Dust Destruction by Charging: A Possible Origin of Gray Extinction Curves of Active Galactic Nuclei

The radiation spectra of AGNs are taken from Nenkova et al. (2008), and the bolometric luminosity is assumed to be ${L}_{\mathrm{AGN}}={10}^{45}\,\mathrm{erg}\,{{\rm{s}}}^{-1}$. For convenience, we define ${L}_{45}=({L}_{\mathrm{AGN}}/{10}^{45}\,\mathrm{erg}\,{{\rm{s}}}^{-1})$. Since we focus on grains at the polar region, where grains are thought to be irradiated by AGN radiation directly, we ignore the attenuation of AGN radiation. At the parsec-scale polar region, radiation-hydrodynamic simulations suggest that the gas temperature and density are about ${T}_{{\rm{g}}}\approx {10}^{4}$ K and ${n}_{{\rm{H}}}\approx 10-{10}^{3}\,{\mathrm{cm}}^{-3}$, respectively (Wada et al. 2016). Hence, we adopt ${T}_{{\rm{g}}}={10}^{4}$ K and ${n}_{{\rm{H}}}={10}^{2}\,{\mathrm{cm}}^{-3}$ as a fiducial set of parameters. For the sake of simplicity, we assume ${T}_{{\rm{g}}}$ and ${n}_{{\rm{H}}}$ are constants.

Dust grains become positively charged in the AGN environments (Weingartner et al. 2006). We compute grain charge ${Z}_{{\rm{d}}}$ (in the electron charge unit) by solving the rate equation (Weingartner et al. 2006; Tazaki & Ichikawa 2020):

$$\begin{eqnarray}&&\displaystyle \frac{{{dZ}}_{{\rm{d}}}}{{dt}}={J}_{\mathrm{pe}}-{J}_{{\rm{e}}}+{J}_{\sec ,\mathrm{gas}}+{J}_{\mathrm{ion}},\end{eqnarray} \tag{ 1 }$$

where ${J}_{\mathrm{pe}}$ is the photoelectric emission rate (Weingartner & Draine 2001b; Weingartner et al. 2006), ${J}_{{\rm{e}}}$ and ${J}_{\mathrm{ion}}$ are the collisional charging rate of electrons and ions, respectively (Draine & Sutin 1987), and ${J}_{\sec ,\mathrm{gas}}$ is the rate for secondary electron emission induced by incident gas-phase electrons (Draine & Salpeter 1979). A typical charging timescale of a neutral grain due to electron collisions is ${J}_{{\rm{e}}}^{-1}\sim 0.9\,{\rm{s}}$ (Draine & Sutin 1987), where ${T}_{{\rm{g}}}={10}^{4}$ K, the grain radius $a=0.1$ μm, the electron density ${n}_{e}={10}^{2}\,{\mathrm{cm}}^{-3}$, and the sticking probability of 0.5 (Draine & Sutin 1987) are used. Since the charging timescale is much shorter than the dynamical timescale, we can assume the steady state in Equation (1). In addition, we can also ignore grain-charge distribution because the single-charge equilibrium approximation gives reliable results for highly charged grains (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 ${Z}_{{\rm{d}}}$.

It is useful to introduce the ionization parameter, ${U}_{\mathrm{ion}}\equiv {n}_{\gamma }/{n}_{{\rm{H}}}$, where n_γ is the total photon number density beyond 13.6 eV. Since the grain charge is mostly determined by the balance of photoelectric emission and electron collisions, U_ion characterizes the grain-charge amount. For the radiation spectra of AGNs used in Nenkova et al. (2008), we obtain

$$\begin{eqnarray}&&{n}_{\gamma }=2.61\times {10}^{6}\,{\mathrm{cm}}^{-3}\left(\displaystyle \frac{{L}_{\mathrm{AGN}}}{{10}^{45}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right){\left(\displaystyle \frac{r}{\mathrm{pc}}\right)}^{-2},\end{eqnarray} \tag{ 2 }$$

where r is the distance from the central engine of the AGN. As a result, the ionization parameter becomes

$$\begin{eqnarray}&&{U}_{\mathrm{ion}}=2.61\times {10}^{4}{\left(\displaystyle \frac{{n}_{{\rm{H}}}}{{10}^{2}{\mathrm{cm}}^{-3}}\right)}^{-1}\left(\displaystyle \frac{{L}_{\mathrm{AGN}}}{{10}^{45}\,\mathrm{erg}\,{{\rm{s}}}^{-1}}\right){\left(\displaystyle \frac{r}{\mathrm{pc}}\right)}^{-2}.\end{eqnarray} \tag{ 3 }$$

The grain temperature, ${T}_{{\rm{d}}}$, is obtained from the radiative equilibrium:

$$\begin{eqnarray}&&\displaystyle \frac{{L}_{\mathrm{AGN}}}{4\pi {r}^{2}}\pi {a}^{2}\langle {Q}_{\mathrm{abs}}{\rangle }_{\mathrm{AGN}}=4\pi {a}^{2}{\sigma }_{\mathrm{SB}}{T}_{{\rm{d}}}^{4}\langle {Q}_{\mathrm{abs}}{\rangle }_{{T}_{{\rm{d}}}},\end{eqnarray} \tag{ 4 }$$

where r is the distance from the AGN, $\langle {Q}_{\mathrm{abs}}{\rangle }_{\mathrm{AGN}}$ and $\langle {Q}_{\mathrm{abs}}{\rangle }_{{T}_{{\rm{d}}}}$ are the AGN-spectrum averaged absorption efficiency and Planck mean absorption efficiency at dust temperature ${T}_{{\rm{d}}}$, respectively, and ${\sigma }_{\mathrm{SB}}$ is the Stefan–Boltzmann constant. The absorption efficiency, ${Q}_{\mathrm{abs}}$, is calculated by using the Mie theory (Bohren & Huffman 1983), and the optical constant of silicate grains was taken from Draine & Lee (1984), Laor & Draine (1993), and Draine (2003). For graphite, the optical constant is calculated by adding the interband and free electron contributions (see also Draine & Lee 1984; Laor & Draine 1993; Draine 2003), where we adopt free electron models of Aniano et al. (2012). If the size parameter $x=2\pi a/\lambda$, 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.

We consider two kinds of dust destruction: Coulomb explosion (Section 2.4.1) and thermal sublimation (Section 2.4.2).

2.4.1. Coulomb Explosion

If a dust grain acquires a large amount of positive charges, Coulomb repulsion force within the grain causes grain fission, so-called Coulomb explosion (Draine & Salpeter 1979). The condition for Coulomb explosion is (Draine & Salpeter 1979)

$$\begin{eqnarray}&&S=\displaystyle \frac{1}{4\pi }{\left(\displaystyle \frac{U}{a}\right)}^{2}\geqslant {S}_{\max },\end{eqnarray} \tag{ 5 }$$

where S is the tensile stress in a charged sphere, $U={Z}_{{\rm{d}}}e/a$ is the electrostatic grain potential, and ${S}_{\max }$ is the tensile strength of the material. In the following, we use the normalized tensile strength defined by ${\hat{S}}_{\max ,9}=({S}_{\max }/{10}^{9}\,\mathrm{erg}\,{\mathrm{cm}}^{-3})$.

If the grain potential satisfies $U(a)\geqslant a{\left(4\pi {S}_{\max }\right)}^{1/2}$, the electric stress exceeds the tensile strength of a grain, and then Coulomb explosion will occur. We define the critical grain radius for Coulomb explosion, ${a}_{\mathrm{CE}}$, such that $U({a}_{\mathrm{CE}})={a}_{\mathrm{CE}}{\left(4\pi {S}_{\max }\right)}^{1/2}$. Grains smaller than ${a}_{\mathrm{CE}}$ are subjected to Coulomb explosion. Coulomb explosion will produce fragments of smaller grains. However, smaller fragments will be also charged enough to cause Coulomb explosion. Hence, we expect that a cascade fragmentation of grains due to Coulomb explosion occurs.

The tensile strength of a dust grain depends on material properties, such as composition and crystallinity. Although the tensile strength of cosmic dust particles is highly uncertain, laboratory measurements give us an estimate (see Hoang et al. 2019 for a summary of tensile strength). The tensile strength of graphite (polycrystalline) is suggested to be about ${\hat{S}}_{\max ,9}=0.5-1$ (Burke & Silk 1974). Hence, in this study, we adopt the conservative value of ${\hat{S}}_{\max ,9}=1$ for graphite grains. The tensile strength of forsterite (silicate) can be as small as ${\hat{S}}_{\max ,9}=1.21$ (Gouriet et al. 2019). Meanwhile, MacMillan (1972) reported the tensile strength of glass rods and fibers is about ${\hat{S}}_{\max ,9}=130$. We adopt the values of ${\hat{S}}_{\max ,9}=10$ for silicate. It is worth noting that the above measurements are based on bulk materials, and therefore small grains may have different values of tensile strength. Since the tensile strength is an uncertain parameter, we discuss how ${S}_{\max }$ changes our results in Section 3.

2.4.2. Thermal Sublimation

We first solve Equation (1) with the steady-state assumption, and the results are presented in Figure 1. As a general tendency, higher ${U}_{\mathrm{ion}}$ and ${T}_{{\rm{g}}}$ give larger grain electrostatic potential. Our calculations quantitatively agree with Weingartner et al. (2006), although assumed radiation spectra are not the same.

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Figure 1 shows that the grain potential depends on the grain radius and becomes maximum at the submicron sizes, while grain charge ${Z}_{{\rm{d}}}$ is a monotonically increasing function of grain radius. This is mainly determined by two competing effects: photoelectric yield and photon absorption efficiency. With decreasing grain radius, the photoelectric yield is increased due to the small particle effect (e.g., Watson 1973; Draine 1978). Hence, smaller grains are more likely to emit photoelectrons once a high-energy photon is absorbed. However, decreasing grain radius reduces the absorption efficiency of photons, once the grain radius is smaller than the incident radiation wavelength, which is typically about $\lambda \sim 0.1$ $\mu {\rm{m}}$. Hence, due to lower photon absorption efficiency, the photoelectric emission rate is decreased, and therefore the grain potential is decreased for smaller grains ($a\lesssim 0.1\,\mu {\rm{m}}$). Because of these two effects, the grain potential is maximized at submicron sizes. Although the grain potential also depends on the grain composition, silicate and graphite grains have almost similar grain potential.

Figure 1 also shows that the critical grain size for Coulomb explosion is about ${a}_{\mathrm{CE}}\sim 0.01-0.1\,\mu {\rm{m}}$ for ${U}_{\mathrm{ion}}={10}^{2}-{10}^{4}$ and ${T}_{{\rm{g}}}={10}^{4}-{10}^{6}$ K at ${\hat{S}}_{\max ,9}=1$.

Next, we compare ${a}_{\mathrm{CE}}$ and ${a}_{\mathrm{sub}}$ as well as their radial dependence from the center of the AGN. Figure 2 shows the critical grain radii ${a}_{\mathrm{sub}}$ and ${a}_{\mathrm{CE}}$ for both silicate and graphite composition as a function of the distance from the AGN. It is found that ${a}_{\mathrm{CE}}$ has shallower radial dependence than ${a}_{\mathrm{sub}}$. The radial dependence of the critical grain radius for sublimation is about ${a}_{\mathrm{sub}}\propto {r}^{-2}$. As long as the Rayleigh approximation is valid, that is, the wavelength of thermal emission is longer than the grain radius, we have $\langle {Q}_{\mathrm{abs}}{\rangle }_{{T}_{{\rm{d}}}}\propto a$. Hence, for a fixed dust temperature (${T}_{{\rm{d}}}={T}_{\mathrm{sub}}$), Equation (4) results in ${a}_{\mathrm{sub}}\propto {r}^{-2}$. In other words, ${a}_{\mathrm{sub}}$ is proportional to the radiative flux from the AGN.

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Grain charging seems to be caused also by the radiation flux, since a grain is charged via photoelectric emission. However, grain potential is not proportional to ${r}^{-2}$. For example, in Figure 1, even if ${U}_{\mathrm{ion}}$ (∝ radiative flux) decreases by an order of magnitude, grain potential U decreases only by a factor of few. These results suggest that Coulomb explosion can be important at larger distances, e.g., parsec-scale distance.

Figure 2 also shows that a difference in ${a}_{\mathrm{CE}}$ between graphite and silicate grains is not so large compared to the difference seen in ${a}_{\mathrm{sub}}$. Thus, Coulomb explosion tends to remove both silicate and graphite grains of almost similar grain radii. Meanwhile, for sublimation, the graphite grains have smaller ${a}_{\mathrm{sub}}$ because (1) the emissivity at near-infrared wavelength is higher, and (2) ${T}_{\mathrm{sub}}$ is higher (see also Laor & Draine 1993; Baskin & Laor 2018). Therefore, this suggests that thermal sublimation preferentially removes small silicate grains rather than graphite grains.

To understand how thermal sublimation or Coulomb explosion change extinction curves, we compute an extinction cross section of grains. We assume that the grain size distribution obeys ${{dn}}_{i}\,\propto \,{{ \mathcal A }}_{i}{a}^{-3.5}{da}$ (${a}_{\min }^{i}\leqslant a\leqslant {a}_{\max }$), where dn_i is the number density of dust grains of type i (silicate or graphite) in a size range $[a,a+{da}]$, and ${{ \mathcal A }}_{i}$ is the abundance of the grain type i (Draine & Lee 1984). We set ${a}_{\min }^{i}=\max ({a}_{\mathrm{sub}}^{i}$, ${a}_{\mathrm{CE}}^{i}$, ${a}_{\min ,\mathrm{MRN}}$), where ${a}_{\min ,\mathrm{MRN}}=0.005$ μm. We treat ${a}_{\max }$ as a free parameter. We also set the abundance of silicate and graphite from Draine & Lee (1984). The extinction magnitude at wavelength λ, A_λ, is

$$\begin{eqnarray}&&{A}_{\lambda }\propto \sum _{i}\int {C}_{\mathrm{ext}}^{i}(\lambda ,a)\displaystyle \frac{{{dn}}_{i}}{{da}}{da},\end{eqnarray} \tag{ 6 }$$

where ${C}_{\mathrm{ext}}^{i}(\lambda ,a)$ is the extinction cross section of a grain of type i with radius a. We define the extinction curve as $E(\lambda -V)/E(B-V)=({A}_{\lambda }-{A}_{V})/({A}_{B}-{A}_{V})$, where A_V and A_B are the extinctions at visual (5500 Å) and blue (4400 Å) wavelengths.

Figure 3 shows the extinction curves with/without Coulomb explosion at $3{L}_{45}^{\tfrac{1}{2}}$ pc away from the nucleus. The extinction curve with Coulomb explosion can successfully reproduce a flat extinction curve as well as the absence of a 2175 Å bump. In addition, the predicted extinction curve is consistent with the observation by Gaskell & Benker (2007). Meanwhile, the extinction curve without Coulomb explosion, or ${a}_{\min }^{i}=\max ({a}_{\mathrm{sub}}^{i},{a}_{\min ,\mathrm{MRN}})$, shows a prominent 2175 Å bump. This is because sublimation does not remove small graphite grains, although small silicate grains are removed (Figure 2). Since observed AGN extinction curves do not show such a bump (Czerny et al. 2004; Gaskell et al. 2004; Gaskell & Benker 2007), the sublimation model is insufficient to reproduce observed extinction curves.

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Figure 4 shows extinction curves at various radial distances. Even if the radial distances are changed, the overall shape of the extinction curve with Coulomb explosion is still similar to the one from Gaskell & Benker (2007) up to $r\simeq 10{L}_{45}^{\tfrac{1}{2}}$ pc, while the thermal sublimation model remains inconsistent with observations. When increasing the distance from the AGN, ${U}_{\mathrm{ion}}$ decreases, and then smaller grains can survive from Coulomb explosion; nevertheless, 2175 Å is still weak for the model with Coulomb explosion.

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Extinction curves with Coulomb explosion are not sensitive to the choice of maximum grain radius as long as it is larger than $0.25\,\mu {\rm{m}}$ (Figure 5). When grain radius is smaller this value, the extinction curve steeply increases with decreasing wavelength.

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Extinction curves with Coulomb explosion are found to be similar to those observed by Gaskell & Benker (2007). In addition, if the maximum grain radius is smaller than $0.25$ μm, the curves become similar to those observed by Czerny et al. (2004). However, our model fails to explain observations by Gaskell et al. (2004). Explaining such extinction curves might require an additional mechanism, such as a reduced-graphite abundance (Gaskell et al. 2004).

We have shown that the Coulomb explosion can be a more important process for dust destruction than thermal sublimation. Meanwhile, near-IR dust reverberation mapping observations for AGNs, which show that the color temperatures of the variable hot dust emission agree with the dust sublimation temperature (1400 − 2000 K), and the innermost radius of the dust torus ${R}_{\mathrm{in}}$ is proportional to the square root of the AGN luminosity (${R}_{\mathrm{in}}\propto {L}_{\mathrm{AGN}}^{1/2}$), strongly suggest that the dust innermost radius is determined by the thermal sublimation (Koshida et al. 2014; Yoshii et al. 2014; Minezaki et al. 2019; Gravity Collaboration et al. 2020 and references therein). This apparent conflict can be attributable to the difference of gas density.

While the ambient gas density of ${n}_{{\rm{H}}}={10}^{2}\,{\mathrm{cm}}^{-3}$ assumed throughout the calculations in this work is a reasonable assumption for the outflowing gas at the polar region of AGNs, the gas density of the equatorial dust torus is expected to be much higher. The gas at the innermost part of the dust torus can be as dense as broad emission line regions, where ${n}_{{\rm{H}}}\sim {10}^{10}\,{\mathrm{cm}}^{-3}$ (e.g., Baskin & Laor 2018; Kokubo & Minezaki 2020); in such a dense gas region, higher dust sublimation temperatures are expected (${T}_{\mathrm{sub}}=1880$ K and 1503 K for graphite and silicate grains, respectively), and the importance of the Coulomb explosion relative to the thermal sublimation is significantly reduced due to inefficient grain charging by the enhanced electron collision rate (see Section 2.2). Therefore, unlike in the case of the polar dust region, the dust destruction at the innermost region of the equatorial dust torus must be governed by the thermal sublimation.

Coulomb explosion depends on the tensile strength assumed. Figure 6 shows how the tensile strength affects extinction curves. In Figure 6, both silicate and graphite grains are assumed to have the same tensile strength. As the tensile strength increases, Coulomb explosion becomes inefficient, and then both small silicate and graphite grains can survive. As a result, extinction curves show an increase toward shorter wavelengths with a prominent 2175 Å bump. Hence, observed extinction curves, e.g., a lack of a 2175 Å bump, seems to be reproduced when ${\hat{S}}_{\max ,9}\lesssim 10$, and this is within a range of measured values for bulk materials (e.g., Section 2.4.1).

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Under the assumption of the tensile strength for bulk materials determined by laboratory experiments, the Coulomb explosion leads to the absence of small dust grains in close vicinity to AGNs and thus can naturally explain the observed flat extinction curve. Conversely, if the tensile strength of the cosmic dust is far stronger than that for the bulk materials due to, e.g., crystallization by annealing for hot small grains, the graphite grains survive even under the large electrostatic potential and a 2175 Å bump feature is unavoidable. Therefore, if our scenario for the flat extinction curve by the Coulomb explosion is true, it also suggests that the tensile strength of the cosmic dust must be close to the value of the bulk materials. However, we should keep in mind that if PAH nanoparticles are the carrier of a 2175 Å bump, the bump might be suppressed by destroying these particles as suggested by observations (e.g., Sturm et al. 2000).