Contribution of the accretion disk, hot corona, and obscuring torus to the luminosity of Seyfert galaxies: Integral and Spitzer observations

We used the InfraRed Spectrograph (IRS, Houck et al. 2004) on Spitzer to obtain low-resolution spectra of our objects. Our program’s observations, for a total of 30 AGNs, were done in spectral mapping mode using the Short-Low (SL) and Long-Low (LL) IRS modules. Each of these modules has first- and second-order sub-slits (SL1, SL2, LL1, and LL2) with widths of 3.7, 3.6, 10.7, and 10.5 arcsec, respectively. We used SL1, LL1, and LL2 for the entire sample and SL2 for a subset of objects. The resulting spectra thus cover a range from either 5.2 or 7.5 $\mu$m up to 38 $\mu$m. Observations with SL1 and SL2 consisted of one cycle of 6 pointings with a ramp duration of 6 s with two 19$\text{arcsec}$ steps in the slit direction and three 1.8$\text{arcsec}$ steps in the dispersion direction. For LL1 and LL2, one cycle of 3 pointings with a ramp duration of 6 s and a step size of 42$\text{arcsec}$ in the slit direction was implemented.

For those sources that were not covered by our Spitzer program, we used archival low-resolution IRS data: mapping mode observations for 14 AGNs and staring mode observations for another 23 AGNs. Almost all of the archival mapping mode observations have the following setup: one cycle of 13 pointings with a step size of 1.8$\text{arcsec}$ in the dispersion direction for SL1 and SL2 and 5 pointings with a step size of 4.85$\text{arcsec}$ in the dispersion direction for LL1 and LL2, the ramp duration being 6 s. For the archival staring mode observations, the ramp times and numbers of cycles vary from one object to another.

In the analysis of mapping mode observations, we used basic calibrated data (BCD), extracted from the Spitzer Science Center pipeline (versions S18.0.1 and newer for our program’s observations and versions S15.3.0 and newer for the archival observations). For each order of a given IRS module and for each position of a given source within the slit, we first produced a background image. For our program’s observations, this was done by averaging over 2D spectra obtained in significantly ($\sim 19\text{arcsec}$ for SL and $\sim 42\text{arcsec}$ for LL) off-source positions, whereas for the archival observations, a similar averaging was done over 2D spectra obtained in the other order of a given IRS module. The background image was then subtracted from the on-source 2D spectrum. Then, a 1D spectum of the source was obtained using the Spitzer IRS Custom Extraction software (SPICE) by applying the regular extraction algorithm, which uses an aperture that gradually increases with wavelength in accordance with the telescope’s point spread function (e.g., the aperture width is 7.2$\text{arcsec}$ at 6 $\mu$m and 36.6$\text{arcsec}$ at 27 $\mu$m). Finally, an averaging over a set of 1D spectra extracted in different source positions within the slit was done. The staring mode spectra were obtained simply by averaging over nod-subtracted post-BCD spectra. These were derived with the same regular extraction algorithm as was used in our analysis of mapping mode observations.

Although the data reduction procedure described above is fairly simplistic and does not fully exploit the potential of mapping mode observations (which are available for nearly two thirds of our sample), it is adequate for the purposes of our HX–MIR cross-correlation study, as confirmed by a comparison with an alternative, more detailed analysis of IRS data for a subset of INTEGRAL sources (see §4.4).

The Compton-thick Seyfert 2 galaxy NGC 1068 is an extremely bright ($\sim 15$ Jy at 15 $\mu$m) infrared source, which caused saturation of IRS. We therefore quote in Table 1 its MIR flux estimate based on a compilation of high angular resolution, infrared observations (Mason et al., 2006).

We used images obtained by the InfraRed Array Camera (IRAC, Fazio et al. 2004) to determine source flux densities at 3.6, 4.5, 5.8, and 8.0 $\mu$m. This wavelength range partially overlaps with that covered by IRS spectroscopy, enabling direct comparison between the spectroscopic and photometric results and providing an extension of spectra to shorter wavelengths.

Our IRAC program included 35 INTEGRAL sources. Observations were done in high dynamic range (HDR) mode. Specifically, a combination of a 0.6 s frame and a 12 s frame was repeated in 9 random dithers for each source. For the rest of the sample, we made use of archival observations, for which the number and duration of frames varied from source to source. Since most of our objects are bright infrared sources, our analysis was in most cases based on stacking the 0.6 s frames. For the 5 weakest sources ($\lesssim 10$ mJy at 3.6–8.0 $\mu$m), including LEDA 138501, IGR J09446$-$2636, ESO 263-G013, IGR J10386$-$4947, and NGC 4395, to improve the accuracy, we determined the fluxes by stacking the long (2, 12, or 30 s) frames. This was also done for 3C 111 and Mrk 3 despite their relative brightness because the observations were not done in HDR mode. We verified that none of the sources was saturated.

We analyzed post-BCD data using the standard point source extraction package APEX, part of the MOPEX software. Post-BCD images are adequate for this work because their main deficiency, poor artifact correction, is unimportant for 0.6-s frames. We estimated source fluxes by integrating the surface brightness in different apertures with radii between 2.4$\text{arcsec}$ to 12$\text{arcsec}$ and correcting for flux leakage outside the aperture under the assumption of a point-like source. Although this procedure is inaccurate for measuring fluxes of spatially extended sources, it is good enough to indicate the presence of extended host galaxy emission as a significant difference between fluxes measured in large and small apertures (see Appendix A).

The existing IRAC photometric measurements for the Compton-thick Seyfert 2 galaxy NGC 1068 proved to be saturated and hence were not used.

We adopted the starburst template from Brandl et al. (2006)¹¹1 http://www.strw.leidenuniv.nl/$\sim$brandl/SB_template.html, which is an average over low-resolution IRS spectra of a dozen nearby ($D\lesssim 100$ Mpc) starburst galaxies. This template is well suited for our analysis because it was obtained by low-resolution IRS spectroscopy, similarly to the spectra studied here.

We normalized starburst components in our objects based on the observed strength of PAH lines, which are believed to be a generic signature of star formation (Roche et al., 1991). The presence of an AGN in a star forming galaxy may lead to a weakening of the PAH spectral features because PAH molecules can be destroyed by hard AGN radiation (Voit, 1992). However, the importance of this effect is controversial (see, e.g., Smith et al. 2007b; O’Dowd et al. 2009; Sales et al. 2010), and we have assumed that the shape of the starburst spectral component is not affected by the presence of a central AGN.

Our analysis consisted of the following steps. First, we fitted the spectra around (typically within $\pm 0.6$ $\mu$m of) the 6.2 $\mu$m and 11.3 $\mu$m PAH lines by a sum of a linear continuum and a Gaussian. We then compared the derived PAH line fluxes with the corresponding values for the starburst template, which yielded two independent estimates of the amplitude of the star-formation component. The average of these two values was then adopted as the normalization of the starburst template. On average, the coefficients implied by the 6.2 $\mu$m and 11.3 $\mu$m features proved to be consistent with each other, although there is $\sim 40$% scatter around the 1:1 ratio of the two coefficients. This indicates that there are $\sim 20$% systematic uncertainties in the derived amplitudes of starburst components for our objects (this issue is further discussed in §4.4 below). If there were no IRS data for the 6.2 $\mu$m feature (i.e., only first-order SL data at $\gtrsim 7.5$ $\mu$m were available), we used the flux of the 11.3 $\mu$m line to normalize the starburst component. If the observed PAH features proved to be strong enough (depending on the source brightness, we required the PAH equivalent widths, EW, to be larger than 0.01–0.02 $\mu$m, as compared to EW=0.45 and 0.55 $\mu$m for the 6.2 $\mu$m and 11.3 $\mu$m bands, respectively, in the starburst template), we subtracted the estimated starburst contribution from the total spectrum to derive the AGN component. Otherwise, we considered star formation contamination insignificant and did not perform any subtraction.

We applied an additional correction to 13 spectra that contained detectable PAH features and exhibited a significant ($>$10–20%, depending on the source brightness) discontinuity near 14 $\mu$m, where the short-wavelength segment measured in the 3.7$\text{arcsec}$-wide SL1 slit connects to the long-wavelength segment measured in the 10.5$\text{arcsec}$-wide LL2 slit. Such a “jump” is mostly likely caused by an extended source, i.e., it cannot be due to the AGN. In these objects, the long-wavelength ($\gtrsim 14$ $\mu$m) part of the starburst component was rescaled to make the AGN component smooth across the SL–LL boundary. We did not make such a correction for NGC 4395, by far the weakest infrared source in our sample (with an estimated flux density of 13 mJy at 15 $\mu$m), despite the apparent presence of a significant SL–LL discontinuity in its spectrum, because of its low statistical quality (see Fig. 1). Furthermore, our comparison with available high-resolution spectroscopy for this object (see §4.2 below) indicates that the SL slit was not positioned sufficiently accurately on the nucleus of NGC 4395, which might have caused an artificial discontinuity at 14 $\mu$m in the low-resolution IRS spectrum.

In principle, we could also use another known strong PAH feature, at 7.7 $\mu$m, for estimating the contribution of star formation. However, this band overlaps with the high-ionization [NeVI] 7.65 $\mu$m line, which can be bright in AGNs (Sturm et al., 2002) and is impossible to separate from the PAH feature in low-resolution IRS spectra. Moreover, the 7.7 $\mu$m feature might be significantly affected by AGN radiation (Smith et al., 2007b; O’Dowd et al., 2009).

In agreement with previous studies (e.g., Weedman et al. 2005; Buchanan et al. 2006; Shi et al. 2006; Deo et al. 2009; Wu et al. 2009), we observe a large variety of infrared spectral shapes among Seyfert galaxies. However, those spectra dominated by dust reprocessed emission generated by black hole accretion (see examples in the left column of Fig. 1), rather than by star formation, almost invariably peak (when plotted in $\nu F_{\nu }$ units) at $\sim$15–20 $\mu$m, in good agreement with models of dusty tori heated by a central source of UV radiation (e.g., Dullemond & van Bemmel 2005; Hönig et al. 2006; Nenkova et al. 2008; Alonso-Herrero et al. 2011). Furthermore, the AGN components of those IRS spectra with inferred significant starburst contamination (see the middle column of Fig. 1) prove to be similar to the spectra of “pure” AGNs. In particular, most of the former also peak at 15–20 $\mu$m. However, since our procedure of estimating the star formation contribution based on the strength of PAH features becomes progressively less reliable with increasing wavelength, there is much uncertainty in the deduced AGN spectral contributions at $\lambda \gtrsim 20$ $\mu$m. All these findings are similar to the results of previous attempts to decompose Spitzer spectra of quasars and Seyfert galaxies into AGN and starburst components (e.g., Netzer et al. 2007; Mullaney et al. 2011).

The IRS spectra of six objects, NGC 1142, ESO 005-G004, IGR J07563$-$4137, NGC 4945, IGR J14561$-$3738, and MCG+04-48-002 (see the right column of panels in Fig. 1) closely resemble the starburst template. We found it practically impossible to distinguish AGN and host galaxy components in these starburst dominated sources and therefore excluded them (see Table 1) from most of the subsequent analysis. Interestingly, most of our starburst dominated objects are strongly X-ray absorbed AGNs ($N_H\sim {10}^{24}$ cm${}^{-2}$). One may speculate that i) large supplies of cold gas and dust associated with starburst activity in a galactic nucleus facilitate the formation of a dense central obscuring torus, and/or ii) part of the X-ray absorption is caused by cold gas tracing star formation in the galaxy and located outside a parsec-scale AGN torus.

The remaining 61 objects (with NGC 1068 excluded for being a Compton-thick source) compose a “clean” sample for our subsequent analysis. As concerns the LLAGN NGC 4395, since its (low signal to noise) low-resolution IRS spectrum leaves doubts as to the presence of a significant starburst contribution (we estimate it at $\sim 14$% at 15 $\mu$m), we have also analyzed available high-resolution IRS data, following the methods described aby Goulding & Alexander (2009). While the derivation of an accurate continuum shape, using only high-resolution IRS, is complicated by the tip-tilt effects of the individual echelle orders, the high-resolution MIR spectrum of NGC 4395 is characterized almost entirely by an AGN-produced broken power law with little or no evidence ($EW\ll 0.1$ $\mu$m) for superposed PAH features. Hence, NGC 4395 is clearly AGN dominated and thus should be part of our clean sample. Nevertheless, as noted before, we still distinguish this “dwarf Seyfert” from the rest of the sample during our HX–MIR cross-correlation analysis because it might represent a physically different class of AGNs.

We have just seen (Fig. 1 and §4.2) that after subtraction of the star formation contribution, AGN MIR continua have an approximately constant shape. This suggests that it should be possible to estimate bolometric luminosities of AGN obscuring tori using their flux densities measured at a single MIR wavelength. We have chosen to use for this purpose the rest-frame $\lambda =15$ $\mu$m. Specifically, $f_{\nu }$ (15 $\mu$m) was determined by averaging a given spectrum over the wavelength range 14.7–15.2 $\mu$m. There are several reasons behind this choice. First, $\lambda =15$ $\mu$m is approximately where AGN torus emission peaks. Second, since IRS SL2 data are not available for some of our sources, we can only use wavelengths $\lambda \gtrsim 8$ $\mu$m for the whole sample. Third, wavelengths $\lambda \gtrsim 20$ $\mu$m are disfavored because cool dust emission associated with star formation becomes more important with increasing wavelength and frequently dominates far-infrared and even mid-infared spectra of Seyferts. Finally, there are no strong emission lines or absorption features within $\sim 0.5$ $\mu$m on either side of 15 $\mu$m.

The last three columns of Table 1 present the total measured flux densities ($f_{\nu }$) and corresponding luminosities ($\nu L_{\nu }$) at 15 $\mu$m as well as estimated fractions of AGN emission in the total flux at 15 $\mu$m, $F_{AGN}$. Statistical uncertainties for the infrared fluxes and luminosities are negligibly small. For the six starburst dominated objects, we assume that $F_{AGN}\lesssim 50$%. In the subsequent analysis, the starburst subtracted MIR flux of AGNs is defined as

The corresponding AGN luminosity is defined as

The main potential source of systematic uncertainty in our estimation of AGN fluxes at 15 $\mu$m is the use of a fixed starburst spectral template from Brandl et al. (2006). In reality, the spectral properties of infrared emission from dust heated by starbursts may vary from one galaxy to another, and some authors (e.g., Mullaney et al. 2011) have attempted to take this diversity into account in separating AGN and host galaxy components for Seyfert galaxies.

A crude estimate of the systematic uncertainty associated with our use of a fixed spectral template was already made in §4.1 using the difference in the normalizations of starburst spectral components determined using the 6.2 and 11.3 $\mu$m PAH features. Namely, starburst component amplitudes could be estimated by our fitting procedure to within $\sim 20$%. This implies that for AGN dominated sources, i.e., those objects with $F_{AGN}\gtrsim 50$%, the AGN fluxes at 15 $\mu$m are also estimated to within $\sim 20$%, i.e., to better than 0.1 dex in log space.

To better understand the uncertainties associated with our estimates of $f_{15\mu m}$, we performed an alternative spectral analysis based on a set of starburst spectral templates for a subsample of our objects. Specifically, we selected a representative subset (16 objects) of AGN-dominated, mixed, and starburst-dominated sources. This comparison sample is equally divided between sources with IRS-staring and mapping data. In §4.1, we assumed that the SL—LL discontinuity observed in the MIR spectra of some of these AGNs arises due to extended host galaxy emission which provides an additional contribution to the larger aperture LL spectrum. However, it has also been suggested in previous studies that this discontinuity between the spectral orders derives from an IRS detector effect; for the purposes of comparison, we imposed this assumption for our sub-sample. Furthermore, for those sources with IRS mapping data, we combined the rogue-pixel cleaned BCD images and extracted nuclear spectra using the 3D spectral reduction program CUBISM (Smith et al., 2007a), which is used widely in recent MIR AGN literature (e.g., Dale et al. 2009; Goulding & Alexander 2009; Diamond-Stanic & Rieke 2010; Petric et al. 2011; Alonso-Herrero et al. 2012). We used DecompIR (Mullaney et al., 2010) to deconvolve the MIR spectra for our comparison sample using a set of empirical and theoretical starburst templates (e.g., Goulding et al. 2011) and assumed an absorbed broken power law to model the AGN component. The derived AGN fractions and fluxes prove to be entirely consistent with the values established by our baseline analysis (§3.1, §4.1) for those sources with $F_{AGN}>0.5$ (the average scatter in the derived $f_{15\mu m}$ is $\pm 8$%). However, for those sources with $F_{AGN}<0.5$ (i.e., the spectra appear starburst dominated), the spectral fits become strongly dependent on the imposed starburst templates, and the scatter in the measured AGN flux increases to a factor of $\sim 2$.

As an additional check of our IRS spectral measurements, we can use the IRAC imaging data available for all of our AGNs. For the vast majority of the sources, the absolute values of flux densities measured by IRAC and IRS in the overlapping spectral region below 8 $\mu$m are in good mutual agreement (see Fig. 1), especially when the smallest (2.4$\text{arcsec}$) IRAC aperture is used (recall that the IRS SL slits have similar widths, 3.6–3.7$\text{arcsec}$). However, there are a few sources for which there is a significant discrepancy between the spectroscopic and photometric fluxes at $\lambda \lesssim 8$ $\mu$m. Since the corresponding IRAC and IRS observations were separated by several years in time, these flux differences probably indicate significant intrinsic variability of AGN torus emission, especially at shorter wavelengths (Suganuma et al., 2006; Kishimoto et al., 2009; Tristram et al., 2009; Kozłowski et al., 2010). Also, as shown in Appendix A, the detection of significant extended emission at 8 $\mu$m by IRAC in many sources is fully consistent with our conclusions about the host-galaxy contamination of IRS spectra.

Finally, we compared our results for 16 AGNs with higher angular resolution observations from Gandhi et al. (2009). We find excellent agreement (see Fig. 5 and discussion in §5.2) between the fluxes derived from the very different observational data sets, which suggests that host galaxy contamination of our measured values of AGN MIR fluxes is minimal.

We conclude that the combined statistical and systematic uncertainty in our AGN 15 $\mu$m-flux estimates for the clean sample (i.e., AGN dominated sources) is probably less than 0.1 dex. This uncertainty proves to be small in comparison with the intrinsic scatter in the HX–MIR flux and luminosity correlations (see §5 below). We further discuss the potential influence of starburst contamination on our derived correlations in §5.1.

The derived $L_{HX}$–$L_{15\mu m}$ relation, eq. (3), makes it possible to predict the HX luminosity for a given MIR luminosity. The combination of three facts, i) that our AGN sample is hard X-ray selected, ii) that this sample is not limited by sensitivity in the MIR band, and iii) that the correlation has been derived by fitting $L_{15\mu m}$ as a function of $L_{HX}$, ensures that this relation reproduces the intrinsic correlation between $L_{HX}$ and $L_{15\mu m}$ for the local AGN population without any bias.

To further test the robustness of the derived trend, we repeated our $L_{HX}$–$L_{15\mu m}$ cross-correlation analysis for various subsamples of AGNs. First, one may ask whether our procedure of separating AGN and starburst spectral components significantly affects the results. To address this issue, we computed the correlation for 46 strongly AGN dominated sources – those objects in which the AGN component accounts for at least 90% of the total emission at 15 $\mu$m (i.e., $F_{AGN}>0.9$). The result (see Table 2) is very close to the correlation found for the total clean sample from which NGC 4395 is excluded (eq. [3]). In addition, we repeated the analysis for the clean sample (without NGC 4395) using total measured 15 $\mu$m fluxes instead of AGN fluxes (i.e., setting $F_{AGN}=1$). The amplitude of the correlation increased by $\sim 20$%, obviously due to the unsubtracted contribution of starburst emission, but the slope changed by less than $1\sigma$ from 0.74 to 0.68. These tests demonstrate that the correlation between $L_{HX}$ and $L_{15\mu m}$ is not significantly affected by the details of our spectral analysis of IRS data.

We next repeated the analysis separately for nearby ($z<0.02$, 35 objects, excluding NGC 4395) and distant ($z>0.02$, 25 objects) sources from the clean sample. The correlation, in particular the slope of $0.69\pm 0.16$, derived for the distant subsample (see Table 2) is fully consistent with the correlation found for the total sample (eq. [3]). Since the $z>0.02$ set, owing to the INTEGRAL detection limit, is represented by luminous AGNs only, with $L_{HX}\sim {10}^{43}$–${10}^{45}$ erg s${}^{-1}$, this result also implies that the slope of the high-luminosity part of the HR–MIR correlation is not significantly different from the trend found over a broader range of luminosities. However, the slope, $0.93\pm 0.10$, determined for the nearby ($z<0.02$) subsample, mainly consisting of lower luminosity AGNs with $L_{HX}\sim {10}^{42}$–${10}^{44}$ erg s${}^{-1}$, is somewhat different from the general trend, but this difference is less than $2\sigma$ significant.

Finally, we repeated our analysis for different types of AGNs, namely Seyfert 1s (Sy1, including the intermediate types 1.2 and 1.5) and Seyfert 2s (Sy2, including the intermediate types 1.8 and 1.9). The derived relations (Table 2) are consistent with each other and with eq. (3). As can be seen from Fig. 4, Sy1s and Sy2s do not distinguish themselves on the $L_{HX}$–$L_{15\mu m}$ diagram, nor do narrow-line Seyfert 1 galaxies occupy a distinct region of this diagram. Finally, there is no significant dependence of the $L_{15\mu m}/L_{HX}$ ratio on the X-ray absorption column density except for the clear separation of the extremely Compton-thick source NGC 1068 from the rest of the sample.

We conclude that the correlation between HX and MIR luminosities described by eq. (3) is robust, although there is a weak indication that the slope of the correlation is not constant and decreases with increasing AGN luminosity.

For some applications, one may also be interested in knowing the distribution of $L_{HX}$ for a given $L_{15\mu m}$, rather than $L_{15\mu m}$ as a function of $L_{HX}$. We have therefore also computed (Appendix B) the inverse linear regression, i.e., $\log L_{HX}$ as a function of $\log L_{15\mu m}$, for our clean sample excluding NGC 4395. As can be seen in Fig. 2, this relation is different from the dependence of $L_{15\mu m}$ on $L_{HX}$.

Mullaney et al. (2011) have studied infrared properties of nearby AGNs detected in the 14–195 keV energy band by Swift/BAT. This sample, although not statistically complete, is similar to our INTEGRAL sample in that it is hard X-ray selected. Using a subsample of AGNs having both Spitzer/IRS spectroscopic and IRAS photometric data, Mullaney et al. (2011) developed and tested a procedure, based on a set of starburst spectral templates, that make it possible to separate AGN and starburst contributions to the infrared flux using IRAS four-band photometry only. They then applied this procedure to a sample of 44 BAT AGNs and found that $L_{12\mu m,43}=(2.4\pm 0.4)L_{14-195keV,43}^{0.74\pm 0.13}$. This result is in excellent agreement with our eq. (3).

On the other hand, Gandhi et al. (2009) have reported a near proportionality between 2–10 keV ($L_X$) and 12 $\mu$m luminosities for Seyfert galaxies using high angular resolution infrared observations: their best estimate is $L_{12\mu m}\propto L_X^{1.11\pm 0.07}$. This result seems to contradict our conclusion that the $L_{15\mu m}/L_{HX}$ ratio decreases with increasing luminosity.

A number of factors might contribute to this discrepancy, but the most important one appears to be the difference in sample luminosities. A difference in galaxy weighting makes at most a minor difference. In their preferred regression procedure, Gandhi et al. (2009) took into account individual uncertainties in X-ray and infrared luminosities. However, the X-ray uncertainties were estimated in a rather arbitrary way, taking into account long-term variability for some sources but not for others. This led to significantly different weights ascribed to different sources in fitting. In our view, for the problem at hand, it is preferable to use a standard linear regression procedure in log–log space giving equal weights to all the sources in a sample. In fact, Gandhi et al. (2009) did perform such an analysis and obtained a somewhat flatter dependence $L_{12\mu m}\propto L_X^{1.02\pm 0.07}$, which is, however, still significantly steeper than the $L_{15\mu m}\propto L_{HX}^{0.74\pm 0.06}$ relation found here.

Another potentially important factor is the use of 2–10 keV X-ray luminosities by Gandhi et al. (2009) vs. our use of HX luminosities. Due to our hard X-ray selection and the resulting insensitivity to absorption effects, we are able to determine HX luminosities directly from 17–60 keV fluxes measured by INTEGRAL. In comparison, the Gandhi et al. (2009) sample contains a large number of significantly absorbed ($N_H\gtrsim {10}^{23}$ cm${}^{-2}$) sources whose intrinsic 2–10 keV luminosities were estimated through model-dependent analysis of X-ray spectra or, in some cases, even using [OIII] optical line fluxes. We have compared the HX and X-ray luminosities for 16 Seyfert galaxies (excluding the Compton thick Seyfert NGC 1068) which are present in both the INTEGRAL and Gandhi et al. (2009) samples. The data, spanning the $L_{HX}$ range from $9\times {10}^{41}$ erg s${}^{-1}$ (Cen A) to $1.4\times {10}^{44}$ erg s${}^{-1}$ (Mrk 509), prove to be consistent with $L_{HX}$ being proportional to $L_X$. The small scatter (0.23 dex) associated with this correlation can be fully attributed to the uncertainties in the $L_X$ values as estimated by Gandhi et al. (2009), whereas the mean ratio $L_{HX}/L_X\approx 1.5$ may be considered typical for AGNs and is only slightly larger than the ratio (1.2) corresponding to a fiducial AGN spectrum used in our analysis below (§6.1). Furtheremore, systematic studies based on AGNs from the INTEGRAL (de Rosa et al., 2012) and Swift/BAT (Winter et al., 2009) hard X-ray surveys have not revealed a significant dependence of the HX/X-ray flux ratio on luminosity. We thus conclude that the use of $L_X$ by Gandhi et al. (2009) vs. our use of $L_{HX}$ is unlikely to lead to a significant difference between the results of the corresponding X-ray–infrared cross-correlation analyses.

Differences in the infrared data analysis might also play a role. Our study is based on Spitzer/IRS spectroscopy and uses a template based separation of AGN and starburst spectral components. Gandhi et al. (2009) use narrow-filter photometry near rest-frame 12 $\mu$m with the VISIR instrument on VLT, which provides a significantly better angular resolution compared to Spitzer/IRS and hence presumably minimizes host galaxy contamination. We have again used the overlapping sample of 16 Seyferts to check if there are any systematic differences associated with these two approaches. Specifically, we used our IRS spectra and derived fluxes within the VISIR filters that were used by Gandhi et al. (2009), which differ from object to object. As shown in Fig. 5, although the total fluxes measured by IRS spectra are somewhat higher than those measured by VISIR, our standard correction for the host galaxy contamination brings both data sets to nearly perfect agreement. A significant difference (a factor of $\sim 2$) is only observed for NGC 3081, but it seems natural to expect some deviations in a sample of 16 objects, given that Spitzer/IRS and VLT/VISIR observations were not simultaneous (for the same reason, the 8 $\mu$m fluxes measured by IRAC and IRS differ significantly for some of our AGNs, see §4.4). Furthermore, there is no trend with either distance or luminosity, although the comparison sample spans distances from 3.6 Mpc (Cen A) to 174 Mpc (ESO 209-G012). We conclude that the differences in the infrared data analysis do not significantly bias our results with respect to those of Gandhi et al. (2009).

Perhaps most importantly, Gandhi et al. (2009) used a heterogeneous sample of 42 Seyfert galaxies, whereas we use a statistically complete and somewhat larger sample. As a result, the Gandhi et al. (2009) sample is significantly shifted to lower luminosities relative to ours (Fig. 6): e.g., the corresponding fractions of AGNs with $L_{HX}>{10}^{44}$ erg s${}^{-1}$ are 12% and 38%. As was dicusssed in §5.1, dividing our sample into two subsets, $z<0.02$ and $z>0.02$, represented by relatively low and high luminosity AGNs, respectively ($\sim {10}^{42}$–${10}^{44}$ vs. $\sim {10}^{43}$–${10}^{45}$ erg s${}^{-1}$), tentatively suggests that the slope of the X-ray–infrared correlation changes from $0.93\pm 0.10$ to $0.69\pm 0.16$ as the AGN luminosity increases. The Gandhi et al. (2009) sample effectively probes the luminosity range $L_{HX}\lesssim {10}^{44}$ erg s${}^{-1}$, similar to our $z<0.02$ subsample, and the slopes inferred for these two data sets are in satisfactory agreement with one another. This suggests that the results of Gandhi et al. (2009) and the present work are actually consistent with each other.

We conclude that further studies using larger, well-defined samples of AGNs are required to clarify if the slope of the X-ray–infrared correlation depends on luminosity, as tentatively suggested by the existing data.

We proceed by determining coefficients for conversion of the measured quantities $L_{HX}$ and $L_{15\mu m}$ to the AGN intrinsic quantities $L_{Corona}$ and $L_{Torus}$, respectively. Hard X-ray spectral shapes do not vary much from one Seyfert galaxy to another, apart from the photoabsorption rollover in type 2 objects below 10 keV. Typically, absorption corrected AGN spectra can be described above 2 keV as a power law with a photon index $\Gamma \sim 1.7$ (e.g., Reeves & Turner 2000) and a rollover above $\sim 100$ keV (e.g., Molina et al. 2009). We adopt that

This relation corresponds to a power-law spectrum with $\Gamma =1.7$ and an exponential cutoff at $E_f=200$ keV and is consistent with an average 3–300 keV spectrum of $\sim 100$ local AGNs detected during INTEGRAL and RXTE surveys of the sky (Sazonov et al., 2008, 2010). Assuming that the values of the power-law index and cutoff energy vary from $\Gamma =1.5$ to 1.9 and from $E_f\sim 50$ to $\sim 500$ keV from one Seyfert galaxy to another (as indicated by numerous studies, e.g., Zdziarski et al. 1995; Molina et al. 2009), we can estimate that the conversion described by eq. (5) can introduce a scatter in $L_{HX}$ values for a given $L_{Corona}$ of $\lesssim 20$%, i.e., $\lesssim 0.1$ dex.

We next introduce a similar correction factor for the reprocessed emission from the torus:

To obtain the above coefficient, we compared the 15 $\mu$m luminosity with that integrated over the rest-frame 6–32 $\mu$m band, $L_{6-32\mu m}$, for those Spitzer/IRS spectra (21 in total) that span this whole wavelength range (i.e., there are available data from the IRS second-order SL module) and do not suffer from significant contamination by MIR emission from dust associated with star formation. For the majority of these objects, the ratio $L_{15\mu m}/L_{6-32\mu m}$ is bounded in the narrow range of 0.65–0.85, only slightly depending on whether AGN emission lines (such as [OIV] 25.9 $\mu$m) are taken into account or not. We therefore adopted the relation $L_{15\mu m}/L_{6-32\mu m}=0.75$ for AGN tori and additionally lowered this ratio by one third in eq. (6) to account for non-negligible ($\sim 50$%) additional torus emission both shortward of 6 $\mu$m and longward of 32 $\mu$m (see, e.g., Nenkova et al. 2008). While the resulting $L_{15\mu m}/L_{Torus}$ ratio (eq. [6]) is determined less strictly than the ratio $L_{HX}/L_{Corona}$ above, the associated scatter in individual $L_{15\mu m}/L_{Torus}$ values around the mean trend given by eq. (6) is probably less than 20%, as suggested by the comparison of IRS spectra for “pure” objects, described above.

The next step in our analysis is to derive the torus solid angle ${\Omega }_{Torus}$. To this end, we assume that, for a given hard X-ray luminosity, ${\Omega }_{Torus}$ is determined by the relative number of obscured (type 2) AGNs of that luminosity, i.e.,

We consider an AGN obscured if its X-ray absorption column density $N_H>{10}^{22}$ cm${}^{-2}$. In this connection, recall (see §2) that the $N_H$ values for our objects are not based on INTEGRAL hard X-ray measurements but have been determined through analysis of high-quality X-ray spectra obtained by various X-ray telescopes.

Dividing our AGN sample into several bins in $L_{HX}$, Fig. 7 shows the fraction of absorbed AGNs as a function of $L_{HX}$. A strong trend of decreasing $N_{type2}/N_{total}$ ratio with increasing luminosity is evident. We can estimate the statistical significance of this trend using a maximum-likelihood estimator,

Here, the summation is over our sample of AGNs and $P_{1,2}$ is the probability for a given object with luminosity $L_{HX,i}$ to be either obscured ($P_2$) or unobscured ($P_1$). We restrict our consideration to the luminosity range $41.5<\log L_{HX}<45$, because there are only two objects in the sample which fall outside this range (one on either side, see Fig. 7). Suppose now that the fraction of obscured AGNs has a power-law dependence on luminosity:

Obviously,

Requiring that $0<P_{1,2}<1$ over the $41.5<\log L_{HX}<45$ range yields constraints on the slope and intercept of the trend:

and

With these definitions, the maximum of the likelihood function proves to be at $p\approx 1$ and $\alpha =0.25$. By integrating $L$ over ($p$, $\alpha$) parameter space with the priors given by eqs. (11) and (12), i.e., using a Bayesian approach, we find that the probability that $\alpha <0$ is 0.999. Hence, the declining luminosity trend of the obscured fraction is ascertained with more than 3$\sigma$ significance.

We thus assume that the phenomenon of decreasing fraction of absorbed AGNs with increasing luminosity reflects an underlying trend of increasing opening angle of the obscuring torus. This is one of the crucial points in our analysis. According to eq. (13), the slope of the ${\Omega }_{Torus}$ ($L_{HX}$) dependence is approximately equal to 0.25 for $L_{HX}$ ranging between $\sim {10}^{41.5}$ and ${10}^{45}$ erg s${}^{-1}$ with the associated uncertainty being small, $\sim 10$% as determined from the dispersion of data points in Fig. 7 and from the Bayesian analysis described above. However, the ${\Omega }_{Torus}$ ($L_{HX}$) dependence holds true in a statistical sense only, and there might be variations in the torus opening angle among AGNs of a given luminosity. Unfortunately, observations do not yet provide reliable information on the distribution of ${\Omega }_{Torus}$ values for a given $L_{HX}$, and hence we cannot predict to what degree this scatter could affect our results below. Furthermore, the exact parameters of the $N_{type2}/N_{total}$ ($L_{HX}$) dependence adopted in eq. (13) should be applied only to the local ($z\sim 0$) population of AGNs, in particular because the fraction of obscured sources among high-luminosity AGNs appears to be larger in the distant ($z\gtrsim 1$) Universe (e.g., Hickox et al. 2007; Hasinger 2008; Treister et al. 2008).

We are now ready to estimate the accretion disk luminosities for our AGNs using eq. (4):

Specifically, we first determine $L_{Corona}$ and $L_{Torus}$ using eqs. (5) and (6), respectively and then use eq. (14) to find $L_{Disk}$. The resulting scatter plot of $L_{Disk}$ vs. $L_{Corona}$ for the clean sample excluding NGC 4395 is shown in Fig. 8.

Fitting the $L_{Disk}$ vs. $L_{Corona}$ data in Fig. 8 with a power law yields the correlation