Dust Destruction by Drift-induced Sputtering in Active Galactic Nuclei

We adopted the radiation spectra of AGNs used in Nenkova et al. (2008) whose parameter is their bolometric luminosity (L_AGN). As a dusty torus, we assumed a flat dust disk geometry (torus component is confined into z = 0 plane) for simplicity; this assumption is also in good agreement with recent results suggesting a geometrically thin dust disk, as discussed in Section 1. Hereafter, we refer to this dusty torus component as an AGN "dusty disk." The radial temperature distribution of the dusty disk is assumed to obey a single power law, ${T}_{{\rm{d}}}{(r)={T}_{\mathrm{in}}(r/{r}_{\mathrm{in}})}^{q}$, where r is the radius and T_in and r_in are the dust sublimation temperature and radius, respectively. The dust sublimation temperature and power-law index were set as T_in = 1500 K and q = −0.5. With the AGN bolometric luminosity L_AGN = 10⁴⁵ erg s⁻¹, the inner and outer radii of the dusty disk are r_in = 0.4 pc (Barvainis 1987; Kishimoto et al. 2011; Koshida et al. 2014) and r_out = 10 pc (Kishimoto et al. 2011; García-Burillo et al. 2016; Imanishi et al. 2016), respectively. In this paper, we adopted M_BH = 10⁸M_⊙, which yields the Eddington ratio of λ_Edd ≃ 0.08 for L_AGN = 10⁴⁵ erg s⁻¹. The SED of the dusty disk is given by

$$\begin{eqnarray}&&\displaystyle \frac{\lambda {L}_{\lambda }}{{L}_{\mathrm{AGN}}}=\epsilon \displaystyle \frac{8{\pi }^{2}\lambda }{{L}_{\mathrm{AGN}}}{\int }_{{r}_{\mathrm{in}}}^{{r}_{\mathrm{out}}}{B}_{\lambda }({T}_{{\rm{d}}}(r)){rdr},\end{eqnarray} \tag{ 1 }$$

where B_λ is the Planck function and λ is the wavelength, and we have introduced the parameter , the ratio of the luminosities of the dusty disk and AGN; is treated as a free parameter. Figure 1 presents the SED of our AGN and dusty disk model for various values of . In terms of energy budget, it is expected  < 1 because energy source of dusty disk emission is radiation from the central engine. Although  > 1 seems to be unrealistic, we also consider such cases as a numerical experiment to study the impact of dusty disk radiation on grain dynamics.

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We study dynamics of a dust grain at the vicinity of AGN by solving the equation of motion:

$$\begin{eqnarray}&&\displaystyle \frac{d{{\boldsymbol{r}}}_{{\rm{p}}}}{{dt}}={{\boldsymbol{v}}}_{{\rm{p}}},\end{eqnarray} \tag{ 2 }$$

$$\begin{eqnarray}&&{m}_{{\rm{d}}}\displaystyle \frac{d{{\boldsymbol{v}}}_{{\rm{p}}}}{dt}={{\boldsymbol{F}}}_{{\rm{g}}{\rm{r}}{\rm{a}}{\rm{v}}}+{{\boldsymbol{F}}}_{{\rm{r}}{\rm{a}}{\rm{d}}}+{{\boldsymbol{F}}}_{{\rm{r}}{\rm{a}}{\rm{d}},{\rm{d}}{\rm{i}}{\rm{s}}{\rm{k}}}+{{\boldsymbol{F}}}_{{\rm{d}}{\rm{r}}{\rm{a}}{\rm{g}}},\end{eqnarray} \tag{ 3 }$$

where ${m}_{{\rm{d}}}$ is the mass of a dust grain, ${{\boldsymbol{r}}}_{{\rm{p}}}$ and ${{\boldsymbol{v}}}_{{\rm{p}}}$ are the position and velocity vectors of the dust grain, ${{\boldsymbol{F}}}_{\mathrm{grav}}$ is the gravitational force due to the central black hole, ${{\boldsymbol{F}}}_{\mathrm{rad}}$ and ${{\boldsymbol{F}}}_{{\rm{r}}{\rm{a}}{\rm{d}},{\rm{d}}{\rm{i}}{\rm{s}}{\rm{k}}}$ are the radiation pressure forces from AGN and a dusty disk, respectively, and ${{\boldsymbol{F}}}_{\mathrm{drag}}$ is the gas drag force. Initially, the dust grain is located at r_init away from AGN at rest (${{\boldsymbol{v}}}_{p}=0$).

2.2.1. Gravity and Radiation Pressure

The gravitational force acting on a grain due to the central black hole is

$$\begin{eqnarray}&&{{\boldsymbol{F}}}_{\mathrm{grav}}=-\displaystyle \frac{{{GM}}_{\mathrm{BH}}{m}_{{\rm{d}}}}{{r}_{{\rm{p}}}^{2}}\displaystyle \frac{{{\boldsymbol{r}}}_{{\rm{p}}}}{{r}_{{\rm{p}}}},\end{eqnarray} \tag{ 4 }$$

where G is the gravitational constant, M_BH is the black hole mass, and ${r}_{{\rm{p}}}=| {{\boldsymbol{r}}}_{{\rm{p}}}|$ is the distance between AGN and the grain.

The radiation pressure force acting on a grain is

$$\begin{eqnarray}&&{{\boldsymbol{F}}}_{\mathrm{rad}}={F}_{\mathrm{rad}}\displaystyle \frac{{{\boldsymbol{r}}}_{{\rm{p}}}}{{r}_{{\rm{p}}}},\,{F}_{\mathrm{rad}}={\sigma }_{{\rm{d}}}{\int }_{0}^{\infty }d\nu {u}_{\nu }{Q}_{\mathrm{pr}}(\nu ),\end{eqnarray} \tag{ 5 }$$

where ${\sigma }_{{\rm{d}}}=\pi {a}^{2}$ is the geometrical cross section of the dust grain with radius a, ν is fRequency of radiation, u_ν is the specific energy density of radiation, and ${Q}_{\mathrm{pr}}={Q}_{\mathrm{abs}}+(1-g){Q}_{\mathrm{sca}}$ is the radiation pressure efficiency; where ${Q}_{\mathrm{abs}}$, ${Q}_{\mathrm{sca}}$, and g are the absorption and scattering efficiencies and the asymmetry parameter, respectively (Bohren & Huffman 1983). ${F}_{\mathrm{rad}}$ can be reduced to more simple expression:

$$\begin{eqnarray}&&{F}_{\mathrm{rad}}={\sigma }_{{\rm{d}}}\displaystyle \frac{{L}_{\mathrm{AGN}}}{4\pi {r}_{{\rm{p}}}^{2}c}\langle {Q}_{\mathrm{pr}}{\rangle }_{\mathrm{AGN}},\end{eqnarray} \tag{ 6 }$$

where c is the speed of light and $\langle {Q}_{\mathrm{pr}}{\rangle }_{\mathrm{AGN}}$ is the spectrum-averaged radiation pressure efficiency.

The ratio of the radiation pressure from AGN radiation and gravity, β_AGN, is given by the following:

$$\begin{eqnarray}&&{\beta }_{\mathrm{AGN}}\equiv \displaystyle \frac{{F}_{\mathrm{rad}}}{{F}_{\mathrm{grav}}}={\lambda }_{\mathrm{Edd}}\left(\displaystyle \frac{{m}_{{\rm{p}}}}{{\sigma }_{{\rm{T}}}}\right)\left(\displaystyle \frac{{\sigma }_{{\rm{d}}}}{{m}_{{\rm{d}}}}\right)\langle {Q}_{\mathrm{pr}}{\rangle }_{\mathrm{AGN}},\end{eqnarray} \tag{ 7 }$$

where m_p is the proton mass, σ_T is the Thomson cross section, and λ_edd is the Eddington ratio of the radiation. Both radiation pressure and gravity are inversely proportional to the square of the distance from the AGN; the ratio, β_AGN, does not depend on the distance.

When the grain radius is sufficiently larger than the incident wavelength, we can expect $\langle {Q}_{\mathrm{pr}}{\rangle }_{\mathrm{AGN}}\simeq 1$ (geometrical optics). Substituting $\langle {Q}_{\mathrm{pr}}{\rangle }_{\mathrm{AGN}}=1$ into Equation (7), we obtain a simple relationship for the radiation pressure force with respect to the gravity:

$$\begin{eqnarray}&&{\beta }_{\mathrm{AGN}}\simeq 54\left(\displaystyle \frac{{\lambda }_{\mathrm{edd}}}{0.01}\right){\left(\displaystyle \frac{a}{1\mu {\rm{m}}}\right)}^{-1}{\left(\displaystyle \frac{{\rho }_{{\rm{s}}}}{3.5{\rm{g}}{\mathrm{cm}}^{-3}}\right)}^{-1},\end{eqnarray} \tag{ 8 }$$

where ρ_s is the material density. We assume ρ_s = 3.5 g cm⁻³ for silicate and 2.24 g cm⁻³ for graphite.

We computed Q_pr using the Mie theory (Bohren & Huffman 1983), and the optical constant of silicate and graphite grains was taken from Draine & Lee (1984), Laor & Draine (1993), Draine (2003). If the size parameter x = 2πa/λ, where λ is the wavelength, is larger than 2 × 10⁴, we use the anomalous diffraction approximation (van de Hulst 1957) instead of using the Mie theory and assume ${Q}_{\mathrm{pr}}\simeq {Q}_{\mathrm{abs}}$.

Figure 2 shows the β-values for silicate and graphite grains with various grain radii. For a ≳ a few × 10⁻² μm, Equation (8) well reproduces the β_AGN-values. As graphite grains have slightly lower material density, they show slightly larger β-values than silicate grains. A dust grain will be gravitationally unbound if β_AGN > 0.5 (Burns et al. 1979). As the β-values are much larger than 0.5, dust grains will be blown out by AGN radiation pressure as long as other forces are negligible.

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2.2.2. Radiation Pressure from Dusty Disk

To calculate the radiation pressure force due to dusty disk emission, we divide the flat dusty disk by a number of patches. Let the position vector of a patch center is ${\boldsymbol{r}}$ and each patch has a width of (Δr, Δϕ), the radiation pressure from each patch may be written as

$$\begin{eqnarray}&&{\rm{\Delta }}{{\boldsymbol{F}}}_{{\rm{r}}{\rm{a}}{\rm{d}},{\rm{d}}{\rm{i}}{\rm{s}}{\rm{k}}}=\displaystyle \frac{{\sigma }_{{\rm{d}}}}{c}{\rm{\Delta }}{\rm{\Omega }}\displaystyle \frac{({{\boldsymbol{r}}}_{{\rm{p}}}-{\boldsymbol{r}})}{| \,{{\boldsymbol{r}}}_{{\rm{p}}}-{\boldsymbol{r}}| }\epsilon {\int }_{0}^{{\rm{\infty }}}{B}_{\nu }({T}_{{\rm{d}}}){Q}_{{\rm{p}}{\rm{r}}}(a,\nu )d\nu ,\end{eqnarray} \tag{ 9 }$$

where B_ν is the Planck function, T_d is the temperature of the dust grains, and ΔΩ is the solid angle of the patch. The solid angle is given by ${\rm{\Delta }}{\rm{\Omega }}=r{\rm{\Delta }}r{\rm{\Delta }}\phi \cos \theta /{d}^{2}$, where $d=| {{\boldsymbol{r}}}_{{\rm{p}}}-{\boldsymbol{r}}|$ is the distance between the disk patch and a grain, and θ is the angle between the unit vector along the z-axis (polar direction) and ${{\boldsymbol{r}}}_{{\rm{p}}}$ − ${\boldsymbol{r}}$. Using the particle coordinate ${{\boldsymbol{r}}}_{{\rm{p}}}=({r}_{{\rm{p}}},{\phi }_{{\rm{p}}},{z}_{{\rm{p}}})$ and the position vector of a patch center ${\boldsymbol{r}}=(r,\phi ,0)$, we obtain $\cos \theta ={z}_{{\rm{p}}}/d$. Integrating Equation (9) from ${r}_{\mathrm{in}}\leqslant r\leqslant {r}_{\mathrm{out}}$ and 0 ≤ ϕ ≤ 2π, the radiation pressure due to disk emission becomes

$$\begin{eqnarray}&&\begin{array}{r}\begin{array}{l}{{\boldsymbol{F}}}_{{\rm{r}}{\rm{a}}{\rm{d}},{\rm{d}}{\rm{i}}{\rm{s}}{\rm{k}}}\\ =\,\displaystyle \frac{{\sigma }_{{\rm{S}}{\rm{B}}}{\sigma }_{{\rm{d}}}}{\pi c}\epsilon {\int }_{{r}_{{\rm{i}}{\rm{n}}}}^{{r}_{{\rm{o}}{\rm{u}}{\rm{t}}}}rdr{\int }_{0}^{2\pi }d\phi \cos \theta {T}_{{\rm{d}}}^{4} \langle {Q}_{{\rm{p}}{\rm{r}}}{ \rangle }_{{T}_{{\rm{d}}}}\displaystyle \frac{({{\boldsymbol{r}}}_{p}-{\boldsymbol{r}})}{{d}^{3}},\end{array}\end{array}\end{eqnarray} \tag{ 10 }$$

where $\langle {Q}_{\mathrm{pr}}{\rangle }_{{T}_{{\rm{d}}}}$ is the Planck mean radiation pressure efficiency.

2.2.3. Gas Drag

A charged grain moving through ionized gas experiences two kinds of gas drag: (i) direct collision and (ii) Coulomb interaction. Because we assume that gas is at rest, the velocity of a dust grain ${v}_{{\rm{p}}}=| {{\boldsymbol{v}}}_{{\rm{p}}}|$ coincides with the relative velocity between gas and dust ${v}_{\mathrm{rel}}$. When a dust grain charged with ${Z}_{{\rm{d}}}$ (in electron charge unit) is moving though gas with the relative velocity v_rel, the drag forces can be expressed as (Draine & Salpeter 1979)

$$\begin{eqnarray}&&{{\boldsymbol{F}}}_{\mathrm{drag}}=-{F}_{\mathrm{drag}}\displaystyle \frac{{{\boldsymbol{v}}}_{{\rm{p}}}}{{v}_{{\rm{p}}}},\end{eqnarray} \tag{ 11 }$$

$$\begin{eqnarray}&&{F}_{\mathrm{drag}}=2{\sigma }_{{\rm{d}}}{k}_{{\rm{B}}}{T}_{{\rm{g}}}{n}_{{\rm{H}}}\left\{\sum _{i}{A}_{i}[{G}_{0}({s}_{i})+{z}_{i}^{2}{\phi }^{2}\mathrm{ln}({\rm{\Lambda }}/{z}_{i}){G}_{2}({s}_{i})\right\},\end{eqnarray} \tag{ 12 }$$

where k_B is Boltzmann's constant, ${T}_{{\rm{g}}}$ is the gas temperature, n_H is the number density of hydrogen atoms, A_i is the abundance of gas species i, z_i is the charge of gas species i (in electron charge unit), and ${s}_{i}={({m}_{i}{v}_{\mathrm{rel}}^{2}/2{k}_{{\rm{B}}}{T}_{{\rm{g}}})}^{1/2}$ is the normalized relative velocity; where m_i is the mass of gas particles of species i. The normalized grain potential ϕ and the Coulomb logarithm Λ are given by

$$\begin{eqnarray}&&\phi =\displaystyle \frac{{eU}}{{k}_{{\rm{B}}}{T}_{{\rm{g}}}},\end{eqnarray} \tag{ 13 }$$

$$\begin{eqnarray}&&{\rm{\Lambda }}=\displaystyle \frac{3}{2{ae}\phi }{\left(\displaystyle \frac{{k}_{{\rm{B}}}{T}_{{\rm{g}}}}{\pi {n}_{e}}\right)}^{1/2},\end{eqnarray} \tag{ 14 }$$

where $U={{eZ}}_{{\rm{d}}}/a$ is the electrostatic potential of a grain and n_e is the number density of electrons. At the right-hand side of Equation (12), the first and the second terms in the parenthesis represent drag forces due to direct collision and Coulomb interaction, respectively. Approximate forms for G₀ and G₂ are (Draine & Salpeter 1979)

$$\begin{eqnarray}&&{G}_{0}(s)\approx \displaystyle \frac{8s}{3\sqrt{\pi }}{\left(1+\displaystyle \frac{9\pi }{64}{s}^{2}\right)}^{1/2},\end{eqnarray} \tag{ 15 }$$

$$\begin{eqnarray}&&{G}_{2}(s)\approx s{\left(\displaystyle \frac{3}{4}\sqrt{\pi }+{s}^{3}\right)}^{-1}.\end{eqnarray} \tag{ 16 }$$

Equations (15)and (16)) are accurate to within 1% and 10% for $0\lt s\lt \infty$, respectively.

The gas phase abundance is defined by ${A}_{i}\equiv {n}_{i}/{n}_{{\rm{H}}}$, where n_i is the number density of species i. The abundance is taken from the standard solar elemental abundances (Grevesse & Sauval 1998). In this study, we consider hydrogen, helium, carbon, nitrogen, and oxygen atoms, and their abundances are A_He = 8.51 × 10⁻², A_C = 3.31 × 10⁻⁴, A_N = 8.32 × 10⁻⁵, and A_O = 6.76 × 10⁻⁴. For the sake of simplicity, every ion is assumed to carry +e charge, then z_i = 1. The electron abundance is A_e = 1.09.

Recently, Hoang (2017) pointed out if drift velocity exceeds a few percent of the speed of light, Equation (11) breaks down because the penetration length of an incident atom becomes comparable to or larger than the dust grain size. Although typical drift speed treated in this study is marginally lower than this threshold, we still use Equation (11) for the sake of simplicity.

To compute the Coulomb drag force, we need to know grain charge. We compute grain charge ${Z}_{{\rm{d}}}$ by solving the rate equation (Weingartner et al. 2006):

$$\begin{eqnarray}&&\displaystyle \frac{{{dZ}}_{{\rm{d}}}}{{dt}}={J}_{\mathrm{pe}}-{J}_{{\rm{e}}}+{J}_{\sec ,\mathrm{gas}}+{J}_{\mathrm{ion}},\end{eqnarray} \tag{ 17 }$$

where ${J}_{\mathrm{pe}}$ is the photoelectric emission rate, ${J}_{{\rm{e}}}$ and ${J}_{\mathrm{ion}}$ are the collisional charging rate of electrons and ions, respectively, and ${J}_{\sec ,\mathrm{gas}}$ is the rate for secondary electron emission induced by incident gas-phase electrons (see Appendix A for more details). Photoelectric emission, ion collisions, and secondary electron emission make grains being positively charged, while electron collisions make them negatively charged. Typical collision timescale of electrons for a neutral grain is ${J}_{{\rm{e}}}^{-1}\sim 0.03\,{\rm{s}}$, where we have used ${T}_{{\rm{g}}}={10}^{5}$ K, a = 0.1 μm, n_e = 10³ cm⁻³, and s_e = 0.5 (see Equations (37) and 38). Because charging timescale is much shorter than dynamical timescale, we can assume the steady state for grain charge. In addition, we also ignore charge distribution because grains will be highly charged in the AGN environments (Weingartner et al. 2006). Thus, in this paper, we solve ${J}_{\mathrm{pe}}-{J}_{{\rm{e}}}+{J}_{\sec ,\mathrm{gas}}+{J}_{\mathrm{ion}}=0$ to find ${Z}_{{\rm{d}}}$. Solutions of Equation (17) are presented in the companion paper (Tazaki et al. 2020) (see also Weingartner et al. 2006).

Dust grains at the vicinity of AGNs may be sputtered by gas either thermally or kinetically (non-thermally). The sputtering rate can be expressed as (e.g., Draine & Salpeter 1979; Dwek & Arendt 1992; Nozawa et al. 2006)

$$\begin{eqnarray}\begin{array}{rcl}\displaystyle \frac{1}{{n}_{{\rm{H}}}}\displaystyle \frac{{da}}{{dt}} & = & -\displaystyle \frac{{m}_{\mathrm{sp}}}{2{\rho }_{{\rm{s}}}}\sum _{i}{\left(\displaystyle \frac{8{k}_{{\rm{B}}}{T}_{{\rm{g}}}}{\pi {m}_{i}}\right)}^{\tfrac{1}{2}}{A}_{i}\displaystyle \frac{{e}^{-{s}_{i}^{2}}}{2{s}_{i}}\\ & & \times {\int }_{{\epsilon }_{\min }}^{\infty }d{\epsilon }_{i}\left(1-\displaystyle \frac{{z}_{i}\phi }{{\epsilon }_{i}}\right)\sqrt{{\epsilon }_{i}}{e}^{-{\epsilon }_{i}}\\ & & \times \sinh (2\sqrt{{\epsilon }_{i}}{s}_{i}){Y}_{\mathrm{sp}}^{i}[({\epsilon }_{i}-{z}_{i}\phi ){kT}],\end{array}\end{eqnarray} \tag{ 18 }$$

where ${\epsilon }_{i}={E}_{i}/{k}_{{\rm{B}}}{T}_{{\rm{g}}}$ is normalized kinetic energy of an impacting ion, ${Y}_{\mathrm{sp}}^{i}$ is the normal-incidence sputtering yield, and m_sp is the average mass of the sputtered atom. The sputtering yield is calculated based on the model described in Nozawa et al. (2006) (see also Appendix B).

Figure 3 shows the sputtering rate as a function of the relative velocity. Sputtering can be classified into two origins: thermal sputtering and kinetic (non-thermal) sputtering (e.g., Dwek & Arendt 1992; Nozawa et al. 2006). The former one is sputtering due to thermal motion of gas particles, whereas the latter is driven by the relative velocity between gas and dust. In Figure 3, the sputtering rate at the low velocity domain (${v}_{\mathrm{rel}}\ll {10}^{2}$ km s⁻¹) corresponds to the rate of thermal sputtering.

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It can be seen from Figure 3 that with increasing grain potential, the thermal sputtering rate decreases. This is because collisions of thermal ions are suppressed due to the Coulomb repulsion force between a grain and the ions. In Equation (18), a collision between a positively charged grain and a hydrogen ion with kinetic energy of about thermal energy (_i ≃ 1) occurs when ${z}_{i}\phi /{\epsilon }_{i}={eU}/{k}_{{\rm{B}}}{T}_{{\rm{g}}}\lesssim 1$. Thus, thermal sputtering occurs when

$$\begin{eqnarray}&&{T}_{{\rm{g}}}\gtrsim 3.5\times {10}^{6}\,{\rm{K}}\left(\displaystyle \frac{U}{300\,{\rm{V}}}\right).\end{eqnarray} \tag{ 19 }$$

As dust grains exposed to the radiation from central engine usually become charged with a few hundreds of volts (Weingartner et al. 2006; Tazaki et al. 2020), thermal sputtering is not likely to occur unless the gas temperature is extremely high.

Similarly, kinetic sputtering due to hydrogen-ion collisions occurs when ${z}_{i}\phi /{\epsilon }_{i}=2{eU}/{m}_{{\rm{H}}}{v}_{\mathrm{rel}}^{2}\lesssim 1$, where we have used ${\epsilon }_{i}={m}_{{\rm{H}}}{v}_{\mathrm{rel}}^{2}/2{k}_{{\rm{B}}}{T}_{{\rm{g}}}$. As a result, we obtain

$$\begin{eqnarray}&&{v}_{\mathrm{rel}}\gtrsim 2.4\times {10}^{2}\,\mathrm{km}\,{{\rm{s}}}^{-1}{\left(\displaystyle \frac{U}{300{\rm{V}}}\right)}^{1/2}.\end{eqnarray} \tag{ 20 }$$

As we will show, typical relative velocity, which is set by the terminal velocity (see Equation (24)), can be much higher than this velocity when the Coulomb drag force is weaker than the radiation pressure force. Hence, we can expect kinetic sputtering at the vicinity of AGNs.

When kinetic sputtering is dominant, Equation (18) can be reduced to more simple form. In the limit of high relative velocity (${s}_{i}\to \infty$), we obtain (Dwek & Arendt 1992; Nozawa et al. 2006),

$$\begin{eqnarray}&&\displaystyle \frac{1}{{n}_{{\rm{H}}}}\displaystyle \frac{{da}}{{dt}}=-\displaystyle \frac{{m}_{\mathrm{sp}}}{2{\rho }_{{\rm{s}}}}{v}_{\mathrm{rel}}\sum _{i}{A}_{i}{Y}_{i}^{0}\left(\displaystyle \frac{{m}_{i}{v}_{\mathrm{rel}}^{2}}{2}\right),\end{eqnarray} \tag{ 21 }$$

where we have ignored grain charge.

The terminal velocity of grains will be set by the balance between radiation pressure and gas drag. Figure 4 shows the gas drag force relative to the radiation pressure force as a function of the relative velocity for a wide range of parameters. As shown in Figure 4(a), the gas drag force consists of two contributions: Coulomb drag and collisional drag (see also Section 2.2.3). At the low velocity regime, the Coulomb drag force dominates, while at the high velocity regime, the collisional drag force dominates.

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First, we study how the Coulomb drag force varies with parameters. The Coulomb drag force becomes maximum when the relative velocity approximately coincides with the thermal velocity of gas. As the thermal velocity is proportional to $\sqrt{{T}_{{\rm{g}}}}$, the peak velocity at which the Coulomb drag force is maximized shifts toward larger velocities for higher gas temperature (Figure 4(b)). In addition, the Coulomb drag force is weaker for higher gas temperature. With increasing the gas temperature, a grain will be more positively charged, which makes Coulomb drag strong; however, in the same time, kinetic energy of gas particles is also increased, which makes Coulomb scattering difficult. Because the latter effect dominates the former one, the Coulomb drag force becomes weak for higher gas temperature.

The Coulomb drag force also increases as the gas density due to more frequent collisions (Figure 4(c)), although the Coulomb drag force is not simply proportional to the gas density. This is because higher gas density provides more electrons in the gas phase, which results in reducing the grain charge.

The Coulomb drag force is weaker for smaller particles because of larger normalized grain potential $\phi ={e}^{2}{Z}_{{\rm{d}}}/{k}_{{\rm{B}}}{T}_{{\rm{g}}}a$ (Figure 4(d)). ϕ usually increases with decreasing grain radius as long as a ≳ 0.1 μm as suggested in Figure 4(d) (see also Weingartner et al. 2006). In Figure 4(d), the ${F}_{\mathrm{drag}}$/${F}_{\mathrm{rad}}$-values suddenly increases for grains smaller than 0.01 μm. This is mainly due to the reduction of the radiation pressure force with respect to the one expected from Equation (8), which happens when a ≲ 0.1 μm (see Figure 2). Silicate grains are slightly more charged than carbonaceous grains, and then, the Coulomb drag force is slightly larger for silicate grains (Figure 4(e)). The ratio also depends on the initial distance, ${r}_{\mathrm{init}}$. As ${r}_{\mathrm{init}}$ increases, the ratio increases due to reduction of radiation pressure forces.

Next, we study the collisional drag force. In the limit of s ≫ 1, the drag force (Equation (12)) due to direct collision can be reduced to

$$\begin{eqnarray}&&{F}_{\mathrm{coll},{\rm{s}}\gg 1}\simeq {\sigma }_{{\rm{d}}}{n}_{{\rm{H}}}{v}_{\mathrm{rel}}^{2}\sum _{i}{m}_{i}{A}_{i}.\end{eqnarray} \tag{ 22 }$$

By using Equation (6), we find

$$\begin{eqnarray}&&\displaystyle \frac{{F}_{\mathrm{coll},{\rm{s}}\gg 1}}{{F}_{\mathrm{rad}}}\simeq \displaystyle \frac{{{cn}}_{{\rm{H}}}{v}_{\mathrm{rel}}^{2}{\sum }_{i}{m}_{i}{A}_{i}}{{F}_{\mathrm{AGN}}\langle {Q}_{\mathrm{pr}}{\rangle }_{\mathrm{AGN}}},\end{eqnarray} \tag{ 23 }$$

where ${F}_{\mathrm{AGN}}={L}_{\mathrm{AGN}}/4\pi {r}_{{\rm{p}}}^{2}$ is the AGN radiation flux. Therefore, as long as $\langle {Q}_{\mathrm{pr}}{\rangle }_{\mathrm{AGN}}\approx 1$, which becomes good approximation for grains larger than ≈0.1 μm (see Figure 2), Equation (23) does not depend on any grain properties, such as size and composition, as well as the gas temperature, as shown in Figures 4(b), (d) and (e). Note that in Figure 4(e), a very subtle difference in ${F}_{\mathrm{coll},{\rm{s}}\gg 1}/{F}_{\mathrm{rad}}$ between silicate and carbonaceous grains can be seen. This is due to the slight difference of $\langle {Q}_{\mathrm{pr}}{\rangle }_{\mathrm{AGN}}$ arising from their difference of optical constants.

Finally, we discuss the terminal velocity of a grain. As mentioned, Coulomb drag is maximized when the relative velocity is similar to the thermal velocity of gas. Therefore, if radiation pressure balances with Coulomb drag, the terminal velocity becomes subsonic. On the other hand, if radiation pressure is stronger than Coulomb drag, and then it finally balances with collisional drag, the terminal velocity will be hypersonic.

In the case of the hypersonic drift, the terminal velocity can be found from Equation (23) by setting the left-hand side as unity, and it becomes

$$\begin{eqnarray}\begin{array}{r}\begin{array}{rcl}{v}_{{\rm{t}}{\rm{e}}{\rm{r}}{\rm{m}}} & = & {\left(\displaystyle \frac{{F}_{{\rm{A}}{\rm{G}}{\rm{N}}} \langle {Q}_{{\rm{p}}{\rm{r}}}{ \rangle }_{{\rm{A}}{\rm{G}}{\rm{N}}}}{{cn}_{{\rm{H}}}{\sum }_{i}{m}_{i}{A}_{i}}\right)}^{1/2},\end{array}\end{array}\end{eqnarray} \tag{ 24 }$$

$$\begin{eqnarray}&&\begin{array}{c}\simeq 3.5\times {10}^{3}\,{\rm{k}}{\rm{m}}\,{{\rm{s}}}^{-1} \langle {Q}_{{\rm{p}}{\rm{r}}}{ \rangle }_{{\rm{A}}{\rm{G}}{\rm{N}}}{\left(\displaystyle \frac{{n}_{{\rm{H}}}}{{10}^{3}{{\rm{c}}{\rm{m}}}^{-3}}\right)}^{-{\textstyle \tfrac{1}{2}}}\\ \,\times \,{\left(\displaystyle \frac{{L}_{{\rm{A}}{\rm{G}}{\rm{N}}}}{{10}^{45}{\rm{e}}{\rm{r}}{\rm{g}}{{\rm{s}}}^{-1}}\right)}^{1/2}{\left(\displaystyle \frac{{r}_{{\rm{p}}}}{1{\rm{p}}{\rm{c}}}\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 25 }$$

For a ≳ 0.1 μm, we have $\langle {Q}_{\mathrm{pr}}{\rangle }_{\mathrm{AGN}}\approx 1$, hence the terminal velocity does not depends on grain radius. As this velocity is higher than the threshold velocity of kinetic sputtering (Equation (20)), kinetic sputtering is found to be important.

Figure 5 shows the parameter range where the sub/hypersonic drift occurs. As mentioned, Coulomb drag becomes weaker for higher gas temperature and/or lower density. The boundary of the sub/hypersonic drift depends on the grain radius. At the pc-scale polar regions of AGN, the gas temperature can be heated higher than 10⁴ K, and the gas density are less than ${10}^{3}\,{\mathrm{cm}}^{-3}$ (e.g., Wada et al. 2016). For example, for ${T}_{{\rm{g}}}={10}^{4}$ K, ${n}_{{\rm{H}}}={10}^{2}\,{\mathrm{cm}}^{-3}$ and ${r}_{\mathrm{init}}=1\,$ pc, grains larger than 0.1 μm will be subjected to the hypersonic drift, whereas grains smaller than 0.1 μm results in the subsonic drift. Although very small grains can avoid destruction by sputtering, they might be disrupted by Coulomb explosion (Tazaki et al. 2020). In addition, the boundary also depends on the initial distance of a grain from the AGN. As shown in Figure 5 (bottom), for ${n}_{{\rm{H}}}\lesssim 80\,{\mathrm{cm}}^{-3}$, grains larger than 0.1 μm initially located within 5 pc from AGN can be accelerated to the hypersonic velocity.

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Although we have treated gas temperature and gas density as individual parameters, those could be related each other. It is important to study how they are related each other in the AGN environment (e.g., Stern et al. 2014) and how this affects grain dynamics. However, this is beyond the scope of this paper.

To study dynamics and dust destruction, we solve the equation of motion (Equations (2) and (3)), grain charge (Equation (17) with steady-state assumption), and sputtering (Equation (18)), simultaneously. Here, we ignore IR radiation from the dusty disk, and therefore, grain motion becomes one-dimensional (radial). The gas density and temperature are assumed to be ${n}_{{\rm{H}}}={10}^{3}\,{\mathrm{cm}}^{-3}$ and ${T}_{{\rm{g}}}={10}^{5}$ K, respectively. Thus, submicron-sized or larger grains at ${r}_{\mathrm{init}}=1\,$ pc are subjected to the hypersonic drift (Figure 5).

Figure 6 shows the time evolution of the grain drift velocity. Initially, dust grains are accelerated by the radiation pressure and reach the terminal velocity (see Equation (24)). The acceleration time to reach the terminal velocity is typically about

$$\begin{eqnarray}&&{t}_{\mathrm{acc}}\equiv \displaystyle \frac{{m}_{{\rm{d}}}{v}_{\mathrm{term}}}{{F}_{\mathrm{rad}}},\end{eqnarray} \tag{ 26 }$$

$$\begin{eqnarray}\begin{array}{rcl} & \simeq & 19\,\mathrm{yr}\langle {Q}_{\mathrm{pr}}{\rangle }_{\mathrm{AGN}}^{-\tfrac{1}{2}}\left(\displaystyle \frac{a}{1\,\mu {\rm{m}}}\right){\left(\displaystyle \frac{{n}_{{\rm{H}}}}{{10}^{3}{\mathrm{cm}}^{-3}}\right)}^{-\tfrac{1}{2}}\\ & & {\left(\displaystyle \frac{{L}_{\mathrm{AGN}}}{{10}^{45}\mathrm{erg}{{\rm{s}}}^{-1}}\right)}^{-1/2}\left(\displaystyle \frac{{r}_{{\rm{p}}}}{1\,\mathrm{pc}}\right).\end{array}\end{eqnarray} \tag{ 27 }$$

Thus, larger grains require more time to reach the terminal velocity. Once the grains reach the terminal velocity, grain radius gradually decreases as they drift because the relative velocity is high enough to cause sputtering (Figure 7). The sputtering timescale is typically about

$$\begin{eqnarray}&&{t}_{\mathrm{sp}}\equiv {\left[\displaystyle \frac{1}{a}\left(\displaystyle \frac{{da}}{{dt}}\right)\right]}^{-1},\end{eqnarray} \tag{ 28 }$$

$$\begin{eqnarray}\begin{array}{rcl} & \simeq & 6.6\times {10}^{2}\,\mathrm{yr}\left(\displaystyle \frac{a}{1\,\mu {\rm{m}}}\right){\left(\displaystyle \frac{{m}_{\mathrm{sp}}}{20{m}_{{\rm{H}}}}\right)}^{-1}\\ & & \times {\left(\displaystyle \frac{{n}_{{\rm{H}}}}{{10}^{3}{\mathrm{cm}}^{-3}}\right)}^{-1}{\left(\displaystyle \frac{{v}_{\mathrm{rel}}}{{10}^{3}\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{-1}{\left(\displaystyle \frac{{Y}_{\mathrm{sp}}}{{10}^{-2}}\right)}^{-1}.\end{array}\end{eqnarray} \tag{ 29 }$$

As long as $\langle {Q}_{\mathrm{pr}}{\rangle }_{\mathrm{AGN}}\approx 1$, the terminal velocity (or the relative velocity) does not depend on grain radius, and then, we have ${t}_{\mathrm{sp}}\propto a$. Therefore, smaller grains are more rapidly eroded compared with larger grains. Once the grain radius becomes below about 0.1 μm, the grain drift velocity rapidly decreases as grain radius decreases. This is because the size-dependent $\langle {Q}_{\mathrm{pr}}{\rangle }_{\mathrm{AGN}}$ is decreased. Finally, dust grains will be disrupted by Coulomb explosion, where grain charge causes grain fission (see Tazaki et al. 2020 for more details).

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Evolution of grain radius as a function of the distance from AGN can be approximately reproduced by using a simplified equation. By using Equations (2) and (21), we can eliminate time derivative, and we obtain a simple equation,

$$\begin{eqnarray}&&\displaystyle \frac{{da}}{{{dr}}_{{\rm{p}}}}=-\displaystyle \frac{{m}_{\mathrm{sp}}}{2{\rho }_{{\rm{s}}}}{n}_{{\rm{H}}}\sum _{i}{A}_{i}{Y}_{i}^{0}\left(\displaystyle \frac{{m}_{i}{v}_{\mathrm{term}}^{2}}{2}\right),\end{eqnarray} \tag{ 30 }$$

where we have substituted the terminal velocity as the relative velocity and ignored grain charge. We solve Equation (30), and the results are shown in Figure 8 with dashed lines. Equation (30) is able to capture overall properties of drift-induced kinetic sputtering. A distance for which a grain can travel depends on the gas density; a lower gas density increases the travel distance (see Figure 8).

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Figure 9 shows the drift distance defined by the distance between initial and final distances from AGN, where the final distance is the location when grains radius becomes 10⁻³ μm (see also the bottom panel of Figure 7). With increasing gas density, drift distance decreases due to efficient sputtering. At ${n}_{{\rm{H}}}\lesssim 20$ cm⁻³, both submicron and micron-sized grains drift longer than pc scales. On the other hand, At n_H ≳ 4 × 10³ cm⁻³, 10 micron-sized grains are disrupted by kinetic sputtering within subpc scales. As a result, intermediate density (n_H ∼ 10³ cm⁻³) is favorable for submicron/micron-sized grains to drift less/more than pc scales as long as gas temperature is higher than 10⁵ K. If gas temperature is lower than that, further lower gas density is necessary to occur drift-induced sputtering because Coulomb drag halts grain drift (see Figure 5).

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Next, we consider IR radiation from the dusty disk, in addition to AGN radiation pressure, and study how dusty disk emission levitates grains. Figure 10 presents the trajectories of dust grains with various initial grain radii. Each line terminates once the grain radius becomes less than 10⁻³ μm. As shown in Section 3.2, larger grains are blown over longer distances than smaller grains. As the disk luminosity () increases, dust grains can levitate to higher altitudes. Additionally, larger grains can be lifted to higher altitudes, due to the larger radiation pressure cross section to IR emission from the dusty disk. For example, Q_pr of disk IR emission for 3 μm grains is larger than that of 0.1 μm grains. In contrast, because both 10 and 3 μm grains have Q_pr ≃ 1, the radiation pressure per unit mass for 10 μm grains is smaller than that of 3 μm grains. As a result, the levitation of 10 μm grains is less efficient than 3 μm grains.

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We also found that the levitation due to disk emission is a minor effect, as AGN luminosity is usually higher than that of a dusty disk (e.g., Elvis et al. 1994; Brown et al. 2006; Richards et al. 2006) with  = 0.1–0.3, indicating that the IR emission from the dusty disk may not be a crucial carrier of dust grains toward polar regions. If dust grains are distributed in the inner polar region, one can readily imagine that they will be blown out to the polar region. Figure 11 presents trajectories of grains initially located in the inner polar regions. Small grains become sputtered after traveling small distances, whereas larger grains can be blown over larger distances. Hence, if the polar dust grains originate from the inner polar region, the drift-induced sputtering scenario predicts a lack of small grains, leading to the suppression of 10 μm silicate in MIR emission spectra.

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In Section 3, we implicitly assumed the presence of micron-sized grains in the inner region of a dusty disk. Here, we discuss the possibility of grain growth in the AGN environments.

4.1.1. Growth by Coagulation in the Dusty Disks

There are two types of grain growth exist: condensation and coagulation.

Condensation (and evaporation) rate of a spherical grain can be estimated by the Hertz–Knudsen equation:

$$\begin{eqnarray}&&\displaystyle \frac{{da}}{{dt}}={s}_{{\rm{a}}}{V}_{0}\displaystyle \frac{(p-{p}_{\mathrm{sat}})}{\sqrt{2\pi {m}_{\mathrm{atom}}{k}_{{\rm{B}}}{T}_{{\rm{g}}}}},\end{eqnarray} \tag{ 31 }$$

where p and p_sat are the vapor pressure and saturated vapor pressure, respectively, s_a is the sticking probability, m_atom is the mass of impinging atoms, and ${V}_{0}={m}_{\mathrm{atom}}/{\rho }_{{\rm{s}}}$ (e.g., Henning 2010; Nozawa & Kozasa 2013). In Equation (31), we have assumed dust temperature equals to gas temperature. If the supersaturation ratio, ${ \mathcal S }=p/{p}_{\mathrm{sat}}$, exceeds unity, condensation happens. The condensation timescale is given by

$$\begin{eqnarray}&&{t}_{\mathrm{cond}}^{-1}\equiv \displaystyle \frac{1}{a}\left(\displaystyle \frac{{da}}{{dt}}\right)={s}_{{\rm{a}}}{V}_{0}{\left(\displaystyle \frac{{k}_{{\rm{B}}}{T}_{{\rm{g}}}}{2\pi {m}_{\mathrm{atom}}}\right)}^{1/2}\left(1-\displaystyle \frac{1}{{ \mathcal S }}\right)\displaystyle \frac{{n}_{\mathrm{atom}}}{a},\end{eqnarray} \tag{ 32 }$$

where n_atom is the number density of impinging atoms. For the case of ${ \mathcal S }=1.1$, typical condensation timescale of silicon atoms is about

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{cond}} & \simeq & \displaystyle \frac{1.6\times {10}^{2}}{{s}_{{\rm{a}}}}\,\mathrm{yr}\ {\left(\displaystyle \frac{{n}_{\mathrm{Si}}}{1.8\times {10}^{6}{\mathrm{cm}}^{-3}}\right)}^{-1}\\ & & \left(\displaystyle \frac{a}{3\ \mu {\rm{m}}}\right){\left(\displaystyle \frac{{T}_{{\rm{g}}}}{1500{\rm{K}}}\right)}^{-1/2},\end{array}\end{eqnarray} \tag{ 33 }$$

where ${n}_{\mathrm{Si}}={A}_{\mathrm{Si}}{n}_{{\rm{H}}}$ is the number density of silicon atoms. We have assumed ${n}_{{\rm{H}}}=5\times {10}^{10}\ {\mathrm{cm}}^{-3}$, which corresponds to the gas density at 0.4 pc away from the black hole with 10⁸ M_⊙ (see Equations (3) and (4) in Ichikawa & Tazaki 2017). As Equation (33) assumes that all silicon atoms are in the gas phase, this timescale should be regarded as the lower limit.

Another possibility of grain growth is coagulation, or the hit-and-stick growth of dust grains. The coagulation timescale of dust grains can be evaluated by

$$\begin{eqnarray}&&{t}_{\mathrm{coag}}\simeq \displaystyle \frac{1}{{s}_{{\rm{c}}}{n}_{\mathrm{gr}}{\sigma }_{\mathrm{col}}{\rm{\Delta }}v},\end{eqnarray} \tag{ 34 }$$

where s_c is the sticking probability, n_gr is the number density of dust grains, σ_col is the collision cross section, and Δv is the relative velocity. Using the dust-to-gas mass ratio f, the number density of grains is estimated to be ${n}_{\mathrm{gr}}={{fn}}_{{\rm{H}}}{m}_{{\rm{H}}}/{m}_{\mathrm{gr}}$. The relative velocity of grains determined by their Brownian motion is given by ${\rm{\Delta }}v={(8{k}_{{\rm{B}}}T/\pi {\mu }_{\mathrm{gr}})}^{1/2}$, where μ_gr is the reduced mass of grains. Suppose two grains are identical and each grain has the mass m_gr, ${\rm{\Delta }}v={(16{k}_{{\rm{B}}}T/\pi {m}_{\mathrm{gr}})}^{1/2}$. If each grain has radius a, the collision cross section is ${\sigma }_{\mathrm{col}}=\pi {(2a)}^{2}=4\pi {a}^{2}$. By assuming homogeneous spherical grains, we obtain

$$\begin{eqnarray}\begin{array}{rcl}{t}_{\mathrm{coag}} & \simeq & \displaystyle \frac{2.6\times {10}^{5}}{{s}_{{\rm{c}}}}\ \mathrm{yr}{\left(\displaystyle \frac{f}{0.01}\right)}^{-1}{\left(\displaystyle \frac{{n}_{{\rm{H}}}}{5\times {10}^{10}{\mathrm{cm}}^{-3}}\right)}^{-1}\\ & & {\left(\displaystyle \frac{a}{3\mu {\rm{m}}}\right)}^{5/2}{\left(\displaystyle \frac{{\rho }_{{\rm{s}}}}{3.5{\rm{g}}{\mathrm{cm}}^{-3}}\right)}^{3/2}{\left(\displaystyle \frac{T}{1500{\rm{K}}}\right)}^{-1/2},\end{array}\end{eqnarray} \tag{ 35 }$$

where ρ_s is the material density of dust grains. Except for very small grains (with a radius less than 0.02 μm for T = 1500 K), the collision velocity is slower than the critical fragmentation velocity of silicate grains (∼m s⁻¹) (Chokshi et al. 1993). Therefore, it is reasonable to assume the sticking probability of micron-sized grains s_c ≃ 1.

As a result, as long as the gas phase is sufficiently supersaturated, which might be expected near the sublimation radius (e.g., Ros & Johansen 2013), condensation may dominates grain growth. Observations suggest the typical AGN lifetime of 10–100 Myr (e.g., Martini 2004) and simulations (e.g., Hopkins et al. 2006) (see also discussions in Inayoshi et al. 2018). Hence, grain growth can potentially form micron-sized grains at the inner region of dusty torus. Even if the condensation is not efficient, coagulation may form large grains within the lifetime of AGN as long as the torus structure is static.

In Section 3, we showed that small dust grains are more readily destroyed due to drift-induced kinetic sputtering. One key observational expectation is the change in strength of the silicate feature, which can be observed in the MIR band.

Figure 12 presents the emission efficiency of silicate particles as a function of wavelength for various dust radii. For particles smaller than 0.3 μm, the silicate feature at ∼10 and ∼18 μm is prominent. In contrast, the silicate feature is attenuated if a dust grain is larger than 1.6 μm ≈ 9.8 μm/ 2π, or the size parameter exceeds unity for the silicate feature wavelength. When the size parameter exceeds unity, a dust grain with a moderate refractive index ($| m-1| \sim 1$, where m is the complex refractive index) becomes optically thick for the radiation; in this case, the emission approaches that of a perfect blackbody. Therefore, the removal of small grains and/or the prevalence of large grains makes the silicate feature less prominent.

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The recent high spatial resolution MIR interferometric observations of the polar dust region often reveal a weak silicate feature (e.g., Hönig et al. 2012; Burtscher et al. 2013) as well as SED decomposition of the dusty polar region (Lyu & Rieke 2018); these observations are consistent with our results suggesting that only large-sized dust grains can survive in such a region.

Drift-induced sputtering can be expected when the grains are exposed to the AGN radiation. Hence, this process may also explain the observed differences in the silicate features of type-1 and type-2 AGNs.

Conventionally, for type-1 (face-on) AGN, one would expect that the silicate feature would be observed in the emission because the heated silicate dust on the surface of the inner dusty disk can be observed directly, i.e., allowing for direct detection of the silicate features in the emission. In contrast, for type-2 (edge-on) AGN, the silicate feature would be visible in absorption spectra, because the external dusty disk wall would block the inner dust emission. Although the silicate feature shows a wide range of variety both in type-1 and type-2 AGN observationally (Levenson et al. 2007; Wu et al. 2009; Hatziminaoglou et al. 2015), it is known that averaged type-1 (face-on) AGN spectra exhibit weak silicate emission, unlike averaged type-2 (edge-on) AGN spectra that exhibit deeper silicate absorption (e.g., Hao et al. 2007).

Such observational issues may be circumvented once the smaller grains in the central few pc are disrupted under AGN irradiation. Here, both the dust in the dusty outflow region and the surface of the dusty disk producing a dust continuum with silicate-free features, as shown in Figure 12, can be observed directly for type-1 AGN. In contrast, the origin of the moderated level of absorption in type-2 AGN would be a combination of (1) the contribution of silicate absorption from the dusty disk, where the small-sized dust grains are still alive, and (2) the contribution of silicate feature-free emission from the partially visible dusty outflow region with a size of a few pc (e.g., Hönig et al. 2013).

If the above discussion is correct, it also naturally explains the dichotomy of silicate features between type-2 AGN and ultra-luminous infrared galaxies (ULIRGs) with a deeper silicate feature (e.g., Hao et al. 2007). Because most of the ULIRGs are considered to be buried AGN obscured by dust in all directions (e.g., Imanishi et al. 2010; Ichikawa et al. 2014), the dusty outflow region is also highly obscured even if it exists. Thus, ULIRGs have a deeper silicate absorption feature, compared with those of type-2 AGN.

It is helpful to compare our results with previous observational and theoretical studies of dust properties in AGNs. Aitken & Roche (1985) first reported that a small-sized grain with a < 0.01 μm would be easily destroyed in AGNs, and only larger grains with a > 0.01 μm could survive due to the longer lifetime. Laor & Draine (1993) also found that a dust grain size of >3 μm is preferable for AGNs with a deficit of silicate features, and Laor & Draine (1993) also discussed that the optically thick dusts also strongly reduces the amplitude of the silicate feature (see Figure 16 in Laor & Draine 1993).

Baskin & Laor (2018) further discussed how the dust sublimation area depends on the dust composition and the grain size; this suggests that the silicate grain size distribution will be shifted to a larger size, for a grain size distribution of up to a ≃ 1 μm. Our results also indicate that a much larger grain size with a > 1 μm would be preferable in the polar dust region at a few pc distance from the central engine; otherwise, the dust would be completely evaporated at that distance. The advantage of our result is that it is no longer necessary to consider the dust composition issue, i.e., how only purely carbon dust can be lifted up to the polar dust region without silicate dust contamination.