Cooling Timescale of Dust Tori in Dying Active Galactic Nuclei

We describe physical properties of the clump in the torus. The idea of the torus clump was first proposed by Krolik & Begelman (1988), and then the model was sophisticated by many authors (e.g., Beckert & Duschl 2004; Hönig & Beckert 2007; Namekata et al. 2014). The basic idea of the torus clump here is mainly compiled in Vollmer et al. (2004).

The clump should be gravitationally bounded, and therefore the mass of the clump M_c is thought to be larger than the Jeans mass M_J,

$$\begin{eqnarray}&&{M}_{{\rm{c}}}\geqslant {M}_{J}\equiv \displaystyle \frac{{\pi }^{5/2}}{6}\displaystyle \frac{{c}_{s}^{3}}{{G}^{3/2}{\rho }_{0}^{1/2}},\end{eqnarray} \tag{ 1 }$$

where G is the gravitational constant, c_s is the speed of sound, and ρ₀ is the gas density of the clump. In addition, the clump radius R_c should be smaller than the tidal radius R_t,

$$\begin{eqnarray}&&{R}_{{\rm{c}}}\leqslant {R}_{{\rm{t}}}\equiv {\left(\displaystyle \frac{{M}_{{\rm{c}}}}{3{M}_{\mathrm{BH}}}\right)}^{\tfrac{1}{3}}r,\end{eqnarray} \tag{ 2 }$$

where r is a distance from the black hole to the clump and M_BH is the black hole mass in the central engine. Using ${\rho }_{0}=3{M}_{{\rm{c}}}/4\pi {R}_{{\rm{c}}}^{3}$, Equation (1) can be reduced to

$$\begin{eqnarray}&&{M}_{{\rm{c}}}=\displaystyle \frac{{\pi }^{2}{c}_{s}^{2}}{3G}{R}_{{\rm{c}}}.\end{eqnarray} \tag{ 3 }$$

Substituting Equation (3) into (2), we obtain

$$\begin{eqnarray}&&{R}_{{\rm{c}}}=\displaystyle \frac{\pi {c}_{s}}{3\sqrt{{{GM}}_{\mathrm{BH}}}}{r}^{3/2}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&=\ 4.8\times {10}^{-3}\left(\displaystyle \frac{{c}_{s}}{3\ \mathrm{km}\,{{\rm{s}}}^{-1}}\right){\left(\displaystyle \frac{r}{1\mathrm{pc}}\right)}^{\tfrac{3}{2}}{\left(\displaystyle \frac{{M}_{\mathrm{BH}}}{{10}^{8}{M}_{\odot }}\right)}^{-\tfrac{1}{2}}\,\mathrm{pc}.\end{eqnarray} \tag{ 5 }$$

It follows from Equations (3) and (4) that the mass of the marginally stable clump is

$$\begin{eqnarray}&&{M}_{{\rm{c}}}=\displaystyle \frac{{\pi }^{3}}{9}\displaystyle \frac{{c}_{s}^{3}}{{G}^{3/2}{M}_{\mathrm{BH}}^{1/2}}\end{eqnarray} \tag{ 6 }$$

$$\begin{eqnarray}&&=\ 33{\left(\displaystyle \frac{{c}_{s}}{3\mathrm{km}{{\rm{s}}}^{-1}}\right)}^{3}{\left(\displaystyle \frac{r}{1\mathrm{pc}}\right)}^{\tfrac{3}{2}}{\left(\displaystyle \frac{{M}_{\mathrm{BH}}}{{10}^{8}{M}_{\odot }}\right)}^{-\tfrac{1}{2}}\,{M}_{\odot }.\end{eqnarray} \tag{ 7 }$$

The average hydrogen number density n_H of one clump is n_H = 3 M_c/4π ${R}_{{\rm{c}}}^{3}$ m_H = 2.6 × 10⁹ cm⁻³, where m_H is the hydrogen mass. In this study, we assume the homogeneous spherical clump for simplicity.

Suppose the size distribution of dust grains obeys the power-law size distribution (Mathis et al. 1977; Draine & Lee 1984),

$$\begin{eqnarray}&&{n}_{i}(a){da}={A}_{i}{n}_{{\rm{H}}}{a}^{p}{da}\ \ ({a}_{\min }\lt a\lt {a}_{\max }),\end{eqnarray} \tag{ 8 }$$

where n(a) is the distribution function of grain size, a is a grain radius, A is a normalization factor and n_H is the number density of H nuclei, and subscript i denotes the silicate or the graphite. Denote E_c,d by the internal energy of a dust clump, then it can be described as follows:

$$\begin{eqnarray}&&{E}_{{\rm{c}},{\rm{d}}}({T}_{{\rm{d}}})={M}_{{\rm{c}},{\rm{g}}}{f}_{{\rm{d}}}\displaystyle \frac{{\sum }_{i}{A}_{i}{\int }_{0}^{{T}_{{\rm{d}}}}{dT}^{\prime} {C}_{i}(T^{\prime} )}{{\sum }_{i}{A}_{i}{\rho }_{i}},\end{eqnarray} \tag{ 9 }$$

where C(T_d) is the heat capacity per unit volume, T is the temperature of a clump, M_c,g is the gas mass of the dust clump, and f_d is a dust-to-gas mass ratio. For the sake of simplicity, we assumed an individual clump has a uniform temperature. That is, noting that the dust temperature at the outer and inner regions within a clump may differ. However, this temperature difference does not significantly alter internal energy and the cooling curve of a clump. Hence, in this paper, we adopt single temperature approximation for an individual clump.

For the model of heat capacities, we adopt Draine & Li (2001),⁵ then

$$\begin{eqnarray}&&{C}_{\mathrm{gra}}=({N}_{C}-2){k}_{B}\left[{f}_{2}^{\prime }\left(\displaystyle \frac{T}{863\,{\rm{K}}}\right)+2{f}_{2}^{\prime }\left(\displaystyle \frac{T}{2504\,{\rm{K}}}\right)\right],\end{eqnarray} \tag{ 10 }$$

$$\begin{eqnarray}&&{C}_{\mathrm{sil}}=({N}_{A}-2){k}_{B}\left[2{f}_{2}^{\prime }\left(\displaystyle \frac{T}{500\,{\rm{K}}}\right)+{f}_{3}^{\prime }\left(\displaystyle \frac{T}{1500\,{\rm{K}}}\right)\right],\end{eqnarray} \tag{ 11 }$$

where

$$\begin{eqnarray}&&{f}_{n}(x)=n{\int }_{0}^{1}\displaystyle \frac{{y}^{n}{dy}}{\exp (y/x)-1},{f}_{n}^{\prime }(x)=\displaystyle \frac{d}{{dx}}{f}_{n}(x).\end{eqnarray} \tag{ 12 }$$

N_C and N_A are the number density of carbon and atoms for graphite and silicate, respectively, assuming ρ_gra = 2.26 g cm⁻³ and ρ_sil = 3.5 g cm⁻³ reduce to N_C = 1.5 × 10²³ cm⁻³ and N_A = 8.5 × 10²² cm⁻³, respectively.

Next, we investigate how the individual clump cools as time goes by. If the dust clump is optically thick, internal energy of the clump will be released through its photosphere, and then the cooling rate will be proportional to the surface area of the clump. Hence, we expect the cooling rate of the optically thick clump to be approximately $4\pi {R}_{{\rm{c}}}^{2}$σ_SBT⁴, where T is the temperature of the clump and σ_SB is a Stefan–Boltzmann constant. On the other hand, if the dust clump is optically thin, the cooling rate should be the same as that of single dust grains. Based on the above consideration, we adopt the energy equation of the individual clump for arbitrary optical depth as follows.

$$\begin{eqnarray}&&\displaystyle \frac{{{dE}}_{{\rm{c}},{\rm{d}}}}{{dt}}=-4\pi {R}_{{\rm{c}}}^{2}\langle {Q}_{{\rm{c}}}{\rangle }_{T}{\sigma }_{\mathrm{SB}}{T}^{4}+{\int }_{{V}_{\mathrm{clump}}}{n}_{{\rm{g}}}{{\rm{\Gamma }}}_{{\rm{g}}-{\rm{d}}}{dV},\end{eqnarray} \tag{ 13 }$$

where R_c is the radius of a clump, ${n}_{{\rm{g}}}{{\rm{\Gamma }}}_{{\rm{g}}-{\rm{d}}}$ is the energy exchange between gas and dust, and $\langle {Q}_{{\rm{c}}}{\rangle }_{T}$ denotes the physical quantity Q_c to be averaged over the Planck function with temperature T. ${Q}_{{\rm{c}}}=[1-{e}^{-{\tau }_{\nu }}]$ is the emission efficiency of a clump, where τ_ν is the optical depth for absorption of a clump defined by

$$\begin{eqnarray}&&{\tau }_{\nu }=\displaystyle \sum _{i}\int {ds}{\int }_{{a}_{\min }}^{{a}_{\max }}{{dan}}_{i}(a){m}_{i}{\kappa }_{i,\nu }(a),\end{eqnarray} \tag{ 14 }$$

where m is the mass of the dust grain, and ${\kappa }_{\nu }(a)$ is the absorption opacity of the dust grain. From Equations (14) and (8), we obtain

$$\begin{eqnarray}&&\displaystyle \frac{{\tau }_{\nu }}{{N}_{{\rm{H}}}}={m}_{{\rm{H}}}{f}_{{\rm{d}}}\displaystyle \frac{{\sum }_{i}{A}_{i}{\rho }_{i}{\overline{\kappa }}_{i,\nu }}{{\sum }_{i}{A}_{i}{\rho }_{i}},\end{eqnarray} \tag{ 15 }$$

where m_H is the hydrogen mass, f_d is the dust-to-gas mass ratio, and

$$\begin{eqnarray}&&{\overline{\kappa }}_{i,\nu }\equiv \displaystyle \frac{\int {{daa}}^{3+p}{\kappa }_{i,\nu }(a)}{\int {{daa}}^{3+p}}.\end{eqnarray} \tag{ 16 }$$

The dust-to-gas mass ratio f_d can be described as, for p$\,\ne$ −4,

$$\begin{eqnarray}&&{f}_{{\rm{d}}}=\displaystyle \frac{4\pi }{3{m}_{{\rm{H}}}}\displaystyle \frac{{a}_{\max }^{4+p}}{4+p}\left[1-{\left(\displaystyle \frac{{a}_{\min }}{{a}_{\max }}\right)}^{4+p}\right]\displaystyle \sum _{i}{A}_{i}{\rho }_{i}.\end{eqnarray} \tag{ 17 }$$

The energy exchange between gas and dust can be written as

$$\begin{eqnarray}&&{n}_{{\rm{g}}}{{\rm{\Gamma }}}_{{\rm{g}}-{\rm{d}}}=\displaystyle \sum _{i}\int {n}_{i}(a){da}4\pi {a}^{2}\displaystyle \frac{{n}_{{\rm{g}}}{v}_{\mathrm{th}}}{4}{\alpha }_{T}(2{k}_{B}{T}_{{\rm{g}}}-2{k}_{B}{T}_{{\rm{d}}}).\end{eqnarray} \tag{ 18 }$$

Using Equation (8), we find

$$\begin{eqnarray}{n}_{{\rm{g}}}{{\rm{\Gamma }}}_{{\rm{g}}-{\rm{d}}} & = & \displaystyle \frac{2\pi }{(3+p)}{k}_{B}{\alpha }_{T}({T}_{{\rm{g}}}-{T}_{{\rm{d}}}){n}_{{\rm{g}}}^{2}{v}_{\mathrm{th}}\\ & & \times {a}_{\min }^{3+p}\left[{\left(\displaystyle \frac{{a}_{\max }}{{a}_{\min }}\right)}^{3+p}-1\right]\displaystyle \sum _{i}{A}_{i},\end{eqnarray} \tag{ 19 }$$

where α_T is the accommodation coefficient, v_th is the thermal velocity of gas, and T_g is the gas temperature. We have averaged over the grain size distribution. In this paper, we assume α_T = 0.15 (e.g., Tielens 2005). Since we have assumed uniform temperature and density structure in the clump, ${n}_{{\rm{g}}}{{\rm{\Gamma }}}_{{\rm{g}}-{\rm{d}}}$ is constant within the cloud. Therefore, we obtain

$$\begin{eqnarray}&&\displaystyle \frac{{{dT}}_{{\rm{d}}}}{{dt}}=\beta \left\{-4\pi {R}_{{\rm{c}}}^{2}\langle {Q}_{{\rm{c}}}{\rangle }_{{T}_{{\rm{d}}}}{\sigma }_{\mathrm{SB}}{{T}_{{\rm{d}}}}^{4}+{n}_{{\rm{g}}}{{\rm{\Gamma }}}_{{\rm{g}}-{\rm{d}}}\cdot \displaystyle \frac{4\pi {R}_{{\rm{c}}}^{3}}{3}\right\}\end{eqnarray} \tag{ 20 }$$

$$\begin{eqnarray}&&\beta =\displaystyle \frac{1}{{M}_{{\rm{c}},{\rm{g}}}{f}_{d}}\displaystyle \frac{{\sum }_{i}{A}_{i}{\rho }_{i}}{{\sum }_{i}{A}_{i}{C}_{i}({T}_{{\rm{d}}})}.\end{eqnarray} \tag{ 21 }$$

The abundance ratio of silicates and graphites is set as A_sil/A_gra = 1.12 (Draine & Lee 1984). This corresponds to 53% of silicate grains, and the other 47% for graphite in volume. The upper and lower cut-off to the size distribution is assumed to be ${a}_{\min }=0.005\,\mu {\rm{m}}$ and a_max = 0.25 μm, and the slope adopted is p = −3.5 (Mathis et al. 1977). Since the dust-to-gas ratio f_d in AGNs is still under debate (e.g., Maiolino et al. 2001), we use the galactic ISM value of f_d = 0.01 (e.g., Tielens 2005) throughout this paper. The dust grain is assumed to be spherical; thus, optical properties can be calculated by using the Mie theory (e.g., Bohren & Huffman 1983). For the dielectric function of silicates and graphites, we adopt Draine & Lee (1984), Laor & Draine (1993), Weingartner & Draine (2001), Draine (2003), Draine & Lee (1984), and Aniano et al. (2012), respectively. Since graphite is a highly anisotropic material, optical properties depend on the direction of incident ${\boldsymbol{E}}$ fields with respect to the basal plane of graphite. We assume randomly oriented graphite grains, and hence, we can use the so-called "$\tfrac{1}{3}-\tfrac{2}{3}$ approximation" (e.g., Draine 1988); κ_gra,ν = 2κ_gra,ν $({\boldsymbol{E}}\perp {\boldsymbol{c}})/3+{\kappa }_{\mathrm{gra},\nu }({\boldsymbol{E}}\ | | \ {\boldsymbol{c}})/3$, where ${\boldsymbol{E}}$ is the incident electric field and ${\boldsymbol{c}}$ is a normal vector to the basal plane of graphite. To estimate the free-electrons contribution to the dielectric function for graphite with ${\boldsymbol{E}}\ | | \ {\boldsymbol{c}}$, we adopt the two-component free-electrons model introduced by Aniano et al. (2012). For free-electron models of ${\boldsymbol{E}}\perp {\boldsymbol{c}}$, we still adopt the model of Draine & Lee (1984).

Since the gas–dust collision term depends on the gas temperature, the energy equation of gas should also be solved. Denote E_c,g by the internal energy of the gas in the clump, then the energy equation for gas can be written as

$$\begin{eqnarray}&&\displaystyle \frac{{{dE}}_{{\rm{c}},{\rm{g}}}}{{dt}}=-{\int }_{{V}_{\mathrm{clump}}}\{{n}_{{\rm{g}}}{{\rm{\Gamma }}}_{{\rm{g}}-{\rm{d}}}+{n}_{{\rm{g}}}^{2}{{\rm{\Lambda }}}_{\mathrm{line}}\}{dV},\end{eqnarray} \tag{ 22 }$$

where ${n}_{{\rm{g}}}{{\rm{\Gamma }}}_{{\rm{g}}-{\rm{d}}}$ and ${n}_{{\rm{g}}}^{2}{{\rm{\Lambda }}}_{\mathrm{line}}$ represent the energy exchange between gas and dust and by line cooling, respectively. Note that ${n}_{{\rm{g}}}{{\rm{\Gamma }}}_{{\rm{g}}-{\rm{d}}}$ has a positive sign when the gas temperature is higher than the dust temperature. We have ignored the compression heating since the free-fall timescale of the clump is longer than the cooling timescale of the clump, e.g., ${t}_{\mathrm{ff}}=\sqrt{3\pi /32G{\rho }_{{\rm{g}}}}=9.6\times {10}^{2}$ year for the clump of R_c ≃ 5 × 10⁻³ pc and M_c ≃ 33 M_☉. The radiative cooling rate of the transition from level u to level l of some species x is written as (e.g., Tielens & Hollenbach 1985)

$$\begin{eqnarray}&&{n}_{{\rm{g}}}^{2}{{\rm{\Lambda }}}_{x}({\nu }_{{ul}})={n}_{u}{A}_{{ul}}h{\nu }_{{ul}}\beta ({\tau }_{{ul}})\{({S}_{x}({\nu }_{{ul}})-P({\nu }_{{ul}}))/{S}_{x}({\nu }_{{ul}})\},\end{eqnarray} \tag{ 23 }$$

where n_u is a population density at level u, $\beta ({\tau }_{{ul}})$ is the espace probability, ${S}_{x}(\nu )$ is the source function, and $P(\nu )$ is the background radiation field. As a background radiation, we assume the ambient thermal radiation from dust grains, and then

$$\begin{eqnarray}&&P({\nu }_{{ul}})=B({\nu }_{{ul}},{T}_{{\rm{d}}})[1-\exp (-{\tau }_{{\rm{d}}})],\end{eqnarray} \tag{ 24 }$$

where τ_d is calculated using Equation (14). Since we have assumed a homogeneous spherical clump, the escape probability averaged over line profile and over the cloud volume can be approximately given by (Draine 2011)

$$\begin{eqnarray}&&\beta ({\tau }_{0})\simeq \displaystyle \frac{1}{1+0.5{\tau }_{0}},\end{eqnarray} \tag{ 25 }$$

where τ₀ is an optical depth at line center, given by

$$\begin{eqnarray}&&{\tau }_{0}=\displaystyle \frac{{g}_{u}}{{g}_{l}}\displaystyle \frac{{A}_{{ul}}{\lambda }_{{ul}}^{3}}{4{(2\pi )}^{3/2}{\sigma }_{V}}{n}_{l}{R}_{{\rm{c}}}\left(1-\displaystyle \frac{{n}_{u}{g}_{l}}{{n}_{l}{g}_{u}}\right),\end{eqnarray} \tag{ 26 }$$

here we assumed that gas motion is a Maxwellian one-dimensional distribution with the velocity dispersion of ${\sigma }_{V}\,={({k}_{B}{T}_{{\rm{g}}}/{m}_{{\rm{H}}})}^{1/2}$.

Gas is assumed to be ideal, and hence, the equation of state becomes ${e}_{{\rm{c}},{\rm{g}}}={n}_{{\rm{g}}}{k}_{B}{T}_{{\rm{g}}}/(\gamma -1)$. As a result, we find

$$\begin{eqnarray}&&\displaystyle \frac{{{dT}}_{{\rm{g}}}}{{dt}}=-\displaystyle \frac{(\gamma -1)}{{n}_{{\rm{g}}}{k}_{B}}\{{n}_{{\rm{g}}}{{\rm{\Gamma }}}_{{\rm{g}}-{\rm{d}}}+{n}_{{\rm{g}}}^{2}{{\rm{\Lambda }}}_{\mathrm{line}}\},\end{eqnarray} \tag{ 27 }$$

where γ is a specific heat ratio and we adopt γ = 5/3. In this paper, we assume the local thermal equilibrium (LTE) to obtain the level populations. As molecular species, we adopt H₂, CO, and H₂O because these molecules are abundant in the AGN dust tori. Note that H₂O becomes abundant at a relatively hot inner region of dust tori, because H₂O molecules form via a neutral–neutral reaction whose energy barrier is ≃1000 K, and the reaction is effective at >300 K (Harada et al. 2010). We adopt the molecular fractional abundances with respect to n_H as ${x}_{{{\rm{H}}}_{2}}=0.5$, x_CO = 3. 0 × 10⁻⁵, and ${x}_{{{\rm{H}}}_{2}{\rm{O}}}=1.4\times {10}^{-4}$ at 3 pc from the black hole (Harada et al. 2010). We adopt the data of line parameters in the Leiden Atomic and Molecular Database LAMDA⁶ (Schöier et al. 2005) and H₂ line parameters in Wagenblast & Hartquist (1988) and Nomura & Millar (2005). We use a part of the RATRAN code⁷ (Hogerheijde & van der Tak 2000) in order to read the molecular data for calculating the line cooling.

In Figure 1, we plot the change of dust and gas temperature of the individual clump. Figure 1 shows that the clump cools down to tens of Kelvin within ∼10 years. Dust temperature drops quickly until t ∼ 10³ s after the heating photon supply ends, and then it reaches a plateau phase, where the radiative cooling and collisional heating of gas particles are balancing. Therefore, dust grains cannot cool down to tens of Kelvin unless the clump gas cools down. The gas temperature starts to decrease at t ∼ 10⁵ s due to molecular line cooling that is mainly dominated by H₂O. Even in the absence of line cooling, the gas clump will be cooled due to dust radiation via gas–dust collisions. As a result, dust temperature drops to tens of Kelvin within ∼10 years. It is worth noting that equilibrium dust temperature at the plateau phase depends on the gas density. This indicates that clumps at the outer region cool down faster than the inner clumps since they have lower gas density. This result indicates that the cooling timescale of dust tori is slightly shorter than the light crossing time with an order of 10 years of the dust torus size. Therefore, we conclude that the dust cooling timescale in the torus is mainly governed by the light crossing time.

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To reproduce the torus SED, we assume smooth distribution of the dust density for simplicity. The SED of AGN torus is obtained by using a two-layer model (Chiang & Goldreich 1997; Chiang et al. 2001). In this model, the optically thick disk is divided into two regions, the surface layer where the dust grains are directly irradiated by the AGN, and the interior layer where the dust grains are indirectly heated by the AGN, in other words, they are heated by the thermal emission of directly irradiated dust grains at the surface layer. The resultant SED can be obtained by superposing the flux from the surface layer (Equation (32)) and interior layer (Equation (33)).

We assume that the torus is axisymmetric, and the surface density of dust grains obeys the simple power-law function, Σ_d = 3.5 × 10⁻² (r/1 pc)⁻¹ g cm⁻². The inner and outer radii were observationally constrained and the AGN luminosity dependence was reported with r_in = 0.4 (L_AGN/10⁴⁵ erg s⁻¹)^0.5 pc (Barvainis 1987; Kishimoto et al. 2011; Koshida et al. 2014), and ${r}_{\mathrm{out}}=10{({L}_{\mathrm{AGN}}/{10}^{45}\mathrm{erg}{{\rm{s}}}^{-1})}^{0.21}$ pc (Kishimoto et al. 2011; García-Burillo et al. 2016; Imanishi et al. 2016), respectively. The opening angle of the torus is assumed to be 45 degrees, and then the grazing angle is $\alpha \approx {r}_{\mathrm{in}}/r$. The opacity of a mixture of silicate and graphite grains is obtained by averaging the absorption efficiency of them weighted for the volume fraction of 53% and 47% for silicate and graphite, respectively.

The temperature of directly irradiated dust grains by the AGN is

$$\begin{eqnarray}&&{T}_{\mathrm{ds}}(r,a)={\left(\displaystyle \frac{{L}_{\mathrm{AGN}}}{16\pi {r}^{2}{\sigma }_{\mathrm{SB}}}\displaystyle \frac{\langle {Q}_{\mathrm{abs}}{\rangle }_{\mathrm{AGN}}}{\langle {Q}_{\mathrm{abs}}{\rangle }_{{T}_{\mathrm{ds}}}}\right)}^{\tfrac{1}{4}},\end{eqnarray} \tag{ 28 }$$

where r is a distance from the AGN, and L_AGN = 10⁴⁵ erg s⁻¹ is the AGN luminosity. $\langle {Q}_{\mathrm{abs}}{\rangle }_{\mathrm{AGN}}$ is an absorption efficiency with respect to the incoming AGN photons, defined by

$$\begin{eqnarray}&&\langle {Q}_{\mathrm{abs}}{\rangle }_{\mathrm{AGN}}=\int {Q}_{\mathrm{abs}}{f}_{\lambda }d\lambda ,\end{eqnarray} \tag{ 29 }$$

f_λ = F_λ/F_AGN, and ${F}_{\mathrm{AGN}}={\int }_{0}^{\infty }{F}_{\lambda }d\lambda$ is the total flux. Using ${F}_{\mathrm{AGN}}={L}_{\mathrm{AGN}}/4\pi {r}^{2}$, the normalized spectrum of the AGN is assumed to be Nenkova et al. (2008),

$$\begin{eqnarray}\lambda {f}_{\lambda }\propto \left\{\begin{array}{lc}{\lambda }^{1.2} & (\lambda \leqslant {\lambda }_{h})\\ {\lambda }^{0} & ({\lambda }_{h}\leqslant \lambda \leqslant {\lambda }_{u})\\ {\lambda }^{-q} & ({\lambda }_{u}\leqslant \lambda \leqslant {\lambda }_{\mathrm{RJ}})\\ {\lambda }^{-3} & ({\lambda }_{\mathrm{RJ}}\leqslant \lambda )\end{array}\right.,\end{eqnarray} \tag{ 30 }$$

where λ_h = 0.01 μm, λ_u = 0.1 μm, λ_RJ = 1 μm, and q = 0.5. The estimated dust temperature at the inner radius is approximately T_ds ≈ 1500 K, which is consistent with the sublimation temperature of the interior of a dust clump. The temperature of interior grain is given by (Chiang et al. 2001)

$$\begin{eqnarray}&&{T}_{\mathrm{di}}(r)={\left\{\displaystyle \frac{{L}_{\mathrm{AGN}}}{8\pi {r}^{2}{\sigma }_{\mathrm{SB}}}\sin \alpha \displaystyle \frac{[1-{e}^{-{{\rm{\Sigma }}}_{{\rm{d}}}\langle \kappa {\rangle }_{{T}_{\mathrm{ds}}}}]}{[1-{e}^{-{{\rm{\Sigma }}}_{{\rm{d}}}\langle \kappa {\rangle }_{{T}_{\mathrm{di}}}}]}\right\}}^{1/4},\end{eqnarray} \tag{ 31 }$$

where κ is a size distribution averaged grain opacity. $\langle \kappa {\rangle }_{{T}_{\mathrm{ds}}}$ is evaluated at the temperature of most luminous grains in the surface. We assume that the temperature of all grains at the midplane is thermally equilibrated.

Once the radial distributions of grain temperature at the surface and interior layer are obtained, the emission spectrum of the AGN torus can be calculated as below. The emission spectrum of the interior layer, ${F}_{\lambda }^{i}$, is

$$\begin{eqnarray}&&4\pi {d}^{2}\lambda {F}_{\lambda }^{i}=8{\pi }^{2}\lambda {\int }_{{r}_{\mathrm{in}}}^{{r}_{\mathrm{out}}}{B}_{\lambda }({T}_{\mathrm{di}})(1-{e}^{-{{\rm{\Sigma }}}_{{\rm{d}}}\kappa }){rdr},\end{eqnarray} \tag{ 32 }$$

where d is the luminosity distance. The emission spectrum of the surface layer, ${F}_{\lambda }^{s}$, is

$$\begin{eqnarray}&&4\pi {d}^{2}\lambda {F}_{\lambda }^{s}=8{\pi }^{2}\lambda {\int }_{{r}_{\mathrm{in}}}^{{r}_{\mathrm{out}}}(1+{e}^{-{{\rm{\Sigma }}}_{{\rm{d}}}\kappa }){S}_{\lambda }(1-{e}^{-{\tau }_{s}}){rdr},\end{eqnarray} \tag{ 33 }$$

where S_λ and τ_s are the source function and optical depth given by (e.g., Chiang et al. 2001)

$$\begin{eqnarray}&&{S}_{\lambda }=\displaystyle \frac{2{\int }_{{a}_{\min }}^{{a}_{\max }}{B}_{\lambda }({T}_{\mathrm{ds}})n(a){a}^{2}{Q}_{\mathrm{abs}}(a,\lambda ){da}}{{\int }_{{a}_{\min }}^{{a}_{\max }}n(a){a}^{2}{Q}_{\mathrm{abs}}(a,\lambda ){da}}\end{eqnarray} \tag{ 34 }$$

$$\begin{eqnarray}&&{\tau }_{s}=\displaystyle \frac{{\int }_{{a}_{\min }}^{{a}_{\max }}n(a){a}^{2}{Q}_{\mathrm{abs}}(a,\lambda ){da}}{{\int }_{{a}_{\min }}^{{a}_{\max }}n(a){a}^{2}\langle {Q}_{\mathrm{abs}}{\rangle }_{\mathrm{AGN}}{da}}\sin \alpha .\end{eqnarray} \tag{ 35 }$$

The dashed gray line in Figure 2 shows the SED for the equilibrium temperature structure of the dust torus.

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The clump is optically thick in the shorter wavelength and optically thin in the longer wavelength than the mid-infrared emission domain. The dust clump at the midplane is heated by the near-infrared emission of the dust clump at the surface layer and reradiates its energy in mid- to far-infrared wavelengths. In the far-infrared wavelength, since a clump is optically thin, most of the photons emitted from the midplane escape without experiencing significant absorption. At the surface optically thin layer, clumps are sparsely distributed, and hence, the radiative interaction between clumps at the surface layer does not frequently occur. Based on the above considerations, we assume that the cooling timescale of the torus is mainly governed by that of each clump.

We calculate the time evolution of the torus SED after the central AGN is quenched. We assume that the temperature of dust tori given by Equations (28) and (31) will be cooled according the characteristic cooling curve of the individual clump defined by Equations (20), (21), and (27). In addition, to take into account the light crossing time, the surface and midplane temperature at radius r starts to decrease at t = r/c and t = 2r/c after the AGN is quenched, respectively. The factor of two in the latter is inserted so that the light crossing time of the vertical direction is taken into account as well as the radial direction. Since cooling timescale depends on the clump density, or the black hole mass, we derive the black hole mass by setting the initial AGN luminosity as 5% of the Eddington luminosity (e.g., Kelly et al. 2010).

Figure 2 shows the time dependence of the AGN torus SED with a face-on view after the quenching of the central engine with the initial AGN luminosity of L_AGN = 10⁴⁵ erg s⁻¹. Once the high-energy photons stop being emitted, individual clumps cool down rapidly, and the light crossing time of the torus, ${r}_{\mathrm{out}}/c\approx 30\,\mathrm{year}$, governs the cooling timescale. In Figure 3, we plot the change of mid-infrared 12 μm emission flux as a function of time since the central engine is quenched. The torus in higher luminosity AGNs shows a longer cooling time because of the larger torus outer radius as shown in Section 2.3.

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To search for the dying AGN candidates discussed above, a longer and more stable timescale AGN indicator than the torus thermal emission is necessary to compare with the torus mid-infrared luminosity. NLR is a promising tool because its size is generally 10^2–4 pc, and the [O iii]λ5007 emission line is one of the good NLR indicators hosting a strong luminosity correlation with AGN indicators (Ueda et al. 2015). A promising method is to cross-match optically selected type-2 AGNs obtained by the Sloan Digital Sky Survey (SDSS; York et al. 2000) with Allwise catalogs (Wright et al. 2010). SDSS type-2 AGNs have prominent [O iii]λ5007 emission and Allwise will give us MIR 12 or 22 μm emission. Based on the luminosity relations of AGNs between [O iii]λ5007 and 12 μm luminosity (Toba et al. 2014), $\mathrm{log}({L}_{[{\rm{O}}{\rm{III}}]}/\mathrm{erg}\,{{\rm{s}}}^{-1})\sim 41$ is equivalent to $\mathrm{log}({L}_{12\mu {\rm{m}}}/\mathrm{erg}\,{{\rm{s}}}^{-1})\sim 44$, which is as luminous as QSO as shown with the gray dashed line in Figure 4. Using the time evolution function of 12 μm luminosity in Figure 3, we calculate the time evolution of the luminosity relation of [O iii]λ5007 and 12 μm luminosity in Figure 4. The AGN located at the bottom right of Figure 4 would be a prominent candidate of those dying AGNs. This study could be achievable for type-2 AGNs with the redshift range of z < 0.3, where the optical line diagnostics of AGNs can be applied (e.g., Kewley et al. 2006). Further studies using the method above will be discussed in a forthcoming paper (K. Ichikawa et al. 2017, in preparation).

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The ongoing Subaru HSC (Miyazaki et al. 2012) SSP deep survey (Aihara et al. 2017a, 2017b), covering 28 deg², will also be a promising region for finding dying AGNs using [O iii] emitters at z ∼ 0.6 and ∼0.8 obtained with a narrow band filter NB816, NB921, respectively (Hayashi et al. 2017). Again, if there are sources with extremely low ratios of ${L}_{\mathrm{12,22}\mu {\rm{m}}}/{L}_{[{\rm{O}}{\rm{III}}]}$, this could be a prominent candidate of high-z dying AGNs, which cannot be covered with the method of SDSS survey above. In this case, optical or near-infrared spectral follow-up is crucial to disentangle the [O iii] emitters into starburst galaxies and AGNs (e.g., Kewley et al. 2006).