our combination of planet formation tracks and a CPD model enables new constraints on planet formation from moon observations. As just one example, the "Grand Tack" (GT) model suggests that Jupiter migrated as close as about 1.5 AU to the Sun before it reversed its migration due to a mutual orbital resonance with Saturn (Walsh et al. 2011). In the proximity of the Sun, however, solar illumination should have depleted the circumjovian accretion disk from water ices during the end stages of Jupiter's accretion (Heller et al. 2015). Thus, Ganymede and Callisto would have formed in a dry environment during the GT, which is at odds with their high H₂O ice contents. They can also hardly have formed over millions of years (Mosqueira & Estrada 2003a) thereafter, because Jupiter's CPD (now truncated from its environment by a gap) still would have been dry. Alternatively, one might suggest that Callisto and Ganymede formed after the GT from newly accreted planetesimals into a still active, gaseous disk around Jupiter. But then Io and Europa might have been substantially enriched in water, too. Tanigawa et al. (2014, see their Figure 8) found that planetesimal accretion via gas drag is most efficient between 0.005 and 0.001 Hill radii (${R}_{\rm{H}}$) or about $4-8\;{R}_{\mathrm{Jup}}$ where gas densities are relatively high.

To come straight to the point, our preliminary studies suggest that in the GT paradigm, the icy Galilean satellites must have formed prior to Jupiter's excursion to the inner solar system (Heller et al. 2015). This illustrates the great potential of moons to constrain planet formation, which is particularly interesting for the GT scenario where the timing of migration and planetary accretion is yet hardly constrained otherwise (Raymond & Morbidelli 2014).

Finally, we must address a technical issue, namely, our choice of the α parameter (10⁻³). While this is consistent with many previous studies, how would a variation of α change our results? Magnetorotational instabilities might be restricted to the upper layers of CPDs, where they become sufficiently ionized (mostly by cosmic high-energy radiation and stellar X-rays). Magnetic turbulence and viscous heating in the disk midplane might thus be substantially lower than in our model (Fujii et al. 2014). On the other hand, Gressel et al. (2013) modeled the magnetic stresses in CPDs with a 3D magnetohydrodynamic model and inferred α values of 0.01 and larger, which would strongly enhance viscous heating. Obviously, sophisticated numerical simulations of giant planet accretion do not yet consistently describe the magnetic properties of the disks and the associated α values.

Given that circumstellar disks are almost certainly magnetized, CPD can be expected to have inherited magnetic fields from this source. This makes it likely that magnetized disk winds can be driven off the CPD (Fendt 2003; Pudritz et al. 2007, p. 277) which can carry significant amounts of angular momentum. Even in the limit of very low ionization, Bai & Stone (2013) demonstrated that magnetized disk winds will transport disk angular momentum at the rates needed to allow accretion onto the central object. However, independent of these uncertainties, the final positions of the H₂O ice line produced in our simulations turn out to depend mostly on planetary illumination, because viscous heating becomes negligible almost immediately following gap opening. Hence, even substantial variations of α by a factor of ten would hardly change our results for the ice line locations at moon formation shutdown since these must develop in radiatively dominated disk structure (but it would alter them substantially in the viscous-dominated regime before and during runaway accretion).

Our assumption of a constant Planck opacity throughout the disk is simplistic and ignores the effects of grain growth, grain distribution within the disk, as well as the evolution of the disk properties. In a more consistent model, ${\kappa }_{\rm{P}}$ depends on both the planetary distance and distance from the midplane, which might entail significant modifications in the temperature distribution that we predict.