11institutetext: INAF - Osservatorio Astronomico di Roma, Via Frascati 33,
00078, Monte Porzio Catone (Roma), Italy 22institutetext: INAF - Istituto di
Astrofisica e Planetologia Spaziali, Via del Fosso del Cavaliere 100, 00133,
Roma, Italy 33institutetext: Dipartimento di Fisica, Università degli Studi di
Roma “Tor Vergata”, via della Ricerca Scientifica 1, 00133 Roma, Italy
44institutetext: Dipartimento di Matematica e Fisica, Università degli Studi
Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy 55institutetext:
Instituto de Estudios Astrofísicos, Facultad de Ingeniería y Ciencias,
Universidad Diego Portales, Avenida Ejercito Libertador 441, Santiago, Chile
66institutetext: Kavli Institute for Astronomy and Astrophysics, Peking
University, Beijing 100871, People’s Republic of China 77institutetext: INAF -
Osservatorio Astrofisico di Arcetri, Largo Enrico Fermi 5, 50125 Firenze,
Italy 88institutetext: CNRS, IPAG, Université Grenoble Alpes, 38000 Grenoble,
France 99institutetext: Space Science Data Center, Agenzia Spaziale Italiana,
Via del Politecnico snc, 00133 Roma, Italy 1010institutetext: INAF -
Osservatorio di Astrofisica e Scienza dello Spazio di Bologna, Via Gobetti,
93/3, 40129 Bologna, Italy 1111institutetext: ASI - Agenzia Spaziale Italiana,
Via del Politecnico snc, 00133 Roma, Italy

# Investigating the interplay between the coronal properties and the hard
X-ray variability of active galactic nuclei with NuSTAR

Roberto Serafinelli , 11<EMAIL_ADDRESS>Alessandra De Rosa 22
Alessia Tortosa 11 Luigi Stella 11 Fausto Vagnetti 3322 Stefano Bianchi 44
Claudio Ricci 5566 Elias Kammoun 4477 Pierre-Olivier Petrucci 88 Riccardo
Middei 9911 Giorgio Lanzuisi 1010
Andrea Marinucci 1111 Francesco Ursini 44 Giorgio Matt 44

(Received XXX; accepted YYY)

Active galactic nuclei (AGN) are extremely variable in the X-ray band down to
very short timescales. However, the driver behind the X-ray variability is
still poorly understood. Previous results suggest that the hot corona
responsible for the primary Comptonized emission observed in AGN is expected
to play an important role in driving the X-ray variability. In this work, we
investigate the connection between the X-ray amplitude variability and the
coronal physical parameters; namely, the temperature ($kT$) and optical depth
($\tau$). We present the spectral and timing analysis of 46 NuSTAR
observations corresponding to a sample of 20 AGN. For each source, we derived
the coronal temperature and optical depth through X-ray spectroscopy and
computed the normalized excess variance for different energy bands on a
timescale of $10$ ks. We find a strong inverse correlation between $kT$ and
$\tau$, with correlation coefficient of $r<-0.9$ and negligible null
probability. No clear dependence was found among the temperature and physical
properties, such as the black hole mass or the Eddington ratio. We also see
that the observed X-ray variability is not correlated with either the coronal
temperature or optical depth under the thermal equilibrium assumption, whereas
it is anticorrelated with the black hole mass. These results can be
interpreted through a scenario where the observed X-ray variability could
primarily be driven by variations in the coronal physical properties on a
timescale of less than $10$ ks; whereas we assume thermal equilibrium on such
timescales in this work, given the capability of the currently available hard
X-ray telescopes. Alternatively, it is also possible that the X-ray
variability is mostly driven by the absolute size of the corona, which depends
on the supermassive black hole mass, rather than resulting from any of its
physical properties.

###### Key Words.:

X-rays:galaxies – galaxies:active – galaxies:Seyfert – black hole physics

##  1 Introduction

Active galactic nuclei (AGN) are bright extragalactic sources powered by the
accretion of matter onto a supermassive black hole (SMBH). In brief, AGN emit
light at all wavelengths and they are characterized by a significantly loud
X-ray emission (e.g., Padovani et al., 2017). The X-ray emission of AGN is
produced by inverse Compton scattering of UV seed photons, emitted by the
accretion disk, off a hot relativistic electron plasma known as the corona
(e.g., Haardt & Maraschi, 1991, 1993). The typical shape of the X-ray spectrum
of an AGN is that of a power law, characterized by a photon index $\Gamma$, up
to a characteristic energy, $E_{c}$, known as the cut-off energy where the
power law breaks. The relation between the cut-off energy and the temperature
can be approximated with $E_{c}\sim 2-3\;kT$ (e.g., Petrucci et al., 2001),
depending on the geometry of the corona and the optical depth, while the
photon index is dependent on both the coronal temperature and the optical
depth. However, more complex relations with both the temperature and optical
depth are needed when considering broader ranges of temperatures and optical
depths (Middei et al., 2019).
Many models have been proposed for the coronal geometry, including slab (e.g.,
Haardt & Maraschi, 1991), spherical (e.g., Frontera et al., 2003) or lamp-post
coronal geometry (e.g., Miniutti & Fabian, 2004). Details of its shape,
location and size are yet largely unknown, though, since spectroscopy alone is
not able to distinguish among different geometries, which can be probed with
polarimetry measurements (e.g., Ursini et al., 2022). Indeed, recent results
with Imaging X-ray Polarimetry Explorer (IXPE, Weisskopf et al., 2016) are
starting to unveil the geometrical properties of the AGN corona. Gianolli et
al. (2023) measured the coronal X-ray polarization for the first time in the
Seyfert galaxy NGC 4151, strongly suggesting a wedge or a slab above the
accretion disk (e.g., Poutanen et al., 2018). For IC4329A (Ingram et al.,
2023), a marginal detection for the X-ray polarimetry also suggests a wedge
coronal geometry. Only upper limits were found for MCG-5-23-16 (Marinucci et
al., 2022; Tagliacozzo et al., 2023), although the results, combined with the
inclination measurement obtained with XMM-Newton and NuSTAR (Serafinelli et
al., 2023b), tentatively favors a wedge geometry as well.
Several measurements of the cut-off energy have been undertaken using many
hard X-ray instruments, like BeppoSAX (e.g., Dadina, 2007), INTEGRAL (e.g.,
Molina et al., 2009; De Rosa et al., 2012), Swift-BAT (e.g., Ricci et al.,
2018) and NuSTAR (e.g., Fabian et al., 2015, 2017; Tortosa et al., 2018b),
including obscured sources (e.g., Baloković et al., 2020; Serafinelli et al.,
2023a). This task is far from trivial, since the hard X-ray spectrum is also
characterized by a reflection component, due to X-ray photons interacting with
the surrounding environment, such as the accretion disk, the broad line region
or the torus, whose parameters are often degenerate with those of the
continuum (see e.g. the review in Reynolds, 2021). The cut-off energy is found
in a large energy range, from $E_{c}\sim 23$ keV (Kammoun et al., 2023) to
$E_{c}\sim 750$ keV (Matt et al., 2015), with average values around $\sim
100-200$ keV (e.g., Ricci et al., 2018; Kamraj et al., 2022). Direct
measurements of the coronal parameters such as the electron temperature $kT$
and the optical depth $\tau$ have also been extensively performed on many AGN,
finding a tight correlation between the temperature and the optical depth
(e.g., Tortosa et al., 2018b; Kamraj et al., 2022).
The X-ray emission of AGN is well-known to be variable on several timescales,
both in amplitude (e.g., Markowitz & Edelson, 2004; Ponti et al., 2012;
Vagnetti et al., 2016; Middei et al., 2017; Serafinelli et al., 2020) and
spectral shape (e.g., Sobolewska & Papadakis, 2009; Serafinelli et al., 2017).
Variability is found on very short timescales down to a few hours (e.g., Ponti
et al., 2012), and this suggests that the X-ray emitting region is compact
(e.g., Mushotzky et al., 1993; De Marco et al., 2013), with a typical radius
of $R_{c}\sim 10R_{g}$ (Ursini et al., 2020a), also supported by microlensing
results (e.g., Chartas et al., 2009; Morgan et al., 2012). Moreover, the X-ray
emission is variable in a wide range of energies, including very hard X-rays
($E>10$ keV) on both long (e.g., years, Soldi et al., 2014; Akylas et al.,
2022; Papadakis & Binas-Valavanis, 2024) and shorter timescales (e.g., hours,
Rani et al., 2019; Akylas et al., 2022).
The X-ray variability of AGN provides crucial insight on the size of the
central source, but its main driver is still poorly understood. We aim here to
investigate the possible relation between the variability of the X-ray
emission coming from the corona and the physical properties of the corona
itself with NuSTAR, which is able to study both coronal parameters and
variability because of its high sensitivity at hard X-rays ($E=3-79$ keV). We
present a study of a sample of 20 nearby ($z<0.2$) Seyfert galaxies for which
we study the coronal parameters using NuSTAR, in order to investigate possible
relations with the X-ray variability. In Sect. 2, we describe our sample, made
up of sources with a wide range of coronal temperatures and optical depths,
and the data reduction of the available X-ray data. In Sect. 3 we describe the
spectral analysis we performed. In Sect. 4 we investigate the X-ray
variability through the computation of the excess variance in different bands
with NuSTAR. Finally our results are discussed in Sect. 5 and present a
summary in Sect. 6. Throughout the paper, we adopt a standard $\Lambda$CDM
cosmology, with $H_{0}=70$ km s${}^{-}1$ Mpc-1, $\Omega_{m}=0.3,$ and
$\Omega_{\Lambda}=0.7.$

##  2 Sample selection and data reduction

Table 1: Sources analyzed in this paper.

Source | $z$ | Type |  $M_{\rm BH}$ ($M_{\odot}$) | $\log L_{\rm bol}/{\rm(erg\;s}^{-1})$ | $\log\lambda_{\rm Edd}$ |
---|---|---|---|---|---|---
1H 0419-577 | $0.104$ | Sy1 | $2.2\times 10^{8}$ | $45.68$ | $-0.84$ |
4C 50.55 | $0.02$ | RLSy1 | $9.3\times 10^{7}$ | $44.91$ | $-1.24$ |
Ark 564${\dagger}$ | $0.0243$ | NLSy1 | $3.2\times 10^{6}$ | $45.00$ | $0.39$ |
ESO 103-G35 | $0.00914$ | Sy1 | $2.3\times 10^{7}$ | $44.54$ | $-1.01$ |
ESO 362-G18 | $0.01244$ | Sy1.5 | $1.3\times 10^{7}$ | $44.07$ | $-1.22$ |
ESO 383-G18 | $0.01241$ | Sy2 | $3.0\times 10^{5}$ | $43.79$ | $0.15$ |
GRS 1734-292 | $0.021$ | Sy1 | $6.9\times 10^{7}$ | $44.95$ | $-1.07$ |
HE 1143-1810 | $0.0329$ | Sy1 | $2.5\times 10^{7}$ | $44.95$ | $-0.62$ |
IC4329A | $0.01613$ | Sy1 | $4.5\times 10^{7}$ | $45.06$ | $-0.77$ |
MCG-5-23-16 | $0.00823$ | Sy1 | $4.5\times 10^{7}$ | $44.34$ | $-1.49$ |
MCG+8-11-11 | $0.02$ | Sy1 | $4.1\times 10^{6}$ | $44.93$ | $0.15$ |
Mrk 6 | $0.01951$ | Sy1.5 | $1.3\times 10^{8}$ | $44.52$ | $-1.76$ |
Mrk 110 | $0.03552$ | Sy1 | $1.9\times 10^{7}$ | $45.06$ | $-0.41$ |
Mrk 509 | $0.01951$ | Sy1 | $1.1\times 10^{8}$ | $45.27$ | $-0.96$ |
NGC 3281 | $0.01073$ | Sy2 | $1.7\times 10^{8}$ | $44.06$ | $-2.04$ |
NGC 5506 | $0.00589$ | NLSy1 | $1.7\times 10^{7}$ | $44.14$ | $-1.24$ |
NGC 5728 | $0.00947$ | Sy2 | $1.8\times 10^{8}$ | $44.14$ | $-2.28$ |
NGC 6814 | $0.00523$ | Sy1 | $1.1\times 10^{7}$ | $43.52$ | $-1.7$ |
SWIFT J2127.4+5654 | $0.0144$ | NLSy1 | $1.4\times 10^{7}$ | $44.11$ | $-1.22$ |
UGC 6728 | $0.00652$ | Sy1 | $3.5\times 10^{5}$ | $43.16$ | $-0.57$ |
111The most common name, the redshift, the AGN type, the black hole mass
($M_{\rm BH}$), the bolometric luminosity ($L_{\rm bol}$), and the Eddington
ratio ($\lambda_{\rm Edd}$) are listed in this table. All values of $M_{\rm
BH}$, $L_{\rm bol}$ and $\lambda_{\rm Edd}$ are extracted from the BASS
catalog published by Koss et al. (2022). ${\dagger}$ The only exception is Ark
564, for which we retrieved a reverberation mapping mass estimate from
Peterson et al. (2004), with $L_{\rm bol}$ estimated applying a bolometric
correction to the X-ray luminosity.

We selected our sample of AGN from the $70$-Month Swift-BAT catalogue
(Baumgartner et al., 2013). Ricci et al. (2017) computed many X-ray properties
of the AGN in the catalog, such as the X-ray flux in several bands, the photon
index, and the cut-off energy. We select all sources where the cut-off has
been measured, namely, excluding the ones with only lower limits. Out of $836$
AGN of the whole Swift-BAT sample, $165$ satisfy this first condition. Among
those AGN, we selected the ones with public NuSTAR observations as of 10th
October 2021, for a total of 229 observations of 110 AGN. Not all NuSTAR
observations have sufficient statistics to compute the coronal parameters;
therefore, we selected only those with enough NuSTAR counts. To this end, we
considered the value of the X-ray flux in the $20-50$ keV, as reported by
Ricci et al. (2017), and we converted such a flux to NuSTAR count rate in the
same band using WebPimms222https://heasarc.gsfc.nasa.gov/cgi-
bin/Tools/w3pimms/w3pimms.pl, adopting an unabsorbed power law with
$\Gamma=1.8$, which is typical for Seyfert galaxies (e.g., Serafinelli et al.,
2017). This count rate was multiplied by the sum of the exposures of each
observation of every source and we selected all sources with at least 1500
counts per FPM module. We note that this selection might exclude very variable
sources in which the NuSTAR count rate may exceed the expected count rate from
the BAT flux, not easily spotted with this criterion. A total of $34$ sources
are selected with these criteria. Six of these sources (IC4329A, MCG-5-23-16,
MCG+8-11-11, NGC 5506, NGC 6814, and SWIFT J2127.4+5654) were already present
in the sample analyzed by Tortosa et al. (2018b). Out of completeness, we
decided to include two more sources from Tortosa et al. (2018b) that are not
included in our selection, Ark 564 (not detected in BAT) and GRS 1734-292, to
consider all non-jetted nearby ($z\leq 0.2$) AGN from their sample. This
resulted in a total of 36 AGN.
Even though our sample might possibly be biased towards the low tail of the
high energy cut-off distributions, this selection provides the best spectral
signal-to-noise ratio (S/N) needed to derive coronal physical properties with
high confidence level. Furthermore, excluding the sources with only lower
limits for the cut-off energy also excludes AGN with additional spectral
complexities, such as multiple ionized absorption gas layers, which could
introduce systematic errors in the measure of $kT$ and $\tau$ due to an
inaccurate continuum fit. For these sources, a simultaneous spectroscopy with
a low energy ($E<3$ keV) bandpass is recommended.
We reduced the NuSTAR observations with NUPIPELINE, available through HEASOFT
v6.30, which is part of the NUSTARDAS software package, calibration files
(CALDB), updated as recently as March 31, 2022. We extracted the FPMA and FPMB
source spectra and light curves from a circular region with $60"$ radius,
centered on the source, using the NUPRODUCTS command. The background spectra
are extracted using two source-free regions of $40"$ each.

Table 2: Coronal parameters of our sample derived with the models described in
Sect. 3.

Source |  $kT_{\rm sl}$ (keV) | $\tau_{\rm sl}$ |  $kT_{\rm sph}$ (keV) | $\tau_{\rm sph}$ | $L_{2-10\;{\rm keV}}$ | $F_{2-10\;{\rm keV}}$ | Model
---|---|---|---|---|---|---|---
1H0419-577 | $14^{+2}_{-1}$ | $2.5^{+0.2}_{-0.3}$ | $14^{+2}_{-1}$ | $5.7^{+0.4}_{-0.5}$ | $35\pm 5$ | $1.3\pm 0.2$ | A
4C50.55 | $18^{+5}_{-2}$ | $2.2^{+0.2}_{-0.3}$ | $18^{+3}_{-2}$ | $5.2^{+0.4}_{-0.5}$ | $10.0\pm 0.1$ | $8.0\pm 0.1$ | A
Ark 564 | $15\pm 2$ | $1.4\pm 0.1$ | $16\pm 1$ | $3.3\pm 0.3$ | $3.5\pm 0.1$ | $6.4\pm 0.1$ | A
ESO 103-G35 | $17^{+18}_{-3}$ | $2.2^{+0.3}_{-0.7}$ | $17^{+7}_{-3}$ | $5.2\pm 0.7$ | $0.8\pm 0.2$ | $1.9\pm 0.5$ | C
ESO 362-G18 | $18^{+14}_{-4}$ | $2.5^{+0.4}_{-0.9}$ | $18^{+15}_{-4}$ | $5.8^{+0.8}_{-1.8}$ | $0.04\pm 0.01$ | $1.1\pm 0.1$ | B
ESO 383-G18 | $7.3^{+0.5}_{-0.4}$ | $4.7\pm 0.3$ | $7.3^{+0.5}_{-0.4}$ | $10.1\pm 0.6$ | $0.03\pm 0.01$ | $0.5\pm 0.1$ | C
GRS 1734-292 | $13^{+2}_{-1}$ | $2.9\pm 0.2$ | $13_{-1}^{+2}$ | $6.5\pm 0.4$ | $0.6\pm 0.1$ | $2.9\pm 0.1$ | A
HE 1143-1810 | $36^{+64}_{-19}$ | $1.2^{+1.1}_{-0.8}$ | $22^{+61}_{-5}$ | $4.4^{+0.9}_{-2.9}$ | $6.0\pm 1.9$ | $2.7\pm 0.9$ | A
IC 4329A | $44^{+20}_{-10}$ | $1.1\pm 0.3$ | $44^{+17}_{-11}$ | $2.8\pm 0.7$ | $5.6\pm 0.4$ | $12.0\pm 0.7$ | A
MCG-5-23-16 | $41\pm 11$ | $0.9\pm 0.2$ | $35^{+10}_{-11}$ | $2.7\pm 0.3$ | $1.80\pm 0.02$ | $10.4\pm 0.2$ | B
MCG+8-11-11 | $110^{+140}_{-40}$ | $0.3\pm 0.2$ | $90^{+125}_{-45}$ | $1.2^{+0.7}_{-0.4}$ | $5.1\pm 0.6$ | $5.6\pm 0.7$ | B
Mrk 6 | $13^{+4}_{-2}$ | $3.4^{+0.4}_{-0.6}$ | $13^{+4}_{-2}$ | $7.5^{+0.7}_{-1.1}$ | $0.99\pm 0.02$ | $0.70\pm 0.01$ | C
Mrk 110 | $35^{+15}_{-10}$ | $1.2^{+0.5}_{-0.4}$ | $24^{+17}_{-5}$ | $4.0^{+0.6}_{-1.4}$ | $9\pm 3$ | $3.1\pm 0.8$ | B
Mrk 509 | $17^{+2}_{-1}$ | $2.2\pm 0.1$ | $17^{+2}_{-1}$ | $5.2^{+0.2}_{-0.3}$ | $12.0\pm 0.3$ | $4.5\pm 0.1$ | A
NGC 3281 | $11^{+4}_{-2}$ | $3.7^{+0.8}_{-0.9}$ | $11^{+4}_{-2}$ | $8\pm 2$ | $0.23\pm 0.04$ | $0.4\pm 0.1$ | C
NGC 5506 | $510^{+250}_{-150}$ | $0.02\pm 0.01$ | $550\pm 250$ | $0.09_{-0.05}^{+0.30}$ | $0.53\pm 0.05$ | $6.2\pm 0.5$ | A
NGC 5728 | $13\pm 1$ | $5^{+2}_{-1}$ | $13\pm 1$ | $10^{+6}_{-3}$ | $1.4\pm 0.6$ | $0.16\pm 0.6$ | C
NGC 6814 | $60^{+24}_{-20}$ | $0.8^{+0.7}_{-0.3}$ | $82^{+80}_{-10}$ | $1.5^{+0.8}_{-0.1}$ | $0.21\pm 0.03$ | $3.8\pm 0.5$ | B
SWIFTJ2127.4+5654 | $33^{+37}_{-15}$ | $1.0_{-0.6}^{+0.8}$ | $24_{-7}^{+24}$ | $3.5_{-1.6}^{+1.0}$ | $1.4\pm 0.4$ | $2.9\pm 0.8$ | A
UGC 6728 | $28^{+16}_{-18}$ | $2.0\pm 1.2$ | $28^{+17}_{-15}$ | $5.0\pm 2.0$ | $0.075\pm 0.007$ | $1.1\pm 0.1$ | B

333Intrinsic luminosities are listed in units of $10^{43}$ erg s-1. Observed
fluxes are in units of $10^{-11}$ erg s-1 cm-2. All values are obtained in
this work with the spectral fits briefly described in Appendix B. All spectra
are shown in Fig. C.

##  3 Spectral analysis

Figure 1: Excess variance in the soft ($3-10$ keV) versus the excess variance
in the full ($3-79$ keV) NuSTAR bands (left). All excess variances are
normalized at $10$ ks. The best-fit line is fully consistent with the bisector
at $90\%$ confidence level, with an angular coefficient of $0.96\pm 0.05$. The
correlation coefficient is $r=0.95$ with a negligible or null probability.
Excess variance in the soft band vs. excess variance in the hard ($10-40$ keV)
X-ray band (right). The best-fit angular coefficient is $0.7\pm 0.1$ with a
correlation coefficient $r=0.90$ and a negligible probability of finding the
correlation by chance.

We performed a quick analysis of 36 selected AGN to verify whether the coronal
temperature is also constrained with NuSTAR. All fits included Galactic
absorption extracted from HI4PI Collaboration et al. (2016) (see Appendix B
for all details). We considered three models. Model A consists of a continuum
plus ionized reflection model, using XILLVERCP, which is the combination of
the reflection model XILLVER (García & Kallman, 2010; García et al., 2011) and
the continuum model NTHCOMP (Zdziarski et al., 1996). The XSPEC equation is:

${\tt ztbabs*xillvercp,}\;\;\;{\rm model\;A}.$

For simplicity, the inclination and iron abundance are fixed to
$\iota=30^{\circ}$, and $A_{\rm Fe}=1$, respectively. The reflection fraction
$\mathcal{R}$ is left free to vary, as XILLVERCP models both continuum and
reflection. Model B is adopted when the Fe K$\alpha$ region is not properly
modeled by model A because of the presence of a broad emission line,
interpreted as due to the presence of a relativistic reflection component
(e.g., Reynolds, 2021). In such cases, we also included a second reflector,
using RELXILLCP, which is the convolution of RELLINE (Dauser et al., 2014) and
XILLVERCP for the reflection component, and NTHCOMP for the continuum. In this
scenario XILLVERCP models reflection from material located farther out from
the black hole, while RELLXILLCP models reflection from the innermost parts of
the accretion disk (e.g., Serafinelli et al., 2023b). The XSPEC equation is:

${\tt ztbabs*(xillvercp+relxillcp),}\;\;\;{\rm model\;B}.$

We assumed a single emissivity index of $-3$ for the accretion disk and we
assumed that RELXILLCP only models reflection ($\mathcal{R}=-1$). Unless
otherwise specified (see Appendix B), we fixed an inner radius of $R_{\rm
in}=10\leavevmode\nobreak\ R_{g}$ and outer radius of $R_{\rm
out}=400\leavevmode\nobreak\ R_{g}$. We assumed a Schwarzschild black hole
($a=0$), while the XILLVERCP parameters are the same as model A, including a
free-to-vary reflection fraction. Finally, model C was adopted when the
reflection component was due to a single neutral reflector, and the model with
one or two ionized reflection components does not fit the data satisfactorily.
In this case, the reflection component was modeled with BORUS, which models
the reflection from spherical distribution of neutral material with polar
cutouts444The BORUS grids are available at
https://sites.astro.caltech.edu/~mislavb/download (see Baloković et al., 2018,
for details). Thus, we added a NTHCOMP component for the continuum, since
BORUS does not have an in-built continuum one and we tied all the reflection
parameters in BORUS that describe the continuum to that of the NTHCOMP. The
XSPEC equation is:

${\tt ztbabs*(nthcomp+borus12)}\;\;\;{\rm model\;C}.$

Since the physical properties of the neutral reflector are not the main goal
of this paper, we always assumed a Compton thick reflector ($\log N_{\rm
H,refl}/{\rm cm^{-2}}=24.5$), with a covering factor $C_{f}=0.5$ and reflector
assumed on the line of sight. All three models include a neutral absorption
component modeled with ZTBABS whenever required (see Appendix B for details).
We did not find any relevant variability of the coronal parameters $kT$ or
$\Gamma$, when considering different observation epochs of the same source.
This is consistent with the recent results obtained by Pal & Stalin (2022),
where evidence of variations in the coronal parameters was found in less than
$5\%$ of their AGN sample. Therefore, during the fit, we kept $kT$ and $\tau$
tied between different observations of the same AGN. We were able to constrain
the coronal parameters in 20 sources; however, we excluded the 16 sources for
which we only obtained lower limits for either the coronal temperature or the
optical depth. The final sample is shown in Table 1, where eblack hole masses
($M_{\rm BH}$), bolometric luminosities ($L_{\rm bol}$), and Eddington ratios
($\lambda_{\rm Edd}$) have been taken from the catalog presented in Koss et
al. (2022); the only exception is Ark 564, which is absent from their list.
Koss et al. (2022) selected masses preferably from reverberation mapping (when
available) followed by single-epoch measurements from H$\alpha$ or H$\beta$
lines and, ultimately, from the host galaxy velocity dispersion. For Ark 564,
we considered the reverberation mapping measurement in Peterson et al. (2004),
$M_{\rm BH}=3.2\times 10^{6}$ $M_{\odot}$, with a bolometric luminosity
computed by applying the bolometric correction proposed by Marconi et al.
(2004) to the unabsorbed X-ray luminosity in the $2-10$ keV energy band, along
with an Eddington ratio computed as $\lambda_{\rm Edd}=L_{\rm bol}/L_{\rm
Edd}$, where $L_{\rm Edd}=1.26\times 10^{38}M/M_{\odot}$ erg s-1 is the
Eddington luminosity. All the values are reported in Table 1.
As we are interested in exploring different coronal geometries and the
interplay between temperature and optical depth, we separated the continuum
and the reflection by including a Comptonization model as continuum, COMPTT
(Magdziarz & Zdziarski, 1995), which is capable of assuming slab-shaped and
spherical-shaped coronae. In models A and B, we simply replaced NTHCOMPT with
COMPTT by setting the reflection parameter to a fixed value of $-1$555We
recall that XILLVERCP models both continuum and reflection, by fitting the
reflection fraction $\mathcal{R}>0$ and assuming NTHCOMP as continuum. When
the reflection fraction is assumed to be frozen to $\mathcal{R}=-1$, XILLVERCP
will only model a pure reflection spectrum, which means that we then need a
second component to model the continuum.; whereas in model C, we explicitly
removed the external NTHCOMP component and replaced it with COMPTT. We started
with the baseline models used for the first round of fits. All the parameters
of the reflection, with the exception of the normalization, are frozen to the
results of the first round of fits, as we are mainly interested in the coronal
parameters. We fix the parameter approx in COMPTT to $0.5$ to model a slab
geometry, and to $1.5$ to model a spherical coronal shape. We checked the
consistency between the two continuum models by comparing the best-fit values
of the temperature obtained before the addition of COMPTT and those obtained
after, assuming a spherical geometry, the only one allowed by NTHCOMP. We find
consistent $kT$ results at $90\%$ confidence level in all cases. Furthermore,
we also simulated spectra in a large range of temperatures and exposures with
NTHCOMP.We fit the simulated spectra with COMPTT, assuming a spherical
geometry, such as the one considered in NTHCOMP model, also finding good
agreement at at least $90\%$ confidence level.
We report the values obtained for $kT$ and $\tau$ for both geometries in Table
2, while details on the fits are available in Appendix B.

##  4 Variability analysis

Table 3:  NuSTAR excess variances in the energy bands $3-79$, $3-10,$ and
$10-40$ keV.

Source | OBSID |  $\sigma_{\rm NXS,10}^{2}$ ($3-79$ keV) |  $\sigma_{\rm NXS,10}^{2}$ ($3-10$ keV) |  $\sigma_{\rm NXS,10}^{2}$ ($10-40$ keV)
---|---|---|---|---
1H 0419-577 | 60101039002 | $1.2\pm 0.2$ | $1.6\pm 0.2$ | $4.6\pm 0.6$
| 60402006002 | $2.7\pm 0.7$ | $3.4\pm 0.9$ | $6\pm 2$
| 60402006004 | $1.8\pm 0.6$ | $2.0\pm 0.7$ | $4\pm 1$
4C 50.55 | 60061305002 | $0.26\pm 0.09$ | $0.2\pm 0.1$ | $0.5\pm 0.2$
| 60301005002 | $0.29\pm 0.08$ | $0.7\pm 0.1$ | $1.0\pm 0.3$
Ark 564 | 60101031002 | $49\pm 7$ | $48\pm 6$ | $66\pm 15$
| 60401031002 | $23\pm 2$ | $26\pm 2$ | $27\pm 5$
| 60401031004 | $33\pm 4$ | $33\pm 4$ | $47\pm 5$
ESO 103-G35 | 60061288002 | $2.8\pm 0.5$ | $3.3\pm 0.6$ | $2.6\pm 0.7$
| 60301004002 | $1.8\pm 0.4$ | $1.6\pm 0.5$ | $3.4\pm 0.8$
ESO 362-G18 | 60201046002 | $5\pm 1$ | $7\pm 2$ | $7\pm 2$
ESO 383-G18 | 60061243002 | $1.5\pm 0.6$ | $1.4\pm 0.9$ | $4\pm 1$
| 60261002002 | $3.3\pm 0.5$ | $4.4\pm 0.7$ | $5.6\pm 0.9$
GRS 1734-292 | 60061279002 | $0.6\pm 0.1$ | $1.0\pm 0.2$ | $0.8\pm 0.3$
| 60301010002 | $1.1\pm 0.2$ | $2.0\pm 0.4$ | $2.0\pm 0.6$
HE 1143-1810 | 60302002002 | $0.6\pm 0.3$ | $0.4\pm 0.3$ | $3\pm 1$
| 60302002004 | $1.5\pm 0.4$ | $2.1\pm 0.6$ | $4\pm 1$
| 60302002006 | $1.9\pm 0.4$ | $2.1\pm 0.5$ | $3\pm 1$
| 60302002008 | $1.0\pm 0.3$ | $0.8\pm 0.3$ | $1.7\pm 0.8$
| 60302002010 | $4.6\pm 0.6$ | $8\pm 1$ | $3\pm 1$
IC 4329A | 60001045002 | $0.8\pm 0.1$ | $0.9\pm 0.3$ | $2.6\pm 0.7$
MCG-5-23-16 | 10002019001 | $2.2\pm 0.2$ | $2.3\pm 0.3$ | $3.1\pm 0.5$
| 60001046002 | $4.7\pm 0.7$ | $5.3\pm 0.8$ | $7\pm 3$
| 60001046004 | $1.9\pm 0.3$ | $2.2\pm 0.4$ | $2.2\pm 0.2$
| 60001046006 | $3.2\pm 0.5$ | $3.7\pm 0.5$ | $2.6\pm 0.5$
| 60001046008 | $2.9\pm 0.4$ | $3.3\pm 0.4$ | $3.0\pm 0.3$
MCG+8-11-11 | 60201027002 | $0.7\pm 0.1$ | $0.9\pm 0.1$ | $1.5\pm 0.2$
Mrk6 | 60102044002 | $1.9\pm 0.7$ | $3\pm 1$ | $3\pm 1$
| 60102044004 | $1.6\pm 0.5$ | $3\pm 1$ | $1.7\pm 0.9$
Mrk 110 | 60201025002 | $1.3\pm 0.6$ | $1.3\pm 0.3$ | $3\pm 2$
| 60502022002 | $0.9\pm 0.3$ | $1.1\pm 0.4$ | $5\pm 1$
| 60502022004 | $1.3\pm 0.4$ | $1.4\pm 0.4$ | $4\pm 1$
Mrk 509 | 60101043002 | $0.56\pm 0.06$ | $0.8\pm 0.1$ | $1.7\pm 0.2$
| 60101043002 | $0.9\pm 0.1$ | $0.8\pm 0.3$ | $3.2\pm 0.8$
NGC 3281 | 60061201002 | $8\pm 2$ | $9\pm 2$ | $14\pm 3$
| 60662003002 | $3.2\pm 0.8$ | $5\pm 1$ | $7\pm 1$
NGC 5506 | 60061323002 | $3.2\pm 0.4$ | $4.2\pm 0.5$ | $3.5\pm 0.7$
NGC 5728 | 60061256002 | $5\pm 1$ | $18\pm 4$ | $3\pm 1$
| 60662002002 | $11\pm 2$ | $23\pm 5$ | $11\pm 2$
NGC 6814 | 60201028002 | $15\pm 4$ | $18\pm 5$ | $11\pm 2$
SWIFT J2127.4+5654 | 60001110002 | $8.1\pm 0.9$ | $9\pm 1$ | $7\pm 2$
| 60001110003 | $4.9\pm 0.6$ | $6.0\pm 0.8$ | $5\pm 1$
| 60001110005 | $8.9\pm 0.8$ | $10\pm 1$ | $8\pm 1$
| 60001110007 | $7.0\pm 0.7$ | $8.4\pm 0.9$ | $5\pm 1$
UGC 6728 | 60160450002 | $34\pm 3$ | $39\pm 4$ | $33\pm 5$
| 60376007002 | $32\pm 2$ | $36\pm 2$ | $26\pm 3$

666All the excess variances are normalized to $10$ ks, as described in Sect.
4. All excess variances are in units of $10^{-3}$.

A straightforward estimator of the X-ray variability is the normalized excess
variance (e.g., Vaughan et al., 2003), defined as:

$\sigma^{2}_{\rm
NXS}=\frac{1}{N\mu^{2}}\sum_{i=1}^{N}\left[\left(x_{i}-\mu\right)^{2}-\sigma_{i}^{2}\right],$
(1)

where $x_{i}$ are the values of the X-ray amplitude of every time bin, $\mu$
is the mean value of the amplitude, $N$ is the number of points, and
$\sigma_{i}$ is the photometric error on the X-ray amplitude. The excess
variance of a random process is the integral of the power spectral density
(PSD) over all frequencies between 0 and infinity. However, with real data it
is limited by $f_{\rm min}=1/t_{\rm max}$ and $f_{\rm max}=1/t_{\rm min}$,
where $t_{\rm min}=2\Delta t$, with $\Delta t$ being the time bin of the light
curves we use (in our case 1 ks), and $t_{\rm max}$ is the length of the
observation segment (10 ks, see below). We note that the excess variance is
not a good estimator when a large range of redshifts is considered, since
same-length light curves in the observer frame represent different rest-frame
lengths at different redshifts (Vagnetti et al., 2016); at the same time, we
would be looking at a different energy range. However, our sample is limited
at $z<0.2$, hence, we were able to avoid these biases among the excess
variance.
Since the excess variance is a quadratic sum over the number of points of a
light curve, two conditions should be satisfied in order to properly compare
these quantities over different sources. First of all, they must have the same
binning, which is ensured in our case by how the light curves were prepared.
Indeed, every NuSTAR light curve is binned at $1$ ks. The other condition is
that the light curves should be equally long, which is not satisfied in our
sample as the exposures are different in the observations of different
sources. To avoid this bias, we normalize every excess variance to the same
length. Given that the smallest NuSTAR exposure in our sample is $\Delta t\sim
17$ ks, we decide to study the variability on a timescale lower than that,
namely: $10$ ks. In order to normalize the excess variance, we divided the
light curve in intervals of $10$ ks and discarded the leftover points up to a
maximum of $9$ ks. We computed $\sigma^{2}_{\rm NXS}$ for each interval and we
adopted the average value between all intervals as the excess variance of the
whole light curve.
Every excess variance value computed for each interval is associated with an
error given by (Vaughan et al., 2003):

${\rm err}(\sigma^{2}_{\rm
NXS})=\sqrt{\left(\sqrt{\frac{2}{N}}\frac{\overline{\sigma^{2}_{\rm
err}}}{\mu^{2}}\right)^{2}+\left(\sqrt{\frac{\overline{\sigma^{2}_{\rm
err}}}{N}}\frac{2F_{\rm var}}{\mu}\right)^{2}},$ (2)

where $\sigma^{2}_{\rm err}=\sum_{i}\sigma^{2}_{i}/N\mu^{2}$ and $F_{\rm
var}=\sqrt{\sigma^{2}_{\rm NXS}}$ is the fractional variability.
Following Ponti et al. (2012), we considered the following three cases. When
only one interval was present for the considered light curve ($\Delta t<20$
ks, one observation in our sample), the error associated with the excess
variance is given by Eq. 2, computed in the single interval available. If the
number of intervals is between $2$ and $9$ (i.e. $20\leq\Delta t<100$ ks, 33
observations), we calculated Eq. 2 for each interval and took its average
value as the global error on the whole light curve. Finally, when the number
of intervals exceeded $10$ (i.e., $\Delta t\geq 100$ ks, 12 observations), the
standard deviation of the excess variance is adopted as error. We caution that
the distribution of $\sigma^{2}_{\rm NXS}$ is Gaussian for an ideal number of
intervals $n\gtrsim 20$ (e.g., Allevato et al., 2013). However, given that
this condition is only satisfied for a total of four observations of two
sources, we opted for the less conservative approach described above.
We computed the excess variance in the full ($3-79$ keV), soft ($3-10$ keV)
and hard ($10-40$ keV) NuSTAR bands, which are listed in Table 3. As shown in
Fig. 1, as expected, the broadband variability in the full band is dominated
by the soft band variability, as most of the signal is detected in this energy
band. Nearly all the points are located in the bisector line. Indeed, we fit
the $\sigma^{2}_{\rm NXS}$ in the two bands in the logarithmic scale, we
obtain a very tight correlation ($r=0.95$ with negligible or null probability)
and we find a slope of $0.96\pm 0.05$, indicating that the variability of the
continuum in the $3-10$ keV band is dominant. We also find a tight correlation
($r=0.90$) between the excess variance between the soft $3-10$ keV and the
hard $10-40$ keV bands, which is however not consistent with the bisector at
$90\%$ confidence level, as we find an angular coefficient of $0.7\pm 0.1$ and
an intercept of $-0.7\pm 0.2$.

Since all three bands are so tightly correlated, in the following sections, we
only used the excess variance derived from the light curves extracted in the
full NuSTAR energy band ($E=3-79$ keV). Full band light curves of every
observation for each AGN of the sample here presented are shown in Fig. D.

##  5 Discussion

###  5.1 Physics of the corona

Figure 2: Temperature versus optical depth plot. The $kT-\tau$ points obtained
assuming a slab geometry are shown in blue, with the blue line indicating the
best-fit line. The red points and line denote the points obtained with a
sphere geometry and their best-fit line, respectively. Confidence intervals
are shown at $90\%$ level. Figure 3: Temperature and optical depth of the
simulated spectra used to validate the correlation found in the NuSTAR data. A
thousand NuSTAR spectra were simulated with random values of $kT$ and $\tau$
(see details in Sect. 5). The left panel show the values used to simulate the
data with average values in bins of temperature (red), while the right panel
shows the temperature and optical depth found when fitting the simulated
spectra, with the average values in temperature bins. Figure 4: We report here
the coronal parameters (kT and $\tau$) $vs$ the black hole mass (left panels)
and the Eddington ratio (right panels) for the sources in our sample (see
Table 1) . The blue dots are the temperatures obtained assuming a slab corona,
while the red dots are those obtained assuming a spherical one. Figure 5:
Excess variance in the $E=3-79$ keV energy band versus the black hole mass of
the AGN (left). The excess variances shown here are averaged among the
different epochs. A clear trend $\sigma^{2}_{\rm NXS}\propto M_{\rm
BH}^{-0.6}$ is found with a strong anticorrelation ($r=-0.7$). Given their
large uncertainties, masses obtained with the velocity dispersion method (red
diamonds) are excluded from the fit. Middle panel gives The average excess
variance vs. the unabsorbed X-ray luminosity $L_{\rm X}$ is shown. We find a
trend $\sigma^{2}_{\rm NXS}\propto L_{\rm X}^{-0.4}$ with moderate
anticorrelation ($r=-0.5$). Right panel shows the average excess variance vs.
the Eddington ratio and no evident correlation is found. Also, in this panel,
the sources with mass measurements obtained with the velocity dispersion
method are drawn as red diamonds. Figure 6: Bolometric (upper panel) and X-ray
(lower panel) luminosities vs. black hole mass of the sources considered in
this work. Both luminosities scale with mass as $M_{\rm BH}^{0.6}$, though the
$L_{\rm bol}$ relation shows a larger correlation coefficient ($r\sim 0.7$)
than the $L_{\rm X}$ relation ($r\sim 0.5$).

We considered the best-fit values of the temperature, $kT,$ and the optical
depth, $\tau,$ for each source, as listed in Table 2. We find a strong
correlation between the coronal temperature and the optical depth, as reported
in Fig. 2. Indeed, the $\tau-kT$ relations have a correlation coefficient of
$r=-0.96$ for the slab geometry, and $r=-0.97$ for the sphere geometry, both
with a negligible probability of finding such correlations by chance. We fit
the linear relation $\log kT=a\log\tau+b$ and found best-fit parameters of
$a=-0.76\pm 0.08$ and $b=1.54\pm 0.05$, assuming a slab geometry. For the
spherical corona, we obtained $a=-0.90\pm 0.09$ and $b=1.91\pm 0.07$. These
results are consistent at $90\%$ confidence level with those obtained by
Tortosa et al. (2018b) with a smaller sample.
It is very important to exclude any possible systematic effect when two
quantities are found to be tightly correlated. To do that, we built a simple
spectrum for an AGN, which is made of neutral absorption modeled with TBABS, a
comptonizing continuum modeled with the physical model COMPTT and a ionized
reflection component modeled with XILLVER, a mock version of model A. We
simulate $1000$ NuSTAR spectra assuming random parameters in predetermined
ranges. We assume that the column density may take any value between $N_{\rm
H}=10^{20}$ cm-2 to $N_{\rm H}=10^{23}$ cm-2. We allowed for a wide range of
photon indices in the reflection spectrum, from a flat spectrum with
$\Gamma=1.45$ to a very steep one with $\Gamma=2.5$. The iron abundance is
taken in the range $A_{\rm Fe}=\left[0.5\;5\right]$, while the ionization
describes a nearly neutral ($\log\xi/({\rm erg\;cm\;s^{-1}})=0$) to a highly
ionized reflector ($\log\xi/({\rm erg\;cm\;s^{-1}})=4.7$). For simplicity, we
tie the reflection normalization to the one of the continuum, assuming a
reflection parameter in the range $\mathcal{R}=\left[-0.1\;-1\right]$, where
the negative values have been adopted in order to simulate pure reflection
spectra. The adopted ranges of coronal temperature ($kT_{e}<500$ keV) and
optical depth ($\tau<10$) are roughly based on the values of Table 2. Finally,
the normalization of the continuum is chosen between $10^{-4}$ counts cm-2 s-1
keV-1 and 5$\times 10^{-2}$ counts cm-2 s-1 keV-1, which corresponds to fluxes
in the range $F_{\rm X}=5\times 10^{-14}-10^{-10}$ erg cm-2 s-1. Each
simulation was run assuming different exposures, between a minimum of $10$ ks
to a maximum of $100$ ks, using simulated background spectra and the latest
NuSTAR response and effective area. As shown in Fig. 3, in the adopted large
range of parameters, the average value of the best-fit parameters, $kT,$ and
$\tau$ (right) are not much different than the simulated ones (left); the
exception is the last bin ($kT>450$ keV), where the optical depth tends to be
slightly overestimated, due to the lower sensitivity of NuSTAR to high-
temperature coronae. The simulations were designed with uncorrelated
$kT-\tau$; therefore, if a spurious correlation was indeed present, it would
also be expected to be present in the right panel of Fig. 3. This suggests
that from a statistical point of view, the correlation found in the data is
not spurious. It is worth noting that the sample has been selected to be not
strongly contaminated from complex, ionized, and multiple soft X-ray absorbers
(see Sect. 2); thus, we did not expect for the model systematics to contribute
to affecting the continuum best-fit parameters.
As discussed by Tortosa et al. (2018b), the observed $kT-\tau$ anticorrelation
is not consistent with a global disk-corona configuration in radiative
balance, which would imply the same heating-to-cooling ratio (HCR) for the
coronae of all the AGN in the sample. One possibility to explain this
anticorrelation is that colder coronae are more compact. In this case, a more
compact corona would imply a larger optical depth, where the number of
scatterings is larger. This will lead to a more efficient cooling (smaller
HCR), and vice versa, a less compact corona would imply a smaller number of
scattering, resulting in a larger HCR. Another possibility is that the disk-
corona configuration is the same for all sources, but the sources might show
different thermal emission due to viscous dissipation over the whole disk
emission, which would also result in a larger cooling efficiency.
We also investigate a possible dependence of the coronal temperature from the
physical properties of the AGN, such as the black hole mass, $M_{\rm BH}$, and
the Eddington ratio, $\lambda=L_{\rm bol}/L_{\rm Edd}$. As described in Sect.
2, the masses and bolometric luminosities were retrieved from the catalog
published by Koss et al. (2022), with the sole exception of Ark 564, whose
mass and bolometric luminosity were retrieved from Peterson et al. (2004) and
adopting the bolometric correction from Marconi et al. (2004) to the X-ray
luminosity obtained from the present data, respectively. As shown in Fig. 4,
there is no indication of a possible correlation between $kT$ or $\tau$ with
either the black hole mass or the Eddington ratio. This is consistent with
past studies of coronal parameters (e.g., Tortosa et al., 2018b; Kamraj et
al., 2022). However, we note that Ricci et al. (2018) found a trend between
$\lambda_{\rm Edd}$ and the cut-off energy, $E_{c}$, when considering a very
large sample of sources.

###  5.2 X-ray variability and coronal properties

Figure 7: Excess variance in the $E=3-79$ keV energy band versus the coronal
temperature (left panel) and the optical depth (right panel) of the sample
analyzed in this work. Blue circles identify temperature and optical depth
values obtained assuming a slab geometry, while the red triangles indicate the
values obtained assuming a spherical geometry.

We investigate a possible correlation between the excess variance in the full
NuSTAR band ($E=3-79$ keV) and the mass of the black hole (Fig. 5, left
panel). For this purpose, we exclude the sources for which Koss et al. (2022)
report the mass value from the velocity dispersion of the stars in the host
galaxy, as they are affected by large uncertainties. We find that the two
quantities are well correlated with a correlation coefficient of $r=-0.7$ and
a null probability of $p_{\rm null}\simeq 10^{-5}$. If we perform a linear fit
on the logarithmic quantities, namely, $\log\sigma^{2}_{\rm NXS}=a\log(M_{\rm
BH}/10^{5}M_{\odot})+b$, we obtain $a=-0.6\pm 0.2$ and $b=-1.2\pm 0.7$. An
anticorrelation was also found in previous variability analyses with XMM-
Newton (e.g., Ponti et al., 2012; Tortosa et al., 2023). We tested whether the
excess variance is dependent on the X-ray luminosity, $L_{\rm X}$, and we
found a moderate anticorrelation $r=-0.5$ ($p_{\rm null}\simeq 0.02$), with
the two quantities related as $\sigma^{2}_{\rm NXS}\propto L_{\rm X}^{a}$
(Fig. 5, middle panel) with $a=-0.4\pm 0.3$. Despite not being highly
significant, this relation is also found in other analyses with a much larger
number of sources (e.g., Vagnetti et al., 2016; Prokhorenko et al., 2024). We
also report an anticorrelation ($r\sim-0.7$ and $p_{\rm null}\sim 10^{-5}$)
with the bolometric luminosity ($\sigma^{2}_{\rm NXS}\propto L_{\rm bol}^{a}$
with $a=-0.6\pm 0.3$). As shown in Fig. 6, though, both $L_{\rm X}$ and
$L_{\rm bol}$ are correlated with the black hole mass, scaling as $\sim M_{\rm
BH}^{0.6}$, with correlation coefficient $r\sim 0.5$ ($p_{\rm null}\sim 0.02$)
and $r\sim 0.7$ ($p_{\rm null}\sim 10^{-5}$), respectively. Therefore it is
likely that the two luminosity relations are degenerate with the black hole
mass relation. Alternatively, the variability-luminosity relation has often
been attributed to a superposition of small flares (e.g., Nandra et al., 1997;
Almaini et al., 2000).
We also test the possible relation between the excess variance and the
Eddington ratio derived in Sect. 5.1 (see Fig. 5, right panel), finding a
correlation coefficient $r=-0.03$, with a probability of finding such
correlation by chance $p_{\rm null}=0.3$. This suggests a lack of correlation
between the Eddington ratio and the X-ray variability. We also tried to remove
the previously described dependency from the mass, checking a possible
correlation between the quantity $\sigma^{2}_{\rm NXS}\times M_{\rm
BH}^{0.6}$, but this quantity is also uncorrelated with the Eddington ratio
values.
We analyzed the possible correlation between the excess variance and the
coronal parameters of the 20 sources analyzed here. We do not find a relevant
dependence between the temperature obtained with both geometries and the
excess variance in the $3-79$ keV energy band, as the correlation coefficient
is $r=-0.02$, with null probability $p_{\rm null}=0.3$, and the angular
coefficient is consistent with a flat line (see Fig. 7, left panel). For
completeness, we also show that the excess variance and the optical depth are
not correlated ($r=0.01$, $p_{\rm null}=0.35$, see Fig. 7, right panel). This
result was expected once the $\sigma-kT$ relation is found to be absent, given
the tight $kT-\tau$ correlations found in Sect. 5.1.
These results suggest that the X-ray variability on timescales of $10$ ks is
not dependent on the coronal temperature nor the optical depth, raising
questions about the origin of the X-ray variability. Indeed, the corona must
introduce variability at timescales shorter than the ones observed for the UV
radiation, which provides the seed photons for the Comptonization.
Additionally, the corona must be responsible for the X-ray variability,
because the UV light curves lag behind the X-ray ones (e.g., Kammoun et al.,
2021).
We note that, so far, we have considered the corona to be in thermal
equilibrium, meaning that we have computed average values of the temperature
and optical depth over the whole observation, which is indeed larger than the
timescale for which we have probed the variability. Therefore, one possibility
is that the X-ray variability could be driven by changes of the temperature
and optical depth at timescales that are consistent with the observed flux
variations (i.e., below $10$ ks). However, measuring time-resolved
temperatures is not yet possible with the currently most advanced hard X-ray
telescope NuSTAR. Moreover, the physical environment could be much more
complex than its simplification above. For instance, geometry is undoubtedly a
parameter that could play a major role in driving the X-ray variability. In
fact, variations in the intrinsic geometry or disk-corona geometry, such as
the ones observed in X-ray binaries (e.g., Kara et al., 2019), could also
drive the X-ray variability. It would be expected that, following a geometry
variation, the temperature and optical depth would change accordingly, but
these variations may also happen at shorter timescales than the ones probed by
the spectral fits, possibly even shorter than 10 ks.
An additional complexity to take into account is a possible spatial (likely
radial) distribution of the temperature of the corona, while we have assumed
an average single value for the whole electron plasma. The spatial and
temporal average values of the temperature may spread out the link between the
best-fit results and the calculations of the variability.
Another possibility is that the variability is mainly driven by the observed
anticorrelation with the black hole mass, which is likely proportional to the
absolute size of the corona. If we consider, for instance, that all coronae of
the AGN in the sample have a typical coronal radius of $R_{c}=10\;R_{g}$
(e.g., Dovčiak & Done, 2016; Ursini et al., 2020a), where $R_{g}=GM_{\rm
BH}/c^{2}$ is the gravitational radius, the coronal size in physical units
would be directly proportional to the supermassive black hole mass. However,
we note that the size of the coronae in $R_{g}$ units could differ from source
to source, as discussed in the previous section; furthermore, the relation
between coronal size and black hole mass, although still increasing, it may be
far from trivial to derive. In any case, even considering more complex
relations between the coronal size and the black hole mass, more massive black
holes correspond to larger coronae, which would imply a larger number of
random scattering of the seed photons in the corona, resulting in a smaller
X-ray variability amplitude.

##  6 Summary and conclusions

We have presented a spectral and timing study of 20 bright AGN with the best
signal-to-noise ratio available basedon NuSTAR data. We measured the
temperature, $kT_{e}$, and optical depth, $\tau,$ of the X-ray emitting corona
under two different geometries (sphere and slab) by modeling the spectrum
continuum with the Comptonization model COMPTT. Additionally, we have studied
the NuSTAR variability in the time range between 1 and 10 ks by means of the
excess variance $\sigma^{2}_{\rm NXS}$. We note that the results of this paper
are related to the sample at hand, which is the highest quality data to date.
We summarize our results in the following

  * •

We report that there is no correlation found between the X-ray variability and
the electron temperature or the optical depth of the corona. This may imply
that the X-ray variability is dependent on $kT$ and $\tau$ variations on
timescales below $10$ ks, which is the timescale probed by the variability in
this paper.

  * •

We did find a strong anticorrelation between $kT$ and $\tau,$ adopting both
slab and spherical geometries. In particular, we find that the optical depth
is related to the temperature by a relation $kT\propto(\tau)^{-a}$ with
$a\sim-0.7\div-1$ (Fig. 2), depending on the assumed geometry. Therefore, we
confirm the trend found by Tortosa et al. (2018b) with our AGN sample size
increased by a factor of 2 with respect to the cited work.

  * •

We did not find any dependence of $kT_{e}$ and $\tau$ with either the black
hole mass or Eddington ratio. This is also consistent with previous results on
other samples of AGN (e.g., Tortosa et al., 2018b; Kamraj et al., 2022).

  * •

There is a strong correlation between the $3-10$ keV and $3-79$ keV, which
implies that the variability of the X-ray emission below $10$ keV is dominant
over the rest of the spectrum. We also find a strong correlation between the
variability of the $10-40$ keV band and the one in $3-10$ keV.

  * •

We found a strong anticorrelation between the X-ray variability and the mass,
following $\sigma^{2}_{\rm NXS}\propto M^{-0.6}$. This correlation is
consistent to the one found by past variability studies with XMM-Newton (e.g.,
Ponti et al., 2012; Tortosa et al., 2023). We also report a moderate
correlation with the X-ray luminosity $\sigma^{2}_{\rm NXS}\propto L_{\rm
X}^{-0.4}$, as well as no correlation seen with the Eddington ratio.

The results of our study show that the main driver of the X-ray continuum
variability produced in the hot-corona remains elusive; furthermore, it is not
even clear whether there is a main driver for the observed variability at all
and it may instead be the product of the superposition of several effects at
work. We have shown here that the variability at $10$ ks does not depend on
the physical properties of the corona, namely, electron temperature and
optical depth. This then raises the question of what drives the X-ray
variability. One possibility is that we might be probing different timescales,
since we are studying relatively fast variability within $10$ ks; on the other
hand, we have averaged the coronal temperature over days, months, and even
years to reach a sufficient signal-to-noise ratio that would allow for an
accurate measurement of the coronal temperature. Variations in the coronal
geometry may also play an important role in producing the observed
variability. In the future, detectors sensitive in a broadband energy range
with much larger effective area, such as the Large Area Detector (LAD; e.g.,
Feroci et al., 2022) proposed for the future enhanced X-ray Timing and
Polarimetry mission (eXTP; Zhang et al., 2019) and the Spectroscopic Time-
Resolving Observatory for Broadband Energy X-rays (STROBE-X; Ray et al.,
2018), will allow us to measure AGN coronal temperatures with high precision
for exposures as short as 10 ks (De Rosa et al., 2019). This would open up the
possibility to probe both the coronal parameters and variability on the same
timescale. Moreover, thanks to the extremely broadband $E=0.2-80$ keV, now
only available with joint NuSTAR observations with other facilities such as
XMM-Newton, the future X-ray telescope HEX-P (Kammoun et al., 2024) will be
able to measure optical depth and temperatures with much more accuracy than
NuSTAR, with much shorter exposures.

###### Acknowledgements.

We thank the referee for their comments, which improved the quality of this
paper. The authors thank Iossif Papadakis and Julien Malzac for useful
discussions on the results of this paper. RS and ADR acknowledge support from
the agreements ASI-INAF n.2017-14-H.0 ”Science case study and scientific
simulations for the enhanced X-ray Timing Polarimetry mission, eXTP”, ASI-INAF
eXTP Fase B-2020-3-HH.1-2021, Bando Ricerca Fondamentale INAF 2022 Large Grant
”Dual and binary supermassive black holes in the multi-messenger era: from
galaxy mergers to gravitational waves” and the INAF-PRIN grant “A Systematic
Study of the largest reservoir of baryons and metals in the Universe: the
circumgalactic medium of galaxies” (No. 1.05.01.85.10). AT acknowledges
financial support from the Bando Ricerca Fondamentale INAF 2022 Large Grant
”Toward an holistic view of the Titans: multi-band observations of z ¿ 6 QSOs
powered by greedy supermassive black holes”. SB acknowledges funding from PRIN
MUR 2022 SEAWIND 2022Y2T94C, supported by European Union - Next Generation EU,
from INAF LG 2023 BLOSSOM, and from the EU grant AHEAD-2020 (GA no. 871158).
CR acknowledges support from the Fondecyt Regular grant 1230345 and ANID BASAL
project FB210003. POP acknowledges financial support from the French space
agency (CNES) and the National High Energy Programme (PNHE) of CNRS. This
research has made use of data and software provided by the High Energy
Astrophysics Science Archive Research Center (HEASARC), which is a service of
the Astrophysics Science Division at NASA/GSFC and the High Energy
Astrophysics Division of the Smithsonian Astrophysical Observatory. This
research has made use of the NuSTAR Data Analysis Software (NUSTARDAS) jointly
developed by the ASI Space Science Data Center (SSDC, Italy) and the
California Institute of Technology (Caltech, USA).

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##  Appendix A Data

Table 4: NuSTAR spectra analyzed in this work.

Source | OBSID | Date | Exp. (s)
---|---|---|---
1H0419-577 | 60101039002 | 2015-06-03 | 169462
| 60402006002 | 2018-05-15 | 64216
| 60402006004 | 2018-11-13 | 48273
4C 50.55 | 60061305002 | 2014-12-13 | 24281
| 60301005002 | 2018-01-02 | 40338
Ark 564 | 60101031002 | 2015-05-22 | 211209
| 60401031002 | 2018-06-09 | 38090
| 60401031004 | 2018-11-28 | 408958
ESO 103-G35 | 60061288002 | 2013-02-24 | 27391
| 60301004002 | 2017-10-15 | 43834
ESO 362-G18 | 60201046002 | 2016-09-24 | 101905
ESO 383-G18 | 60061243002 | 2014-09-11 | 17342
| 60261002002 | 2016-01-20 | 106576
GRS 1734-292 | 60061279002 | 2014-09-16 | 20288
| 60301010002 | 2018-05-28 | 26020
HE 1143-1810 | 60302002002 | 2017-12-16 | 20960
| 60302002004 | 2017-12-18 | 20838
| 60302002006 | 2017-12-20 | 23096
| 60302002008 | 2017-12-22 | 20716
| 60302002010 | 2017-12-24 | 22378
IC 4329A | 60001045002 | 2012-08-12 | 162390
MCG-5-23-16 | 10002019001 | 2012-07-11 | 33925
| 60001046002 | 2013-06-03 | 160469
| 60001046004 | 2015-02-15 | 210887
| 60001046006 | 2015-02-21 | 98459
| 60001046008 | 2015-03-13 | 220835
MCG+8-11-11 | 60201027002 | 2016-08-16 | 97921
Mrk 6 | 60102044002 | 2015-04-21 | 62472
| 60102044004 | 2015-11-09 | 43816
Mrk 110 | 60201025002 | 2017-01-23 | 184563
| 60502022002 | 2019-11-16 | 86772
| 60502022004 | 2020-04-05 | 88674
Mrk 509 | 60101043002 | 2015-04-29 | 165885
| 60101043004 | 2015-06-02 | 36474
NGC 3281 | 60061201002 | 2016-01-22 | 22986
| 60662003002 | 2020-07-15 | 24616
NGC 5506 | 60061323002 | 2014-04-01 | 56585
NGC 5728 | 60061256002 | 2013-01-0 | 24357
| 60662002002 | 2020-07-13 | 24923
NGC 6814 | 60201028002 | 2016-07-04 | 148428
SWIFT |  |  |
J2127.4+5654 | 60001110002 | 2012-11-04 | 49200
| 60001110003 | 2012-11-05 | 28764
| 60001110005 | 2012-11-06 | 74578
| 60001110007 | 2012-11-08 | 42106
UGC 6728 | 60160450002 | 2016-07-10 | 22615
| 60376007002 | 2017-10-13 | 58077
777The table lists source names with OBSID, observation date, and exposure.

##  Appendix B Spectral fits

1H0419-577
This is a Seyfert 1 galaxy at $z=0.104$, with a black hole mass of $M_{\rm
BH}=3.8\times 10^{8}M_{\odot}$. We binned the spectra at a minimum of $100$
counts per energy bin over the whole $3-79$ keV energy range. A Galactic
absorption with column density $N_{\rm H,Gal}=1.14\times 10^{20}$ cm-2 is
adopted. The spectra are well fitted by an absorbed continuum plus non-
relativistic reflection (model A, ztbabs*(comptt+xillvercp)). We let the
intrinsic absorption column density $N_{\rm H}$ to vary between observations.
We find $N_{\rm H}=(6\pm 4)\times 10^{21}$ cm-2 for the 2015 observation and
$N_{\rm H}=(1.0\pm 0.5)\times 10^{22}$ cm-2 for both the 2018 observations.
When we assume a slab geometry (parameter approx$=0.5$) we obtain
$kT=14^{+2}_{-1}$ keV and an optical depth $\tau=2.5^{+0.2}_{-0.3}$, while
when a sphere geometry (approx$=1.5$) is adopted we find the same temperature
and an optical depth of $\tau=5.7^{+0.4}_{-0.5}$. We find a goodness of fit
$\chi^{2}/{\rm dof}=1566/1588$. The results are consistent with the broadband
analysis performed by Turner et al. (2018), although we note that in that work
the OPTXAGNF model (Done et al. 2012) was used.

4C 50.55
4C 50.55 is a radio-loud Seyfert 1 galaxy at redshift $z=0.02$, yielding a
black hole mass of $6.4\times 10^{7}$ $M_{\odot}$. We bin the NuSTAR spectra
at a minimum of 100 counts per energy bin and consider the full $E=3-79$ keV
energy band. We note that Tazaki et al. (2010) ruled out a significant role of
the jet in the X-ray spectrum of this source using Suzaku. We consider a
Galactic column of $N_{\rm H,Gal}=9.45\times 10^{21}$ cm-2. The source is well
fitted by an absorbed continuum plus a single disk reflector, i.e.
ztbabs*(comptt+xillvercp) (model A). The column densities of the two
observations are $N_{\rm H}=(3.0\pm 0.4)\times 10^{22}$ cm-2 and $N_{\rm
H}=(2.0\pm 0.3)\times 10^{22}$ cm-2 for the 2014 and 2018 observations,
respectively. We find a temperature of $kT=18^{+5}_{-2}$ keV
($kT=18^{+3}_{-2}$ keV) and optical depth of $\tau=2.2^{+0.2}_{-0.3}$
($\tau=5.2^{+0.4}_{-0.5}$) assuming a slab (sphere) geometry. The statistic is
$\chi^{2}/{\rm dof}=1325/1243$. These results are consistent with the analysis
of Buisson et al. (2018).

Ark 564
Ark 564 is a Narrow-Line Seyfert 1, with mass $M_{\rm BH}=3.2\times
10^{6}M_{\odot}$ (Peterson et al. 2004). The NuSTAR spectra have been binned
at a minimum of $100$ counts per energy bin and only considered in the
$E=3-30$ keV energy range. A Galactic column of $N_{\rm H,Gal}=5\times
10^{20}$ cm-2 is assumed. A simple power law model unveils a single narrow Fe
K$\alpha$ component and a reflection component in the residual spectra.
Therefore, the reflection is modeled with model A (compTT+xillvercp), with no
absorption needed. Assuming a slab (sphere) geometry we obtain $kT=15\pm 2$
keV ($16\pm 1$ keV) and an optical depth of $\tau=1.4\pm 0.1$ ($3.3\pm 0.3$).
The statistic is $\chi^{2}/{\rm dof}=1267/1127$. A detailed broadband spectral
analysis, which is consistent with the results presented here, can be found in
Kara et al. (2017).

ESO 103-G35
ESO 103-G35 is a Seyfert 2 galaxy at redshift $z=0.00914$ with a black hole
mass of $M_{\rm BH}=1.3\times 10^{7}$ $M_{\odot}$. The four FPM spectra were
binned at a minimum of 100 counts per energy bin in the $E=3-50$ keV energy
band. We consider a Galactic absorption of $N_{\rm H,Gal}=5.8\times 10^{20}$
cm-2. The source X-ray spectra appear reflection-dominated due to a heavy
obscuration, therefore we adopt a model consisting of a distant absorber plus
continuum, with intrinsic absorption on the line of sight, i.e.
ztbabs*(comptt+borus12) (model C). The two observations are characterized by
nearly the same Compton-thin absorbing column density, which is consequently
kept tied between epochs, of $N_{\rm H}=(1.65\pm 0.08)\times 10^{23}$ cm-2.
The temperature and optical depth assuming a slab geometry are
$kT=17^{+18}_{-3}$ keV and $\tau=2.2^{+0.3}_{-0.7}$, respectively, while the
two parameters assuming a sphere geometry are $kT=17^{+7}_{-3}$ keV and
$\tau=5.2\pm 0.7$. The goodness of fit is $\chi^{2}/{\rm dof}=1320/1302$. The
results are consistent with the analysis of Buisson et al. (2018), in which a
different reflection model was used.

ESO 362-G18
ESO 362-G18 is a Seyfert 1.5 galaxy at redshift $z=0.01244$ with a black hole
mass $M_{\rm BH}=4.5\times 10^{7}$ $M_{\odot}$. The spectra were binned at a
minimum of $50$ photon counts per energy bin in the full $E=3-79$ keV NuSTAR
energy band. We consider a Galactic absorption of $N_{\rm H,Gal}=1.35\times
10^{20}$ cm-2. The X-ray spectrum has a typical Seyfert 1 shape with no
evident intrinsic absorption. Following Agís-González et al. (2014) we model
the spectrum with two reflection components, i.e. comptt+xillvercp+relxillcp
(model B). For a slab (sphere) coronal geometry we obtain a temperature of
$kT=18^{+14}_{-4}$ keV ($kT=18^{+15}_{-4}$ keV) and an optical depth of
$\tau=2.5^{+0.4}_{-0.9}$ ($\tau=5.2\pm 0.7$). The goodness of fit is given by
$\chi^{2}/{\rm dof}=464/641$.

ESO 383-G18
ESO 383-G18 is a Seyfert 2 galaxy with $z=0.01241$, with $M_{\rm BH}=3\times
10^{5}$ $M_{\odot}$. We bin the spectra of OBSID 60061243002 at a minimum of
50 counts per energy bin and the spectra of OBSID 60261002002 at a minimum of
100 counts per energy bin. We consider the $E=3-50$ keV energy band for both
observations. We assume a Galactic absorption of $N_{\rm H,Gal}=3.8\times
10^{20}$ cm-2. The X-ray spectrum appears severely absorbed by a intervening
cold material, which suggests that the reflector is best modeled by distant
cold material, i.e. ztbabs*(comptt+borus12) (model C). The absorbing column
density of the two observations is consistent within the error at 90$\%$
confidence level, therefore we decided to keep them tied and obtain $N_{\rm
H}=(1.2\pm 0.2)\times 10^{23}$ cm-2. The temperature is identical assuming
both assuming a slab or a spherical corona, i.e. $kT=7.3^{+0.5}_{-0.4}$ keV,
while instead we find $\tau=4.7\pm 0.3$ for a slab geometry and $\tau=10.1\pm
0.6$ for a spherical one. The goodness of fit is $\chi^{2}/{\rm
dof}=1035/1054$.

GRS 1734-292
GRS 1734-292 is a Seyfert 1 galaxy, having mass $M_{\rm BH}=2.5\times
10^{8}M_{\odot}$. We binned each FPM module of the two NuSTAR epochs at a
minimum of 100 counts per energy bin, and considered the full $E=3-79$ keV
energy band. We adopt a Galactic absorption of $N_{\rm H,Gal}=6.5\times
10^{21}$ cm-2. The source is also intrinsically moderately absorbed, with
$N_{\rm H}=4^{+4}_{-3}\times 10^{21}$ cm-2 for OBSID 60061279002 and $N_{\rm
H}=(15\pm 4)\times 10^{21}$ cm-2 for OBSID 60301010002. An absorbed powerlaw
only show narrow Fe K$\alpha$ residuals, therefore we adopt model A plus
absorption (ztbabs(compTT+xillvercp)). We obtain a coronal temperature of
$kT=13^{+2}_{-1}$ keV for both slab and sphere geometries. An optical depth of
$\tau=2.9\pm 0.2$ ($6.5\pm 0.4$) is recovered in a slab (sphere) geometry. We
obtain a statistic of $\chi^{2}/{\rm dof}=1103/983$. These results are
consistent with the broadband analysis presented in Tortosa et al. (2017), in
which OBSID 60061279002 was studied.

HE 1143-1810
HE 1143-1810 is a Seyfert 1 galaxy at redshift $z=0.0329$ with an estimated
black hole mass $M_{\rm BH}=4\times 10^{7}$ $M_{\odot}$. The 10 FPM spectra
were all binned at a minimum of 100 photon counts per energy bin in the
$E=3-79$ keV energy band. The Galactic absorption is fixed at $N_{\rm
H,Gal}=3\times 10^{20}$ cm-2. The spectrum has a typical unabsorbed shape,
therefore no neutral absorption is needed. The source is nicely modeled by a
single reflector plus continuum, i.e. comptt+xillvercp (model A). Assuming a
slab (spherical) coronal geometry we find a temperature of $kT=36^{+64}_{-19}$
keV ($kT=22^{+61}_{-5}$ keV) and an optical depth of $\tau=1.2^{+1.1}_{-0.8}$
($\tau=4.4^{+0.9}_{-2.9}$). The goodness of fit is $\chi^{2}/{\rm
dof}=2639/2665$. These values are largely in agreement with the broadband
analysis presented in Ursini et al. (2020b).

IC 4329A
IC 4329A is a Seyfert 1 galaxy with mass $M_{\rm BH}=1.3\times 10^{8}$. We bin
both FPM modules of the analyzed observation (OBSID 60001045002) at a minimum
of 100 counts per energy bin. We consider the full NuSTAR band, i.e. $E=3-79$
keV. The Galactic absorption is $N_{\rm H,Gal}=4\times 10^{20}$ cm-2. A modest
absorption of $N_{\rm H}=(5\pm 2)\times 10^{21}$ cm-2 is included. A single
distant reflector is needed, therefore we adopt model A. The coronal
temperature for a slab (sphere) coronal geometry is $kT=44^{+20}_{-10}$ keV
($44^{+17}_{-11}$ keV) and the optical depth is $\tau=1.1\pm 0.3$ ($2.8\pm
0.7$). The goodness of fit is $\chi^{2}/{\rm dof}=1411/1298$. A broadband
spectral analysis is presented in Brenneman et al. (2014) and Ingram et al.
(2023), with consistent results with our NuSTAR fits.

MCG-5-23-16
MCG-5-23-16 is a Seyfert 1.9 galaxy with a mass of $M_{\rm BH}=1.8\times
10^{7}M_{\odot}$. All FPMA and FPMB spectra of the five available observations
are binned at $100$ counts per energy bin and the $3-79$ keV energy range is
considered. We assume a Galactic absorption of $N_{\rm H,Gal}=8\times 10^{20}$
cm-2 and we include cold intrinsic absorption with the model ztbabs, finding a
column density $N_{\rm H}=(1.4\pm 0.1)\times 10^{22}$ cm-2. We the source is
modeled with a single reflector plus continuum with xillvercp, we find
significant residuals on the Fe K$\alpha$ region, strongly hinting at the
presence of a broad component, likely a reflection component from the
accretion disk. Therefore, we adopt the absorbed model B,
ztbabs(compTT+xillvercp+relxillcp). Following Serafinelli et al. (2023b), we
assume an emissivity index of $-3$, an inner radius $R_{\rm in}=40R_{g}$,
inclination $45^{\circ}$, and a non-spinning black hole. Assuming a slab
geometry, we find a coronal temperature $kT=44\pm{11}$ keV and an optical
depth $\tau=0.9\pm 0.2$, while $kT=35^{+10}_{-11}$ keV and $\tau=2.7\pm 0.3$
are found if a spherical coronal geometry is assumed. The goodness of fit is
$\chi^{2}/{\rm dof}=6995/5776$. These values are consistent at $99\%$
confidence level with the broadband analyses of Baloković et al. (2015) and
Serafinelli et al. (2023b) performed on this source.

MCG+8-11-11
MCG+8-11-11 is a Seyfert 1 galaxy with mass $M_{\rm BH}=1.6\times
10^{7}M_{\odot}$. The FPMA and FPMB spectra were binned at 100 counts per
energy bin over the whole energy range ($E=3-79$ keV), and a Galactic column
density of $N_{\rm H}=1.75\times 10^{21}$ cm-2 is assumed. No intrinsic cold
absorption is present. Two reflection components are required in the model, as
the Fe K$\alpha$ emission line is not adequately fitted by a single narrow
component. We adopt therefore model B, compTT+xillvercp+relxillcp. We assume a
non-spinning black hole and a disk with inner radius $R_{\rm in}=2R_{g}$,
where $R_{g}=GM/c^{2}$ is the gravitational radius, and emissivity index $-3$.
We find a coronal temperature of $kT=110^{+140}_{-40}$ keV and an optical
depth $\tau=0.3\pm 0.2$ when assuming a slab coronal geometry. When a
spherical geometry is assumed for the corona, we obtain $kT=90^{+125}_{-45}$
keV and $\tau=1.2^{+0.7}_{-0.4}$. The goodness of fit is $\chi^{2}/{\rm
dof}=873/851$. These values are consistent with the results of Tortosa et al.
(2018a), where this data was analyzed with contemporaneous Swift-XRT data.

Mrk 6
Mrk 6 is a Seyfert 1.5 galaxy at $z=0.01951$ with mass $M_{\rm BH}=1.9\times
10^{7}$ $M_{\odot}$. All four FPM spectra were binned at a minimum of 100
counts per energy bin. We consider the $E=3-50$ keV energy band. The X-ray
spectrum appear severely absorbed as in previous analyses of this source
(e.g., Molina et al. 2019). Therefore, once the Galactic absorption is modeled
with a fixed $N_{\rm H,Gal}=7.6\times 10^{20}$ cm-2, we model the spectra with
an absorbed continua and a distant neutral reflector, i.e.
ztbabs*(comptt+borus12) (model C). For OBSID 60102044002 and 60102044004 we
find an intrinsic column density $N_{\rm H}=(1.2\pm 0.1)\times 10^{23}$ cm-2
and $N_{\rm H}=(1.0\pm 0.1)\times 10^{23}$ cm-2, respectively. The temperature
is $kT=13^{+4}_{-2}$ keV both assuming a slab or a spherical coronal geometry,
while the optical depth is $\tau=3.4^{+0.4}_{-0.6}$ and
$\tau=7.5^{+0.7}_{-1.1}$, respectively. The statistic is $\chi^{2}/{\rm
dof}=1283/1335$. A few moderate residuals are present below $5$ keV, possibly
due to the presence of a warm absorber (Kayanoki et al. 2023), but given the
low energy resolution of NuSTAR in the $3-5$ keV energy range, we did not
model such a component.

Mrk 110
Mrk 110 is a Seyfert 1 galaxy with redshift $z=0.03552$ and mass $M_{\rm
BH}=1.8\times 10^{7}$ $M_{\odot}$. The FPM spectra are all binned at a minimum
of 100 counts per energy bin in the full $E=3-65$ keV energy range. We adopt a
Galactic absorption of $N_{\rm H,Gal}=1.3\times 10^{20}$ cm-2, while no
intrinsic absorption is present confirming its nature as a ”bare” AGN (Reeves
et al. 2021; Porquet et al. 2021). We model the X-ray spectra with two
reflectors, i.e. comptt+xillvercp+relxillcp (model B) finding
$kT=35^{+15}_{-10}$ keV ($kT=24^{+17}_{-5}$ keV) and $\tau=1.2^{+0.5}_{-0.4}$
($\tau=4.0^{+0.6}_{-1.4}$) when assuming a slab (sphere) coronal geometry. The
goodness of fit is $\chi^{2}/{\rm dof}=2600/2305$. The results agree with the
broadband analysis presented in Porquet et al. (2021).

Mrk 509
Mrk 509 is a Seyfert 1 galaxy at $z=0.01951$ with a black hole mass of $M_{\rm
BH}=2\times 10^{8}$ $M_{\odot}$. The FPMA and FPMB spectra for the two epochs
were binned at a minimum of 100 photon counts per energy bin in $E=3-65$ keV.
The Galactic absorption is fixed at $N_{\rm H,Gal}=3.9\times 10^{20}$ cm-2. We
model the X-ray spectra with an unabsorbed continuum with a single ionized
reflector, i.e. comptt*xillvercp (model A). We find a temperature of
$kT=17^{+2}_{-1}$ keV for both slab and sphere geometries, while the optical
depth is $\tau=2.2\pm 0.1$ ($\tau=5.2^{+0.2}_{-0.3}$) when assuming a slab
(sphere) geometry. The statistic is $\chi^{2}/{\rm dof}=1949/1647$.

NGC 3281
NGC 3281 is a Seyfert 2 galaxy with redshift $z=0.01073$, with mass $M_{\rm
BH}=1.7\times 10^{8}$ $M_{\odot}$. We bin the FPM spectra with a minimum of 50
counts per energy bin in the energy band $E=3-60$ keV. We assume a Galactic
absorption of $N_{\rm H,Gal}=6.6\times 10^{20}$ cm-2. The shape of the X-ray
spectrum is that of a severely absorbed source, therefore we model the spectra
with an absorbed continuum plus neutral reflection, i.e.
ztbabs*(comptt+borus12) (model C). We find that the intrinsic column density
is moderately variable, by a factor of $\sim 4$, since we find $N_{\rm
H}=8^{+6}_{-5}\times 10^{22}$ cm-2 for OBSID 60061201002 and $N_{\rm
H}=(3.1\pm 0.5)\times 10^{23}$ cm-2 for OBSID 60662003002. Assuming either a
slab or a spherical geometry we find a temperature $kT=11^{+4}_{-2}$ keV, with
optical depth $\tau=3.7^{+0.8}_{-0.9}$ and $\tau=8\pm 2$, respectively. The
goodness of fit is $\chi^{2}/{\rm dof}=693/612$.

NGC 5506
NGC 5506 is a Seyfert 1 galaxy with mass $M=8.8\times 10^{7}$ $M_{\odot}$. The
NuSTAR Focal Plane Modules were both binned at a minimum of 100 counts per
energy bin in the full NuSTAR band $E=3-79$ keV. The Galactic cold absorption
is $N_{\rm H,Gal}=4.2\times 10^{20}$ cm-2, and we recover an intrinsic
absorption of $N_{\rm H}=(2.4\pm 0.3)\times 10^{22}$ cm-2. A single reflector
is able to model the hard X-ray spectrum of this AGN, therefore we adopt model
A with neutral absorption (ztbabs(compTT+xillvercp)). When a slab geometry is
assumed, we obtain $kT=510^{+250}_{-150}$ keV and $\tau=0.02\pm 0.01$, while
assuming a sphere geometry the best-fit parameters are $kT=550\pm 250$ keV and
$\tau=0.09^{+0.30}_{-0.05}$. The goodness of fit is $\chi^{2}/{\rm
dof}=819/738$ and the best-fit values are largely in agreement with the
broadband analysis performed in Matt et al. (2015).

NGC 5728
NGC 5728 is a Seyfert 2 galaxy at $z=0.00947$ with an estimated black hole
mass of $M_{\rm BH}=3.4\times 10^{7}$ $M_{\odot}$. We bin all the X-ray
spectra at a minimum of 50 counts per energy bin. We consider We adopt a fixed
Galactic absorption column density of $N_{\rm H,Gal}=7.5\times 10^{20}$ cm-2.
The spectrum appears as that of a typical absorbed source, therefore we model
it with an absorbed continuum plus neutral reflection, i.e.
ztbabs*(comptt+borus12) (model C). The intrinsic absorption is not variable
and therefore we keep it tied between the two epochs, finding $N_{\rm
H}=4.6^{+1.6}_{-1.8}\times 10^{23}$ cm-2. We find a coronal temperature of
$kT=13\pm 1$ keV for both slab and spherical geometry and an optical depth of
$\tau=5^{+2}_{-1}$ ($\tau=10^{+6}_{-3}$) for a slab (spherical) coronal
geometry. The statistic is given by $\chi^{2}/{\rm dof}=480/524$.

NGC 6814
NGC 6814 is a Seyfert 1 galaxy with mass $M=2.7\times 10^{6}$ $M_{\odot}$. We
bin both FPMA and FPMB detectors at a minimum of 100 counts per energy bin in
the $E=3-60$ keV energy band. We assume a Galactic absorption with column
density $N_{\rm H,Gal}=8\times 10^{20}$ cm-2, while no intrinsic cold
absorption is found in this source. The spectra need to be modeled with model
B (comptt+xillvercp+relxillcp), having two reflection components, since the Fe
K$\alpha$ emission line is broadened. For simplicity, we assume a non-spinning
black hole, and a disk with emissivity index $-3$ from a disk with inner
radius $R_{\rm in}=2R_{g}$. The coronal temperature for a slab (spherical)
coronal geometry is $kT=60^{+24}_{-20}$ keV ($kT=82^{+80}_{-10}$ keV) and the
optical depth is $\tau=0.8^{+0.7}_{-0.3}$ ($1.5^{+0.8}_{-0.1}$). The goodness
of fit is $\chi^{2}/{\rm dof}=897/846$. Though with larger errors due to the
sole use of NuSTAR, the values are consistent with those found by Tortosa et
al. (2018a) in the broad ($E=0.5-60$ keV) band.

SWIFT J2127.4+5654
The Narrow-line Seyfert 1 galaxy SWIFT J2127.4+5654 is characterized by a mass
of $M=1.5\times 10^{7}$ $M_{\odot}$ at redshift $z=0.014$. Both FPM spectra A
and B are binned at 100 min counts per energy bin in the whole NuSTAR band
$E=3-79$ keV. The Galactic absorption column density is $N_{\rm H,Gal}=7\times
10^{20}$ cm-2 and negligible intrinsic absorption. The source is well modeled
by model A with a single non-relativistic reflector, i.e. compTT+xillvercp, we
obtain for a slab (spherical) coronal geometry a temperature of
$kT=33^{+37}_{-15}$ keV ($kT=24^{+24}_{-7}$ keV) and an optical depth of
$\tau=1.0^{+0.8}_{-0.6}$ ($\tau=3.5^{+1.0}_{-1.6}$). The statistic is
$\chi^{2}/{\rm dof}=1751/1775$, with values consistent at $99\%$ confidence
level with those of the broadband study by Marinucci et al. (2014).

UGC 6728
UGC 6728 is a Seyfert 1 galaxy with redshift $z=0.00652$ with a black hole
mass of $M_{\rm BH}=7.1\times 10^{5}$ $M_{\odot}$. The four FPM spectra were
binned at a minimum of 50 photon counts per energy bin. We consider data in
the $E=3-40$ keV energy band. We assume a Galactic absorption of $N_{\rm
H}=4.5\times 10^{20}$ cm-2. No evident intrinsic absorption is present. The
spectra are best modeled with continuum and two reflectors, i.e.
comptt+xillvercp+relxillcp (model B). We find a temperature of
$kT=28^{+16}_{-18}$ keV ($kT=28^{+17}_{-15}$ keV) and an optical depth of
$\tau=2.0\pm 1.2$ ($\tau=5.0\pm 2.0$) when assuming a slab (sphere) coronal
geometry. The goodness of fit is $\chi^{2}/{\rm dof}=1321/1183$.

##  Appendix C Spectra

c

Spectra and data-to-model ratios of the sources analyzed in this work.
Multiple observation spectra are also shown together. FPMA and FPMB are
plotted together.

##  Appendix D Light curves

c

Background-subtracted NuSTAR light curves. Each panel represents a source,
while each sub-panel represents one epoch of each AGN. Source names and OBSIDs
are printed in the figures. FPMA and FPMB light curves are combined together.