# Probing the Extent of Fe K$\alpha$ Emission in Nearby Active Galactic Nuclei
using Multi-Order Analysis of Chandra High Energy Transmission Grating Data

Megan Masterson MIT Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Institute of
Astronomy, University of Cambridge, Madingley Road, Cambridge CB3 0HA, UK
Christopher S. Reynolds Institute of Astronomy, University of Cambridge,
Madingley Road, Cambridge CB3 0HA, UK Megan Masterson<EMAIL_ADDRESS>
###### Abstract

We present a study of the narrow Fe K$\alpha$ line in seven bright, nearby AGN
that have been observed extensively with the Chandra High Energy Transmission
Grating (HETG). The HETG data reveal a wider Fe K$\alpha$ line in the first
order spectrum than in the second and third order spectra, which we interpret
as the result of spatially extended Fe K$\alpha$ emission. We utilize these
differences in narrow Fe K$\alpha$ line widths in the multi-order Chandra HETG
spectra to determine the spatial extent and intrinsic velocity width of the
emitting material in each object. We find that there is modest evidence for
spatially extended emission in each object, corresponding to extension of
$r\sim 5-100$ pc. These distances are significantly larger than those inferred
from velocity widths assuming gravitational motions, which give $r\sim 0.01-1$
pc. This implies that either the gas is emitting at a range of radii, with
smaller radii dominating the velocity width and larger radii dominating the
spatial extent, or that the gas is exhibiting non-gravitational motions, which
we suggest would be outflows due to slight excess redshift in the line and
velocities that exceed the freefall velocity. We also use the spatial extent
information to estimate the mass of the emitting gas by counting fluorescing
iron atoms, finding masses on the order of $M_{\mathrm{gas}}\sim
10^{5}-10^{8}\,M_{\odot}$. Future work with observatories like XRISM will be
able to extend this study to a larger number of AGN and decrease uncertainties
that arise due to the low signal-to-noise of the higher order HETG data.

Active galactic nuclei (16)–High energy astrophysics (739)–Seyfert galaxies
(1447)–Supermassive black holes (1663)–X-ray active galactic nuclei (2035)

††software: CIAO (Fruscione et al., 2006), XSPEC (Arnaud, 1996), MARX (Davis
et al., 2012), Astropy (Astropy Collaboration et al., 2013, 2018), Matplotlib
(Hunter, 2007), NumPy (van der Walt et al., 2011)

##  1 Introduction

Active galactic nuclei (AGN) are powered by accretion onto a central
supermassive black hole (SMBH) with $M_{\mathrm{BH}}\sim
10^{6}-10^{9.5}M_{\odot}$. Canonically, the key components of the environment
around the SMBH are an accretion disk, the broad-line region (BLR) in which
fast moving gas clouds produce broad optical emission lines, a parsec-scale
torus responsible for large amounts of obscuration and reprocessing of
emission from the central engine (see Hickox & Alexander, 2018, for a review),
and the narrow-line region (NLR), which extends from beyond the torus into the
host galaxy and produces narrow optical emission lines. The torus is an
important factor in the unified model of AGN (e.g. Antonucci, 1993; Urry &
Padovani, 1995; Netzer, 2015), but it is still not well understood how this
structure connects with the gas reservoir in the host galaxy through feedback
and fueling of the central SMBH.

The X-ray spectrum of an AGN is dominated by two components, a power law
continuum with a high energy cut-off, which is a result of UV photons from the
accretion disk being Compton up-scattered by the hot corona (T $\sim
10^{8}-10^{9}$ K; e.g. Fabian et al., 2015; Kara et al., 2017; Ricci et al.,
2017; Tortosa et al., 2018; Ursini et al., 2019), and a reflection spectrum,
which arises due to reprocessing of coronal emission in the surrounding
circumnuclear material (Haardt & Maraschi, 1991, 1993). The reflection
spectrum is a powerful probe of the geometry, kinematics, and ionization state
of the surrounding accretion flow and circumnuclear material (George & Fabian,
1991).

One of the most prominent features in the X-ray reflection spectrum is the Fe
K$\alpha$ emission line, as iron has a high cosmic abundance and fluorescent
yield. The shape of the neutral iron line probes the location and dynamics of
the emitting material. Reflection from the inner accretion disk produces a
broad iron line (FWHM $\gtrsim 50,000$ km s-1), which is asymmetric due to
relativistic effects such as gravitational redshift and the transverse Doppler
effect (Fabian et al., 1989). The redward extent of this line can be modeled
to determine the innermost stable circular orbit (ISCO) and can therefore be
used to probe black hole spin (e.g. Tanaka et al., 1995; Brenneman & Reynolds,
2006; Risaliti et al., 2013). To date, modeling the broad iron line is our
best way of measuring black hole spins in AGN (see Reynolds, 2021, for a
recent review). Reflection from more distant material produces the narrow core
of the neutral iron line (FWHM $\lesssim 10,000$ km s-1), which is a
ubiquitous feature of AGN X-ray spectra with sufficient signal-to-noise (e.g.
Kaspi et al., 2001; Yaqoob et al., 2001; Zhou & Wang, 2005; Fukazawa et al.,
2011; Hitomi Collaboration et al., 2018). The exact origin of the narrow Fe
K$\alpha$ line is still uncertain, with possible origins including the outer
accretion disk, BLR, or torus. Various studies utilizing high resolution X-ray
spectroscopy of nearby AGN have revealed a range of Fe K$\alpha$ line widths
and no correlation with the optical BLR (Yaqoob & Padmanabhan, 2004; Nandra,
2006; Shu et al., 2010), whereas others utilizing both spectroscopy and
variability have found that most of the Fe K$\alpha$ emission is produced
within the dust sublimation radius (Gandhi et al., 2015; Minezaki &
Matsushita, 2015; Andonie et al., 2022). However, imaging data from the
Chandra X-ray Observatory has shown spatially resolved and significantly
extended Fe K$\alpha$ emission on scales of the order $\sim 10$s-$100$s pc in
a few nearby AGN (e.g. Young et al., 2001; Marinucci et al., 2013; Fabbiano et
al., 2017; Kawamuro et al., 2019; Ma et al., 2020; Yi et al., 2021).

Liu (2016) used data from the Chandra High Energy Transmission Grating (HETG)
to analyze the narrow Fe K$\alpha$ line in seven nearby AGN, chosen from the
Chandra data archive based on the requirement of having a measurable neutral
iron line in their second and third order spectra. The second and third order
HETG spectra have higher spectral resolution, but effective areas that are
approximately 15 times lower than the first order effective area at 6.4 keV,
the energy of the neutral iron line in question (Canizares et al., 2005).
Thus, using higher order spectra requires bright sources and significantly
longer exposures to have a useful signal-to-noise. For each of the objects
studied, Liu (2016) found that the narrow Fe K$\alpha$ line width was
consistently larger in the first order spectrum than in the combined second
and third order spectrum. The author attributed this unexpected discrepancy to
the first order spectra being overestimated due to miscalibration of the HETG.
In response to the Liu (2016) claim, Marshall (2017) noted that the larger
first order line widths were likely a result of spatially extended Fe
K$\alpha$ emission in these nearby sources.

For a simple Gaussian emission line, there are three key factors which can
affect the observed line width in dispersive spectroscopy, the intrinsic
instrument line spread function ($\sigma_{i}$), the Doppler effect
($\sigma_{v}$), and spatial extent ($\sigma_{\theta}$). Assuming these all
have Gaussian profiles, the total broadening of the line in detector space for
a simple linear diffraction grating is given by

$\sigma_{x}^{2}=\sigma_{i}^{2}+\left(\frac{Rm\lambda}{Pc}\right)^{2}\sigma_{v}^{2}+F^{2}\sigma_{\theta}^{2},$
(1)

where $m$ is the spectral order, $R$ is the Rowland distance of the HETG, $P$
is the grating period, and $F$ is the focal length of the High Resolution
Mirror Assembly (HRMA; see Marshall, 2017). The dependence on the spectral
order arises from the application of the one-dimensional grating equation,
which describes the dispersion angle, $\beta$, by

$\sin\beta=\frac{m\lambda}{P}\approx\frac{x}{R}.$ (2)

Thus, spatial extent is important if the quantity $m\sigma_{v}$ is small and
can only be ignored if $\sigma_{v}\gg 3400\frac{\sigma_{\theta}}{m\lambda}$ km
s-1, where $\sigma_{\theta}$ is in arcseconds and $\lambda$ is in Å (Marshall,
2017). Thus, at lower intrinsic velocities, the first order data will be more
sensitive to spatial extent than the second and third order data. This
suggests that the effect seen in Liu (2016) is the result of spatial extent
contributing to first order line widths.

However, actually measuring this spatial extent directly is difficult with
current X-ray imaging studies. The best spatial resolution at X-ray
wavelengths is $\sim 0.5"$ with Chandra. Hence, unlike with recent advances in
the sub-millimeter regime allowing for the central few parsecs around nearby
AGN to be resolved and mapped in detail, X-ray emission from the smallest
scales around the central engine is unresolved. Furthermore, when used in
spectral imaging mode, bright sources which enable a detailed study of the
neutral iron line at 6.4 keV will cause significant pile-up on the Chandra
detectors, in which multiple photons are counted as a single event. Pile-up
causes distortion of the CCD-determined spectrum and widening of the point
spread function (PSF) of the image. Thus, in practice, it is difficult to
perform direct imaging studies of extended Fe K$\alpha$ emission with Chandra
on scales less than a few arcseconds.

In this work, we utilize the differences in the narrow Fe K$\alpha$ line
widths for different spectral orders to obtain spatial information about the
X-ray fluorescent gas around nearby AGN on scales smaller than can be probed
with X-ray imaging studies. In Section 2, we outline our data reduction and
analysis techniques for the sample of AGN. This includes a discussion of how
the methodology was calibrated using simulated Chandra observations with MARX.
In Section 3, we present the results of the fits to all three spectral orders
and calculate distances to the black holes, cold gas masses, and mass outflow
rates. In Section 4, we discuss the implications of our findings, including
two possible explanations for discrepancies between the radius of emission
measured with spatial extents and velocity widths. Finally, our results are
summarized in Section 5. Throughout this paper, we adopt $H_{0}=70$ km s-1
Mpc-1, $\Omega_{\Lambda}=0.73$, and $\Omega_{M}=0.27$. All quoted errors
correspond to the 90% confidence level, unless otherwise noted.

##  2 Methods

###  2.1 Observations and Data Reduction

The Chandra HETG consists of two grating arrays, the high energy grating (HEG)
for 0.8-10.0 keV and medium energy grating (MEG) for 0.4-5.0 keV. Observations
with the HETG are done in combination with the Advanced CCD Imaging
Spectrometer (ACIS), which can determine the energy of detected photons. The
CCD-determined energy from ACIS-S allows for order sorting to be performed to
obtain high-resolution HETG spectra of multiple orders.

For this work, we analyzed the same sample of seven objects from Liu (2016),
NGC 1068, NGC 3783, NGC 4151, NGC 4388, NGC 4507, Mrk 3, and the Circinus
galaxy. These are the AGN in the Chandra data archive which have sufficient
HETG spectral data to probe the line width of the Fe K$\alpha$ line in second
and third order spectra. While the second and third order spectra have higher
resolution than the first order spectrum, their effective areas are smaller by
a factor of approximately $1/15$ (Canizares et al., 2005). Thus, requiring
high signal-to-noise second and third order spectra limits the sample to
bright, nearby AGN that have been extensively observed with the Chandra HETG.
Table 2.1 summarizes the Chandra Observation IDs (ObsIDs) used for each source
in this analysis. For Circinus, we do not include ObsID 374 in our analysis as
it was approximately 7 ks long, contributing negligibly to the total
observation time and lacking sufficient data above the iron line to properly
constrain the fit. For NGC 4151, we only utilize the first and second order
data due to issues identified in the third order data across different
observations. Given that NGC 4151 has one of the strongest narrow Fe K$\alpha$
features and we only need two spectral widths to fully classify the velocity
width, $\sigma_{v}$ and spatial extent, $\sigma_{\theta}$, we only use the
first and second order data in this object to avoid issues with the third
order spectrum.

HETG data were obtained from the Chandra archive and reprocessed using
CIAOv4.11 (Fruscione et al., 2006) and CALDBv4.8.5. We followed the standard
data reduction process for grating data with one exception. We decreased the
width of the masks on the grating arms used to extract the spectra to 18
pixels (approximately half the default value), which decreases the overlap
between the HEG and MEG arms and hence allows us to extend our HEG analysis to
higher energies. First, second, and third order spectra were extracted from
each observation, and the positive and negative spectra for each order were
combined to increase signal-to-noise using combine_grating_spectra. These
combined spectra were then grouped to have a minimum of 1 count per bin using
the grppha tool from HEASOFTv6.26.1. This binning is required in order to
correctly employ use of the $C$ statistic during spectral fitting. Spectra
were analyzed with the X-ray spectral-fitting package, XSPECv12.10.1f (Arnaud,
1996). Fits were completed using only HEG data, as it has a larger effective
area and better resolution at 6.4 keV, the energy of the neutral Fe K$\alpha$
line under investigation. All fits were performed in the 4-9 keV range and
with $C$ statistic used for minimization, which is suitable for Poisson-
distributed data (Cash, 1979).

Table 1: Source observation information Object | $z$ | $D$ | Ref. bbfootnotemark:  | ObsIDs Used | Total Exposure Time
---|---|---|---|---|---
|  | (Mpc) |  |  | (ksec)
NGC 1068 | 0.003793 | 14.4 | 1 | 332, 9148, 9149, 9150, 10815, 10816, 10817, | 438.8
|  |  |  | 10823, 10829, 10830 |
NGC 3783 | 0.00973 | 38.5 | 2 | 2090, 2091, 2092, 2093, 2094, 14991, 15626, | 1145.7
|  |  |  | 18192, 19694 |
NGC 4151 | 0.00332 | 16.1aafootnotemark:  | 3 | 335, 3052, 3480, 7829, 7830, 16089, 16090 | 630.7
NGC 4388 | 0.00842 | 18.1 | 4 | 9276, 9277 | 268.9
NGC 4507 | 0.0118 | 50.5 | – | 2150 | 138.2
Mrk 3 | 0.013509 | 57.9 | – | 873, 12874, 12875, 13254, 13261, 13263, 13264, | 389.3
|  |  |  | 13406, 14331 |
Circinus | 0.001448 | 4.2 | 5 | 4770, 4771, 10223, 10224, 10225, 10226, 10832, 10833, | 660.2
|  |  |  | 10842, 10843, 10844, 10850, 10872, 10873, 62877 |

###  2.2 Spectral Modeling

For each object in our sample, we model the 4-9 keV spectra with Galactic
absorption, neutral absorption at the source, a power law continuum, and the
neutral Fe K$\alpha$ emission line. All abundance values were adopted from
Wilms et al. (2000). Galactic absorption was modeled with tbabs with column
densities adopted from HI4PI Collaboration et al. (2016). We chose to model
absorption at the source with the ztbabs model, which accounts for a single
neutral absorber, redshifted to the source redshift. There are objects in our
sample whose absorption has been studied extensively and found to be more
complex than our simple model. For example, NGC 3783 has been shown to have
complex ionized absorption (e.g. Reynolds, 1997; Kaspi et al., 2002; Krongold
et al., 2003; Netzer et al., 2003). However, complex absorption in the X-ray
spectrum is far more prevalent at softer energies, below about 2 keV. Since we
are fitting from 4-9 keV, we do not include such complex absorption models as
our fit is not sensitive to them. Our continuum power law and Fe K$\alpha$
emission line are both redshifted to the source redshifts given in Table 2.1.
To model the Fe K$\alpha$ line, we use two Gaussian emission lines. The Fe
K$\alpha$ transition is a doublet due to a difference in energy released based
on the spin of the electron involved in the transition. The two lines,
K$\alpha_{1}$ and K$\alpha_{2}$, are at energies of at 6.404 keV and 6.391
keV, respectively. The laboratory observed flux ratio between the two
transitions is 2:1 (Bearden, 1967). Hence, we set the normalization of the
K$\alpha_{1}$ line to be twice the K$\alpha_{2}$ normalization and set their
widths to be the same.

When modeling the Fe K$\alpha$ line as seen by the HETG, we write the observed
width, $\sigma_{E\text{,measured}}$, directly in terms of the velocity width,
$\sigma_{v}$, and spatial extent, $\sigma_{\theta}$, outlined in Equation 1.
The instrumental width, $\sigma_{i}$, is accounted for in the data reduction
pipeline via the response matrix. Thus, we can write Equation 1 in terms of
the observed energy width by applications of the grating equation and Doppler
physics. Approximating the energy of the line to be much greater than the
width yields

$\sigma_{E\text{,measured}}^{2}=\left(\frac{E_{0}}{c}\right)^{2}\sigma_{v}^{2}+\left(\frac{FPE_{0}^{2}}{hcmR}\right)^{2}\sigma_{\theta}^{2},$
(3)

where $E_{0}$ is the energy of the line and the other quantities are as
defined in Equation 1. We approximate $E_{0}$ as the average of the
K$\alpha_{1}$ and K$\alpha_{2}$ energies. The advantage of writing the model
this way is that we can obtain more accurate error estimates on $\sigma_{v}$
and $\sigma_{\theta}$, which would be difficult otherwise because of how close
we are to the instrument resolution. This yields a final XSPEC model of
tbabs(ztbabs(zpower + zgauss + zgauss)). In fitting, we fix the Galactic
column density, redshift, and centroid energies of the Gaussian emission lines
and fit for the source column density, power law normalization, photon index,
line normalization, velocity width, and spatial extent.

For each object, we fit simultaneously to first, second, and third order
spectra for each observation. We allow the source absorption column density,
power law normalization, power law photon index, and line normalization to
vary for every ObsID. This accounts for changes in the source obscuration over
time and the intrinsic variability of the source. These same values are fixed
across all three spectral orders for a given observation when fitting, as the
normalizations are corrected for the effective area with response files. We
fix the value of $z$ for all model components to the values given in Table
2.1. Lastly, we fix $\sigma_{v}$ and $\sigma_{\theta}$ to be the same for all
observations and spectral orders of a given source. The difference in Fe
K$\alpha$ width that we are measuring then is accounted for in the dependence
on the spectral order, $m$, in the spatial extent in Equation 3. For a
constant $\sigma_{v}$ and $\sigma_{\theta}$, the velocity width contributes a
constant amount across all orders, but the spatial width contribution
decreases with increasing spectral order. This behavior should give the
observed behavior of decreasing total line width with spectral order from Liu
(2016), and permit independent measurements of $\sigma_{v}$ and
$\sigma_{\theta}$.

###  2.3 Calibration with MARX

Figure 1: Left: Simulation results for 10 individual simulations with
$\sigma_{v}=750$ km s-1 and $\sigma_{\theta}=0.4$”. Blue dotted lines show the
input values of $\sigma_{v}$ and $\sigma_{\theta}$. Black data points show the
fit values for $\sigma_{v}$ and $\sigma_{\theta}$ for each simulation. The red
data point shows the fit value $\sigma_{v}$ and $\sigma_{\theta}$ for a
simultaneous fit to all 10 simulations with $\sigma_{v}$ and $\sigma_{\theta}$
fixed to be the same between all simulations. All error bars show 1$\sigma$
(68%) confidence. Right: Calibration factor fits for the same 10 individual
simulations with $\sigma_{v}=750$ km s-1 and $\sigma_{\theta}=0.4$” as in left
figure. The blue dotted line shows $f=1$. Black data points show the fit
values for $f$ for each simulation. The red data dashed line shows the fit
value for $f$ for a simultaneous fit to all 10 simulations. The shaded red
area shows $1\sigma$ error on this combined fit value for $f$. All error bars
show 1$\sigma$ (68%) confidence.

We expect that the simple, one-dimensional grating equation used to obtain
Equation 3 may be insufficient to describe the complexity of the HRMA-HETG
geometry in detail. Hence, we use MARX (version 5.5.0) to calibrate our model
by simulating Chandra observations and fitting these simulated observations
with the methodology described previously. MARX is a Monte Carlo, ray-tracing
program used to create simulated Chandra observations (Davis et al., 2012). We
simulated a power law source with point geometry, since the corona has been
shown to be compact and quite close to the central supermassive black hole.
The Fe K$\alpha$ emission however, is spatially extended. Hence, we modelled
the Fe K$\alpha$ doublet with some intrinsic velocity width, $\sigma_{v}$, and
extended the emission over a Gaussian source with width $\sigma_{\theta}$. The
two components, the corona and the line source, were concatenated to make a
single simulated observation using marxcat and then processed in the standard
way with MARX and CIAO tools.

Simulations were run for a range of $\sigma_{v}$ and $\sigma_{\theta}$ values,
spanning a range of reasonable narrow Fe K$\alpha$ line widths, including
cases where $\sigma_{v}$ dominates $\sigma_{\theta}$, $\sigma_{\theta}$
dominates $\sigma_{v}$, and the two contribute similarly. The left panel of
Figure 1 shows an example of the fit that uses Equation 3 to relate the total
line width in each order to $\sigma_{v}$ and $\sigma_{\theta}$ for 10
simulations using input values of $\sigma_{v}=750$ km s-1 and
$\sigma_{\theta}=0.4$”. Each simulation was run for 10 Ms, giving us excellent
data with which to test the model. Following the reprocessing of the data
using standard CIAO tools, the first, second, and third order spectra were fit
using the same procedure as described in Section 2.1 and with the same model
as described in Section 2.2. The 10 simulations were each fit individually and
then fit simultaneously, with $\sigma_{v}$ and $\sigma_{\theta}$ fixed across
all simulations, similarly to how we fit simultaneously to all observations
for a given object.

The left panel of Figure 1 shows that while the model fits the velocity width
accurately, the use of Equation 3 leads to a spatial width that is
significantly underestimated. This trend also holds for the other combinations
of $\sigma_{v}$ and $\sigma_{\theta}$ that we tested. It is thus apparent that
Equation 3 does not fully capture the response of the HETG to finite spatial
extent. We adopt a calibration factor, $f$, to attain the true value of
$\sigma_{\theta}$ when fitting. This is implemented by inserting this
calibration factor $f$ into the spatial width term of Equation 3, giving

$\sigma_{E\text{,measured}}^{2}=\left(\frac{E_{0}}{c}\right)^{2}\sigma_{v}^{2}+\left(f\frac{FPE_{0}^{2}}{hcmR}\right)^{2}\sigma_{\theta}^{2}.$
(4)

The value of $f$ is then determined by fixing $\sigma_{v}$ and
$\sigma_{\theta}$ to the input values and fitting the simulated data to $f$.
The right panel of Figure 1 shows the result of fitting to $f$ for the same
simulations as in the left panel. We adopt the average value of the
simultaneous fits for five different input combinations of $\sigma_{v}$ and
$\sigma_{\theta}$, $f=0.73$, for the analysis of the data.

We stress that this is a first order correction to Equation 3. While the
calibration factors show general consistency across a range of velocity and
spatial widths, there is some hint of small deviations across different input
values of $\sigma_{v}$ and $\sigma_{\theta}$. We suspect that there is a
second order correction due to a deviation from gaussianity that is on the
order of 10% of $f$ and potentially dependent on $\sigma_{v}$,
$\sigma_{\theta}$, and the signal-to-noise. For the purposes of this work, we
simply adopt the first order correction, $f=0.73$, as these small, second
order corrections are a negligible source of uncertainty.

###  2.4 MCMC Fitting of AGN Data

Adopting a calibration factor of $f=0.73$, we performed fits to the spectral
model described in Section 2.2 using the Markov Chain Monte Carlo (MCMC)
method implemented in XSPEC. We utilized the Goodman-Weare algorithm (Goodman
& Weare, 2010), with 5 independent chains for each object, where each chain
has $10^{6}$ steps and 500 walkers. For NGC 3783, NGC 4151, NGC 4388, NGC
4507, and Circinus, the MCMC chains were started based on the covariance
matrix derived from the initial fit to the data. In the Circinus chains, the
covariance matrix used to populate the initial distribution of walkers was
scaled by $10^{-1}$ to allow the walkers to fully explore parameter space. The
burn-in length for these objects was $2\times 10^{5}$ chain steps. For NGC
1068 and Mrk 3, the MCMC chains were populated from a uniform distribution for
all parameters across a wide range of reasonable values. This was done because
the initial fit performed in XSPEC using $C$ statistic minimization collapsed
to essentially zero spatial extent, resulting in subsequent chains not fully
exploring $\left(\sigma_{v},\sigma_{\theta}\right)$ parameter space. These
objects required a longer burn-in length of $10^{6}$ chain steps for
convergence. For each object, all 5 chains are then combined into a single
composite chain with a total of $5\times 10^{6}$ chain steps and used for
analysis. Reported best fit values are median values from the combined MCMC
chains and uncertainties are the 5th and 95th percentile chain values.

For each object, we checked convergence with the Geweke diagnostic (Geweke,
1992) and the Gelman-Rubin scale reduction factor, $\hat{R}$ (Gelman & Rubin,
1992). The Geweke diagnostic, which compares the mean of a given parameter at
the beginning of the chain (first 10%) and at the end of the chain (last 50%),
was found to be $\left|z_{G}\right|<1$ for each fit parameter in each chain.
The Gelman-Rubin scale reduction factor measures convergence across
independent chains, and satisfies $\hat{R}\lesssim 1.2$ for each fit
parameter. Together, these two convergence factors indicate that the chains
have converged to the posterior distribution.

##  3 Results

###  3.1 Velocity and Spatial Widths

For each object in the sample, we obtained a fit to $\sigma_{v}$ and
$\sigma_{\theta}$ with uncertainty determined using the MCMC posterior
distributions. These fit values are given in Table 2, along with the line
width in energy space for each order as calculated with Equation 4. The values
and errors on the line width in each order are computed directly from the MCMC
chains, taking $\sigma_{v}$ and $\sigma_{\theta}$ for each chain step and
converting these to a total energy width for each order. We find that the
unexpected behavior from Liu (2016), in which first order line widths are
larger than their second and third order counterparts, can indeed be described
by the intrinsic velocity width and spatial extent, as proposed in Marshall
(2017).

Table 2: Fits to $\sigma_{v}$ and $\sigma_{\theta}$, resulting Fe K$\alpha$ line widths for different order spectra, and Fe K$\alpha$ equivalent widths  Object | $\sigma_{v}$ | $v_{\mathrm{FWHM}}$ | $r_{v}$ | $\sigma_{\theta}$ | $r_{\theta}$ | $\sigma(\pm 1)$ | $\sigma(\pm 2)$ | $\sigma(\pm 3)$ | EWunabs |  $C$ / dof
---|---|---|---|---|---|---|---|---|---|---
| (km s-1) | (km s-1) | (pc) | (arcsec) | (pc) | (eV) | (eV) | (eV) | (eV) |
NGC 1068 | $898_{-432}^{+231}$ | $2110_{-1020}^{+540}$ | $0.013_{-0.005}^{+0.035}$ | $0.31_{-0.28}^{+0.41}$ | $21.8_{-19.6}^{+28.4}$ | $21.6_{-3.6}^{+4.0}$ | $19.7_{-6.1}^{+4.6}$ | $19.4_{-7.7}^{+4.8}$ | 10 - 510 | $2189.7/2617$
NGC 3783 | $592_{-245}^{+177}$ | $1390_{-580}^{+420}$ | $0.088_{-0.036}^{+0.168}$ | $0.30_{-0.26}^{+0.25}$ | $55.5_{-49.0}^{+46.0}$ | $15.5_{-2.7}^{+2.9}$ | $13.3_{-3.4}^{+3.3}$ | $12.9_{-4.2}^{+3.6}$ | $79.5_{-23.9}^{+29.8}$ | $12331.4/13673$
NGC 4151 | $894_{-359}^{+185}$ | $2100_{-850}^{+440}$ | $0.059_{-0.019}^{+0.106}$ | $0.39_{-0.34}^{+0.33}$ | $30.3_{-26.6}^{+25.5}$ | $22.1_{-2.3}^{+2.4}$ | $19.8_{-5.0}^{+3.4}$ | $19.4_{-6.4}^{+3.7}$ | $108.6_{-57.8}^{+55.2}$ | $7925.8/8501$
NGC 4388 | $619_{-506}^{+413}$ | $1460_{-1190}^{+970}$ | $0.023_{-0.015}^{+0.672}$ | $0.57_{-0.49}^{+0.33}$ | $49.9_{-42.8}^{+29.2}$ | $21.1_{-5.3}^{+5.8}$ | $15.5_{-5.2}^{+7.0}$ | $14.2_{-6.7}^{+8.0}$ | $101.3_{-19.7}^{+20.4}$ | $2087.7/2367$
NGC 4507 | $561_{-458}^{+379}$ | $1320_{-1080}^{+890}$ | $0.806_{-0.519}^{+22.844}$ | $0.36_{-0.33}^{+0.38}$ | $89.1_{-79.6}^{+94.1}$ | $16.7_{-6.0}^{+7.2}$ | $13.3_{-5.8}^{+7.3}$ | $12.5_{-6.8}^{+7.7}$ | $45.5_{-9.6}^{+10.8}$ | $1091.2/1117$
Mrk 3 | $976_{-556}^{+259}$ | $2300_{-1310}^{+610}$ | $0.485_{-0.182}^{+2.142}$ | $0.36_{-0.33}^{+0.45}$ | $101.0_{-91.5}^{+126.3}$ | $23.7_{-3.9}^{+4.4}$ | $21.4_{-7.5}^{+5.2}$ | $21.1_{-9.7}^{+5.4}$ | $157.9_{-81.4}^{+103.8}$ | $2138.7/2824$
Circinus | $341_{-108}^{+106}$ | $800_{-260}^{+250}$ | $0.015_{-0.006}^{+0.017}$ | $0.37_{-0.10}^{+0.07}$ | $7.5_{-2.0}^{+1.4}$ | $12.4_{-1.1}^{+1.1}$ | $8.9_{-1.3}^{+1.5}$ | $8.0_{-1.7}^{+1.9}$ | 20 - 770 | $6452.9/7400$

First, second, and third order spectra for all seven objects in our sample are
shown in Figure 2. These spectra contain the combined data from all
observations listed in Table 2.1. Each spectrum is plotted with error bars,
and the model for each order containing fit parameters averaged over all
observations is over-plotted in bold. The first order spectrum is reduced in
normalization by a factor of 15 to roughly account for the difference in
effective areas between the orders, and all spectra are binned for plotting
purposes only. However, we stress that this is not how we performed the fits,
as we fit each observation for each object to its own absorption, continuum
parameters, and line normalization to account for variability (see Section 2.2
for details). This is simply a representation of the data we are working with
and the features of the spectrum at 6.4 keV.

Figure 2: Spectra for all seven objects in our sample with all ObsIDs combined
together and corrected for redshift. Black lines with error bars indicate the
first order spectrum, decreased by a factor of 15 to roughly account for the
difference in effective areas. Blue and orange lines with error bars indicate
the second and third order spectra, respectively. For NGC 4151, only the first
and second order spectra are shown. All first order spectra are binned to a
minimum signal-to-noise of 3. For NGC 3783 and NGC 4151, the higher order
spectra are also binned to a minimum signal-to-noise of 3. For Circinus, the
higher order spectra are binned to a minimum signal-to-noise of 2. Due to low
signal-to-noise in the remaining objects, the higher order spectra in the
remaining objects are binned in groups of 5 bins. All binning is done solely
for visual purposes in plotting. The models with fit parameters from Tables 2
and 4 averaged over all observations are over-plotted in bold.

We note that our errors on the second and third order line widths are
significantly smaller than those in Liu (2016), which show that the measured
line width in the combined higher order spectrum is an upper limit for four of
the seven sources. Our errors are smaller because we did not fit the second
and third order data individually. Rather, we fit all three orders
simultaneously and linked them through writing the line widths in terms of
$\sigma_{v}$ and $\sigma_{\theta}$ and tying these values across all three
orders. Given that the first order data has significantly higher signal-to-
noise than the second and third order data, our fit to $\sigma_{v}$ and
$\sigma_{\theta}$ is most sensitive to first order data. The second and third
order data allows us to break the degeneracy between $\sigma_{v}$ and
$\sigma_{\theta}$ and estimate where in this parameter space each object lies.

Figure 3: Corner plots for each object in the sample. For each corner plot,
the two histogram plots show the posterior distributions for $\sigma_{v}$ and
$\sigma_{\theta}$. The dashed vertical line shows the median chain value,
reported in Table 2. The two dotted vertical lines show the 90% confidence
bounds on the velocity widths and spatial extents. The other plot shows the
density of chain steps in $(\sigma_{v},\sigma_{\theta})$ space. The lines show
lines of constant line width, $\sigma_{E\text{,measured}}$, in each order at
the median values given in Table 2. The red dashed line corresponds to first
order, blue solid line to second order, and orange dot-dashed line to third
order.

However, we also see less spatial extent in NGC 1068 and Mrk 3 than would be
expected given the fits from Liu (2016). That is, our second and third order
line widths for these objects are significantly larger than those from Liu
(2016), which are quite small (upper limits of $\sim 5-7$ eV). We investigate
this discrepancy by fitting the second and third order data of both of these
objects individually and fixing the continuum parameters to those from our
overall fit. In the third order spectrum of NGC 1068, we find anomalously high
and poorly constrained energy widths on the order of $\sim 100-400$ eV, a
consequence of the limited data quality in the third order spectrum. In the
second order spectrum for NGC 1068 and in both second and third order spectra
for Mrk 3, we find larger line widths than those from Liu (2016), on the order
of $\sim 25-50$ eV, albeit it with quite large error bars ranging from $\sim
10-25$ eV. With slightly different fitting methods and more recent calibration
files implemented, we find no evidence for such small line widths in higher
order HETG data for NGC 1068 and Mrk 3.

Corner plots for each object in our sample are shown in Figure 3. In each
corner plot, the two histograms show the posterior distributions of
$\sigma_{v}$ and $\sigma_{\theta}$. The bottom left plot shows the density of
chain steps in $(\sigma_{v},\sigma_{\theta})$ space. Lines of constant total
line width are over-plotted, where the energy is equal to the values given in
Table 2, the median $\sigma_{\mathrm{E,\,measured}}$ from Equation 4. The red
dashed line, which shows constant first order line width, tends to trace the
shape of the distribution of $(\sigma_{v},\sigma_{\theta})$ values quite well.
This illustrates that our fitting is most sensitive to first order data and is
picking out a roughly constant first order line width over the entire chain.
The blue and green dashed lines show constant line width in second and third
order, respectively. These values are breaking the degeneracy between
$\sigma_{v}$ and $\sigma_{\theta}$ that would not be possible with first order
data alone. From the plots, we can see that with increasing order, the effect
of $\sigma_{\theta}$ decreases. In third order, the green dashed line is
nearly vertical, indicating that the third order data is relatively
insensitive to the value of $\sigma_{\theta}$ compared to the first order
data.

For each object and each observation, additional fit parameters from the MCMC
analysis are given in Section A of the Appendix. These include the source
absorption column density ($N_{H}$), photon index ($\Gamma$), power law
normalization ($K_{\mathrm{power}}$), and the line normalization
($K_{\mathrm{line}}$). We note that for NGC 1068 and Circinus, our source
absorption column densities are significantly lower than those reported in the
literature from more complex modeling of the absorption (e.g. Matt et al.,
1997, 1999; Arévalo et al., 2014; Bauer et al., 2015; Zaino et al., 2020).
These two sources are known Compton-thick AGN, whereby the obscuring material
is optically thick to electron scattering (with $N_{H}\geq 1.5\times 10^{24}$
cm-2). Similarly, Mrk 3 has been shown to be near the Compton-thick limit
along the line of sight (e.g. Cappi et al., 1999; Yaqoob et al., 2015;
Guainazzi et al., 2016). When the nucleus is this obscured, it can be
difficult to obtain information about the intrinsic X-ray properties. This is
addressed further in Section 4.2, where we discuss how this impacts our
results and outline prior work that has been done in an attempt to unveil the
nuclei of these objects and determine their intrinsic X-ray properties.

###  3.2 Extended Fe K$\alpha$ Emission

From both $\sigma_{v}$ and $\sigma_{\theta}$, we can estimate the distance of
the gas from the black hole. Assuming the motion is purely gravitational with
the potential dominated by the supermassive black hole, and the emitting gas
is far enough away from the black hole to ignore general relativistic effects,
this is given by the simple Keplerian formula, $r=GM_{\mathrm{BH}}/\langle
v^{2}\rangle$. Adopting the assumption that the velocity dispersion is related
to the FWHM velocity by $\langle
v^{2}\rangle=\frac{3}{4}v_{\mathrm{FWHM}}^{2}$ (Netzer et al., 1990), the
distance from the black hole is

$r_{v}=\frac{4GM_{\mathrm{BH}}}{3v_{\mathrm{FWHM}}^{2}}.$ (5)

We adopt black hole masses from the literature, given in Table 3.3. For our
fit values to $\sigma_{v}$, this gives distances on the order of $r_{v}\sim
0.01-1$ pc, or emission coming from the BLR or inner parts of the torus.

The spatial extent, $\sigma_{\theta}$, provides another measurement for the
distance of the material from the black hole. With the distance to the galaxy,
$D$, the spatial extent in angular units is converted to a physical distance
simply by

$r_{\theta}=D\sigma_{\theta}.$ (6)

The two most distant galaxies, NGC 4507 and Mrk 3, can be well-described by
Hubble flow distances, $D=cz/H_{0}$. For the remainder of the galaxies in our
sample, we use distances determined from other measures in the literature,
given in Table 2.1. For our fits to $\sigma_{\theta}$, we obtain distances on
the order of $r_{\theta}\sim 5-100$ pc, indicating that the emission is more
extended than the typical BLR or inner torus.

These distances for each object are given in Table 2 and a comparison of the
distances measured using the spatial extent and using the velocity width is
shown in Figure 4. We find that distances measured with the velocity width are
significantly smaller than the distances from spatial extent. Further
discussion of how this informs our understanding of the origin of the narrow
Fe K$\alpha$ line follows in Section 4.1.

Figure 4: Comparison of the distance of the emitting gas from the black hole,
using two different measurements. The black dashed line shows equality between
the two estimates. The distances measured by the spatial extent, $r_{\theta}$,
are consistently larger than the distances than the distance measured by the
velocity width, $r_{v}$ under the assumption of gravitational motions for all
seven objects in the sample. The values in this figure are reported in Table 2
and two possible explanations for the discrepancy are discussed in Section
4.1.

###  3.3 Mass Estimates

Following the steps outlined in Hitomi Collaboration et al. (2018), which
builds off of the equivalent width analysis originally presented in Reynolds
et al. (2000), we estimate the mass of the X-ray irradiated gas. The average,
unabsorbed equivalent width measured from the spectral model is used to
calculate the quantity $f_{\mathrm{cov}}N_{H}$ via Equation (1) of Hitomi
Collaboration et al. (2018),

$EW_{\mathrm{Fe}}\approx
65\,f_{\mathrm{cov}}\left(\frac{Z_{\mathrm{Fe}}}{Z_{\odot}}\right)\left(\frac{N_{H}}{10^{23}\,\mathrm{cm}^{-2}}\right)\,\mathrm{eV}.$
(7)

Here $f_{\mathrm{cov}}$ is the covering fraction of the material, allowing for
a non-uniform, clumpy or patchy distribution of material, and $N_{H}$ is the
column density. This calculation stems from counting fluorescing iron atoms
based on the photoionization probability, fluorescent yield, and incident
flux, and is scaled from the initial calculation of
$EW_{\mathrm{Fe,\,max}}\approx 65$ eV with $Z_{\mathrm{Fe}}=Z_{\odot}$ and
$N_{H}=10^{23}$ cm-2 for NGC 4258 from Reynolds et al. (2000). The details
behind this calculation are given in full in Section B of the Appendix. Then,
using the quantity $f_{\mathrm{cov}}N_{H}$ and assuming the gas is located in
a shell at radius $r$, the gas mass can be estimated using
$M_{\mathrm{gas}}=4\pi r^{2}f_{\mathrm{cov}}N_{H}\mu m_{p}$, where $\mu$ is
the mean molecular weight. We adopt $\mu=1.3$ for neutral, solar metallicity
material, $Z_{\mathrm{Fe}}=Z_{\odot}$ for solar iron abundance, and use the
distance measured from our fit to the spatial extent, $r_{\theta}$.

This calculation requires the total equivalent width of the Fe K$\alpha$ line,
which is found by adding the equivalent widths for the K$\alpha_{1}$ and
K$\alpha_{2}$ transitions from the first order spectrum and averaging over all
observations for a given object. However, the equivalent width in Equation 7
is the unabsorbed equivalent width. To compute the unabsorbed equivalent width
for gas mass estimates, we first fit to the model tbabs(zgauss + zgauss +
ztbabs*zpower), where only the continuum is being obscured at the source.
Then, we remove the ztbabs component to compute the Fe K$\alpha$ equivalent
width measured against the unabsorbed incident X-ray flux. To fit this model,
we fix the fit parameters $N_{H}$, $\Gamma$, $K_{\mathrm{power}}$,
$\sigma_{v}$ and $\sigma_{\theta}$, leaving only the line normalization
($K_{\mathrm{line}}$) free to vary as we expect this to be the only affected
parameter in changing the model. This assumption was explicitly verified for
NGC 3783, NGC 4151, NGC 4388, and NGC 4507. The normalization of the Fe
K$\alpha$ line decreases in this new model because the line emission is no
longer attenuated by the absorption. The amount of decrease in
$K_{\mathrm{line}}$ is positively correlated with column density of the
absorber, $N_{H}$, as expected.

These unabsorbed equivalent width values are given in Table 2. We report the
Compton-thin equivalent widths with errors that account for the uncertainties
in the original fit parameters as well as those from the new fit to
$K_{\mathrm{line}}$. To assess these errors, we took 100 random draws from the
MCMC chains for the fixed fit parameters and followed the above procedure to
fit to $K_{\mathrm{line}}$ for each draw. Then for each draw, we took 100
random draws for $K_{\mathrm{line}}$ from a Gaussian distribution centered at
the fit value of $K_{\mathrm{line}}$ with standard deviation based on the fit
error. This gives us a distribution of unabsorbed equivalent width values from
which we compute uncertainties.

For the two Compton-thick sources in our sample (NGC 1068 and Circinus),
accurately determining the intrinsic continuum is extremely difficult as the
majority of the observed emission in the 4-9 keV range is scattered and
reprocessed emission. The intrinsic nuclear continuum is best determined by
including hard X-ray data in the modeling, which contains more transmitted
emission than the soft energy band. Such modeling is beyond the scope of this
work and hence, for the Compton-thick sources in our sample, we report a range
of equivalent width values, which represent the extremes of possible
equivalent width values depending on different assumptions about the continuum
emission.

As an upper bound on the equivalent width, we assume that the observed
continuum is the continuum that the Fe K$\alpha$ emitting material sees. In
reality, this is likely all scattered emission, but this provides a lower
bound on the continuum flux and hence an upper bound on the equivalent width.
To compute the lower bound for the equivalent width in our Compton-thick
sources, we use the nuclear continuum estimated from detailed, broad-band
modeling including NuSTAR data from prior work (e.g. NGC 1068–Bauer et al.
2015; Marinucci et al. 2016, Circinus–Arévalo et al. 2014; Uematsu et al.
2021). In this scenario, the Fe K$\alpha$ emitting material sees a much
stronger continuum, leading to a lower equivalent width measurement. The
picture is certainly more complicated than these two extremes, especially
given that the Fe K$\alpha$ emitting material likely emits at a range of
radii, thus seeing different illuminating radiation and being obscured
differently depending on the location of the emission. However, the ranges
provided in Table 2 for the equivalent width of the Compton-thick sources
capture the uncertainty in determining the intrinsic continuum that the Fe
K$\alpha$ emitting material sees and the location of the Fe K$\alpha$ emitting
material relative to the obscurer. We further discuss the uncertainty
associated with modeling Compton-thick sources in Section 4.2.

Adopting the unabsorbed equivalent widths from Table 2, we compute the mass of
the cold, X-ray irradiated gas. The masses and their uncertainties are given
in Table 3.3. As with the equivalent widths, we report a range of gas masses
corresponding to the equivalent width ranges for the Compton-thick sources in
our sample. For the Compton-thin sources, the uncertainty in the gas masses
includes the uncertainty in both the spatial extent and the unabsorbed
equivalent width by using the distribution of unabsorbed equivalent width
values and corresponding $r_{\theta}$ chain value from each equivalent width
measurement. Most of our mass measurements are close to upper limits, with
large uncertainty due to the large uncertainty on the spatial extent of the
emitting material. It is important to also note that we assumed solar
abundance for iron, as our fit does not constrain this value, but that the gas
mass is proportional to $Z_{\mathrm{Fe}}^{-1}$. Having non-solar abundances
could change the result by a factor of a few, but we expect that our results
are accurate to within an order of magnitude.

In Section 4.1, we explore two possibilities for the discrepancy between the
distance measurements from Section 3.2, one of which is that the Fe K$\alpha$
emitting material is outflowing. Using the mass measurements, we can thus
estimate the mass outflow rates of the cold gas. Adopting the velocity width,
$\sigma_{v}$, to be the characteristic outflow velocity of the gas and
$r_{\theta}$ to be the distance of the emitting gas from the black hole, the
mass outflow rate is given by

$\dot{M}_{\mathrm{out}}=\frac{M_{\mathrm{gas}}v_{\mathrm{out}}}{r}\sim\frac{M_{\mathrm{gas}}\sigma_{v}}{r_{\theta}}.$
(8)

The outflow rates for each object is given in Table 3.3. We note that, like
the mass estimates from earlier in the section, many of these values are
highly uncertain. The similarities between these outflow rates and molecular
outflow rates are discussed further in Section 4.3.

Table 3: Cold Gas Mass, Luminosity, and Mass Outflow Rates Object | $M_{\mathrm{gas}}$ | $\dot{M}_{\mathrm{out}}$ | $\log L_{\mathrm{bol}}$ | Ref. | $\log M_{\mathrm{BH}}$ | Ref. | $\lambda_{\mathrm{Edd}}$
---|---|---|---|---|---|---|---
| ($10^{7}M_{\odot}$) | ($M_{\odot}$ yr-1) | (erg s-1) |  | ($M_{\odot}$) |  |
NGC 1068 | 0.06-5$\dagger$$\dagger$footnotemark:  | 20-2100$\dagger$$\dagger$footnotemark:  | $44.7$ | 1 | $7.0$ | 5 | $0.399$
NGC 3783 | $5.3_{-5.2}^{+12.7}$ | $500_{-420}^{+480}$ | $44.2$ | 2 | $7.47$ | 6 | $0.042$
NGC 4151 | $1.8_{-1.8}^{+5.3}$ | $440_{-380}^{+570}$ | $43.7$ | 3 | $7.66$ | 7 | $0.008$
NGC 4388 | $4.9_{-4.7}^{+8.7}$ | $460_{-400}^{+620}$ | $43.7$ | 2 | $6.93$ | 8 | $0.047$
NGC 4507 | $6.3_{-6.2}^{+23.8}$ | $360_{-300}^{+600}$ | $44.3$ | 2 | $8.39$ | 9 | $0.006$
Mrk 3 |  $33.8_{-33.6}^{+135.9}$$\dagger$$\dagger$footnotemark:  |  $2870_{-2530}^{+3940}$$\dagger$$\dagger$footnotemark:  | $44.8$ | 2 | $8.65$ | 10 | $0.011$
Circinus | 0.02-0.9$\dagger$$\dagger$footnotemark:  | 10-410$\dagger$$\dagger$footnotemark:  | $43.6$ | 4 | $6.23$ | 11 | $0.186$

###  3.4 Trends with AGN Properties

We investigated trends between our results for $\sigma_{v}$ and
$\sigma_{\theta}$ and AGN properties such as X-ray luminosity, obscuration,
and Eddington ratio. However, due to the large uncertainties on all objects
except for Circinus and our relatively small sample size, we are unable to
find any significant trends with AGN properties. Likewise, we are not able to
draw any firm conclusions on the lack of correlations, but suggest that this
is best left as work that could be investigated with future X-ray missions,
such as XRISM.

##  4 Discussion

###  4.1 Origin of the Narrow Fe K$\alpha$ Line

The origin of the narrow Fe K$\alpha$ line has long been in debate, with the
typical location thought to be in either the outer accretion disc, BLR, or
torus. Nearby AGN show a wide variety of measured line widths and no
connection with the optical BLR as measured by the width of the H$\beta$ line,
which suggests no universal location for the narrow Fe K$\alpha$ line (Yaqoob
& Padmanabhan, 2004; Nandra, 2006; Shu et al., 2010). In our sample of seven
nearby AGN, we use a new technique to probe the spatial extent of the emitting
material, which provides an alternative measurement of the location of the
narrow Fe K$\alpha$ line. We find that distance estimates from measurements of
the intrinsic velocity width suggest BLR or torus origin, but that these
measurements consistently underestimate the inferred spatial extent of the
emitting material.

Many of the sources in our sample have previously shown evidence for extended
Fe K$\alpha$ emission. Circinus, the closest object in our sample, shows Fe
K$\alpha$ emission (Marinucci et al., 2013; Arévalo et al., 2014) extending up
to $\sim 3"$ ($\sim 60$ pc) from the nucleus, which is anti-correlated with
molecular gas in the circumnuclear environment (Kawamuro et al., 2019). In NGC
1068, Young et al. (2001) revealed spatially extended Fe K$\alpha$ emission
out to approximately 2.2 kpc from the nucleus, and Bauer et al. (2015) found
that approximately 30% of the Fe K$\alpha$ emission is extended on scales
larger than $\sim 140$ pc. These two sources were also recently found to have
the most extended emission in a sample of many bright AGN, using a combination
of variability and imaging analyses (Andonie et al., 2022). Likewise,
Guainazzi et al. (2012) found that the Fe K$\alpha$ emitting material extended
up to 300 pc from the nucleus in Mrk 3, another highly obscured AGN in our
sample.

Extended emission is easier to detect in Compton-thick AGN because of the
heavy obscuration of the nuclear emission, but it has also been detected in
the Compton-thin AGN NGC 4388. Faint extended Fe K$\alpha$ emission was first
detected out as far as 2.5 kpc in one direction by Iwasawa et al. (2003), and
more recently, additional observations have allowed for more detailed
morphological studies of the extended Fe K$\alpha$ emission on scales of
$\lesssim 1$ kpc (Yi et al., 2021). In NGC 4151, another Compton-thin AGN, the
amount of extended Fe K$\alpha$ emission has been debated, with first evidence
coming early on in the Chandra mission (Ogle et al., 2000). However, later
Wang et al. (2011) argued with updated MARX PSF calculations that the extended
Fe K$\alpha$ emission could only account for $\sim 5\%$ of the total line
emission. Likewise, Miller et al. (2018) and Szanecki et al. (2021) suggest
that the narrow Fe K$\alpha$ line is produced at several hundred gravitational
radii through modeling of first order HETG and Suzaku/NuSTAR spectra,
respectively. Time delays also suggest a BLR origin, with Zoghbi et al. (2019)
finding time delays between the Fe K$\alpha$ line and the continuum in the
XMM-Newton data on the order of $\sim 3$ days.

We note also that a recent study focused on the developing a consistent
picture of the torus of Circinus has implemented a similar analysis involving
the first, second, and third order Chandra HETG data (Uematsu et al., 2021).
Their analysis revealed a spatial extent of
$\sigma_{\theta}=0.66_{-0.21}^{+0.33}$ arcseconds, which is slightly larger
than our results for Circinus of $\sigma_{\theta}=0.37_{-0.10}^{+0.07}$
arcseconds. We suspect that our value is lower because Uematsu et al. (2021)
considered both point-like emission from closer to the black hole and extended
emission. They found that roughly 20% of emission from the Fe K$\alpha$ line
is extended in Circinus. Thus, our smaller extent is likely because our
methodology can be interpreted as providing a spatially-averaged measurement
of the extent of the Fe K$\alpha$ emitting material.

Our analysis provides another measure of spatial extent, avoiding the issues
of pile-up and PSF contributions that must be corrected for and properly
modeled in imaging analysis. Given that these issues greatly affect the
nucleus in imaging data, our method is more sensitive to smaller extension
than can be properly observed with imaging data. We find that the bulk of the
Fe K$\alpha$ emitting material is extended on scales of $\sim 5-100$ pc across
the seven sources in our sample, although with relatively large uncertainty
due to the quality of the higher order HETG spectra. However, in most of our
objects there is a discrepancy between the spatial extent of the material
measured by $\sigma_{\theta}$ and the location of the gas measured with the
intrinsic velocity width, $\sigma_{v}$, assuming motion is governed by the
gravitational potential of the central SMBH. The intrinsic velocity of the gas
indicates that the material is much closer to the black hole than we measure
with the spatial extent. Here we present two possible explanations for the
discrepancy between these distances.

Through modeling emission from both a point-like source and extended emission,
Uematsu et al. (2021) found that extended emission in Circinus accounted for
roughly 20% of the Fe K$\alpha$ line emission. Our fitting method does not
include these two features as separate model components, and instead can be
interpreted as a spatial average of the two in a single model component. It is
thus possible that our fits are being driven by gas closer to the BLR for the
velocity width and by gas in and beyond the torus for the spatial extent. This
suggests that the Fe K$\alpha$ emitting material responsible for the narrow Fe
K$\alpha$ line in nearby AGN is present at a range of radii, from the inner
BLR out to in and beyond the putative torus. This interpretation is consistent
with recent findings that the variability and widths of the Fe K$\alpha$ line
in a sample of almost 40 nearby AGN do not show a universal picture for the
location of the narrow Fe K$\alpha$ line (Andonie et al., 2022) and can help
explain the discrepancy between different measurements of spatial extent.

Another possibility is that these large velocity widths are the result of non-
gravitational motions driving the intrinsic velocity width to larger values.
We investigate the nature of possible non-gravitational motions by allowing
the redshift of the Fe K$\alpha$ line to vary, finding that six of the seven
sources prefer slightly redshifted emission at roughly the 1$\sigma$ level.
Assuming the Fe K$\alpha$ emitting material is Compton-thick so that our view
is dominated by clouds on the far side of the nucleus, excess redshift would
imply outflowing material. We note that this excess redshift could be the
result of contamination from the Compton shoulder, which we have not included
in our modeling. However, the intrinsic velocity widths we measure are much
larger than the freefall velocities expected of inflowing gas in the galactic
potential. Outflows can be driven to higher velocities via radiative, thermal,
or magnetic launching mechanisms, and have been observed in UV and X-ray
spectra with velocities ranging from $v\sim 100$s km s-1 (warm absorbers and
their UV counterparts; e.g. Reynolds, 1997; Crenshaw et al., 1999) up to
$\gtrsim 0.1c$ (ultra-fast outflows; e.g. Tombesi et al., 2010, 2011, 2013;
Gofford et al., 2013). Outflows, with a variety of phases, scales, and
velocities, have been detected in all seven objects in our sample (e.g. NGC
1068–Cecil et al. 1990; García-Burillo et al. 2014; NGC 3783–Kaspi et al.
2001; Mehdipour et al. 2017; NGC 4151–Crenshaw & Kraemer 2007; Storchi-
Bergmann et al. 2010; NGC 4388–Rodríguez-Ardila et al. 2017; Domínguez-
Fernández et al. 2020; NGC 4507–Fischer et al. 2013; Mrk 3–Ruiz et al. 2001;
Gnilka et al. 2020; Circinus–Greenhill et al. 2003; Zschaechner et al. 2016;
Stalevski et al. 2019). Thus, we suggest that if the distance discrepancy is
resolved by non-gravitational motions driving the intrinsic velocity widths
higher, then it is likely that the Fe K$\alpha$ emitting material is
outflowing from the central engine.

###  4.2 Compton-Thick Sources

Two of the sources in our sample, NGC 1068 and Circinus, are well-known
Compton-thick AGN (with $N_{H}\geq 1.5\times 10^{24}$ cm-2), and Mrk 3 is
close to the Compton-thick limit. Extracting the intrinsic X-ray properties
from Compton-thick AGN is difficult because the amount of emission transmitted
directly from the AGN is often quite low compared to the amount of reprocessed
emission. The additional scattering component that must be included in the
modeling of Compton-thick sources is strongly dependent on the geometry of the
system, which introduces a large amount of uncertainty. The best treatment of
Compton-thick AGN to date involves analyzing data over a wide energy range. In
particular, including hard X-ray data from observatories such as NuSTAR and
Swift BAT is important to accurately model Compton-thick AGN because almost
all of the emission with energy $\lesssim 10$ keV is scattered and
reprocessed, leaving no intrinsic emission to detect in the soft X-ray band.
Recent studies of NGC 1068 and Circinus have revealed estimates of the
intrinsic AGN properties by fitting detailed models for various different
geometries of the obscuring material to broad-band X-ray data from many
different observatories (Arévalo et al., 2014; Bauer et al., 2015; Marinucci
et al., 2016; Uematsu et al., 2021).

Our simple absorption modeling of these sources fails to account for the
additional scattering component and introduces some biases in the continuum
parameter estimation. For NGC 1068 and Circinus, the column densities we find
from our simple absorption model significantly underestimate those from the
literature with a full model including both absorption and electron scattering
(see Table 4). Thus, for these sources, we cannot properly estimate parameters
like the equivalent width of the Fe K$\alpha$ line using our simple continuum
modeling of the relatively narrow 4-9 keV band.

Properly modeling the absorption and scattering in these Compton-thick sources
is beyond the scope of this work. Hence, we quote a range of values for the
properties which depend on the intrinsic continuum, including the equivalent
width and the subsequent gas mass and mass outflow rate measurements, for the
two truly Compton-thick AGN in our sample (NGC 1068 and Circinus). The range
is determined based on two extreme continuum estimates and utilizing the
nuclear emission estimates from past multi-wavelength modeling, details of
which are given in Section 3.3. For Mrk 3, we find an average column density
of $N_{H}\sim 0.8\times 10^{24}$ cm-2, which is at the low end of the line of
sight values typically found in the literature (e.g. Yaqoob et al., 2015;
Guainazzi et al., 2016), but is in line with the finding that Mrk 3 is close
to, but does not exceed the Compton-thick limit along our line of sight. Thus,
we report values for Mrk 3, albeit with relatively high uncertainties, but
urge caution in the interpretation of values that depend on the intrinsic
continuum, such as the equivalent width, gas mass, and mass outflow rates,
which may be overestimated due to our simple phenomenological modeling.

###  4.3 Comparison to Molecular Gas

The Fe K$\alpha$ emission line traces cold, neutral gas around AGN, regardless
of whether the gas is in atomic or molecular form. Molecular gas around AGN
can also be traced via specific line transitions at submillimeter and infrared
wavelengths, which have the advantage of high spatial resolution and velocity
map information from spatially resolved spectroscopy. Studies of the molecular
gas around nearby AGN have probed the circumnuclear region around the black
hole in great detail, revealing significant outflows, indications of inflowing
material (see Storchi-Bergmann & Schnorr-Müller, 2019, and references
therein), and complexities of the molecular torus (e.g. García-Burillo et al.,
2016; Imanishi et al., 2018; Combes et al., 2019). Molecular outflows can be
quite massive, with mass outflow rates on the order of 100s-1000s $M_{\odot}$
yr-1 (e.g. Feruglio et al., 2010; Sturm et al., 2011; Veilleux et al., 2013;
Cicone et al., 2014; Tombesi et al., 2015), which are similar to our derived
values from the Fe K$\alpha$ emitting material.

Two of the objects in our sample, NGC 1068 and Circinus, have been extensively
studied with ALMA, allowing us to directly compare our results to the
molecular gas properties. ALMA observations of NGC 1068 reveal significant
molecular gas outflows on a variety of scales and gas densities with various
molecular line tracers (e.g. García-Burillo et al., 2014; Gallimore et al.,
2016; Impellizzeri et al., 2019; García-Burillo et al., 2019; Imanishi et al.,
2020). García-Burillo et al. (2014) found significant molecular outflows in
the circumnuclear disk (CND). On these scales of approximately 100 pc, the
projected outflow velocities were on the order of $v\sim 100$ km s-1 and the
molecular outflow rate was $\dot{M}_{\mathrm{out}}\sim 63M_{\odot}$ yr-1. This
is consistent with, but falls on the low end of, our large range of mass
outflow rates for the neutral Fe K$\alpha$ emitting material in NGC 1068 (see
Table 3.3). On smaller scales, similar to those traced by the neutral iron
line in this study ($\sim 20$ pc), García-Burillo et al. (2019) showed that
about half of the mass in the molecular torus outflowing, although this
portion of the outflow is significantly smaller in mass, momentum, and energy
compared to the outflow in the CND. On even smaller scales of roughly a few
parsecs, fast molecular outflows have been detected in both Gallimore et al.
(2016) and Impellizzeri et al. (2019), with projected velocities on the order
of $v\sim 400-450$ km s-1. Although with higher uncertainties, our results for
NGC 1068 reveal similar mass outflow rates and higher velocities of the
potentially outflowing Fe K$\alpha$ emitting material compared to those traced
by molecular line transitions with ALMA.

Previous ALMA observations of the Circinus galaxy have revealed a molecular
and atomic outflows, although the geometry of Circinus makes it difficult to
distinguish between outflow and disk motions. Zschaechner et al. (2016) found
that the molecular outflow in the central kiloparsec of Circinus had a
consistent velocity with the ionized outflow ($v\sim 150$ km s-1 Veilleux &
Bland-Hawthorn, 1997). However, this molecular outflow has significantly lower
energetics than some of the massive molecular outflows (e.g. Cicone et al.,
2014), with an estimated mass outflow rate of $\dot{M}_{\mathrm{out}}\sim
0.35-12.3M_{\odot}$ (Zschaechner et al., 2016). Like in NGC 1068, this is on
the low end of our reported range of mass outflow rates for the Fe K$\alpha$
emitting material. On smaller scales, Izumi et al. (2018) found streaming
motions indicative of an inflow to the AGN and an atomic outflow with a
velocity on the order of $v\sim 60$ km s-1. This outflow is thought to give
rise to the thickness of the diffuse atomic gas and provides evidence for a
multiphase, dynamic torus in Circinus. Additionally, Kawamuro et al. (2019)
found that the spatial extent of the molecular line emission in Circinus was
anti-correlated with the Fe K$\alpha$ emitting material, but had similar
extents in the central 100 pc of the galaxy.

###  4.4 Possibilities with Future X-ray Missions

This study opens the door for further work that can be done in tandem with
future X-ray missions, such as XRISM, Athena, and Lynx. These missions will
have microcalorimeters to perform non-dispersive, high resolution X-ray
spectroscopy. The power of microcalorimeters in X-ray spectroscopy is evident
from the Hitomi observations of NGC 1275 in the Perseus cluster, which
provided the ability to resolve the narrow Fe K$\alpha$ line in NGC 1275 for
the first time (Hitomi Collaboration et al., 2018). In relation to this work,
microcalorimeter spectroscopy will provide an independent measure of the
intrinsic velocity width of the narrow Fe K$\alpha$ line and avoid
contributions from the spatial extent, an effect that we have just shown has
increased the line widths in dispersed Chandra HETG spectra. In principle, the
spatial extent of the emitting material can be extracted by comparing Chandra
HETG first order line widths with those from a microcalorimeter spectrum. This
will be more prevalent in the future with X-ray missions, such as XRISM,
Athena, and Lynx, but Reynolds et al. (2021) have already demonstrated the
possibilities of this technique by estimating the extent of the Fe K$\alpha$
emitting material in NGC 1275 with Hitomi SXS data and first order Chandra
HETG data.

The higher order HETG spectra have significantly lower signal-to-noise
compared to the first order spectrum due to their smaller effective areas, and
thus are the limiting factor in our analysis and the cause of the large
uncertainty on the spatial extent reported in this work. Therefore, being able
to disentangle the spatial extent and intrinsic velocity width solely with the
first order HETG spectrum and one from an X-ray microcalorimeter should
greatly decrease the uncertainty in our current estimates of the spatial
extent of the Fe K$\alpha$ emitting material in these objects, most of which
are close to upper limits. Likewise, requiring adequate signal-to-noise in the
higher order HETG spectra restricts the sample of AGN for which this analysis
can be done to bright, nearby sources with an extensive amount of observation.
As already shown in Reynolds et al. (2021), it is possible to obtain
constraints on the Fe K$\alpha$ emitting material in NGC 1275, a difficult
target because of the surrounding X-ray emitting intracluster medium of the
Perseus cluster, by applying this technique to microcalorimeter and dispersive
spectroscopy together. Thus, together with archival first order Chandra HETG
data, future X-ray microcalorimeter spectroscopy will will also allow for this
study to be extended to many more AGN.

##  5 Conclusions

We have presented fits to the narrow Fe K$\alpha$ line in seven nearby AGN
that include the contribution of spatial extent on the line width in HETG
spectra. These AGN were chosen based on the criteria of having usable second
and third order HETG spectra, which have effective areas which are
approximately a factor of 15 lower than the first order data. Thus, in order
to have sufficient data, these sources are bright, relatively local, and have
been observed extensively with the HETG. Given their proximity, these sources
are spatially extended on the Chandra ACIS-S detectors. The spatial extent has
a larger contribution to the first order HETG data, making the first order
line widths larger than the second and third order widths, as originally seen
in Liu (2016). By fitting to all three orders simultaneously, we were able to
break the degeneracy between the velocity width and spatial extent and fit
directly to these values.

We find generally that the X-ray irradiated gas is extended on scales of
approximately $~{}0.3-0.6$”, corresponding to physical distances of the order
$5-100$ pc. This indicates that the narrow Fe K$\alpha$ line is the result of
reflection off of distant material. However, this conflicts with our
measurements of distance using the velocity width, which gives distances on
the order of $0.01-1$ pc. We suggest two possibilities to remedy this
discrepancy. First, we propose that the difference could be the result of
radially stratified emission with the emission closest to the black hole
dominating the velocity width and the emission furthest from the black hole
dominating the spatial extent. This scenario would imply that there is no
universal location for the narrow Fe K$\alpha$ emitting material around AGN,
but rather that the emission region extends from the broad line region out
beyond the torus. Another possibility is that the discrepancy is due to non-
gravitational motions driving the velocity width to larger values, namely in
the form of outflows from the central engine. The argument for outflows is
supported by large velocities relative to freefall in the galactic potential
and slight excess redshift in the Fe K$\alpha$ line in the majority of our
sources.

Using our measurement of the spatial extent of the emitting material, we
estimate the mass of the emitting gas to be in the range of
$10^{5}-10^{8}M_{\odot}$. This gives potential mass outflow rates on the order
10-3000 $M_{\odot}$ yr-1, assuming the velocity width of the Fe K$\alpha$ line
is characteristic of the outflow velocity. These outflow rates, although
highly uncertain, are comparable to molecular mass outflow rates as traced by
specific line transitions observed with submillimeter observatories. We also
investigate trends in our derived AGN properties, but find no significant
trends given the large uncertainties on our results. In order to accurately
determine if any trends exist between the extent of the Fe K$\alpha$ emission
and various AGN properties, a larger sample of AGN is necessary. With future
X-ray observatories, such as XRISM, coming online soon and providing non-
dispersive, high resolution X-ray spectra and independent measures of the
intrinsic velocity width of the narrow Fe K$\alpha$ line, this study could be
extended to many more AGN with first order Chandra HETG spectra.

We would like to thank the anonymous referee for helpful comments that
improved the manuscript. We would also like to thank Dom Walton and Christos
Panagiotou for their insightful feedback and discussion. M.M. is grateful for
support from the Gates Cambridge Trust. C.S.R. thanks the STFC for support
under the Consolidated Grant ST/S000623/1, as well as the European Research
Council (ERC) for support under the European Union’s Horizon 2020 research and
innovation programme (grant 834203). CXO (HETG)

## References

  * Andonie et al. (2022) Andonie, C., Bauer, F. E., Carraro, R., et al. 2022, arXiv e-prints, arXiv:2204.09469.  https://arxiv.org/abs/2204.09469
  * Antonucci (1993) Antonucci, R. 1993, ARA&A, 31, 473, doi: 10.1146/annurev.aa.31.090193.002353
  * Arévalo et al. (2014) Arévalo, P., Bauer, F. E., Puccetti, S., et al. 2014, ApJ, 791, 81, doi: 10.1088/0004-637X/791/2/81
  * Arnaud (1996) Arnaud, K. A. 1996, Astronomical Society of the Pacific Conference Series, Vol. 101, XSPEC: The First Ten Years, ed. G. H. Jacoby & J. Barnes, 17
  * Astropy Collaboration et al. (2013) Astropy Collaboration, Robitaille, T. P., Tollerud, E. J., et al. 2013, A&A, 558, A33, doi: 10.1051/0004-6361/201322068
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Bauer et al. (2015) Bauer, F. E., Arévalo, P., Walton, D. J., et al. 2015, ApJ, 812, 116, doi: 10.1088/0004-637X/812/2/116
  * Bearden (1967) Bearden, J. A. 1967, Reviews of Modern Physics, 39, 78, doi: 10.1103/RevModPhys.39.78
  * Bentz et al. (2006) Bentz, M. C., Denney, K. D., Cackett, E. M., et al. 2006, ApJ, 651, 775, doi: 10.1086/507417
  * Brenneman & Reynolds (2006) Brenneman, L. W., & Reynolds, C. S. 2006, ApJ, 652, 1028, doi: 10.1086/508146
  * Canizares et al. (2005) Canizares, C. R., Davis, J. E., Dewey, D., et al. 2005, PASP, 117, 1144, doi: 10.1086/432898
  * Cappi et al. (1999) Cappi, M., Bassani, L., Comastri, A., et al. 1999, A&A, 344, 857.  https://arxiv.org/abs/astro-ph/9902022
  * Cash (1979) Cash, W. 1979, ApJ, 228, 939, doi: 10.1086/156922
  * Cecil et al. (1990) Cecil, G., Bland, J., & Tully, R. B. 1990, ApJ, 355, 70, doi: 10.1086/168742
  * Cicone et al. (2014) Cicone, C., Maiolino, R., Sturm, E., et al. 2014, A&A, 562, A21, doi: 10.1051/0004-6361/201322464
  * Combes et al. (2019) Combes, F., García-Burillo, S., Audibert, A., et al. 2019, A&A, 623, A79, doi: 10.1051/0004-6361/201834560
  * Crenshaw & Kraemer (2007) Crenshaw, D. M., & Kraemer, S. B. 2007, ApJ, 659, 250, doi: 10.1086/511970
  * Crenshaw et al. (1999) Crenshaw, D. M., Kraemer, S. B., Boggess, A., et al. 1999, ApJ, 516, 750, doi: 10.1086/307144
  * Davis et al. (2012) Davis, J. E., Bautz, M. W., Dewey, D., et al. 2012, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8443, Raytracing with MARX: x-ray observatory design, calibration, and support, 84431A, doi: 10.1117/12.926937
  * Domínguez-Fernández et al. (2020) Domínguez-Fernández, A. J., Alonso-Herrero, A., García-Burillo, S., et al. 2020, A&A, 643, A127, doi: 10.1051/0004-6361/201936961
  * Duras et al. (2020) Duras, F., Bongiorno, A., Ricci, F., et al. 2020, A&A, 636, A73, doi: 10.1051/0004-6361/201936817
  * Fabbiano et al. (2017) Fabbiano, G., Elvis, M., Paggi, A., et al. 2017, ApJ, 842, L4, doi: 10.3847/2041-8213/aa7551
  * Fabian et al. (2015) Fabian, A. C., Lohfink, A., Kara, E., et al. 2015, MNRAS, 451, 4375, doi: 10.1093/mnras/stv1218
  * Fabian et al. (1989) Fabian, A. C., Rees, M. J., Stella, L., & White, N. E. 1989, MNRAS, 238, 729, doi: 10.1093/mnras/238.3.729
  * Feruglio et al. (2010) Feruglio, C., Maiolino, R., Piconcelli, E., et al. 2010, A&A, 518, L155, doi: 10.1051/0004-6361/201015164
  * Fischer et al. (2013) Fischer, T. C., Crenshaw, D. M., Kraemer, S. B., & Schmitt, H. R. 2013, ApJS, 209, 1, doi: 10.1088/0067-0049/209/1/1
  * Fruscione et al. (2006) Fruscione, A., McDowell, J. C., Allen, G. E., et al. 2006, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 6270, CIAO: Chandra’s data analysis system, 62701V, doi: 10.1117/12.671760
  * Fukazawa et al. (2011) Fukazawa, Y., Hiragi, K., Mizuno, M., et al. 2011, ApJ, 727, 19, doi: 10.1088/0004-637X/727/1/19
  * Gallimore et al. (2016) Gallimore, J. F., Elitzur, M., Maiolino, R., et al. 2016, ApJ, 829, L7, doi: 10.3847/2041-8205/829/1/L7
  * Gandhi et al. (2015) Gandhi, P., Hönig, S. F., & Kishimoto, M. 2015, ApJ, 812, 113, doi: 10.1088/0004-637X/812/2/113
  * García-Burillo et al. (2014) García-Burillo, S., Combes, F., Usero, A., et al. 2014, A&A, 567, A125, doi: 10.1051/0004-6361/201423843
  * García-Burillo et al. (2016) García-Burillo, S., Combes, F., Ramos Almeida, C., et al. 2016, ApJ, 823, L12, doi: 10.3847/2041-8205/823/1/L12
  * García-Burillo et al. (2019) —. 2019, A&A, 632, A61, doi: 10.1051/0004-6361/201936606
  * Gelman & Rubin (1992) Gelman, A., & Rubin, D. B. 1992, Statistical Science, 7, 457, doi: 10.1214/ss/1177011136
  * George & Fabian (1991) George, I. M., & Fabian, A. C. 1991, MNRAS, 249, 352, doi: 10.1093/mnras/249.2.352
  * Geweke (1992) Geweke, J. 1992, in Bayesian Statistics 4 (Oxford University Press), 169–193
  * Gnilka et al. (2020) Gnilka, C. L., Crenshaw, D. M., Fischer, T. C., et al. 2020, ApJ, 893, 80, doi: 10.3847/1538-4357/ab8000
  * Gofford et al. (2013) Gofford, J., Reeves, J. N., Tombesi, F., et al. 2013, MNRAS, 430, 60, doi: 10.1093/mnras/sts481
  * Goodman & Weare (2010) Goodman, J., & Weare, J. 2010, Communications in Applied Mathematics and Computational Science, 5, 65, doi: 10.2140/camcos.2010.5.65
  * Greenhill et al. (1996) Greenhill, L. J., Gwinn, C. R., Antonucci, R., & Barvainis, R. 1996, ApJ, 472, L21, doi: 10.1086/310346
  * Greenhill et al. (2003) Greenhill, L. J., Booth, R. S., Ellingsen, S. P., et al. 2003, ApJ, 590, 162, doi: 10.1086/374862
  * Guainazzi et al. (2012) Guainazzi, M., La Parola, V., Miniutti, G., Segreto, A., & Longinotti, A. L. 2012, A&A, 547, A31, doi: 10.1051/0004-6361/201219946
  * Guainazzi et al. (2016) Guainazzi, M., Risaliti, G., Awaki, H., et al. 2016, MNRAS, 460, 1954, doi: 10.1093/mnras/stw1033
  * Haardt & Maraschi (1991) Haardt, F., & Maraschi, L. 1991, ApJ, 380, L51, doi: 10.1086/186171
  * Haardt & Maraschi (1993) —. 1993, ApJ, 413, 507, doi: 10.1086/173020
  * HI4PI Collaboration et al. (2016) HI4PI Collaboration, Ben Bekhti, N., Flöer, L., et al. 2016, A&A, 594, A116, doi: 10.1051/0004-6361/201629178
  * Hickox & Alexander (2018) Hickox, R. C., & Alexander, D. M. 2018, ARA&A, 56, 625, doi: 10.1146/annurev-astro-081817-051803
  * Hitomi Collaboration et al. (2018) Hitomi Collaboration, Aharonian, F., Akamatsu, H., et al. 2018, PASJ, 70, 13, doi: 10.1093/pasj/psx147
  * Hunter (2007) Hunter, J. D. 2007, Computing in Science and Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Imanishi et al. (2018) Imanishi, M., Nakanishi, K., Izumi, T., & Wada, K. 2018, ApJ, 853, L25, doi: 10.3847/2041-8213/aaa8df
  * Imanishi et al. (2020) Imanishi, M., Nguyen, D. D., Wada, K., et al. 2020, ApJ, 902, 99, doi: 10.3847/1538-4357/abaf50
  * Impellizzeri et al. (2019) Impellizzeri, C. M. V., Gallimore, J. F., Baum, S. A., et al. 2019, ApJ, 884, L28, doi: 10.3847/2041-8213/ab3c64
  * Iwasawa et al. (2003) Iwasawa, K., Wilson, A. S., Fabian, A. C., & Young, A. J. 2003, MNRAS, 345, 369, doi: 10.1046/j.1365-8711.2003.06857.x
  * Izumi et al. (2018) Izumi, T., Wada, K., Fukushige, R., Hamamura, S., & Kohno, K. 2018, ApJ, 867, 48, doi: 10.3847/1538-4357/aae20b
  * Kara et al. (2017) Kara, E., García, J. A., Lohfink, A., et al. 2017, MNRAS, 468, 3489, doi: 10.1093/mnras/stx792
  * Karachentsev et al. (2013) Karachentsev, I. D., Makarov, D. I., & Kaisina, E. I. 2013, AJ, 145, 101, doi: 10.1088/0004-6256/145/4/101
  * Kaspi et al. (2001) Kaspi, S., Brandt, W. N., Netzer, H., et al. 2001, ApJ, 554, 216, doi: 10.1086/321333
  * Kaspi et al. (2002) Kaspi, S., Brandt, W. N., George, I. M., et al. 2002, ApJ, 574, 643, doi: 10.1086/341113
  * Kawamuro et al. (2019) Kawamuro, T., Izumi, T., & Imanishi, M. 2019, PASJ, 71, 68, doi: 10.1093/pasj/psz045
  * Krongold et al. (2003) Krongold, Y., Nicastro, F., Brickhouse, N. S., et al. 2003, ApJ, 597, 832, doi: 10.1086/378639
  * Kuo et al. (2011) Kuo, C. Y., Braatz, J. A., Condon, J. J., et al. 2011, ApJ, 727, 20, doi: 10.1088/0004-637X/727/1/20
  * Liu (2016) Liu, J. 2016, MNRAS, 463, L108, doi: 10.1093/mnrasl/slw164
  * Lopez-Rodriguez et al. (2018) Lopez-Rodriguez, E., Fuller, L., Alonso-Herrero, A., et al. 2018, ApJ, 859, 99, doi: 10.3847/1538-4357/aabd7b
  * Ma et al. (2020) Ma, J., Elvis, M., Fabbiano, G., et al. 2020, ApJ, 900, 164, doi: 10.3847/1538-4357/abacbe
  * Marinucci et al. (2013) Marinucci, A., Miniutti, G., Bianchi, S., Matt, G., & Risaliti, G. 2013, MNRAS, 436, 2500, doi: 10.1093/mnras/stt1759
  * Marinucci et al. (2016) Marinucci, A., Bianchi, S., Matt, G., et al. 2016, MNRAS, 456, L94, doi: 10.1093/mnrasl/slv178
  * Marshall (2017) Marshall, H. 2017, Chandra News, 24, 21
  * Matt et al. (1997) Matt, G., Guainazzi, M., Frontera, F., et al. 1997, A&A, 325, L13.  https://arxiv.org/abs/astro-ph/9707065
  * Matt et al. (1999) Matt, G., Guainazzi, M., Maiolino, R., et al. 1999, A&A, 341, L39.  https://arxiv.org/abs/astro-ph/9811301
  * Mehdipour et al. (2017) Mehdipour, M., Kaastra, J. S., Kriss, G. A., et al. 2017, A&A, 607, A28, doi: 10.1051/0004-6361/201731175
  * Miller et al. (2018) Miller, J. M., Cackett, E., Zoghbi, A., et al. 2018, ApJ, 865, 97, doi: 10.3847/1538-4357/aadbaa
  * Minezaki & Matsushita (2015) Minezaki, T., & Matsushita, K. 2015, ApJ, 802, 98, doi: 10.1088/0004-637X/802/2/98
  * Moorwood et al. (1996) Moorwood, A. F. M., Lutz, D., Oliva, E., et al. 1996, A&A, 315, L109
  * Nandra (2006) Nandra, K. 2006, MNRAS, 368, L62, doi: 10.1111/j.1745-3933.2006.00158.x
  * Netzer (2015) Netzer, H. 2015, ARA&A, 53, 365, doi: 10.1146/annurev-astro-082214-122302
  * Netzer et al. (1990) Netzer, H., Maoz, D., Laor, A., et al. 1990, ApJ, 353, 108, doi: 10.1086/168594
  * Netzer et al. (2003) Netzer, H., Kaspi, S., Behar, E., et al. 2003, ApJ, 599, 933, doi: 10.1086/379508
  * Ogle et al. (2000) Ogle, P. M., Marshall, H. L., Lee, J. C., & Canizares, C. R. 2000, ApJ, 545, L81, doi: 10.1086/317890
  * Peterson et al. (2004) Peterson, B. M., Ferrarese, L., Gilbert, K. M., et al. 2004, ApJ, 613, 682, doi: 10.1086/423269
  * Reynolds (1997) Reynolds, C. S. 1997, MNRAS, 286, 513, doi: 10.1093/mnras/286.3.513
  * Reynolds (2021) —. 2021, ARA&A, 59, doi: 10.1146/annurev-astro-112420-035022
  * Reynolds et al. (2000) Reynolds, C. S., Nowak, M. A., & Maloney, P. R. 2000, ApJ, 540, 143, doi: 10.1086/309327
  * Reynolds et al. (2021) Reynolds, C. S., Smith, R. N., Fabian, A. C., et al. 2021, MNRAS, 507, 5613, doi: 10.1093/mnras/stab2507
  * Ricci et al. (2017) Ricci, C., Trakhtenbrot, B., Koss, M. J., et al. 2017, ApJS, 233, 17, doi: 10.3847/1538-4365/aa96ad
  * Risaliti et al. (2013) Risaliti, G., Harrison, F. A., Madsen, K. K., et al. 2013, Nature, 494, 449, doi: 10.1038/nature11938
  * Rodríguez-Ardila et al. (2017) Rodríguez-Ardila, A., Mason, R. E., Martins, L., et al. 2017, MNRAS, 465, 906, doi: 10.1093/mnras/stw2642
  * Ruiz et al. (2001) Ruiz, J. R., Crenshaw, D. M., Kraemer, S. B., et al. 2001, AJ, 122, 2961, doi: 10.1086/323917
  * Shu et al. (2010) Shu, X. W., Yaqoob, T., & Wang, J. X. 2010, ApJS, 187, 581, doi: 10.1088/0067-0049/187/2/581
  * Stalevski et al. (2019) Stalevski, M., Tristram, K. R. W., & Asmus, D. 2019, MNRAS, 484, 3334, doi: 10.1093/mnras/stz220
  * Storchi-Bergmann et al. (2010) Storchi-Bergmann, T., Lopes, R. D. S., McGregor, P. J., et al. 2010, MNRAS, 402, 819, doi: 10.1111/j.1365-2966.2009.15962.x
  * Storchi-Bergmann & Schnorr-Müller (2019) Storchi-Bergmann, T., & Schnorr-Müller, A. 2019, Nature Astronomy, 3, 48, doi: 10.1038/s41550-018-0611-0
  * Sturm et al. (2011) Sturm, E., González-Alfonso, E., Veilleux, S., et al. 2011, ApJ, 733, L16, doi: 10.1088/2041-8205/733/1/L16
  * Szanecki et al. (2021) Szanecki, M., Niedźwiecki, A., & Zdziarski, A. A. 2021, ApJ, 909, 205, doi: 10.3847/1538-4357/abe380
  * Tanaka et al. (1995) Tanaka, Y., Nandra, K., Fabian, A. C., et al. 1995, Nature, 375, 659, doi: 10.1038/375659a0
  * Tombesi et al. (2013) Tombesi, F., Cappi, M., Reeves, J. N., et al. 2013, MNRAS, 430, 1102, doi: 10.1093/mnras/sts692
  * Tombesi et al. (2011) —. 2011, ApJ, 742, 44, doi: 10.1088/0004-637X/742/1/44
  * Tombesi et al. (2010) —. 2010, A&A, 521, A57, doi: 10.1051/0004-6361/200913440
  * Tombesi et al. (2015) Tombesi, F., Meléndez, M., Veilleux, S., et al. 2015, Nature, 519, 436, doi: 10.1038/nature14261
  * Tortosa et al. (2018) Tortosa, A., Bianchi, S., Marinucci, A., Matt, G., & Petrucci, P. O. 2018, A&A, 614, A37, doi: 10.1051/0004-6361/201732382
  * Tully (1988) Tully, R. B. 1988, Nearby galaxies catalog
  * Tully et al. (2016) Tully, R. B., Courtois, H. M., & Sorce, J. G. 2016, AJ, 152, 50, doi: 10.3847/0004-6256/152/2/50
  * Tully & Fisher (1988) Tully, R. B., & Fisher, J. R. 1988, Catalog of Nearby Galaxies
  * Uematsu et al. (2021) Uematsu, R., Ueda, Y., Tanimoto, A., et al. 2021, ApJ, 913, 17, doi: 10.3847/1538-4357/abf0a2
  * Urry & Padovani (1995) Urry, C. M., & Padovani, P. 1995, PASP, 107, 803, doi: 10.1086/133630
  * Ursini et al. (2019) Ursini, F., Bassani, L., Malizia, A., et al. 2019, A&A, 629, A54, doi: 10.1051/0004-6361/201936273
  * van der Walt et al. (2011) van der Walt, S., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science and Engineering, 13, 22, doi: 10.1109/MCSE.2011.37
  * Vasudevan et al. (2010) Vasudevan, R. V., Fabian, A. C., Gandhi, P., Winter, L. M., & Mushotzky, R. F. 2010, MNRAS, 402, 1081, doi: 10.1111/j.1365-2966.2009.15936.x
  * Veilleux & Bland-Hawthorn (1997) Veilleux, S., & Bland-Hawthorn, J. 1997, ApJ, 479, L105, doi: 10.1086/310588
  * Veilleux et al. (2013) Veilleux, S., Meléndez, M., Sturm, E., et al. 2013, ApJ, 776, 27, doi: 10.1088/0004-637X/776/1/27
  * Wang et al. (2011) Wang, J., Fabbiano, G., Elvis, M., et al. 2011, ApJ, 736, 62, doi: 10.1088/0004-637X/736/1/62
  * Wilms et al. (2000) Wilms, J., Allen, A., & McCray, R. 2000, ApJ, 542, 914, doi: 10.1086/317016
  * Winter et al. (2009) Winter, L. M., Mushotzky, R. F., Reynolds, C. S., & Tueller, J. 2009, ApJ, 690, 1322, doi: 10.1088/0004-637X/690/2/1322
  * Woo & Urry (2002) Woo, J.-H., & Urry, C. M. 2002, ApJ, 579, 530, doi: 10.1086/342878
  * Yaqoob et al. (2001) Yaqoob, T., George, I. M., Nandra, K., et al. 2001, ApJ, 546, 759, doi: 10.1086/318315
  * Yaqoob & Padmanabhan (2004) Yaqoob, T., & Padmanabhan, U. 2004, ApJ, 604, 63, doi: 10.1086/381731
  * Yaqoob et al. (2015) Yaqoob, T., Tatum, M. M., Scholtes, A., Gottlieb, A., & Turner, T. J. 2015, MNRAS, 454, 973, doi: 10.1093/mnras/stv2021
  * Yi et al. (2021) Yi, H., Wang, J., Shu, X., et al. 2021, ApJ, 908, 156, doi: 10.3847/1538-4357/abcec3
  * Young et al. (2001) Young, A. J., Wilson, A. S., & Shopbell, P. L. 2001, ApJ, 556, 6, doi: 10.1086/321561
  * Yuan et al. (2020) Yuan, W., Fausnaugh, M. M., Hoffmann, S. L., et al. 2020, ApJ, 902, 26, doi: 10.3847/1538-4357/abb377
  * Zaino et al. (2020) Zaino, A., Bianchi, S., Marinucci, A., et al. 2020, MNRAS, 492, 3872, doi: 10.1093/mnras/staa107
  * Zhou & Wang (2005) Zhou, X.-L., & Wang, J.-M. 2005, ApJ, 618, L83, doi: 10.1086/427871
  * Zoghbi et al. (2019) Zoghbi, A., Miller, J. M., & Cackett, E. 2019, ApJ, 884, 26, doi: 10.3847/1538-4357/ab3e31
  * Zschaechner et al. (2016) Zschaechner, L. K., Walter, F., Bolatto, A., et al. 2016, ApJ, 832, 142, doi: 10.3847/0004-637X/832/2/142

##  Appendix A Fit Parameters for Each ObsID

Table 4 shows the fit parameters for each ObsID for each object. The fit
parameters are the column density of the source absorption, the photon index,
the power law normalization, and the Gaussian line normalization. These are
all fixed across all three orders.

Table 4: Fit parameters for each object and ObsID Object | ObsID | $N_{H}$ | $\Gamma$ | $K_{\mathrm{power}}$ | $K_{\mathrm{line}}$
---|---|---|---|---|---
|  | ($10^{22}$ cm-2) |  | ($10^{-2}$ photons keV-1 cm-2 s-1) | ($10^{-4}$ photons cm-2 s-1)
NGC 1068$\dagger$$\dagger$footnotemark:  | 332 | $63.4_{-16.7}^{+18.5}$ | $2.41_{-0.49}^{+0.39}$ | $1.16_{-0.73}^{+1.05}$ | $0.85_{-0.23}^{+0.31}$
| 9148 | $72.6_{-15.6}^{+15.4}$ | $2.29_{-0.43}^{+0.35}$ | $1.13_{-0.66}^{+0.98}$ | $0.61_{-0.16}^{+0.20}$
| 9149 | $58.8_{-13.8}^{+14.9}$ | $2.30_{-0.38}^{+0.39}$ | $0.87_{-0.47}^{+0.95}$ | $0.70_{-0.16}^{+0.19}$
| 9150 | $73.8_{-21.0}^{+25.0}$ | $2.33_{-0.54}^{+0.47}$ | $1.08_{-0.73}^{+1.25}$ | $0.52_{-0.19}^{+0.30}$
| 10815 | $85.5_{-35.6}^{+49.5}$ | $2.33_{-0.70}^{+0.67}$ | $1.24_{-0.89}^{+1.53}$ | $0.94_{-0.45}^{+0.84}$
| 10816 | $103.4_{-47.3}^{+73.7}$ | $2.07_{-0.77}^{+0.72}$ | $1.16_{-0.88}^{+1.55}$ | $0.81_{-0.52}^{+1.21}$
| 10817 | $73.5_{-21.0}^{+24.1}$ | $2.26_{-0.50}^{+0.42}$ | $1.13_{-0.69}^{+1.05}$ | $0.92_{-0.33}^{+0.46}$
| 10823 | $78.0_{-19.7}^{+23.6}$ | $2.24_{-0.50}^{+0.41}$ | $1.14_{-0.72}^{+1.09}$ | $0.78_{-0.28}^{+0.40}$
| 10829 | $64.2_{-20.8}^{+25.3}$ | $2.49_{-0.56}^{+0.49}$ | $1.10_{-0.73}^{+1.32}$ | $0.55_{-0.19}^{+0.30}$
| 10830 | $53.1_{-18.1}^{+21.5}$ | $2.57_{-0.52}^{+0.45}$ | $1.11_{-0.71}^{+1.22}$ | $0.64_{-0.20}^{+0.29}$
NGC 3783 | 2090 | $3.1_{-2.2}^{+2.1}$ | $1.67_{-0.12}^{+0.11}$ | $1.46_{-0.30}^{+0.34}$ | $0.38_{-0.07}^{+0.07}$
| 2091 | $8.6_{-2.2}^{+2.1}$ | $1.97_{-0.12}^{+0.11}$ | $2.71_{-0.56}^{+0.64}$ | $0.40_{-0.07}^{+0.08}$
| 2092 | $3.7_{-2.2}^{+2.1}$ | $1.75_{-0.12}^{+0.11}$ | $1.68_{-0.35}^{+0.39}$ | $0.43_{-0.07}^{+0.07}$
| 2093 | $7.4_{-1.9}^{+1.9}$ | $2.03_{-0.10}^{+0.10}$ | $4.09_{-0.77}^{+0.88}$ | $0.47_{-0.08}^{+0.08}$
| 2094 | $8.1_{-2.0}^{+2.0}$ | $1.99_{-0.11}^{+0.10}$ | $3.36_{-0.67}^{+0.75}$ | $0.38_{-0.07}^{+0.08}$
| 14991 | $8.0_{-3.7}^{+3.5}$ | $1.89_{-0.17}^{+0.15}$ | $1.76_{-0.51}^{+0.61}$ | $0.42_{-0.11}^{+0.12}$
| 15626 | $7.7_{-3.1}^{+3.0}$ | $1.73_{-0.15}^{+0.13}$ | $1.24_{-0.32}^{+0.39}$ | $0.36_{-0.08}^{+0.09}$
| 18192 | $9.8_{-3.2}^{+3.1}$ | $2.07_{-0.15}^{+0.14}$ | $2.69_{-0.73}^{+0.84}$ | $0.39_{-0.10}^{+0.12}$
| 19694 | $15.3_{-3.6}^{+3.3}$ | $2.24_{-0.17}^{+0.15}$ | $3.28_{-0.95}^{+1.11}$ | $0.37_{-0.10}^{+0.11}$
NGC 4151 | 335 | $22.9_{-3.6}^{+3.4}$ | $1.67_{-0.17}^{+0.16}$ | $2.89_{-0.86}^{+1.08}$ | $1.77_{-0.27}^{+0.29}$
| 3052 | $8.3_{-1.7}^{+1.6}$ | $1.61_{-0.09}^{+0.09}$ | $4.74_{-0.82}^{+0.91}$ | $1.32_{-0.15}^{+0.15}$
| 3480 | $9.1_{-2.0}^{+2.0}$ | $1.54_{-0.11}^{+0.10}$ | $4.37_{-0.87}^{+0.99}$ | $1.39_{-0.20}^{+0.21}$
| 7829 | $20.6_{-4.5}^{+4.3}$ | $1.68_{-0.21}^{+0.17}$ | $1.51_{-0.53}^{+0.62}$ | $1.00_{-0.19}^{+0.21}$
| 7830 | $14.6_{-2.5}^{+2.3}$ | $2.02_{-0.13}^{+0.12}$ | $11.51_{-2.64}^{+3.07}$ | $1.21_{-0.26}^{+0.29}$
| 16089 | $24.7_{-2.2}^{+2.2}$ | $1.93_{-0.11}^{+0.11}$ | $4.79_{-1.01}^{+1.15}$ | $1.51_{-0.14}^{+0.15}$
| 16090 | $24.4_{-3.3}^{+3.1}$ | $1.68_{-0.16}^{+0.14}$ | $2.84_{-0.79}^{+0.95}$ | $1.38_{-0.20}^{+0.22}$
NGC 4388 | 9276 | $37.2_{-5.1}^{+4.8}$ | $1.38_{-0.25}^{+0.24}$ | $0.75_{-0.31}^{+0.48}$ | $0.67_{-0.10}^{+0.11}$
| 9277 | $38.5_{-6.0}^{+5.7}$ | $1.71_{-0.30}^{+0.28}$ | $1.62_{-0.75}^{+1.28}$ | $0.76_{-0.14}^{+0.16}$
NGC 4507 | 2150 | $64.7_{-5.8}^{+5.8}$ | $1.70_{-0.27}^{+0.27}$ | $2.61_{-1.14}^{+1.99}$ | $0.75_{-0.14}^{+0.16}$
Mrk 3 | 873 | $72.1_{-10.4}^{+10.8}$ | $2.03_{-0.35}^{+0.33}$ | $0.99_{-0.51}^{+0.89}$ | $0.85_{-0.17}^{+0.20}$
| 12874 | $76.0_{-11.6}^{+12.0}$ | $1.98_{-0.37}^{+0.31}$ | $1.09_{-0.59}^{+0.88}$ | $0.86_{-0.19}^{+0.23}$
| 12875 | $81.7_{-18.4}^{+19.7}$ | $1.94_{-0.45}^{+0.40}$ | $1.08_{-0.65}^{+1.09}$ | $0.81_{-0.30}^{+0.41}$
| 13254 | $83.1_{-19.1}^{+22.6}$ | $1.98_{-0.50}^{+0.39}$ | $1.14_{-0.72}^{+1.10}$ | $0.76_{-0.28}^{+0.42}$
| 13261 | $68.7_{-20.8}^{+23.9}$ | $2.22_{-0.57}^{+0.46}$ | $1.20_{-0.81}^{+1.34}$ | $0.58_{-0.27}^{+0.40}$
| 13263 | $82.0_{-24.9}^{+32.7}$ | $1.98_{-0.57}^{+0.51}$ | $1.08_{-0.73}^{+1.38}$ | $1.03_{-0.44}^{+0.69}$
| 13264 | $82.7_{-16.1}^{+17.9}$ | $1.89_{-0.43}^{+0.37}$ | $1.03_{-0.60}^{+1.02}$ | $0.59_{-0.24}^{+0.32}$
| 13406 | $97.0_{-25.0}^{+30.2}$ | $1.85_{-0.52}^{+0.44}$ | $1.14_{-0.74}^{+1.20}$ | $1.42_{-0.51}^{+0.76}$
| 14331 | $76.0_{-15.5}^{+16.5}$ | $1.86_{-0.41}^{+0.38}$ | $0.85_{-0.49}^{+0.94}$ | $0.83_{-0.23}^{+0.29}$
Circinus$\dagger$$\dagger$footnotemark:  | 10223 | $51.4_{-5.3}^{+5.4}$ | $1.26_{-0.17}^{+0.17}$ | $0.41_{-0.13}^{+0.15}$ | $3.87_{-0.33}^{+0.36}$
| 10224 | $50.4_{-6.1}^{+6.4}$ | $1.35_{-0.20}^{+0.18}$ | $0.46_{-0.16}^{+0.18}$ | $3.96_{-0.38}^{+0.43}$
| 10225 | $58.1_{-6.8}^{+7.0}$ | $1.56_{-0.21}^{+0.20}$ | $0.72_{-0.26}^{+0.31}$ | $4.43_{-0.47}^{+0.51}$
| 10226 | $79.5_{-13.7}^{+15.3}$ | $2.27_{-0.33}^{+0.30}$ | $3.66_{-1.78}^{+2.35}$ | $4.83_{-0.96}^{+1.20}$
| 10832 | $46.8_{-13.3}^{+14.3}$ | $0.64_{-0.33}^{+0.31}$ | $0.11_{-0.05}^{+0.07}$ | $3.92_{-0.74}^{+0.88}$
| 10833 | $59.1_{-11.0}^{+11.6}$ | $1.55_{-0.29}^{+0.27}$ | $0.77_{-0.35}^{+0.44}$ | $4.52_{-0.73}^{+0.86}$
| 10842 | $73.8_{-11.4}^{+12.4}$ | $2.01_{-0.28}^{+0.27}$ | $1.94_{-0.83}^{+1.12}$ | $5.30_{-0.82}^{+0.99}$
| 10843 | $54.2_{-8.0}^{+8.5}$ | $1.19_{-0.23}^{+0.21}$ | $0.35_{-0.13}^{+0.16}$ | $3.88_{-0.47}^{+0.52}$
| 10844 | $56.4_{-11.2}^{+11.7}$ | $1.73_{-0.29}^{+0.28}$ | $0.90_{-0.41}^{+0.50}$ | $3.40_{-0.60}^{+0.69}$
| 10850 | $53.0_{-16.5}^{+18.4}$ | $1.00_{-0.36}^{+0.36}$ | $0.25_{-0.13}^{+0.18}$ | $2.98_{-0.75}^{+0.94}$
| 10872 | $75.9_{-16.3}^{+17.6}$ | $1.55_{-0.34}^{+0.34}$ | $0.98_{-0.48}^{+0.64}$ | $5.54_{-1.22}^{+1.51}$
| 10873 | $67.4_{-13.2}^{+15.2}$ | $2.30_{-0.35}^{+0.33}$ | $2.88_{-1.45}^{+1.87}$ | $4.35_{-0.86}^{+1.09}$
| 62877 | $63.3_{-7.5}^{+7.7}$ | $1.67_{-0.23}^{+0.21}$ | $0.98_{-0.36}^{+0.44}$ | $3.84_{-0.46}^{+0.51}$
| 4770 | $64.6_{-7.9}^{+8.2}$ | $1.79_{-0.23}^{+0.22}$ | $1.19_{-0.46}^{+0.56}$ | $4.05_{-0.49}^{+0.54}$
| 4771 | $70.4_{-8.1}^{+8.6}$ | $1.64_{-0.23}^{+0.22}$ | $0.98_{-0.37}^{+0.46}$ | $4.61_{-0.56}^{+0.65}$

##  Appendix B Details of Equivalent Width Estimate in Equation 7

The equivalent width of an emission line is defined as

$\mathrm{EW}=\frac{\int_{E_{\mathrm{min}}}^{E_{\mathrm{max}}}\left(F-F_{0}\right)\,dE}{F_{0}},$
(B1)

where $F$ is the flux of the total spectrum (line and continuum), $F_{0}$ is
the flux of the continuum at the line energy, and $E_{\mathrm{min}}$ and
$E_{\mathrm{max}}$ bound the emission line. For the neutral Fe K$\alpha$ line,
$F_{0}$ is the flux at 6.4 keV. The total flux of the Fe K$\alpha$ photons is
given by $F-F_{0}$.

We can calculate the rate of Fe K$\alpha$ photon production, and therefore the
photon flux of the Fe K$\alpha$ line, by counting fluorescing neutral iron
atoms around the AGN. Consider an AGN with incident flux, $F_{\mathrm{inc}}$,
on neutral iron atoms in the circumnuclear material. The incident photon flux
is $F_{\mathrm{inc}}/h\nu=F_{\mathrm{inc}}/E$. If these photons have energy
greater than $E_{\mathrm{thresh}}\approx 7.1$ keV, then they can photoionize
neutral iron, removing an inner K shell electron. Then the rate of
photoionization of an iron atom is given by

$R_{\mathrm{Fe,\,ion}}=\int_{E_{\mathrm{thresh}}}^{\infty}\frac{F_{\mathrm{inc}}}{E}\sigma(E)\,dE,$
(B2)

where $\sigma(E)$ is the photoionization cross section. Not all photoionized
iron atoms will fluoresce and produce Fe K$\alpha$ photons. The probability of
fluorescence is given by the fluorescent yield, $Y$, which for iron is
$Y_{\mathrm{Fe}}=0.33$. Then the rate of Fe K$\alpha$ photon production around
the AGN is given by

$R_{\mathrm{Fe\,K}\alpha}=N_{\mathrm{Fe}}Y_{\mathrm{Fe}}R_{\mathrm{Fe,\,ion}},$
(B3)

where $N_{\mathrm{Fe}}$ is the total number of iron atoms. Approximating the
iron in a shell of radius $R$, the number of iron atoms is

$N_{\mathrm{Fe}}=Z_{\mathrm{abs,\,Fe}}N_{H}4\pi R^{2},$ (B4)

where $Z_{\mathrm{abs,\,Fe}}$ is the abundance of iron relative to hydrogen
and $N_{H}$ is the column density of hydrogen. Therefore the rate of Fe
K$\alpha$ photon production can be written as

$R_{\mathrm{Fe\,K}\alpha}=4\pi
R^{2}Z_{\mathrm{abs,\,Fe}}Y_{\mathrm{Fe}}N_{H}\int_{E_{\mathrm{thresh}}}^{\infty}\frac{F_{\mathrm{inc}}}{E}\sigma(E)\,dE.$
(B5)

The photon flux of Fe K$\alpha$ photons at a radius $R$ is simply the rate of
Fe K$\alpha$ photon production divided by $4\pi R^{2}$. This can be related to
the energy flux of Fe K$\alpha$ photons by

$F_{\mathrm{Fe\,K}\alpha}=E_{\mathrm{Fe\,K}\alpha}\frac{R_{\mathrm{Fe\,K}\alpha}}{4\pi
R^{2}}=E_{\mathrm{Fe\,K}\alpha}Z_{\mathrm{abs,\,Fe}}Y_{\mathrm{Fe}}N_{H}\int_{E_{\mathrm{thresh}}}^{\infty}\frac{F_{\mathrm{inc}}}{E}\sigma(E)\,dE,$
(B6)

which is used to determine the equivalent width of the Fe K$\alpha$ line. We
then assume that the photon index of the incident radiation from the AGN is
$\Gamma=2$, which yields a spectral index of $\alpha=1$. This implies that the
incident flux can be written as $F_{\mathrm{inc}}=AE^{-\alpha}\propto E^{-1}$,
where $A$ is a constant. Then the equivalent width of the Fe K$\alpha$ line
can be written

$EW_{\mathrm{Fe\,K}\alpha}=\frac{E_{\mathrm{Fe\,K}\alpha}Z_{\mathrm{abs,\,Fe}}Y_{\mathrm{Fe}}N_{H}\int_{E_{\mathrm{thresh}}}^{\infty}A\frac{\sigma(E)}{E^{2}}\,dE}{AE_{\mathrm{Fe\,K}\alpha}^{-1}}=E_{\mathrm{Fe\,K}\alpha}^{2}Z_{\mathrm{abs,\,Fe}}Y_{\mathrm{Fe}}N_{H}\int_{E_{\mathrm{thresh}}}^{\infty}\frac{\sigma(E)}{E^{2}}\,dE.$

This is Equation (1) of Reynolds et al. (2000), for which the authors
calculate $EW_{\mathrm{Fe,\,max}}\approx 65$ eV for NGC 4258. This calculation
was based on a column density of $N_{H}=10^{23}$ cm-2 and solar value of
$Z_{\mathrm{abs,\,Fe}}$. Hence, Equation 7 (Equation (1) from Hitomi
Collaboration et al., 2018) is simply a scaled version of Equation B, which
also accounts for possible clumpy or patchy material with the inclusion of the
covering fraction, $f_{\mathrm{cov}}$.