Planet Formation around Supermassive Black Holes in the Active Galactic Nuclei

The dusty gas around SMBHs extends beyond the dust sublimation radius r_sub, where the dust temperature is higher than the sublimation temperature of the dust grains (T_sub ∼1500 K). The radius depends on the AGN luminosity: ${r}_{\mathrm{sub}}\,=1.3\,\mathrm{pc}{\left(\tfrac{{L}_{{UV}}}{{10}^{43}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{0.5}$ ${\left(\tfrac{{T}_{\mathrm{sub}}}{1500{\rm{K}}}\right)}^{-2.8}\,{\left(\tfrac{{a}_{d}}{1\mu {\rm{m}}}\right)}^{-1/2}$, where L_UV is the ultraviolet luminosity of the AGN and a_d is the dust size (Barvainis 1987). The temperature of the gas and dust beyond r_sub, especially at the midplane of the dusty torus, should be cold ≲100 K (Schartmann et al. 2014), because the radiation originated from the accretion disk is weaker in the direction of the disk plane, and it is further attenuated by the dense dusty gas. Interestingly, even for X-rays, a large fraction of AGNs are Compton-thick, i.e., the hydrogen column density is N_H >10²⁴ cm⁻² (Buchner et al. 2014). Although near-infrared and mid-infrared interferometer observations of AGNs show the presence of hot dusts (several 100 K) around AGNs (Tristram et al. 2014), colder dust particles are also present in this dense media around AGNs. The total amount of dust in the central 6–27 pc around SMBHs estimated from recent molecular lines (e.g., CO) observations of nearby AGNs by ALMA (Combes et al. 2019) is enormous, e.g., ∼(0.7–3.9) × 10⁵M_⊙ for the dust-to-gas mass ratio of ∼0.01 (Draine 2011). This number could be even larger for the high metallicity environment around AGNs (Groves et al. 2006; Rémy-Ruyer et al. 2014). The internal dynamics and structure of the molecular tori are still observationally unclear. However, since the mass feeding to the AGN through the circumnuclear disk is necessary during their lifetime (∼10⁷–10⁸ yr), the turbulent viscosity works in the dusty gas disk. Here, we model the turbulent disk based on the α-viscosity formalism (Shakura & Sunyaev 1973; Shlosman & Begelman 1987).

The dust grains in the circumnuclear disks around SMBHs are in a qualitatively similar situation as the ones in protoplanetary disks. The major differences between the two systems are summarized in Table 1. The "snow line" for a_d = 0.1 μm dust irradiated by X-ray around an AGN with an SMBH of 10⁷M_⊙ for the Eddington ratio (γ_Edd) of 0.1 is

$$\begin{eqnarray}\begin{array}{rcl}{r}_{\mathrm{snow}} & \approx & 4.7\,\mathrm{pc}{\left({L}_{X}/1.3\times {10}^{42}\mathrm{erg}{{\rm{s}}}^{-1}\right)}^{1/2}\\ & & \times \,{\left({T}_{\mathrm{ice}}/170{\rm{K}}\right)}^{-2.8}\,{a}_{d,0.1}^{-1/2}\end{array}\end{eqnarray} \tag{ 1 }$$

(Barvainis 1987).⁶ Moreover, AGNs are often heavily obscured (Compton-thick) even for hard-X-rays (Buchner et al. 2014), suggesting that a cold dusty layer exits around SMBHs. Therefore, it is expected that the dust in most of the circumnuclear disk is icy.

Table 1.  Differences between the Protoplanetary Disk and AGN

	Protoplanetary Disk	Circumnuclear Disk
mass of the central object	M⋆ ∼ M⊙	MBH ∼ 106–9 M⊙
luminosity of the central source	∼L⊙	1010–12 L⊙
spectrum of the central source	blackbody	power law
size of the dusty disk	10–100 au	0.1–100 pc
inner edge of the dusty disk	∼0.1 pca	dust sublimation radius (sub pc ∼ pc)
gas mass	∼0.01 M⋆	∼0.1 MBH
dust mass	∼10−4 M⊙	∼103–106M⊙
rotational period	∼100 yr	∼106–108 yr
lifetime	∼106 yr	107–108 yr
drag law	Epstein/Stokes	Epstein
mean free path of gas	∼1–100 cm	$\sim {10}^{12}{\left(\tfrac{n}{{10}^{3}\mathrm{cm}{{\rm{s}}}^{-3}}\right)}^{-1}\,\mathrm{cm}$

Note.

^aEisner et al. (2007), Suzuki et al. (2010).

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We then apply recent models of coagulation of dust particles and their aggregates outside the snow line in the protoplanetary disk (Okuzumi et al. 2012; Suyama et al. 2012; Kataoka et al. 2013; Michikoshi & Kokubo 2016, 2017) to the dust around AGNs. The coagulation of "fluffy icy dust" is one of the plausible solutions to avoid the theoretical obstacles that prevent the growth of dust grains (monomers) to "planetesimal" such as the "radial drift barrier" (Okuzumi et al. 2012). In this scenario, the evolution of the dust can be divided into four stages: (1) the "hit-and-stick" phase, (2) the collisional or gas pressure compression phase, (3) the gravitationally compression phase, and (4) the gravitational instability (GI) phase (e.g., Goldreich & Ward 1973). We investigated each phase in the circumnuclear disk as discussed below (see also the Appendix).

We track the evolution of icy monomers, whose size and density are a₀ = 0.1 μm and ρ₀ = 1 g cm⁻¹, and their aggregates. Here, we investigate the evolution of a representative dust particle size, using the single-size approximation (Sato et al. 2016). In the hit-and-stick phase, when two monomers/aggregates collide, the internal structure of the aggregates becomes porous (i.e., average internal density is smaller than ρ₀) with internal voids (Suyama et al. 2012). This "fluffy dust" formation is also examined by numerical experiments (Dominik & Tielens 1997; Wada et al. 2008). The internal density and size of the aggregates are ρ_int ∼ (m_d/m₀)^−1/2ρ₀ and a_d ∼ (m_d/m₀)^1/2a₀, where m_d and m₀ are the masses of the aggregate and monomers, for the fractal dimension of 2. When the collision energy exceeds a critical value, the porous aggregates start to get compressed, and the evolution of the internal density changes beyond this point (Suyama et al. 2012). During this compression phase, the aggregates' mass rapidly increase; however, their internal densities gradually increase as well, from ρ_int ∼ 10⁻⁶ g cm⁻³ to ∼10⁻⁵ g cm⁻³.

In contrast to the dust coagulation process in the protoplanetary disks (Weidenschilling 1977), the drag between dust particles and gas obeys the Epstein law only. The aggregate's size (a_d) is always much smaller than the mean free path of the gas (${\lambda }_{\mathrm{mfp}}\sim {10}^{12}\,\mathrm{cm}{\left(\tfrac{{\sigma }_{\mathrm{mol}}}{{10}^{-15}{\mathrm{cm}}^{2}}\right)}^{-1}{\left(\tfrac{{n}_{\mathrm{mol}}}{{10}^{3}{\mathrm{cm}}^{-3}}\right)}^{-1}$, where σ_mol and n_mol are the collisional cross-section and number density of the gas). At all times the radial drift velocity of the dust is negligibly small compared to the Kepler velocity v_K (i.e., 10⁻⁴–10⁻⁵v_K).

In both the protostellar and the circumnuclear disks, the dust-gas coupling is characterized by the normalized stopping time, i.e., the Stokes number, S_t ≡ Ω_K t_stop, where t_stop is the timescale of the dust particles to reach the terminal velocity due to the gas drag. In the Epstein law, t_stop is proportional to ${\rho }_{\mathrm{int}}\,{a}_{d}$, then S_t is

$$\begin{eqnarray}\begin{array}{rcl}{S}_{t} & = & \displaystyle \frac{\pi {\rho }_{\mathrm{int}}\,{a}_{d}}{2{{\rm{\Sigma }}}_{g}}\\ & = & \displaystyle \frac{\pi {\rho }_{\mathrm{int}}\,{a}_{d}(\pi {{GQ}}_{g})}{2{c}_{s}{{\rm{\Omega }}}_{K}}\\ & \sim & 1.5\times {10}^{-5}\,{\rho }_{\mathrm{int},1}\,{a}_{d,0.1}\,{c}_{s,1}^{-1}\,{r}_{1}^{3/2}\,{M}_{\mathrm{BH},6}^{-1/2}\,{Q}_{g},\end{array}\end{eqnarray} \tag{ 2 }$$

where Q_g is the Toomre's Q-value for a gas disk and Q_g ≡ c_sΩ_K/(πGΣ_g), with the surface density of the gas disk Σ_g, and a_d,0.1 ≡ a_d/(0.1 μm), the sound velocity of the gas c_s,1 = c_s/(1 km s⁻¹) and ρ_int,1 ≡ ρ_int/(1 g cm⁻³). Q_g < 1 is a necessary condition for the ring-mode GI.

The radial velocity of the dust v_r,d relative to the gas (Weidenschilling 1977; Tsukamoto et al. 2017) is

$$\begin{eqnarray}&&{v}_{r,d}=\displaystyle \frac{2{S}_{t}}{1+{S}_{t}^{2}}\,\eta \,{v}_{K},\end{eqnarray} \tag{ 3 }$$

where η is a parameter that determines the sub-Keplerian motion of the gas,

$$\begin{eqnarray}&&\eta \equiv -\displaystyle \frac{1}{2}\displaystyle \frac{{c}_{s}^{2}}{{v}_{K}^{2}}\displaystyle \frac{d\mathrm{ln}P}{d\mathrm{ln}r}\sim 2\times {10}^{-5}{M}_{\mathrm{BH},7}^{-1}\,{c}_{s,1}^{2},\end{eqnarray} \tag{ 4 }$$

where M_BH,7 ≡ M_BH/10⁷M_⊙. Therefore, both in the early stage of the dust evolution (S_t ≪ 1) and in the late phase (S_t ∼ 1) in the circumnuclear disk, v_r,d is much smaller than v_K; thus, we can ignore the radial drift of the aggregates in the circumnuclear disk during the whole evolution. The "radial drift barrier," i.e., the dust growth is limited by infalling to the central stars before dust particles obtain large enough mass as planetesimals, is not a serious problem in the circumnuclear disk.

The growth time of the aggregate during the hit-and-stick phase can be estimated as in Tsukamoto et al. (2017):

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{grow}} & \equiv & {\left(d\mathrm{ln}{m}_{d}/{dt}\right)}^{-1}\\ & = & \displaystyle \frac{4\sqrt{2\pi }}{3}\displaystyle \frac{{H}_{d}\,{\rho }_{\mathrm{int}}\,{a}_{d}}{{\rm{\Delta }}v\,{{\rm{\Sigma }}}_{d}}=\displaystyle \frac{8{\left(2\pi \right)}^{3/2}}{3}\\ & & \times \,\displaystyle \frac{{H}_{g}}{\sqrt{\alpha }\,{R}_{e}^{1/4}\,{c}_{s}\,{f}_{{dg}}}\\ & \sim & 6.3\times {10}^{7}\,[\mathrm{yr}]\,{c}_{s,1}^{-1}{\left(\displaystyle \frac{{f}_{{dg}}}{0.01}\right)}^{-1}\\ & & \times \,\left(\displaystyle \frac{{H}_{g}}{0.1\,\mathrm{pc}}\right){\left(\displaystyle \frac{\alpha }{0.1}\right)}^{-1/2}{\left(\displaystyle \frac{{R}_{e}}{{10}^{4}}\right)}^{-1/4},\end{array}\end{eqnarray} \tag{ 5 }$$

where, f_dg is the dust-to-gas ratio and H_d and H_g are scale heights of dust and gas disks, and ${H}_{d}\approx {H}_{g}\propto {M}_{\mathrm{BH}}^{-1/2}{r}^{3/2}$ in the circumnuclear disk. The Reynolds number R_e and the relative velocity of the dust particles Δv are given in the Appendix. This growth time is comparable to the AGN lifetime (see Section 3 and the Appendix in more details).

During the dust compression phase due to collisions, the kinematics of the dust aggregates are dominated by eddies of turbulence in the gas disk. Therefore, relative velocity of the aggregates, Δv, which is important for both growth and destruction of them, is determined by the property of the turbulence, i.e., R_e, and S_t (Ormel & Cuzzi 2007). The dimensionless parameter α is a parameter to determine the kinematic viscosity in the turbulent disk (Shakura & Sunyaev 1973):

$$\begin{eqnarray}&&\alpha \equiv \displaystyle \frac{\dot{{M}_{g}}}{3\pi {{\rm{\Sigma }}}_{g}{c}_{s}^{2}/{{\rm{\Omega }}}_{K}}=\displaystyle \frac{\dot{{M}_{g}}G}{3{c}_{s}^{3}}{Q}_{g}\simeq 0.3\,{Q}_{g}\left(\displaystyle \frac{{\gamma }_{\mathrm{Edd},6}}{0.01}\right){c}_{s,1}^{-3},\end{eqnarray} \tag{ 6 }$$

where $\dot{{M}_{g}}$ is the radial mass accretion rate in the disk, and γ_Edd,6 is the Eddington rate for the BH mass with M_BH = 10⁶M_⊙ for the energy conversion efficiency of 0.1. Here, we assume α is a constant, smaller than unity throughout the disk. In this phase, S_t gradually increases, and eventually the phase ends when S_t ∼ 1, then the aggregates decouple from the turbulent gas. At this time, the mass and size of the aggregates become m_d ∼ 10⁵ g and a_d ∼ 100 m, respectively. There internal density is still very low (i.e., "fluffy").

In the next gravitational compression phase, the aggregates are compressed owing to their self-gravity, and their internal density increases as ${\rho }_{\mathrm{int}}\propto {m}_{d}^{0.4}$ (Kataoka et al. 2013). The relative velocity of the aggregates in this phase is determined by the energy balance among gravitational scattering, collisional energy loss, turbulent stirring, turbulent scattering, and gas drag (Michikoshi & Kokubo 2016, 2017; see also the Appendix). The aggregates finally grow to ∼kilometer-sized bodies (i.e., planetesimals).

The value of the critical velocity for collisional destruction of centimeter- to kilometer-sized dust aggregates is not clear. Numerical experiments of collisions between aggregates (Wada et al. 2009) showed that the critical velocity is 50–100 m s⁻¹ for ∼10⁴ monomers (m_d ∼ 10⁻¹¹ g), and this scales with the mass of the aggregates as $\propto {m}_{d}^{1/4}$. If the critical velocity simply scales, it should exceed 1 km s⁻¹ for kilometer-sized "planetesimals."

In the regime with S_t > 1, if the Toomore Q-value for the disk of aggregates becomes smaller than ∼2, the GI takes place (e.g., Goldreich & Ward 1973), and spiral density enhancements are formed, and it leads to rapid growth of planetesimals in a rotational period t_K ≡ 2π/Ω_K ≃ 9.5 × 10⁴ yr (M_BH/10⁶M_⊙)^−1/2 (r/1 pc)^3/2 (see also Michikoshi & Kokubo 2017). In fact, we found that the aggregate disks become gravitationally unstable soon after S_t reaches unity in most cases.

If the dust aggregates grow by ballistic cluster–cluster aggregation (BCCA), the internal structure of the aggregate should be porous (i.e., ρ_int ≪ ρ₀ ∼ 1 g cm⁻³), and its fractal dimension is D ≃ 1.9 (Mukai et al. 1992; Okuzumi et al. 2009). In this case, the internal density of the aggregates in the hit-and-stick phase (Okuzumi et al. 2012; Kataoka et al. 2013) evolves as

$$\begin{eqnarray}&&{\rho }_{\mathrm{int}}={\left({m}_{d}/{m}_{0}\right)}^{1-3/D}{\rho }_{0},\end{eqnarray} \tag{ 7 }$$

where m_d is the mass of the aggregate, and m₀ and ρ₀ are the monomer's mass and density, respectively. We assume that m₀ = 10⁻¹⁵ g and ρ₀ = 1 g cm⁻³. In contrast to the protoplanetary disk, the radial motions of the gas and dust are small. For example, the radial velocity of the gas disk for the Eddington ratio γ_Edd = 0.1 and the black hole mass M_BH = 10⁷M_⊙ is 10⁻⁴v_K − 10⁻³v_K, where v_K is the Keplerian rotational velocity (see also Equations (3) and (4)). Therefore, as the first approximation, we assume that the gas and dust surface density distribution (Σ_d(r) = f_dgΣ_g(r)) does not change during the evolution.

The growth rate for m_d is then

$$\begin{eqnarray}&&\displaystyle \frac{{{dm}}_{d}}{{dt}}=\displaystyle \frac{2\sqrt{2\pi }\,{{\rm{\Sigma }}}_{d}\,{a}_{d}^{2}\,{\rm{\Delta }}v}{{H}_{d}},\end{eqnarray} \tag{ 8 }$$

where a_d is the size of the dust aggregate, Δv is collisional velocity between the aggregates and H_d is the scale height of the dust disk given as (Youdin & Lithwick 2007; Tsukamoto et al. 2017).

$$\begin{eqnarray}&&{H}_{d}={\left(1+\displaystyle \frac{{S}_{t}}{\alpha }\displaystyle \frac{1+2{S}_{t}}{1+{S}_{t}}\right)}^{-1/2}{H}_{g},\end{eqnarray} \tag{ 9 }$$

where H_g = c_s/Ω_K is the scale height of the gas disk.

The relative velocity between aggregates ${\rm{\Delta }}v$ for S_t < 1 can be divided into two regimes (Ormel & Cuzzi 2007): regime (I) t_s ≪ t_η = t_L Re^−1/2, and regime (II) t_η ≪ t_s ≪ Ω⁻¹. The Reynolds number, ${R}_{e}\equiv \alpha {c}_{s}^{2}/({\nu }_{\mathrm{mol}}{\rm{\Omega }})$ with the molecular viscosity ${\nu }_{\mathrm{mol}}\sim \tfrac{1}{2}{c}_{s}{\lambda }_{g}$ is

$$\begin{eqnarray}&&{R}_{e}\approx 3\times {10}^{4}\,{M}_{\mathrm{BH},6}^{-1/2}\,{r}_{1}^{3/2}\,{c}_{s,1}^{-1}\,{Q}_{g}\,{\gamma }_{\mathrm{Edd},0.01},\end{eqnarray} \tag{ 10 }$$

where Q_g is the Toomre's Q-value for the gas disk. The eddy turnover time t_L is ${t}_{L}\sim {{\rm{\Omega }}}_{K}^{-1}$, and t_η ∼ t_L for the smallest eddy. For the hit-and-stick phase, S_t ≪ R_e, then Δv obeys the regime I, and

$$\begin{eqnarray}&&{\rm{\Delta }}{v}_{I}\sim \sqrt{\alpha }{c}_{s}{R}_{e}^{1/4}| {S}_{t,1}-{S}_{t,2}| \sim \displaystyle \frac{1}{2}\sqrt{\alpha }{c}_{s}{R}_{e}^{1/4}{S}_{t}\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&\sim 0.5\,[{\rm{m}}\,{{\rm{s}}}^{-1}]{\left(\displaystyle \frac{{S}_{t}}{1.5\times {10}^{-4}}\right)}^{1/2}{\left(\displaystyle \frac{\alpha }{0.3}\right)}^{1/2}\left(\displaystyle \frac{{R}_{e}}{3\times {10}^{4}}\right)\,{c}_{s,1},\end{eqnarray} \tag{ 12 }$$

where S_t,1 and S_t,2 are Stokes numbers of two colliding particles. We here assume S_t,2 ∼ S_t,1/2 (Sato et al. 2016). For regime II, on the other hand,

$$\begin{eqnarray}&&{\rm{\Delta }}{v}_{{II}}\sim {v}_{L}\sqrt{{t}_{\mathrm{stop}}/{t}_{L}}\sim \sqrt{\alpha {S}_{t}}\,{c}_{s}\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&\approx 6.7\,[{\rm{m}}\,{{\rm{s}}}^{-1}]{\left(\displaystyle \frac{{S}_{t}}{1.5\times {10}^{-4}}\right)}^{1/2}{\left(\displaystyle \frac{\alpha }{0.3}\right)}^{1/2}\,{c}_{s,1}\end{eqnarray} \tag{ 14 }$$

where v_L is velocity of the largest eddy.

The size of dust aggregates determines how they interact with the gas (e.g., the Stokes parameter is proportional to ρ_inta_d for the Epstein law). Dynamics of the aggregates is affected by their cross sections, which depend on the internal inhomogeneous structure. The radius of BCCA cluster a_BCCA for large numbers of monomers N is given as a_BCCA ≃ N^0.5a₀ (Mukai et al. 1992; Wada et al. 2008, 2009), and this was also confirmed by N-body simulations (Suyama et al. 2012).

The hit-and-stick phase ends, when the rolling energy E_roll, which is the energy required to rotate a particle around a connecting point by 90°, is comparable to the impact energy, E_imp between the porous aggregates. Beyond this point, the aggregates start to get compressed by mutual collisions. Here, we assume E_roll = 4.37 × 10⁻⁹ erg (Suyama et al. 2012). When the number of monomers in the aggregates exceeds a critical number ${N}_{\mathrm{crit}}\equiv \beta \tfrac{8{E}_{\mathrm{roll}}}{{m}_{0}{\rm{\Delta }}{v}^{2}}$ with β = 0.5 (Suyama et al. 2012), they are compressed, and the internal density no longer decreases during the coagulation process.

The collisional velocity Δv between aggregates is determined by the interaction with the turbulence for S_t < 1, depending on S_t and R_e (Ormel & Cuzzi 2007): for S_t ≤ R_e,

$$\begin{eqnarray}&&{\rm{\Delta }}v=\displaystyle \frac{1}{2}\sqrt{\alpha }\,{c}_{s}\,{{Re}}^{1/4}\,{S}_{t},\end{eqnarray} \tag{ 15 }$$

or for S_t > R_e,

$$\begin{eqnarray}&&{\rm{\Delta }}v=\sqrt{\alpha {S}_{t}}\,{c}_{s},\end{eqnarray} \tag{ 16 }$$

where ${R}_{e}\equiv \alpha {c}_{s}^{2}/{\nu }_{\mathrm{mol}}{{\rm{\Omega }}}_{K}$ with the sound velocity of the gas disk c_s.

The internal density of the aggregated ρ_int,f formed of two equal-mass aggregates, with density ρ_int, is calculated according to Suyama et al. (2012):

$$\begin{eqnarray}&&{\rho }_{\mathrm{int},f}^{4}={\left({\rho }_{\mathrm{int}}^{4}+{\rho }_{0}^{4}\displaystyle \frac{{E}_{\mathrm{imp}}}{0.15{{NE}}_{\mathrm{roll}}}\right)}^{1/4}.\end{eqnarray} \tag{ 17 }$$

As the aggregates become more massive for S_t < 1, they start getting compressed by their self-gravity, and the internal density evolves as ${\rho }_{\mathrm{int}}\propto {({\rm{\Delta }}v)}^{3/5}\,{m}_{d}^{-1/5}$ (Okuzumi et al. 2012). This phase ends when the Stokes parameter becomes unity (S_t ∼ 1). Then the aggregates are decoupled from the turbulent gas, and they evolve as an N-body system.

When S_t > 1, the collisional velocity between the aggregates is determined by a balance between heating and cooling processes as the N-body particles. According to Michikoshi & Kokubo (2016, 2017), we solve the following equation to get equilibrium random velocity of the dust aggregates v_d,

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{{{dv}}_{d}^{2}}{{dt}} & = & {\left(\displaystyle \frac{{{dv}}_{d}^{2}}{{dt}}\right)}_{\mathrm{grav}}+{\left(\displaystyle \frac{{{dv}}_{d}^{2}}{{dt}}\right)}_{\mathrm{turb},\mathrm{stir}}+{\left(\displaystyle \frac{{{dv}}_{d}^{2}}{{dt}}\right)}_{\mathrm{turb},\mathrm{grav}}\\ & & -\,{\left(\displaystyle \frac{{{dv}}_{d}^{2}}{{dt}}\right)}_{\mathrm{coll}}-{\left(\displaystyle \frac{{{dv}}_{d}^{2}}{{dt}}\right)}_{\mathrm{drag}}=0.\end{array}\end{eqnarray} \tag{ 18 }$$

The first three heating terms are due to the gravitational scattering of the particles, stirring by turbulence, and gravitational scattering by turbulent fluctuation, respectively. The two cooling terms in Equation (18) represent the collisional damping and the gas drag.

We investigate GI of the disk that consisted of the dust aggregate at S_t > 1 based on the Toomre's Q-value defined as ${Q}_{d}\equiv ({v}_{d}/\sqrt{3}){{\rm{\Omega }}}_{K}/3.36G{{\rm{\Sigma }}}_{d}$. For the axisymmetric mode, Q_d < 1 is the necessary condition for the liner GI, but the nonaxisymmetric mode can be developed for Q_d ≲ 2, and the spiral-like density enhancements are formed followed by fragmentation of the spirals (Michikoshi & Kokubo 2017), which leads to the formation of planets. The mass of the fragments can be estimated as ${m}_{{pl}}\simeq {\lambda }_{\mathrm{GI}}^{2}{{\rm{\Sigma }}}_{d}$, where the critical wavelength for GI ${\lambda }_{\mathrm{GI}}=4{\pi }^{2}G{{\rm{\Sigma }}}_{d}/{{\rm{\Omega }}}_{K}^{2}$. The number of "planets" then can be estimated as N_pl ∼ 2πr/λ_GI. We found that the velocity dispersion of the aggregates drops rapidly due to the cooling terms in Equation (18). As a result, the system becomes gravitationally unstable after S_t = 1 in a rotational period.