FORMATION OF MULTIPLE-SATELLITE SYSTEMS FROM LOW-MASS CIRCUMPLANETARY PARTICLE DISKS

We examine accretion of satellites from circumplanetary particle disks in a gas-free environment by performing global N-body simulation. Collision and gravitational interaction between particles are taken into account. As the initial condition, we consider particle disks confined within the planet's Roche limit given by

$$\begin{equation} a_{\rm R} = 2.456 \left(\frac{\rho }{\rho _{\rm c}} \right)^{-1/3} R_{\rm c}, \end{equation} \tag{ 1 }$$

where ρ is the material density of particles, and ρ_c and R_c are the central planet's density and radius, respectively. Initially, particles are distributed in an annulus with 0.4a_R ⩽ r ⩽ a_R, and their surface density follows a power-law distribution Σ∝r^β; we examine cases with β = −1, −3, and −5. Initial orbital eccentricities and inclinations of particles are assumed to have small values that follow a Rayleigh distribution. In the following, we use the non-dimensional angular momentum of the disk given by

$$\begin{equation} j_{\rm disk,ini} = L_{\rm disk,ini} / \sqrt{{GM}_{\rm c}R_{\rm c}}, \end{equation} \tag{ 2 }$$

where L_{disk, ini} is the angular momentum of the initial disk, and M_c is the mass of the central planet. In the case of surface density distribution given above, we have j_{disk, ini} ≃ 0.83, 0.78, and 0.73 for β = −1, −3, and −5, respectively. We examine cases of various initial disk masses, with M_{disk, ini}/M_c = 0.01 − 0.06. The initial disks are described by 3 × 10⁴ to 5 × 10⁴ particles (Table 1).

We set the mass of the central planet to be the Earth's mass and assume that the densities of the particles and the central planet are ρ = 3.3 g cm⁻³ and ρ_c = 5.5 g cm⁻³, respectively, so that the case of massive disks corresponds to that of lunar accretion examined by previous works (Ida et al. 1997; Kokubo et al. 2000). However, evolution of self-gravitating particle disks is governed by their surface density rather than the density or mass of individual particles, and the mass of the initial disk relative to the planet (M_{disk, ini}/M_c) and the initial angular momentum distribution of the disk are important parameters that control the outcome of satellite accretion (Ida et al. 1997; Kokubo et al. 2000; Takeda & Ida 2001). Also, R_c/a_R ≃ 0.34 when the above values are assumed for the densities of the central planet and the particles, while we have R_c/a_R ≃ 0.4–0.5 if the densities of the central planet and satellites corresponding to the gas giant planets are assumed. However, as we show later, dynamical behavior near the disk outer edge is important for satellite accretion from a disk initially confined within the Roche limit (see also Ida et al. 1997; Kokubo et al. 2000). Therefore, our results can be applied to satellite accretion from particle disks in various cases, including ones around giant planets.

In our model, we assume a disk of condensed particles as the initial condition, but this may not be appropriate for massive disks. In fact, Salmon & Canup (2012) showed that vaporization of constituent particles would be important in the protolunar disk around the Earth when M_disk/M_⊕ ≳ 0.003, because in this case the amount of energy released during the viscous spreading of the disk is comparable to the latent heat of vaporization of silicate and the rate of radiative cooling from the disk surfaces is too small compared to the viscous heating rate. By generalizing their estimates to the case of particle disks around a central planet with mass M_c, we find that the amount of energy released during disk spreading is comparable to the latent heat of vaporization of silicate when M_c/M_⊕ ≳ 0.3. On the other hand, the rate of radiative cooling from the disk surfaces becomes smaller than the viscous heating rate when M_disk/M_⊕ ≳ 0.003 × (M_c/M_⊕)^−1/3. If we compare the latter condition with the range of disk-to-planet mass ratios examined in the present work (0.01  ⩽  M_{disk, ini}/M_c ⩽ 0.06), we find that vaporization would be important at least near the midplane even in the case of the lightest disk (M_{disk, ini}/M_c = 0.01) when M_c/M_⊕  ≳  0.03. Salmon & Canup (2012) showed that the timescales of disk spreading and satellite accretion become significantly longer when the fluid disk inside the Roche limit is considered. Also, because of the prolonged period of interaction with the fluid inner disk, the formed satellite tends to have a larger semi-major axis compared to the pure N-body simulations. However, other basic characteristics of the final outcomes were quite similar to those of pure N-body simulations. For example, they found that the relationship between the mass of the formed satellite and the initial disk angular momentum can be explained by the semi-analytic relationship derived from pure N-body simulations (Ida et al. 1997) with the above revised estimate of the semi-major axis of the formed satellite. While we acknowledge the importance of vaporization, in the present work we will focus on disks of solid bodies, as a first step of full N-body simulation of the multiple-satellite system formation.

The orbits of particles are calculated by numerically integrating the following equation of motion,

$$\begin{equation} \frac{d\boldsymbol {v}_{i}}{dt} = -{GM}_{\rm c} \frac{\boldsymbol {x}_{i}}{|\boldsymbol {x}_{i}|^3} - \sum ^{N}_{j \ne i} G m_{j} \frac{\boldsymbol {x}_{i} - \boldsymbol {x}_{j}}{|\boldsymbol {x}_{i} - \boldsymbol {x}_{j}|^3}, \end{equation} \tag{ 3 }$$

where $\boldsymbol {x}_i$, $\boldsymbol {v}_i$, and m_i denote the position relative to the center of the planet, the velocity, and the mass of particle i, respectively, and G is the gravitational constant. We use the modified fourth-order Hermite scheme (Kokubo & Makino 2004) with shared time steps for the integration. Gravitational forces between particles and the planet are calculated using GRAPE-DR, which is a special purpose hardware for gravity calculation.

Collision between particles is taken into account, assuming that particles are smooth spheres with normal restitution coefficient ε_n = 0.1. Previous works show that dynamical evolution is hardly affected by ε_n, as long as ε_n < 0.6 (e.g., Takeda & Ida 2001). For the search of colliding pairs, we adopt the octree method implemented in REBOUND (Rein & Liu 2012). When a collision is detected, velocity change is basically calculated based on the hard-sphere model (Richardson 1994). Particles that collide with the planet are removed from the system. When a collision between particles takes place outside the Roche limit, colliding particles can become gravitationally bound and form aggregates. Whether colliding pairs become gravitationally bound or not is judged by the accretion criteria that takes account of the tidal effect (Ohtsuki 1993; Kokubo et al. 2000); we calculate the energy for the relative motion of the colliding pair using their positions and velocities, and the pair is regarded as being gravitationally bound when their centers are within their mutual Hill radius and the energy is negative (Kokubo et al. 2000). Gravitational aggregates formed outside the Roche limit can be tidally disrupted when they are scattered back inside the Roche limit by a forming satellite or another aggregate. In order to correctly describe such phenomena, we basically adopt the so-called rubble-pile model (Kokubo et al. 2000) for aggregates formed near the Roche limit, and follow the orbits of constituent particles.

However, if we adopt this method for all the bodies for the entire simulation, a large portion of computing time has to be used for resolving collisions among constituent particles that form aggregates, and it is difficult to handle long-term orbital integration of formed aggregates. On the other hand, as we will show below, a satellite formed by particle accretion near the disk edge gradually migrates outward due to gravitational interaction with the disk. Once the satellite migrates sufficiently far from the Roche limit, it does not need to be treated as a rubble-pile, because it will not be scattered back inside the Roche limit any more. Therefore, in the present work, for the largest and the second largest aggregates in the system, we adopt the following treatment for the merger of constituent particles into a single body. The conditions for this treatment are: (1) the aggregate (the largest or the second largest) is sufficiently massive, and (2) the radial distance of its center of mass from the planet is sufficiently larger than a_R so that it is ensured that it does not enter the Roche limit any more. When both the above two conditions are satisfied, the aggregate is replaced by a single sphere that has the same mass and center-of-mass velocity as those of the aggregate and the density equal to that of the constituent particles, ρ. The critical values of the mass and semi-major axis for this treatment are determined empirically; typically, in the case of the first satellite, the critical mass corresponds to about 70% of its final mass (the final mass is estimated empirically from previous works on lunar accretion as well as our simulations with and without the above treatment; see Figure 11), and the critical distance is about 1.15a_R. When the newly formed body (satellite) further accretes other particles, the mass of the satellite is increased accordingly.

Each simulation is continued until the system reaches a quasi-steady state, although formed satellites continue slow outward migration after their formation. In the case of the formation of a single satellite from massive disks, simulations are typically continued for 500 T_K (T_K is the orbital period at r = a_R). On the other hand, in the case of low-mass disks where the evolution of the system is slow, we continue our simulations for 10³ T_K.

Next, we present results for accretion from lighter particle disks, where the second satellite is formed. In our simulations with j_{disk, ini} = 0.775, the second satellite was formed when 0.015 ⩽ M_{disk, ini}/M_c ⩽ 0.03 (Table 1). In the marginal case of M_{disk, ini}/M_c = 0.03, the second satellite was not formed in one simulation (Run 7), but it was formed in another simulation where initial conditions were generated using a different set of random numbers (Run 7b). On the other hand, in the case with M_{disk, ini}/M_c = 0.01, the evolution of the system was so slow that we had to stop the simulation before the second satellite was formed, although its formation is expected if we continue the simulation. Figure 5 shows time series of the evolution of the system in the case of M_{disk, ini}/M_c = 0.015. Initial evolution of the disk is similar to the case of more massive disks shown in Section 3. First, spiral structures are formed as a result of gravitational instability (t = 4 T_K). Since the critical wavelength of the instability (λ_c) is proportional to the disk surface density, we find that there are a larger number of arms in the present case compared to the case shown in Figure 1 (t = 5 T_K). As a result, the mass of each aggregate formed from particles transferred outside the Roche limit, which is roughly proportional to $\Sigma \lambda _{\rm c} ^2$ (Kokubo et al. 2000), is smaller compared to the case of more massive disks (compare Figure 5, t = 32 T_K with Figure 1, t = 17 T_K). Also, since the surface density of the disk is smaller, the outward mass flux across the Roche limit is also smaller. Collisions between these aggregates outside the Roche limit produce the first satellite (t = 69 T_K), whose mass (∼0.002 × M_c) is smaller than the case of more massive disks. When this first satellite satisfies the conditions described in Section 2.2, it is replaced by a single sphere (t = 102 T_K).

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After its formation, the first satellite gradually migrates outward via gravitational interaction with the disk, as in the case of more massive disks (t = 69, 102 T_K). On the other hand, particles near the disk outer edge are pushed inward by the satellite, and a gap is formed between the satellite and the disk outer edge (t = 102 T_K). The growth of the satellite almost stalls by this stage, but a significant amount of mass corresponding to 3.3 times the satellite mass remains in the disk. In the case of more massive disks shown in Section 3, the mass of the first satellite was so large that it gravitationally scatters most of disk particles to lead to collision with the planet. However, in the present case of the lighter disk, the first satellite is too small to scatter disk particles onto the planet. Instead, the satellite migrates outward significantly through gravitational interaction with the remaining disk, and then the disk outer edge also gradually expands (t = 565 T_K).

The torque from the satellite exerts on disk particles at resonant locations (Goldreich & Tremaine 1980), and there are a number of m: m − 1 inner mean motion resonances with the first satellite between the orbit of the first satellite and the disk outer edge. Because of this resonant effect, particles cannot spread across the Roche limit immediately after the first satellite is formed and starts outward migration. However, these resonances vanish beyond the radial location of the 2:1 inner mean motion resonance, which is located at r_{2: 1} = 0.63a_s (r_{2: 1} is the radial location of the 2:1 inner mean motion resonance with the first satellite, and a_s is the semi-major axis of the first satellite). Therefore, when r_{2: 1} becomes larger than a_R as a result of outward migration of the first satellite, particles near the disk outer edge pile up outside the Roche limit and the accretion of the second satellite begins (Figure 5, t = 690 T_K; Goldreich & Tremaine 1978; Takeda 2002). As in the case of the first satellite, the second satellite is replaced by a single sphere when its radial location is sufficiently far from the disk outer edge (Figure 5, t = 986 T_K). Formation of the second satellite at the location of the 2:1 resonance with the first one was also found in some cases in the hybrid simulations of Salmon & Canup (2012), but they mostly focused on the case of the formation of single-satellite systems because they were primarily interested in the lunar formation.

Figure 6 shows the evolution of the mass and the semi-major axis of the first and the second satellites formed in the case of Run 10. The mass growth of the first satellite is similar to the case of the formation of a single satellite from more massive disks described in Section 3; it undergoes rapid growth just outside the Roche limit, then repels the disk outer edge and begins outward migration. In the present case, the timescale of the viscous evolution of the particle disk is longer because of the smaller disk surface density, thus the growth timescale of the satellite is also longer. On the other hand, the second satellite is formed from particles piled up near the 2:1 resonance with the first satellite, as mentioned above. Exactly speaking, the satellite seed, which is formed near the location of the 2:1 resonance with the first satellite, grows by accreting particles spreading from the disk outer edge, while the orbits of many of these particles are interior to the resonance location and have specific angular momentum smaller than that of the satellite seed. As a result, the specific angular momentum of the second satellite can decrease during its rapid growth, and its semi-major axis stays near the disk outer edge (Figure 6). When the second satellite becomes sufficiently massive to repel the disk outer edge, its growth stalls and outward migration begins. Then the second satellite is captured into the 2:1 mean motion resonance with the first satellite. Since the angular momentum that the second satellite receives from the disk is partly transferred to the outer first satellite through resonant interaction (Peale 1986), the two satellites continue outward migration, with their radial locations being locked in the resonance.

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A similar evolution can be found in other cases of the formation of two satellites from disks with different initial masses. Figure 7 shows results for two cases with slightly more massive disks than the case shown above, with the same initial non-dimensional angular momentum of the disk. Because of the larger surface density of the initial disks, the viscous evolution of the disk and satellite formation proceeds faster, but the general feature of the evolution is similar to the case shown in Figure 6. In the case of M_{disk, ini}/M_c = 0.025 (Run 8) shown in the right panel of Figure 7, a co-orbital satellite whose mass is 7% of the first satellite is formed on the orbit of the first satellite. This small companion was captured into the tadpole orbit when the first satellite was still near the disk, and migrates outward together with the first satellite. Then, when the eccentricity of the first satellite grew significantly, its orbit became unstable and collided with the first satellite at t = 833 T_K. Out of our four simulations where two satellites (excluding co-orbitals) were formed, capture of a co-orbital satellite by the first satellite was found in two cases (Runs 3 and 8), which suggests that the formation of co-orbital satellites would be more common than the case of satellite formation from more massive disks (Kokubo et al. 2000). In the other case (Run 3), the mass of the co-orbital satellite was 7% of the first satellite.

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The first and the second satellites continue outward migration while their orbits are kept locked in the 2:1 mean motion resonance. In the present work, we do not examine their subsequent longer-term evolution in detail. If there is sufficient amount of mass still available in the disk when the second satellite migrates sufficiently far from the disk outer edge, viscous diffusion of the disk is expected to lead to the formation of the third and the fourth satellites (Crida & Charnoz 2012). On the other hand, when the first two satellites are rather massive and are locked in the 2:1 resonance during their outward migration, as in the case examined in the present work, orbital eccentricities of the two satellites can grow during their migration, which would influence their subsequent dynamical evolution. Figure 8 shows the evolution of the eccentricities of the satellites in the cases where two satellites were formed. The eccentricity of the first satellite before the formation of the second satellite is generally small. However, after the second satellite is formed and is locked into the 2:1 mean motion resonance, the eccentricity of the second satellite oscillates and takes on rather large values (0.1–0.25), and the eccentricity of the first satellite increases accordingly (∼0.05). For comparison, we performed three-body orbital integration for the two satellites and the planet but without disk particles. The initial conditions were taken from those at t = 800 T_K of our Run 10, and the integration was continued for 5000 T_K. We confirmed that the eccentricities of the first satellite (0.02–0.04) and that of the second satellite (0.1–0.13) did not grow as large as the cases shown in Figure 8, which suggests that the interaction between the satellites and the disk as well as their migration play an important role for the eccentricity evolution of the satellites.

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In the present work, sufficiently massive aggregates that migrated far from the disk edge are replaced by single spheres. However, if the eccentricity of the second satellite becomes so large that its peri-center gets inside the Roche limit, the satellite would undergo tidal disruption, which is not taken into account in our simulation after the satellites are replaced by single spheres. Figure 9 shows results of an additional simulation we performed to examine such effects.³ The simulation is the same as Run 8 until the formation of the second satellite, but we did not replace it with a single sphere in the present case and continued treating it as a rubble pile. We find that the mass of the second satellite grows after its formation and then decreases significantly; this is because the satellite gets inside the Roche limit after its eccentricity grows significantly, and it undergoes tidal disruption. After that, the fragments of the first-generation second satellite accrete again to form the second satellite of the second generation. Such a cycle of tidal disruption and re-accumulation would be repeated. Interestingly, a part of the fragments of the disrupted second satellite reach the orbit of the first satellite, and contribute to its additional growth. If the first satellite migrates sufficiently outward during such a cycle, the orbit of the second satellite would move outward and, eventually, its peri-center would avoid getting inside the Roche limit. In any case, orbital evolution of multiple-satellite systems is important to understand the final outcomes of satellite accretion from particle disks.

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Figure 10 shows the evolution of the mass and semi-major axis of the first satellite for various initial disk masses. From Figure 10(a) we confirm that more massive disks lead to more rapid evolution and produce more massive first satellites. In general, the satellite grows mostly during the early phase of rapid growth, and then its growth slows down. Figure 10(b) shows the plots of the evolution of the semi-major axis of the first satellite for the three cases out of the six shown in Figure 10(a). In all these cases, we can see that the satellite undergoes rapid growth when its semi-major axis is smaller than about 1.2a_R by accreting particles spreading across the Roche limit. When the satellite grows large enough to repel the disk outer edge, its growth almost stalls. At this stage, its semi-major axis is about 1.2–1.3a_R, and its outward migration is rather smooth afterward.

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Figure 11 shows the plots of the mass of the first satellites as a function of the initial mass of the disk (j_{disk, ini} = 0.775). We find that the dependence changes at M_{disk, ini}/M_c ≃ 0.03. In the case of massive disks that produce single-satellite systems, Ida et al. (1997) analytically showed from the conservation of mass and angular momentum that the satellite mass is proportional to the mass of the initial disk (see also Kokubo et al. 2000). In the derivation of this relationship, it is assumed that particles initially in the disk either accrete to form the satellite, collide with the planet, or escape from the system through gravitational scattering by the satellite, and that the disk does not remain at the final stage. Ida et al. (1997) and Kokubo et al. (2000) confirmed that their numerical results of N-body simulations agree well with the analytic relationship.

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The dashed line in Figure 11 is the fit to our numerical results for M_{disk, ini}/M_c > 0.03, assuming that the satellite mass is proportional to the mass of the initial disk. We confirm that our results for the case of the formation of single-satellite systems agree well with this relationship as in the previous works. On the other hand, in the case of lighter disks with M_{disk, ini}/M_c < 0.03, a significant mass still remains in the disk after the formation of the first satellite, and the second satellite is formed from the remaining disk. Thus, the assumption of the complete depletion of the initial disk at the time of the formation of the first satellite that was made in the derivation of the above analytic linear relationship between the satellite mass and the initial disk mass is not valid anymore. In fact, in the case of such low-mass disks, the dependence of the mass of the first satellite on the initial disk mass is found to be stronger, and the satellite mass is approximately proportional to the square of the initial disk mass (Takeda 2002). In this case, the accretion efficiency of incorporation of disk material into the satellite (Kokubo et al. 2000) decreases with decreasing disk mass.

On the other hand, satellite accretion from particle disks with much lower initial mass was studied by Crida & Charnoz (2012), as we mentioned before. They investigated satellite accretion in the limit where the mass of the satellites accreted from the disk is negligible compared to the disk mass, and assumed that the disk surface density and the mass flow rate across the Roche limit are constant during satellite accretion. In this case, if we define the timescale of the viscous spreading of the disk with viscosity ν as $T_{\rm \nu } \equiv a_{\rm R}^2/\nu$, the rate of outward mass flow across the disk outer edge can be written as F = M_disk/T_ν. Using the expression of the viscosity for the self-gravitating collisional disks (Equation 4) and $M_{\rm disk} = \pi a_{\rm R}^2\Sigma$, we then have

$$\begin{equation} F=\pi C G^2 \Sigma ^3/\Omega ^3. \end{equation} \tag{ 5 }$$

In the model of Crida & Charnoz (2012), satellites (or satellite seeds) formed near the disk outer edge grow by directly accreting particles spreading from the disk outer edge when the satellites are still near the edge ("continuous regime"), or by capturing moonlets formed by accretion of such spreading particles when the satellites somewhat migrate outward ("discrete regime"). In both cases, the growth rate of the mass of the satellites (M_s) is determined by the mass flow rate F, and we have

$$\begin{equation} M_{\rm s} \propto F \propto \Sigma ^3 \propto M_{\rm disk}^3. \end{equation} \tag{ 6 }$$

Crida & Charnoz (2012) showed that the mass of the formed satellite at the end of the discrete regime is given by M_s/M_c ≃ 2200 (M_disk/M_c)³. These satellites continue outward migration due to torques from the particle disk and the planet, and grow further by mutual collisions in the course of the migration ("pyramidal regime").

In Figure 12, we compile the fitting results to our simulations as well as those of previous works for satellite accretion from massive disks (Ida et al. 1997; Kokubo et al. 2000) and from light disks (Crida & Charnoz 2012). As we mentioned above, in the case of the formation of single-satellite systems from massive disks, the disk is cleared out quickly due to gravitational scattering by the massive first satellite. In this case, the mass of the first satellite is proportional to the initial disk mass (Ida et al. 1997; Kokubo et al. 2000). On the other hand, the assumption of a constant disk surface density seems to be reasonable when the mass of the formed satellites is much smaller than the disk mass. In this case, the mass of the satellites is proportional to the cube of the disk mass, as we have shown above. The case of the formation of the first and the second satellites in our simulation is intermediate between the above two cases. In this case, the mass of the first satellite is not large enough to clear out the remaining disk, while it is too massive to neglect its influence on the remaining disk; the mass of the disk significantly decreases as a result of the formation of the first satellite, and the disk outer edge is shepherded by the first satellite after its formation. As a result, the dependence of the mass of the first satellite on the initial disk mass becomes also intermediate between the above two cases.

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In the case of the disks with nondimensional angular momentum j_{disk, ini} = 0.775 shown above, the second satellite was formed when 0.015 ⩽ M_{disk, ini}/M_c ⩽ 0.03, while single-satellite systems are formed from more massive disks (Section 3). However, evolution of particle disks depends not only on the mass but also on the angular momentum distribution of the initial disk (Ida et al. 1997; Kokubo et al. 2000). Figure 13 shows results of a series of simulations where the mass and angular momentum of the initial disk are varied. The filled circles show the cases where the second satellite was formed, while the open circles represent the cases where single-satellite systems are formed. The open triangle indicates the marginal case where whether the second satellite is formed or not depends on the choice of random numbers for generating initial conditions. From Figure 13, we find a tendency that the critical mass of the initial disk that produces multiple satellites decreases with increasing angular momentum, indicating that higher angular momentum of the initial disk facilitates the formation of single-satellite systems. Figure 14 shows the dependence of the mass of the first satellite on the disk angular momentum. When the disk angular momentum is larger and more mass is located in the outer part of the disk, particles in the disk can be transported outward across the Roche limit more easily, thus the mass of the first satellite tends to be larger. Such a massive first satellite easily clears out the remaining disk, and the second satellite cannot be formed. On the other hand, in the case of compact disks with smaller angular momentum, the outward mass flux across the Roche limit is rather small and the mass of the first satellite tends to be also small, which facilitates the formation of the second satellite.

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Figure 15 shows the mass of the second satellites together with that of the first satellites as a function of the initial disk mass. As we mentioned in Section 5.2, the mass of satellites produced from lighter disks tends to be smaller. At the time of the accretion of the second satellite, the mass and the surface density of the disk are smaller compared to the initial disk due to the formation of the first satellite. Therefore, the mass of the second satellite tends to be smaller than that of the first satellite.

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On the other hand, as we mentioned above, the efficiency of incorporation of disk material into the first satellite decreases with decreasing disk mass in the case of the formation of multiple satellites. When the mass of the first satellite is much smaller than the initial disk mass, the mass of the remaining disk at the time of the formation of the second satellite is still similar to that of the initial disk. As a result, the mass of the second satellite formed from such a remaining disk becomes similar to that of the first satellite. From Figure 15, we can confirm that the difference between the masses of the first and the second satellites becomes smaller with decreasing mass of the initial disk. This is also consistent with Crida & Charnoz (2012), who considered the case with much lighter disks. In this case, the mass of the formed satellite at the end of the discrete regime is given by M_s/M_c ≃ 2200 (M_disk/M_c)³ as we mentioned above, which is constant as the disk mass is assumed to be constant.

As we have shown above, the typical outcome of satellite accretion from massive particle disks is single-satellite systems, while multiple-satellite systems are produced from lighter disks. Also, when multiple satellites are formed, outer satellites tend to be more massive, because of a larger surface density of the disk at the time of accretion and/or as a result of outward migration and merger of formed satellites. In both cases of single- and multiple-satellite systems, co-orbital satellites can be formed occasionally.

On the other hand, dynamical evolution of self-gravitating particle disks inevitably involves stochastic nature arising from gravitational scattering between aggregates as well as their collisional disruption and subsequent re-accumulation, which can result in diversity in outcomes of disk evolution. As an example, Figures 16 and 17 show results of Run 7b; in this case, the parameter values of disk mass and angular momentum are the same as those assumed in Run 7 (M_{disk, ini}/M_c = 0.03, j_disk = 0.775), but a different set of random numbers are used to generate initial positions and velocities of particles. In the case of Run 7, only one satellite was formed. On the other hand, two satellites are formed in Run 7b; unlike the above-mentioned typical cases, the inner satellite is more massive than the outer one. Also, the two satellites are found to be locked in the 1:2 mean motion resonance.

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This system with an inner larger satellite and an outer smaller one was formed in the following way. First, radial spreading of particles from the disk leads to the formation of a number of satellite seeds just outside of the Roche limit, and they grow through mutual collision and accretion. However, accretion efficiency between bodies in the strong tidal field is not 100% (Ohtsuki 1993; Canup & Esposito 1995; Ohtsuki et al. 2013), and collision between gravitational aggregates can lead to complete or partial disruption (Karjalainen 2007; Hyodo & Ohtsuki 2014). In the case of M_{disk, ini}/M_c = 0.03 shown in Figures 16 and 17, the mass of satellite seeds is significantly large. When such aggregates are disrupted by collision, they are often elongated due to the tidal effect and ejected particles are spread in a rather wide range of radial locations. Also, gravitational interaction between aggregates as well as between aggregates and dispersed particles leads to significant changes in semi-major axes and eccentricities of aggregates. In the case shown in Figure 16, a relatively massive aggregate (which eventually becomes the outer smaller satellite in the final state) undergoes collision with another aggregate, which leads to partial disruption of the colliding bodies (Figure 16, t = 29 T_K). The mass of the aggregate shows abrupt increase at the time of the collision, because the temporarily combined object is regarded as a single body (the green line at t ≃ 28 T_K in Figure 17(a)). The largest remnant body produced by this collision and disruption has a rather large semi-major axis, and its eccentricity is also increased (the green line in Figures 17(b) and (c)). Then, at t ≃ 33 T_K (Figure 16), another aggregate experiences scattering and partial disruption that leads to outward displacement, when the aggregate loses significant mass (the red line in Figures 17(a) and (b)). Then, this aggregate undergoes a close encounter with the aggregate that was scattered outward before, and the former is scattered inward to the vicinity of the outer edge of the disk (t ≃ 43 T_K; the red line in Figure 17(b)). Afterwards, this satellite seed grows by accreting particles spreading from the disk outer edge, and eventually becomes the largest satellite (Figure 16, t = 59, 146 T_K; Figure 17(a)). During the growth of this largest satellite, particles located in the region exterior to the satellite's orbit are scattered by the satellite to outer orbits, and then captured by the outer smaller satellite. Thus, the outer satellite grows significantly during this phase. When the mass of the inner satellite becomes large enough to shepherd the outer edge of the disk, the satellite begins outward migration. The radial distance between the inner larger satellite and the outer smaller one gradually shrinks and, eventually, they get captured into the 1:2 mean motion resonance (see also Salmon & Canup 2012). Then they continue outward migration while being locked in the resonance. The eccentricity of the outer smaller satellite becomes large (∼0.2) after it is captured into the resonance (the green line in Figure 17(c)). The eccentricity of the inner satellite is rather small initially, but it grows significantly in the course of the outward migration due to the resonant effect (the red line in Figure 17(c)).

In the above evolution, gravitational scattering between aggregates and their imperfect accretion at collision in the tidal environment plays an essential role in producing the final outcome that is different from the typical case. When the disk is not too light, as in the case shown above, those aggregates formed near the disk outer edge have significant mass, and gravitational scattering between them and their disruption plays an important role in delivering significant mass to the outer region, which facilities formation of satellites with large orbits. Thus, stochastic nature of gravitational scattering and collisional disruption in the tidal field can lead to significant diversity in the final outcome of satellite accretion.