In Situ Formation of Icy Moons of Uranus and Neptune

We ran three-three hydrodynamic simulations for Uranus and Neptune with the JUPITER code (Szulágyi et al. 2016) developed by F. Masset and J. Szulágyi. The code is three-dimensional, grid-based, solving the Euler equations together with radiative dissipation via the flux the limited diffusion approximation method (Kley 1989; Commerçon et al. 2011). In each case there is a spherical coordinate system with a star in the center, surrounded by a gas circumstellar disk. The planets were treated as point masses at the location of the current Uranus (19.2 au) and Neptune (30.1 au). The circumstellar disk radius ranged between 7.7 au and 45.8 au for Uranus, and 12.0 au and 71.8 au for Neptune. Radially 215 cells, azimuthally 618 cells were used on the base mesh. The initial disk opening angle was 568 (from the midplane to the disk surface, using 20 cells), but the disk got thinner after reaching thermal equilibrium with heating and cooling effects. The initial surface density was $16.29\,{\rm{g}}\,{\mathrm{cm}}^{-2}$ at the location of Uranus and $6.63\,{\rm{g}}\,{\mathrm{cm}}^{-2}$ at Neptunes, considering flat disks. This setup corresponds to roughly half of the Minimum Mass Solar Nebula (Hayashi 1981) densities, mimicking the very last stage of planet formation within the solar system when the satellites were formed. Given that real circumstellar disks are dissipating in a nearly exponential fashion at the end of their lifetime, our choice on setting half of the Minimum Mass Solar Nebula was just an educational guess for a mean value that is definitely a mass that the Solar Nebula eventually had at the late stage of its lifetime.

The equation of state in these simulations were of an ideal gas: $p=(\gamma -1){E}_{\mathrm{int}}$ connecting the (p) pressure with the internal energy (E_int) via the adiabatic exponent (γ = 1.43). The code solves the viscous stress tensor for a constant, kinematic viscosity that is equal to $1.95\times {10}^{14}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-1}$ in the case of Uranus and $2.44\times {10}^{14}\,{\mathrm{cm}}^{2}\,{{\rm{s}}}^{-1}$ for Neptune. These values corresponds to a reasonable, low alpha-like viscosity that scales with the semimajor axis from the star, which includes that planets form in dead zones. In the simulations the gas can heat up through viscous heating, adiabatic compression and cool through radiative dissipation and adiabatic expansion. The Rosseland-mean-opacities used in the radiation-hydrodynamics simulations were constructed self-consistently from frequency-dependent dust opacities computed with a version of the Mie code from Bohren & Huffman (1984). The dust consisted of 40% water, 40% silicates, and 20% carbonaceous material (Warren 1984; Zubko et al. 1996; Draine 2003), assuming spherical, compact, micron-sized grains. The star was assumed to have solar properties and the dust-to-gas ratio was chosen to be 1%. The opacity table contains the evaporation of the different dust components: 170 K for water, 1500 K for silicates, and 2000 K for carbon, respectively. Above 2000 K the gas opacities were used from Bell & Lin (1994). This method ensures that even though the dust is not treated explicitly in the gas hydrodynamic simulations, its effect on the temperature is taken into account through the dust opacities. The mean molecular weight was set to 2.3 to be consistent with the solar composition.

In order to have sufficient resolution in the CPDs, we increased the resolution in this area with nested meshes. We placed seven additional grids around the ice giants, reaching a final resolution of 1.38 × 10⁻³ au (8.1 R_p) for Uranus and 2.17 × 10⁻³ au (13.1 R_p) for Neptune. For the planets, we fixed their mass, radius, and temperature similarly to Szulágyi (2017). The eight cells around the point masses were the extent of the planet, where the temperatures were fixed to 1000, 500, and 100 K in the three different simulations for Uranus and for Neptune, corresponding to the late surface temperatures of these planets during their formation around 3–4 million years (Fortney & Nettelmann 2010; Nettelmann et al. 2016). This temperature sequence was set in order to mimic how the planet radiates away its formation heat, up until it is fully formed. This is only true in the very late stage of CPD evolution, close to the time when the circumstellar disk has dissipated away.

To examine satellite formation within the CPDs of the hydrodynamical simulations, we used population synthesis. The technique was already successfully used in our previous work (Cilibrasi et al. 2018) for moon formation around Jupiter, hence we only summarize here the main points of our method.

We azimuthally averaged the density and temperature filed on the midplane from the hydrodynamic simulations for the coldest planet case (100 K) when a CPD formed. Satellites can only form in disks, not in envelopes; therefore, we only considered the disk cases. In the semi-analytical model the CPDs have the same properties as in the hydrodynamic simulation: same viscosity, adiabatic index, and mean molecular weight. The angular velocity of the gas, the scale height of the disk, and the sound speed was then computed from the local temperature and density values of the disk and using the common one-dimensional model for disks (Pringle 1981). Due to the fact that the hydrodynamic simulation only computes the gas distribution, we assumed that the dust distribution is the same, but only a fraction of it in mass controlled by the dust-to-gas ratio parameter. The temperature of the dust was assumed to be the same as the gas temperature, assuming perfect thermal equilibrium. The CPD ranged between 1 R_p and 500 R_p, according to the hydrodynamical simulation's CPD radius. The population synthesis includes the CPD evolution (that it cools through radiative dissipation and changes is mass), the continuous mass infall from the circumstellar disk (Szulágyi et al. 2014) as well as the mass loss: accreting to satellites and to the planet. The net mass infall was also computed from the hydrodynamic simulations: $8.4\times {10}^{-8}\,{M}_{\mathrm{Sun}}\,{\mathrm{yr}}^{-1}$ for Uranus, and $7.4\times {10}^{-8}\,{M}_{\mathrm{Sun}}\,{\mathrm{yr}}^{-1}$ for Neptune. These infall rates then decreased exponentially, as the circumstellar disk also dissipates exponentially in the last phase of disk evolution and the feeding to the CPD changes accordingly.

We ran 25,000 individual calculations where in each case varied four parameters.

• 1.  
  Disk dispersion timescale: different circumstellar disks dissipate on different timescales based on observations, hence we chose to vary this parameter between 0.1 and 1 million years, which is roughly 1/10th of the total lifetime of the circumstellar disk and hence probably of the CPD as well.
• 2.  
  Dust-to-gas ratio: it has a wide range in circumstellar disks (Ansdell et al. 2016) and according to dust-coagulation studies the CPD can be quite dust rich (Drazkowska & Szulágyi 2018). We therefore varied this parameter between 1% and 50%.
• 3.  
  Dust-refilling timescale: how quickly the dust is accreted and depleted within the CPD, and how quickly it reaches again its equilibrium profile. This parameter was varied between 100 yr and 1 million years (Cilibrasi et al. 2018).
• 4.  
  Distance from the planet where the first seeds of satellites are forming. It was varied randomly between 50 and 150 planetary-radius.

At the beginning of simulation, the algorithm creates a new protosatellite with a mass of 10⁻⁷ M_p, which is a seed size that quickly forms from the incoming micron-sized dust via coagulation (Drazkowska & Szulágyi 2018). Then it accretes mass from the dust disk with a common analytical prescription (Greenberg et al. 1991). This accreted mass will be then subtracted from the dust disk density. The protosatellites that form will feel the torque exerted on them by the gas disk and start to migrate. As long as they are small and do not open a gap in the gas distribution of the CPD, they migrate according to the type I regime, computed with the Paardekooper formula (Paardekooper et al. 2010, 2011). In rare cases when they grow to such large mass that can open a gas-gap, they then enter the type II regime of migration and change their orbit on the viscous timescale (Duffell 2015). Sometimes the migration is so fast that the protosatellites will be accreted into the planet. In these cases, we kept track of the lost satellite-mass, in order to know what amount of solids the ice giants were "polluted." The population synthesis simulation stopped when the CPD dissipated, because then the migration of the satellites stopped the and the moon system that formed was considered final.

Owing to the high resolution that we could achieve in the planet vicinity within the hydrodynamic simulations, we were able to determine that disks are forming when the planet temperature dropped below 500 K (Figure 1). When steady state of the simulation was reached, the mass of the CPD was only $7.2\times {10}^{-4}\,{M}_{\mathrm{Uranus}}$, measured based on the rotational profile and a corresponding isodensity surface. This is only slightly larger than the integrated mass of the moons ($\sim {10}^{-4}\,{M}_{\mathrm{Uranus}}$). However, as it was described earlier, the CPD is not a closed reservoir of mass, it is continuously fed by the circumstellar disk with gas and micron-sized dust (Szulágyi et al. 2014; Fung & Chiang 2016). Some of this mass is accreted to the planet and to the moons, and some is flowing back to the circumstellar disk, maintaining a sort of equilibrium with the circumstellar disk mass (Szulágyi 2017). We measured the net influx rate to the CPDs from the hydrodynamic simulations by defining a box around the CPD and calculating the sum of the inflowing and outflowing gas, finding $8.4\times {10}^{-8}\,{M}_{\mathrm{Sun}}\,{\mathrm{yr}}^{-1}$, therefore over the years, there is enough mass flowing through the CPD to eventually build the current satellites of Uranus.

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We tested whether or not the CPD with planetary surface temperature of 100 K could form moons with satellite population synthesis. We found that in most cases of the population, there is a satellite system forming. Most of the moons formed over only ∼500,000 yr (Figure 2, left panel). While some of the satellites migrated inward in the disk up until they got accreted into the planet, the migration velocities were slow enough, that only 25% of the population was lost. Often sequential satellite formation happened: after the lost satellites new ones could still form until the CPD has dissipated. All of the satellites were formed at a location of the disk where the temperature was below the water freezing point, hence the building blocks of these moons can contain a significant fraction of ices to make icy moons, such as the current ones around Uranus. The masses of the formed satellites spread across several orders of magnitude, mainly in the smaller mass regime between 10⁻⁷ and ${10}^{-4}\,{M}_{\mathrm{Uranus}}$ (Figure 2, right panel). On Figure 2 we also mark with red stripes the current masses of the five major satellites. Clearly, the moon system of Uranus as we see today can be reproduced by our model. In 5.8% of the cases we can see systems with four or five satellites and with a total mass between 0.5 and 2 times the current Uranian satellites. Considering the distances from the central planet, we found that in about 18% of the cases they are comparable to the orbital radii of the current five Uranian satellites. Both conditions together occur in 5.1% of all the cases.

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We investigated the impact of different physical ingredients in the model by varying only one parameter at a time. We found that the masses of the final moons basically scales linearly with the dust-to-gas ratio, hence in order to reproduce the major five satellites, a dust-to-gas ratio of 7% or higher is needed within the CPD. Moreover, the dust-refilling timescale has to be 10,000 yr or shorter, meaning relatively quick dust evolution, which can easily happen according to a recent study of dust-coagulation in CPDs (Drazkowska & Szulágyi 2018).

Tests were made also on the distance where the satellite seeds were formed. In this case, when we fixed one location in the disk where satellites can form (e.g., at $50{M}_{\mathrm{Uranus}}$, at $100{M}_{\mathrm{Uranus}}$, and at $150{M}_{\mathrm{Uranus}}$), we found the number of survived satellites are larger when the seeds were all placed at $150{M}_{\mathrm{Uranus}}$ than at $50{M}_{\mathrm{Uranus}}$. This is just because excluding the outer CPD will of course result in more compact moon system that can accommodate fewer satellites. Because one has no a priori knowledge on where the current satellites were formed, we randomized the satellite-seed positions in our model presented in this Letter.

As in the case of Uranus, the nearly equal-mass Neptune could also form a CPD when ${T}_{{\rm{p}}}\lt 500$ K (Figure 1). The CPD in this case was $1.7\times {10}^{-3}\,{M}_{\mathrm{Neptune}}$. The mass infall rate to the CPD was found to be $7.4\times {10}^{-8}\,{M}_{\mathrm{Sun}}\,{\mathrm{yr}}^{-1}$.

From the population synthesis, we found similar trends for Neptune as in the case of Uranus. This CPD could also form relatively massive moons in most of the cases. The majority of the population formed over a couple of 10⁵ yr (Figure 3, left panel). The entire CPD had a temperature below water freezing point, so this disk also likely forms only icy satellites. The masses of the formed moons were again often in the low-mass regime between 10⁻⁷ and ${10}^{-4}\,{M}_{\mathrm{Neptune}}$, i.e., smaller than the mass of Triton (Figure 3, right panel). Note that Triton is most likely a captured Kuiper-object (McKinnon et al. 1995), so it did not form around Neptune; our mass comparison in this case is simply for scaling purposes. However, our result is that a similar moon system could have formed around Neptune like around Uranus, before the capture of Triton happened. It was already suggested by a dynamical study (Rufu & Canup 2017) that the capture of Triton is only possible, if the original moon system was similar in mass than the Uranian satellite system.

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