# Black Hole Spin Measurements Based on a Thin Disc Model with Finite
Thickness I. An example study of MCG$-$06-30-15

Jiachen Jiang,1 Askar B. Abdikamalov2,3,4, Cosimo Bambi2 and Christopher S.
Reynolds1
1Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge
CB3 0HA, UK
2Department of Physics, Fudan University, 2005 Songhu Road, Shanghai 200438,
China
3Ulugh Beg Astronomical Institute, Tashkent 100052, Uzbekistan
4Institute of Fundamental and Applied Research, National Research University
TIIAME, Kori Niyoziy 39, Tashkent 100000, Uzbekistan E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a re-analysis of the XMM-Newton and NuSTAR observing campaign for
the well-studied, X-ray-bright AGN MCG$-$06-30-15. In particular, we consider
a disc model with finite thickness. By fitting the disc reflection spectra in
the data, we obtain a black hole spin of 0.87–0.99 (90% confidence range)
after taking the thickness of the disc into consideration. Spectral models
with a grid of mass accretion rate from $0$ to $30\%\dot{M}_{\rm Edd}$ are
calculated for MCG$-$06-30-15. This result is obtained by considering a free
disc reflection fraction parameter $f_{\rm refl}$ and is consistent with
previous measurements based on razor-thin disc models. Besides, an isotropic,
point-like geometry, i.e. the ‘lamppost’ geometry, is assumed for the corona
in our model. We find that such a geometry overestimates $f_{\rm refl}$ in the
data. Therefore, thin disc models with consistent ‘lamppost’ values of $f_{\rm
refl}$ provide a worse fit than ones with a free $f_{\rm refl}$ parameter. We
discuss possible reasons for the discrepancy between the observed and
theoretical values of $f_{\rm refl}$ at the end of the paper. Modifications
for the over-simplified lamppost model might be needed when the thickness of
the thin disc is considered in future work.

###### keywords:

accretion, accretion discs - black hole physics, X-ray: galaxies, galaxies:
Seyfert

††pubyear: 2021††pagerange: Black Hole Spin Measurements Based on a Thin Disc
Model with Finite Thickness I. An example study of MCG$-$06-30-15–Black Hole
Spin Measurements Based on a Thin Disc Model with Finite Thickness I. An
example study of MCG$-$06-30-15

##  1 Introduction

###  1.1 Disc Reflection Spectroscopy and Black Hole Spin

The X-ray continuum emission of Seyfert active galactic nuclei (AGN) shows a
power-law shape, which originates in the up-scattering Comptonisation
processes of lower energy disc photons in the black hole (BH) corona (e.g.
Maisack et al., 1993). The optically thick disc is illuminated by the hot
corona and produces a reprocessed spectrum on its surface, which is often
referred to as the disc ‘reflection’ spectrum. The most prominent features of
the disc reflection spectrum are the Fe K$\alpha$ emission line at 6.4 keV
(e.g. Tanaka et al., 1995) and the back-scattering Compton hump above 10 keV
(e.g. Nandra et al., 1990). The Fe K$\alpha$ emission line of the disc shows a
distinctive shape rather than a narrow line, which results from the strong
relativistic effects in the vicinity of a BH (Fabian et al., 2004). Such a
broad Fe K$\alpha$ emission line has been seen in not only a handful of AGN
(e.g. Larsson et al., 2008; Brenneman et al., 2011; Tan et al., 2012; Risaliti
et al., 2013; Parker et al., 2014b; Walton et al., 2013; Jiang et al., 2019a)
but also X-ray binaries (e.g. Fabian et al., 1989; Cackett & Miller, 2013;
Miller et al., 2013).

The modelling of disc reflection spectra involves many detailed scientific
calculations. For instance, the properties of the accretion disc (e.g. García
et al., 2016; Jiang et al., 2018, 2019c), the geometry of the corona (e.g.
Wilkins et al., 2014; Gonzalez et al., 2017) and the spacetime around a BH
(e.g. Berti et al., 2015; Bambi, 2017) all have to be taken into
consideration. The disc reflection model is applied accordingly to study
different topics.

One of the most frequent applications of disc reflection spectroscopy is the
measurement of BH spin. The extent of the broad Fe K$\alpha$ emission line to
low energies, the ‘red wing’, is determined by how close the line emission
region of the disc is to the BH (Fabian et al., 1989). When the inner edge of
the disc is closer to the BH, the disc Fe K$\alpha$ emission shows a more
extended red wing due to gravitational redshift. Assuming the inner edge of
the disc is at the innermost stable circular orbit (ISCO), one can measure the
spin of the BH by using disc reflection spectropy based on the simple
correlation between BH spin and the radius of ISCO (see latest reviews in
Reynolds, 2019; Bambi et al., 2021).

###  1.2 The Thin Disc Model with Finite Thickness

Previous disc reflection models assumed a simple thin disc model with
infinitely small height for simplification. There have been few attempts to
fit the X-ray data of AGN using a model with finite disc thickness as expected
in reality because of the difficulty of modelling more complex geometries.

In the standard thin disc model, the pressure scale height of a radiation
pressure-dominated disc at $r$ is

$H=\frac{3}{2}\frac{\dot{m}}{\epsilon}[1-(\frac{r_{\rm
ISCO}}{r\sin(\theta)})^{1/2}],$ (1)

where $\dot{m}$ is mass accretion rate in units of Eddington accretion rate
($\dot{m}=\dot{M}/\dot{M}_{\rm Edd}$), $\epsilon$ is the radiative efficiency
of the disc and $r\sin(\theta)$ is the pseudo-cylindrical radius (Shakura &
Sunyaev, 1973). Materials at the same $r\sin(\theta)$ share the same angular
velocity. This solution is calculated assuming the flat Newtonian spacetime
but very similar with the relativistic solution (e.g. Novikov & Thorne, 1973;
Pariev & Bromley, 1998).

As in previous reflection model in Pariev & Bromley (1998); Taylor & Reynolds
(2018), we assume the X-ray reflection atmosphere of the disc is at $z=2H$. We
show the profiles of $z/r$ for various $\dot{m}$ and $a_{*}$ in Fig. 1. At
$\dot{m}=5\%$, $z/r$ reaches 4% in the inner disc region and starts decreasing
in the outer disc. In comparison, $z/r$ reaches a higher value of 30% in the
inner disc region, indicating a thicker disc at higher $\dot{m}$. When
$\dot{m}$ is higher than 30%, the disc may not hold the thin disc geometry as
in Eq. 1 (Shakura & Sunyaev, 1973). We, therefore, only discuss the cases of
$\dot{m}<30\%$ in this work.

In Fig. 1, we show that the inner radius of the disc, which is assumed to be
at ISCO, is closer to the BH at the maximum BH spin than a modest BH spin. The
disc is also thinner at a higher spin than a lower spin, suggesting that
$\dot{m}$ has a smaller impact on disc thickness when the BH spin is high
because of a higher radiative efficiency. For instance, the height of the disc
is up to 5 $r_{\rm g}$ within a radius of 100 $r_{\rm g}$ for $a_{*}=0.9$
while the height reaches only 2.5 $r_{\rm g}$ at most for $a_{*}=0.998$.

Figure 1: Disc shapes at different $\dot{m}$. The left two panels show the
disc thickness profile $z/r$ for $a_{*}=0.90$ (top) and $a_{*}=0.998$
(bottom). The inner radius of the disc is assumed to be at ISCO. At higher
$\dot{m}$, the disc is thicker. The right two panels show schematic images of
the thin disc models for different $\dot{m}$ and $a_{*}$. The black circles
show that the corona is located on the spinning axis of the BH in the lamppost
geometry.

The thickness of the disc in the standard disc model (Shakura & Sunyaev, 1973)
is comparable to the observed size of the coronal region in AGN. X-ray
reverberation studies suggest that the corona has to be very compact within a
region of approximately 10 $r_{\rm g}$ (e.g. Fabian et al., 2009; De Marco et
al., 2013; Kara et al., 2017; Alston et al., 2020). So the thickness of the
disc may play an important role in disc reflection modelling.

The effects of disc thickness on resulting reflection spectra were calculated
in Pariev & Bromley (1998); Wu & Wang (2007); Tripathi et al. (2021).
Numerical ray-tracing techniques were used. They calculated $g$-factors, the
energy shift from the disc to the observer, on the surface of the inner
accretion disc accordingly. However, the disc emissivity profile was
approximated as a power law. Further studies by Taylor & Reynolds (2018) show
that the observed emissivity profile is also an essential indicator of disc
thickness. For example, we compare the emissivity profiles of a razor-thin
disc and ones with finite thickness in Fig. 2. The emissivity profile of the
disc is flatter than the one for a razor-thin disc as shown in Fig. 1.

To calculate the emissivity profile of a thin disc, one needs to assume the
coronal geometry first. We consider the same geometry as in Taylor & Reynolds
(2018) by adapting the simplest and most understood ‘lamppost’ geometry in our
model, where the corona is a point source located on the spinning axis of the
BH (Martocchia & Matt, 1996). In this set-up, the corona emits isotropic
emission in its rest frame. The lamppost geometry successfully explains the
X-ray data of many AGN (e.g. Miniutti & Fabian, 2004; Martocchia et al., 2002;
Niedźwiecki & Życki, 2008) and is thought to be related to the base of the jet
(e.g. Ghisellini et al., 2004) or pair productions in magnetosphere around the
BH (e.g. Hirotani & Okamoto, 1998; Chen & Yuan, 2020). Although the lamppost
model is used in this work, we will also investigate whether the lamppost
geometry is sufficient to explain the data at the end of this paper.

Figure 2: Disc emissivity profiles for different values of $\dot{m}$ when the
thickness of the disc is considered. The corona is at $h=4$ $r_{\rm g}$ and
the BH spin is $a_{*}=0.95$ in the calculations.

###  1.3 MCG$-$06-30-15

To investigate the effects of disc thickness on the reflection modelling of
AGN spectra, we start with the most studied X-ray bright AGN in MCG$-$06-30-15
(MCG$-$06 hereafter). MCG$-$06 is a Seyfert 1.2 galaxy (Bennert et al., 2006)
at $z=0.008$ (Fisher et al., 1995). The BH mass of MCG$-$06 is estimated to be
$1.6\pm 0.4\times 10^{6}M_{\odot}$ based on optical reverberation measurements
(Bentz et al., 2016).

In the X-ray band, MCG$-$06 is the first AGN where broad Fe K$\alpha$ emission
was studied in detail (Tanaka et al., 1995; Iwasawa et al., 1996; Fabian et
al., 2002), allowing measurements of its BH spin using disc reflection
spectroscopy. Multiple studies indicate that the spin of the BH in MCG$-$06 is
high (e.g. $a_{*}$>0.90, Brenneman & Reynolds, 2006; Vaughan & Fabian, 2004;
Young et al., 2005; Reynolds et al., 2005; Miniutti et al., 2007; Marinucci et
al., 2014). Detailed modelling suggests two components in the Fe K emission of
MCG$-$06, one from the innermost accretion region close to the ISCO and one
from a distant neutral reflector (Ballantyne et al., 2003). Soft X-ray
reverberation lags are also discovered in MCG$-$06 (De Marco et al., 2013;
Kara et al., 2014). They are related to the light travel time difference
between the disc reflected light and the coronal X-ray continuum. However, no
significant evidence of Fe K reverberation lags has been found in MCG$-$06
(Kara et al., 2014).

Previously, Tripathi et al. (2021) applied the thin disc model with finite
thickness to the X-ray data of MCG$-$06\. A broken power law was used to model
the emissivity profile of its disc. They achieved consistent BH spin
measurements for MCG$-$06 as in previous work. However, the emissivity profile
is also affected by the geometry of the disc as introduced above. We,
therefore, reanalyse the data of MCG$-$06 and consider the same geometry set-
up as in Taylor & Reynolds (2018)–a thin disc with finite thickness shouldered
by the lamppost corona.

In Section 2, we introduce our data reduction processes. In Section 3, we
present a thin disc model with finite thickness for the spectral data of
MCG$-$06\. The reflection fraction parameter is treated as a free parameter
during the fit. In Section 4, we further discuss the effects of disc thickness
on the reflection fraction parameter of the model. Models with consistent
values of reflection fraction assuming the lamppost geometry are used to fit
the data of MCG$-$06\. In Section 6, we discuss and conclude our results.

##  2 Data Reduction

In this work, we consider the XMM-Newton (Jansen et al., 2001) and NuSTAR
(Harrison et al., 2013) observing campaign of MCG$-$06 in 2013. The same
observations were analysed in Marinucci et al. (2014); Tripathi et al. (2021).
A full list of the observations is in Table 1.

The EPIC data are reduced using V19 of the XMM-Newton Science Analysis System
(SAS) software package. The version of the calibration files is v.20201028. We
first generate a clean event file by running EPPROC. Then, we select good time
intervals by filtering out the intervals that are dominated by flaring
particle background. These high-background intervals are where the single
event (PATTERN=0) count rate in the >10 keV band is larger than 0.4 counts s-1
for pn data. By running the EVSELECT task, we select single and double event
lists from a circular source region of 35 arcsec. Background spectra are
extracted from a nearby circular region of 60 arcsec. No obvious evidence of
pile-up effects has been found in pn data. We do not consider MOS data because
they have significant pile-up effects. The pile-up effects are checked using
the EPATPLOT tool in SAS. Last, we create redistribution matrix files and
ancillary response files by running RMFGEN and ARFGEN. The spectra are grouped
to have a minimum of 20 counts per bin and oversample by a factor of 3.

We reduced the NuSTAR data using the NuSTAR Data Analysis Software (NuSTARDAS)
package and calibration data of V20211103. The energy spectra of MCG$-$06 were
extracted for both the FPMA and FPMB detectors from a 100′′ radius circle
centered on the source, while the background spectra were extracted from a
nearby circular region of the same size. We consider the 3–50 keV band of the
two FPM spectra. The FPM data above 50 keV are dominated by the background.

In the soft X-ray band, MCG$-$06 is known to show complex absorption features
from dust and warm absorbers (e.g. Lee et al., 2002; Turner et al., 2003;
Young et al., 2005; Chiang & Fabian, 2011). Besides, the inclusion of soft
X-ray data may introduce extra uncertainty to our spin measurements (e.g. by
modelling the soft excess emission, Jiang et al., 2019c). Therefore, we focus
on only the hard X-ray data of MCG$-$06 above 3 keV, where the broad Fe
K$\alpha$ emission line and the Compton hump are. We show in Section 3 that
consistent measurements as in previous work can still be achieved when we
consider only the hard X-ray data.

NuSTAR Obs ID | Date | Time (ks) |  XMM-Newton Obs ID | Date | Time (ks)
---|---|---|---|---|---
60001047002 | 2013-01-29 | 23 | 0693781201 | 2013-01-29 | 134
60001047003 | 2013-01-30 | 127 | 0693781301 | 2013-01-31 | 134
60001047005 | 2013-02-02 | 30 | 0693781401 | 2013-02-02 | 49
Table 1: NuSTAR and XMM-Newton observations of MCG$-$06 analysed in this work.

##  3 Spectral Analaysis

We use XSPEC V.12 (Arnaud et al., 1985) for spectral analysis and $\chi^{2}$
to estimate goodness of fit. We start our analysis with the simple razor-thin
disc model. Then we consider a thin disc model with a grid of $\dot{m}$ at 5%,
10%, 20% and 30%. We will also compare our results with previous spin
measurements.

###  3.1 The Razor-Thin Disc Model

We model the spectra of MCG$-$06 by following previous conclusions (e.g. Young
et al., 2005; Ballantyne et al., 2003; Marinucci et al., 2014): the Fe
K$\alpha$ emission of MCG$-$06 consists of two components, one from a distant
neutral reflector and one from the innermost accretion disc. The former is
calculated using the xillver model (García & Kallman, 2010). The ionisation
parameter111The ionisation parameters are reported in units of erg cm s-1
hereafter. of this component is fixed at $\log(\xi$/erg cm s${}^{-1})=0$. The
reflection spectrum from the inner accretion disc is calculated using the
relxilllpth_nk (Abdikamalov et al., 2020), an extended version222Tripathi et
al. (2021) applied the relxillth_nk model to the data of MCG$-$06\. A more
general metric than the Kerr metric was used, where an extra deformation
parameter was considered. When the deformation parameter is zero, this metric
is identical to the Kerr metric (Kerr, 1963). It has been found that the data
of MCG$-$06 are consistent with the Kerr metric (Tripathi et al., 2021). So we
only consider the Kerr metric in this work for simplicity. of the relxill_nk
package (Bambi, 2017; Abdikamalov et al., 2019). To model a razor-thin disc,
the $\dot{m}$ parameter of the model is set to be zero. Free parameters
include the dimensionless spin of the BH ($a_{*}$), the inclination angle of
the disc ($i$), the ionisation and the iron abundance of the disc ($\xi$ and
$Z_{\rm Fe}$) and the height of the corona ($h$). The inner radius of the disc
is set to be the ISCO. The illuminating spectrum of relxilllpth_nk is a power
law ($\Gamma$) with an exponential high-energy cutoff at Ecut. The emissivity
profile of the disc is calculated consistently according to the the geometry
of the disc and the height of the corona. Note that the reflection fraction
parameter $f_{\rm refl}$ is a free parameter in this section333The reflection
fraction $f_{\rm refl}$ is defined as the ratio between the flux that reaches
the disc and infinity (Dauser et al., 2016). We will consider consistent
values of $f_{\rm refl}$ in Section 4..

A constant model is used to account for cross-calibration uncertainties
between instruments. The tbabs model is used to account for Galactic
absorption. We consider the interstellar medium abundances calculated by Wilms
et al. (2000) in the tbabs model. The Galactic column density along the line
of sight towards MCG$-$06 is estimated to be $4.7\times 10^{20}$ cm-2
(Willingale et al., 2013). The effect of such a low column density is confined
in the soft X-ray band, e.g. <2 keV. Since we do not include soft X-ray data,
we therefore fix $N_{\rm H}$ at this value in the following analysis.

Figure 3: Top: best-fit razor-thin disc model ($\dot{m}=0$) for the pn and FPM
spectra of MCG$-$06\. Solid: total model; dashed line: relativistic reflection
component and coronal emission; dotted line: distant reflector. A zoom-in of
the spectra in the iron emission band is in Fig. 4. Bottom: corresponding
data/model ratio plot (red circles: pn; blue: FPMA; green: FPMB). The
reflection fraction parameter is allowed to vary during the fit.

The best-fit razor-thin disc model parameters are shown in Table 2. The best-
fit model is shown in Fig. 3. The razor-thin disc model provides a very good
fit to the data of MCG$-$06 with $\chi^{2}/\nu=903.99/765$. No significant
residuals are found (see the lower panel of Fig. 3). As shown in Fig. 4, the
iron emission of MCG$-$06 has two component as suggested by previous analyses.
The narrow emission line at 6.4 keV is from the neutral reflector. The broad
emission line is from the inner accretion disc.

Figure 4: Best-fit razor-thin disc ($\dot{m}=0$) model overlaid with the data
of MCG$-$06 (red circles: pn; blue: FPMA; green: FPMB). The dashed line shows
the relxilllpth_nk component.

We checked the constraints of all the parameters in this analysis by using the
Markov chain Monte Carlo (MCMC) algorithm. We use 500 walkers with a total
length of 5000000, burning the first 20000. A convergence test has been
conducted and the Gelman-Rubin scale-reduction factor $R<1.3$ for every
parameter. No obvious degeneracy was found. In particular, we show the
measurement uncertainties of $a_{*}$, $f_{\rm refl}$ and $h$ in Fig. 5.

Figure 5: Output distributions for the MCMC analysis of the best-fit razor-
thin disc model for MCG$-$06.

The best-fit razor-thin disc model infers a high BH spin of
$0.94^{+0.02}_{-0.04}$ in MCG$-$06, which is consistent with previous
measurements by the same observations (e.g. $a_{*}=0.91^{+0.06}_{-0.07}$,
Marinucci et al., 2014). Similar high BH spins were also achieved by other
observations (e.g. Young et al., 2005). Our model also suggests a compact
coronal region above the accretion disc ($h\approx 3r_{\rm g}$). The disc
requires an iron abundance approximately 3.7 times the solar value similar to
previous measurements (Marinucci et al., 2014). The total unabsorbed flux of
the coronal emission and the relativistic disc reflection component is around
$9.2\times 10^{-11}$ erg cm-2 s-1 in the 3-50 keV band. The distant, neutral
reflection component has an unabsorbed flux of around $5\times 10^{-13}$ erg
cm-2 s-1 in the same energy band, which is only 0.6% of the total X-ray flux.

So far, we have obtained a good fit using a razor-thin disc model. Our disc
reflection parameters are consistent with previous results, although we do not
include the soft X-ray data as in Marinucci et al. (2014); Tripathi et al.
(2021). We, therefore, continue focusing on only the Fe K$\alpha$ emission and
the Compton hump of MCG$-$06 in the following analysis.

Figure 6: Top: the solid line shows the best-fit model based on the razor-thin disc model assuming $\dot{m}=0$. The dashed and dotted line shows the disc models for $\dot{m}=5\%,10\%$ respectively. Other parameters of the $\dot{m}=5\%,10\%$ models shown in this figure are set to be the same as the ones in the best-fit $\dot{m}=0$ model. The difference between $\dot{m}=0$ and $\dot{m}=5\%$ models is small in the 3–50 keV band. The $\dot{m}=10\%$ model, however, shows a significant different spectral shape–a double-peak Fe K$\alpha$ emission is shown due to a flatter disc emissivity profile (see Fig. 2). Bottom: data/model ratio plot for the spectra of MCG$-$06 fit to the $\dot{m}=10\%$ model in the upper panel. Models | Parameters | Units | $\dot{m}=0$ | $\dot{m}=5\%$ | $\dot{m}=10\%$ | $\dot{m}=20\%$ | $\dot{m}=30\%$
---|---|---|---|---|---|---|---
relxilllpth_nk | h | $r_{\rm g}$ | $3.0^{+0.7}_{-0.5}$ | $3.6^{+0.6}_{-0.8}$ | $2.4^{+0.3}_{-0.2}$ | $2.3^{+0.7}_{-0.3}$ | $2.4\pm 0.4$
| $i$ | deg | $39.8\pm 1.7$ | $39.1^{+1.3}_{-1.0}$ | $40.4^{+0.6}_{-0.8}$ | $41.2\pm 0.6$ | $37.9\pm 1.7$
| $a_{*}$ | - | $0.94^{+0.02}_{-0.04}$ | $0.92^{+0.04}_{-0.05}$ | $0.95^{+0.02}_{-0.03}$ | $0.97\pm 0.02$ | $0.97\pm 0.02$
| $\log(\xi)$ | erg cm s-1 | $1.69^{+0.06}_{-0.20}$ | $1.69^{+0.04}_{-0.07}$ | $1.70^{+0.04}_{-0.20}$ | $1.70^{+0.10}_{-0.20}$ | $1.69^{+0.06}_{-0.20}$
| $Z_{\rm Fe}$ | $Z_{\odot}$ | $3.7^{+0.5}_{-0.2}$ | $3.4^{+0.6}_{-0.3}$ | $3.5\pm 0.2$ | $3.97^{+0.10}_{-0.08}$ | $3.7^{+0.2}_{-0.3}$
| $\Gamma$ | - | $1.94\pm 0.02$ | $1.94\pm 0.02$ | $1.95^{+0.02}_{-0.04}$ | $1.93\pm 0.02$ | $1.94\pm 0.02$
| Ecut | keV | $100\pm 12$ | $106^{+12}_{-9}$ | $103\pm 11$ | $97^{+8}_{-4}$ | $100^{+13}_{-10}$
| $f_{\rm refl}$ | - | $2.1^{+0.5}_{-0.4}$ | $1.25^{+0.10}_{-0.18}$ | $1.09^{+0.10}_{-0.20}$ | $0.84^{+0.17}_{-0.04}$ | $0.45^{+0.11}_{-0.07}$
| $\log(F_{\rm disc})$ | erg cm-2 s-1 | $-10.035\pm 0.007$ | $-10.003^{+0.007}_{-0.004}$ | $-10.032^{+0.007}_{-0.009}$ | $-10.039\pm 0.005$ | $-10.036\pm 0.007$
xillver | $\log(F_{\rm dis})$ | erg cm-2 s-1 | $-12.26\pm 0.06$ | $-12.28\pm 0.08$ | $-12.31^{+0.07}_{-0.04}$ | $-12.28^{+0.05}_{-0.03}$ | $-12.4\pm 0.2$
| $\chi^{2}/\nu$ | - | 903.99/765 | 904.44/765 | 899.41/765 | 900.58/765 | 907.73/765
Table 2: Best-fit parameters assuming different $\dot{m}$. $F_{\rm disc}$ and
$F_{\rm dis}$ are the 3-50 keV band flux of the disc reflection component and
the distant reflector respectively.

###  3.2 Thin Disc Models with Finite Thickness

To demonstrate how the thickness of the disc affects the shape of Fe K$\alpha$
emission line in the reflection model, we show spectral models for different
$\dot{m}$ in Fig. 6. The black solid line shows the best-fit razor-thin disc
model for MCG$-$06\. Then we only change the mass accretion rate parameter and
keep other parameters the same. For instance, the dashed line shows the model
for $\dot{m}=5\%$, which is almost consistent with the razor-thin disc model.
Their difference is less than 5% in the 3-50 keV band. However, the broad Fe
K$\alpha$ emission line in the model starts showing a double-peak shape when
$\dot{m}>10\%$. The $\dot{m}=10\%$ model shows a weaker red wing than the
$\dot{m}=0,5\%$ models. Because the region of the accretion disc around 5
$r_{\rm g}$, which is less affected by gravitational redshift, is more
illuminated when the disc is thicker. This leads to a flatter disc emissivity
profile for $r<5$ $r_{\rm g}$ as shown in Fig.2 and thus changes the shape of
the disc emission lines.

We then apply the thin disc model with finite thickness to the same spectra
analysed in Section 3.1. The $\dot{m}$ parameter is set at 5%, 10%, 20% and
30% respectively444Note that the $\dot{m}$ parameter in relxilllpth_nk is
defined as $\dot{M}/\dot{M_{\rm Edd}}$, where $\dot{M}$ is the mass accretion
rate and $L_{\rm Bol}=\epsilon\dot{M}c^{2}$. $\epsilon$ is the radiative
efficiency, the value of which for MCG$-$06 is uncertain due to observational
limit and Galactic obscuration (Raimundo et al., 2012). We do not further
discuss the accretion efficiency of MCG$-$06 as this is beyond the purpose of
this work.. This model has the same free parameters as the razor-thin disc
model introduced in Section 3.1.

Best-fit models are shown in Fig. 7 and best-fit parameters are shown in Table
2. These thin disc models with finite thickness provide similar good fits to
the data of MCG$-$06 as the razor-thin disc model. The model with the highest
mass accretion rate ($\dot{m}=30\%$) offers a slightly worse fit than others
by $\Delta\chi^{2}=3-8$ with the same number of free parameters. Although disc
thickness is considered in these models, most of the parameters have
consistent best-fit values as the razor-thin disc model, e.g. the photon index
and the high-energy cutoff of the continuum emission, the iron abundance and
the ionisation of the disc. We conducted similar MCMC analysis as in Section
3.1. The output distributions of the same key parameters in Fig. 5 are shown
in Fig. A14.

We note that a slightly lower $h$ is needed to fit the data when a higher
$\dot{m}$ is assumed, although the values are consistent within their 90%
confidence ranges. This is because the emissivity profile is flatter when and
the disc becomes thicker at high $\dot{m}$ (see Fig 2 and Fig. 6). A more
compact coronal region is required in the model. So the inner disc produces
more reflected light to fit the observed red wing of the broad Fe K$\alpha$
emission line. Similarly, a slightly higher BH spin is suggested by the
high-$\dot{m}$ models, although they are all consistent within their 90%
confidence ranges. A higher BH spin at high $\dot{m}$ indicates a smaller
inner disc radius, which also produces an extended red wing in the broad Fe
K$\alpha$ emission to fit the data.

Figure 7: The same as Fig. 3 but for different $\dot{m}$. The reflection
fraction parameter is allowed to vary during the fit. The $\dot{m}=30\%$ model
provides a slightly worse fit to the data than other models
($\Delta\chi^{2}\approx 3-8$ and the same number of free parameters).

In particular, we compare the spin measurements obtained in this work (black
squares) with previous results (black circles) based on razor-thin disc models
in Fig. 8. Among previous work, Young et al. (2005) analysed the grating data
of MCG$-$06 from Chandra while others all considered CCD-resolution data. The
highest value of BH spin was achieved by Brenneman & Reynolds (2006) where
Suzaku data of MCG$-$06 were analysed and a close-to-maximum BH spin of
$a_{\\*}=0.989^{+0.009}_{-0.002}$ was obtained. It is important to note that
Brenneman & Reynolds (2006) considered a relativistic line model instead of a
full disc reflection spectral model. A lower BH spin of
$a_{*}=0.91^{+0.06}_{-0.07}$ was achieved by Marinucci et al. (2014), where
the combination of relconv * xillver was used instead of relxill. Tripathi et
al. (2020) studied the difference between the relconv * xillver and relxill
models for MCG$-$06\. The latter model in particular considers the angle
dependence of the emission at different regions of the disc while the former
does not. This effect may play an important role in reflection modelling and
is considered in our thin disc model. Nevertheless, our spin measurements at
all $\dot{m}$ are consistent with previous results in Young et al. (2005);
Reynolds et al. (2005); Miniutti et al. (2007); Marinucci et al. (2014).

In summary, the lower limit of the BH spin in MCG$-$06 is 0.87 for
$\dot{m}=5\%$ and the higher limit is 0.99 for $\dot{m}=20\%$. The mass
accretion rate of MCG$-$06 is uncertain. We, therefore, conclude that the 90%
confidence range of the BH spin in MCG$-$06 is 0.87–0.99 after taking the
thickness of the disc into consideration. This result is consistent with
previous measurements based on razor-thin disc models. Besides, a slightly
higher $a_{*}$ and a more compact coronal region is needed in a
higher-$\dot{m}$ disc model to fit the red wing of the observed broad Fe
K$\alpha$ emission. We will present spectral simulations for further
discussion in the following paper of this series.

Figure 8: Best-fit spin parameters in this work (squares) in comparison with
previous measurements (circles) in the literature (Young et al., 2005;
Reynolds et al., 2005; Brenneman & Reynolds, 2006; Miniutti et al., 2007;
Marinucci et al., 2014). The data analysed in their work are noted behind the
references. Nu: NuSTAR; XMM: XMM-Newton; Ch: Chandra grating data; Su: Suzaku.

##  4 Reflection Fraction in the thin disc model

We have obtained a good fit for the spectra of MCG$-$06 using a thin disc
model with finite thickness. A grid of $\dot{m}$ up to 30% is considered. They
provide similar good fits to the data and achieve consistent BH spin
measurements as before.

A free reflection fraction parameter is used in the models. The reflection
fraction parameter in relxilllpth_nk is defined as the ratio of the coronal
intensity illuminating the disk to the coronal intensity that reaches the
observer555This parameter is referred to as the system reflection fraction in
Ingram et al. (2019). (Dauser et al., 2016). The value of this parameter can
easily reach a very high value significantly larger than 1 due to the light
bending effects in a compact coronal region. The best-fit reflection fraction
parameter shows an anti-correlation with the assumed $\dot{m}$ values (see
Fig. 9 or Table 2): the razor-thin disc model has $f_{\rm refl}\approx 2.1$
and the $\dot{m}=30\%$ model has $f_{\rm refl}\approx 0.45$. This discrepancy
needs further discussion and calculations as following.

We first fit the data of MCG$-$06 to thin disc reflection models where $f_{\rm
refl}$ is calculated consistently according to the height of the lamppost
corona and the thickness of the disc. We then compare the fits of such models
to our previous fits with a free $f_{\rm refl}$ parameter. In the end, we
calculate the theoretical values of $f_{\rm refl}$ in the thin disc model and
discuss our results.

Figure 9: Best-fit reflection fraction parameters when different $\dot{m}$ is
used in the model.

###  4.1 Spectral Models with Consistent Reflection Fraction

To adapt consistent values of $f_{\rm refl}$, we apply the same model to the
data of MCG$-$06 in XSPEC as in Section 3.1. The only difference is that
relxilllpth_nk calculates $f_{\rm refl}$ according to the height of the
lamppost corona and the thickness of the disc in the following analysis. This
is achieved by numerical ray-tracing techniques.

Best-fit values of thin disc models with linked $f_{\rm refl}$ are shown in
Table 3 and corresponding models are shown in Fig. 11. The $\chi^{2}$
distribution against $\dot{m}$ is shown in Fig. 10.

Figure 10: $\chi^{2}$ vs. $\dot{m}$. The circles show the $\chi^{2}$ values of
models with a free $f_{\rm refl}$ parameter. The squares show the values of
models where $f_{\rm refl}$ is calculated consistently for a thin disc
illuminated by the lamppost corona.

After including consistent values of $f_{\rm refl}$, we find that the models
provide worse fits to the data than the ones in Section 3. For example, the
razor-thin disc model with linked $f_{\rm refl}$ provides a worse fit than the
one with a free $f_{\rm refl}$ parameter by $\Delta\chi^{2}=10$. The former
has one fewer free parameter, $f_{\rm refl}$, than the latter. As shown in
Fig. 11, positive residuals above 20 keV are seen. The razor-thin disc model
with linked $f_{\rm refl}$ cannot fit the Compton hump of the disc reflection
component so well as the one with free $f_{\rm refl}$, although the goodness
of the fit is acceptable.

As shown in Fig. 10, $\chi^{2}$ is particularly high at large $\dot{m}$ when a
linked $f_{\rm refl}$ parameter is used. When $\dot{m}=30\%$, the model fails
to fit not only the hard X-ray band but also the iron emission band. A low BH
spin of $a_{*}=0.60$ is suggested by this fit. It is important to note that
this model provides a much worse fit to the data than other models by
$\Delta\chi^{2}\approx 90$. We, therefore, reject this measurement. The
inferred low BH spin in this fit suggests that a large inner disc radius is
needed in the model to produce the right amount of disc reflection in the
model. However, by doing so, the model fails to reproduce the observed Fe
K$\alpha$ line shape as shown in the right bottom panel of Fig. 11.

Figure 11: Same as Fig. 3 but with a linked $f_{\rm refl}$ parameter. Significant residuals are seen in the iron emission band and the hard X-ray band, e.g. >20 keV when $\dot{m}>10\%$. The bottom right panel shows the zoom-in of the best-fit models in the iron emission band. The models are shifted vertically for clarification. See text for more details. Models | Parameters | Units | $\dot{m}=0$ | $\dot{m}=5\%$ | $\dot{m}=10\%$ | $\dot{m}=20\%$ | $\dot{m}=30\%$
---|---|---|---|---|---|---|---
relxilllpth_nk | h | $r_{\rm g}$ | $3.2^{+0.3}_{-0.4}$ | $2.6\pm 0.3$ | $2.0^{+0.3}_{-0.2}$ | $2.3^{+0.3}_{-0.2}$ | $2.0\pm 0.2$
| $i$ | deg | $38.8^{+0.7}_{-1.2}$ | $41.0^{+0.4}_{-1.6}$ | $40.7^{+0.8}_{-0.9}$ | $39.2^{+0.7}_{-0.2}$ | $42^{+3}_{-2}$
| $a_{*}$ | - | $0.92^{+0.05}_{-0.04}$ | $0.95^{+0.04}_{-0.02}$ | $0.983^{+0.008}_{-0.010}$ | $0.92\pm 0.02$ | $0.60^{+0.08}_{-0.02}$
| $\log(\xi)$ | erg cm s-1 | $1.69^{+0.07}_{-0.13}$ | $1.69^{+0.10}_{-0.03}$ | $1.69^{+0.02}_{-0.03}$ | $1.69^{+0.02}_{-0.03}$ | $1.69\pm 0.03$
| $Z_{\rm Fe}$ | $Z_{\odot}$ | $2.5\pm 0.2$ | $2.8\pm 0.2$ | $2.6^{+0.4}_{-0.3}$ | $2.15^{+0.12}_{-0.14}$ | $2.1\pm 0.3$
| $\Gamma$ | - | $2.030^{+0.010}_{-0.008}$ | $2.025^{+0.014}_{-0.011}$ | $2.080^{+0.010}_{-0.019}$ | $2.054^{+0.012}_{-0.002}$ | $2.049\pm 0.002$
| Ecut | keV | $123^{+7}_{-11}$ | $111^{+3}_{-5}$ | $125^{+12}_{-10}$ | $135^{+5}_{-13}$ | $124\pm 16$
| $f_{\rm refl}$ | - | $l$ | $l$ | $l$ | $l$ | $l$
| $\log(F_{\rm disc})$ | erg cm-2 s-1 | $-10.022\pm 0.007$ | $-10.029^{+0.007}_{-0.010}$ | $-10.0243^{+0.007}_{-0.006}$ | $-10.024\pm 0.008$ | $-10.012\pm 0.005$
xillver | $\log(F_{\rm dis})$ | erg cm-2 s-1 | $-12.22\pm 0.04$ | $-12.17\pm 0.04$ | $-12.23^{+0.05}_{-0.04}$ | $-12.31^{+0.05}_{-0.04}$ | <-13.2
| $\chi^{2}/\nu$ | - | 913.17/766 | 923.65/766 | 945.94/766 | 969.05/766 | 989.42/766
Table 3: Best-fit parameters for different $\dot{m}$. $f_{\rm refl}$ is
calculated consistently by given $a_{*}$ and $h$ in the model. $F_{\rm disc}$
and $F_{\rm dis}$ are the 3-50 keV band flux of the disc reflection component
and the distant reflector respectively. See text for more details.

So far, we found that the thin disc model provides significantly worse fits to
the data of MCG$-$06 when considering lamppost values of $f_{\rm refl}$. Among
the fits with linked $f_{\rm refl}$ parameters, the razor-thin disc model
provides the best fit to the data, although its fit is still worse than the
one with a free $f_{\rm refl}$ parameter.

In the following section, we argue that the reason why models with a linked
$f_{\rm refl}$ parameter provide worse fits to the data is that they
overestimate the reflection fraction in MCG$-$06.

###  4.2 Reflection Fraction of a Thin Disc Illuminated by the Lamppost Corona

Figure 12: Top panels: the black solid lines show $f_{\rm refl}$ of a razor-
thin disc ($\dot{m}=0$) illuminated by the lamppost corona. The three panels
are for $a_{*}=0.998,0.95,0.90$ from left to right. The crosses show the
observed $h$ and $f_{\rm refl}$ for $\dot{m}=0,5\%,10\%,20\%,30\%$ from top to
bottom respectively. Bottom panels: difference in $f_{\rm refl}$ for different
$\dot{m}$. For instance, the increase of $f_{\rm refl}$ is expected to be less
than 1% when the thickness of the disc is considered for maximum BH spin.
Significant difference up to a few per cent is only seen when the mass
accretion rate is high, spin has an intermediate value and the corona is very
compact, e.g. $a_{*}=0.90$, $\dot{m}=30\%$ and $h<3r_{\rm g}$. We show the
reflection fraction of a disc with $\dot{m}=30\%$ in purple in the top right
panel in comparison. Note that the best-fit values of $f_{\rm refl}$ are all
lower than the expected reflection fraction by given $h$, which explains why
models with a consistent $f_{\rm refl}$ parameter provide worse fits to the
data of MCG$-$06 than the ones with a free $f_{\rm refl}$ parameter.
Modifications to the simple lamppost corona might be needed when a thin disc
model with finite thickness is considered.

The reflection fraction parameter in the lamppost geometry was calculated in
detail by Dauser et al. (2016) where a razor-thin disc was used. We show three
examples for different BH spins in Fig. 12. The reflection fraction of the
disc increases with decreasing $h$. Because the inner disc is more illuminated
when the corona is closer. The spin of the BH also plays an important role:
the disc has a lower reflection fraction for an intermediate spin than a near
maximum spin. The theoretical values of the reflection fraction parameter in
the lamppost model can easily go over 1 due to the strong light-bending
effects near a BH. Only when $h$ is very large, the flux that reaches the disc
and infinity becomes similar and the reflection fraction approaches 1.

We show the difference of reflection fraction in disc models with infinite
small height and finite thickness in the lower panels of Fig. 12. When the
disc becomes thicker at higher $\dot{m}$, a higher reflection fraction is
expected. However, the overall increase of reflection fraction is less than
10%. When the central BH has a maximum spin, the difference is as small as
less than 1%.

Fig. 12 also shows that the reflection fraction of the disc is less affected
by mass accretion rate when the BH spin is high. Because $\dot{m}$ has a
smaller impact on the thickness of the disc when the BH spin is high (see Fig.
1). Only when 1) the BH has a modest spin; 2) the mass accretion rate of the
disc is very high; 3) the corona is very compact, the difference between the
reflection fraction in two models is larger than 1%. For example, we show the
reflection fraction for $a_{*}=0.90$ and $\dot{m}=30\%$ in purple in the top
right panel of Fig. 12. The difference between the black and purple curves is
up to 10% only when the corona is very close to the event horizon of the BH.

We show the observed values of $h$ and $f_{\rm refl}$ obtained in Section 3 in
Fig. 12. Note that the best-fit BH spin of MCG$-$06 are 0.92–0.97 (see Table
2). The observed values of $f_{\rm refl}$ are all lower than the theoretical
values in the lamppost geometry. Only when $a_{*}=0.90$, the expected
reflection fraction by the razor thick disc model is consistent with the
observed value. The difference between observed and theoretical values of
$f_{\rm refl}$ is larger when the thin disc model with thickness is
considered. The overestimation of reflection fraction in the lamppost model
explains the residuals in the hard X-ray band where Compton hump is (see Fig.
11).

In summary, we apply a thin disc model, which takes the thickness of the disc
into consideration and calculates consistent $f_{\rm refl}$ assuming the
lamppost geometry, to the data of MCG$-$06\. We find that the razor-thin disc
model provides a better fit to the data than other models with finite
thickness. Because the lamppost geometry overestimates the reflection fraction
of the disc when the thickness of the disc is considered. Although this razor-
thin disc model with linked $f_{\rm refl}$ offers a slightly worse fit than
the models with free $f_{\rm refl}$ by $\Delta\chi^{2}=10$, their BH spin
measurements are still consistent.

##  5 Other Systematic Uncertainties in the Disc Reflection Model

In this work, we study the effects of thin disc geometry on disc reflection
spectral modelling. Our analysis is based on the XMM-Newton and NuSTAR
observing campaign of the well-studied X-ray bright AGN MCG$-$06\. We ignore
the soft X-ray data below 3 keV to focus only on the broad Fe K emission and
the Compton hump in the spectra, which are the prominent features of disc
reflected emission. A certain geometry has to be chosen for the corona in our
model as it determines the emissivity profile of the disc as well as the
thickness of the disc (e.g. Wilkins & Fabian, 2012; Gonzalez et al., 2017).
One has to consider a specific coronal geometry to separately study the
systematic effects of disc thickness in the reflection model. We, therefore,
consider the best-understood lamppost geometry for the coronal region in our
model to start with.

By fitting the spectra of MCG$-$06, we find that the thin disc model provides
a similarly good fit to the data and a similar constraint on BH spin as the
razor-thin disc model. However, as argued in Dauser et al. (2016) and Section
4.2, the reflection fraction of the disc is also an indicator of the geometry
of the innermost accretion region. This parameter is expected to increase in
the thin disc model compared to the simplified razor-thin disc model. But the
increase of reflection fraction is usually less 10% unless in extreme cases
(see Fig. 12), while the observed values of this parameter in the thin disc
model are significantly lower than the expectations of the lamppost model.

The discrepancy in the observed and expected values of disc reflection
fraction leads to the discussion in this section: is the lamppost geometry
valid? In particular, is the lamppost geometry applicable to the X-ray data of
MCG$-$06? Are there other systematic uncertainties in the disc reflection
model, e.g. the properties of the disc in MCG$-$06, which we did not take into
account? We also refer interested readers to the latest review by Bambi et al.
(2021) for a similar but more detailed discussion.

###  5.1 Is the Lamppost Geometry Valid in MCG-06-30-15?

Following the same approach in Taylor & Reynolds (2018), we implement the
best-understood lamppost geometry (Martocchia & Matt, 1996) in our thin disc
model for simplicity in Section 3. The best-fit thin disc mode suggests a
compact coronal region of 2–4 $r_{\rm g}$ and a relatively lower Ecut666The
temperature of the thermal Comptonisation corona is usually half or a third of
the value of Ecut. Ecut=100 keV corresponds to a temperature of $kT_{\rm
e}=30-50$ keV. at around 100 keV (see Table 2) in comparison with the sample
in Fabian et al. (2015). If the coronal region is stationary and very compact
in MCG$-$06 as suggested by the lamppost model, one might need to consider the
strong relativistic effects, e.g. gravitational redshifts, on the observed
values of coronal temperatures and luminosities (Niedźwiecki et al., 2016).
The relation between the observed luminosity and the source-frame luminosity
is $L_{\rm obs}=g^{4}\ell L_{\rm s}$ (ignoring the small cosmological
redshift), where $g$ is the redshift factor and $\ell$ is the lensing factor.
For the lamppost model, $g=\sqrt{1-2hr_{\rm g}/(h^{2}+a^{2}_{*}r^{2}_{\rm
g})}$ (Bambi et al., 2021) and $\ell$ can be found, for instance, in Fig. 3 in
Ingram et al. (2019). For $h=4r_{\rm g}$ and an observed X-ray luminosity of
4% of its Eddington limit for MCG$-$06 (see Section 5.2 for detailed
calculations of X-ray luminosities), we have $g\approx 0.73$, $\ell\approx
0.4$, and $L_{\rm s}\approx 0.35L_{\rm Edd}$. The temperature of the corona in
MCG$-$06 is around $kT_{\rm e}/$g$\approx$20–30/$g$=40–70 keV. On the other
hand, for $h=2r_{\rm g}$, $g\approx 0.45$, $\ell\approx 0.2$, $L_{\rm
s}\approx 5L_{\rm Edd}$ and we find a super-Eddington luminosity in the rest-
frame of the source. Coronae very close to a black hole can also be affected
by the runaway $e^{\pm}$ pair production (Fabian et al., 2015; Niedźwiecki et
al., 2016).

However, we note that our value of $h$ is to be taken with caution as inferred
within the ideal lamppost coronal geometry (point-like source, exactly along
the rotational axis of the black holes, with isotropic emission). A realistic
corona is extended and non-isotropic and this motivates the development of
more physically consistent models (see ,e.g., the discussion in Niedźwiecki et
al., 2016; Bambi et al., 2021). A more physical model might be required for
MCG$-$06 especially when the thin disc model is considered. For example, we
find a negative correlation between the inferred disc reflection fraction
parameter and the value of $\dot{m}$ used in the model, which is driven by the
spectral fitting of the broad Fe K emission in the data. The inferred
reflection fraction parameter of the razor-thin disc model is consistent with
the expected value for $a_{*}=0.9$. But all values are lower than the expected
values in the thin disc models, making the thin disc models with the
reflection fraction parameter consistently calculated for the lamppost
geometry fail to fit the data so well as the ones with a free reflection
fraction parameter (see Fig. 10 for comparison of $\chi^{2}$).

Based on the discrepancy between the observed and expected values of
reflection fraction in the thin disc model for MCG$-$06, a more complex
coronal region with a more extended shape, e.g. an outflow/jet-like corona
(Wilkins et al., 2014), may exist in MCG$-$06\. An outflowing corona explains
the overestimation of reflection fraction in the lamppost model.
Unfortunately, complex coronal geometries have not been implemented in the
thin disc model yet due to the complexity of calculations.

The X-ray time-series data analysis of MCG$-$06 supports a similar conclusion
of a potentially more complex coronal geometry in MCG$-$06\. The reduced Root-
Mean-Square (RMS) variability in the iron emission band of MCG$-$06 was
noticed in its ASCA data (Matsumoto et al., 2003). No significant evidence of
correlation among the variability of the blue and red wings of the Fe K
emission line and the primary X-ray continuum was found either (Matsumoto et
al., 2003). Similar conclusions were found through the Principal Component
Analysis of the XMM-Newton observations of this object (Parker et al., 2014a).
The disc reflection component is responsible for less than 1% of the X-ray
variability. Although the light-bending model (Miniutti et al., 2003) explains
the weak variability of the reflection component, the lack of highly variable
reflection in MCG$-$06 leads to the reduced correlated variability in the iron
emission band (Kara et al., 2014). Not all disc reflection is correlated with
the variability of the corona in MCG$-$06 at all Fourier frequencies,
including the frequency range where Fe K reverberation lags are usually found
in other AGN. Thus no obvious evidence of reverberation lag has been
detected777Chainakun et al. (2022) tried to find a correlation between the
amplitudes of Fe K reverberation lags, BH masses, X-ray variability and some
other parameters in a sample of 22 reverberating AGN based on neural networks.
No assumptions for coronal geometries were made in this work. The neural
network output suggested that the fractional excess variance of MCG$-$06 might
be too low for a BH as small as the one in MCG$-$06 to show reverberation
lags. in MCG$-$06 (Kara et al., 2014).

One possible reason for the insignificant evidence of Fe K lag and reduced
covariance variability in the iron emission band might be a moving corona with
changing line-of-sight velocity (Matsumoto et al., 2003). When the corona
moves towards the observer with a speed of a significant fraction of the light
speed, the primary continuum emission is highly beamed and boosted due to
relativistic effects. Meanwhile, the disc is less illuminated than in the
stationary lamppost geometry. A lower reflection fraction and disconnection
between the Fe K emission and the primary continuum are thus expected.
Calculations for complex coronal geometries in combination with the thin disc
model are needed for MCG$-$06\. They may help explain both the observed values
of reflection fraction in its energy spectra and the decrease of correlated
variability in its covariance spectra.

###  5.2 The Geometry of the Inner Accretion Disc of MCG-6-30-15

Figure 13: The critical mass accretion ($\dot{m}/\epsilon$) where gas and
radiation pressure equals as a function radius $r$ in the standard thin disc
model (Shakura & Sunyaev, 1973, in red solid curve). We use $M_{\rm
BH}=1.6\times 10^{6}$ $M_{\odot}$ (Bentz et al., 2016) and $\alpha=0.1$. Note
that this solution is for the flat space time with the inner disc radius at
$6$ $r_{\rm g}$. This function has a lowest point at 11.44 $r_{\rm g}$
corresponding to $\dot{m}/\epsilon=0.083$. When $\dot{m}/\epsilon$ is lower
than this value, there is no radiation pressure-dominated region in the disc.
When $\dot{m}/\epsilon$ is higher than 0.083, the radiation pressure-dominated
region expands both outwards and inwards. Assuming the highest accretion
efficiency of $\epsilon=32\%$ for the maximum BH spin and $\dot{m}=20\%$ for
MCG$-$06, the radiation pressure will dominate in the region of $r=6.1-117$
$r_{\rm g}$ (the shaded region). A higher $\dot{m}$ or a lower $\epsilon$
would lead to an even larger radiation pressure-dominated region in MCG$-$06.

In the model presented in this work, we assume the thickness of the disc to be
scaled with the pressure scale height in the radiation pressure-dominated
region of the standard geometrically thin, optically thick disc (Shakura &
Sunyaev, 1973). In this disc theory, the radiation pressure-dominated region
exists in the innermost accretion region of a disc when the mass accretion
rate is higher than a critical value and is thermally unstable (e.g. Lightman
& Eardley, 1974; Honma et al., 1991). Other approximations have been made in
Shakura & Sunyaev (1973) too: the thin disc model was based on the flat
spacetime but very similar to the relativistic solutions (Novikov & Thorne,
1973); the effects of the energy transformation from the disc to the corona
were not considered either in this model (Haardt & Maraschi, 1991, 1993;
Svensson & Zdziarski, 1994).

The thickness profiles $z/r$ for different BH spins used in this work are
given in Equation 1 and Fig. 1. BH spins change the $z/r$ profile by affecting
$r_{\rm isco}$ and $\epsilon$. Our model also assumes the radiation pressure
dominates the region where disc reflection mostly comes from. We review the
radiation and gas pressure solutions in Shakura & Sunyaev (1973) and argued
that the radiation pressure solutions are a good approximation. To do so, we
calculate the the radius where gas pressure ($P_{\rm gas}$) and radiation
pressure ($P_{\rm rad}$) are equal as a function of mass accretion rate
$\dot{M}c^{2}/L_{\rm Edd}=\dot{m}/\epsilon$. The results are given in Fig. 13.
The calculations are based on Equation 29 in Svensson & Zdziarski (1994). We
assume a viscosity parameter of $\alpha=0.1$, a BH mass of $1.6\times
10^{6}M_{\odot}$ as in MCG$-$06 (Bentz et al., 2016) and zero coronal power
$f=0$ to recover the solutions in Shakura & Sunyaev (1973). The inner radius
of the disc is assumed to be 6 $r_{\rm g}$ in Shakura & Sunyaev (1973).

When $\dot{m}/\epsilon$ is below 0.083, there is no solution where $P_{\rm
gas}=P_{\rm rad}$ in the disc as shown in Fig. 13 and the whole disc is
dominated by $P_{\rm gas}$. At $\dot{m}/\epsilon=0.083$, $P_{\rm
rad}$-dominated region appears at 11.4 $r_{\rm g}$. When $\dot{m}/\epsilon$ is
higher 0.083, the $P_{\rm rad}$-dominated region expands both inwards and
outwards.

We estimate the size of the $P_{\rm rad}$-dominated region in MCG$-$06 by
starting with the estimation of its Eddington ratio. The unabosrbed flux of
MCG$-$06 is $4.6\times 10^{-11}$ ($3.5\times 10^{-11}$) erg cm-2 s-1 in the
2–10 keV (3–10 keV) band, corresponding a luminosity of $8.2\times 10^{44}$
erg s-1 which is 4% of the Eddington luminosity for $1.6\times
10^{6}M_{\odot}$. Assuming a very low bolometric correction factor of 5
(Vasudevan & Fabian, 2007; Netzer, 2019), we estimate the Eddington ratio of
MCG$-$06 to be at least 20%. For sources that are accreting at a significant
fraction of the Eddington limit, a higher bolometric correction factor for the
2–10 keV luminosity might be needed, which would only lead to a higher
Eddington ratio estimation. $\dot{M}/\dot{M_{\rm Edd}}$ is approximately
$L_{\rm Bol}/L_{\rm Edd}$ assuming that the accretion efficiency is the same
at $\dot{M_{\rm Edd}}$ and a significant fraction of $\dot{M_{\rm Edd}}$. To
estimate the lower limit of $\dot{m}/\epsilon$ for MCG$-$06, we consider the
highest value of $\epsilon$ at 32% for the maximum BH spin. The horizontal
dashed line in Fig. 13 shows that the lower limit of the size of the $P_{\rm
rad}$-dominated region in MCG$-$06 (6.1-117 $r_{\rm g}$). In this calculation,
we consider the lowest bolometric correction factor and the highest radiative
efficiency. A higher bolometric correction factor or a lower radiative
efficiency would lead to a larger $P_{\rm rad}$-dominated region size in
MCG$-$06\. In conclusion, we find that the radiation-pressure solutions are a
good approximation of the inner disc region in MCG$-$06 where most reflection
originates in according to the standard thin accretion disc model. A more
accurate model, e.g. for the Kerr spacetime or inclusion of the outer $P_{\rm
gas}$-dominated region, introduces higher order corrections.

Recently, optically thick outflows have also been proposed to be the origin of
the broad Fe K emission in AGN. In AGN with a high Eddington ratio, winds may
form from the innermost accretion region (Chartas et al., 2002; Pounds et al.,
2003; Reeves et al., 2003). Different mechanisms were proposed to explain the
formation of the winds (e.g. King & Pounds, 2003; Mizumoto et al., 2021).
Associated Fe K emissions from these flows due to line scattering or
recombination in the outflow have been discussed to explain the P-Cygni line
profiles in the data(e.g. Done et al., 2007). Such an emission feature may
contribute to part of the Fe K emission observed in AGN suggested by Parker et
al. (2022) where simulations based on a hybrid model of both razor-thin discs
(Dauser et al., 2010) and winds (Sim et al., 2008) are discussed. In Sim et
al. (2008), the cone-shaped optically thick wind model has a finite thickness
too with the inner boundary in a simple linear correlation with $r$ (see Fig.1
in Sim et al. (2008)). However, MCG$-$06 does not show significant evidence of
disc winds as in other AGN (Tombesi et al., 2010; Parker et al., 2022),
including either blueshifted Fe xxv–xxvi (Parker et al., 2016) or blueshifted
narrow emissions in the middle energy band (Jiang et al., 2019b). Detailed
soft X-ray spectral analysis of Chandra grating data of this object also
suggests no significant evidence of fast winds forming from the inner region
of the disc (Lee et al., 2001), at least along the line of sight.

###  5.3 The Properties of the Inner Accretion Disc in MCG-06-30-15

In this section, we discuss other uncertainties concerning the properties of
the accretion disc itself in MCG$-$06 which we have not taken into account.

First, the returning radiation of the inner accretion disc has not been
considered in our model but may be important in MCG$-$06\. X-ray emission from
the innermost region of the accretion disc is subject to the strong
gravitational light bending in the strong gravitational field close to the BH.
A significant fraction of disc emission may be returned to the disc to be
reflected/reprocessed for a second time or multiple times. Such radiation is
often referred to as returning radiation.

Returning radiation was studied by Cunningham (1976) and calculated for the
disc thermal emission in the X-ray band of stellar-mass BH X-ray binaries. The
effects of returning radiation may degenerate with mass accretion rate in the
continuum-fitting method (Li et al., 2005), however, play an important role in
polarisation measurements of disc thermal emission (Schnittman & Krolik,
2009).

Similar effects of returning radiation are expected for the non-thermal
reflected emission of the disc. Wilkins et al. (2020) found that the number
fraction of returned photons is expected to be the highest ($\approx 39\%$) at
$a_{*}=0.998$ and the lowest ($6\%$) at $a_{*}=0$ based on ray-tracing
calculations. Both the thin disc model and the razor-thin disc model suggest a
high BH spin of $a_{*}>0.87$ in MCG$-$06\. So, at least $\approx 20\%$ of the
reflected photons of the disc may return to the disc in MCG$-$06 (Wilkins et
al., 2020).

Moreover, we consider a single-zone model with a constant density of $n_{\rm
e}=10^{15}$ cm-3 for the disc reflection spectrum in the rest frame of the
disc. Recent reflection modelling suggests a higher disc density is required
to explain the X-ray data of many AGN and X-ray binaries (Ross & Fabian, 2007;
García et al., 2016). At a high disc density, the soft X-ray band of the disc
reflection spectrum shows a blackbody-like emission due to stronger free-free
absorption in the reflection slab on the disc. The effects of high disc
density are confined in the soft X-ray band. The previous assumption of
$n_{\rm e}=10^{15}$ cm-3 is appropriate for massive supermassive BHs with a
high Eddington ratio (e.g. Jiang et al., 2019a). On the contrary, stellar-mass
BHs require a very high disc density of $10^{20}-10^{21}$ cm-3 (e.g. Tomsick
et al., 2018; Jiang et al., 2019b; Jiang et al., 2020a). Mixed results have
been found for narrow-line Seyfert 1 galaxies (e.g. Mallick et al., 2018;
Jiang et al., 2020b), which are believed to host a smaller supermassive BH
like the one in MCG$-$06\. Various results found for the discs of low-mass
supermassive BHs might be due to their different accretion rates (Jiang et
al., 2019c).

In the high-density disc reflection model, the soft excess emission commonly
seen in AGN is explained as part of disc reflection. The smooth shape of the
soft excess emission often requires extreme relativistic effects in the
reflection spectrum, e.g. resulting from a high BH spin. Jiang et al. (2019b)
compared the spin measurements obtained by different reflection models for the
broadband spectra of AGN and found that they were all consistent. However,
different spin measurements were found in a few cases where previous work only
applied disc reflection models to the Fe K emission band (e.g. Ton S180,
Walton et al., 2013; Parker et al., 2018; Jiang et al., 2019b) and other
alternative models were used for the soft excess (e.g. Petrucci et al., 2018).
Due to the ambiguous origin of the soft excess emission, we ignore the soft
X-ray band of our data and focus on the prominent features of the disc
reflection spectrum of MCG$-$06, the broad Fe K emission and the Compton hump.
Moreover, the soft X-ray band of MCG$-$06 is known to show a mixture of
complex dust extinction and warm absorption (Lee et al., 2001). Nevertheless,
we are able to reproduce the similar BH spin measurement using only >3 keV
data when considering the razor-thin disc model as before (see Section 3.1).

One final missing piece in the calculation of rest-frame disc reflection
spectra is the radial profile of disc properties, e.g. ionisation. The
ionisation parameter is defined as $F/n_{\rm e}$ where $F$ is the illuminating
flux that reaches the disc. One may consider the distribution of disc
ionisation to be caused by the different illuminating flux in different
regions of the disc (e.g. Svoboda et al., 2012). Such a model considers a
constant density $n_{\rm e}$ across the disc. Spectral simulations suggest
that such an ionisation distribution may increase the inferred coronal size
obtained by the single-zone model (Kammoun et al., 2019). However, both $F$
and $n_{\rm e}$ are expected to change with radius (Shakura & Sunyaev, 1973;
Jiang et al., 2020a). Therefore, efforts have also been made to calculate
radial profiles of disc densities: some consider a simple, phenomenological
power law for the radial profile of disc ionisation (Abdikamalov et al., 2021)
while some consider the same solutions for $P_{\rm rad}$-dominated regions of
the standard thin disc in Shakura & Sunyaev (1973) (Ingram et al., 2019).

The calculations of multi-zone disc reflection models are very difficult
because 1) the exact density profile of the disc is uncertain. The power of
the corona may play an important role in the density profile (Svensson &
Zdziarski, 1994). Observationally, Haardt & Maraschi (1993) found that a
significant fraction of the disc power may be transferred to the corona, which
was also supported by the inferred disc densities of AGN (Jiang et al.,
2020b); 2) detailed ray-tracing is required to calculate the illuminating flux
$F$ in different regions of the disc. To do so, a certain geometry of the
coronal region, which introduces additional systematic uncertainties as
discussed in Section 5.1, has to be considered in the calculation (Ballantyne,
2017); 3) the existence of returning radiation will lead to a steeper radial
profile of disc ionisation due to additional illuminating flux from self
irradiation (Wilkins et al., 2020).

##  6 Concluding remarks

In this work, we apply the thin disc model relxilllpth_nk with finite
thickness to the hard X-ray data of MCG$-$06\. The mass accretion rate of the
disc in MCG$-$06 is uncertain because of Galactic absorption. We, therefore,
consider a grid of mass accretion rates in the model ranging from $\dot{m}=0$
to 30%. The lamppost geometry is assumed for the corona of MCG$-$06\. The
relxilllpth_nk calculates consistent disc emissivity profiles during the fit.

We first consider a model with the reflection fraction as a free parameter. By
doing so, we obtain consistent BH spin measurements as in previous work,
although a slightly higher BH spin and a more compact coronal region are
inferred by a higher-$\dot{m}$ model. After taking the disc thickness into
consideration, we conclude that the BH spin of MCG$-$06 is between 0.87-0.99
(90% confidence range).

For the first time, we also consider a thin disc model with the reflection
fraction consistently calculated for the lamppost geometry. A higher
reflection fraction is expected in a thicker disc than in a razor-thin disc.
But the difference between the two models is usually less than 1%. Only in
some extreme cases, the difference can reach 10% at most.

We argue that the observed values of reflection fraction in MCG$-$06 are lower
than the theoretical values of this parameter in the lamppost geometry. This
explains why significant worse fits are achieved when the lamppost values of
$f_{\rm refl}$ are used instead of being treated as a free parameter. Among
fits with a linked $f_{\rm refl}$ parameter, the razor-thin disc model
provides a better fit to the data than other models with finite thickness.
After considering the lamppost value of $f_{\rm refl}$, the razor-thin disc
model offers a slightly worse fit than when having a free $f_{\rm refl}$
parameter by $\Delta\chi^{2}=10$.

The overestimation of the reflection fraction parameter in the model suggests
that modifications to the over-simplified lamppost geometry are needed for
MCG$-$06 in future especially when the thickness of the disc is considered.
For instance, an outflowing or jet-like corona may exist in MCG$-$06 (e.g.
Wilkins et al., 2014; Gonzalez et al., 2017). When the corona is moving away
from the BH along the spinning axis, the coronal emission is beamed and the
reflection fraction of the disc can be significantly reduced.

## Acknowledgements

This paper was written during the worldwide COVID-19 pandemic in 2020–2022. We
acknowledge the hard work of all the health care workers around the world. We
would not be able to finish this paper without their protection. We
acknowledge constructive discussion with Michael Parker and Andrew Fabian.
J.J. acknowledges support from the Leverhulme Early Career Fellowship, the
Isaac Newton Trust and St Edmund’s College, University of Cambridge. The
research is supported in part by Grants F-FA-2021-432 and MRB-2021-527 of the
Uzbekistan Ministry for Innovative Development.

## Data Availability

All the data can be downloaded from the HEASARC website at
https://heasarc.gsfc.nasa.gov. The thin disc reflection model used in this
work will be available at https://github.com/ABHModels.

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Figure 14: Output distributions for the MCMC analysis of $\dot{m}$=5% (top
left), 10% (top right), 20% (bottom left) and 30% (bottom right) for MCG$-$06.