Model description

There is extensive evidence that dust exists around Active Galactic Nuclei (AGN) and forms a toroidal shape. This material is most likely distributed in clumps, instead of homogeneously filling the torus volume. Numerous efforts to model the torus as a homogeneous dust distribution could not explain some puzzling features of the observed dust continuum emission (please see the corresponding references in the supplied publications).

Here we present our first attempts to model a distribution of dusty clumps in an AGN environment. The model is described in these publications.

Geometry

Torus Geometry

We consider a clumpy distribution with a radial extent given by \(Y = R_o / R_d\), the ratio of the outer to inner radii. The inner radius \(R_d\) is defined by dust sublimation temperature.

The radial density profile is a power law \(r^{-q}\). Various angular distributions, characterized by a width parameter \(m\), are considered. For smooth boundaries, the parameter \(m\) determines the smoothness. E.g. \(m=2\) is a Gaussian \(\propto \exp{(-|\beta/\sigma|^m)}\), with a width \(\sigma\), where the angle \(\beta\) is measured from the equatorial plane. The total number of clouds, on average, along any radial equatorial ray, is parametrized by \(N_0\), and the variation of the total number of clouds along rays at angle \(\beta\) from the equator is \(N_T(\beta) = N_0\, F(\beta)\). \(F\) is a step function at \(\beta = \sigma\) for a sharp-edged torus, and \(\exp{(-|\beta/\sigma|^m)}\) for soft-edged tori.

Each clump is specified by its optical depth, and all clumps are assumed to have the same optical depth \(\tau_V\), the depth at \(0.55\mu m\). The dust extinction profile is taken as for standard MRN Galactic dust.

The clumps, of a given dust composition, are heated by an AGN with given spectral shape and luminosity L. For such a configuration, the inner radius \(R_d\) is uniquely determined by the luminosity and the chosen dust sublimation temperature:

\begin{equation*} R_d = 0.4 \left( \frac{L}{10^{45} \mathrm{erg\ s^{-1}}}\right)^{1/2} \left(\frac{\mathrm{T_{sub}}}{1500\ \mathrm{K}}\right)^{-2.6}\ \mathrm{pc} \end{equation*}

Dust & heating

The emission of directly and indirectly heated clumps at various positions around the AGN is calculated by our SFN program using calculations of both slabs and spherical shells modeled by our radiative transfer code DUSTY https://faculty.washington.edu/ivezic/dusty_web/. These two components of the source function are used to calculate the overall emission from the distribution of clumps by the separate code CLUMPY.

We provide our current model spectra of a clumpy torus. We assume standard ISM chemical dust composition with 53% standard Galactic silicate dust after Ossenkopf, Henning & Mathis (1992), and 47% graphites after Draine (2003). The grains are spherical with size distribution from Mathis, Rumpl & Nordsieck (1977). The dust sublimation temperature of this composition is \(T_{\mathrm{sub}} = 1500\ \mathrm{K}\).

The spectral slope of the heating AGN radiation \(\lambda f_\lambda\) follows a piecewise (broken) power-law

\begin{equation*} \lambda f_\lambda \propto \begin{cases} \lambda^{1.2} & \lambda \le \lambda_h\\ \mathrm{const} & \lambda_h \le \lambda \le \lambda_u\\ \lambda^{-p} & \lambda_u \le \lambda \le \lambda_{RJ}\\ \lambda^{-3} & \lambda_{RJ} \le \lambda \end{cases} \end{equation*}

We provide models with and without the contribution of the AGN heating spectrum.