# Redshifted iron emission and absorption lines in the Chandra X-ray spectrum
of Centaurus A

David Bogensberger Department of Astronomy, The University of Michigan, 1085
South University Avenue, Ann Arbor, MI 48103, USA  Jon Miller Department of
Astronomy, The University of Michigan, 1085 South University Avenue, Ann
Arbor, MI 48103, USA  Elias Kammoun IRAP, Université de Toulouse, CNRS, UPS,
CNES, 9, Avenue du Colonel Roche, BP 44346, F-31028, Toulouse Cedex 4, France
INAF – Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, I-50125
Firenze, Italy  Dipartimento di Matematica e Fisica, Università Roma Tre, via
della Vasca Navale 84, I-00146 Rome, Italy  Richard Mushotzky  Department of
Astronomy, University of Maryland, College Park, MD 20742, USA  Laura
Brenneman  Center for Astrophysics, Observatory Building E, 60 Garden St,
Cambridge, MA 02138, USA  W. N. Brandt  Department of Astronomy and
Astrophysics, 525 Davey Lab, The Pennsylvania State University, University
Park, PA 16802, USA  Institute for Gravitation and the Cosmos, The
Pennsylvania State University, University Park, PA 16802, USA  Department of
Physics, 104 Davey Laboratory, The Pennsylvania State University, University
Park, PA 16802, USA  Edward M. Cackett Department of Physics and Astronomy,
Wayne State University, 666 W. Hancock St, Detroit, MI, 48201, USA  Andrew
Fabian  Institute of Astronomy, University of Cambridge, Madingley Road,
Cambridge, CB3 0HA, United Kingdom  Jelle Kaastra  SRON Netherlands Institute
for Space Research, Niels Bohrweg 4, 2333 CA Leiden, Netherlands  Leiden
Observatory, Leiden University, PO Box 9513, 2300 RA Leiden, The Netherlands
Shashank Dattathri  Department of Astronomy, Yale University, Kline Tower 266
Whitney Avenue, New Haven, CT 06511, USA  Ehud Behar  Faculty of Physics,
Technion - Israel Institute of Technology, Haifa, 3200003, Israel  Abderahmen
Zoghbi  Department of Astronomy, University of Maryland, College Park, MD
20742  HEASARC, Code 6601, NASA/GSFC, Greenbelt, MD 20771  CRESST II, NASA
Goddard Space Flight Center, Greenbelt, MD 20771

###### Abstract

Cen A hosts the closest active galactic nucleus to the Milky Way, which makes
it an ideal target for investigating the dynamical processes in the vicinity
of accreting supermassive black holes. In this paper, we present 14 _Chandra_
HETGS spectra of the nucleus of Cen A that were observed throughout 2022. We
compared them with each other, and contrasted them against the two previous
_Chandra_ HETGS spectra from 2001. This enabled an investigation into the
spectral changes occurring on timescales of months and 21 years. All _Chandra_
spectra could be well fitted by an absorbed power law with a strong and narrow
Fe K$\alpha$ line, a leaked power law feature at low energies, and Si and S
K$\alpha$ lines that could not be associated with the central engine. The flux
of the continuum varied by a factor of $2.74\pm 0.05$ over the course of the
observations, whereas the Fe line only varied by $18.8\pm 8.8\%$. The photon
index increased over 21 years, and the Hydrogen column density varied
significantly within a few months as well. The Fe K$\alpha$ line was found at
a lower energy than expected from the Cen A redshift, amounting to an excess
velocity of $326^{+84}_{-94}~{}\mathrm{km}~{}\mathrm{s}^{-1}$ relative to Cen
A. We investigated warped accretion disks, bulk motion, and outflows as
possible explanations of this shift. The spectra also featured ionized
absorption lines from Fe XXV and Fe XXVI, describing a variable inflow.

Active Galactic Nuclei (16) — X-ray astronomy(1810) — Spectroscopy(1558) —
Black holes(162)

##  1 Introduction

The distance to the galaxy Centaurus A (commonly abbreviated as Cen A, and
also known as NGC 5128) is merely $3.8\pm 0.1~{}\mathrm{Mpc}$ (Harris et al.,
2010). It hosts the nearest active galactic nucleus (AGN) of type Seyfert 2
(Beckmann et al., 2011), which is powered by a supermassive black hole (SMBH)
with a mass of $5.5\pm 3.0\times 10^{7}~{}\mathrm{M}_{\odot}$ (Cappellari et
al., 2009; Koss et al., 2022). Bowyer et al. (1970) detected X-ray emission
from Cen A for the first time, and it has subsequently played a crucial role
in the developing scientific understanding of AGNs.

AGN X-ray spectra are commonly described by an absorbed power law, with a
strong Fe K$\alpha$ emission line, and reflection features such as the Compton
hump. Other features include a soft excess, and additional, but weaker
emission or absorption lines, which can trace the composition, ionization, and
dynamics of the system.

To better understand the processes at work in the central engine of an AGN, we
decided to investigate how X-ray spectral features vary over the course of
months and decades in the AGN of Cen A. AGNs feature large flux variability on
these timescales, which is mostly associated with changes in the accretion
rate, the absorbing column density (Risaliti et al., 2002), or the power law
slope (Connolly et al., 2016). Some AGNs have strongly variable column
densities, which change between Compton-thin and Compton-thick regimes, and
are known as changing-look AGNs (e.g. Ricci et al., 2016). Correlation between
variations in photon index and AGN luminosity have been observed in several
AGNs (Fanali et al., 2013). For bright AGNs, there is a positive correlation
between the two parameters. However, low luminosity AGNs, with
$2-10~{}\mathrm{keV}$ luminosities below $10^{-3}~{}L_{\rm Edd}$ were found to
exhibit the opposite correlation (Gu & Cao, 2009; Yang et al., 2015). The
parameter $L_{\rm Edd}$ refers to the Eddington luminosity.

The variability of X-ray spectral lines in AGNs is, however, less well
established. The Fe K band generally shows a lesser degree of variability than
other components of the spectrum (Markowitz et al., 2003). However, it has
been possible to detect reverberation lags between the continuum and the Fe
K$\alpha$ line (Zoghbi et al., 2012).

The shape of spectral lines can further be used to investigate the structure
and kinematics of the accretion disk. For instance, the shape, and width of
the line are indicative of the inner radius of the emitting region. The
centroid energy of a spectral line can also reflect bulk motion properties.
For example, Doppler shifted emission and absorption lines can indicate the
presence of outflows from the disk (Marziani et al., 2017; Waters et al.,
2021; Mizumoto et al., 2021).

The X-ray spectrum of Cen A has been observed, and studied over five decades
with many different telescopes. Over this interval, the main spectral shape
has not been observed to strongly vary, despite significant luminosity
variability (Rothschild et al., 2011). In this entire interval, the power law
photon index has been found at $\Gamma\approx 1.8$ (Culhane, 1978; Grandi et
al., 2003; Rothschild et al., 2006, 2011; Fürst et al., 2016). Rothschild et
al. (2011) analysed _RXTE_ observations in the interval from 1996 to 2009, and
found a consistent $\Gamma=1.822\pm 0.004$. However, some spectra were also
observed to be shallower than this. For instance, Mushotzky et al. (1978)
found $\Gamma=1.66\pm 0.03$, and Baity et al. (1981) observed it to vary from
$1.68\pm 0.03$ to $1.62\pm 0.03$ over the course of six months.

The power law component extends unbroken to high energies, with a consistent
slope. However, there is disagreement regarding the energy at which a break to
a steeper power law occurs. Baity et al. (1981) detected a break to
$\Gamma\approx 2.0\pm 0.2$ at an energy of $E_{\rm break}=140~{}\mathrm{keV}$,
using broadband observations from _HEAO 1_. In contrast, Kinzer et al. (1995)
found a break between $300-700~{}\mathrm{keV}$ with _CGRO_. A break of
$\approx 180~{}\mathrm{keV}$ was found by Miyazaki et al. (1996) with _Ginga_.
Grandi et al. (2003) fitted _Beppo-SAX_ data, and estimated a folding energy
of $E_{\rm fold}\approx 600~{}\mathrm{keV}$. Rothschild et al. (2006, 2011)
found a lower limit for the break in the power law of $E_{\rm
break}>2~{}\mathrm{MeV}$. A _NuSTAR_ spectrum provided a lower limit of:
$E_{\rm fold}>1~{}\mathrm{MeV}$ (Fürst et al., 2016).

The $\gamma$-ray emission from Cen A has also been detected, and studied with
_Fermi_ and _H.E.S.S_ (Aharonian et al., 2009; Abdo et al., 2010; H. E. S. S.
Collaboration et al., 2018). The spectrum was observed to follow a steepening
power law, with $\Gamma=2.52\pm 0.13_{\rm stat}\pm 0.20_{\rm sys}$. No
significant variability was detected over the course of eight years of
observation.

In contrast, the Hydrogen column density has been observed to vary. Rothschild
et al. (2006, 2011) found a variation between about $10-26\times
10^{22}~{}\mathrm{cm}^{-2}$.

The Fe K$\alpha$ line was prominently observed in all X-ray spectra since it
was first detected by Mushotzky et al. (1978). There is disagreement over
whether the line varies significantly. Rothschild et al. (2006, 2011) found
the flux of the line to have a consistent value of $4.55\pm 0.14\times
10^{-4}~{}\mathrm{photons}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$. In contrast,
Fukazawa et al. (2011) detected a $20-30\%$ variation, and Andonie et al.
(2022) found it to vary by a factor of 10.

Most spectral analyses did not detect any significant reflection features like
the Compton hump. Rothschild et al. (2011) found an upper limit on the
reflection fraction of $R<0.005$. Fürst et al. (2016) described $R<0.01$, and
Markowitz et al. (2007) found that the spectra were best fit with $R=0$. This
is contrasted by Fukazawa et al. (2011), who detected a significant Compton
hump, with a reflection fraction of $R=0.19$.

Evans et al. (2004) analysed two _Chandra_ -HETGS spectra of Cen A that were
observed in 2001. Their results are mostly consistent with the results of
other spectral analyses of Cen A, except that they fitted a comparatively
shallow power law slope of $\Gamma=1.64\pm 0.07$. They detected excess X-ray
emission at $\approx 2~{}\mathrm{keV}$, which they fitted by including a
second power law with a different absorption, and a photon index of $\Gamma=2$
in their spectral model. Si and S K$\alpha$ lines were also detected at
$1.74~{}\mathrm{keV}$ and $2.30~{}\mathrm{keV}$, respectively. They further
discussed that the $20~{}\mathrm{eV}$ width of the Fe K$\alpha$ line indicates
that it originates from a cold, neutral medium far from the SMBH.

Andonie et al. (2022) analysed archival non-grating _Chandra_ ACIS spectra of
Cen A. They also estimated the radius of the region emitting the Fe K$\alpha$
line ($0.10\pm 0.05~{}\mathrm{pc}$), as well as the dust sublimation radius
($0.04\pm 0.02~{}\mathrm{pc}$).

The redshift of Cen A had been independently, and consistently measured by
different groups, using different methods. Graham (1978) first measured a
heliocentric redshift for the entire galaxy of $1.825\pm 0.017\times 10^{-3}$
from optical emission and absorption lines. Wilkinson et al. (1983), and
Skrutskie et al. (2006) later measured systemic velocities corresponding to
redshifts of $z=1.73\times 10^{-3}$, and $z=1.826\pm 0.017\times 10^{-3}$,
respectively. Hui et al. (1995) and Walsh et al. (2015) studied the kinematics
of planetary nebulae in Cen A, and found $z=1.805\pm 0.023\times 10^{-3}$, and
$z=1.798\times 10^{-3}$. Woodley et al. (2007) investigated the globular
clusters in Cen A, and measured a systemic velocity corresponding to a
redshift of $1.821\pm 0.023\times 10^{-3}$. The average of the redshift
measurements with known uncertainties is $1.819\pm 0.010\times 10^{-3}$. We
will henceforth be using this value for the Cen A redshift.

This paper is structured as follows. Section 2 describes the observations that
were analysed in this paper, as well as the data reduction, and methodology
that was used. The spectral analysis utilizing two different models is
described in Section 3. Section 4 discusses the results of this analysis, and
investigates different interpretations of them. Finally, section 5 summarises
and concludes the paper.

##  2 Observations and data analysis

We performed 14 observations of Cen A with the _Chandra X-ray Observatory_
(Chandra; Weisskopf et al., 2000), using the Advanced CCD Imaging Spectrometer
(ACIS; Garmire et al., 2003) optimized for spectroscopy (ACIS-S), and the High
Energy Transmission Grating Spectrometer (HETGS; Canizares et al., 2005), from
January to September 2022. The observations were performed with a reduced sub-
array size, to further reduce pileup of the bright central source.

Figure 1: _Swift_ /BAT light curve of Cen A with overplotted
$2-10~{}\mathrm{keV}$ _Chandra_ fluxes for each of the observations listed in
Table 1. The inset shows the observations obtained throughout 2022. The
_Swift_ /BAT daily light curve was rebinned by a factor of 10, for display
clarity. The colors of the _Chandra_ datapoints indicate the three distinct
groups into which the spectra were merged. The error bars for the _Chandra_
fluxes are smaller than the size of the data points.

These results were compared with the two previous _Chandra_ -HETGS
observations of Cen A that were performed in 2001, which have already been
described by Evans et al. (2004). These observations also used the ACIS-S, but
with the full array size. Table 1 lists the properties of all the 16 _Chandra_
observations analysed in this paper. The set of _Chandra_ datasets used can be
found in the https://doi.org/10.25574/cdc.167 (catalog Chandra Data Collection
(CDC) 167).

We did not find any evidence that the different observing modes affected any
of the results discussed in the following sections. We selected consistent
energy ranges to minimise the impact of the different array sizes on the
spectral analysis, as will be discussed in Section 3.

For all of these observations, we generated type-2, first order _Chandra_
HETGS spectra111https://space.mit.edu/CXC/analysis/AGfCHRS/AGfCHRS.html, using
CIAO version 4.14.0, and HEASOFT version 6.30.1. First, we identified the
position of the zeroth order image, and created region files for the grating
source and background spectrum sky boundaries, by running tg_detect with the
default parameters, followed by tg_create_mask, with an HETG width factor of
18. Next, we assigned grating events to spectral orders using
tg_resolve_events, with a pixel randomization half-width of 0.5. We created
$+1$ and $-1$ grating order type II PHA spectral files for the source and
background using tgextract, with the default parameters. We generated RMF and
ARF files for the spectra with mkgrmf and fullgarf, using the default
parameters. Finally, we combined the positive and negative orders of
individual, or groups of observations using combine_grating_spectra. We kept
the HEG and MEG spectra separate.

Fig. 1 shows the _Swift_ /BAT light curve of Cen A, with overplotted _Chandra_
fluxes at the times of the observations. Cen A was brighter in 2001 than in
2022, but it had been even brighter during the intervening interval.

We used XSPEC (Arnaud, 1996) version 12.12.1 to fit all _Chandra_ spectra. We
assumed solar abundances, as described by Wilms et al. (2000), and the cross-
sections defined by Verner et al. (1996). The spectra were not rebinned prior
to fitting, but were subsequently rebinned for visual clarity in the following
figures. The best fits were found by minimising the C-statistic (Cash, 1979).

ObsID | Date | $\boldsymbol{T}(\mathrm{ks})$ | $\boldsymbol{F_{2-10}}$ | $\boldsymbol{C}(10^{3})$
---|---|---|---|---
1600 | 2001-05-09 | 46.85 | $2.96\pm 0.01$ | $37.724$
1601 | 2001-05-21 | 51.51 | $3.43\pm 0.01$ | $48.047$
24322 | 2022-01-25 | 25.32 | $1.25\pm 0.02$ | $8.287$
23823 | 2022-04-04 | 27.81 | $1.87\pm 0.03$ | $13.462$
24321 | 2022-05-04 | 13.95 | $1.56\pm 0.02$ | $5.879$
26405 | 2022-05-04 | 13.95 | $1.41\pm 0.03$ | $5.332$
24319 | 2022-06-02 | 28.27 | $1.97\pm 0.03$ | $14.576$
24325 | 2022-07-09 | 29.66 | $2.25\pm 0.01$ | $17.477$
24323 | 2022-07-11 | 20.51 | $2.38\pm 0.02$ | $12.509$
26453 | 2022-07-12 | 9.33 | $1.92\pm 0.04$ | $5.021$
24318 | 2022-07-19 | 28.79 | $2.49\pm 0.03$ | $18.053$
24324 | 2022-08-10 | 29.66 | $2.40\pm 0.03$ | $18.097$
24320 | 2022-09-07 | 13.02 | $2.19\pm 0.04$ | $7.181$
24326 | 2022-09-07 | 13.02 | $1.78\pm 0.04$ | $6.378$
27344 | 2022-09-08 | 13.02 | $2.07\pm 0.04$ | $6.919$
27345 | 2022-09-09 | 13.02 | $1.91\pm 0.04$ | $6.757$
Table 1: The list of _Chandra_ observations of Cen A that we used. $T$ denotes
the total exposure time of each observation, and $F_{2-10}$ refers to the
absorbed $2-10~{}\mathrm{keV}$ HEG flux, and is listed in units of
$10^{-10}~{}\mathrm{erg}~{}\mathrm{cm}^{-2}~{}\mathrm{s}$. The parameter $C$
refers to the sum of the total number of source counts of the MEG from
$1.65-3.5~{}\mathrm{keV}$, and the HEG from $3.5-10.0~{}\mathrm{keV}$. The
horizontal lines distinguish between the 2001, early 2022, and late 2022
groups of observations.

##  3 Spectral analysis

Initially, we fitted the spectra of individual observations. As they have a
limited sensitivity, we fitted them with a comparatively simple
phenomenological model that only described the three most important features
of the spectrum; the power law, the Fe K$\alpha$ line, and the Hydrogen
absorption. In XSPEC notation, the model we used is:
constant*ztbabs*(powerlaw+gauss).

The HETGS MEG and HEG spectra were fitted jointly for each observation. The
constant component was used to account for slight normalization differences
between them, with a fixed value of $1.0$ for the HEG spectrum. The
normalizations of the HEG and MEG spectra differed by at most $7\%$ in
individual observations. This agrees well with previous results that found
approximately $8\%$ difference between the spectral normalizations of HEG and
MEG222https://space.mit.edu/ASC/calib/heg_meg/. The ztbabs component describes
the total absorption, featuring contributions from Cen A and the Milky Way.
Since we expect most of the absorption to take place in Cen A, we set the
redshift in ztbabs to the value for the host galaxy. The component gauss is
used to describe the Fe K$\alpha$ line, on top of the powerlaw continuum.
Other emission or absorption lines were not resolved in the spectra of most of
the individual observations.

Table 1 also lists the $2-10~{}\mathrm{keV}$ HEG fluxes measured in each of
the observations. This absorbed continuum flux varied by a factor of $2.74\pm
0.05$, between $1.25\pm 0.02\times
10^{-10}~{}\mathrm{erg}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$ and $3.43\pm
0.01\times 10^{-10}~{}\mathrm{erg}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$. The
corresponding unabsorbed fluxes range from $2.33\pm 0.05\times
10^{-10}~{}\mathrm{erg}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$ to $5.97\pm
0.02\times 10^{-10}~{}\mathrm{erg}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$.
Cappellari et al. (2009) measured the Cen A SMBH to have a mass of $5.5\pm
3.0\times 10^{7}~{}\mathrm{M}_{\odot}$, with the error quoted at the $3\sigma$
level. As we are using $1\sigma$ errors throughout this paper, we will assume
that the $1\sigma$ errors of the mass are $\pm 1.0\times
10^{7}~{}\mathrm{M}_{\odot}$. Using this mass, the measured distance of
$3.8\pm 0.1~{}\mathrm{Mpc}$, and the assumption of an isotropic flux, we can
calculate the source luminosity to range from $4.43\pm 0.16\times
10^{41}~{}\mathrm{erg}~{}\mathrm{s}^{-1}=6.4\pm 1.2\times 10^{-5}~{}L_{\rm
Edd}$, to $1.03\pm 0.03\times 10^{42}~{}\mathrm{erg}~{}\mathrm{s}^{-1}=1.49\pm
0.27\times 10^{-4}~{}L_{\rm Edd}$, respectively. These luminosities do not
include a bolometric correction.

Individual spectra lacked the sensitivity to constrain various spectral
parameters, investigate weaker features, and fit more complex physical models.
We found that the main difference between the spectral fits of individual
observations was the normalization of the power law. This caused most of the
variation in the $2-10~{}\mathrm{keV}$ flux shown in Table 1. Most other
spectral parameters had consistent values for spectra of individual
observations. Furthermore, we did not detect any indication of significant
variation in the spectra obtained within a few weeks or even months of each
other.

Therefore, to better investigate the Cen A spectra, we merged the data from
all observations into one of three groups, corresponding to the two
observations in 2001, the first five observations in 2022, and the subsequent
nine observations in the same year. The observations in 2022 were divided into
two groups to investigate changes occurring on timescales of a few months. The
selection of observations to include in the early and late 2022 grouped
spectra was based on the close proximity in time of the final nine
observations, as well as the lower flux detected in most of the earlier five
observations (see Fig. 1).

To investigate the differences occurring between 2001 and 2022 as best as
possible, we also created a spectrum that combined all observations from 2022.
Finally, to obtain the best constraints on some spectral properties that were
found to be consistent over this 21-year interval, we also analyzed the
spectrum created by grouping all data together. We will subsequently refer to
these five groups of merged spectra as the 2001, early 2022, late 2022, 2022,
and total spectra. The total exposure time of these five groups of
observations is: $98.36~{}\mathrm{ks}$, $109.30~{}\mathrm{ks}$,
$170.03~{}\mathrm{ks}$, $279.33~{}\mathrm{ks}$, and $377.69~{}\mathrm{ks}$,
respectively. The total number of counts from $1.65-3.5~{}\mathrm{keV}$ in the
MEG, and $3.5-10.0~{}\mathrm{keV}$ in the HEG is: $85771$, $47536$, $98392$,
$145928$, and $231699$, respectively. The three non-overlapping groups of
observations are indicated via different colors in Fig. 1. Their merged
spectra are shown in Fig. 2, indicating their qualitatively similar spectral
shapes. This is essential for fitting the total spectrum.

The background spectrum is mostly flat across the energy range for both the
MEG and HEG. It is consistently at least one order of magnitude below the
source spectrum between $2-10~{}\mathrm{keV}$. Even at the lowest energy we
investigated, of $1.65~{}\mathrm{keV}$, the background spectrum is still
fainter than the source spectrum by a factor of at least 3.

Figure 2: $2-10~{}\mathrm{keV}$ spectra of the grouped 2001, early 2022, and
late 2022 spectra. This figure contains both HEG and MEG spectra, which are
depicted with the same color, unlike the following spectra. It also shows the
best fit to the data, using model B. The spectra have been rebinned for visual
clarity.

The part of the spectrum below $2.5~{}\mathrm{keV}$ is brighter than expected
from extrapolating the higher energy spectral shape. This feature was first
described by Turner et al. (1997), and studied in greater detail in the 2001
_Chandra_ observations by Evans et al. (2004). To accurately describe it, we
included a leaked power law component that is only weakly absorbed. This is
still a nuclear emission which is either the result of a leaky absorber or
emission from the innermost part of the jet. Therefore, all of the following
spectral fits include the additional terms constant*tbabs*powerlaw. The
constant was set to a low initial value, in order to only describe the slight
discrepancies observed at low energies, rather than the main spectral shape.
The Hydrogen column density in tbabs was set equal to the weighted average of
the Milky Way absorption in the direction of Cen A, of $N_{\rm H}=2.35\times
10^{20}~{}\mathrm{cm}^{-2}$(HI4PI Collaboration et al., 2016). Finally, the
slope and normalization of the powerlaw component were set equal to those of
the power law describing the main part of the spectrum.

Nevertheless, the MEG spectrum is too faint to accurately describe below about
$1.65~{}\mathrm{keV}$. At lower energies, the source is also comparable to the
background level. Therefore, we only investigated the spectra at greater
energies. The smaller size of the sub-array used for the 2022 observations
limits the HEG energy range to above about
$2.6~{}\mathrm{keV}$333https://cxc.harvard.edu/proposer/POG/html/chap8.html.
Furthermore, the $+1$ and $-1$ arms of all the HEG spectra significantly
deviated at about $3.3~{}\mathrm{keV}$, with the appearance of an apparent
emission line in the -1 arm. This was caused by a large drop in the response
efficiency, possibly due to a chip gap. Above $3.5~{}\mathrm{keV}$, the HEG
spectra have a greater sensitivity than the MEG spectra, but the two are
consistent with each other, when correcting for their slightly different
normalizations. Therefore, we selected an energy range of
$1.65-3.5~{}\mathrm{keV}$ for the co-added MEG spectra, and
$3.5-10.0~{}\mathrm{keV}$ for the co-added HEG spectra.

The low energy part of the spectra also exhibit several peaks away from the
continuum shape. Evans et al. (2004) detected Si and S K$\alpha$ lines, which
have rest frame energies of $1.740~{}\mathrm{keV}$, and
$2.307~{}\mathrm{keV}$, respectively. In the merged spectrum of all
observations, we saw other features that might be interpreted as emission
lines.

In order to determine the statistical significance of these lines, we fitted a
segment of the total spectrum in the energy range around each line with both
ztbabs*powerlaw, and ztbabs*(powerlaw+gauss). We calculated the Bayesian
Information Criterion (BIC; Schwarz, 1978) of the two fits, and only selected
lines as statistically significant if the addition of the gauss component
resulted in a lower BIC. Of all the deviations from the spectrum that we
observed, only the Si and S K$\alpha$ lines satisfied this condition, so we
will subsequently only describe them.

We fitted the five groups of _Chandra_ spectra with two main types of models,
that each include the features described above. Figs. 3, 4, as well as Figs. 8
and 9 in Appendix A show the best fits to the five spectra, using models A and
B.

###  3.1 Model A

Model A is a phenomenological model describing the spectrum as a set of
absorbed power laws, absorption features, and emission lines. In XSPEC
notation, it is written as: constant1*(tbabs*ztbabs*(powerlaw+diskline+
gauss1+gauss2)+constant2*tbabs*powerlaw). The parameter constant1 describes
slight normalization differences between the HEG and MEG spectra. The Milky
Way absorption is parameterized by tbabs, with a fixed column density of
$N_{\rm H}=2.35\times 10^{20}~{}\mathrm{cm}^{-2}$(HI4PI Collaboration et al.,
2016). The Cen A absorption in the line of sight is described by ztbabs, with
a redshift of $1.819\times 10^{-3}$, and a column density that is free to
vary. The diskline model is a physically accurate description of an emission
line profile in an accretion disk, which we used to fit the Fe K$\alpha$ line.
The two gauss components represent the emission lines of Si, and S K$\alpha$.
The parameters of components with identical names were linked. Components with
different numbers describe different features, so were not linked. Table 2
lists the best fitting parameters for model A applied to the five groups of
spectra, as well as their $2-10~{}\mathrm{keV}$ fluxes. The results are also
depicted in Fig. 5.

The merged 2001 spectrum was the brightest, and featured a lower Hydrogen
column density and power law index than both the early and late spectra from
2022. The $2-10~{}\mathrm{keV}$ flux dropped by a factor of $1.89\pm 0.01$
from 2001 to early 2022. Meanwhile, the Hydrogen column density increased by
$1.59^{+0.07}_{-0.08}\times 10^{22}~{}\mathrm{cm}^{-2}$ from 2001 to late
2022. The fits also found a slight difference of $0.46^{+0.09}_{-0.10}\times
10^{22}~{}\mathrm{cm}^{-2}$ between the column density for the early and late
2022 spectra. The photon index of the power law also increased by $0.165\pm
0.003$ from 2001 to late 2022. Even though the increase in the Hydrogen column
density and the power law index also contributed to the reduction in flux,
that was predominantly caused by a decrease in the power law normalization.

There is a degeneracy between these two parameters, so we sought to determine
whether the observed variation in the photon index could instead be attributed
to variations in the Hydrogen column density, and other fit parameters. For
this purpose, we fitted all five spectra again with model A, but with the
photon index set to the fixed value of $\Gamma=1.815$. That is the photon
index found by Fürst et al. (2016) for a _NuSTAR_ spectrum, which also agrees
well with the results of several other X-ray spectral analyses of Cen A
(Culhane, 1978; Grandi et al., 2003; Rothschild et al., 2006, 2011).

Comparing the results of the two spectral fits, we found that the three merged
spectra that were generated from the 2022 observations all had a lower BIC
when the photon index was fixed at $\Gamma=1.815$. The reason for this, is
that fits with a free $\Gamma$ already found it to be close to $1.815$.
However, the best fit photon indices for the late 2022, and entire 2022
spectra with a free $\Gamma$ are inconsistent with a value of $1.815$, within
$1\sigma$ errors. This indicates that the errors in the parameter may be
underestimated.

In contrast, the BIC increased by $35.36$ for the 2001 spectrum when freezing
the photon index at $\Gamma=1.815$. This demonstrates that the best fit value
for it, of $1.646\pm 0.002$ is indeed inconsistent with $\Gamma=1.815$.
Therefore, the photon index did significantly change from 2001 to 2022. The
variation of $N_{\rm H}$ and $\Gamma$ between 2001 and 2022 cause the best fit
values for these two parameters in the total spectrum to be unreliable.

Figure 3: The best fit spectra, and the ratio of the data to the folded model,
for the grouped 2001 spectrum. The first panel shows the spectrum, and the
best fit using model B. The subsequent two panels depict the residuals
normalized by the folded model, for spectral models A and B. The spectra were
rebinned for visual clarity.

The leaked power law component has a variable strength, but nevertheless
remains close to $c_{2}\approx 2.8\times 10^{-3}$ in all spectra. It can vary
on short timescales, as it was found to decrease from $3.06\pm 0.27\times
10^{-3}$ to $2.54^{+0.18}_{-0.16}\times 10^{-3}$ from early to late 2022. This
may, however, be due to a degeneracy between the leaked power law strength and
the Hydrogen column density.

Component |  | Units | 2001 | Early 2022 | Late 2022 | 2022 | Total
---|---|---|---|---|---|---|---
phabs | $\boldsymbol{N_{\rm H}}$ | $10^{22}~{}\mathrm{cm}^{-2}$ | $15.40\pm 0.05$ | $16.53\pm 0.08$ | $16.99^{+0.05}_{-0.06}$ | $16.76\pm 0.04$ | $16.25^{+0.03}_{-0.04}$
powerlaw | $\boldsymbol{\Gamma}$ |  | $1.646\pm 0.002$ | $1.809^{+0.003}_{-0.019}$ | $1.811\pm 0.002$ | $1.803\pm 0.002$ | $1.764\pm 0.001$
| $\boldsymbol{N_{\rm PL}}$ | $\mathrm{photons}~{}\mathrm{keV}^{-1}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$ | $0.1250\pm 0.0004$ | $0.0908^{+0.0031}_{-0.0027}$ | $0.1253\pm 0.0004$ | $0.1102\pm 0.0003$ | $0.1164^{+0.0002}_{-0.0003}$
diskline | $\boldsymbol{E_{\rm l}}$ | $\mathrm{keV}$ | $6.382^{+0.004}_{-0.002}$ | $6.378\pm 0.004$ | $6.379^{+0.004}_{-0.002}$ | $6.378^{+0.003}_{-0.002}$ | $6.381\pm 0.002$
Fe K$\alpha$ | $\boldsymbol{z}$ | $10^{3}$ | $2.75^{+0.33}_{-0.61}$ | $3.37\pm 0.68$ | $3.26^{+0.39}_{-0.63}$ | $3.37^{+0.39}_{-0.44}$ | $2.95^{+0.28}_{-0.31}$
| $\boldsymbol{q}$ |  | $-1.96\pm 0.15$ | $-2.34^{+0.14}_{-0.15}$ | $-2.28^{+0.12}_{-0.18}$ | $-2.40^{+0.14}_{-0.09}$ | $-2.15^{+0.06}_{-0.11}$
| $\boldsymbol{R_{\rm in}}$ | $10^{3}~{}r_{\rm g}$ | $4.6^{+3.5}_{-2.6}$ | $4.2^{+1.5}_{-1.1}$ | $6.0^{+1.8}_{-1.5}$ | $6.1^{+1.3}_{-1.1}$ | $4.8^{+0.9}_{-1.2}$
| $\boldsymbol{N_{\rm L}}$ | $10^{-4}~{}\mathrm{photons}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$ | $2.58\pm 0.23$ | $2.09\pm 0.20$ | $2.10\pm 0.17$ | $2.08^{+0.14}_{-0.12}$ | $2.23^{+0.1}_{-0.13}$
| $\boldsymbol{F_{\rm L}}$ | $10^{-12}~{}\mathrm{erg}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$ | $2.63\pm 0.24$ | $2.13\pm 0.20$ | $2.15\pm 0.17$ | $2.14\pm 0.13$ | $2.26\pm 0.11$
| $\boldsymbol{EW_{\rm L}}$ | $\mathrm{eV}$ | $43.4\pm 3.9$ | $63.5\pm 6.3$ | $48.1\pm 3.8$ | $53.1\pm 3.2$ | $50.1\pm 2.5$
gauss1 | $\boldsymbol{E_{\rm l1}}$ | $\mathrm{keV}$ | $1.7371^{+0.0006}_{-0.0007}$ | $1.736\pm 0.001$ | $1.737\pm 0.001$ | $1.7362^{+0.0009}_{-0.0005}$ | $1.7368^{+0.0004}_{-0.0007}$
Si K$\alpha$ | $\boldsymbol{z_{2}}$ | $10^{-3}$ | $1.52^{+0.41}_{-0.35}$ | $2.15^{+0.45}_{-0.60}$ | $1.82^{+0.59}_{-0.55}$ | $2.08^{+0.30}_{-0.52}$ | $1.72^{+0.37}_{-0.22}$
| $\boldsymbol{\sigma_{1}}$ | $\mathrm{eV}$ | $3.04^{+0.79}_{-0.89}$ | $2.9^{+1.0}_{-0.8}$ | $4.7^{+1.2}_{-1.0}$ | $3.94^{+0.82}_{-0.67}$ | $3.71^{+0.47}_{-0.64}$
| $\boldsymbol{N_{\rm L1}}$ | $10^{-4}~{}\mathrm{photons}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$ | $33.9\pm 4.7$ | $40.4^{+8.5}_{-7.8}$ | $62.0^{+9.7}_{-9.0}$ | $51.0^{+6.6}_{-5.8}$ | $44.7\pm 4.1$
| $\boldsymbol{F_{\rm L1}}$ | $10^{-12}~{}\mathrm{erg}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$ | $9.4\pm 1.3$ | $11.2^{+2.4}_{-2.2}$ | $17.3\pm 2.6$ | $14.3\pm 1.7$ | $12.4\pm 1.1$
gauss2 | $\boldsymbol{E_{\rm l2}}$ | $\mathrm{keV}$ | $2.304^{+0.002}_{-0.001}$ | $2.300\pm 0.005$ | $2.313\pm 0.006$ | $2.313\pm 0.006$ | $2.304\pm 0.002$
S K$\alpha$ | $\boldsymbol{z_{2}}$ | $10^{-3}$ | $1.72^{+0.60}_{-0.68}$ | $3.2^{+2.2}_{-1.9}$ | $-2.3\pm 2.8$ | $0.6^{+1.8}_{-1.6}$ | $1.30\pm 0.78$
| $\boldsymbol{\sigma_{2}}$ | $\mathrm{eV}$ | $2.1^{+2.2}_{-2.1}$ | $12.2^{+4.3}_{-2.9}$ | $14.7^{+6.8}_{-4.0}$ | $11.3^{+3.9}_{-3.0}$ | $6.5^{+2.6}_{-2.0}$
| $\boldsymbol{N_{\rm L2}}$ | $10^{-4}~{}\mathrm{photons}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$ | $3.23^{+0.9}_{-0.77}$ | $4.2^{+1.3}_{-1.2}$ | $4.1^{+1.3}_{-1.2}$ | $3.55^{+0.92}_{-0.74}$ | $3.36^{+0.41}_{-0.78}$
| $\boldsymbol{F_{\rm L2}}$ | $10^{-12}~{}\mathrm{erg}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$ | $1.20\pm 0.31$ | $1.55\pm 0.47$ | $1.54\pm 0.47$ | $1.34\pm 0.30$ | $1.17\pm 0.21$
constant2 | $\boldsymbol{c_{2}}$ | $10^{-3}$ | $2.81^{+0.19}_{-0.17}$ | $3.06\pm 0.27$ | $2.54^{+0.18}_{-0.16}$ | $2.69^{+0.18}_{-0.11}$ | $2.96^{+0.13}_{-0.10}$
| $\boldsymbol{F_{2-10}}$ | $10^{-10}~{}\mathrm{erg}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$ | $3.20\pm 0.01$ | $1.689\pm 0.008$ | $2.286\pm 0.007$ | $2.054\pm 0.005$ | $2.363\pm 0.005$
| $\boldsymbol{C}$ |  | $1781.63$ | $1817.27$ | $1816.20$ | $1895.73$ | $1927.01$
| $\boldsymbol{BIC}$ |  | $1893.32$ | $1928.96$ | $1927.89$ | $2007.42$ | $2038.70$
Table 2: Best fit properties of the fits to the spectra with XSPEC model A,
constant1*(tbabs*ztbabs*(powerlaw+diskline+
gauss1+gauss2)+constant2*tbabs*powerlaw). The parameter $N_{\rm H}$ describes
the Hydrogen column density. The power law component is parameterized by the
index ($\Gamma$) and its normalization ($N_{\rm PL}$). The diskline model
describing the Fe K$\alpha$ line has an energy of $E_{\rm l}$, an emissivity
power law index of $q$, an inner accretion disk radius of $R_{\rm in}$, and a
normalization of $N_{\rm L}$. The flux of the line is denoted as $F_{l}$, and
its equivalent width is $EW_{\rm L}$. The two other emission lines are
described by Gaussian functions with a standard deviation of $\sigma$. The
redshift, $z$, describes the shift of the centroid of the emission line
relative to its rest frame energy. The strength of the leaked component of the
power law is described by the constant, $c_{2}$. $F_{2-10}$ denotes the
absorbed $2-10~{}\mathrm{keV}$ flux of the spectrum. The Cash statistic ($C$),
and its corresponding BIC are listed at the bottom of the table. For all of
these fits, there are 1713 bins, and 1699 degrees of freedom.

As Fig. 2 shows, the Fe K$\alpha$ line is narrow, which means that either the
inclination of the disk is low, or its inner radius is large, or both. The
inclination of the AGN is not known. Although Neumayer et al. (2007) found a
mean inclination of the warped gas disk of $\approx 34\degree$, inconsistent
values have been found for the inclination of the jet. For instance, Dufour et
al. (1979) found an inclination of $72\pm 3\degree$, and Skibo et al. (1994)
measured it to be $61\pm 5\degree$. However, Hardcastle et al. (2003) argued
for an inclination of $\approx 15\degree$, and Müller et al. (2014) found it
to be $12-45\degree$.

A too small inclination is at odds with the identification of Cen A as a
Seyfert 2 galaxy (Beckmann et al., 2011), given the AGN unification model
(Antonucci, 1993). Furthermore, the galaxy has a high inclination, and both
the jet and counterjet from the AGN are visible, even though the counterjet
appears noticeably weaker.

We investigated whether _Chandra_ spectra could distinguish between different
inclinations, as the shape of the emission line described by diskline is
inclination dependent. For this purpose, we jointly fitted the spectra of all
individual observations simultaneously with model A. The inclination in each
of the fits was fixed to a particular value between $10\degree$ and
$90\degree$, in steps of $10\degree$. We allowed the inner radius and
emissivity to vary. In all cases, comparably good spectral fits were found, so
we concluded that these _Chandra_ spectra were insensitive to different
inclinations.

An inclination of $60\degree$ was assumed in previous studies of the Cen A
X-ray spectrum (Fukazawa et al., 2011; Fürst et al., 2016), as it is equally
likely to find a higher or a lower inclination, given a uniform, isotropic
distribution of angles. As we were unable to constrain the inclination, we
will also assume it to be $60\degree$.

This choice of inclination requires a large inner and outer radius of the disk
for an accurate description of the shape of the Fe K$\alpha$ line. The fits
can only constrain the outer radius to within an order of magnitude, so we set
it to a value of $10^{6}~{}r_{\rm g}$, where $r_{\rm g}=GM/c^{2}$ is the
gravitational radius. The best fit values found for the inner radius depend on
the choice of inclination for the fit. At a lower inclination, comparable fits
are found with smaller inner and outer disk radii. The errors quoted for the
inner radii do not incorporate the uncertainty of the inclination of the
system. The redshift of the Fe K$\alpha$ line is unaffected by the inclination
selected for the fits.

We allowed the emissivity index, $q$, to vary freely. The resulting fits were
significantly better than those obtained by freezing it at a value of $q=-3$.

We compared the measured centroid energies of the three fluorescent emission
lines with their rest frame energies. The K$\alpha_{1}$ and K$\alpha_{2}$
lines are so close in energy, that they merge to form a single emission line
in the observed spectra. The laboratory-measured rest frame energies are found
by calculating the weighted average of the K$\alpha_{1}$ and K$\alpha_{2}$
lines, using a 2:1 ratio of intensities. The resulting Fe, Si, and S K$\alpha$
rest frame energies are $6.3996796\pm 0.0000074$, $1.739788\pm 0.000017$, and
$2.307490\pm 0.000026~{}\mathrm{keV}$,
respectively444https://physics.nist.gov/PhysRefData/XrayTrans/Html/search.html.
The redshifts associated with the difference between the measured centroid
energy, and these rest frame energies have also been listed in Table 2.

The Fe K$\alpha$ line is consistently found at a higher redshift that is
inconsistent with that of the galaxy as a whole, as is shown in Fig. 6. The
redshifts found in the 2022 spectra are larger than the one from the 2001
spectrum, but they are nevertheless all still consistent with each other,
within $1\sigma$ errors. The best fit parameters of the Fe K$\alpha$ line
remained mostly consistent between 2001 and 2022. This means that the total
spectrum provides the best estimate of its properties. The 1, 2, and $3\sigma$
errors of the Fe K$\alpha$ line energy for the total spectrum correspond to
redshifts of: $2.95^{+0.28}_{-0.31}\times 10^{-3}$, $\pm 0.60\times 10^{-3}$,
and $\pm 0.90\times 10^{-3}$, respectively. This is also depicted in Fig. 6.
We found that the spectra could not be well fitted by setting the centroid
energy to the value expected from the Cen A redshift. Despite the decrease in
fit parameters, the BIC increased by 6.8 for the total spectrum, which further
demonstrates the inconsistency between the Cen A redshift, and that of the Fe
K$\alpha$ line.

Figure 4: The best fit spectra, and the ratio of the data to the folded model,
for the grouped 2022 spectrum. The layout of the spectra is identical to that
of Fig. 3.

Figure 5: Variation of the best fit parameters of models A and B between the
2001, early 2022, late 2022, 2022, and total spectra. The value of the best
fit of a parameter in the total spectrum is unreliable, if it varied
significantly from 2001 to 2022.

In contrast, the Si K$\alpha$ line in the total spectrum was fitted with a
centroid energy that is redshifted by $1.68^{+0.43}_{-0.19}\times 10^{-3}$
relative to its rest frame energy. All the measured reshifts of this line are
consistent with this value, and also with the Cen A redshift. However, the
redshifts of the Si K$\alpha$ line are inconsistent with those found for Fe
K$\alpha$. The width of the Si K$\alpha$ line remained constant, within
errors, throughout all observations. It increased in amplitude, from a
normalization of $33.9\pm 4.7\times
10^{-4}~{}\mathrm{photons}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$ in 2001, to
$62.0^{+9.7}_{-9.0}\times
10^{-4}~{}\mathrm{photons}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$ in late 2022.

Of the three fluorescent emission lines, the S K$\alpha$ line is the least
well constrained in the spectra. Its centroid energy was fit with a wide range
of different redshifts, some of which were not consistent with each other. The
total spectrum was best fit with a Gaussian function that was redshifted by
$z=1.40^{+0.71}_{-0.92}\times 10^{-3}$. This value is consistent with the Cen
A redshift, the redshift of the Si K$\alpha$ line, but inconsistent with the
redshift of the Fe K$\alpha$ line. There is also significant variation in the
best fit width of the S K$\alpha$ line. However, it is unclear if this is
caused by a variation in the line, or instead reflects the complex shape of
the spectrum at these energies. The amplitude remained consistent within
errors.

We investigated if the Si K$\alpha$ and S K$\alpha$ lines could be self-
consistently described by an AGN fluorescent emission line spectrum.
Regardless of the ionization degree, it was not possible to get an accurate
fit to the spectral shape that included these lines, especially not when also
including the Fe K$\alpha$ line. Furthermore, the Si and S lines are too
bright compared to the expected ratio of their fluxes relative to Fe (Rahin &
Behar, 2020). Rather than the expected $0.104:0.069:1.0$ ratio of the Si to S
to Fe fluxes, we found ratios of $5.38\pm 0.56:0.506\pm 0.095:1.0$ for the
total spectrum. Therefore, we conclude that the Si K$\alpha$ and S K$\alpha$
lines in these spectra are not produced by the central engine around the SMBH.
This is further supported by the Si and S lines having redshifts that are
inconsistent with those measured for Fe K$\alpha$. In contrast, the two lines
are consistent with the Cen A redshift, which indicates an accurate energy
calibration of the _Chandra_ instruments. Therefore, the excess redshift of
the Fe K$\alpha$ line is unlikely to have been caused by an error in the
energy calibration.

This could also explain the variation in the unabsorbed Si K$\alpha$ flux,
which correlated with the Hydrogen column density. It might have been caused
by the wrong assumption that the line is absorbed by the same column density
as the rest of the spectrum. When fitting the Si line without the ztbabs
absorption component, its flux was found to remain consistent across all
spectra. Using a different absorption for the Si and S lines affects their
amplitude, but the measured width remains consistent within errors. The lines
are shifted to slightly higher energies, which slightly reduces their measured
redshifts, but does not affect any of the above discussion of their
properties, as they are still consistent with the Cen A redshift.

The normalization of the Fe K$\alpha$ line was found to vary slightly between
2001 and 2022. This corresponds to a flux decrease of $18.8\pm 8.8\%$. In
contrast, the continuum flux varied significantly more, by $47.2\pm 0.2\%$. As
a result of this, the equivalent width of the line is largest when the
continuum flux is lowest, so in the early 2022 grouped spectrum. The
normalization of the Fe K$\alpha$ line remained stable between early and late
2022, indicating that it varies more slowly, and possibly to a lesser degree
than the continuum flux.

We also fitted spectral model A, but with the diskline component replaced with
gauss, a Gaussian line profile. The results of these fits are very similar to
those shown in Table 2. In particular, the redshifts for the Fe K$\alpha$ line
were identical, within errors, and were also all inconsistent with the Cen A
redshift. In these fits, we found the width of the line, as parameterized
through the standard deviation of the Gaussian to increase from
$\sigma=18.8^{+4.6}_{-2.8}~{}\mathrm{eV}$ in 2001, to
$\sigma=28.9^{+2.9}_{-2.3}~{}\mathrm{eV}$ in 2022. This corresponds to a
velocity dispersion for the half width at half maximum of between
$1.04^{+0.25}_{-0.16}\times 10^{6}~{}\mathrm{m}~{}\mathrm{s}^{-1}$ and
$1.59^{+0.16}_{-0.13}\times 10^{6}~{}\mathrm{m}~{}\mathrm{s}^{-1}$. This is
broader than the energy resolution of the HETG, and can therefore not be
purely attributed to it. The shift, and varying width of the Fe K$\alpha$ line
can be seen in Fig. 7, which zooms into the $6.15-7.15~{}\mathrm{keV}$
interval of the spectra. It also shows the asymmetry of the line, which is
expected.

The variable width of the Fe K$\alpha$ line is described by a variable
emissivity, when fit with diskline. The emissivity power law index decreased
from $-1.96\pm 0.15$ in 2001, to $-2.40^{+0.14}_{-0.09}$ in 2022. The inner
radius of the accretion disk was found to be consistent within errors from
2001 to 2022, possibly due to a degeneracy with the emissivity.

Motivated by the energy shift of the Fe K$\alpha$ line, we also investigated
whether the redshift used by ztbabs differs from the Cen A value. In so doing,
we investigated to what extent the absorption edges were shifted in energy.
Allowing this redshift to vary freely only slightly reduced the C statistic,
but always increased the BIC. Although the best fit redshifts of ztbabs were
slightly larger than the Cen A redshift, they had large errors, which made
them consistent with it. Therefore, we kept the ztbabs redshift frozen at a
value of $1.819\times 10^{-3}$ in the best fit results shown in Tables 2 and
3, as well as Fig. 5.

Figure 6: Redshift of the Fe K$\alpha$ line, as found from the best fits using
model A to the 2001, early 2022, late 2022, 2022, and total spectra. The red
dashed line indicates the redshift of the systemic velocity of Cen A. The
redshift for the total spectrum is shown alongside its $1\sigma$ (black),
$2\sigma$ (dark grey), and $3\sigma$ (light grey) error bars.

###  3.2 Model B

In the interval between the Fe K$\alpha$ line and the Fe edge, we detected
ionized absorption features corresponding to Fe XXV and Fe XXVI (see Fig. 7).
To model these absorption lines self-consistently, we used the “xstar2xspec”
functionality within the XSTAR suite (Kallman & McCray, 1982; Bautista &
Kallman, 2001) to create a table model of photoionized absorption spectra. The
input spectrum was assumed to be typical of an AGN, composed of a
$T=25,000~{}\mathrm{K}$ blackbody and a $\Gamma=1.7$ power law. The power law
component was bent to zero flux below 0.3 keV to avoid infinite flux at low
energy. Solar elemental abundances relative to Hydrogen and a gas turbulent
velocity of $v_{\rm turb}=300~{}{\rm km}~{}{\rm s}^{-1}$ were also assumed.
The grid includes 6400 individual XSTAR realizations, with 100 grid points in
the ionization range $1\leq{\rm log}~{}\xi\leq 6$ and 64 grid points in the
column density range $1.0\times 10^{21}~{}{\rm cm}^{-1}\leq N_{\rm H}\leq
6.0\times 10^{23}~{}{\rm cm}^{-2}$. XSPEC is able to interpolate between these
models to derive the overall best fit model. The relatively high resolution of
the grid ensures that important gas effects tied to ionization are not lost
owing to coarse sampling. In the following, we will refer to this
multiplicative model as xstar_abs.

We also replaced the powerlaw and diskline components of model A with
comparable MYTorus (Murphy & Yaqoob, 2009) components. The zeroth order power
law continuum in model B is described by zpowerlw. The scattered continuum is
parameterized by constants*mytorusS, and the fluorescent Fe K$\alpha$ and Fe
K$\beta$ lines are described by constantl*rdblur*mytorusL. The component
mytorusL includes the Compton shoulder, and therefore allows us to test its
impact on the measured excess redshift of the Fe K$\alpha$ line. Therefore,
model B is described as: constant1*(tbabs*ztbabs*(xstar_abs*zpowerlaw+
constants*mytorusS+constantl*rdblur*mytorusL+
gauss+gauss)+constant2*tbabs*zpowerlaw), in XSPEC notation. The parameters of
the three MYTorus models were linked, and the photon index, redshift, and
normalization were set equal to the values in the zpowerlaw model. As in model
A, the two gauss components represent the Si and S K$\alpha$ lines, and
constant2*tbabs*zpowerlaw describes the leaked power law. Components of this
model with identical names had linked parameters, but different subscripts
differentiate components with independent parameters. We again assumed an
inclination of $60\degree$.

Table 3, and Fig. 5 show the best fit results when fitting the five spectra
with model B. Identical parameters had mostly the same values, within errors,
between models A and B. The best fit parameters of the Si and S K$\alpha$
lines are consistent between the two models, and have not been repeated in
Table 3.

Figure 7: Zoomed in spectra of the $6.15-7.15~{}\mathrm{keV}$ energy range,
featuring the Fe K$\alpha$ and Fe K$\beta$ emission lines, the Fe XXV and Fe
XXVI absorption lines, as well as the Fe edge. The best fits of model B to the
spectra are also depicted as continuous lines through the data. The vertical
blue line describes the expected centroid energy of the Fe K$\alpha$ line,
based on the Cen A systemic velocity. The spectra were rebinned for display
clarity. The 2022 and total spectra were shifted upwards in this figure, to
distinguish them from the other spectra.

In model B, we assumed that the ionized absorber, xstar_abs, is located closer
to the central engine than the origin of the reflection and fluorescence
spectrum, and therefore only acts on the coronal spectrum, zpowerlaw*mytorusZ.
To account for velocity differences and emission and absorption at different
radii, we allowed the redshift of the ionized absorber to vary.

The properties of the ionized absorber varied significantly from 2001 to 2022.
The ionized column density increased alongside the Hydrogen column density,
from $0.50^{+0.30}_{-0.24}\times 10^{22}~{}\mathrm{cm}^{-2}$, to
$1.38^{+0.38}_{-0.32}\times 10^{22}~{}\mathrm{cm}^{-2}$. At the same time, its
redshift increased by a factor of $3.5\pm 1.4$, from $4.2\pm 1.7\times
10^{-3}$, to $15.0^{+0.9}_{-0.7}\times 10^{-3}$. This redshift even exceeds
that of the Fe K$\alpha$ line, in all spectra.

The Fe XXV absorption line could not be constrained in the early 2022 spectrum
(see Fig. 7), so the best fit identified a higher ionization degree for it
than for the other spectra. Therefore, the ionization degree and ionized
column density of the early 2022 spectrum might be overestimated. The
ionization degrees of all other spectra are consistent, within errors. Due to
significant variation in the properties of the ionized absorber from 2001 to
2022, its best fit parameters in the fit to the total spectrum are unreliable.

The redshifts of the zpowerlaw component, which are also used by all MYTorus
components, are lower than the corresponding redshifts found for the Fe
K$\alpha$ line with diskline in model A. This may be the consequence of the
use of this redshift to describe many spectral features, not just the Fe
K$\alpha$ line. Therefore, the redshifts found in model A are still the most
reliable for estimating the shift of the Fe K$\alpha$ line itself. In the fits
with model B, the redshifts for the 2001 and early 2022 spectra are
comparatively low and high, and have strongly asymmetric uncertainties.
Nevertheless, the redshifts found by model B are still consistent with the
values found by model A, and are inconsistent with the Cen A redshifts, for
all five spectra. This further supports the notion that the Fe K$\alpha$ line
is indeed found at a lower energy than expected from the Cen A redshift, even
when including a Compton shoulder. The components of the rdblur model are
consistent, yet less well constrained than the comparable parameters of the
diskline model from spectral model B.

The MYTorus $N_{\rm H}$ describes the column density in the equatorial plane
of the torus. At an inclination of $60\degree$, it does not contribute to the
absorption seen along the line of sight, but instead affects the fluorescent
and reflection features. That is why a mytorusZ component was not included, as
it has no effect at this inclination. The MYTorus Hydrogen column density
appeared to vary significantly from 2001 to 2022. However, the reason for this
is that it could not be well constrained for the 2001 spectra, and was likely
to have been significantly overestimated, as it only has a small impact on the
spectra at an inclination of $60\degree$.

Of particular note is that the strength of the scattering component,
parameterized by the constant_s factor, was found to be 0 for all spectra. We
could only place an upper limit on this reflection fraction, of $c_{s}<0.027$
for the total spectrum. The variation of the constant_l factor is the result
of changes in other parameters, and does not reflect the small variation in
the Fe K$\alpha$ line flux (see Table 2).

The spectral fits using model B have a lower C statistic than the ones using
model A, but have a higher BIC. Model A is the simplest possible model to
describe the main spectral shape. Model B is more complex, and uses more
parameters to describe finer features of the spectra.

Component |  | Units | 2001 | Early 2022 | Late 2022 | 2022 | Total
---|---|---|---|---|---|---|---
ztbabs | $\boldsymbol{N_{\rm H1}}$ | $10^{22}~{}\mathrm{cm}^{-2}$ | $15.30\pm 0.05$ | $16.53^{+0.08}_{-0.07}$ | $16.94\pm 0.07$ | $16.78\pm 0.05$ | $16.05\pm 0.04$
xstar_abs | $\boldsymbol{N_{\rm I}}$ | $10^{22}~{}\mathrm{cm}^{-2}$ | $0.50^{+0.30}_{-0.24}$ | $2.7^{+1.7}_{-1.5}$ | $1.57^{+0.35}_{-0.48}$ | $1.38^{+0.38}_{-0.32}$ | $0.56^{+0.28}_{-0.25}$
| $\boldsymbol{\log(\xi)}$ |  | $4.22^{+0.11}_{-0.21}$ | $4.84^{+0.37}_{-0.19}$ | $4.31^{+0.08}_{-0.07}$ | $4.38^{+0.09}_{-0.07}$ | $4.39^{+0.17}_{-0.10}$
| $\boldsymbol{z_{\mathrm{I}}}$ | $10^{-3}$ | $4.2\pm 1.7$ | $14.6^{+1.5}_{-1.3}$ | $15.6^{+0.5}_{-1.1}$ | $15.0^{+0.9}_{-0.7}$ | $14.4^{+1.4}_{-1.1}$
zpowerlaw | $\boldsymbol{\Gamma}$ |  | $1.636^{+0.003}_{-0.002}$ | $1.811\pm 0.003$ | $1.804\pm 0.003$ | $1.805^{+0.003}_{-0.002}$ | $1.743^{+0.002}_{-0.001}$
| $\boldsymbol{z}$ | $10^{-3}$ | $2.04^{+0.55}_{-0.04}$ | $3.29^{+0.05}_{-0.94}$ | $2.60^{+0.69}_{-0.57}$ | $2.82^{+0.48}_{-0.26}$ | $2.59^{+0.68}_{-0.54}$
| $\boldsymbol{N_{\rm PL}}$ | $\mathrm{photons}~{}\mathrm{keV}^{-1}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$ | $0.1232^{+0.0004}_{-0.0005}$ | $0.0916^{+0.0035}_{-0.0047}$ | $0.1245^{+0.0005}_{-0.0006}$ | $0.1113^{+0.0003}_{-0.0005}$ | $0.1125^{+0.0003}_{-0.0002}$
mytorus | $\boldsymbol{N_{\rm H2}}$ | $10^{22}~{}\mathrm{cm}^{-2}$ | $240^{+180}_{-70}$ | $32^{+28}_{-6}$ | $41^{+26}_{-9}$ | $40.8^{+8.4}_{-6.8}$ | $33.6^{+5.0}_{-3.7}$
constants | $\boldsymbol{c\textsubscript{s}}$ |  | $0.00^{+0.088}_{-0.000}$ | $0.00^{+0.10}_{-0.00}$ | $0.00^{+0.05}_{-0.00}$ | $0.00^{+0.035}_{-0.000}$ | $0.000^{+0.027}_{-0.000}$
constantl | $\boldsymbol{c\textsubscript{l}}$ |  | $0.726^{+0.061}_{-0.068}$ | $1.12\pm 0.11$ | $0.747^{+0.075}_{-0.070}$ | $0.839\pm 0.055$ | $0.817\pm 0.045$
rdblur | $\boldsymbol{q}$ |  | $-1.98^{+0.38}_{-0.15}$ | $-2.25\pm 0.16$ | $-2.13^{+0.18}_{-0.19}$ | $-2.18^{+0.13}_{-0.12}$ | $-2.04^{+0.09}_{-0.10}$
| $\boldsymbol{R_{\rm in}}$ | $10^{3}~{}r_{\rm g}$ | $8^{+13}_{-3}$ | $4.4^{+2.0}_{-1.4}$ | $6.3^{+3.7}_{-2.2}$ | $5.6^{+1.9}_{-1.3}$ | $4.6^{+2.1}_{-1.0}$
constant2 | $\boldsymbol{c_{2}}$ | $10^{-3}$ | $2.77^{+0.20}_{-0.17}$ | $3.05\pm 0.27$ | $2.54\pm 0.17$ | $2.73\pm 0.14$ | $2.93\pm 0.12$
| $\boldsymbol{C}$ |  | $1775.28$ | $1810.77$ | $1786.09$ | $1863.14$ | $1904.46$
| $\boldsymbol{BIC}$ |  | $1924.20$ | $1959.69$ | $1935.01$ | $2012.06$ | $2053.38$
Table 3: Best fit properties of fitting the spectra with XSPEC model B,
constant1*(tbabs*ztbabs*
(xstar_abs*zpowerlaw+constants*mytorusS+constantl*rdblur*mytorusL+gauss+gauss)+constant2*tbabs*zpowerlaw).
The ionized absorber model, xstar_abs, has parameters of $N_{\rm I}$, the
ionized column density, and $\xi$, the ionization degree of the accretion
disk. The rdblur model has an emissivity power law index of $q$, and an inner
accretion disk radius of $R_{\rm in}$. All the MYTorus models use the same
Hydrogen column density, $N_{\rm H2}$, as well as the parameters of the
zpowerlaw component. The remaining parameters are described in Table 2. The
parameters of the two gauss components are equivalent to those listed in Table
2. For all of these fits, there are 1713 bins, and 1694 degrees of freedom.

##  4 Discussion

###  4.1 Neutral absorption, power law

Comparing the different spectral fits, we investigated how variable individual
components were over the course of months and decades. The main spectral shape
remained mostly consistent throughout all observations, and the largest
variation was observed in the amplitude of the power law.

The Hydrogen column density was observed to vary from early to late 2022, and
featured even larger differences when compared over a 21-year interval. This
result is expected, and consistent with previous findings.

The photon index of the power law was found to be consistent at a value of
$\approx 1.81$ throughout 2022. This is also consistent with many previous
spectral analyses that measured a similar photon index (Culhane, 1978; Grandi
et al., 2003; Rothschild et al., 2006, 2011; Fürst et al., 2016). However, the
merged 2001 spectrum was found to be significantly shallower, with
$\Gamma=1.646\pm 0.002$, using model A. This result is in agreement with that
of Evans et al. (2004), and corresponds to a similarly shallow slope as the
one found by Mushotzky et al. (1978) and Baity et al. (1981). The inability to
fit all _Chandra_ spectra from 2001 and 2022 with the same photon index
provides an indication that it is variable, and can become significantly
shallower than it usually is, albeit on timescales of years or decades.

Furthermore, we detected an anticorrelation between the photon index and the
luminosity of Cen A. The 2001 spectrum was both the hardest, with a photon
index of $\Gamma=1.646\pm 0.002$ (for model A), and the brightest, with an
unabsorbed $2-10~{}\mathrm{keV}$ luminosity of $1.40\pm 0.26\times
10^{-4}~{}L_{\rm Edd}$. In contrast, the 2022 spectrum was best fit with
$\Gamma=1.803\pm 0.002$, and an unabsorbed luminosity of $9.7\pm 1.8\times
10^{-5}~{}L_{\rm Edd}$. This anticorrelation agrees with the results of Yang
et al. (2015) and Connolly et al. (2016), who found a decreasing photon index
with an increasing luminosity for AGN accreting with a $2-10~{}\mathrm{keV}$
luminosity between $10^{-6.5}-10^{-3}~{}L_{\rm Edd}$. A possible explanation
for this effect is that the synchrotron power law from the jet becomes
stronger than that from the advection-dominated accretion flow (Yuan & Cui,
2005).

###  4.2 Fe K$\alpha$ line

Several previous studies of the Cen A X-ray spectrum concluded that the disk
generating the fluorescent Fe K$\alpha$ line had to have a large extent to
account for the apparent stability of its flux over intervals of several
years, compared with a significant variability of the continuum (Rothschild et
al., 2006, 2011; Fürst et al., 2016). We found the Fe K$\alpha$ flux to only
vary slightly, by $18.8\pm 8.8\%$ between 2001 and 2022. This is comparable to
the $20-30\%$ variation found by Fukazawa et al. (2011). However, it should be
noted that the flux we found for the line in the total spectrum,
$2.08^{+0.14}_{-0.12}\times
10^{-4}~{}\mathrm{photons}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$, is less than
half as large as the consistent $4.55\pm 0.14\times
10^{-4}~{}\mathrm{photons}~{}\mathrm{cm}^{-2}~{}\mathrm{s}^{-1}$ flux found by
Rothschild et al. (2011) using RXTE data from 1996 to 2009.

The lack of large variation in the Fe K$\alpha$ line flux is contrasted by
Andonie et al. (2022), who observed a variation of about an order of magnitude
between two non-grating _Chandra_ ACIS observations of Cen A. However, we re-
analysed the spectra from those two observations, and measured a consistent Fe
K$\alpha$ line flux.

We also found the width of the Fe K$\alpha$ line to vary from 2001 to 2022,
but remain consistent on short time scales. The increased standard deviation
of the best fit gauss model corresponds to an increased emissivity and a
slightly larger, albeit still consistent inner radius, when fitting with
diskline.

The centroid energy of the Fe K$\alpha$ line was observed to be significantly
offset from its expected energy based on the known recession velocity of Cen
A. In Model A, we measured a redshift of $2.95^{+0.28}_{-0.31}\times 10^{-3}$
for the total spectrum, compared to the Cen A redshift of $1.819\pm
0.010\times 10^{-3}$. This excess redshift was observed by both spectral
models A and B, as well as other models we investigated, involving gauss, and
pexmon components. The excess redshift also does not depend on the assumption
made in the spectral fits, such as the selection of an inclination of
$60\degree$. The Fe K$\alpha$ line energy was found to only vary within the
respective errors from 2001 to 2022.

AGN spectral lines between $6$ and $7~{}\mathrm{keV}$ with significant
redshifts relative to the systemic velocity of the galaxy, have been
previously found by _Chandra_ in $\mathrm{M81}^{\ast}$ (Young et al., 2007;
Shi et al., 2021). However, these concerned the Fe XXVI Ly$\alpha$ emission
line, and also featued a blueshifted line.

The absolute energy calibration of _Chandra_ ’s HETGS has a systematic error
on the scale of $\approx 100~{}\mathrm{km}~{}\mathrm{s}^{-1}$
555https://cxc.harvard.edu/proposer/POG/html/chap8.html. Doppler shifts caused
by velocities of as low as $\approx 50~{}\mathrm{km}~{}\mathrm{s}^{-1}$ have
been detected (Zhang et al., 2012). Furthermore, both the Si and S K$\alpha$
lines, which probably do not originate in the accretion disk, were redshifted
by an amount slightly smaller, but still consistent with the Cen A redshift,
and inconsistent with the Fe K$\alpha$ redshift. This leads us to conclude
that the excess redshift of the Fe K$\alpha$ line is likely not the result of
an offset of the _Chandra_ energy calibration.

The redshift to Cen A is measured relative to a heliocentric reference frame.
The motion of _Chandra_ relative to this reference frame might slightly offset
the measured energies. We assume that the Doppler shift caused by the
_Chandra_ orbit around Earth, averaged over several observations, is small
compared with the shift caused by the orbit of Earth around the Sun. The
magnitude of the component of the orbital velocity of Earth in the direction
of Cen A, as observed in the heliocentric reference frame, is at most
$25.4~{}\mathrm{km}~{}\mathrm{s}^{-1}$. The average orbital velocity component
in the direction of Cen A, weighted by the exposure time, was calculated to be
$25.3$, $15.8$, $4.5$, $7.9$, and $13.2~{}\mathrm{km}~{}\mathrm{s}^{-1}$, for
the 2001, early, and late 2022, 2022, and total observations, respectively.
Subtracting these from the measured redshifts, we find the Fe K$\alpha$ line
to have an average radial velocity relative to the Cen A systemic velocity, of
$250^{+100}_{-180}$, $450\pm 200$, $430^{+120}_{-190}$, $460^{+120}_{-130}$,
and $326^{+84}_{-94}~{}\mathrm{km}~{}\mathrm{s}^{-1}$, respectively. Using the
XSPEC error command, we found the line to be offset from the Cen A redshift by
$3.62\sigma$, equivalent to a p-value of $0.0145\%$, for the total spectrum,
fit with model A. We found a similar significance with model B as well.

These velocity shifts are still up to one order of magnitude smaller than the
component of the orbital velocity in the line of sight at the inner radius.
The width of the line is far greater than the shift of its center away from
the Cen A systemic velocity.

Studies of the varying kinematics in the images of infrared and radio spectral
lines of the circumnuclear disk within several hundred pc of the SMBH revealed
complex structures that have been explained via a warped disk model (Quillen
et al., 2006; Neumayer et al., 2007; Espada et al., 2017; McCoy et al., 2017).
These studies traced red-, and blueshifted regions, but did not find any clear
indication of large bulk motion relative to the Cen A systemic velocity.
However, McCoy et al. (2017) found two absorption complexes, one of which
moved at the systemic velocity of Cen A, the other was redshifted by
$60~{}\mathrm{km}~{}\mathrm{s}^{-1}$.

One way to interpret the excess redshift of the Fe K$\alpha$ line, is if the
warped structure is still present at much smaller radii. For simulations of
warped accretion disks around black holes, see e.g. Ogilvie (1999); Tremaine &
Davis (2014); Liska et al. (2023a, b) In that case, the excess redshift could
be caused by a greater visibility of the redshifted side of the disk. The
blueshifted side would occupy a smaller solid angle, and parts of it may be
blocked by the warp. Abarr & Krawczynski (2021) simulated the Fe K$\alpha$
line profile for a warped disk, and found that it can appear to be shifted to
lower energies.

In this model, we would expect the warp to propagate around the disk, which
would change the size of the excess redshift identified. At a radius of
$5\times 10^{3}~{}r_{\rm g}$, the orbital period is 19 years. Most of the Fe
line originates at greater distances from the SMBH, and the warp is expected
to have a longer precession than orbital period (Inoue, 2012). Therefore, this
model can account for the consistency of the excess redshift over 21 years,
but does require it to vary sinusoidally on longer timescales.

A theoretical study by Pringle (1996) argued that AGN disks would only show
warps at radii of $\geq 10^{6}~{}r_{\rm g}$. However, in a followup study, the
critical radius above which disks can develop warps was set at $r_{\rm
crit}\approx 4\times 10^{3}~{}r_{\rm g}$ (Pringle, 1997). Nevertheless, large
amplitude warps were only found to develop at radii orders of magnitude larger
than the critical radius. The characteristic timescale of variability of the
warp was calculated to be of order $10^{6}~{}\mathrm{yr}$. Therefore, it
remains unclear if significant warps can develop at the required radii to
account for the observed excess redshift.

A different class of explanations for the observed excess redshift can be
found by arguing that it reflects the bulk motion of the central engine
relative to the center of the galaxy. This motion might be in the form of an
oscillation with a period that is many times longer than the span of the
observations. Assuming a constant velocity at the value found from the total
spectrum, the central engine would have travelled $7.0^{+1.8}_{-2.0}\times
10^{-3}~{}\mathrm{pc}$ along the line of sight over the 21-year span of
observations.

This would be an unusually large velocity of the SMBH relative to its host
galaxy. However, even larger velocities have been inferred in a number of
systems with off-center SMBHs (Menezes et al., 2014, 2016; Shen et al., 2019;
Reines et al., 2020; Chu et al., 2023). Velocities like these can be caused
from the merger of two SMBHs. Asymmetries in the spin and mass of merging
SMBHs result in an asymmetric gravitational wave emission, which produces a
recoil of the merged SMBH (Peres, 1962; Gualandris & Merritt, 2008; Blecha et
al., 2011, 2016), with velocities of up to
$4000~{}\mathrm{km}~{}\mathrm{s}^{-1}$ (Campanelli et al., 2007). This recoil
results in a damped oscillation around the galactic center that can last for
more than $1~{}\mathrm{Gyr}$ (Gualandris & Merritt, 2008). The kinematics,
metallicities, and halo features of Cen A suggest that it had a major merger
$\approx 2~{}\mathrm{Gyr}$ ago (Wang et al., 2020). For a large velocity of
the SMBH relative to the galaxy to be maintained for $2~{}\mathrm{Gyr}$ would
require a large initial velocity, and weak dynamical friction.

The main drawback of this interpretation is that the AGN spectral lines
observed at other wavelengths do not feature a comparably large offset from
the Cen A systemic velocity. Additionally, it is unusual for an SMBH to have a
large velocity relative to the galaxy, but still be found at its center.

One way of maintaining a large velocity at small radii, but only minimal bulk
motion at distances of hundreds of parsecs, is if the velocity shift is caused
by the orbit of the SMBH around another massive body. Given the consistency of
the measured excess redshift, this would have to be a large orbit, with an
orbital period much longer than 21 years. Given the size of the velocity
shift, the secondary body would also have to be comparably massive, so it
would have to be another SMBH. A close binary SMBH in Cen A was also suggested
by Cosandey (2022) to explain peculiarities in the _EHT_ image of the galactic
center of Cen A (Janssen et al., 2021). However, Cosandey (2022) argued for
orbital periods in the $10^{-1}-10^{1}~{}\mathrm{yrs}$ order of magnitude
range, which might be too small to justify the consistency of the excess
redshift observed.

A third type of explanation for this shift, is that it reflects an outflow of
material from the disk. Cen A hosts prominent jets that present one avenue for
an outflow, albeit at a higher velocity (Tingay et al., 1998; Snios et al.,
2019). If the excess redshift of the Fe K$\alpha$ line is caused by an
outflow, it would require the line emission to predominantly originate from
the far side of the black hole. The emission from the near side would either
have to be suppressed, or be from non-outflowing material. In this scenario,
we might expect the excess redshift to vary significantly over the course of a
few years. It remains unclear whether such a model could account for the
consistency observed over 21 years. Asymmetric AGN outflows resulting in a
redshifted Fe XXVI emission line have previously been described by Young et
al. (2007). However, Shi et al. (2021) later observed both redshifted and
blueshifted components of the line for the same source. Similarly, the excess
redshift could alternatively represent a inflow of material towards the SMBH
on the closer side. It is similarly unclear if such an inflow could remain
consistent over 21 years.

Another possibility is that the Fe K$\alpha$ line is produced by gas
illuminated by the counterjet, and accelerated by it to the measured recession
velocity. This could account for the excess redshift, the equivalent width of
the line, and the consistency in time. However, it has difficulty explaining
the velocity dispersion observed in the line, which is up to an order of
magnitude larger than the shift. This model would also require that a similar
region does not exist in the direction of the jet, or that the emission from
it is strongly absorbed. This could be possible if it has a high column
density, thereby obscuring the emission.

Further studies of the excess redshift of the Fe K$\alpha$ line, and its
variability over years and decades are required to distinguish between the
different potential explanations that were discussed here. A greater spectral
resolution could also help to identify possibly suppressed blueshifted wings
of the line, and line profiles inconsistent with a planar disk. Investigating
the cause of the excess redshift could help improve our understanding of the
kinematics in accretion disks around AGNs, and expand the ways in which it can
be probed.

The size of the disk emitting the Fe K$\alpha$ line photons can be estimated
from the spectral fits of the line, under the assumption of a particular
inclination. When fixing it at a value of $60\degree$, and setting the outer
radius to be $10^{6}~{}r_{\rm g}$, we find consistent inner radii for all
grouped spectra. The total spectrum was best fit with $R_{\rm
in}=4.8^{+0.9}_{-1.2}\times 10^{3}~{}r_{\rm g}$. Alternatively, when equating
the half width at half maximum of the Fe K$\alpha$ line with the Doppler shift
of a stable orbit, for an inclination of $60\degree$, we find radii of
$4.1^{+1.0}_{-1.6}\times 10^{4}~{}r_{\rm g}$, and $2.66^{+0.43}_{-0.54}\times
10^{4}~{}r_{\rm g}$ for the 2001, and 2022 grouped spectra, respectively.

###  4.3 Fe XXV and Fe XXVI absorption lines

The Fe XXV and Fe XXVI absorption lines at $6.697~{}\mathrm{keV}$ and
$6.966~{}\mathrm{keV}$ changed significantly from 2001 to 2022. Not only did
the ionized absorber column density increase from $0.58\pm 0.28\times
10^{22}~{}\mathrm{cm}^{-2}$ to $1.62\pm 0.37\times
10^{22}~{}\mathrm{cm}^{-2}$, the redshift also increased from $4.2\pm
1.6\times 10^{-3}$ to $15.0^{+0.9}_{-0.7}\times 10^{-3}$. In contrast, the
ionization degree remained consistent over the 21-year interval.

These redshifts are significantly larger than those found for the Fe K$\alpha$
line. This could be indicative of an inflow of the ionized material. Assuming
that the SMBH moves with the systemic velocity of Cen A, rather than with the
velocity indicated by the excess redshift of the Fe K$\alpha$ line, the inflow
velocities equate to: $690\pm 460~{}\mathrm{km}~{}\mathrm{s}^{-1}$ and
$3950^{+260}_{-220}~{}\mathrm{km}~{}\mathrm{s}^{-1}$, for the 2001, and the
2022 spectra, respectively. If this redshift is caused by an inflow, the
properties reflect the variable inflow.

The mass accretion rate can be estimated from the properties of the ionized
absorber. We extrapolated from the Fe ion column density to describe the total
composition of the inflowing material. Furthermore, we assumed an isotropic
accretion, and assumed that the inflow starts at the inner radius of the disk.
However, the mass accretion rate estimated in this way is six orders of
magnitude too large for the measured luminosities. There is further
inconsistency with this association due to the system being brighter at a time
when a lower ionized absorber column density and inflow velocity was measured.
This indicates that this description might be too simplistic. The redshift of
the absorption lines may derive a component from the orbital motion, if the
blueshifted components are blocked from view.

The spectral fits did not require the addition of a broad Fe line at
$6.8~{}\mathrm{keV}$, as suggested by Grandi et al. (2003). There was no
indication of a break in the power law. The best fit reflection strength, not
including any fluorescent features, was 0 in all spectra, which agrees with
the results of Markowitz et al. (2007); Rothschild et al. (2011); Fürst et al.
(2016), but contrasts those of Fukazawa et al. (2011). It is unclear why these
spectra lack reflection features, as the optical depth should be sufficient to
produce these.

###  4.4 XRISM

The Resolve instrument on the X-Ray Imaging and Spectroscopy Mission (_XRISM_
; XRISM Science Team, 2020) will be able to determine the properties of the
Cen A spectrum with far greater sensitivity than is possible with previous
X-ray spectrometers. In particular, it will enable an investigation into the
exact properties of the Compton shoulder of the Fe K$\alpha$ line. We
simulated a $100~{}\mathrm{ks}$ _XRISM_ spectrum based on the best fit to the
total _Chandra_ spectrum, using model B. Due to its higher energy resolution,
and greater effective area, it will be able to constrain the redshift of the
Fe K$\alpha$ line, and the ionized absorber up to one order of magnitude
better than was possible when combining $378~{}\mathrm{ks}$ of _Chandra_ HETGS
exposure. Furthermore, it will be possible to constrain the inclination of the
disk to within $10\degree$, by fitting the exact shape of the Fe K$\alpha$
line at a greater energy resolution. These spectra will also be much more
sensitive to the Hydrogen column density in the torus, and the inner radius of
the disk.

##  5 Conclusions

The AGN at the center of Cen A was observed by the _Chandra_ HETGS twice in
2001, and 14 times in 2022. All spectra were well described by an absorbed
power law with a strong and narrow Fe K$\alpha$ line. We grouped spectra from
different observations together, to analyse weaker spectral features, and
investigate spectral variability on timescales of months and years. Variation
in the flux on short and long timescales was predominantly caused by a
variation in the amplitude of the entire spectrum.

Within these observations, the AGN was brightest in 2001, and also had the
hardest power law slope and the smallest Hydrogen column density. The Hydrogen
column density was found to vary on timescales of months. The power law slope
varied as well from 2001 to 2022, but remained consistent throughout 2022.

There was no indication of any reflection features, or any break in the power
law. To fit the part of the spectra below $2~{}\mathrm{keV}$ required the
addition of a leaked power law component, with an amplitude of about $0.3\%$
of the main power law feature.

The Fe K$\alpha$ line increased in width from 2001 to early 2022, and became
dimmer by $18.8\pm 8.8\%$, but remained consistent throughout 2022. Si and S
K$\alpha$ lines were also detected, but could not be associated with the
accretion disk.

The energy of the Fe K$\alpha$ line was measured to be lower than expected
from the Cen A redshift. This excess redshift was consistently found in
different spectral models of the line profile, and in all spectra. The total
spectrum was best fit with an excess velocity of
$326^{+84}_{-94}~{}\mathrm{km}~{}\mathrm{s}^{-1}$, and is inconsistent with
the Cen A redshift with a significance of $3.62\sigma$.

We interpret this result as possibly indicative of a warped accretion disk on
sub-parsec scales, which enhances the redshifted, but reduces or obscures part
of the blueshifted wing of the line emitting region. We also consider the
possibility of motion of the SMBH relative to the center of the galaxy, as
well as an outflow or inflow from the disk.

The spectra also featured absorption lines of Fe XXV and Fe XXVI. The
properties of these lines varied significantly from 2001 to 2022, with a
higher ionized column density, and a significantly higher redshift in the
latest observations. These result may be interpreted as a variable inflow with
velocities of $690\pm 460~{}\mathrm{km}~{}\mathrm{s}^{-1}$ and
$3950^{+260}_{-220}~{}\mathrm{km}~{}\mathrm{s}^{-1}$ in 2001 and 2022.

##  6 Acknowledgements

This research has made use of data obtained from the _Chandra_ Data Archive
and the _Chandra_ Source Catalog, and software provided by the _Chandra_ X-ray
Center (CXC) in the application packages CIAO and Sherpa. This work made use
of the software packages astropy (https://www.astropy.org/; Astropy
Collaboration et al., 2013, 2018, 2022), numpy (https://www.numpy.org/; Harris
et al., 2020), matplotlib (https://matplotlib.org/; Hunter, 2007), and scipy
(https://scipy.org/; Virtanen et al., 2020). EB is partially supported by a
Center of Excellence of THE ISRAEL SCIENCE FOUNDATION (grant No. 2752/19). EK
acknowledges financial support from the Centre National d’Etudes Spatiales
(CNES). SRON is supported financially by NWO, The Netherlands Organization for
Scientific Research. AZ is supported by NASA under award number 80GSFC21M0002.

We thank the anonymous referee for their insightful comments.

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##  Appendix A Further figures of spectra fitted with models A and B

Figure 8: The best fit spectra, and the ratio of the data to the folded model,
for the grouped early (left), and late 2022 (right) spectra. The layout of the
spectra is identical to that of Fig. 3.

Figure 9: The best fit spectra, and the ratio of the data to the folded model,
for the grouped total spectrum. The layout of the spectra is identical to that
of Fig. 3.