THE DUST SUBLIMATION RADIUS AS AN OUTER ENVELOPE TO THE BULK OF THE NARROW Fe Kα LINE EMISSION IN TYPE 1 AGNs

Our starting sample is the Chandra High Energy Grating (HEG) sample of Shu et al. (2010), the largest sample to date of high spectral resolution data covering the ∼6.4 keV energy regime. This sample includes 36 unique galaxies below z = 0.3. In 27 of these, a measurement of the velocity full widths at half maximum (v_FWHM) of the narrow line core was reported by the authors. This includes positive FWHM measurements, as well as unresolved line upper limits. The mean v_FWHM for this sample was found to be 2060 ± 230 km s⁻¹. The HEG spectral resolution is 0.012 Å, corresponding to a velocity resolution of ≈1860 km s⁻¹ at 6.4 keV, and is accounted for in the line modeling.

We have chosen measurements that are likely to best represent the narrow line core. In this respect, note that several new observations for NGC 4051 have become available since the work of Shu et al. (2010). We use the results discussed by Lobban et al. (2011) and Shu et al. (2012). The former work found that the presence of an unresolved line component at 6.4 keV significantly improved the fit of the line complex in the time-averaged data from 12 observations totaling ∼300 ks of exposure. Because this component is the one most likely associated with distant material, we treat the core as being unresolved. This leads to an upper limit on v_FWHM and a corresponding lower limit on R_Fe.

IC 4329A is also treated as upper limit on v_FWHM. Although a broad component to the line is detected by Shu et al. (2010; see also McKernan & Yaqoob 2004) with v_FWHM = ${18830}_{-9620}^{+18590}$ km s⁻¹, the line peak energy in this case is found to be E_peak = ${6.305}_{-0.096}^{+0.139}$ keV, which is lower than the neutral line energy at 68% confidence. An alternative fit with an unresolved line by the authors yielded a more plausible value of E_peak = ${6.399}_{-0.005}^{+0.006}$ keV, and the authors stress that these are more reliable measurements of the true narrow core.

For measurement of the Fe Kα emission radii (R_Fe), we assume virial motion for the emitting clouds. Then,

$$\begin{eqnarray}&&{R}_{{\rm{Fe}}}=\displaystyle \frac{{{GM}}_{{\rm{BH}}}}{{v}^{2}}\end{eqnarray} \tag{ 1 }$$

where M_BH is the black hole mass, and v = $\sqrt{3}/2$ v_FWHM. The correction factor of $\sqrt{3}/2$ arises under the assumption of an isotropic velocity distribution and accounts for the fact that the line of sight velocity dispersion is half of the FWHM (cf. Netzer 1990; Peterson et al. 2004). Uncertainties related to unknown geometric projection factors are discussed below and in the appendix. Where possible, black hole mass values based upon reverberation mapping measurements are used, mostly based upon the Hβ time lag. This was the case for all sources except IRAS 13349+2438, which is derived from constraints on modeling its spectral energy distribution. In this case, we assumed a large uncertainty equal to the mass measurement itself in order to account for the lack of reverberation mapping. In addition, for NGC 4151, we use a recent measurement obtained from dynamical modeling. References are given in Table 1. Uncertainties on R_Fe were determined by joint Monte Carlo sampling of M_BH and v in log space.

Table 1.  Sample Properties

Source	MBH	Ref. (MBH)	τdust	Rdust,rev	Rdust,intf	Ref. (Rdust)	RBLR	LBol	${v}_{{\rm{FWHM}}}^{{\rm{Fe}}}$	RFe
	× 106 M⊙		(days)	× 0.1 pc	× 0.1 pc		× 0.01 pc	(1045 erg s−1)	(km s−1)	× 0.1 pc
Fairall 9	255.0 ± 56.0	i	400.0 ± 100.0	3.21 ± 0.80	⋯	1	${1.40}_{-0.35}^{+0.26}$	1.90	${18100}_{-12390}^{+76840}$	${0.045}_{-0.045}^{+0.135}$
Mrk 590	15.8 ± 3.4	ii	32.2 ± 4.0	0.26 ± 0.03	⋯	2	${2.09}_{-0.19}^{0.16}$	0.17	${4350}_{-2030}^{+6060}$	${0.048}_{-0.045}^{+0.077}$
NGC 3516	42.7 ± 14.6	i	71.5 ± 5.8	0.59 ± 0.05	⋯	2	${0.56}_{-0.32}^{0.57}$	0.025	${3180}_{-670}^{+880}$	${0.242}_{-0.113}^{+0.177}$
NGC 3783	29.8 ± 5.4	i	76.3 ${}_{-17.2}^{+10.9}$	${0.64}_{-0.14}^{+0.09}$	1.60 ± 0.50	3, 4	${0.84}_{-0.19}^{+0.27}$	0.17	${1750}_{-360}^{+360}$	${0.558}_{-0.203}^{+0.314}$
NGC 4051	1.9 ± 0.8	i	15.8 ± 0.5	0.13 ± 0.004	0.32 ± 0.05	2, 5	${0.49}_{-0.15}^{+0.22}$	0.0059	<1860	>0.032
NGC 4151	48.5 ± 20.0	iii	36.0 ${}_{-7.0}^{+9.0}$	${0.30}_{-0.06}^{+0.08}$	0.41 ± 0.04	6, 5	${0.55}_{-0.07}^{+0.09}$	0.050	${2250}_{-360}^{+400}$	${0.549}_{-0.228}^{+0.377}$
3C 273	6590.0 ± 1300.0	iv	⋯	⋯	8.80 ± 4.70	7	${22.23}_{-6.59}^{+4.96}$	58.80	${5900}_{-5830}^{+8640}$	${10.857}_{-10.240}^{+67.554}$
NGC 4593	5.4 ± 9.4	i	42.1 ± 0.9	0.35 ± 0.01	⋯	2	${0.31}_{-0.07}^{+0.07}$	0.042	${2230}_{-1100}^{+8180}$	${0.062}_{-0.062}^{+0.346}$
IRAS 13349+2438	1000.0 ± 1000.0	v	⋯	⋯	8.20 ± 3.40	8	⋯	11.16	${5150}_{-2810}^{+60200}$	${2.162}_{-2.162}^{+7.217}$
IC 4329A	9.9 ± 17.9	i	⋯	⋯	2.90 ± 0.40	8	${0.12}_{-0.15}^{+0.22}$	0.79	<1860	>0.164
NGC 5548	67.1 ± 2.6	i	61.3 ± 0.3	0.51 ± 0.002	⋯	2	${1.49}_{-0.05}^{+0.05}$	0.07	${2540}_{-820}^{+1140}$	${0.596}_{-0.358}^{+0.545}$
Mrk 509	143.0 ± 12.0	i	120.3 ± 1.1	0.98 ± 0.01	3.00 ± 0.50	2, 9	${6.46}_{-0.44}^{+0.50}$	1.04	${2910}_{-1250}^{+2590}$	${0.968}_{-0.801}^{+1.337}$
NGC 7469	12.2 ± 1.4	i	78.1 ± 0.1	0.65 ± 0.0004	<1.90	2, 8	${0.37}_{-0.07}^{+0.06}$	0.19	${4890}_{-1700}^{+2770}$	${0.029}_{-0.020}^{+0.029}$

Notes. Uncertainties are quoted for 68% confidence. M_BH ref.: (i) Peterson et al. (2004), (ii) Kaspi et al. (2000), (iii) Hönig et al. (2014), (iv) Paltani and Türler (2005), (v) Lee et al. (2013). R_dust ref.: (1) Clavel et al. (1989), (2) Koshida et al. (2014), (3) Lira et al. (2011), (4) Weigelt et al. (2012), (5) Kishimoto et al. (2009), (6) Hönig et al. (2014), (7) Kishimoto et al. (2011a), (8) Kishimoto et al. (2015, submitted), (9) Kishimoto et al. (2013).

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One systematic uncertainty in converting velocities to sizes comes from the unknown geometric projection, which may result in underestimation of the true space velocities, leading to overestimated sizes. This is a well-known effect in BLR reverberation mapping and single-epoch estimates of M_BH, and is captured in the f-factor (e.g., Peterson et al. 2004). If we posit that conservation of angular momentum of the accreting matter leads to a flattened geometry, then we can expect that the projection effects are very similar for the X-ray emitting region and the BLR. In this case, size measurements of these regions will be affected in a similar way for any individual object, so comparisons between these quantities will not be affected. On the sample level, the distribution of geometric inclinations will lead to a widening of any size-luminosity relation above the intrinsic dispersion by a factor of the order unity. Given that our sample covers four decades in luminosity and two decades in sizes, we can expect that the essence of the relation will be preserved even for unknown projection effects in individual sources. In the appendix, we will discuss alternate size comparisons using reverberation-independent M_BH values—a test that avoids the above assumption.

Near-IR (NIR) observations probe the emission region of the sublimation zone in the torus at temperatures of ∼1500 K. The size of this region can be inferred either from reverberation mapping or directly from interferometry. Although both types of measurements are qualitatively similar, the small quantitative differences between both are probably related to the detailed dust distribution (e.g., Kishimoto et al. 2009; Hönig & Kishimoto 2011; Hönig et al. 2014; Koshida et al. 2014). We collect and use both types of data from the literature (collectively referred to as R_dust). Note that R_dust measurements are available mostly for Type 1 AGNs, because in Type 2 AGNs the innermost hot dust is not easily visible.

The dust reverberation mapping radii (R_dust,rev) are the result of a long-term monitoring campaign with the MAGNUM telescope by Koshida et al. (2014) and have been inferred from V- and K-band light curve time lags (τ_dust) as R_dust,rev = cτ_dust/(1 + z), where c denotes the speed of light and the (1 + z) factor corrects for cosmological time dilation. The interferometric sizes (R_dust,intf) are based on Keck Interferometer data and represent the radii of a thin-ring model (Kishimoto et al. 2009, 2011a, 2013).

Of the 27 sources with v_FWHM from the sample of Shu et al. (2010), 7 have R_dust,intf measurements, and 10 have published τ_dust values. Our final sample of objects with measured values of R_Fe as well as R_dust comprises 13 unique sources. These are listed in Table 1.

For these objects, we also compute the radii of their optical BLRs (R_BLR) for comparison to R_Fe. Measurements of the reverberation time lag of the Hβ emission line (τ_Hβ) for 12 AGNs of our sample (i.e., all except IRAS 13349+2438) have been tabulated in Bentz et al. (2009), which we use here. Then, R_BLR = cτ_Hβ/(1 + z).

We follow this by investigating any possible relation between the locations of emitting regions and the continuum emission. As a proxy of the latter, we consider the monochromatic continuum luminosities at 5500 Å (L₅₅₀₀), whose values are gathered from Kishimoto et al. (2011a), Suganuma et al. (2006), and Bentz et al. (2009). These are based upon fitting their spectral energy distributions and corrected for starlight contamination. A mean uncertainty of 0.1 dex is assumed for L₅₅₀₀. These are also listed in Table 1.

Our key result is that the dust sublimation radius forms an outer envelope to the Fe Kα emission zone in Type 1 AGNs, with uncertainties allowing R_Fe at most a factor of a few times R_dust for our sample. This is not an obvious prediction given that the gas distribution is likely to extend continuously on scales both larger and smaller than the innermost dust radius, and that tori may be clumpy with low optical depth "holes" allowing radiation to penetrate well into its body. Our work is thus constraining for AGN torus models, as follows.

For axisymmetric gas distributions, the emitted Kα line intensity at any radius r depends upon the local solid angle Ω_Kα(r) of gas illuminated by the AGN (Krolik & Kallman 1987). The absence of sources with R_Fe ≫ R_dust then argues against geometries in which ${{\rm{\Omega }}}_{{\rm{K}}\alpha }(r\gg {R}_{{\rm{dust}}})$ > ${{\rm{\Omega }}}_{{\rm{K}}\alpha }({R}_{{\rm{dust}}})$ (i.e., where the line-emitting area increases strongly with radius beyond R_dust). Line emission will not cease exactly at R_dust, which would be an unphysical scenario. Figure 1 shows that the R_Fe values of up to a factor of a few times R_dust are allowed within the uncertainties. But, how do typical torus models match up to these constraints? In the geometry adopted by the widely used mytorus model (Murphy & Yaqoob 2009), the torus is assumed to have a donut shape with a circular cross-section and a half opening angle of 60^◦. Radiation from the nucleus that directly illuminates the torus would impinge upon the donut surface out to distances of the surface tangent line. In the mytorus geometry, this distance is 1.7 R_dust, where R_dust is the inner edge of the torus. This is consistent within the constraints from Figure 1. On the other hand, tori with strong flaring in height as a function of radius would present large illuminated surface solid angles at large nuclear distances, and would not easily satisfy the above constraints on R_Fe relative to R_dust.

Radiation would also penetrate below the torus surface, but if the integrated column density through the torus³ is Compton-thick, as is generally thought to be the case, then absorption and Compton scattering would quickly deplete line photons in the body of the torus. Thus, Ω_Kα refers to the solid angle of the reflecting torus surface down to an effective optical depth τ_Kα ∼ 1. A natural prediction of this scenario is that sources in which the obscuring tori are likely to be Compton-thin (e.g., NGC 2110, Marinucci et al. 2015; NGC 7213, Ursini et al. 2015) need not follow the trend of Figure 1 (i.e., their Fe Kα emission need not be restricted to lie around R_dust).

We also emphasize that our results do not exclude the presence of gas on extended scales altogether. If more distant gas is not excited by AGN radiation, it will not fluoresce and will remain invisible. Furthermore, studies based upon line width measurements will selectively target the strongest emission regions present on the innermost (fastest) scales. More distant (and fainter) emitting components could become apparent if our direct line of sight to the inner torus were obscured by material optically thick to the line photons, as may be expected in Type 2 and Compton-thick AGNs (cf. the detection of the extended emission in NGC 4945 by Marinucci et al. 2012 and in NGC 1068 by Bauer et al. 2014). Monte Carlo radiative transfer simulations can place detailed constraints on the radial gas geometry in the torus.

There are several potential complexities related to modeling the Fe Kα line and sample selection, which must be investigated to test the robustness of the above constraints. These caveats are discussed at length in Section 4.4.

Conversely, the scatter in R_Fe to scales much smaller than R_dust argues for Fe Kα emission from extended regions. This result is not new (cf. Yaqoob & Padmanabhan 2004; Nandra 2006; Shu et al. 2010), but is now inferred from the direct size comparison of Figure 1. The absence of any obvious luminosity scaling in Figure 2 supports this scenario, and Figure 3 shows that any Eddington rate-driven physical components (e.g., accretion disk outflows) do not control the origin of Fe Kα, at least over the range of L_Bol/L_Edd that we sample. The outer accretion disk and BLR clouds have been invoked as Fe Kα emitters in several objects. For example, a rapidly variable narrow Fe Kα line from small scales has been found in Mrk 509 (Ponti et al. 2013), Mrk 841 (Petrucci et al. 2002), and NGC 7314 (Yaqoob et al. 2003). For NGC 7213, Bianchi et al. (2008) found that the resolved Fe Kα line width matches the optical Hα line width, and argued for a BLR origin in this case. Shu et al. (2010) found that a significant fraction of their sample showed Fe Kα line widths that are consistent with, or broader than, the widths of typical optical and infrared photoionized lines, concluding that the Fe Kα emission zone location (relative to that of the BLR) genuinely varies from source to source. A BLR origin is also consistent for several sources in our sample.

Figure 1 shows that in many of the sources with the best constraints on R_Fe (e.g., NGC 4151, NGC 5548, NGC 3783), R_Fe is well matched to R_dust. However, this does not imply that low grating data signal-to-noise ratio (S/N) is the cause of mismatching R_Fe and R_dust values in other objects. It is certainly true that the largest Fe Kα error bars in Figures 1 and 2 must be unphysical and are likely to be a result of the line fits modeling the underlying continuum (see, for example, the discussion on Fairall 9 by Shu et al. 2010). In addition, some of the sources with ill-constrained R_Fe also have relatively short Chandra exposure times (t_exp)—for example, both Fairall 9 and NGC 4593 were observed for 80 ks with the HEG, as compared to a median t_exp = 150 ks for the sample and a maximum of 890 ks in the case of NGC 3783. But as emphasized by Yaqoob & Padmanabhan (2004), the uncertainties are not simply a function of S/N, with the grating data showing that the peak Fe Kα energy of 6.4 keV is not dominated by a narrow core in these sources.⁴ Moreover, if we use stricter confidence intervals on v_FWHM (e.g., the 90% range from Shu et al. 2010), the sources with an R_Fe significantly less than R_dust still remain discrepant in all cases. The large uncertainties then appear to reflect an origin from multiple regions in the AGN environment, including the (fast) clouds present on many scales. Longer grating observations of the sources with large uncertainties on v_FWHM will help to confirm the absence of dominant narrow cores, or to disentangle fainter core components.

In two other recent works, Jiang et al. (2011) and Minezaki & Matsushita (2015) have investigated the use of the Fe Kα line for the measurement of black hole masses in AGNs. Jiang et al. (2011) assumed a torus origin for the Fe Kα line with radii derived from infrared reverberation, and found a consistent trend between the masses predicted assuming isotropic motion of the Fe Kα clouds with the masses estimated from optical reverberation mapping. However, a scaling factor was required for the two sets of masses to match each other—with the scaling being equivalent to an average line core v_FWHM that is ${2.6}_{-0.4}^{+0.9}$ times broader than predicted from the initial assumption of a torus origin.

Minezaki & Matsushita (2015) studied a restricted AGN sample with "best constraints" on the FWHM of the Fe Kα line, and found that the Fe Kα emission zone is generally located between the reverberation radii of the broad Hβ emission line and the dust torus emission, the latter being estimated statistically (i.e., using a fixed ratio of R_dust to R_BLR based on K-band reverberation measurements). The criteria that constitute "best constraints" are not quantified by Minezaki & Matsushita (2015), but their selection excludes most objects with the broadest Fe Kα lines, as well as sources with unresolved Fe Kα lines. The former category of sources includes objects in which the Fe Kα emission zone is consistent with, or significantly smaller than the BLR (as we discussed in Section 4.2), whereas the unresolved sources could potentially show Fe Kα emission from outside the torus reverberation radius. Jiang et al. (2011), on the other hand, do include two sources with upper limits on velocity FWHM,⁵ and note the possibility that the narrow line core originates from radii smaller than the infrared radiation.

Our main result was derived entirely independent of the above works. While being consistent, a key distinction is that we go beyond average comparisons on a statistical basis. Importantly, from our object-by-object comparison, we make a prediction that R_dust forms an approximate outer envelope for normal Type 1 AGNs. This prediction can be easily falsified if R_Fe were observed to be much larger than R_dust in future observations (see also discussion in next section). In this sense, it is much more powerful than inferences based upon statistical trends. This holds, despite the fact that we have not restricted the base sample of Shu et al. (2010) according to FWHM constraints, and have also included upper limits. We then explored the consequences of such a postulation for the structure of AGN tori (Section 4.1)—aspects not mentioned in these other works, whose focus is M_BH measurement. We note that since Minezaki & Matsushita (2015) do not include sources in which the peak Fe Kα energy is not dominated by a narrow core,⁶ they do not probe the smallest scales that we discuss in Section 4.2.

Next, we will address various potential caveats which could affect our main result.

Several potential caveats need to be kept in mind regarding the current analyses of Chandra Fe Kα line profiles, and these are discussed in detail below.

4.4.1. Combining Multiple HEG Datasets

4.4.2. The Limited HEG Spectral Resolution

4.4.3. Biases Due to Multiple Unmodeled Line Components

4.4.4. Sample Representativeness

4.4.5. Alternate Black Hole Mass Estimates

Finally, one may question the robustness and suitability of our adopted M_BH measurements. As discussed in Section 2.1, certain assumptions about the geometric correction (f-factor) are implicit in virial M_BH estimates based upon reverberation mapping, and we have assumed that similar correction factors apply for the Fe Kα emission zone.

The influence of making such an assumption can be judged by using alternate black hole mass measurements, which are entirely independent of reverberation mapping. In the appendix, we compiled the measurements that are shown in Table 2. We present a figure showing the resultant comparison of R_Fe with R_dust in this case, and demonstrate that our main inference of R_dust forming an envelope to R_Fe remains unchanged. We refer the reader to the appendix for this test.

Table 2.  Alternate Black Hole Mass Estimates

Source	Method (MBH)	Measurement	Reference	log MBH	Δlog MBH	RFe
				M⊙	M⊙	× 0.1 pc
(1)	(2)	(3)	(4)	(5)	(6)	(7)
Fairall 9	MBH–σ*	σ* = 228 ± 18 km s−1	Oliva et al. (1999)	8.36 ± 0.46	+0.05 ± 0.47	${0.040}_{-0.040}^{+0.186}$
Mrk 590	MBH–σ*	σ* = 189 ± 6 km s−1	Nelson et al. (2004)	8.02 ± 0.44	–0.82 ± 0.45	${0.314}_{-0.298}^{+0.890}$
NGC 3516	MBH–σ*	σ* = 181 ± 5 km s−1	Nelson et al. (2004)	7.94 ± 0.44	–0.31 ± 0.47	${0.490}_{-0.335}^{+1.016}$
NGC 3783	MBH–σ*	σ* = 95 ± 10 km s−1	Onken et al. (2004)	6.75 ± 0.48	+0.73 ± 0.49	${0.105}_{-0.073}^{+0.241}$
NGC 4051	MBH–σ*	σ* = 86 ± 3 km s−1	Nelson et al. (2004)	6.57 ± 0.44	–0.28 ± 0.47	>0.061
NGC 4151	MBH–σ*	σ* = 116 ± 3 km s−1	Onken et al. (2014)	7.12 ± 0.44	+0.57 ± 0.48	${0.148}_{-0.098}^{+0.288}$
3C 273	X-ray variance	log(NVA) = 0.20 ± 0.08	McHardy (2013)	8.86 ± 0.08	+0.96 ± 0.12	${1.193}_{-1.129}^{+7.775}$
NGC 4593	MBH–σ*	σ* = 135 ± 6 km s−1	Nelson et al. (2004)	7.40 ± 0.45	–0.67 ± 0.88	${0.287}_{-0.287}^{+0.903}$
IRAS 13349+2438	L5100–RBLR	L5100 = 1044.64 erg s−1	Wang et al. (2004), Grupe et al. (2004)	7.74 ± 0.50	+1.26 ± 0.66	${0.119}_{-0.119}^{+0.474}$
IC 4329A	MBH–σ*	σ* = 225 ± 9 km s−1	Oliva et al. (1999)	8.24 ± 0.44	–1.25 ± 0.90	>2.911
NGC 5548	MBH–σ*	σ* = 195 ± 13 km s−1	Woo et al. (2010)	8.07 ± 0.46	–0.25 ± 0.46	${1.053}_{-0.788}^{+2.659}$
Mrk 509	X-ray variance	log(${\sigma }_{{\rm{rms}}}^{2}$) = −3.24 ± 0.59	Zhou et al. (2010)	8.21 ± 0.62	−0.05 ± 0.62	${1.098}_{-0.993}^{+4.413}$
NGC 7469	MBH–σ*	σ* = 131 ± 5 km s−1	Nelson et al. (2004)	7.34 ± 0.45	–0.25 ± 0.45	${0.053}_{-0.042}^{+0.122}$

Notes. Column (2) denotes the method used to determine M_BH. Three methods are used: the M_BH–σ_* relation from Gültekin et al. (2009); the X-ray variance scaling with M_BH either from McHardy (2013, who characterizes this in terms of the Normalized Variability Amplitude or NVA) or from Zhou et al. (2010, who provide measurements of the rms variability ${\sigma }_{{\rm{rms}}}^{2}$); the empirical relation of L₅₁₀₀ with R_BLR from Vestergaard (2002). Column (3) lists the relevant observable used in the method. Column (4) gives the reference for the measurement of the observable. Column (5) lists the resultant M_BH value, and Column (6) the difference in log M_BH with respect to the value quoted and used in the main paper (Table 1). Finally, Column (7) lists the corresponding value of R_Fe using the alternate M_BH estimate. For IRAS 13349+2438, we assume a large error of 0.5 dex in M_BH as in De Marco et al. (2013). Note that for this source ${v}_{{\rm{FWHM}}}^{{\rm{H}}\beta }$ = 2800 ± 180 km s⁻¹ is reported in Grupe et al. (2004).

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To summarize, we have discussed and quantified some caveats to our work, including those from combining multiple data sets, limited instrumental resolution, and biases related to simplistic (single Gaussian) modeling of a likely complex line feature. None of these issues appears to cause a bias strong enough to overwhelm our main result of R_dust acting as an outer envelope to R_Fe. We point out current shortfalls in terms of sample completeness, although the apparent heterogeneous nature of sample selection argues against any strong selection bias. We also explore the use of alternate black hole masses and their influence on our main result, which do not change our main result.

Although we cannot rule out the presence of a combination of these various biases, it would be surprising if they were to conspire to produce a limit of R_Fe around R_dust, which is a completely independent quantity. However, we acknowledge the current limited sample size and the paucity of line variability measurements. More complete sampling of AGNs in both X-rays and in the infrared are clearly important for drawing robust conclusions, and we discuss future possibilities in the next section.

This field is expected to leap forward with the imminent launch of the Astro-H mission (Takahashi et al. 2014). With a spectral resolution of ≈4–7 eV, the Soft X-ray Spectrometer (SXS) employing calorimetric photon detection will provide an improvement in spectral resolution of ≈6–10 as compared to present grating instruments. Because R_Fe has an inverse quadratic scaling with line velocity, SXS will extend the radius out to which the Kα emission can be localized by a factor of ≳40 in any individual object. This assumes a single Gaussian feature. Individual ionization stages will not all be necessarily separable by the SXS, although some deconvolution will be possible. In terms of the overall sensitivity to line parameter measurement, this can be quantified in terms of a "figure of merit," which combines the effective area and the spectral resolution. Relative to HEG, the overall SXS sensitivity to line detection is expected to be improved by a factor of ≈7, extending our reach to correspondingly faint (and more distant) systems. However, the improvement in the corresponding figure of merit for the detection of line broadening (which is key to our work) is about two orders of magnitude relative to the HEG.⁹

In the next decade, the Athena mission (Nandra et al. 2013) will have a much larger collecting area than Astro-H, pushing such studies out to high redshift. Reverberation of the narrow Fe Kα core for more objects would also be an independent constraint on R_Fe. Variable narrow lines in some cases have already been mentioned. In addition, Liu et al. (2010) found Fe line reverberation on a timescale of ∼20–40 day in NGC 5548. This would push R_Fe a factor of ∼1.5–3 lower than our estimate in Table 1, but still consistent within the uncertainty on R_Fe.

And in a similar vein, enlarged NIR coverage will greatly help to better fill the R_Fe versus R_dust parameter space. There are currently a total of 31 AGNs with measurements of R_dust. This includes sources from the references cited in Table 1 as well as a few other published and unpublished sources (e.g., GQ Com; Sitko et al. 1993). We have included 13 of these (42%) in our X-ray cross-matched sample. Although the Keck Interferometer is no longer available, more size measurements will be available soon (Kishimoto et al. 2015, submitted) from the three-telescope beam combiner AMBER at the VLTI. This will boost the present sample and address projection effects by invoking closure phase data. And although the MAGNUM telescope is now decommissioned, there are many other world-wide efforts ongoing to obtain AGN NIR time lags (e.g., Pozo Nuñez et al. 2015). LSST could also serve as a dust reverberation machine providing thousands of time lag measurements (Hönig 2014). Finally, in the distant future, we can expect to directly resolve the torus in many more sources with larger interferometers, such as the Planet Formation Imager (Monnier et al. 2014).