What Can We Learn about Compton-Thin AGN Tori from Their X-ray Spectra? *

Abstract

We have developed a Monte Carlo code for simulation of X-ray spectra of active galactic nuclei (AGN) based on a model of a clumpy obscuring torus. Using this code, we investigate the diagnostic power of X-ray spectroscopy of obscured AGN with respect to the physical properties and orientation of the torus, namely: the average column density, $\langle N_{\textrm{H}}\rangle$, the line-of-sight column density, $N_{\textrm{H}}$, the abundance of iron, $A_{\textrm{Fe}}$, the clumpiness (i.e., the average number of gas clouds along the line of sight), $\langle N\rangle$, and the viewing angle, $\alpha$. In this first paper of a series, we consider the Compton-thin case, where both $\langle N_{\textrm{H}}\rangle$ and $N_{\textrm{H}}$ do not exceed $10^{24}$ cm${}^{-2}$. To enable quantitative comparison of the simulated spectra, we introduce five measurable spectral characteristics: the low-energy hardness ratio (ratio of the continuum fluxes in the 7–11 and 2–7 keV energy bands), the high-energy hardness ratio (ratio of the continuum fluxes in the 10–100 and 2–10 keV energy bands), the depth of the iron K absorption edge, the equivalent width of the Fe K$\alpha$ line, and the fraction of the Fe K$\alpha$ flux contained in the Compton shoulder. We demonstrate that by means of X-ray spectroscopy it is possible to tightly constrain $\langle N_{\textrm{H}}\rangle$, $N_{\textrm{H}}$, and $A_{\textrm{Fe}}$ in the Compton-thin regime, while there is degeneracy between clumpiness and viewing direction.

Appendices

APPENDIX A

1.1 IRON $K$ ABSORPTION EDGE

We have found that the shape of the spectral continuum near the Fe$K$ absorption edge can be well described (see examples in Fig. 14) by the simple expression

$$F_{E}(E)=\begin{cases}F_{E,0}(E),\quad E\lessapprox E_{K}\\ F_{E,0}(E)N_{\textrm{Fe}},\quad E\gtrapprox E_{K},\end{cases}$$

(A1)

where $E_{K}=7124$ eV is the energy of the Fe$K$ edge (Verner et al. 1996) and $F_{E,0}(E)$ is a linear fitting function (which depends on the particular spectrum).

We fit the spectral continuum with this function (using the $\chi^{2}$ criterion) simultaneously on both sides of the edge in the energy ranges from 6500 to 6925 eV and from 7140 to 7410 eV. These bands were chosen by trial and error so that (i) they are sufficiently broad to provide enough simulated photons for a precise determination of $1-N_{\textrm{Fe}}$, (ii) to exclude the innermost region around $E_{K}$, where the spectral shape can be significantly affected by Compton recoil (the effective width of this region below $E_{K}$ can be estimated as $\Delta E\sim 2E_{K}(E_{K}/511\,{\rm keV})\sim 200$ eV) and also slightly by inverse Compton scattering (within $\Delta E\sim E_{K}\sqrt{2kT_{\textrm{e}}/511\,{\rm keV}}\sim 15$ eV around $E_{K}$, given the adopted electron temperature $kT_{\textrm{e}}=1$ eV; see, e.g., Sazonov and Sunyaev 2000), and (iii) fluorescent lines of Fe and Ni do not fall into these bands. The last issue is not important for our numerical results (since we can readily separate spectral continuum from line emission) but can be important in applying a similar fitting procedure to actual X-ray spectroscopy data.

APPENDIX B

1.1 FITTING FUNCTIONS

It is useful to describe some of the trends that we have found for the key spectral characteristics by approximate analytical functions of parameters of the obscurer. Specifically, in application to the $1-N_{\textrm{Fe}}$ vs. $H_{\textrm{low}}$ diagram (Fig. 10), we can define the following iso-$A_{\textrm{Fe}}$ and iso-$N_{\textrm{H}}$ relations:

$$1-N_{\textrm{Fe}}=H_{\textrm{low}}^{v_{1}(A_{\textrm{Fe}})}+0.5-$$

(B1)

$${}-v_{2}(A_{\textrm{Fe}})\left(H_{\textrm{low}}^{v_{1}(A_{\textrm{Fe}})}+0.5\right)^{2}-v_{3}(A_{\textrm{Fe}}),$$

$$1-N_{\textrm{Fe}}=w_{1}(N_{\textrm{H}})H_{\textrm{low}}+w_{2}(N_{\textrm{H}}),$$

where

$$v_{1}(A_{\textrm{Fe}})=0.21\sqrt{A_{\textrm{Fe}}}+0.47A_{\textrm{Fe}}-0.07,$$

$$v_{2}(A_{\textrm{Fe}})=-0.19\sqrt{A_{\textrm{Fe}}}+0.07A_{\textrm{Fe}},$$

$$v_{3}(A_{\textrm{Fe}})=0.48\sqrt{A_{\textrm{Fe}}}-1.52,$$

$$w_{1}(N_{\textrm{H}})=32.3\sqrt{\eta}+2.8\eta^{0.3}-11.14\eta-27,$$

$$w_{2}(N_{\textrm{H}})=122.9\sqrt{\eta}-162.7\eta^{0.3}-14.9\eta+57.6,$$

$$\eta=N_{\textrm{H}}/10^{22}\,{\rm cm}^{-2}.$$

Table 1. Numerical values of $b_{0}(N_{\textrm{H}},\langle N_{\textrm{H}}\rangle)$

$\langle N_{\textrm{H}}\rangle\backslash N_{\textrm{H}}$	$2\times 10^{23}$	$5\times 10^{23}$	$10^{24}$
$10^{23}$	$-3.52$	Not-fitted	Not-fitted
$2\times 10^{23}$	$-3.54$	$-4.5$	Not-fitted
$5\times 10^{23}$	$-3.63$	$-4.6$	$-5.36$
$10^{24}$	$-3.88$	$-4.8$	$-5.56$

Table 2. Numerical values of $b_{1}(N_{\textrm{H}},\langle N_{\textrm{H}}\rangle)$

$\langle N_{\textrm{H}}\rangle\backslash N_{\textrm{H}}$	$2\times 10^{23}$	$5\times 10^{23}$	$10^{24}$
$10^{23}$	$-0.87$	Not-fitted	Not-fitted
$2\times 10^{23}$	$-0.91$	$-0.14$	Not-fitted
$5\times 10^{23}$	$-0.83$	$-0.11$	$0.03$
$10^{24}$	$-0.32$	$-0.07$	$0.05$

Similarly, the location on the $1-N_{\textrm{Fe}}$ vs. $H_{\textrm{high}}$ diagram (Fig. 11) can be approximately given as follows:

$$1-N_{\textrm{Fe}}=b_{0}(N_{\textrm{H}},\langle N_{\textrm{H}}\rangle)+6h-0.9h^{3}$$

(B2)

$${}+b_{1}(N_{\textrm{H}},\langle N_{\textrm{H}}\rangle)h^{5},$$

where $h=\log_{10}(H_{\textrm{high}})$, and $b_{0}$ and $b_{1}$ are two functions that depend on both column densities, $N_{\textrm{H}}$ and $\langle N_{\textrm{H}}\rangle$. We do not provide explicit analytical expressions for $b_{0}(N_{\textrm{H}},\langle N_{\textrm{H}}\rangle)$ and $b_{1}(N_{\textrm{H}},\langle N_{\textrm{H}}\rangle)$, which would be cumbersome, and just provide in Tables 1 and 2 their numerical values for a number of representative values of $N_{\textrm{H}}$ and $\langle N_{\textrm{H}}\rangle$ (where ‘‘not-fitted’’ means that we did not fit the functions for those combinations of $N_{\textrm{H}}$ and $\langle N_{\textrm{H}}\rangle$).

Finally, the dependence of the strength of the Fe$K\alpha$ Compton shoulder on $\langle N_{\textrm{H}}\rangle$ (Fig. 13) can be approximated as follows:

$$C_{{\rm K}\alpha}=214.59-165.42\log_{10}(\langle N_{\textrm{H}}\rangle)^{0.23}$$

(B3)

$${}+16.89\log_{10}(\langle N_{\textrm{H}}\rangle)^{0.64}.$$

All these fitting functions were found for the $\langle N_{\textrm{H}}\rangle$ and $N_{\textrm{H}}$ between $10^{23}$ and $10^{24}$ cm${}^{-2}$ and $A_{\textrm{Fe}}$ between 0.5 and 2.