# Estimating black hole masses in obscured AGN using X-rays

Mario Gliozzi1 and James K. Williams1 and Dina A. Michel1
1 Department of Physics and Astronomy, George Mason University, 4400
University Drive, Fairfax, VA 22030
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Determining the black hole masses in active galactic nuclei (AGN) is of
crucial importance to constrain the basic characteristics of their central
engines and shed light on their growth and co-evolution with their host
galaxies. While the black hole mass ($M_{\mathrm{BH}}$) can be robustly
measured with dynamical methods in bright type 1 AGN, where the variable
primary emission and the broad line region (BLR) are directly observed, a
direct measurement is considerably more challenging if not impossible for the
vast majority of heavily obscured type 2 AGN. In this work, we tested the
validity of an X-ray-based scaling method to constrain the $M_{\mathrm{BH}}$
in heavily absorbed AGN. To this end, we utilized a sample of type 2 AGN with
good-quality hard X-ray data obtained by the NuSTAR satellite and with
$M_{\mathrm{BH}}$ dynamically constrained from megamaser measurements. Our
results indicate that, when the X-ray broadband spectra are fitted with
physically motivated self-consistent models that properly account for
absorption, scattering, and emission line contributions from the putative
torus and constrain the primary X-ray emission, then the X-ray scaling method
yields $M_{\mathrm{BH}}$ values that are consistent with those determined from
megamaser measurements within their respective uncertainties. With this method
we can therefore systematically determine the $M_{\mathrm{BH}}$ in any type 2
AGN, provided that they possess good-quality X-ray data and accrete at a
moderate to high rate.

###### keywords:

Galaxies: active – Galaxies: nuclei – X-rays: galaxies

††pubyear: 2015††pagerange: Estimating black hole masses in obscured AGN using
X-rays–B

##  1 Introduction

Historically, radio-quiet active galactic nuclei (AGN) have been divided into
two main categories based on their optical spectroscopy: type 1 AGN, whose
spectra are characterized by the presence of broad permitted lines (with full
width at half maximum FWHM $>2000\,\mathrm{km~{}s^{-1}}$) along with narrow
forbidden lines, and type 2 AGN, where only narrow forbidden lines are
detected (e.g., Khachikian & Weedman, 1974; Antonucci, 1983).

According to the basic AGN unification model, type 2 AGN can be considered as
the obscured counterpart of type 1 AGN and their main differences can be
simply ascribed to different viewing angles, due to the presence of an
obscuring toroidal structure made of gas and dust surrounding the AGN (e.g.,
Osterbrock, 1978; Antonucci, 1993; Tadhunter, 2008; Urry & Padovani, 1995).
However, over the years, theoretical and observational studies have revealed
that the simplest version of the unification model, based on a smooth donut-
shaped torus, is unable to explain several observations, favoring instead a
scenario where the torus is clumpy, with a covering factor depending on
various AGN properties, and where the overall obscuration occurs on different
scales with significant contribution from the galaxy itself. See Netzer (2015)
and Ramos Almeida & Ricci (2017) for recent comprehensive reviews on the
unification model of AGN.

Regardless of the nature of the obscuration, in type 2 AGN, the central engine
– an optical/UV emitting accretion disk, coupled with an X-ray emitting
Comptonization corona – and the broad line region (BLR) are not directly
accessible to observations. This makes it more difficult to determine the
properties of obscured AGN, which represent the majority of the AGN population
and thus play a crucial role in our understanding of the AGN activity, census,
and cosmological evolution (see Hickox & Alexander 2018 for a recent review on
obscured AGN).

Table 1: Properties of the sample

Source | Distance | $M_{\mathrm{BH}}$ | $\lambda_{\mathrm{Edd}}$ | NuSTAR | Exposure
---|---|---|---|---|---
name | (Mpc) | ($10^{6}$ M☉) | ($L_{\mathrm{bol}}$/$L_{\mathrm{Edd}}$) | observation ID | (ks)
(1) | (2) | (3) | (4) | (5) | (6)
NGC 1068 | $14.4^{\textrm{a}}$ | $8.0\pm 0.3$ | $0.210\pm 0.053$ | 60002033002 | 52.1
NGC 1194 | $53.2^{\textrm{b}}$ | $65.0\pm 3.0$ | $0.007\pm 0.002$ | 60061035002 | 31.5
NGC 2273 | $25.7^{\textrm{b}}$ | $7.5\pm 0.4$ | $0.132\pm 0.034$ | 60001064002 | 23.2
NGC 3079 | $17.3^{\textrm{c}}$ | $2.4_{-1.2}^{+2.4}$ | $0.011\pm 0.009$ | 60061097002 | 21.5
NGC 3393 | $50.0^{\textrm{d}}$ | $31.0\pm 2.0$ | $0.062\pm 0.016$ | 60061205002 | 15.7
NGC 4388 | $19.0^{\textrm{b}}$ | $8.5\pm 0.2$ | $0.035\pm 0.009$ | 60061228002 | 21.4
NGC 4945 | $3.7^{\textrm{e}}$ | $1.4\pm 0.7$ | $0.135\pm 0.075$ | 60002051004 | 54.6
IC 2560 | $26.0^{\textrm{f}}$ | $3.5\pm 0.5$ | $0.175\pm 0.050$ | 50001039004 | 49.6
Circinus | $4.2^{\textrm{g}}$ | $1.7\pm 0.3$ | $0.143\pm 0.044$ | 60002039002 | 53.9

Columns: 1 = megamaser AGN name. 2 = distance used computing the
$M_{\mathrm{BH}}$ from the maser measurements. References for the distances
and black hole masses are (a) Lodato & Bertin (2003), (b) Kuo et al. (2011),
(c) Kondratko, Greenhill, & Moran (2005), (d) Kondratko, Greenhill, & Moran
(2008), (e) Greenhill et al. (1997), (f) Yamauchi et al. (2012), and (g)
Greenhill et al. (2003). 3 = black hole mass. 4 = Eddington ratio with
Brightman’s bolometric correction of $10\times$ to $L_{\mathrm{X}}$ from
Brightman et al. (2016). 5 = NuSTAR observation ID. 6 = exposure time.

In order to shed light on the properties of the AGN central engine and its
accretion state, we need to accurately determine the black hole mass
($M_{\mathrm{BH}}$). In type 1 AGN, a reliable dynamical method frequently
used is the so-called reverberation mapping method, where intrinsic changes in
the continuum emission of the central engine, measured with some time delay in
the line emission produced by the BLR, are used to constrain the
$M_{\mathrm{BH}}$, modulo a geometric factor (Blandford & McKee, 1982;
Peterson et al., 2004). On the other hand, in type 2 AGN, by definition the
BLR is not visible and hence the reverberation mapping technique cannot be
applied. Nevertheless, there is a small fraction of heavily obscured AGN for
which it is still possible to measure the $M_{\mathrm{BH}}$ in a reliable way
via a dynamical method. These are the sources that display water megamaser
emission; if this emission is located in the accretion disk and is
characterized by the Keplerian motion, then the $M_{\mathrm{BH}}$ can be
constrained with great accuracy (e.g. Kuo et al., 2011).

In this work, we use a sample of heavily obscured type 2 AGN with
$M_{\mathrm{BH}}$ constrained by megamaser measurements and with good-quality
hard X-ray spectra obtained with the Nuclear Spectroscopic Telescope Array
(NuSTAR), a focusing hard X-ray telescope launched in 2012 with large
effective area and excellent sensitivity in the energy range 3–78 keV, where
the signatures of absorption and reflection are most prominent. Our main goal
is to test whether an X-ray scaling method that yields $M_{\mathrm{BH}}$
values broadly consistent with those obtained from reverberation mapping in
type 1 AGN can be extended to type 2 AGN.

The paper is structured as follows. In Section 2, we describe the sample
properties and the X-ray data reduction. In Section 3, we report on the
spectral analysis of NuSTAR data. The application of the X-ray scaling method
and the comparison between the $M_{\mathrm{BH}}$ values derived with this
method and those obtained from megamaser measurements are described in Section
4. We discuss the main results and draw our conclusions in Section 5.

##  2 Sample Selection and Data Reduction

We chose our sample of type 2 AGN based on the following two criteria: these
objects must have 1) the $M_{\mathrm{BH}}$ dynamically determined by megamaser
disk measurements, and 2) good-quality hard X-ray data. The former criterion
is crucial to quantitatively test the validity of the X-ray scaling method
applied to heavily obscured AGN, whereas the latter criterion is necessary to
robustly constrain the properties of the primary X-ray emission by accurately
assessing the contributions of absorption and reflection caused by the
putative torus. These criteria are fulfilled by the sample described by
Brightman et al. (2016), which is largely based on the sample of megamasers
analyzed by Masini et al. (2016) and spans a range in X-ray luminosity between
$10^{42}~{}\mathrm{erg~{}s^{-1}}$ and a few units in
$10^{43}~{}\mathrm{erg~{}s^{-1}}$. The general properties of this sample,
including the distance used to determine the $M_{\mathrm{BH}}$ from maser
measurements, the $M_{\mathrm{BH}}$ itself, and the Eddington ratio
$\lambda_{\mathrm{Edd}}=L_{\mathrm{bol}}/L_{\mathrm{Edd}}$, are reported in
Table 1.

The archival NuSTAR data of these nine objects were calibrated and screened
using the NuSTAR data analysis pipeline nupipeline with standard filtering
criteria and the calibration database CALDB version 20191219. From the
calibrated and screened event files we extracted light curves and spectra,
along with the RMF and ARF files necessary for the spectral analysis, using
the nuproduct script. The extraction regions used for both focal plane
modules, FPMA and FPMB, are circular regions of radii ranging from 40″ to 100″
depending on the brightness of the source, and centered on the brightest
centroid. Background spectra and light curves were extracted by placing
circles of the same size used for the source in source-free regions of the
same detector. No flares were found in the background light curves. All
spectra were binned with a minimum of 20 counts per bin using the HEASoft task
grppha 3.0.1 for the $\chi^{2}$ statistics to be valid.

Figure 1: The top panels show the NuSTAR spectra (black data points indicate
FPMA data whereas the red ones indicate FPMB data) with the best-fit models,
whereas the bottom panels show the data-to-model ratios.

##  3 Spectral analysis

Table 2: Spectral Results

Source | $N_{{\textrm{H}}_{\textrm{Gal}}}$ | $\log(N_{{\textrm{H}}_{\textrm{bor}}})$ | $\cos{\theta}$ | CFtor | $A_{\textrm{Fe}}$ | $N_{{\textrm{H}}_{\textrm{mytz}}}$ | $\Gamma$ | $N_{\mathrm{BMC}}$ | $\log{A}$ | $f_{\textrm{s}}$ | ($\chi^{2}/$dof)
---|---|---|---|---|---|---|---|---|---|---|---
name | ($10^{20}$ cm-2) |  |  | (%) |  | ($10^{24}$ cm-2) |  |  |  | (%) |
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | (9) | (10) | (11) | (12)
NGC 1068 | $2.59$ | $23.3\pm 0.1$ | $0.1$ | 15 | 1.0 | $3.5\pm 0.1$ | $1.98_{-0.01}^{+0.01}$ | $2.2_{-0.1}^{+0.1}\times 10^{-3}$ | 0.08 | 1.6 | 754.9/713
NGC 1194 | $5.53$ | $23.9\pm 0.1$ | $0.1$ | 91 | 3.2 | $0.8\pm 0.1$ | $1.62_{-0.05}^{+0.05}$ | $6.4_{-0.8}^{+1.1}\times 10^{-5}$ | 0.57 | 4.0 | 194.3/167
NGC 2273 | $5.80$ | $25.0\pm 0.7$ | $0.1$ | 15 | 1.0 | $6.8\pm 0.4$ | $1.95_{-0.05}^{+0.05}$ | $3.1_{-0.2}^{+0.2}\times 10^{-3}$ | 2.0 | … | 54.0/57
NGC 3079 | $0.87$ | $24.5\pm 0.1$ | $0.1$ | 20 | 1.0 | $2.9\pm 0.1$ | $1.91_{-0.06}^{+0.05}$ | $1.1_{-0.2}^{+0.2}\times 10^{-3}$ | 0.8 | 0.3 | 80.1/75
NGC 3393 | $6.13$ | $25.2\pm 0.2$ | $0.1$ | 15 | 1.0 | $2.4\pm 0.1$ | $1.86_{-0.10}^{+0.10}$ | $9.6_{-1.2}^{+3.1}\times 10^{-4}$ | 0.22 | … | 54.3/65
NGC 4388 | $2.57$ | $23.6\pm 0.1$ | $0.1$ | 91 | 1.0 | $0.4\pm 0.1$ | $1.66_{-0.04}^{+0.04}$ | $3.3_{-0.4}^{+0.5}\times 10^{-4}$ | -0.55 | 17.0 | 435.2/420
NGC 4945 | $14.0$ | $24.4\pm 0.1$ | $0.1$ | 91 | 0.8 | $3.0\pm 0.7$ | $1.74_{-0.05}^{+0.05}$ | $1.6_{-0.1}^{+0.1}\times 10^{-3}$ | 2.15 | 0.5 | 1699.8/1716
IC 2560 | $6.51$ | $25.1\pm 0.1$ | $0.1$ | 15 | 2.3 | $6.9\pm 0.1$ | $2.08_{-0.08}^{+0.08}$ | $1.8_{-0.3}^{+0.4}\times 10^{-3}$ | 2.0 | … | 88.3/64
Circinus | $52.5$ | $23.6\pm 0.1$ | $0.1$ | 24 | 1.7 | $1.6\pm 0.1$ | $2.17_{-0.01}^{+0.01}$ | $9.8_{-0.2}^{+0.1}\times 10^{-3}$ | -0.43 | 3.3 | 1713.2/1714

Columns: 1 = megamaser AGN name. 2 = Galactic column density from NASA’s
HEASARC. 3 = column density calculated with the Borus model. 4 = cosine of the
inclination angle. 5 = covering factor. 6 = iron abundance relative to the
solar value. 7 = column density calculated with the MYTorus model. 8 = photon
index. 9 = normalization of the BMC model. 10 = logarithm of $A$, where
$A=(f+1)/f$ and $f$ is the fraction of seed photons that are scattered. 11 =
fraction of the primary emission scattered along the line of sight by an
extended ionized reflector. 12 = $\chi^{2}$ divided by degrees of freedom.

The X-ray spectral analysis was performed using the xspec v.12.9.0 software
package (Arnaud, 1996), and the errors quoted on the spectral parameters
represent the 1$\sigma$ confidence level.

The NuSTAR spectra of this sample have already been reasonably well fitted
with self-consistent physically motivated models such as MYTorus (Murphy &
Yaqoob, 2009) and Torus (Brightman & Nandra, 2011) to account for the
continuum scattering and absorption, as well as the fluorescent line emission
produced by the torus, whereas the primary emission was parametrized with a
phenomenological power-law model. However, in order to apply the X-ray scaling
method (whose key features are described in the following section), the
primary emission needs to be parametrized by the Bulk Motion Comptonization
model (BMC), which is a generic Comptonization model that convolves thermal
seed photons producing a power law (Titarchuk, Mastichiadis, & Kylafis, 1997).
This model, which can be used to parametrize both the bulk motion and the
thermal Comptonization, is described by four spectral parameters: the
normalization $N_{\mathrm{BMC}}$, the spectral index $\alpha$, the temperature
of the seed photons $kT$, and $\log A$, where $A$ is related to the fraction
of scattered seed photons $f$ by the relationship $A=(f+1)/f$. Unlike the
phenomenological power-law model, the BMC parameters are computed in a self-
consistent way, and the power-law component produced by the BMC does not
extend to arbitrarily low energies.

We carried out a homogeneous systematic reanalysis of the NuSTAR spectra of
these sources. We started from the best-fit models reported in the literature
but utilized the Borus model (Baloković et al., 2018), which can be considered
as an evolution of the previous torus models. Specifically, Borus has the same
geometry implemented in Torus but can also be used in a decoupled mode, where
the column density $N_{\mathrm{H}}$ responsible for the continuum scattering
and fluorescent line emission is allowed to be different from the
$N_{\mathrm{H}}$ responsible for the attenuation of the primary component.
Additionally, unlike Torus, this model correctly accounts for the absorption
experienced by the photons backscattered from the far side of the inner torus.
With respect to MYTorus, Borus contains additional emission lines, has a
larger range for $N_{\mathrm{H}}$, and directly yields the value of the
covering fraction. However, since Borus only parametrizes the scattered
continuum and the fluorescent line components associated with the torus, to
account for the absorption and scattering experienced by the primary emission,
we utilized the zeroth-order component of MYTorus (MYTZ), which properly
includes the effects of the Klein-Nishina Compton scattering cross section
that are relevant in heavily absorbed AGN at energies above 10 keV. In
summary, our procedure can be summarized in three steps: 1) we started from
the spectral best fits reported in the literature; 2) we then substituted
Borus (more specifically, we used the borus02_v170323a.fits table) for either
Torus or MYTorus to account for the scattered and line components, and used
the zeroth-order component of MYTorus for the transmitted one; 3) finally, we
substituted BMC for the power-law model used for the primary emission.

In the spectral fitting, in order to preserve the self-consistency of these
physically motivated torus models, which are created by Monte Carlo
simulations using a power law to parametrize the X-ray primary emission, one
needs to link the primary emission parameters – the photon index $\Gamma$ and
the normalization $N_{\mathrm{PL}}$ – to the input parameters of the scattered
continuum and emission-line components. In the case of the BMC model, the
power-law slope is described by the spectral index $\alpha$, which is related
to the photon index by the relationship $\Gamma=\alpha+1$. However, there is
not a known mathematical equation linking the normalizations
$N_{\mathrm{BMC}}$ and $N_{\mathrm{PL}}$. We therefore derived this
relationship empirically by using a sample of clean type 1 AGN (i.e., AGN
without cold or warm absorbers), whose details are described in Williams,
Gliozzi, & Rudzinsky (2018); Gliozzi & Williams (2020). We fitted the 2–10 keV
XMM-Newton spectra twice, first with the BMC model and then with a power law.
The results of this analysis are illustrated in Fig. 2, where
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ is plotted versus $N_{\mathrm{BMC}}$,
showing that, regardless of the value of $N_{\mathrm{BMC}}$, the normalization
ratios cluster around the average value, $30.8\pm 0.9$, represented by the
longer-dashed line, with moderate scattering of $\sigma=7.2$, represented by
the shorter-dashed lines. Fig. 2, where the data point’s size and color
provide information about the photon index, also reveals a tendency for the
AGN with steeper spectra to have larger values of
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$. This trend is formally confirmed by a
least-squares best fit of $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ vs. $\Gamma$,
which yields $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =-18.9 + 26.3$\Gamma$, with a
Pearson’s correlation coefficient of 0.85.

These results are in agreement with those obtained from a series of
simulations carried out with the fakeit command in xspec. Simulating spectra
of the BMC model with the parameters varying over a broad range, and then
fitting them with a power-law model, we found that
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ shows a horizontal trend when plotted vs.
$N_{\mathrm{BMC}}$ with an average value consistent with 30 for $\Gamma=1.9$,
whereas the horizontal trend is consistent with an average value of 24 for
$\Gamma=1.6$ and 33 for $\Gamma=2.2$.

Based on these findings, in our spectral fitting of the megamaser sample we
forced $N_{\mathrm{BMC}}$ to be equal to $N_{\mathrm{PL}}$/30 by linking these
parameters to reflect this relationship. For completeness, and to take into
account the weak dependence of $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ on $\Gamma$,
we have also carried out the spectral analysis assuming
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 24 (i.e., the average value minus one
standard deviation) for flat spectrum sources and
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 38 (average $+\sigma$) for steep spectrum
sources.

We note that, compared to $\Gamma$ and $N_{\mathrm{BMC}}$, the remaining BMC
parameters $kT$ and $\log A$ play a marginal role in the shape of the spectrum
and in the determination of the $M_{\mathrm{BH}}$, as explicitly assessed in
Gliozzi et al. (2011). Therefore, to limit the number of free parameters, we
fixed $kT$ to 0.1 keV, which is consistent with the values generally obtained
when the BMC model is fitted to X-ray AGN spectra (e.g., Gliozzi et al., 2011;
Williams, Gliozzi, & Rudzinsky, 2018), whereas $\log A$ was fixed to the best-
fit value obtained in the first fitting iteration.

Figure 2:  $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ plotted vs. $N_{\mathrm{BMC}}$
for a sample of “clean” type 1 AGN (i.e., AGN with negligible warm or cold
absorbers). The black longer-dashed line represents the average value, whereas
the shorter-dashed lines indicate the one standard deviation levels from the
average. Both the data point’s size and color provide information on the
source’s photon index $\Gamma$: the larger the symbol and the darker the
color, the steeper the $\Gamma$.

Our baseline model for all type 2 AGN fitted in this work is expressed in the
xspec syntax as follows:

    phabs * (atable(Borus) + MYTZ * BMC + const * BMC)

where the first absorption model phabs accounts for our Galaxy contribution,
the Borus table model parametrizes the continuum scattering and fluorescent
emission line components associated with the torus, and MYTZ models the
absorption and Compton scattering acting on the transmitted primary emission,
which is described by the Comptonization model BMC. The last additive
component const*BMC parametrizes the fraction of primary emission directly
scattered towards the observer by a putative optically thin ionized medium,
which is often observed below 5 keV in spectra of heavily obscured AGN (e.g.,
Yaqoob, 2012).

Depending on the source and the complexity of its X-ray spectrum, additional
components (such as the host galaxy contribution, individual lines, additional
absorption and scattering components, or models describing off-nuclear sources
contained in the NuSTAR extraction region) are included and described in the
individual notes of each source reported in the Appendix.

The spectral parameters obtained by fitting this baseline model are reported
in Table 2, and the best fits and model-to-data ratios are shown in Fig 1.

##  4 Black hole masses

###  4.1 $M_{\mathrm{BH}}$ from the X-ray scaling method

The X-ray scaling method was first introduced by Shaposhnikov & Titarchuk
(2009), who showed that the BH mass and distance $D$ of any stellar mass BH
can be obtained by scaling these properties from those of an appropriate
reference source (i.e., a BH system with $M_{\mathrm{BH}}$ dynamically
determined and distance tightly constrained). In its original form this
technique exploits the similarity of the trends displayed by different BH
systems in two plots – the photon index $\Gamma$ vs. quasi-periodic
oscillation (QPO) frequency plot and the $N_{\mathrm{BMC}}$–$\Gamma$ diagram –
to derive their $M_{\mathrm{BH}}$ and $D$.

Based on the assumption that the process leading to the ubiquitous emission of
X-rays – the Comptonization of seed photons produced by the accretion disk –
is the same in all BH systems regardless of their mass, this method can in
principle be extended to any BH including the supermassive BHs at the cores of
AGN. In the latter case, since the detection of QPOs is extremely rare but the
distance is generally well constrained by redshift or Cepheid measurements,
only the $N_{\mathrm{BMC}}$–$\Gamma$ diagram is used to determine the
$M_{\mathrm{BH}}$. Indeed, over the years, this method has been successfully
applied to stellar mass BHs (e.g., Seifina, Titarchuk, & Shaposhnikov 2014;
Titarchuk & Seifina 2016) and to ultraluminous X-ray sources (e.g., Titarchuk
& Seifina 2016; Jang et al. 2018), as well as to a handful of AGN that showed
high spectral and temporal variability during deep X-ray exposures (e.g.,
Gliozzi et al. 2010; Giacché, Gilli, & Titarchuk 2014; Seifina, Chekhtman, &
Titarchuk 2018.)

Although the vast majority of AGN do not possess long-term X-ray observations
and do not show strong intrinsic spectral variability (i.e., variability
described by substantial changes of $\Gamma$ not caused by obscuration
events), the X-ray scaling method can be extended to any type 1 AGN with one
good-quality X-ray observation. Indeed, Gliozzi et al. (2011) demonstrated
that the $M_{\mathrm{BH}}$ values determined with this method are fully
consistent with the corresponding values obtained from the reverberation
mapping technique. The reference sources, used in that study and then also in
this work, are three stellar mass BHs residing in X-ray binaries – GRO
J1655-40, GX 339-4, and XTE J1550-564 – with $M_{\mathrm{BH}}$ dynamically
determined and spectral evolution during the rising and decaying phases of
their outbursts mathematically parametrized by Shaposhnikov & Titarchuk
(2009). The physical properties of the stellar references and the mathematical
description of their spectral trends, as well as the details of the method,
are reported in Gliozzi et al. (2011).

In summary, all the reference trends yielded $M_{\mathrm{BH}}$ measurements
consistent with the reverberation mapping values within their nominal
uncertainties, with the decaying trends showing a slightly better agreement
than the rising trends, which have a tendency to underestimate
$M_{\mathrm{BH}}$ to a moderate degree. Unfortunately, the most reliable
reference source – GRO J1655-40 during the 2005 decaying phase (hereafter
GROD05) – has a fairly small range of $\Gamma$ during its spectral transition
limiting its application to sources with relatively flat photon indices. Using
the reverberation mapping values as calibration, it was determined that for
AGN with steep spectra ($\Gamma>2$) the best estimate of $M_{\mathrm{BH}}$ is
obtained using the value derived from the rising phase of the 1998 outburst of
XTE J1550-564 multiplied by a factor of 3 (hereafter 3*XTER98). Below, we
summarize the general principles at the base of this technique; a more
detailed explanation can be found in Shaposhnikov & Titarchuk (2009) and
Gliozzi et al. (2011). For completeness, in the Appendix we report the basic
information on the reference sources, including the mathematical expression of
their spectral trends, which is necessary to derive $M_{\mathrm{BH}}$ using
the equation reported below.

The scaling method assumes that all BH systems accreting at a moderate or high
rate undergo similar spectral transitions, characterized by the “softer when
brighter” trend (i.e., the X-ray spectrum softens when the accretion and hence
the luminosity increases). These spectral transitions are routinely observed
in stellar BHs (e.g., Remillard & McClintock, 2006) and often found in samples
of AGN (e.g., Shemmer et al. 2008; Risaliti, Young, & Elvis 2009; Brightman et
al. 2013, 2016), which are characterized by considerably longer dynamical
timescales, making it nearly impossible to witness a genuine state transition
in a supermassive BH system, although a few long monitoring studies have
observed this spectral trend in individual AGN (e.g., Sobolewska & Papadakis,
2009). The “softer when brighter” trend, usually illustrated by plotting the
photon index versus the Eddington ratio $\lambda_{\mathrm{Edd}}$, is seen with
some scattering in numerous type 1 AGN samples and also in the heavily
absorbed type 2 AGN, which are the focus of our work (Brightman et al., 2016).
This lends support to the hypothesis that the photon index $\Gamma$ is a
reliable indicator of the accretion state of any BH.

Indeed, this is the fundamental assumption of the X-ray scaling method:
$\Gamma$ is indicative of the accretion state of the source, and BH systems in
the same accretion state are characterized by the same accretion rate (in
Eddington units) and the same radiative efficiency $\eta$. As a consequence,
when we compare the accretion luminosity ($L\propto\eta
M_{\mathrm{BH}}\dot{m}$) in BH systems that are in the same accretion state
(i.e., with the same $\Gamma$), we are directly comparing their
$M_{\mathrm{BH}}$. This explains why the comparison of the values of the
normalization of the BMC model, $N_{\mathrm{BMC}}$ (which is defined as the
accretion luminosity in units of $10^{39}$ erg s-1 divided by the distance
squared in units of 10 kpc), computed at the same value of $\Gamma$ between
the AGN and a known stellar BH reference source, yields the $M_{\mathrm{BH}}$.
This is illustrated in Fig. 3 and mathematically described by

$M_{\mathrm{BH,AGN}}=M_{\mathrm{BH,ref}}\times\left(\frac{N_{\mathrm{BMC,AGN}}}{N_{\mathrm{BMC,ref}}}\right)\times\left(\frac{d_{\mathrm{AGN}}^{2}}{d_{\mathrm{ref}}^{2}}\right)$

where $N_{\mathrm{BMC,ref}}$ and $d_{\mathrm{ref}}$ are the BMC model
normalization and distance of the stellar mass BH system used as a reference.

Figure 3: $N_{\mathrm{BMC}}$–$\Gamma$ plot, showing the data point
corresponding to NGC 4945 and two reference patterns; the darker trend refers
to GROD05, the spectral evolution of GRO 1655-40 during the decay of an
outburst that occurred in 2005, and the lighter color trend indicates GROR05,
the spectral evolution shown by the same source during the outburst rise. The
dashed lines indicate the uncertainties in the reference spectral trends,
whereas the error bars represent the uncertainties of the AGN spectral
parameters.

Fig. 3 illustrates the X-ray scaling method and its inherent uncertainties
that are related to the statistical errors on the spectral parameters $\Gamma$
and $N_{\mathrm{BMC}}$ and on the uncertainty of the reference source spectral
trend (shown by the dashed lines), as well as on the specific reference source
trend utilized. Although similar in shape, the reference spectral trends show
some differences (e.g., in their plateau levels and slopes), leading to
slightly different $M_{\mathrm{BH}}$ values. From Fig. 3, it is clear that
these differences exist also between the rise and decay phases of the same
reference source.

It is important to note that at very low accretion rates both stellar mass and
supermassive BHs show an anti-correlation between $\Gamma$ and
$\lambda_{\mathrm{Edd}}$ (e.g., Constantin et al. 2009; Gu & Cao 2009;
Gültekin et al. 2012). Since the X-ray scaling method is based on the positive
correlation between these two quantities, it cannot be applied to determine
the $M_{\mathrm{BH}}$ of objects in the very low-accretion regime. This was
explicitly demonstrated by the work of Jang et al. (2014), who analyzed a
sample of low-luminosity low-accreting AGN.

In the following, we systematically estimate the $M_{\mathrm{BH}}$ using all
the reference sources available (depending on the AGN’s $\Gamma$, not all
reference sources can be used since their photon index ranges vary from
reference source to reference source) and then compute the $M_{\mathrm{BH}}$
average value and its uncertainty $\sigma/\sqrt{n}$ (where $\sigma$ is the
standard deviation and $n$ is the number of reference trends utilized). As
already explained above, for AGN with steep spectra, the most reliable
estimate of $M_{\mathrm{BH}}$ is obtained using the 3*XTER98 reference trend;
therefore, we also include this value in Table 3. All $M_{\mathrm{BH}}$ values
listed in this table were computed assuming $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$
= 30; however, for completeness, we also report the $M_{\mathrm{BH}}$ obtained
assuming $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 24 and 38 for flat- and steep-
spectrum sources, respectively. We note that such changes in
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ lead to $M_{\mathrm{BH}}$ values that are
consistent with the values obtained with the original assumption
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 30, within the respective
$M_{\mathrm{BH}}$ uncertainties that are of the order of 10%–40%.

The $M_{\mathrm{BH}}$ values obtained with the different reference sources and
their average are illustrated in Fig. 4. As already found in Gliozzi et al.
(2011) for the reverberation mapping AGN sample, the reference trends of
decaying outbursts yield systematically larger $M_{\mathrm{BH}}$ values
compared to those obtained from the rising trends. For each obscured AGN,
several $M_{\mathrm{BH}}$ values obtained from different reference sources and
their average appear to be broadly consistent with the value obtained from
megamaser measurements (a quantitative comparison is carried out in the next
subsection). The only noticeable exception is NGC 1194, for which the X-ray
scaling method yields values significantly lower than the maser one. This
discrepancy however is not surprising, since this source has a fairly low
accretion rate and in that regime the X-ray scaling method cannot be safely
applied.

Figure 4: $M_{\rm BH}$ values obtained with the X-ray scaling method using
different reference sources, compared with $M_{\rm BH}$ obtained from
megamaser measurements, which are represented by the black symbols at the top
of each panel. Table 3: Black hole masses with the X-ray scaling method

Source | $M_{{\textrm{BH}}_{\textrm{GROD05}}}$ | $M_{{\textrm{BH}}_{\textrm{GROR05}}}$ | $M_{{\textrm{BH}}_{\textrm{GXD03}}}$ | $M_{{\textrm{BH}}_{\textrm{GXR04}}}$ | $M_{{\textrm{BH}}_{\textrm{XTER98}}}$ | $M_{{\textrm{BH}}_{\textrm{aver}}}$ | $M_{{\textrm{BH}}_{\textrm{3*XTER98}}}$
---|---|---|---|---|---|---|---
name | ($10^{6}$ M☉) | ($10^{6}$ M☉) | ($10^{6}$ M☉) | ($10^{6}$ M☉) | ($10^{6}$ M☉) | ($10^{6}$ M☉) | ($10^{6}$ M☉)
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
NGC 1068 | … | $1.6_{-0.2}^{+0.3}$ | $8.5_{-1.0}^{+0.8}$ | $4.1_{-0.2}^{+0.3}$ | $1.9_{-0.5}^{+0.8}$ | $4.0_{-1.6}^{+1.6}$ | $5.6_{-1.5}^{+2.4}$
NGC 1194 | $5.6_{-0.7}^{+1.0}$ | $1.7_{-0.4}^{+1.0}$ | $9.0_{-2.3}^{+5.7}$ | $2.0_{-0.1}^{+0.2}$ | $1.2_{-0.4}^{+0.6}$ | $3.9_{-1.5}^{+1.5}$ | $3.7_{-1.1}^{+1.9}$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $6.9_{-1.0}^{+1.5}$ | $2.2_{-0.6}^{+1.6}$ | $12_{-3.5}^{+13}$ | $2.4_{-0.1}^{+0.4}$ | $1.5_{-0.4}^{+0.7}$ | $5.0_{-2.0}^{+2.0}$ | $4.4_{-1.3}^{+2.2}$
NGC 2273 | $22.8_{-3.4}^{+5.2}$ | $7.3_{-1.0}^{+1.3}$ | $39.6_{-4.8}^{+4.0}$ | $19.0_{-0.8}^{+1.4}$ | $8.6_{-2.3}^{+3.8}$ | $19.5_{-5.8}^{+5.8}$ | $25.8_{\ -6.9}^{+11.3}$
NGC 3079 | $4.8_{-0.6}^{+0.6}$ | $1.3_{-0.2}^{+0.3}$ | $7.1_{-0.9}^{+0.8}$ | $3.2_{-0.1}^{+0.2}$ | $1.5_{-0.4}^{+0.7}$ | $3.6_{-1.1}^{+1.1}$ | $4.5_{-1.2}^{+2.0}$
NGC 3393 | $40.1_{-4.2}^{+4.8}$ | $10.7_{-1.6}^{+2.3}$ | $57.1_{-7.5}^{+7.3}$ | $23.3_{-1.0}^{+1.8}$ | $11.6_{-3.1}^{+5.2}$ | $28.5_{-8.9}^{+8.9}$ | $34.7_{\ -9.4}^{+15.7}$
NGC 4388 | $3.3_{-0.4}^{+0.5}$ | $1.0_{-0.2}^{+0.4}$ | $5.0_{-1.1}^{+2.0}$ | $1.3_{-0.1}^{+0.1}$ | $0.8_{-0.2}^{+0.4}$ | $2.3_{-0.8}^{+0.8}$ | $2.3_{-0.7}^{+1.1}$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $4.1_{-0.5}^{+0.7}$ | $1.2_{-0.3}^{+0.5}$ | $6.3_{-1.4}^{+2.5}$ | $1.6_{-0.1}^{+0.2}$ | $1.0_{-0.3}^{+0.5}$ | $2.9_{-1.0}^{+1.0}$ | $2.9_{-0.8}^{+1.4}$
NGC 4945 | $0.5_{-0.05}^{+0.06}$ | $0.14_{-0.03}^{+0.04}$ | $0.7_{-0.1}^{+0.2}$ | $0.2_{-0.01}^{+0.01}$ | $0.13_{-0.04}^{+0.06}$ | $0.3_{-0.1}^{+0.1}$ | $0.4_{-0.1}^{+0.2}$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $0.6_{-0.1}^{+0.1}$ | $0.17_{-0.03}^{+0.05}$ | $0.9_{-0.1}^{+0.2}$ | $0.3_{-0.01}^{+0.02}$ | $0.16_{-0.05}^{+0.08}$ | $0.4_{-0.1}^{+0.1}$ | $0.5_{-0.1}^{+0.2}$
IC 2560 | … | $3.2_{-0.4}^{+0.5}$ | $16.1_{-3.0}^{+2.0}$ | $10.2_{-0.5}^{+1.0}$ | $4.2_{-1.1}^{+1.7}$ | $8.4_{-3.0}^{+3.0}$ | $12.5_{-3.2}^{+5.2}$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =38 | … | $2.8_{-0.3}^{+0.4}$ | $14.6_{-2.0}^{+1.6}$ | $8.8_{-0.6}^{+0.7}$ | $3.6_{-0.9}^{+1.5}$ | $7.4_{-2.7}^{+2.7}$ | $10.8_{-2.8}^{+4.6}$
Circinus | … | $0.4_{-0.1}^{+0.1}$ | … | … | $0.5_{-0.1}^{+0.2}$ | $0.4_{-0.1}^{+0.1}$ | $1.6_{-0.4}^{+0.6}$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =38 | … | $0.3_{-0.1}^{+0.1}$ | … | … | $0.4_{-0.1}^{+0.2}$ | $0.6_{-0.4}^{+0.4}$ | $1.2_{-0.3}^{+0.5}$

Columns: 1 = AGN name. 2–8 = black hole masses determined with the X-ray
scaling method. Subscripts denote GROD05 = reference source GRO J1655-40 in
the decreasing phase; GROR05 = reference source GRO J1655-40 in the rising
phase; GXD03 = reference source GX 339-4 in the decreasing phase; GXR03 =
reference source GX 339-4 in the rising phase; XTER98 = reference source XTE
J1550-564 in the rising phase; 3*XTER98 = reference source XTE J1550-564 in
the rising phase with a multiplicative correction of a factor 3 applied. Note,
the average value (in column 7) is obtained averaging all the
$M_{\mathrm{BH}}$ obtained from all the reference sources but excluding
3*XTER98. Note: For each source the first line reports the $M_{\mathrm{BH}}$
values obtained using $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 30 in the spectral
fitting; the second line (present only for sources with relatively flat or
steep spectra) explicitly states the different value of
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ used.

Figure 5: Plots showing the difference between the BH mass determined from
megamaser measurements and the values obtained with the X-ray scaling method
for the different reference sources, divided by the uncertainty of the
difference, $\Delta M_{\rm BH}/\sigma_{\rm diff}$. The horizontal dashed lines
enclose the region where the difference between the BH masses is within
3$\sigma$.

###  4.2 Black hole mass comparison

To compare the $M_{\mathrm{BH}}$ values obtained from the X-ray scaling method
with the maser ones in a quantitative way, we computed, using all the
available reference trends, the difference $\Delta
M_{\mathrm{BH}}=M_{\mathrm{BH,maser}}-M_{\mathrm{BH,scaling}}$ and its
uncertainty $\sigma_{\mathrm{diff}}$, obtained by adding the respective errors
in quadrature. As explained before, the error on the $M_{\mathrm{BH}}$
inferred from the scaling method includes the uncertainties on the spectral
AGN parameters and on the reference trends in the $N_{\mathrm{BMC}}$–$\Gamma$
diagram. Depending on the reference trend utilized, the percentage
uncertainties range from 10%–15% for GROD05 and GXD03 to 30%–40% for XTER98,
which is also the percentage uncertainty of the average $M_{\mathrm{BH}}$.

The error on the $M_{\mathrm{BH}}$ obtained with megamaser measurements
accounts for the uncertainties associated with the source position and with
the fitting of the Keplerian rotation curve (Kuo et al., 2011). For the
uncertainties on the $M_{\mathrm{BH}}$ determined via megamaser measurements
we used the errors quoted in the literature with the exception of NGC 4388,
for which we multiplied the quoted uncertainty by a factor of 10; this yields
a percentage error of $\sim$24%, which better reflects the actual uncertainty
on the $M_{\mathrm{BH}}$ in this source, where there is no systemic maser
detected and the five maser spots detected are not sufficient to demonstrate
that the rotation is Keplerian (Kuo et al., 2011). Note that both methods
explicitly depend on the sources’ distances and hence, in principle, their
total uncertainties should account also for the distance uncertainties (indeed
some of the sources of this maser sample are fairly close and thus their
distances cannot be obtained from the redshift and Hubble’s law). However,
since our goal is to compare the two methods, we can avoid the uncertainty
associated with the distance by assuming the exact same distance used in the
maser papers.

Figure 6: $M_{\rm BH,X}$, the BH mass obtained with the scaling method plotted
versus $M_{\rm BH,maser}$ obtained from the megamaser. The top left panel
shows the X-ray scaling values derived from the GROD05 reference, the top
right panel those from GXD03, the bottom left the values from 3*XTER98, and
the bottom right panel the $M_{\mathrm{BH}}$ values obtained from the average
of all the available reference sources. The longer-dashed line represents the
perfect one-to-one correspondence between the two methods, i.e., a ratio
$M_{\rm BH,maser}/M_{\rm BH,X}=1$, whereas the shorter-dashed lines indicate
the ratios of 3 and 1/3, respectively.

We used the criterion $\Delta M_{\mathrm{BH}}/\sigma_{\mathrm{diff}}<3$ to
assess whether the $M_{\mathrm{BH}}$ values derived with these two methods are
statistically consistent. In other words, the X-ray scaling measurements of
the $M_{\mathrm{BH}}$ are considered formally consistent with the
corresponding megamaser values if their difference is less than three times
the uncertainty $\sigma_{\mathrm{diff}}$.

The results of these comparisons are summarized in Table 4 and illustrated in
Fig. 5, where the dashed lines represent the 3$\sigma$ levels. From this
figure it is evident that every source has at least one $M_{\mathrm{BH}}$
scaling value that is consistent with the maser one, with GROD05 and GXD03
being the most reliable ones, along with the average $M_{\mathrm{BH}}$ and the
value obtained with 3*XTER98. The latter ones are always within 3$\sigma$ from
the megamaser value, also by virtue of their slightly larger uncertainties.

An alternative way to compare the two methods is offered by the ratio
$M_{\mathrm{BH,maser}}/M_{\mathrm{BH,scaling}}$. The ratios, obtained by
dividing the megamaser $M_{\mathrm{BH}}$ by each of the available reference
sources, as well as by the $M_{\mathrm{BH}}$ average and by 3*XTER98, are
reported in Table 5 and illustrated in Fig. 6, where the $M_{\mathrm{BH}}$
values obtained with the scaling method for the most reliable references
(GROD05, GXD03, 3*XTER98) and the average values are plotted versus their
respective megamaser values. From this figure, one can see that, for GROD05
(top left panel), 3*XTER98 (bottom left panel), and the average (bottom right
panel), all values are consistent with the ratio of 1 within a factor of 3,
and a good agreement is found also with GXD03 (top right panel) with two
sources (IC 2560 and NGC 2273) that have slightly larger values.

Based on the values reported in Table 5, all ratios obtained from these
reference trends are consistent with unity at the 3$\sigma$ limit (i.e., their
ratio $\pm 3\sigma$ is consistent with 1) confirming the statistical agreement
between the two methods. Finally, we note that using
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 24 (for flat spectrum sources) and 38
(for steep spectrum sources) confirms and reinforces the conclusions derived
from the original assumption $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 30.

Table 4: $\Delta{M_{\mathrm{BH}}}/\sigma_{\mathrm{diff}}$: Comparison between
$M_{\textrm{BH}}$ from maser and X-ray scaling

Source | $\Delta{M_{\mathrm{BH}}}/\sigma$
---|---
name | GROD05 | GROR05 | GXD03 | GXR04 | XTER98 | average | 3$*$XTER98
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
NGC 1068 | … | 16.7 | $-0.5$ | 10 | 8.6 | 2.5 | 1.2
NGC 1194 | $19.0$ | 20.5 | $11.3$ | $21.0$ | $21.0$ | $18.3$ | $20.4$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $17.9$ | 19.7 | $6.0$ | $20.8$ | $20.8$ | $16.7$ | $20.0$
NGC 2273 | $-3.5$ | 0.1 | $-7.3$ | $-9.7$ | $-0.4$ | $-2.0$ | $-2.0$
NGC 3079 | $-1.2$ | 0.6 | $-2.4$ | $-0.4$ | 0.5 | $-0.6$ | $-0.9$
NGC 3393 | $-1.8$ | 7.3 | $-3.4$ | 3.2 | 4.2 | 0.3 | $-0.3$
NGC 4388 | $2.5$ | 3.7 | $1.4$ | 3.6 | 3.8 | 2.9 | 2.8
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $2.1$ | 3.6 | $0.8$ | 3.4 | 3.7 | 2.5 | 2.4
NGC 4945 | $1.3$ | 1.8 | $1.0$ | 1.7 | 1.8 | 1.5 | 1.4
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $1.1$ | 1.8 | $0.6$ | 1.6 | 1.8 | 1.4 | 1.3
IC 2560 | … | 0.5 | $-4.9$ | $-7.5$ | $-0.4$ | $-1.6$ | $-2.1$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =38 | … | 1.1 | $-5.7$ | $-6.3$ | $-0.1$ | $-1.4$ | $-2.0$
Circinus | … | 4.4 | … | … | 3.4 | 4.1 | 0.3
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =38 | … | 4.7 | … | … | 4.0 | 2.3 | 1.0

Columns: 1 = AGN name. 2–8 = Change in black hole mass over error for each
reference source. Reference sources: GROD05 = reference source GRO J1655-40 in
the decreasing phase; GROR05 = reference source GRO J1655-40 in the rising
phase; GXD03 = reference source GX 339-4 in the decreasing phase; GXR03 =
reference source GX 339-4 in the rising phase; XTER98 = reference source XTE
J1550-564 in the rising phase; 3*XTER98 = reference source XTE J1550-564 in
the rising phase with a multiplicative correction of a factor 3 applied. Note,
the average value (in column 7) is obtained averaging all the
$M_{\mathrm{BH}}$ obtained from all the reference sources but excluding
3*XTER98. Note: For each source the first line reports the values obtained
using $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 30 in the spectral fitting; the
second line (present only for sources with relatively flat or steep spectra)
explicitly states the different value of $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$
used.

Table 5: Ratio between $M_{\mathrm{BH}}$ values obtained from maser
measurements and the X-ray scaling method:
$M_{\mathrm{BH,maser}}/M_{\mathrm{BH,scaling}}$

Source | Ratio
---|---
name | GROD05 | GROR05 | GXD03 | GXR04 | XTER98 | average | 3$*$XTER98
(1) | (2) | (3) | (4) | (5) | (6) | (7) | (8)
NGC 1068 | … | $5.1\pm 0.8$ | $0.9\pm 0.1$ | $1.9\pm 0.1$ | $4.3\pm 1.5$ | $2.0\pm 0.8$ | $1.4\pm 0.5$
NGC 1194 | $12\pm 2$ | $38\pm 16$ | $7\pm 3$ | $32\pm 3$ | $52\pm 21$ | $17\pm 7$ | $17\pm 7$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $10\pm 2$ | $30\pm 15$ | $6\pm 4$ | $27\pm 3$ | $44\pm 18$ | $13\pm 6$ | $15\pm 6$
NGC 2273 | $0.3\pm 0.1$ | $1.0\pm 0.2$ | $0.20\pm 0.02$ | $0.40\pm 0.03$ | $0.9\pm 0.3$ | $0.4\pm 0.1$ | $0.3\pm 0.1$
NGC 3079 | $0.5\pm 0.4$ | $1.8\pm 1.4$ | $0.3\pm 0.3$ | $0.8\pm 0.6$ | $1.6\pm 1.3$ | $0.7\pm 0.5$ | $0.5\pm 0.4$
NGC 3393 | $0.8\pm 0.1$ | $2.9\pm 0.6$ | $0.5\pm 0.1$ | $1.3\pm 0.1$ | $2.7\pm 1.0$ | $1.1\pm 0.3$ | $0.9\pm 0.3$
NGC 4388 | $2.6\pm 0.7$ | $8.8\pm 3.6$ | $1.7\pm 0.7$ | $6.5\pm 1.6$ | $10.9\pm 5.0$ | $3.7\pm 1.6$ | $3.6\pm 1.7$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $2.1\pm 0.6$ | $7.0\pm 2.8$ | $1.3\pm 0.5$ | $5.2\pm 1.3$ | $8.7\pm 4.0$ | $3.0\pm 1.3$ | $2.9\pm 1.3$
NGC 4945 | $2.8\pm 1.5$ | $10.3\pm 5.7$ | $2.0\pm 1.1$ | $6.2\pm 3.1$ | $11\pm 6.9$ | $4.1\pm 2.5$ | $3.7\pm 2.3$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =24 | $2.3\pm 1.2$ | $8.2\pm 4.6$ | $1.6\pm 1.2$ | $4.8\pm 2.4$ | $8.7\pm 5.4$ | $3.3\pm 2.0$ | $2.9\pm 1.8$
IC 2560 | … | $1.1\pm 0.2$ | $0.2\pm 0.1$ | $0.3\pm 0.1$ | $0.8\pm 0.3$ | $0.4\pm 0.2$ | $0.3\pm 0.1$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =38 | … | $1.3\pm 0.2$ | $0.2\pm 0.1$ | $0.4\pm 0.1$ | $1.0\pm 0.4$ | $0.5\pm 0.2$ | $0.3\pm 0.1$
Circinus | … | $4.6\pm 0.8$ | … | … | $3.3\pm 1.2$ | $3.9\pm 0.9$ | $1.1\pm 0.4$
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ =38 | … | $5.9\pm 1.3$ | … | … | $4.3\pm 1.6$ | $2.7\pm 1.6$ | $1.4\pm 0.5$

Columns: 1 = AGN name. 2–8 = Ratio of maser to X-ray scaling for each
reference source. Reference sources: GROD05 = reference source GRO J1655-40 in
the decreasing phase; GROR05 = reference source GRO J1655-40 in the rising
phase; GXD03 = reference source GX 339-4 in the decreasing phase; GXR03 =
reference source GX 339-4 in the rising phase; XTER98 = reference source XTE
J1550-564 in the rising phase; 3*XTER98 = reference source XTE J1550-564 in
the rising phase with a multiplicative correction of a factor 3 applied. Note,
the average value (in column 7) is obtained averaging all the
$M_{\mathrm{BH}}$ obtained from all the reference sources but excluding
3*XTER98. Note: For each source the first line reports the values obtained
using $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 30 in the spectral fitting; the
second line (present only for sources with relatively flat or steep spectra)
explicitly states the different value of $N_{\mathrm{PL}}/N_{\mathrm{BMC}}$
used.

##  5 Discussion and Conclusions

Constraining the $M_{\mathrm{BH}}$ in AGN is of crucial importance, since it
determines the space and temporal scales of BHs, constrains their accretion
rate via the Eddington ratio, and plays an essential role in our understanding
of the BH growth and co-evolution with the host galaxy. The most reliable ways
to determine the $M_{\mathrm{BH}}$ are direct dynamical methods, which measure
the orbital parameters of “test particles”, whose motion is dominated by the
gravitational force of the supermassive BH. For example, the mass of the
supermassive BH at the center of our Galaxy has been tightly constrained by
detailed studies of the orbits of a few innermost stars observed over several
years (e.g., Ghez et al. 2008; Gillessen et al. 2009). In nearby weakly active
galaxies, the $M_{\mathrm{BH}}$ is determined by the gas dynamics within the
sphere of influence of the BH (e.g., Gebhardt et al. 2003). On the other hand,
in bright type 1 AGN, the $M_{\mathrm{BH}}$ measurement is obtained from the
dynamics of the BLR via the reverberation mapping technique (e.g., Peterson et
al. 2004). Finally, in heavily absorbed type 2 AGN, where the BLR is
completely obscured, the only possible direct measurement of the
$M_{\mathrm{BH}}$ is based on megamaser measurements (e.g., Kuo et al. 2011
and references therein).

The main problem with direct dynamical methods is that they are fairly limited
in their application. For instance, direct measurements of $M_{\mathrm{BH}}$
via gas dynamics are limited to nearby weakly active galaxies, where the
sphere of influence is not outshined by the AGN and are sufficiently close to
be resolvable at the angular resolution of ground-based observatories.
Similarly, the reverberation mapping technique, which is heavily time and
instrument consuming, is limited to type 1 AGN with small or moderate masses.
Finally, the megamaser emission in type 2 AGN is relatively rare, and only
when the megamaser originates in the accretion disk (as opposed to the jet and
outflows) can this technique be used to constrain the $M_{\mathrm{BH}}$ (e.g.,
Panessa et al. 2020 and references therein).

Fortunately, there are a few robust indirect methods that make it possible to
constrain the $M_{\mathrm{BH}}$ beyond the range of applicability of the
direct dynamical ones. For example, the tight correlation between
$M_{\mathrm{BH}}$ and the stellar velocity dispersion in the bulge
$\sigma_{\ast}$, observed in nearby nearly quiescent galaxies (e.g., Tremaine
et al. 2002), can be extrapolated to constrain the $M_{\mathrm{BH}}$ in many
distant and more active galaxies. Similarly, the empirical relationship
between the BLR radius and optical luminosity makes it possible to determine
the mass of numerous type 1 AGN with only one spectral measurement without the
need of long monitoring campaigns (e.g., Kaspi et al. 2000).

Although indirect methods have proven to be very useful to derive general
results for large samples of AGN, caution must be applied when these methods
are extrapolated well beyond the original range of applicability of the direct
methods. To check for consistency and avoid potential biases associated with
the various assumptions inherent in these indirect methods, it is important to
develop and utilize alternative techniques to constrain the $M_{\mathrm{BH}}$.
In this perspective, X-ray-based methods may offer a useful complementary way
to the more commonly used optically based ones, since X-rays that are produced
very close to the BH are less affected by absorption and by star and galaxy
contamination. Indeed, model-independent methods based on X-ray variability
yielded $M_{\mathrm{BH}}$ values broadly consistent with those obtained with
dynamical methods (e.g. Papadakis 2004; Nikołajuk et al. 2006; McHardy et al.
2006; Ponti et al. 2012). In a previous work focused on a sample of AGN with
reverberation mapping measurements and good quality XMM-Newton data, we
demonstrated that the X-ray scaling method also provides results in agreement
with reverberation mapping within the respective uncertainties (Gliozzi et
al., 2011).

It is important to bear in mind that the X-ray scaling method is not
equivalent to making some general assumptions on the accretion rate and the
bolometric correction and deriving the BH mass from the X-ray luminosity using
the formula $M_{\mathrm{BH}}=\kappa_{\mathrm{bol}}L_{\mathrm{X}}/(1.3\times
10^{38}\lambda_{\mathrm{Edd}})$, where $\kappa_{\mathrm{bol}}$ is the
bolometric correction that may range from 15 to 150 depending on the accretion
rate of the source (Vasudevan & Fabian, 2009), and $L_{\mathrm{X}}$ the X-ray
luminosity in erg/s. With this simple equation, without an a priori knowledge
of the accretion rate of the source, one could at best obtain the order of
magnitude of the $M_{\mathrm{BH}}$. Since $\lambda_{\mathrm{Edd}}$ can vary
over a broad range (for example, for this small sample of obscured AGN, the
Eddington ratio varies from 0.01 to 0.3), it is not possible to obtain a
specific value of $M_{\mathrm{BH}}$ that can be quantitatively compared with
the value obtained from the dynamical method and find a good agreement, as we
did with the scaling method.

One may then argue that the only important parameter in the scaling method is
$N_{\mathrm{BMC}}$ (because of its direct dependence on the accretion
luminosity and distance) and that it is still possible to obtain a good
agreement with the dynamically estimated $M_{\mathrm{BH}}$ with any value of
the photon index. To test this hypothesis, we have selected the two sources
with the flattest spectra of our sample (NGC 4388 and NGC 4945) and the two
sources with the steepest spectra (IC 2560 and Circinus), and recalculated
their $M_{\mathrm{BH}}$ with the scaling method assuming $\Gamma=2.17$ for the
flattest sources and $\Gamma=1.66$ for the steepest sources. This led to
changes of $M_{\mathrm{BH}}$ by a factor slightly larger than 2 (note that
considerably larger changes of $M_{\mathrm{BH}}$ would have resulted if we had
used a larger difference in the photon indices instead of the minimum and
maximum values of this small sample). If the photon index did not play any
role, then these $M_{\mathrm{BH}}$ changes should have not made a difference
in the agreement with the values obtained via the dynamical method, with some
objects showing a slightly better agreement and others a slightly worse
agreement. Instead, all four sources, which were originally consistent with
their maser respective estimates based on the mass ratio criterion described
above (see Table 5 and Figure 6), showed a clear departure from the dynamical
$M_{\mathrm{BH}}$ values with three sources (NGC 4388, NGC 4945, and IC 2560)
that were not formally consistent with the maser values anymore (their new
mass ratios were 8.0, 7.1, and 6.7, respectively) and only Circinus (ratio of
0.5) still consistent, but only by virtue of the fact that the original ratio
was basically 1. We therefore conclude that the scaling method works because
the photon index accurately characterizes the accretion state of accreting
black holes and allows the correct selection of the reference source’s
$N_{\mathrm{BMC}}$ value to be compared with the AGN’s value.

In this study, we have extended the X-ray scaling method to a sample of
heavily obscured type 2 AGN with $M_{\mathrm{BH}}$ already constrained by
megamaser measurements. This dynamical method is rightly considered one of the
most reliable; however, the accuracy of the $M_{\mathrm{BH}}$ derived with
this technique depends on the quality of the radio data, on the assumption
that the megamaser emission is produced in an edge-on disk, and that its
rotation curve is strictly Keplerian. Additionally, one should bear in mind
that this technique measures the mass enclosed within the megamaser emission.
As a consequence, the actual $M_{\mathrm{BH}}$ may be slightly smaller if the
measured enclosed mass encompasses a nuclear cluster or the inner part of a
massive disk, or alternatively slightly larger if radiation pressure (not
included in the $M_{\mathrm{BH}}$ derivation) plays an important role (Kuo et
al., 2011).

Specifically, for the sources of our sample, the rotation curve traced by the
megamaser in NGC 1068 is non-Keplerian; the $M_{\mathrm{BH}}$ was derived
assuming a self-gravitating accretion disk model (Lodato & Bertin, 2003). NGC
1194 displays one of the largest maser disks (with inner and outer radii of
0.54 and 1.33 parsecs) which appears to be slightly bent and is consistent
with Keplerian rotation (Kuo et al., 2011). NGC 2273 also shows indications of
a warped but much smaller disk (with inner and outer disk radii of 0.028 and
0.084 pc) with Keplerian rotation (Kuo et al., 2011). In NGC 3079 the disk
appears to be thick and flared (Kondratko, Greenhill, & Moran, 2005), whereas
in NGC 3393 the maser seems to describe a flat disk perpendicular to the kpc
radio jet, and the positions of the maser points have substantial
uncertainties (Kondratko, Greenhill, & Moran, 2008). NGC 4388, located in the
Virgo cluster, has only five megamaser spots, which make it impossible to
demonstrate that they lie on a disk or that the rotation is Keplerian (Kuo et
al., 2011). For this reason, to reflect the actual uncertainty on the
$M_{\mathrm{BH}}$ derived by megamaser measurements, we have increased the
statistical error by a factor of 10, leading to an uncertainty of $\sim$24%.
The megamaser in NGC 4945 has been modeled as an edge-on thin disk, although
this is not the only possible interpretation of the data; the non-Keplerian
rotation of the blue-shifted emission and the substantial position errors lead
to a relatively large uncertainty in the $M_{\mathrm{BH}}$ of $\sim$50%
(Greenhill et al., 1997). In IC 2560 the megamaser emission has been
attributed to an edge-on thin disk with Keplerian rotation with some
additional contribution from a jet (Yamauchi et al., 2012). Finally, the
megamaser emission in Circinus appears to be associated with a warped
accretion disk and a wide-angle outflow (Greenhill et al., 2003). In summary,
because of the presence of outflows, jets, disk warps, or non-Keplerian
rotation curves, we should consider the $M_{\mathrm{BH}}$ values determined
from megamaser measurements as robust estimates but not as extremely accurate
values, and the errors reported in Table 1 are likely lower limits on their
actual uncertainties.

With respect to type 1 AGN, the main difficulty of applying the X-ray scaling
method to heavily obscured AGN is the need to properly constrain the
parameters of the primary emission in sources whose X-ray spectra are
dominated by absorption and reflection. However, the NuSTAR spectra of these
specific sources, often complemented with Chandra and XMM-Newton data, were
the object of very detailed analyses, which led to the disentanglement and a
careful characterization of the different contributions of the AGN direct and
reprocessed emission, of the host galaxy, and of the off-nuclear sources
located in the spectral extraction region (e.g., Yaqoob 2012; Puccetti et al.
2014; Arévalo et al. 2014; Bauer et al. 2015). Guided by these findings, we
were able to parametrize the torus contribution using the physically motivated
self-consistent model Borus (Baloković et al., 2018) instead of the MYTorus or
Torus models used in the previous analyses. To characterize the primary
emission, instead of the phenomenological power-law model, we utilized the BMC
Comptonization model, since the scaling method directly scales the
normalization of this model $N_{\mathrm{BMC}}$ between AGN and an appropriate
stellar reference to determine $M_{\mathrm{BH}}$.

With our baseline spectral model, where we assumed
$N_{\mathrm{PL}}/N_{\mathrm{BMC}}$ = 30, as described in detail in Section 3
(see also the Appendix for details on the spectral fittings of individual
sources), and applying the scaling technique summarized in Section 4.1, we
obtained the following results:

  * •

Many of the $M_{\mathrm{BH}}$ values, obtained with different reference
trends, are broadly in agreement with the corresponding megamaser ones. In
particular, the estimates derived using GROD05, 3*XTER98, and the ones
obtained by averaging the values inferred from all the available reference
sources, are consistent at the 3$\sigma$ level, based on measurements of
$\Delta
M_{\mathrm{BH}}/\sigma_{\mathrm{diff}}=(M_{\mathrm{BH,maser}}-M_{\mathrm{BH,scaling}})/\sigma_{\mathrm{diff}}$,
which are reported in Table 4 and shown in Fig. 5.

  * •

The agreement between the two methods is confirmed by the
$M_{\mathrm{BH,maser}}/M_{\mathrm{BH,scaling}}$ ratio: for all type 2 AGN of
our sample $(M_{\mathrm{BH,maser}}/M_{\mathrm{BH,scaling}})\pm 3\sigma\leq 1$,
when using the best reference sources or the average $M_{\mathrm{BH}}$, as
summarized in Table 5. Fig. 6 illustrates the good agreement between the two
methods, showing that GROD05, GXD04 (partially), 3*XTER98, and the average
obtained from all reference patterns are all consistent with the one-to-one
ratio within a factor of three.

  * •

The only object of our sample for which the $M_{\mathrm{BH}}$ inferred from
the X-ray scaling method is statistically inconsistent with the megamaser
value is NGC 1194, which is the AGN with the lowest accretion rate
($\lambda_{\mathrm{{Edd}}}\simeq 7\times 10^{-3}$). However, this discrepancy
is expected, since the X-ray scaling method cannot be applied in this regime,
where $\Gamma$ generally shows an anti-correlation with
$\lambda_{\mathrm{Edd}}$.

In conclusion, our work demonstrates that the same X-ray scaling method works
equally well for type 1 AGN (given the formal agreement with the reverberation
mapping sample) and type 2 AGN (based on the agreement with the megamaser
sample). We thus conclude that this method can be safely applied to any type
of AGN regardless of their level of obscuration, provided that these sources
accrete above a minimum threshold and that their primary X-ray emission can be
robustly characterized via spectral analysis. This also proves that this
method is robust and can be used to complement the various indirect methods,
especially when they are applied well beyond the range of validity of the
direct methods, from which they were calibrated. Finally, the X-ray scaling
method offers the possibility to investigate in a systematic and homogeneous
way the existence of any intrinsic difference in the fundamental properties of
the central engines in type 1 and type 2 AGN. We plan to carry out this type
of investigation in our future work.

## Acknowledgements

We thank the anonymous referee for constructive comments and suggestions that
improved the clarity of the paper and helped strengthen our conclusions. This
research has made use of data, software, and/or web tools obtained from the
High Energy Astrophysics Science Archive Research Center (HEASARC), a service
of the Astrophysics Science Division at NASA/GSFC and of the Smithsonian
Astrophysical Observatory’s High Energy Astrophysics Division, and of the
NuSTAR Data Analysis Software (NuSTARDAS) jointly developed by the ASI Science
Data Center (ASDC, Italy) and the California Institute of Technology (Caltech,
USA).

## Data Availability

The data underlying this article are available in the High Energy Astrophysics
Science Archive Research Center (HEASARC) Archive at
https://heasarc.gsfc.nasa.gov/docs/archive.html.

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##  Appendix A Additional Spectral Results

NGC 1068: A detailed analysis of the NuSTAR, XMM-Newton, and Chandra spectra
of this source was carried out by Bauer et al. (2015). Thanks to the excellent
sensitivities of XMM-Newton and NuSTAR over broad complementary energy ranges,
and to the sub-arcsecond spatial resolution of Chandra, the authors were able
to disentangle the contributions of the host galaxy and off-nuclear sources
from the AGN emission within the NuSTAR extraction region. The overall best-
fit model is fairly complex and comprises several Fe and Ni emission lines, a
Bremsstrahlung component to account for the radiative recombination continuum
and lines, a cutoff power-law model to account for the off-nuclear X-ray
sources, in addition to the AGN-related emission, which is parametrized by two
different MYTorus scattered and line components, in addition to the
transmitted one described by the zeroth-order component of that model. In our
fitting, in addition to our baseline model we added the Bremsstrahlung and
cutoff power law with all parameters fixed at the values provided by Bauer et
al. (2015), and a Gaussian line to roughly model the excess around 6.5 keV. To
account for the multiple absorption components, we also added a second Borus
model, whose best-fit parameters are
$\log(N_{{\textrm{H}}_{\textrm{bor}}})=24.9\pm 0.1$, CFtor = 83%, and
$A_{\textrm{Fe}}=1$. Our best-fit parameters are broadly consistent with the
results presented by Bauer et al. (2015). The observed flux in the 2–10 keV
energy band is $5.4\times 10^{-12}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and the
intrinsic one (i.e., corrected for absorption) $1.3\times
10^{-10}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.

NGC 1194: The starting model for the spectral fit of this source is provided
by the work of Masini et al. (2016), who fitted the NuSTAR spectrum with the
MYTorus model in the decoupled mode, with the addition of a Gaussian line at
6.8 keV, and a scattering fraction of the primary continuum of
$f_{\mathrm{s}}\sim 3\%$. In our fitting, we used our baseline model and found
the main parameters ($\Gamma$, $N_{\mathrm{H}}$, and $f_{\mathrm{s}}$) to be
fully consistent with their best-fit results. The 2–10 keV observed flux is
$1.2\times 10^{-12}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and the intrinsic one
$1.0\times 10^{-11}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.

NGC 2273: The starting spectral model for this source is again provided by the
work of Masini et al. (2016), who fitted the NuSTAR spectrum with the Torus
model that favored a heavily absorbed scenario with $N_{\mathrm{H}}>7\times
10^{24}\,{\mathrm{cm^{-2}}}$. In our fitting, we used our baseline model,
which yielded a best fit broadly consistent with their results. The 2–10 keV
observed flux is $9.2\times 10^{-13}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and the
intrinsic one $3.6\times 10^{-10}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.

NGC 3079: The starting spectral model for this source is again provided by the
work of Masini et al. (2016), who fitted the NuSTAR spectrum with the MYTorus
model in a coupled mode. The results obtained with our baseline model are
consistent within the respective uncertainties with their results. The 2–10
keV observed flux is $6.4\times 10^{-13}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and
the intrinsic one $1.2\times 10^{-10}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.

NGC 3393: The starting spectral model for this source is provided by the work
of Koss et al. (2015) and Masini et al. (2016), who fitted the NuSTAR spectrum
with both MYTorus and Torus models. The results obtained with our baseline
model are broadly consistent with the results presented by these authors with
a slightly larger value of $N_{\mathrm{H}}$ ($10^{25}$ vs. $2.2\times
10^{24}\,{\mathrm{cm^{-2}}}$). The 2–10 keV observed flux is $4.4\times
10^{-13}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and the intrinsic one $8.3\times
10^{-11}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.

NGC 4388: The starting spectral model for this source is once more provided by
the work of Masini et al. (2016), who fitted the NuSTAR spectrum with the
MYTorus and Torus models, which favor a Compton-thin scenario with a
substantial scattered primary emission that dominates below 5 keV. The results
from our baseline model are fully consistent with their results. The 2–10 keV
observed flux is $7.9\times 10^{-12}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and the
intrinsic one $1.4\times 10^{-11}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.

NGC 4945: A detailed analysis of the NuSTAR, Suzaku, and Chandra spectra of
this source was carried out by Puccetti et al. (2014), who in turn, were
guided by the results obtained by Yaqoob (2012) based on a comprehensive
analysis of all the hard X-ray spectra available at that time. The wealth of
high-quality broad-band spectra obtained with several observatories made it
possible to parametrize separately the different contributions of the host
galaxy, the AGN, and contaminating sources within the NuSTAR extraction
region. The best-fit model is fairly complex and comprises several emission
lines, the galaxy optically thin thermal continuum, which is described by the
APEC model, the contamination from off-nuclear sources parametrized by a power
law, and the AGN emission seen through a torus described by the MYTorus model
in the decoupled mode. In our fitting procedure, in addition to our baseline
model we included the APEC and power-law models with all parameters fixed at
the values provided by Yaqoob. Our results are broadly consistent with those
obtained by both Yaqoob and Puccetti. The 2–10 keV observed flux is $3.7\times
10^{-12}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and the intrinsic one $2.7\times
10^{-10}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.

IC 2560: The starting spectral model for this source is again provided by the
work of Masini et al. (2016), who fitted the NuSTAR spectrum with the Torus
model, which favors a heavily absorbed primary emission characterized by a
steep photon index. The results from our baseline model are broadly consistent
with their results. The 2–10 keV observed flux is $3.7\times
10^{-13}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and the intrinsic one $1.7\times
10^{-10}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.

Circinus: A detailed analysis of the NuSTAR, XMM-Newton, and Chandra spectra
of this source was carried out by Arévalo et al. (2014). Combining the
complementary properties of these observatories (i.e., the high sensitivities
of XMM-Newton and NuSTAR over broad energy ranges and the sub-arcsecond
spatial resolution of Chandra), the authors were able to disentangle the
contributions of different contamination sources (diffuse emission from the
host galaxy, supernova remnant contribution, and off-nuclear X-ray binary
sources) from the AGN emission within the NuSTAR extraction region. The
overall best-fit model is complex and comprises several emission lines, an
APEC model for the diffuse emission, three Mekal models to parametrize the
supernova remnant, and a power-law model to account for the off-nuclear point-
like sources, in addition to two different MYTorus models used in the
decoupled mode. In our fitting, in addition to our baseline model we added all
the contamination models with all the parameters fixed at the values provided
by Arévalo et al. (2014) and three Gaussian lines to roughly model the line
excess in the 5.5–7.5 keV range. To account for the multiple absorption
components, we also added a second Borus model, whose best-fit parameters are
$\log(N_{{\textrm{H}}_{\textrm{bor}}})=24.6\pm 0.1$, CFtor = 10%, and
$A_{\textrm{Fe}}=1$. Our best-fit parameters are broadly consistent with their
results. The 2–10 keV observed flux is $2.0\times
10^{-11}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$, and the intrinsic one $2.1\times
10^{-10}~{}\mathrm{erg~{}cm^{-2}s^{-1}}$.

##  Appendix B The X-ray scaling method

The X-ray scaling method for determining the mass of a black hole
($M_{\mathrm{BH}}$) was first described by Shaposhnikov & Titarchuk (2009) and
first applied to AGN by Gliozzi et al. (2011), where the method is described
in detail. Here, we only report the essential information on the stellar
reference sources – their $M_{\mathrm{BH}}$ values and distances (Table 6) and
the mathematical expression of the spectral trend with the best fit parameters
for the different sources (Table 7) – that is needed to reproduce the
$M_{\mathrm{BH}}$ values. The two steps below accomplish the scaling described
in Section 4.1.

Step 1. Use the following equation to solve for $N_{\textrm{BMC,r}}$, the BMC
normalization the reference source would have at the same photon index as the
target AGN. The reference source is a Galactic, stellar-mass black hole with
known mass and distance.

$N_{\textrm{BMC,r}}(\Gamma)=N_{\textrm{tr}}\times\left\\{1-\ln\left[\exp\left(\frac{A-\Gamma}{B}\right)-1\right]\right\\}^{(1/\beta)}$
(1)

where $\Gamma$ is the photon index of the target AGN as determined by the
spectral fit, and $A$, $B$, $N_{\mathrm{tr}}$, and $\beta$ are given in Table
7. Note: this equation was first presented by Jang et al. (2018) with an
error: there should be a minus sign before the logarithm.

Step 2. Use the equation presented in Section 4.1 to solve for
$M_{\textrm{BH,t}}$.

$M_{\textrm{BH,t}}=M_{\textrm{BH,r}}\times\left(\frac{N_{\textrm{BMC,t}}}{N_{\textrm{BMC,r}}}\right)\times\left(\frac{d_{\textrm{t}}}{d_{\textrm{r}}}\right)^{2}$
(2)

where $M_{\mathrm{BH}}$ is the black hole mass, $N_{\mathrm{BMC}}$ is the BMC
normalization, and $d$ is the distance. The $t$ subscript denotes the target
AGN and the $r$ subscript denotes the reference source.

Table 6: Characteristics of reference sources

Name | $M_{\mathrm{BH}}$ | $d$
---|---|---
| (M☉) | (kpc)
GRO J1655-40 | $6.3\pm 0.3$ | $3.2\pm 0.2$
GX 339-4 | $12.3\pm 1.4$ | $5.7\pm 0.8$
XTE J1550-564 | $10.7\pm 1.5$ | $3.3\pm 0.5$

Source: Gliozzi et al. (2011)

Table 7: Parametrization of $\Gamma$–$N_{\mathrm{BMC}}$ reference patterns

Transition | $A$ | $B$ | $N_{\mathrm{tr}}$ | $\beta$
---|---|---|---|---
(1) | (2) | (3) | (4) | (5)
GRO J1655-40 D05 | $1.96\pm 0.02$ | $0.42\pm 0.02$ | $0.023\pm 0.001$ | $1.8\pm 0.2$
GRO J1655-40 R05 | $2.35\pm 0.04$ | $0.74\pm 0.04$ | $0.131\pm 0.001$ | $1.0\pm 0.1$
GX 339-4 D03 | $2.13\pm 0.03$ | $0.50\pm 0.04$ | $0.0130\pm 0.0002$ | $1.5\pm 0.3$
GX 339-4 R04 | $2.10\pm 0.03$ | $0.46\pm 0.01$ | $0.037\pm 0.001$ | $8.0\pm 1.5$
XTE J1550-564 R98 | $2.96\pm 0.02$ | $2.8\pm 0.2$ | $0.055\pm 0.010$ | $0.4\pm 0.1$

Columns: 1 = reference source spectral transition. 2 = parameter that
determines the rigid translation of the spectral pattern along the y-axis. 3 =
parameter characterizing the lower saturation level of the pattern. 4 =
parameter that determines the rigid translation of the spectral pattern along
the x-axis. 5 = slope of the spectral pattern. Source: Gliozzi et al. (2011)