# Centaurus A: Exploring the Nature of the Hard X-ray/Soft Gamma-ray Emission
with INTEGRAL

James Rodi INAF - Istituto di Astrofisica e Planetologia Spaziali; via Fosso
del Cavaliere 100; 00133 Roma, Italy  E. Jourdain Université de Toulouse; UPS-
OMP; IRAP; Toulouse, France  CNRS; IRAP; 9 Av. Colonel Roche, BP 44346,
F-31028 Toulouse cedex 4, France  M. Molina Istituto di Astrofisica Spaziale e
Fisica Cosmica di Milano; Milan, Italy  J. P. Roques Université de Toulouse;
UPS-OMP; IRAP; Toulouse, France  CNRS; IRAP; 9 Av. Colonel Roche, BP 44346,
F-31028 Toulouse cedex 4, France

###### Abstract

The question of the origin of the hard X-ray/soft gamma-ray emission in
Centaurus A (Cen A) persists despite decades of observations. Results from
X-ray instruments suggest a jet origin since the implied electron temperature
($kT_{e}$) would cause pair production runaway in the corona. In contrast,
instruments sensitive to soft gamma-rays report electron temperatures
indicating a corona origin may be possible. In this context, we analyzed
archival INTEGRAL/IBIS-ISGRI and SPI data and observations from a 2022 Cen A
monitoring program. Our analysis did not find any spectral variability. Thus
we combined all observations for long-term average spectra, which were fit
with a NuSTAR observation to study the 3.5 keV $-$ 2.2 MeV spectrum. Spectral
fits using a CompTT model found $kT_{e}\sim 550$ keV, near pair-production
runaway. The spectrum was also well described by a log-parabola to model
synchrotron self-Compton emission from the jet. Additionally, a spectral fit
with the 12-year catalog Fermi/LAT spectrum using a log-parabola can explain
the data up to $\sim 3$ GeV. Above $\sim 3$ GeV, a power-law excess is
present, which has been previously reported in LAT/H. E. S. S. analysis.
However, including a corona spectral component can also describe the data
well. In this scenario, the hard X-rays/soft gamma-rays are due the corona and
the MeV to GeV emission is due to the jet.

gamma rays: General —Galaxies: active — X-rays: individual (Centaurus A)

##  1 Introduction

As one of the brightest radio-loud active galactic nuclei (AGNs) in the hard
X-ray sky, Centaurus A (Cen A, $z=0.001826$) has been observed by numerous
missions since its first detection in 1969 (Bowyer et al., 1970). Results have
shown that the spectral shape does not vary with flux (Jourdain et al., 1993;
Rothschild et al., 2006; Beckmann et al., 2011; Rothschild et al., 2011; Burke
et al., 2014). Consequently, one would expect reported spectral parameters for
Cen A to be similar across time and instruments. Yet that has not been the
case as there has been a lack of consistency in the values, which has made
interpreting the origin of the hard X-ray/soft gamma-ray emission difficult.

The presence of spectral curvature and where it occurs have implications on
the emission process. Often, the hard X-ray continuum in AGNs is interpreted
as thermal inverse Comptonization of UV seed photons from the accretion disk
(Haardt & Maraschi, 1993; Merloni & Fabian, 2003). The spectra typically have
emission up to a few 100 keV (Molina et al., 2013), and the spectral cutoff
energy is related to the temperature of electrons in the corona. In the
corona, photon-photon collisions can create electron-positron pairs when the
sum of the photon energies is above $\sim 1$ MeV (Svensson, 1984; Stern et
al., 1995; Fabian et al., 2015). Thus if the electron temperature is too high,
the production of electron-positron pairs can become a runaway process that
reduces the corona temperature. Additional pairs form and gain energy from the
corona, which lowers the amount of energy per particle (Fabian et al., 2015,
2017).

The maximum temperature is related to the geometry of the source, which is
typically given by the compactness parameter, $\ell$, which is the ratio of
the source luminosity and its size (Guilbert et al., 1983). The limits on
electron temperature range from $\sim 500$ keV in the case of very low
compactness and $\sim 70$ keV in the case of very high compactness (Fabian et
al., 2017). For Cen A, $\ell\sim 10$ (Fabian et al., 2017), suggesting a
maximum corona temperature of a few 100 keV.

Such disagreements over the curvature in Cen A can be seen comparing results
from the past $\sim 30$ years with the main disagreement over the location of
spectral curvature. Multiple analyses have found spectra well fit with a
power-law model with similar spectral indexes. _GRANAT_ /SIGMA observations in
$1990-1991$ found a power-law spectrum with a slope of $1.9\pm 0.3$ ($35-200$
keV) with only a weak indication of a spectral break (Jourdain et al., 1993).
Later observations from _RXTE_ and _INTEGRAL_ in the $3-240$ keV energy range
found a power-law spectrum with a slope of $1.83\pm 0.01$ and no clear
indication of a spectral break (Rothschild et al., 2006). Also, Rothschild et
al. (2011) analysis using approximately 12 years of _RXTE_ data supported
earlier findings with a slope of $1.822\pm 0.004$ and no evidence of
curvature, constraining the cutoff to $>2$ MeV (Rothschild et al., 2011). More
recently, XMM-Newton and NuSTAR observations found an index of 1.81 with no
need for a cutoff $<1$ MeV (Furst et al. 2016). These high-energy spectral
cutoff energies are consistent with the hard X-rays as part of the inverse
Compton hump of the spectral energy distribution (SED), which Chiaberge et al.
(2001) interpreted as synchrotron self-Compton (SSC) emission. However, they
do find the data is consistent with a thermal Comptonization model with an
electron temperature $\sim 100-300$ keV. All of these proposed spectral
turnovers in the above results are well outside the energy ranges of the
considered spectra and thus are not able to constrain the spectral break.

Unfortunately, observations at soft gamma-ray energies have not fared better
at determining where the spectral break is. Joint spectral fits from the OSSE
and COMPTEL instruments (50 keV $-$ 30 MeV) found a spectral break at $\sim
150$ keV and a low energy power-law slope of 1.74 and high energy power-law
slope of 2.0 (Steinle et al., 1998). _INTEGRAL_ observations covering $3-1000$
keV analyzed by Beckmann et al. (2011) can be fit with a cutoff power-law with
a photon index of $\sim 1.7\textrm{ and E}_{c}\sim 400$ keV and excluding
cutoff energies below 300 keV and above 700 keV at $3\sigma$ so significantly
different from the spectral cutoff above $1-2$ MeV predicted from lower energy
instruments. Analysis of _INTEGRAL_ /SPI with _Chandra_ in the $2-500$ keV
energy range also found spectral curvature. Their analysis, which included a
reflection component, found a cutoff energy above 700 keV with a confidence of
$>95\%$ (Burke et al., 2014).

Skibo et al. (1994) proposed an interesting alternative where the emission is
the result of beamed photons Compton scattered into our line-of-sight. Using
OSSE data in the $50-1000$ keV range, they found a photon index of $1.68\pm
0.03$ and an angle of $\sim 61^{\circ}$ between the jet and our line-of-sight
assuming the material scattering the beamed emission is non-relativistic.

In this work, we report on additional INTEGRAL observations of Cen A spanning
nearly 20 years, which allows us to explore the long-term temporal variability
and study the soft gamma-ray spectrum to investigate the possible high-energy
cutoff, which provides key information on the emission process at these
energies.

##  2 Instruments and Observations

###  2.1 INTEGRAL

The INTEGRAL satellite was launched in October 2002 from Baikonur, Kazachstan
with an eccentric orbit (Jensen et al., 2003) that has a period of $\sim
2.5-3$ days. We analyzed data from both SPI (SPectrometer on-board INTEGRAL
(Vedrenne et al., 2003)) and IBIS/ISGRI (Imager on Board INTEGRAL/INTEGRAL
Soft Gamma-Ray Imager (Ubertini et al., 2003)). SPI spans the $20$ keV $-$ 8
MeV energy range with an energy resolution of $2-8$ keV (Roques et al., 2003),
and ISGRI covers energies from $15-1000$ keV.

The SPI data were analyzed using the SPI Data Analysis Interface
(SPIDAI)111Publicly available interface developed at IRAP to analyze SPI data.
Available by contacting<EMAIL_ADDRESS>See description in Burke et al.
(2014) to generate light curves and spectra, which consist of 39 channels over
the 20 keV $-$ 2.2 MeV range. The data above 400 keV were corrected for Pulse
Shape Discriminator efficiency following the method outlined in Roques &
Jourdain (2019). The ISGRI data were analyzed with the Offline Scientific
Analysis 11.2 (OSA
11.2)222https://www.isdc.unige.ch/integral/analysis#Software to produce light
curves and spectra, which span the $30-630$ keV range in 43 channels with a 1%
systematic error included.

The observations used in this work span from revolution 48 to 2590 (MJD 52705
$-$ 59948 or 2003-03-07 to 2023-01-04) when the source was within $10^{\circ}$
from the INTEGRAL pointing direction and incorporates archival data and data
from our current observing campaign during INTEGRAL’s AO-19 observation cycle.
Each observation or science window (scw) is $\sim 1800-3600$ s. Observations
contaminated by solar activity and Earth’s radiation belt have been removed.
Thus the total exposure time is 5.3 Ms for SPI and 4.6 Ms for ISGRI.

###  2.2 NuSTAR

NuSTAR data (from both focal plane detectors, FPMA and FPMB) were reduced
using the nustardas_04May21_v2.1.1 and CALDB version 20220118\. In this work,
we reduced observation 60001081002, taken Aug. 6, 2013, with a cleaned
exposure of $\sim$51.3 ksec; spectral extraction and the subsequent production
of response and ancillary files were performed using the nuproducts task with
an extraction radius of $\sim$136′′; to maximise the signal-to-noise ratio;
background spectra were extracted from circular regions of 70′′ radius in
source-free areas of the detectors.

Figure 1: Cen A SPI (black diamonds) and ISGRI (red triangles) long-term light
curves on an INTEGRAL revolution timescale. The date of the NuSTAR observation
is shown with a vertical green line. The solid black line denotes a flux of 0
mCrab. The dashed black line denotes the mean flux.

##  3 Results

###  3.1 Temporal Variability

To study the temporal behavior of Cen A, we constructed the SPI and ISGRI
light curves on an INTEGRAL revolution timescale ($\sim 2.5-3$ days). Figure 1
shows the temporal variability in the $20-40$, $40-60$, $60-100$, and
$100-200$ keV energy bands. The SPI data are shown as black diamonds, and the
ISGRI data are shown as red triangles. The time of the NuSTAR observation is
denoted by a vertical green line. Both SPI and ISGRI instruments show similar
temporal evolutions with significantly higher fluxes above the mean between
MJD 54500 and 55000, which are present in the data up to the $100-200$ keV
range. The source is significantly detected up to the $200-400$ keV band in
SPI at $11.3\sigma$ and in the $200-450$ keV band in ISGRI at $10.0\sigma$
during the mission.

###  3.2 Spectral Variability

Numerous authors have reported that Cen A’s spectral shape does not change
with flux (Jourdain et al., 1993; Rothschild et al., 2006; Beckmann et al.,
2011; Rothschild et al., 2011; Burke et al., 2014). However, recent results
from Swift/BAT reported spectral softening when the flux increased after 2013
(Rani et al., 2022). Therefore, we searched for spectral variability in the
INTEGRAL data by fitting the SPI $25-70$ keV and ISGRI $34-68$ keV spectra
independently to a power-law model. To reduce uncertainties in the fit
parameters, we grouped observations close in time to generate average spectra.
Table 3.2 lists the revolution groupings. Isolated revolutions were excluded
from the comparison.

Figure 2 shows the photon index vs 50 keV flux for each grouping in SPI (top)
and ISGRI (bottom). The best-fit photon index values from all the groups were
fit to a constant value. The best-fit average values are $\Gamma=1.73\pm 0.04$
for SPI and $\Gamma=1.83\pm 0.02$ for ISGRI, compared to $1.75\pm 0.04$ and
$1.80\pm 0.01$ for the long-term average spectra of SPI and ISGRI,
respectively. The errors are at the $1-\sigma$ level (as are subsequent
errors). The reduced $\chi^{2}/\nu=0.509/13$ for SPI and $1.08/13$ for ISGRI.
Therefore, the photon indexes are consistent with a constant value, with the
best-fit indexes for both instruments presenting a marginal agreement.

Table 1: Spectral Groupings

INTEGRAL
---
Revolutions
$48-49$ | $1309-1311$
$149-163$ | $1370-1386$
$514-523$ | $1437-1439$
$574-588$ | $2313-2315$
$648-651$ | $2457-2470$
$702-705$ | $2517-2539$
$764-768$ | $2581-2590$

We note that the last observing period (rev $2581-2590$) showed relatively
soft spectra with photon indexes of roughly 2.1 (SPI) and 2.0 (ISGRI), though
the errors are large. During this period, the source was at its lowest flux
level observed by INTEGRAL, which does not confirm the behavior reported by
Rani et al. (2022).

Figure 2: Plot of photon index versus 50 keV flux for spectral groups for SPI
(top) and ISGRI (bottom). The best-fit photon index for each instrument is
shown as a red dashed line.

###  3.3 Average Spectrum

####  3.3.1 Phenomenological Models

Since no significant spectral variability was found, the groupings were
combined for long-term average spectra to study the high-energy spectrum of
Cen A. Also, we included NuSTAR observations, which extends to lower energies,
that can test the presence of an Fe line and reflection. As shown in previous
works (Burke et al., 2014), a reflection component in the Cen A spectrum can
result in larger high-energy cutoff energy.

The NuSTAR data were fit first with an absorbed powerlaw model
(phabs*phabs*(zpowerlaw+zgauss)) with absorption components for both the
galactic absorption (fixed to $2.36\times 10^{20}$ cm-2 (HI4PI Collaboration
et al., 2016)) and an intrinsic absorption. The best parameters from this fit
are $\Gamma=1.798\pm 0.002$ with $n_{H}=8.73\pm 0.07\times 10^{22}$ cm-2. The
Fe line energy is $6.36\pm 0.01$ keV with width of $0.08\pm 0.2$ keV with
reduced $\chi^{2}=2581.36/2533=1.02$. The calculated equivalent width from
XSPEC is $\sim 32$ eV. The best-fit continuum parameters are listed in Table
2. In subsequent fits, the Fe line energy and width were fixed to these
values.

To investigate the presence of a spectral cutoff, we next fit the NuSTAR
spectra to a zcutoffpl model. The model better describes the data with a
reduced $\chi^{2}=2469.56/2534=0.97$ with a $\Delta\chi^{2}=111.8$. The
intrinsic absorption was $8.0\pm 0.1\times 10^{22}$ cm-2. The fit parameters
are given in Table 2.

Next we applied the zpowerlw model to the SPI and ISGRI data independently.
The best-fit photon indexes are in perfect agreement between the two
instruments for the powerlaw model. However, neither spectrum was well
described by the model, which over-predicted the fluxes above $\sim 150$ keV.
The resulting $\chi^{2}/\nu$ values were $63.18/34$ for SPI and $64.20/43$ for
ISGRI.

Due to the implied presence of curvature in the spectra, we next fit the data
with a cutoff power-law model (zcutoffpl). The $\chi^{2}/\nu$ values improved
to $31.15/33$ and $38.66/42$ for SPI and ISGRI, respectively. The best-fit
parameters in Table 2 are in agreement between the two instruments. Thus we
performed a joint fit with the two instruments. A cross-instrument
normalization was added to the model with the SPI normalization fixed. The
model was still an acceptable fit to the data with a $\chi^{2}/\nu$ of
$77.81/79$. The cross-instrument normalization was $0.873\pm 0.008$.

Subsequently, we performed a joint fit between the NuSTAR, SPI, and ISGRI data
covering $3.5-2200$ keV. A power-law fit to the data is statistically
acceptable with a reduced $\chi^{2}=2795.87/2610=1.07$, but it is
unsurprisingly a worse fit due to the high-energy cutoffs seen in the three
instruments. As expected, including a high-energy cutoff improves the quality
of the fit (reduced $\chi^{2}=2625.40/2609=1.01$), but with a relatively high
cutoff energy of $\sim 650$ keV. The cross-normalization values for the NuSTAR
spectra were $1.81\pm 0.02$ for FPMA and $1.80\pm 0.02$ for FPMB. Subsequent
spectral fits result in similar cross-normalization values. These values are
large compared to the long-term average SPI spectrum because the NuSTAR
observation occurs while Cen A is in a high-flux state.

To investigate the presence of a reflection component, which can result in
artificially high cutoff energies (Burke et al., 2014), we used the pexrav
continuum model. The best-fit reflection fraction was $2\times 10^{-7}\pm
1\times 10^{-2}$. We also tested the presence of reflection with relativistic
smearing using a xion*(pexrav+zgauss) model. We assumed a geometry with a
central hot sphere and a cold outer disk. The model describes the data well
(red. $\chi^{2}=2503.97/2527=0.991$), but most of the fit parameters are
unconstrained, including the reflection fraction ($0.0\pm 0.2$). Thus we
consider that no significant reflection is required by the data. Consequently,
the reflection fraction was fixed to 0. The spectrum was refit, and the best-
fit parameters (Table 2 and subsequent fit parameters;
$\chi^{2}/\nu=2594.66/2608=0.99$) are the same with those of the zcutoffpl
model.

Table 2: Average Spectrum Parameters

| ZPowerlw | Cutoff ZPowerlaw |
---|---|---|---
Inst. | $n_{H}$ | $\Gamma$ | $\chi^{2}/\nu$ | $n_{H}$ | $\Gamma$ | $E_{cut}$ | $\chi^{2}/\nu$ |
| ($\times 10^{22}$ cm-2) |  |  | ($\times 10^{22}$ cm-2) |  | (keV) |  |
NuSTARa | $8.73\pm 0.07$ | $1.798\pm 0.002$ | 1.02 | $8.0\pm 0.1$ | $1.725\pm 0.007$ | $246\pm 24$ | 0.97 |
SPI | $-$ | $1.86\pm 0.02$ | $1.86$ | $-$ | $1.62\pm 0.05$ | $381\pm 85$ | 0.94 |
ISGRI | $-$ | $1.851\pm 0.007$ | $1.49$ | $-$ | $1.72\pm 0.03$ | $517\pm 112$ | 0.92 |
SPI/ISGRI | $-$ | $1.859\pm 0.007$ | $1.51$ | $-$ | $1.72\pm 0.02$ | $543\pm 94$ | 0.98 |
NuSTAR/SPI/ISGRI111zgauss model included in fit with fixed parameters to model Fe line. (See Sec. 3.3.1.) | $8.94\pm 0.07$ | $1.807\pm 0.002$ | 1.071 | $8.37\pm 0.08$ | $1.767\pm 0.004$ | $638\pm 54$ | 1.00 |

####  3.3.2 Physically Motivated Models

To better understand the emission mechanism(s) in this energy range, we fit
the data using more physically motivated models. We fit the data to a CompTT
model. As pointed out in Fürst et al. (2016), the seed photon temperature is
unknown thus they fit their data to a range of seed photon temperatures with
extremes of $0.05$ keV and $0.5$ keV. Therefore, we fit our data to minimum
and maximum values to explore the range of electron temperatures. Both fits
find comparable values of $kT_{e}\sim 520$ keV and $\tau\sim 0.03$. The data
prefer the lower $kT_{0}$ (0.05 keV) with $\Delta\chi^{2}=21.64$, though both
models acceptably describe the data. We also tried a fit with $kT_{0}$ free,
finding a best-fit seed photon temperature of $0.4\pm 0.1$ keV and a reduced
$\chi^{2}=2588.54/2607=0.99$, $\Delta\chi^{2}=5.34$ for one additional free
parameter compared to the fit with $kT_{0}=0.05$ keV. The best-fit parameters
for each fit are listed in Table 3. We note that implied electron temperatures
above 500 keV disfavor a thermal origin (Fabian et al., 2017).

As discussed above, the hard X-ray/soft gamma-ray emission is potentially due
to SSC jet emission. Thus, we fit the spectra to a log parabola model, which
can model the inverse Comptonization found in Fermi blazars (See Esposito et
al. (2015) and references within). The $\alpha$ value in the log-parabola is
the slope observed at the pivot energy ($F(E)=N(E/E_{pivot})^{-\alpha-\beta
log(E/E_{pivot})}$). Thus, we fixed the $E_{pivot}$ to 50 keV to compare this
slope with previously reported power-law results. This model also provides a
good fit to the data (reduced $\chi^{2}=2553.05/2608=0.98$) with
$\Delta\chi^{2}=41.45$ for the same number of free parameters compared to the
pexrav model and $\Delta\chi^{2}=35.49$ for 1 fewer free parameter compared to
the CompTT model with $kT_{0}$ free. Figure 3 shows the NuSTAR spectrum in
black diamonds (FPMA) and red triangles (FPMB), SPI spectrum in green squares,
and the ISGRI spectrum in blue stars with the CompTT with $kT_{e}=0.5$ keV
(cyan dot-dashed line) and log parabola (black dashed line) models. Therefore,
we find that statistically both thermal Comptonization and SSC emission can
acceptably describe the joint NuSTAR/INTEGRAL spectrum. Below, we discuss the
implications of the parameters to potentially differentiate the two origins.

Table 3: Joint NuSTAR/SPI/ISGRI Fit Spectral Parameters

CompTT | Log Parabola
---|---
$N_{H}$ | $kT_{0}$ | $kT_{e}$ | $\tau$ | $\chi^{2}/\nu$ | $N_{H}$ | $\Gamma$ | $b$ | $\chi^{2}/\nu$
($\times 10^{22}$ cm-2) | (keV) | (keV) |  |  |  |  |  |
$8.38\pm 0.08$ | 0.05 (fixed) | $535\pm 53$ | $0.029\pm 0.006$ | 1.01 | $7.8\pm 0.1$ | $1.830\pm 0.003$ | $0.069\pm 0.005$ | $0.98$
$8.66\pm 0.08$ | 0.5 (fixed) | $525\pm 53$ | $0.029\pm 0.006$ | 0.99 |  |  |  |
$8.5\pm 0.2$ | $0.4\pm 0.1$ | $556\pm 62$ | $0.026\pm 0.006$ | 0.99 |  |  |  |

Figure 3: Cen A long-term average spectra from INTEGRAL SPI (green square) and
ISGRI (blue stars) with NuSTAR/FPMA (black diamonds) and FPMB (red triangles).
The best-fit CompTT with $kT_{e}=0.5$ keV model (cyan dot-dash line) and log
parabola model (black dashed line) are overplotted.

##  4 Discussion

###  4.1 Implications of Model Parameters

####  4.1.1 Iron Line and Reflection

A Fe K$\alpha$ line is suggestive of the presence of photons from an X-ray
source reflecting off of a surrounding medium (George & Fabian, 1991). The
width of the line profile provides an indication of the distance between the
source and the reflecting medium (Evans et al., 2004). The width of $\leq 8$
eV found in this work is similar to Fürst et al. (2016) indicating that the
reflecting medium is far from the central black hole.

Additionally, the column density and the equivalent width of the line are
related to the geometry and elemental abundances of the material. The
calculated equivalent width from XSPEC is $\sim 32$ eV. This value combined
with the fit column densities ($\sim 7-9\times 10^{22}$ cm-2) is compatible
with a face-on torus interpretation with no significant Compton hump expected
(Matt et al., 2003), which is consistent with the reflection value from our
pexrav fit. Such an interpretation is in agreement with MYtorus and BNtorus
fit results in Fürst et al. (2016) and infrared observations from Ramos
Almeida et al. (2009). Absence of reflection has also been reported in other
radio-loud AGN (Sambruna et al., 1999; Eracleous et al., 2000; Reeves &
Turner, 2000; Grandi et al., 2001; Ballantyne et al., 2002; Ogle et al., 2005;
Ballantyne, 2007; Sambruna et al., 2009; Evans et al., 2010; Walton et al.,
2013; Ballantyne et al., 2014; Madsen et al., 2015). There are other potential
explanations for the lack of Compton hump: a truncated inner disk from a
retrograde spin (Garofalo, 2009), an ionized accretion disk (Ballantyne et
al., 2002), an outflowing corona (Beloborodov, 1999; Malzac et al., 2001),
relativistic jet dilution (Reeves & Turner, 2000), changing geometry of the
inner disk (Eracleous et al., 2000), or central accretion flow obscuration by
the jet (Sambruna et al., 2009). All of these models are consistent compared
to result in Cen A.

####  4.1.2 Continuum

When including data up to the soft gamma-rays, we find spectral curvature is
present in Cen A. Even the cutoff power-law fit to the NuSTAR spectrum is
preferred over the power-law fit. This contrasts with spectral fits based on
hard X-ray instruments that reported no spectral cutoff until the MeV region
(Rothschild et al., 2011; Fürst et al., 2016). The cutoff energy value varies
across the instruments ($\sim 250-520$ keV) in this work. This behavior is
similar to what has been reported in other works using soft gamma-ray data,
since the parameter is sensitive the energy domain of the instruments.

Our joint NuSTAR/SPI/ISGRI fit has a best-fit cutoff energy of roughly 650 keV
or an electron temperature of $\sim 520-550$ keV. This implies a corona near
the runaway pair-production zone of the compactness-temperature diagram for a
normal corona size of $3-10$ $r_{g}$ (Fabian et al., 2015, 2017). These
results together with the lack of significant reflection suggest that the
continuum in the hard X-ray is due to SSC emission. However, we are not able
to exclude coronal emission as the dominant emission process in the hard
X-ray/soft gamma-ray band. For example, 3C 273 showed similar Fe line and
reflection characteristics and analysis of the hard X-ray/soft gamma-ray
spectrum continuum was found to be a combination of coronal plus jet emission
(Grandi & Palumbo, 2004; Madsen et al., 2015).

###  4.2 Joint Fit With Fermi/LAT

Following Madsen et al. (2015), we included Fermi/LAT data using the 12-year
Data Release 3 catalog fluxes in 8 channels over the 50 MeV $-$ 1 TeV energy
range (Abdollahi et al., 2022). Joint LAT/HESS analysis found a spectral break
at $\sim 3$ GeV (H. E. S. S. Collaboration et al., 2018). Thus, we included a
power-law model to fit the data above 3 GeV for a continuum model of zpowerlw
in fits to the LAT data.

####  4.2.1 Jet Only Scenario

Initially, we tested a model where the keV to GeV emission is due to the jet
(log-parabola + zpowerlw) with the LAT cross-normalization parameter
($C_{LAT}$) free and fixed to 1. The parameters for this fit are $n_{H}=8.1\pm
0.1\times 10^{22}$ cm-2, $\alpha=1.83\pm 0.009$ ($E_{pivot}=50$ keV),
$\beta=0.115\pm 0.008$, and $\Gamma_{HE}=2.17\pm 0.02$ with reduced
$\chi^{2}=2490.84/2613=0.99$ and $C_{LAT}=0.38\pm 0.09$. When the cross-
normalization is fixed to 1, the parameters are $N_{H}=8.3\pm 0.1\times
10^{22}$ cm-2, $\alpha=1.826\pm 0.007$ ($E_{pivot}=50$ keV), $\beta=0.156\pm
0.003$, and $\Gamma_{HE}=2.250\pm 0.005$ with reduced
$\chi^{2}=2611.81/2614=1.00$. We can describe the data with both $C_{LAT}$
values, but the free value implies a LAT flux significantly higher than the
flux ($\times\sim 2.6$) reported by the LAT team (Abdollahi et al., 2022).
Thus we fixed $C_{LAT}$ to 1 in subsequent fits.

####  4.2.2 Jet and Corona Scenario

Also, we tested a model where the coronal emission dominates the hard
X-ray/soft gamma-ray emission and the jet component dominates in the MeV to
GeV range, similar to the scenario presented for 3C 273 in Grandi & Palumbo
(2004) and Madsen et al. (2015). We used a (CompTT + log-parabola + zpowerlw)
continuum model. However, some parameters are poorly constrained due to the
lack of data in the MeV range and the GeV power law component.

We found $n_{H}=8.40\pm 0.09\times 10^{22}$ cm-2, $kT_{e}=481\pm 121$ keV,
$\tau=0.03\pm 0.01$, $\alpha=2.6\pm 0.1$ at an $E_{pivot}=1\times 10^{5}$ keV,
$\beta=0.3\pm 0.3$, and $\Gamma_{HE}=2.1\pm 0.2$ with reduced
$\chi^{2}=2611.60/2612=1.00$. We report only the results with $kT_{0}$ fixed
to 0.5 keV as the fit to $kT_{0}=0.05$ keV results in similar poorly
constrained values. We obtain a $kT_{e}$ near the runaway limit, but with a
large error.

The fit well describes the data, but the high-energy power-law over-predicts
the observed high-energy flux seen by LAT and H. E. S. S. Thus, we fixed the
photon index value found in the jet only scenario (2.25) and refit the
spectra. The new model results in comparable parameters without much change in
constraints and in goodness of fitwhen including a expabs to the zpowerlw to
limit its low-energy contribution. $\chi^{2}=2623.95/2613=1.00$. The low-
energy cutoff energy was fixed to $10^{5}$ keV.

Figure 4 shows the NuSTAR spectrum in black diamonds (FPMA) and red triangles
(FPMB), SPI spectrum in green squares, the ISGRI spectrum in blue stars, and
LAT spectrum in magenta x’s with both logparabola+zpowerlw (left) and
CompTT+logparabola+zpowerlw (right; where the hard X-rays are due to the
corona and the MeV to GeV flux is from the jet) models. The differences
between the models can be seen in the MeV region where the jet only scenario
smoothly connects from hard X-rays to high-energy gamma-rays. In contrast, the
other scenario predicts a drop in flux above a few MeV. Thus, using the long-
term average spectra from INTEGRAL/SPI and ISGRI and LAT, we are not able to
differentiate between the possibilities without observations in the medium MeV
region.

Note that Rani et al. (2022) proposed a scenario where the peak energy of the
jet overlaps with the corona component at $\sim 500$ keV, thus hiding the
corona component. However, we were not able to disentangle the presence of the
two components with our data.

####  4.2.3 Comparison with Previous Works

In their work on Cen A, Rani et al. (2022) also showed that the 230 GHz flux
correlates with the hard X-ray flux with no time lag. (Israel et al. (2008)
found similar results.) In addition, they found the 230 GHz flux is higher
than what is predicted for an advection dominated accretion flow (ADAF), and
the radio spectrum is inconsistent with the slope predicted for an ADAF. Thus
the authors conclude a jet component is enough to explain the hard X-ray
emission. Our work supports that conclusion including X-ray to GeV data.

The source 3C 273 provides an interesting comparison since it was the first
AGN where both corona and jet components have been reported (Grandi & Palumbo,
2004). However, the Cen A jet component peaks at $\sim 50$ MeV compared to
roughly 2 MeV for 3C 273. Additionally, for Cen A, the SED peak is dominated
by the corona emission (around 500 keV) while the jet component dominates the
SED peak in 3C 273 (Madsen et al., 2015). Therefore, it seems probable that
even if a corona exists and contributes to the hard X-ray emission in Cen A,
it would be hard to observe and distinguish from the jet emission.

Figure 4: Cen A long-term average spectra from INTEGRAL SPI (green square),
ISGRI (blue stars) with NuSTAR/FPMA (black diamonds) and FPMB (red triangles)
and LAT (magenta x’s). The left panel shows the logparabola+zpowerlw model
(black dashed line) with the individual continuum components shown as red
dotted lines. The right panel shows the CompTT+logparabola+zpowerlw model
(black dashed line) with individual continuum components as red dotted lines.

##  5 Conclusion

In this work, we analyzed INTEGRAL/ISGRI and SPI observations of Cen A
covering roughly 20 years. We first searched for spectral variability on a
timescale of $\sim$ months, and did not find any significant deviations from a
power-law with a constant value. Building from this result, we constructed a
long-term average spectrum up to 2.2 MeV. We included an observation from
NuSTAR, which provides coverage down to 3.5 keV, to study the X-ray to soft
gamma-ray spectrum.

Analysis of this spectrum found a narrow Fe line ($<8$ eV) and no significant
reflection component, which are consistent with previous results (apart from
Burke et al. (2014) that report marginal reflection). The joint
NuSTAR/ISGRI/SPI spectral fit has a cutoff energy of $\sim 650$ keV or an
electron temperature of $\sim 520-550$ keV, depending on the seed temperature.
The physical interpretation of the hard X-ray/soft gamma-ray emission with
these values is ambiguous, as the $kT_{e}$’s are near the runaway pair-
production region for the corona. Another possible scenario is synchrotron
self-Compton emission from the jet, which can also well describe the spectrum
using a log-parabola model.

To complete our overview of the source, the LAT spectrum in the 50 MeV to 100
GeV range was included to the spectral fit. A fit to a log-parabola model for
the jet emission explains the spectrum from NuSTAR to LAT below $\sim 3$ GeV.
However, the joint spectrum can also be adequately described by the inclusion
of a CompTT component to investigate the presence of coronal emission.

The results with this component are degenerate, however, but the best-fit
parameters indicate two general scenarios. One where the X-ray emission is
completely due to the corona and the jet emission dominates at MeV/GeV
energies. The other is that the jet flux can explain the spectrum from X-rays
to the GeV region. Our work favors the jet interpretation as one component can
explain the whole emission from keV to LAT and due to the high electron
temperature implied by the corona scenario. This interpretation is consistent
with the polarization results by IXPE, which did not detect significant X-ray
polarization (Ehlert et al., 2022). In their scenario, the X-rays can be
explained by SSC emission with electrons accelerated in magnetic fields that
are highly disorganized around the innermost jet.

To definitively exclude/constrain the presence of a corona component, several
possibilities can be explored. As discussed above, the high-energy gamma-ray
spectrum has two components, which limits the contraints on the curvature of
the jet spectrum from LAT observations that can be extrapolated to lower
energies. A detailed analysis of the Cen A long-term spectrum in high-energy
gamma-rays (potentially including H. E. S. S. data) may help to better
constrain the hard X-ray/soft gamma-ray emission.

Another constraint can come from a significant Cen A detection at energies
between SPI and LAT to differentiate between the jet only model and the corona
$+$ jet scenario as the predicted fluxes are significantly different in the
MeV range (Figure 4). Future observations by COSI may provide clarification,
but the energy range is expected to extend up to only 5 MeV (Tomsick et al.,
2019). Thus observations from a proposed mission such a AMEGO-X will be the
best tool to study the full MeV energy range (Caputo et al., 2022).

We thank the referee for their comments and input. The research leading to
these results has received funding from the European Union’s Horizon 2020
Programme under the AHEAD2020 project (grant agreement n. 871158). J.R.
acknowledges financial support under the INTEGRAL ASI/INAF No. 2019-35.HH.0
and from an INAF 2022 mini-grant. M.M. acknowledges financial support from ASI
under contract n. 2019-35-HH.0. The INTEGRAL SPI project has been completed
under the responsibility and leadership of CNES. We are grateful to ASI, CEA,
CNES, DLR, ESA, INTA, NASA and OSTC for support.

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the constituents is not sufficient for an efficient application of the
spectral tool (Lemma 1.14). However, we expect that a result of this flavor
would introduce important techniques to the analysis.

We also would like to mention that there should be a reasonable hope to prove
Conjecture 1.7 for the case when no color is overwhelmingly dominant. The
following result easily follows from minimal distinguishing number analysis.
In particular, in the case of bounded rank it gives an $\Omega(n)$ bound on
the motion.

######  Proposition 7.2.

Fix $0<\delta<1$ and an integer $r\geq 3$. Let $\mathfrak{X}$ be a primitive
coherent configuration of rank $r$ on $n$ vertices. Assume that each
constituent $X_{i}$ has degree $k_{i}\leq\delta n$. Then

$\operatorname{motion}(\mathfrak{X})\geq
D_{\min}(\mathfrak{X})\geq\frac{\min(\delta,1-\delta)}{6(r-1)}n.$

###### Proof.

The condition $k_{i}\leq\delta n$, for all $i$, implies that there exists a
set $I$ of colors such that $\sum\limits_{i\in I}k_{i}=\alpha n$ for some
$\min(\delta,1-\delta)/2\leq\alpha\leq 1/2$. Fix any vertex $u$ of
$\mathfrak{X}$. We want to show that for some vertex $v$ the inequality
$D(u,v)\geq\alpha n/3$ holds.

Assume this is not true. Denote $N_{I}(u)=\\{z|\,c(u,z)\in I\\}$. Let us count
the number of pairs $(v,z)$ with $c(u,z)\in I$ and $c(v,z)\in I$ in two
different ways. Since $\sum\limits_{i^{*}\in I}k_{i}=\alpha n$ and $z\in
N_{I}(u)$, there are $\alpha^{2}n^{2}$ such pairs. On the other hand, for
every $v$ we have $D(u,v)\leq\alpha n/3$, so at least $2\alpha n/3$ vertices
$z\in N_{I}(u)$ are paired with $v$. Therefore, the number of pairs in
question is at least $n\cdot\dfrac{2\alpha}{3}n$. This contradicts the
condition $0<\alpha\leq\dfrac{1}{2}$.

Therefore, there exists a pair of vertices with $D(u,v)\geq\alpha n/3$.
Finally, the configuration $\mathfrak{X}$ is primitive, so by Lemma 2.13 we
get that $\operatorname{motion}(\mathfrak{X})\geq
D_{\min}(\mathfrak{X})\geq\dfrac{\alpha}{3(r-1)}n$. ∎

However, when the rank is unbounded, this seemingly simple case of Conjecture
1.7 (every constituent has degree $\leq\delta n$) is still open. To avoid
exceptions we relax the conjectured lower bound to $\Omega(n/\log(n))$.

######  Conjecture 7.3.

Fix $0<\delta<1$. Let $\mathfrak{X}$ be a primitive coherent configuration on
$n$ vertices. Assume that every constituent has degree $\leq\delta n$. Then
$\operatorname{motion}(\mathfrak{X})=\Omega(n/\log(n))$.

Next, we observe that Cameron schemes satisfy Conjecture 7.3.

######  Proposition 7.4.

Fix $0<\delta<1$. Consider a Cameron group $(A_{m}^{(k)})^{d}\leq G\leq
S_{m}\wr S_{d}$ acting on $n=\binom{m}{k}^{d}$ points and let
$\mathfrak{X}=\mathfrak{X}(G)$ be the corresponding Cameron scheme. Assume
that every constituent of $\mathfrak{X}$ has degree $\leq\delta n$. Then
$\operatorname{motion}(\mathfrak{X})=\Omega(n/\log(n))$.

###### Proof.

We can assume $k\leq m/2$. Note that then the rank of $\mathfrak{X}$ is equal
to $kd+1$.

Case 1. Suppose that $k\leq m/3$. Then

$n=\binom{m}{k}^{d}\geq\left(\frac{m-k}{k}\right)^{kd}\geq\left(\frac{2k}{k}\right)^{kd}=2^{kd}$

Thus, $kd\leq\log(n)$ in this case, and the statement follows from Proposition
7.2.

Case 2. Suppose that $m/3<k\leq m/2$. By Lemma 2.27, we have that as
$m\rightarrow\infty$ the inequality
$\operatorname{motion}(\mathfrak{X})\geq\alpha n$ holds for some $\alpha>0$.
At the same time, by the proof of Lemma 2.27 we know that the motion of
$\mathfrak{X}$ does not depend on $d$. Thus as
$\operatorname{motion}(X)\geq\alpha n$ is violated just by finite number of
pairs $(m,k)$, we still have $\operatorname{motion}(\mathfrak{X})=\Omega(n)$
in this case. ∎

We observe that the bound in Conjecture 7.3, if true, is nearly tight, for
$\delta\in(1/e,1)$, as the example of Hamming schemes $\mathfrak{H}(tm,m)$
with $t=-\lfloor\log(\delta)m\rfloor/m$ shows. Note that for $m\geq 3$ the
Hamming scheme $\mathfrak{H}(k,m)$ is primitive.

######  Proposition 7.5.

Consider Hamming scheme $\mathfrak{H}(tm,m)$ with
$\displaystyle{t=-\frac{\lfloor\log(\delta)m\rfloor}{m}}$ on $n=m^{tm}$
points, for $\delta\in(1/e,1)$. Then its maximum constituent degree satisfies
$k_{\max}\leq\delta n$ and the motion satisfies

$\operatorname{motion}(\mathfrak{H}(tm,m))=O\left(\frac{n\log\log(n)}{\log(n)}\right).$

###### Proof.

Note that since $tm<m$ the maximum degree is $k_{\max}=(m-1)^{tm}$. Then

$k_{\max}=\left(\frac{m-1}{m}\right)^{mt}n\leq\mathrm{e}^{-t}n\leq\delta n.$

The motion of $\mathfrak{H}(tm,m)$ is realized by a 2-cycle in the first
coordinate, and is equal to $2n/m$. The number of vertices is
$n=m^{mt}=\mathrm{e}^{tm\log(m)}$, so $m\log(m)=\log(n)/t$. Thus
$m>\log(n)/(t\log\log(n))$. Hence,

$\operatorname{motion}(\mathfrak{H}(tm,m))\leq\frac{2n\log\log(n)t}{\log(n)}=O\left(\frac{n\log\log(n)}{\log(n)}\right).$

∎

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