11institutetext: Indian Institute of Astrophysics, Bangalore, India
11email<EMAIL_ADDRESS>22institutetext: Inter University Centre for
Astronomy and Astrophysics, Pune, India
33institutetext: Cahill Center for Astronomy and Astrophysics, California
Institute of Technology, Pasadena, CA 91125, USA

#  NuSTAR spectral analysis of three Seyfert galaxies: NGC 3227, NGC 5548 and
MR 2251$-$178

Indrani Pal  C. S. Stalin 11 L. Mallick and Priyanka Rani 11331122

(Received October 15, 2021; accepted February 28, 2022)

###### Abstract

Context. The observed nuclear X-ray emission in the radio-quiet category of
active galactic nuclei (AGN) is believed to be from a compact region, the
corona situated in the vicinity of the central supermassive black holes
(SMBH). The shape of the X-ray continuum, among other factors, depends on the
temperature of the corona ($\rm{kT_{e}}$). The launch of the Nuclear
Spectroscopic Telescope Array (NuSTAR) has led to the determination of the
high energy cut-off ($\rm{E_{cut}}$; and thereby $\rm{kT_{e}}$) in many AGN.
In a handful of sources, multiple observations with NuSTAR have also revealed
changes in $\rm{E_{cut}}$.

Aims. In this work we aimed to investigate the variation in $\rm{kT_{e}}$ in
three AGN, namely NGC 3227, NGC 5548 and MR 2251$-$178 using more than one
epoch of data on a source from NuSTAR.

Methods. We carried out spectral analysis of multiple epochs of data acquired
using NuSTAR on the three sources including a few new observations not
published so far. By fitting Comptonization model to the data we determined
the temperature of the corona and also investigated changes in $\rm{kT_{e}}$
if any in these sources.

Results. In NGC 3227, we found evidence for variation in $\rm{kT_{e}}$. We
found no correlation of $\rm{kT_{e}}$, photon index ($\Gamma$), reflection
fraction ($R$) and optical depth ($\tau$) with flux, while, $\tau$ is found to
anti-correlate with $\rm{kT_{e}}$. These could be due to more than one
physical process at work in the source that causes the change in
$\rm{kT_{e}}$. Conclusive evidence for the variation in $\rm{kT_{e}}$ is not
found in MR 2251$-$178 and NGC 5548.

###### Key Words.:

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

##  1 Introduction

Active galactic nuclei (AGN) are amongst the most luminous objects (L = 1042
$-$ 1046 erg s-1; Fabian 1999) in the Universe that emit radiation over a wide
range of wavelengths. They are believed to be powered by accretion of matter
onto SMBH (105 $-$ 109 $M_{\odot}$) situated at the center of galaxies (Rees
1984; Dewangan et al. 2008). The SMBH is generally supposed to be surrounded
by an optically thick and geometrically thin accretion disk (Shakura & Sunyaev
1973). The observed X-ray emission from the nuclear region of the radio-quiet
category of AGN is believed to be produced by inverse Compton process, caused
by the interaction of the seed ultra-violet (UV) photons from the accretion
disk with the thermal electrons in a hot ($\sim$108-9 K) region called the
corona that is situated close to the accretion disk (Haardt & Maraschi 1991;
Haardt et al. 1994). This X-ray continuum gets reprocessed in the accretion
disk giving rise to the reflection hump at around 15$-$30 keV and also the
broad FeK$\alpha$ line at 6.4 keV (George & Fabian 1991; Matt et al. 1993).
Soft excess between 0.1 $-$ 2 keV is ubiquitously observed in Type I AGN
(Magdziarz et al. 1998; Fabian et al. 2002; Crummy et al. 2006; Bianchi et al.
2009; Gliozzi & Williams 2020), although the physical origin of this component
remains highly debated (García et al. 2019; Xu et al. 2021), different
analyses showed that a two temperature Comptonization process agrees well with
such a component either from an observational (e.g. Jin et al. 2012; Porquet
et al. 2018; Petrucci et al. 2018; Middei et al. 2020; Matzeu et al. 2020) or
theoretical (e.g. Różańska et al. 2015; Petrucci et al. 2020; Ballantyne 2020;
Ballantyne & Xiang 2020) point of view. Analysis of these spectral features
(reflection, FeK$\alpha$ line, soft excess) will help in providing strong
constraints on the nature of the X-ray emitting region. From X-ray
reverberation studies (Fabian et al. 2009; Zoghbi et al. 2012) AGN corona is
believed to be a compact region situated above the accretion disk typically
within 3$-$10 $R_{G}$, where $R_{G}$ is the gravitational radius and it is
defined as, $R_{G}=GM_{BH}/c^{2}$, here MBH is the SMBH mass and $G$ is the
gravitational constant. However, there are strong debates concerning the
geometry of the corona. The lamp post is one such possibility but other models
also exist (e.g. Haardt et al. 1994; Done et al. 2012; Petrucci et al.
2013).Also, rapid X-ray flux variability studies (McHardy et al. 2005), the
observed small time scales of X-ray eclipses (Risaliti et al. 2005, 2011) and
microlensing studies (Chartas et al. 2009) point to the small size of the
X-ray corona of 5$-$10 $R_{G}$.

The observed shape of the X-ray continuum can be described by a power law with
an exponential cut-off ($\rm{E_{cut}}$), and the spectral shape depends on the
optical depth ($\tau$), temperature of the coronal plasma ($\rm{kT_{e}}$),
seed photon temperature and the viewing angle. From a study of the Seyfert
galaxy NGC 5548, Petrucci et al. (2000) showed the existence of an approximate
relation between $\rm{E_{cut}}$ and $\rm{kT_{e}}$ as $\rm{E_{cut}}$ = 2$-$3
$\rm{kT_{e}}$. On analysis of a sample of Seyfert galaxies, according to
Petrucci et al. (2001), for an optically thin corona with $\tau$ $<$ 1;
$\rm{E_{cut}}$ $\approx$ 2 $\rm{kT_{e}}$, while for an optically thick corona
with $\tau$ $>$ 1, $\rm{E_{cut}}$ $\approx$ 3 $\rm{kT_{e}}$. However, by
fitting Comptonized spectra simulated using a range of $\tau$ and
$\rm{kT_{e}}$ with a power law with an exponential cut-off model, Middei et
al. (2019), showed that the commonly adopted relation of $\rm{E_{cut}}$ =
2$-$3 $\rm{kT_{e}}$ is not valid for all values of $\tau$ and $\rm{kT_{e}}$,
instead valid for only low values of $\tau$ and $\rm{kT_{e}}$.

Table 1: Details of the sources analyzed in this work. The columns are (1) name of the source, (2) right ascension (h:m:s), (3) declination (d:m:s), (4) redshift, (5) galactic hydrogen column density $\rm{N_{H}}$ in units of $10^{22}$ atoms $\rm{cm^{-2}}$ obtained from Willingale et al. (2013), (6) type of the source, (7) observation ID, (8) epoch, (9) date of observation, and (10) exposure time in sec. The information such as the right ascension, declination and $z$ are from Véron-Cetty & Véron (2010). The OBSIDs that are analysed for the first time are shown in bold. Name | $\alpha_{2000}$ | $\delta_{2000}$ | $z$ | $\rm{N_{H}}$ | Type | OBSID | Epoch | Date | Exposure
---|---|---|---|---|---|---|---|---|---
NGC 3227 | 10:23:30.60 |  $+$19:51:56 | 0.004 | 0.021 | Sy1.5 | 60202002002 | A | 09-11-2016 | 49800
|  |  |  |  |  | 60202002004 | B | 25-11-2016 | 42457
|  |  |  |  |  | 60202002006 | C | 29-11-2016 | 39685
|  |  |  |  |  | 60202002008 | D | 01-12-2016 | 41812
|  |  |  |  |  | 60202002010 | E | 05-12-2016 | 40887
|  |  |  |  |  | 60202002012 | F | 09-12-2016 | 39277
|  |  |  |  |  | 60202002014 | G | 21-01-2017 | 47602
|  |  |  |  |  | 80502609002 | H | 15-11-2019 | 28782
|  |  |  |  |  | 80502609004 | I | 05-12-2019 | 27690
NGC 5548 | 14:17:59.53 |  $+$25:08:12 | 0.017 | 0.017 | Sy1.5 | 60002044002 | A | 11-07-2013 | 24096
|  |  |  |  |  | 60002044003 | B | 12-07-2013 | 27272
|  |  |  |  |  | 60002044005 | C | 23-07-2013 | 49521
|  |  |  |  |  | 60002044006 | D | 10-09-2013 | 51460
|  |  |  |  |  | 60002044008 | E | 20-12-2013 | 50102
|  |  |  |  |  | 90701601002 | F | 26-01-2021 | 38719
MR 2251$-$178 | 22:54:05.90 |  $-$17:34:55 | 0.064 | 0.027 | Sy1.5 | 60102025002 | A | 18-05-2015 | 23112
|  |  |  |  |  | 60102025004 | B | 17-06-2015 | 23185
|  |  |  |  |  | 60102025006 | C | 10-11-2015 | 20588
|  |  |  |  |  | 60102025008 | D | 11-12-2015 | 21707
|  |  |  |  |  | 90601637002 | E | 16-12-2020 | 23620

Observations from high energy X-ray missions such as CGRO (Zdziarski et al.
2000; Johnson et al. 1997), BeppoSAX (Nicastro et al. 2000; Dadina 2007),
INTEGRAL (Malizia et al. 2014; Lubiński et al. 2010, 2016; Ricci et al. 2011),
Swift-BAT (Vasudevan et al. 2013b; Ricci et al. 2017); and Suzaku (Tazaki et
al. 2011) showed that the corona in Seyfert galaxies has a wide range of
temperature with $\rm{E_{cut}}$ ranging from 50 $-$ 500 keV. However,
observations from those missions are limited to bright and nearby sources.
Thus, it is very clear that many efforts were put to measure $\rm{E_{cut}}$ in
the X-ray spectra of several AGN. However, a major transformation in the
studies of the Comptonization spectrum of AGN to determine $\rm{E_{cut}}$ from
an epoch of observation happened after the launch of the Nuclear Spectroscopic
Telescope Array (NuSTAR; Harrison et al. 2013) in the year 2012, owing to its
broad spectral coverage of 3$-$79 keV and its high sensitivity beyond 10 keV.
Since the launch of NuSTAR, $\rm{E_{cut}}$ values were obtained for many AGN
(Fabian et al. 2017; Tortosa et al. 2018; Rani & Stalin 2018b, a; Rani et al.
2019; Lanzuisi et al. 2019; Baloković et al. 2020; Reeves et al. 2021; Kang et
al. 2021). Importantly, in addition to the determination of $\rm{E_{cut}}$
values (and thereby constraining $\rm{kT_{e}}$), in less than half-a-dozen
sources there are also reports of variation in the $\rm{E_{cut}}$ values
pointing to variations in $\rm{kT_{e}}$.

For example in seven sources, namely MCG-5-23-16 (Zoghbi et al. 2017), 3C 382
(Ballantyne et al. 2014), NGC 4593 (Ursini et al. 2016), NGC 5548 (Ursini et
al. 2015), Mrk 335 (Keek & Ballantyne 2016), NGC 3227 and SWIFT J2127.4+5654
(Kang et al. 2021) variations in the $\rm{E_{cut}}$ values are available in
the literature. Recently, from a reanalysis of the NuSTAR spectra of five
sources using a model independent approach, Zhang et al. (2018) confirmed the
$\rm{E_{cut}}$ variation in three of the five sources namely 3C 382, NGC 5548
and Mrk 335. Most of these inferences were based on variations in
$\rm{E_{cut}}$ obtained from phenomenological model fits to the data. However,
to know changes in $\rm{kT_{e}}$ it is imperative to fit physical models to
the data, because, it is known recently, that the relation $\rm{E_{cut}}$ =
2-3 $\rm{kT_{e}}$ does not always hold true (Middei et al. 2019). Therefore it
is imperative to fit physical model fits to the observed spectra to get
$\rm{kT_{e}}$. Though $\rm{E_{cut}}$ is known to vary, we do not yet know the
causes for its variation. Despite that, it is important to increase the number
of sources that show variation in the temperature of the corona. This is now
achievable owing to the multiple-epochs of observation available on a large
number of AGN in the NuSTAR archives111https://heasarc.gsfc.nasa.gov/cgi-
bin/W3Browse/w3browse.pl. The primary motivation here is, therefore to
increase the number of AGN that show variation in $\rm{kT_{e}}$. We are in the
process of a careful and systematic investigation of $\rm{kT_{e}}$ variation
in a large number of AGN. Here, we present the results from the multiple-epoch
spectral analysis of three AGN, namely NGC 3227, NGC 5548 and MR 2251$-$178\.
This also includes few new observations not published so far.

NGC 3227,at a redshift of z = 0.004 and powered by a black hole of mass 4.79
$\times$ $10^{6}$ $M_{\odot}$ (Bentz & Katz 2015), has been extensively
studied in the X-ray band. Signatures of warm absorbers are evident in this
source from observations with ASCA (Netzer et al. 1994; George et al. 1998),
ROSAT (Komossa & Fink 1997) and XMM-Newton (Markowitz et al. 2009). In the
XMM-Newton observations, FeK$\alpha$ line was evident (Markowitz et al. 2009).
It has also been recently studied by Lobban et al. (2020) for flux variations
combining XMM-Newton and NuSTAR observations. It has complex absorption
features, which are also variable (Turner et al. 2018). Recently Mehdipour et
al. (2021) has reported the broad band spectral modeling of the source using
multi-wavelength data from XMM-Newton, NuSTAR and Hubble Space Telescope
(HST). NGC 5548 is a galaxy located at $z$ = 0.017 with a black hole of mass
5.0 $\times$ 107 $M_{\odot}$ (Bentz & Katz 2015). It has been extensively
studied in the X-ray band using data from various satellites and has also been
found to be strongly absorbed in soft X-rays (Cappi et al. 2016; Ursini et al.
2015; Mehdipour et al. 2015; Kaastra et al. 2014). MR 2251$-$178, with a black
hole mass of 2.0 $\times$ 108 $M_{\odot}$ (Wang et al. 2009), first discovered
by its strong X-ray emission (Ricker et al. 1978) was found to be a low
redshift AGN at $z$ = 0.06 (Bergeron et al. 1983). From VLA observations,
Macchetto et al. (1990) found the source to show weak radio emission with
elongated morphology resembling a FRI source. Details of the observations and
reduction of the data are presented in Section 2, analysis is presented in
Section 3, results and discussion are presented in Section 4 followed by the
summary in the final section.

##  2 Observations and Data Reduction

Table 2: Results of the fit to the spectra of NGC 3227. The models are Model I: const*TBabs*zTBabs(zpo+zgauss), Model II: const*TBabs*zTBabs(pexrav+zgauss), Model III: const*TBabs*zTBabs(xillver) and Model IV: const*TBabs*zTBabs(xillverCP). The zTBabs component is added with all the models to fit the epoch G, H and I spectra. The fluxes are in units of 10-10 erg cm-2 s-1 in the 4$-$60 keV band except in epoch I where the flux was calculated in 4$-$50 keV band. Energy (E) of the FeK$\alpha$ line, equivalent width (EW) of the line, $\rm{E_{cut}}$ and $\rm{kT_{e}}$ are expressed in units of keV, model normalization is in units of 10-4 photons keV-1 cm-2s-1 and $\rm{N_{H}}$(zTBabs) is the host galaxy hydrogen column density in units of $10^{22}$ atoms $\rm{cm^{-2}}$. $\rm{C_{FPMA/FPMB}}$ is the cross-calibration constant. The width of the FeK$\alpha$ line was fixed to 0.1 keV during the fitting. Parameter |  epoch A |  epoch B |  epoch C |  epoch D |  epoch E |  epoch F |  epoch G |  epoch H |  epoch I
---|---|---|---|---|---|---|---|---|---
Model I : const*TBabs*zTBabs(zpo+zgauss)
$\Gamma$ |  1.56${}^{+0.01}_{-0.01}$ |  1.55${}^{+0.01}_{-0.01}$ |  1.61${}^{+0.01}_{-0.01}$ |  1.63${}^{+0.01}_{-0.01}$ |  1.66${}^{+0.01}_{-0.01}$ |  1.63${}^{+0.01}_{-0.01}$ |  1.61${}^{+0.02}_{-0.02}$ |  1.64${}^{+0.04}_{-0.04}$ |  1.48${}^{+0.07}_{-0.07}$
$\rm{N_{H}}$(zTBabs) |  - |  - |  - |  - |  - |  - |  2.91${}^{+1.04}_{-1.03}$ |  4.66${}^{+1.83}_{-1.80}$ |  4.57${}^{+3.73}_{-3.62}$
E |  6.35${}^{+0.03}_{-0.03}$ |  6.38${}^{+0.03}_{-0.04}$ |  6.33${}^{+0.04}_{-0.04}$ |  6.32${}^{+0.04}_{-0.03}$ |  6.30${}^{+0.05}_{-0.05}$ |  6.34${}^{+0.05}_{-0.05}$ |  6.23${}^{+0.04}_{-0.04}$ |  6.28${}^{+0.07}_{-0.07}$ |  6.38${}^{+0.05}_{-0.05}$
EW |  149${}^{+23}_{-32}$ |  188${}^{+32}_{-42}$ |  142${}^{+30}_{-35}$ |  129${}^{+22}_{-24}$ |  86${}^{+26}_{-30}$ |  115${}^{+26}_{-38}$ |  115${}^{+23}_{-23}$ |  123${}^{+44}_{-40}$ |  269${}^{+66}_{-81}$
norm |  71${}^{+2}_{-2}$ |  58${}^{+2}_{-2}$ |  73${}^{+2}_{-2}$ |  90${}^{+3}_{-3}$ |  101${}^{+3}_{-3}$ |  92${}^{+3}_{-3}$ |  122${}^{+7}_{-6}$ |  76${}^{+8}_{-7}$ |  17${}^{+4}_{-3}$
$\chi^{2}/dof$ |  930/826 |  752/706 |  834/704 |  909/777 |  815/790 |  684/717 |  1081/935 |  600/595 |  257/247
$\rm{C_{FPMA/FPMB}}$ |  1.04${}^{+0.01}_{-0.01}$ |  1.02${}^{+0.01}_{-0.01}$ |  1.01${}^{+0.01}_{-0.01}$ |  1.01${}^{+0.01}_{-0.01}$ |  1.02${}^{+0.01}_{-0.01}$ |  1.00${}^{+0.01}_{-0.01}$ |  0.99${}^{+0.01}_{-0.01}$ |  1.04${}^{+0.01}_{-0.01}$ |  1.06${}^{+0.03}_{-0.03}$
Model II : const*TBabs*zTBabs(pexrav+zgauss)
$\Gamma$ |  1.68${}^{+0.05}_{-0.05}$ |  1.64${}^{+0.06}_{-0.05}$ |  1.77${}^{+0.05}_{-0.05}$ |  1.83${}^{+0.04}_{-0.04}$ |  1.88${}^{+0.05}_{-0.05}$ |  1.82${}^{+0.05}_{-0.05}$ |  1.92${}^{+0.01}_{-0.07}$ |  1.85${}^{+0.08}_{-0.08}$ |  1.70${}^{+0.04}_{-0.11}$
$\rm{N_{H}}$(zTBabs) |  - |  - |  - |  - |  - |  - |  6.43${}^{+0.92}_{-1.42}$ |  7.71${}^{+1.93}_{-1.95}$ |  8.52${}^{+4.30}_{-5.18}$
$\rm{E_{cut}}$ |  212${}^{+140}_{-63}$ |  163${}^{+118}_{-50}$ |  $>$254 |  $>$1147 |  $>$411 |  $>$406 |  $>$571 |  $>$775 |  $>$126
R |  0.52${}^{+0.15}_{-0.13}$ |  0.48${}^{+0.18}_{-0.15}$ |  0.53${}^{+0.18}_{-0.15}$ |  0.55${}^{+0.16}_{-0.14}$ |  0.73${}^{+0.19}_{-0.17}$ |  0.55${}^{+0.18}_{-0.16}$ |  0.83${}^{+0.16}_{-0.15}$ |  0.45${}^{+0.20}_{-0.18}$ |  0.38${}^{+0.49}_{-0.34}$
E |  6.35${}^{+0.03}_{-0.03}$ |  6.38${}^{+0.04}_{-0.04}$ |  6.33${}^{+0.04}_{-0.04}$ |  6.32${}^{+0.04}_{-0.04}$ |  6.31${}^{+0.06}_{-0.06}$ |  6.34${}^{+0.05}_{-0.05}$ |  6.23${}^{+0.04}_{-0.04}$ |  6.28${}^{+0.07}_{-0.07}$ |  6.38${}^{+0.06}_{-0.06}$
EW |  137${}^{+32}_{-20}$ |  177${}^{+37}_{-42}$ |  129${}^{+26}_{-33}$ |  116${}^{+25}_{-23}$ |  74${}^{+26}_{-21}$ |  102${}^{+32}_{-22}$ |  85${}^{+23}_{-19}$ |  102${}^{+34}_{-31}$ |  231${}^{+82}_{-74}$
norm |  84${}^{+6}_{-6}$ |  66${}^{+6}_{-5}$ |  91${}^{+8}_{-7}$ |  119${}^{+8}_{-7}$ |  139${}^{+10}_{-10}$ |  120${}^{+9}_{-9}$ |  207${}^{+19}_{-25}$ |  112${}^{+18}_{-16}$ |  25${}^{+9}_{-9}$
$\chi^{2}/dof$ |  856/824 |  711/704 |  784/702 |  831/775 |  714/788 |  625/715 |  920/933 |  577/593 |  253/245
$\rm{C_{FPMA/FPMB}}$ |  1.04${}^{+0.01}_{-0.01}$ |  1.02${}^{+0.01}_{-0.01}$ |  1.01${}^{+0.01}_{-0.01}$ |  1.01${}^{+0.01}_{-0.01}$ |  1.02${}^{+0.01}_{-0.01}$ |  1.00${}^{+0.01}_{-0.01}$ |  0.99${}^{+0.01}_{-0.01}$ |  1.04${}^{+0.01}_{-0.01}$ |  1.06${}^{+0.03}_{-0.03}$
Model III : const*TBabs*zTBabs(xillver)
$\Gamma$ |  1.69${}^{+0.01}_{-0.01}$ |  1.69${}^{+0.01}_{-0.01}$ |  1.77${}^{+0.02}_{-0.01}$ |  1.83${}^{+0.01}_{-0.01}$ |  1.80${}^{+0.01}_{-0.01}$ |  1.79${}^{+0.01}_{-0.01}$ |  1.77${}^{+0.01}_{-0.01}$ |  1.87${}^{+0.02}_{-0.02}$ |  1.91${}^{+0.03}_{-0.03}$
$\rm{N_{H}}$(zTBabs) |  - |  - |  - |  - |  - |  - |  4.73${}^{+0.57}_{-0.56}$ |  8.58${}^{+0.98}_{-1.08}$ |  12.12${}^{+1.86}_{-1.82}$
$\rm{E_{cut}}$ |  127${}^{+14}_{-12}$ |  92${}^{+10}_{-8}$ |  204${}^{+46}_{-34}$ |  $>$439 |  378${}^{+152}_{-91}$ |  326${}^{+123}_{-78}$ |  248${}^{+67}_{-34}$ |  $>$510 |  $>$198
R |  0.86${}^{+0.12}_{-0.12}$ |  1.09${}^{+0.17}_{-0.16}$ |  0.85${}^{+0.14}_{-0.13}$ |  0.76${}^{+0.14}_{-0.12}$ |  0.61${}^{+0.12}_{-0.09}$ |  0.71${}^{+0.14}_{-0.11}$ |  0.68${}^{+0.08}_{-0.08}$ |  0.64${}^{+0.16}_{-0.14}$ |  1.36${}^{+0.47}_{-0.40}$
norm |  1.98${}^{+0.02}_{-0.03}$ |  1.46${}^{+0.02}_{-0.05}$ |  2.01${}^{+0.03}_{-0.03}$ |  2.98${}^{+0.04}_{-0.04}$ |  2.97${}^{+0.03}_{-0.03}$ |  2.75${}^{+0.04}_{-0.04}$ |  3.73${}^{+0.04}_{-0.04}$ |  2.72${}^{+0.03}_{-0.04}$ |  0.68${}^{+0.03}_{-0.03}$
$\chi^{2}/dof$ |  868/826 |  728/706 |  785/704 |  837/777 |  735/790 |  627/717 |  976/935 |  579/595 |  260/247
$\rm{C_{FPMA/FPMB}}$ |  1.04${}^{+0.01}_{-0.01}$ |  1.02${}^{+0.01}_{-0.01}$ |  1.01${}^{+0.01}_{-0.01}$ |  1.01${}^{+0.01}_{-0.01}$ |  1.02${}^{+0.01}_{-0.01}$ |  1.00${}^{+0.01}_{-0.01}$ |  0.99${}^{+0.01}_{-0.01}$ |  1.04${}^{+0.01}_{-0.01}$ |  1.06${}^{+0.03}_{-0.03}$
Model IV : const*TBabs*zTBabs(xillverCP)
$\Gamma$ |  1.78${}^{+0.01}_{-0.01}$ |  1.80${}^{+0.01}_{-0.01}$ |  1.83${}^{+0.01}_{-0.01}$ |  1.84${}^{+0.01}_{-0.01}$ |  1.83${}^{+0.01}_{-0.01}$ |  1.83${}^{+0.01}_{-0.01}$ |  1.82${}^{+0.01}_{-0.01}$ |  1.87${}^{+0.02}_{-0.02}$ |  1.91${}^{+0.04}_{-0.03}$
$\rm{N_{H}}$(zTBabs) |  - |  - |  - |  - |  - |  - |  4.97${}^{+0.59}_{-0.55}$ |  8.17${}^{+0.99}_{-0.82}$ |  12.21${}^{+1.73}_{-1.65}$
$\rm{kT_{e}}$ |  33${}^{+9}_{-7}$ |  28${}^{+9}_{-6}$ |  56${}^{+131}_{-18}$ |  $>$80 |  $>$45 |  $>$47 |  50${}^{+39}_{-10}$ |  $>$85 |  $>$36
R |  0.79${}^{+0.11}_{-0.11}$ |  0.96${}^{+0.18}_{-0.10}$ |  0.83${}^{+0.15}_{-0.15}$ |  0.78${}^{+0.13}_{-0.12}$ |  0.62${}^{+0.11}_{-0.11}$ |  0.71${}^{+0.14}_{-0.13}$ |  0.67${}^{+0.09}_{-0.10}$ |  0.65${}^{+0.16}_{-0.15}$ |  1.28${}^{+0.60}_{-0.30}$
norm |  1.94${}^{+0.02}_{-0.02}$ |  1.52${}^{+0.02}_{-0.02}$ |  1.89${}^{+0.03}_{-0.03}$ |  2.56${}^{+0.02}_{-0.03}$ |  2.55${}^{+0.03}_{-0.05}$ |  2.50${}^{+0.04}_{-0.04}$ |  3.32${}^{+0.03}_{-0.07}$ |  2.24${}^{+0.07}_{-0.05}$ |  0.66${}^{+0.02}_{-0.02}$
$\chi^{2}/dof$ |  881/826 |  746/706 |  790/704 |  838/777 |  733/790 |  629/717 |  970/935 |  581/595 |  261/247
$\rm{C_{FPMA/FPMB}}$ |  1.04${}^{+0.01}_{-0.01}$ |  1.02${}^{+0.01}_{-0.01}$ |  1.01${}^{+0.01}_{-0.01}$ |  1.01${}^{+0.01}_{-0.01}$ |  1.02${}^{+0.01}_{-0.01}$ |  1.00${}^{+0.01}_{-0.01}$ |  0.99${}^{+0.01}_{-0.01}$ |  1.04${}^{+0.01}_{-0.01}$ |  1.06${}^{+0.03}_{-0.03}$
Flux |  1.04${}^{+0.02}_{-0.01}$ |  0.86${}^{+0.01}_{-0.01}$ |  0.94${}^{+0.01}_{-0.02}$ |  1.11${}^{+0.01}_{-0.01}$ |  1.15${}^{+0.01}_{-0.02}$ |  1.13${}^{+0.01}_{-0.01}$ |  1.55${}^{+0.01}_{-0.01}$ |  0.88${}^{+0.01}_{-0.01}$ |  0.27${}^{+0.01}_{-0.01}$

###  2.1 Data reduction

We reduced NuSTAR data in the 3$-$79 keV band using the standard NuSTAR data
reduction software
NuSTARDAS222https://heasarc.gsfc.nasa.gov/docs/nustar/analysis/nustar
swguide.pdf distributed by HEASARC within HEASoft v6.29. Considering the
passage of the satellite through the South Atlantic Anomaly (SAA) we selected,
SAACALC 2, SAAMODE optimized and also excluded the tentacle region. The
calibrated, cleaned, and screened event files were generated by running the
nupipeline task using the CALDB release 20210701. To extract the source counts
we chose a circular region of radius 60 arcsec centered on the source.
Similarly, to extract the background counts, we selected a circular region of
the same radius away from the source on the same chip to avoid contamination
from source photons. We then used the nuproducts task to generate energy
spectra, response matrix files (RMFs) and auxiliary response files (ARFs), for
both the hard X-ray detectors housed inside the corresponding focal plane
modules FPMA and FPMB. For spectral analysis, using XSPEC version 12.12.0
(Arnaud 1996), we fitted the background subtracted spectra from FPMA and FPMB
simultaneously (without combining them) allowing the cross normalization
factor to vary freely during spectral fits. The spectra were binned to have a
S/N ratio greater than 5 in each spectral channel using the NuSTAR-specific
Python script snrgrppha333https://sites.astro.caltech.edu/$\sim$mislavb/. To
get an estimate of the model parameters that best describe the observed data,
we used the chi-square ($\chi^{2}$) statistics and for calculating the errors
in the model parameters we used the $\chi^{2}$ = 2.71 criterion i.e. 90 per
cent confidence range in XSPEC.

Table 3: Results of the model fits to the spectra of MR 2251$-$178\. The models are Model I: const*TBabs*zTBabs(zpo), Model II: const*TBabs*zTBabs(pexrav), Model III: const*TBabs*zTBabs(xillver) and Model IV: const*TBabs*zTBabs(xillverCP). The zTBabs component is added with all the models to fit the epoch E spectra. In epoch D the width of the Fek$\alpha$ line was fixed to 0.1 keV. In the case where no line was detected, the upper limit on the EW was calculated by fixing the line energy to 6.4 keV.The fluxes are in units of 10-10 erg cm-2 s-1 in the 4$-$60 keV band except in epoch E where the flux was calculated in 4$-$50 keV band. Columns and parameters have the same meaning as given in Table 2. Parameter |  epoch A |  epoch B |  epoch C |  epoch D |  epoch E
---|---|---|---|---|---
Model I : const*TBabs*zTBabs(zpo+zgauss)
$\Gamma$ |  1.75${}^{+0.02}_{-0.02}$ |  1.79${}^{+0.01}_{-0.01}$ |  1.79${}^{+0.02}_{-0.02}$ |  1.79${}^{+0.02}_{-0.02}$ |  1.83${}^{+0.05}_{-0.05}$
$\rm{N_{H}}$(zTBabs) |  - |  - |  - |  - |  7.82${}^{+2.99}_{-2.93}$
E |  - |  - |  - |  6.49${}^{+0.37}_{-0.27}$ |  -
EW |  $<$35 |  $<$49 |  $<$34 |  $<$70 |  $<$46
norm |  142${}^{+5}_{-5}$ |  177${}^{+6}_{-6}$ |  162${}^{+6}_{-6}$ |  162${}^{+6}_{-6}$ |  84${}^{+13}_{-11}$
$\chi^{2}/dof$ |  583/601 |  656/633 |  574/551 |  516/576 |  446/400
$\rm{C_{FPMA/FPMB}}$ |  1.02${}^{+0.02}_{-0.02}$ |  1.00${}^{+0.01}_{-0.01}$ |  1.03${}^{+0.02}_{-0.02}$ |  1.02${}^{+0.02}_{-0.02}$ |  1.05${}^{+0.02}_{-0.02}$
Model II : const*TBabs*zTBabs(pexrav+zgauss)
$\Gamma$ |  1.65${}^{+0.05}_{-0.05}$ |  1.72${}^{+0.05}_{-0.03}$ |  1.76${}^{+0.07}_{-0.07}$ |  1.79${}^{+0.06}_{-0.06}$ |  1.82${}^{+0.11}_{-0.10}$
$\rm{N_{H}}$(zTBabs) |  - |  - |  - |  - |  7.66${}^{+3.26}_{-3.02}$
$\rm{E_{cut}}$ |  125${}^{+96}_{-39}$ |  185${}^{+200}_{-69}$ |  110${}^{+70}_{-32}$ |  193${}^{+417}_{-80}$ |  $>$175
R |  $<$0.07 |  $<$0.11 |  0.29${}^{+0.22}_{-0.18}$ |  0.19${}^{+0.20}_{-0.17}$ |  $<$0.23
E |  - |  - |  - |  6.48${}^{+0.67}_{-0.31}$ |  -
EW |  $<$49 |  $<$43 |  $<$33 |  $<$63 |  $<$50
norm |  111${}^{+8}_{-8}$ |  144${}^{+11}_{-6}$ |  139${}^{+15}_{-13}$ |  144${}^{+15}_{-13}$ |  75${}^{+19}_{-14}$
$\chi^{2}/dof$ |  569/599 |  647/631 |  552/549 |  509/574 |  446/398
$\rm{C_{FPMA/FPMB}}$ |  1.02${}^{+0.02}_{-0.02}$ |  1.00${}^{+0.01}_{-0.01}$ |  1.03${}^{+0.02}_{-0.02}$ |  1.02${}^{+0.02}_{-0.02}$ |  1.05${}^{+0.02}_{-0.02}$
Model III : const*TBabs*zTBabs(xillver)
$\Gamma$ |  1.65${}^{+0.02}_{-0.02}$ |  1.72${}^{+0.02}_{-0.02}$ |  1.70${}^{+0.02}_{-0.02}$ |  1.77${}^{+0.02}_{-0.02}$ |  1.82${}^{+0.02}_{-0.03}$
$\rm{N_{H}}$(zTBabs) |  - |  - |  - |  - |  7.56${}^{+1.69}_{-1.38}$
$\rm{E_{cut}}$ |  124${}^{+22}_{-18}$ |  169${}^{+45}_{-30}$ |  103${}^{+18}_{-14}$ |  163${}^{+46}_{-30}$ |  $>$366
R |  $<$0.10 |  $<$0.16 |  0.17${}^{+0.14}_{-0.13}$ |  0.22${}^{+0.15}_{-0.13}$ |  $<$0.11
norm |  2.92${}^{+0.02}_{-0.04}$ |  3.54${}^{+0.03}_{-0.05}$ |  2.79${}^{+0.05}_{-0.05}$ |  3.05${}^{+0.02}_{-0.05}$ |  2.10${}^{+0.03}_{-0.52}$
$\chi^{2}/dof$ |  569/599 |  647/631 |  556/549 |  510/576 |  447/398
$\rm{C_{FPMA/FPMB}}$ |  1.02${}^{+0.02}_{-0.02}$ |  1.00${}^{+0.01}_{-0.01}$ |  1.03${}^{+0.02}_{-0.02}$ |  1.02${}^{+0.02}_{-0.02}$ |  1.05${}^{+0.02}_{-0.02}$
Model IV : const*TBabs*zTBabs(xillverCP)
$\Gamma$ |  1.76${}^{+0.01}_{-0.01}$ |  1.79${}^{+0.01}_{-0.01}$ |  1.80${}^{+0.02}_{-0.01}$ |  1.83${}^{+0.02}_{-0.01}$ |  1.83${}^{+0.02}_{-0.02}$
$\rm{N_{H}}$(zTBabs) |  - |  - |  - |  - |  7.54${}^{+1.42}_{-1.45}$
$\rm{kT_{e}}$ |  25${}^{+26}_{-6}$ |  35${}^{+149}_{-11}$ |  21${}^{+8}_{-4}$ |  35${}^{+67}_{-11}$ |  $>$32
R |  $<$0.06 |  $<$0.11 |  $<$0.25 |  0.17${}^{+0.14}_{-0.13}$ |  $<$0.09
norm |  2.68${}^{+0.02}_{-0.03}$ |  3.26${}^{+0.03}_{-0.05}$ |  2.66${}^{+0.04}_{-0.04}$ |  2.85${}^{+0.04}_{-0.04}$ |  1.71${}^{+0.02}_{-0.04}$
$\chi^{2}/dof$ |  576/599 |  650/631 |  555/549 |  511/576 |  447/398
$\rm{C_{FPMA/FPMB}}$ |  1.02${}^{+0.02}_{-0.02}$ |  1.00${}^{+0.01}_{-0.01}$ |  1.03${}^{+0.02}_{-0.02}$ |  1.02${}^{+0.02}_{-0.02}$ |  1.05${}^{+0.02}_{-0.02}$
Flux |  1.07${}^{+0.01}_{-0.01}$ |  1.22${}^{+0.01}_{-0.01}$ |  1.08${}^{+0.01}_{-0.01}$ |  1.09${}^{+0.01}_{-0.01}$ |  0.47${}^{+0.02}_{-0.01}$

Figure 1: The unfolded spectra of the nine observations for NGC 3227 (left),
six observations for NGC 5548 (middle) and five observations for MR 2251$-$178
(right) fitted with a simple power-law. The bottom panels show the ratio of
the observed spectra to the model. For clarity, we used only FPMA data. The
spectra are rebinned for visualization purpose only.

Figure 2: Ratio of data to the model for the model fits
const*TBabs(zpo+zgauss), const*TBabs(pexrav+zgauss) and const*TBabs(xillverCP)
to the FPMA (blue triangle) and FPMB (yellow triangle) spectra of OBID
60202002002 of NGC 3227. The spectra are rebinned for visualization purpose.

Figure 3: Ratio plots for the model fits const*TBabs*zTBabs*(zpo+zgauss),
const*TBabs*zTBabs*(pexrav+zgauss) and const*TBabs*zTBabs(xillverCP) to the
FPMA (blue star) and FPMB (yellow star) spectra of OBID 60002044006 of NGC
5548. The spectra are rebinned for visualization purpose only.

Figure 4: Ratio of data to model for the model fits const*TBabs*zTBabs*(zpo),
const*TBabs*zTBabs*(pexrav) and const*TBabs*zTBabs(xillverCP) to the FPMA
(blue dot) and FPMB (yellow dot) spectra of OBID 90601637002 of MR
2251$-$178\. We re-binned the spectra for visualization purpose only.

##  3 Analysis of the data

For few epochs of the sources studied in this work, we do have observations in
the soft band from telescopes such as XMM-Newton for NGC 3227 and XMM-Newton,
Chandra for NGC 5548 and XMM-Newton for MR 2251$-$178\. However, for this
work, we decided to use only NuSTAR data as (a) the good sensitivity of NuSTAR
over the 3$-$79 keV energy band captures all the key reflection features of an
AGN spectrum and as the main goal of this work is to model the high energy
rollover of the Comptonized spectra we did not want the absorption in the soft
band affecting our analysis in the determination of $\rm{kT_{e}}$ and (b)
observations in the soft band are not available simultaneous to the NuSTAR
observations for all the epochs and for all the sources. However we note here,
the inclusion of soft X-ray data in the fitting might have an effect on the
photon index ($\Gamma$) obtained from using the NuSTAR data alone. However, a
simplest approach to constrain $\rm{kT_{e}}$ (which is the aim of this work)
is the use of only NuSTAR data, but for better constraining the other physical
characteristics of the sources broad band spectral analysis including data
from UV to hard X-ray band is more appropriate. While analyzing only NuSTAR
data, we ignored the 3$-$4 keV band to limit the effect of absorption if any
and also did not consider data in the energy range beyond 60 keV due to lack
of source photons. Thus, we carried out spectral fits to the NuSTAR data in
the 4$-$60 keV energy band for all the OBSIDS except for epoch I of NGC 3227,
epoch F of NGC 5548 and epoch E of MR 2251$-$178\. Due to the unavailability
of photons beyond 50 keV we restricted the spectral fit in the 4$-$50 keV
energy band to the epoch I and epoch E spectra of NGC 3227 and MR 2251$-$178
respectively. Similarly, for the epoch F spectrum of NGC 5548, we used the
FPMA/FPMB data in the 4$-$55 keV range. (see Fig 1).

###  3.1 Phenomenological spectral fits

For our spectral fits, to model the line of sight galactic absorption, the
value of the neutral hydrogen column density ($\rm{N_{H}}$) for all the
sources were frozen to the values obtained from Willingale et al. (2013).
These $\rm{N_{H}}$ values are given in Table 1. Similarly, the redshifts of
the sources were frozen to their corresponding values given in Table 1. Also,
we used the solar abundances from Wilms et al. (2000) and the photoelectric
cross sections from Verner et al. (1996). For models that require inclination
angle ($i$) , we used $i$ = 50∘ for NGC 3227 (Schmitt et al. 1997, Alonso-
Herrero et al. 2019, Middleton et al. 2016), $i$ = 30∘ for NGC 5548 (Ursini et
al. 2015) and $i$ = 60∘ (i.e., the default value) for MR 2251$-$178.

####  3.1.1 Absorbed power law

Firstly, to understand the continuum emission in our sample of sources we
fitted the observed X-ray spectra with the baseline phenomenological absorbed
power law model and having the following form in XSPEC

$const*TBabs(zpo)$ (1)

The first component of this model is the constant used to calibrate the two
focal plane modules of NuSTAR. The second component, TBabs (Wilms et al. 2000)
was used to model the line of sight galactic absorption. The parameters that
were kept free are $\Gamma$ and the normalization (i.e. photons keV-1 cm-2
s-1). We found evidence of intrinsic absorption present in the lower energy
end for all the epochs in NGC 5548, epoch G, H and I in NGC 3227 and epoch E
in MR 2251$-$178\. We therefore included a zTBabs component with the absorbed
power law model to fit their spectra and the model looks like

$const*TBabs*zTBabs(zpo)$ (2)

in XSPEC. For all epochs in NGC 5548, two epochs in NGC 3227 and one epoch of
MR 2251$-$178 non-inclusion of zTBabs returned a poor fit with $\chi^{2}/dof$
larger than 1.2. Inclusion of zTBabs with $\rm{N_{H}}$(zTBabs) kept free
improved the fit with $\chi^{2}/dof$ close to unity.

Figure 5: The unfolded spectra with the ratio of data to model for the model
fits const*TBabs(xillverCP) to the FPMA (blue) and FPMB (yellow) spectra of
OBID 60202002014 (epoch G) of NGC 3227.

From the residual spectra obtained from the simple absorbed power law model
fit to the observations, we noticed the presence of the fluorescent
FeK$\alpha$ line in NGC 3227, NGC 5548 and in epoch D spectra of MR
2251$-$178, but in the other observations of MR 2251$-$178, the residual
spectra do not show conspicuous FeK$\alpha$ line. The spectral fits are shown
in Fig. 1 for NGC 3227 and NGC 5548 and MR 2251$-$178\. Though this line is
common in most of the X ray spectra of AGN (Mushotzky et al. 1993; Nandra et
al. 2007; Della Ceca et al. 2010) there are exceptions (Bhayani & Nandra
2011). The apparent non-detection of FeK$\alpha$ line in the spectra of MR
2251$-$178 could be due to weaker reflection owing to larger viewing angle
(Bhayani & Nandra 2011), low signal-to-noise ratio spectra, very high ionized
accretion disk (Ross & Fabian 1993; Zycki & Czerny 1994) or a combination of
all. However, the presence of a weak FeK$\alpha$ line in epoch D of MR
2251$-$178 could point to the physical characteristic of MR 2251$-$178 being
different from the other two sources. A thorough analysis is needed to know
the exact reasons for the absence/weakness of this line but this issue is
beyond the scope of this work as we are in this work mainly interested in the
changes in the temperature of the corona. To model the FeK$\alpha$ line seen
in the residual spectra in NGC 3227, NGC 5548 and in one observation (epoch D)
of MR 2251$-$178, we included a gaussian component and with this inclusion the
model takes the form const*TBabs(zpo + zgauss) and the quality of the fit
improved.

After the inclusion of a gaussian component with the power law in the case of
NGC 3227, in all nine epochs the $\chi^{2}$ value reduced in the range between
60 and 242 for a reduction of 2 dof. In epochs C, D, G and I, non-inclusion of
the gaussian component, resulted in a reduced $\chi^{2}$ greater than 1.30.
After the inclusion of the gaussian component the reduced $\chi^{2}$ ranged
between 1.04$-$1.16 in these epochs. For the other epochs, the $\chi^{2}/dof$
was $>$ 1.1 before and it became $\sim$ 1.0 after the line inclusion. For
MR2251$-$178, gaussian component was used to fit the FeK$\alpha$ line only in
epoch D. Inclusion of the line component lead to a change in $\chi^{2}$ of 5
for a reduction of 2 dof. This negligible change in $\chi^{2}$ over 2 dof did
not improve the fit quality significantly in this case. For NGC 5548, on
inclusion of gaussian line component in all the epochs, the value of the
$\chi^{2}$ reduced in the range 23 to 105 for a reduction of 2 dof. Just an
absorbed power law fit to the data produced the reduced $\chi^{2}$ greater
than 1.2 for the epochs C, D, E and F. Adding the Gaussian with the power law
lead to a reduced $\chi^{2}$ of $\sim$ 1.1 for epoch C and D and $\sim$ 1.0
for epoch E and F. Similarly, for epochs A and B, the $\chi^{2}/dof$ changed
from 1.10 to 1.06 and 1.02 to 0.97 respectively. The width of the FeK$\alpha$
line was frozen to the value of 0.1 keV during the fitting, letting it free to
vary did not improve the fit significantly. The best fit parameters for the
sources are given in Table 2, 3 and 4 for NGC 3227, MR 2251$-$178 and NGC
5548.

####  3.1.2 Pexrav

In the residuals of the simple power law fit to all the spectra of the sources
(see Fig. 1) we found the signature of a high energy turn over and a
reflection hump beyond 15 keV. To appropriately model both the high energy cut
off and the reflection feature present in the spectra we replaced the zpo
component in our earlier model with pexrav and the new model has the form

$const*TBabs(pexrav).$ (3)

While modelling the reflection component of NGC 5548, epoch G, H and I spectra
of NGC 3227 and epoch E spectrum of MR 2251$-$178 an intrinsic absorption
component, zTBabs was added with the above model. The intrinsic hydrogen
column density, $\rm{N_{H}}$(zTBabs) was kept free during the fit.

Figure 6: Variation of $\rm{E_{cut}}$ and $\rm{kT_{e}}$ with the observation
epochs as obtained from xillver and xillverCP model fit respectively to the
spectra NGC 3227. The plotted errors were calculated using the $\chi^{2}$ =
2.71 criterion i.e. 90% confidence range. The black dashed lines in each panel
are fits of constant (mean of $\rm{E_{cut}}$ and $\rm{kT_{e}}$) to the data
points. For the fitting, epochs in which we were unable to constrain
$\rm{E_{cut}}$ and $\rm{kT_{e}}$ were dropped (indicated with red point).

Figure 7: The unfolded spectra along with const*TBabs(xillverCP) model fits
and the data to model ratio for the epoch A (top left panel), epoch B (top
middle panel), epoch C (top right panel), epoch D (bottom left panel), epoch E
(bottom middle panel) and epoch F (bottom right panel) for the source NGC
3227. blue and yellow data points refer to FPMA and FPMB respectively.

Figure 8: The unfolded spectra and model fits along with the data to model
ratio for const*TBabs(xillverCP) fits to the epoch G (left), epoch H (middle)
and epoch I (right) of NGC 3227. For epoch G, epoch H and epoch I zTBabs was
added to the model. blue and yellow data points refer to FPMA and FPMB
respectively.

This model implements both photoelectric absorption and Compton scattering,
however, it does not include fluorescence. Therefore to model the FeK$\alpha$
line seen in the residual spectra in NGC 3227, NGC 5548 and epoch D in MR
2251$-$178, we included a gaussian component. So, the model takes the
following form,

$const*TBabs(pexrav+zgauss)$ (4)

in XSPEC. The model pexrav improved the fitting compared to zpo as evident in
the reduced $\chi^{2}$ in Tables 2, 3 and 4. This model includes primary
emission having a power law form with an exponential cut-off and a reflection
component. The reflector is considered to be an optically thick medium in an
infinite plane geometry and covering a larger fraction of the X-ray source. A
parameter that comes as an output in the model fit is the reflection parameter
$R$. This parameter gives a measure of the reflection component present in the
observed spectra of the sources. For an isotropic source, this parameter is
related to the solid angle ($\Omega$) as $R\sim\Omega/2\pi$, and it is
dependent on the angle of inclination ($i$) between the perpendicular to the
accretion disk and the line of sight to the observer. The width of the
FeK$\alpha$ line was fixed at 0.1 keV, treating the parameter free did not
improve the fit. For all the sources, the parameters that were left free in
the model fits are $\rm{E_{cut}}$, $\Gamma$, R and normalization.

Using (pexrav+zgauss) in all the nine epochs of NGC 3227, the $\chi^{2}$
reduced in the range between 4 and 161 for a reduction of 2 dof compared to
the (zpo+zgauss) fit. In MR 2251$-$178, the reduction in $\chi^{2}$ is in the
range between 0 and 22 with a reduction of 2 in dof. For all the six epochs in
NGC 5548, the (pexrav+zgauss) fit produced a reduction in the $\chi^{2}$
values between 9$-$60 with a reduction of 2 dof compared to (zpo+zgauss). The
best fit parameters are given in Tables 2 , 3 and 4.

Figure 9: The unfolded spectra with the const*TBabs*zTBabs(xillverCP) model
fits and the data to model ratio to the spectra of NGC 5548. Here epoch A is
shown in the left panel, epoch B is shown in the middle panel and epoch C is
shown in the right panel. Data from FPMA and FPMB are shown in blue and yellow
respectively.

Figure 10: The unfolded spectra with the const*TBabs*zTBabs(xillverCP) model
fits and the data to model ratio to the spectra of NGC 5548. Here the left,
middle and right panels show the results for epochs D, E and F respectively.
Data from FPMA and FPMB are shown in blue and yellow respectively.

###  3.2 Physical model fits

####  3.2.1 xillver and xillverCP

Accretion disk reflection features in the form of narrow FeK$\alpha$ line was
conspicuously present in the residuals of NGC 3227, NGC 5548 and epoch D
spectra of MR 2251$-$178 and the line was modelled using a gaussian component
in pexrav model fits. For MR 2251$-$178 we have seen signatures of reflection
(though weak) in few epochs. This is evident from the best fit values of the
reflection fraction obtained using pexrav (see Table 3) and in the residuals
of the zpo model fits (see Fig. 4: top panel). We therefore modelled the
spectra of the sources with the self consistent ionized reflection model
xillver(García & Kallman 2010; García et al. 2013). The emergent reflected
spectrum from the surface of the X-ray illuminated accretion disk is
calculated in the model by solving the equations of radiative transfer, energy
balance and ionization equilibrium in a Compton thick plane parallel medium
(García et al. 2013). The model fit to the spectra has the following form in
XSPEC,

$const*TBabs(xillver)$ (5)

Here also, to account for the effect of intrinsic absorption, zTBabs was used
for all epochs of NGC 5548, three epochs (G, H and I) of NGC 3227 and epoch E
of MR 2251$-$178\. The intrinsic hydrogen column densities,$N_{\rm H}$(zTBabs)
were kept as free parameters. In this model fit, the parameters describing the
properties of the corona such as $\Gamma$, $\rm{E_{cut}}$ and R were made to
vary, while the inclination angle was frozen to the value of 50∘ for NGC 3227,
30∘ for NGC 5548 and 60∘ for MR 2251$-$178\. The other parameter that was kept
free is the normalization. To account for a fair comparison between the
$\rm{E_{cut}}$ values obtained from pexrav and xillver the reflector was
assumed to be neutral i.e. log$\zeta$ = 0 erg cm s-1. Treating the ionization
as a free parameter too, returned results consistent with the ones obtained
using log$\zeta$ = 0 erg cm s-1. The value of Fe abundance was fixed to the
solar value. The results of the model fits are given in Tables 2, 3 and 4.
Fitting the spectra using xillver yielded values of $\rm{E_{cut}}$ similar to
that obtained using pexrav, although the errors in the $\rm{E_{cut}}$
measurements using xillver are relatively smaller.

As we are interested in the determination of $\rm{kT_{e}}$ and its variation
we decided to use the thermal Comptonization model xillverCP (García et al.
2014). This model that takes into account the emission lines assuming it to be
originating from disk reflection includes the Comptonization model nthcomp
(Zdziarski et al. 1996; Życki et al. 1999) plus the ionized reflection model
xillver(García & Kallman 2010; García et al. 2013). The model fit to the
spectra has the following form in XSPEC,

$const*TBabs(xillverCP)$ (6)

We used xillverCP to model the primary continuum and the reflection spectrum
simultaneously. Also, to take care of the intrinsic absoption present in all
epochs of NGC 5548, epoch G, H and I of NGC 3227 and epoch E of MR 2251$-$178
spectra the zTBabs component was added with the described model. The results
of the model fits are given in Tables 2, 3 and 4.

##  4 Results and Discussion

Here, we aimed to find changes of $\rm{kT_{e}}$ in NGC 3227, NGC 5548 and MR
2251$-$178 . We discuss below the results obtained on them.

Table 4: Results of the model fits to the spectra of NGC 5548. The models are
Model I: const*TBabs*zTBabs(zpo+zgauss), Model II:
const*TBabs*zTBabs(pexrav+zgauss), Model III: const*TBabs*zTBabs(xillver) and
Model IV: const*TBabs*zTBabs(xillverCP). The width of the FeK$\alpha$ line was
frozen to the value of 0.1 keV during the fitting. The fluxes are in units of
10-10 erg cm-2 s-1 in the 4$-$60 keV band except in epoch F where the flux was
derived in 4$-$55 keV band. Columns and parameters have the same meaning as
given in Table 2

.  Parameter epoch A epoch B epoch C epoch D epoch E epoch F Model I :
const*TBabs*zTBabs(zpo+zgauss) $\Gamma$ 1.62${}^{+0.03}_{-0.03}$
1.60${}^{+0.03}_{-0.03}$ 1.59${}^{+0.02}_{-0.02}$ 1.64${}^{+0.02}_{-0.02}$
1.57${}^{+0.03}_{-0.02}$ 1.57${}^{+0.03}_{-0.01}$ $\rm{N_{H}}$(zTBabs)
3.89${}^{+1.76}_{-1.73}$ $<3.21$ 6.29${}^{+1.40}_{-1.32}$
2.70${}^{+1.28}_{-1.27}$ 8.97${}^{+1.65}_{-1.63}$ $<$1.28 E
6.37${}^{+0.09}_{-0.09}$ 6.33${}^{+0.08}_{-0.08}$ 6.32${}^{+0.04}_{-0.04}$
6.38${}^{+0.04}_{-0.04}$ 6.66${}^{+0.04}_{-0.04}$ 6.31${}^{+0.04}_{-0.04}$ EW
70${}^{+42}_{-23}$ 89${}^{+27}_{-31}$ 117${}^{+34}_{-34}$ 107${}^{+28}_{-29}$
114${}^{+26}_{-44}$ 137${}^{+24}_{-45}$ norm 100${}^{+10}_{-9}$
86${}^{+11}_{-11}$ 79${}^{+8}_{-8}$ 93${}^{+7}_{-6}$ 77${}^{+6}_{-5}$
67${}^{+5}_{-2}$ $\chi^{2}/dof$ 684/643 612/634 880/811 890/813 840/810
692/692 $\rm{C_{FPMA/FPMB}}$ 1.02${}^{+0.02}_{-0.02}$ 1.05${}^{+0.01}_{-0.01}$
1.02${}^{+0.01}_{-0.01}$ 1.05${}^{+0.01}_{-0.01}$ 1.02${}^{+0.01}_{-0.01}$
1.05${}^{+0.01}_{-0.01}$ Model II : const*TBabs*zTBabs(pexrav+zgauss) $\Gamma$
1.75${}^{+0.02}_{-0.10}$ 1.81${}^{+0.02}_{-0.08}$ 1.62${}^{+0.10}_{-0.10}$
1.79${}^{+0.09}_{-0.09}$ 1.61${}^{+0.10}_{-0.11}$ 1.81${}^{+0.08}_{-0.09}$
$\rm{N_{H}}$(zTBabs) 6.06${}^{+1.58}_{-2.46}$ 4.77${}^{+1.31}_{-2.05}$
5.61${}^{+2.34}_{-2.36}$ 4.46${}^{+2.13}_{-2.16}$ 6.91${}^{+2.45}_{-2.47}$
3.66${}^{+1.18}_{-2.10}$ $\rm{E_{cut}}$ $>$345 $>$502 152${}^{+158}_{-54}$
$>$170 160${}^{+178}_{-58}$ $>$414 R 0.19${}^{+0.14}_{-0.11}$
0.35${}^{+0.16}_{-0.14}$ 0.28${}^{+0.11}_{-0.10}$ 0.37${}^{+0.12}_{-0.11}$
0.33${}^{+0.12}_{-0.10}$ 0.42${}^{+0.14}_{-0.13}$ E 6.37${}^{+0.10}_{-0.10}$
6.32${}^{+0.09}_{-0.10}$ 6.32${}^{+0.04}_{-0.04}$ 6.38${}^{+0.05}_{-0.05}$
6.35${}^{+0.04}_{-0.04}$ 6.31${}^{+0.05}_{-0.05}$ EW 55${}^{38}_{-27}$
64${}^{+31}_{-46}$ 114${}^{+24}_{-38}$ 89${}^{+25}_{-20}$ 113${}^{+29}_{-34}$
104${}^{+44}_{-27}$ norm 126${}^{+7}_{-23}$ 125${}^{+15}_{-19}$
79${}^{+18}_{-15}$ 118${}^{+23}_{-20}$ 73${}^{+17}_{-14}$ 103${}^{+3}_{-17}$
$\chi^{2}/dof$ 675/641 588/632 834/809 840/811 780/808 652/690
$\rm{C_{FPMA/FPMB}}$ 1.02${}^{+0.02}_{-0.02}$ 1.05${}^{+0.01}_{-0.01}$
1.02${}^{+0.01}_{-0.01}$ 1.05${}^{+0.01}_{-0.01}$ 1.02${}^{+0.01}_{-0.01}$
1.05${}^{+0.01}_{-0.01}$ Model III : const*TBabs*zTBabs(xillver) $\Gamma$
1.75${}^{+0.02}_{-0.02}$ 1.78${}^{+0.02}_{-0.04}$ 1.68${}^{+0.02}_{-0.01}$
1.77${}^{+0.01}_{-0.01}$ 1.66${}^{+0.01}_{-0.02}$ 1.81${}^{+0.01}_{-0.01}$
$\rm{N_{H}}$(zTBabs) 6.47${}^{+0.98}_{-0.91}$ 4.83${}^{+0.95}_{-0.91}$
7.13${}^{+0.73}_{-0.73}$ 4.52${}^{+0.69}_{-0.67}$ 8.16${}^{+0.76}_{-0.75}$
4.25${}^{+0.83}_{-0.81}$ $\rm{E_{cut}}$ $>$487 $>$480 129${}^{+15}_{-13}$
179${}^{+40}_{-23}$ 133${}^{+15}_{-13}$ $>$395 R 0.25${}^{+0.08}_{-0.08}$
0.36${}^{+0.10}_{-0.09}$ 0.49${}^{+0.09}_{-0.08}$ 0.49${}^{+0.08}_{-0.08}$
0.51${}^{+0.09}_{-0.08}$ 0.54${}^{+0.11}_{-0.10}$ norm
4.35${}^{+0.10}_{-0.06}$ 3.61${}^{+0.47}_{-0.14}$ 2.20${}^{+0.03}_{-0.03}$
2.47${}^{+0.03}_{-0.03}$ 2.07${}^{+0.04}_{-0.04}$ 2.70${}^{+0.04}_{-0.04}$
$\chi^{2}/dof$ 677/643 595/634 851/811 849/813 792/810 666/692
$\rm{C_{FPMA/FPMB}}$ 1.02${}^{+0.02}_{-0.02}$ 1.05${}^{+0.01}_{-0.01}$
1.02${}^{+0.01}_{-0.01}$ 1.05${}^{+0.01}_{-0.01}$ 1.02${}^{+0.01}_{-0.01}$
1.05${}^{+0.01}_{-0.01}$ Model IV : const*TBabs*zTBabs(xillverCP) $\Gamma$
1.77${}^{+0.02}_{-0.02}$ 1.80${}^{+0.02}_{-0.03}$ 1.79${}^{+0.01}_{-0.01}$
1.85${}^{+0.01}_{-0.01}$ 1.77${}^{+0.01}_{-0.01}$ 1.84${}^{+0.01}_{-0.02}$
$\rm{N_{H}}$(zTBabs) 6.47${}^{+0.99}_{-0.91}$ 4.63${}^{+0.94}_{-0.92}$
8.72${}^{+0.73}_{-0.74}$ 5.66${}^{+0.69}_{-0.68}$ 9.79${}^{+0.76}_{-0.75}$
4.30${}^{+0.87}_{-0.92}$ $\rm{kT_{e}}$ $>$53 $>$54 39${}^{+14}_{-10}$
65${}^{+147}_{-24}$ 38${}^{+12}_{-9}$ $>$65 R 0.25${}^{+0.08}_{-0.09}$
0.35${}^{+0.11}_{-0.08}$ 0.43${}^{+0.08}_{-0.08}$ 0.46${}^{+0.09}_{-0.08}$
0.45${}^{+0.08}_{-0.08}$ 0.55${}^{+0.11}_{-0.11}$ norm
3.46${}^{+0.05}_{-0.05}$ 2.95${}^{+0.34}_{-0.04}$ 2.23${}^{+0.03}_{-0.03}$
2.46${}^{+0.03}_{-0.04}$ 2.07${}^{+0.03}_{-0.03}$ 2.34${}^{+0.08}_{-0.03}$
$\chi^{2}/dof$ 677/643 595/634 855/811 852/813 795/810 667/692
$\rm{C_{FPMA/FPMB}}$ 1.02${}^{+0.02}_{-0.02}$ 1.05${}^{+0.01}_{-0.01}$
1.02${}^{+0.01}_{-0.01}$ 1.05${}^{+0.01}_{-0.01}$ 1.02${}^{+0.01}_{-0.01}$
1.05${}^{+0.01}_{-0.01}$ Flux 1.23${}^{+0.01}_{-0.01}$
1.11${}^{+0.01}_{-0.01}$ 1.02${}^{+0.01}_{-0.01}$ 1.05${}^{+0.01}_{-0.01}$
0.96${}^{+0.01}_{-0.01}$ 0.92${}^{+0.01}_{-0.01}$

Figure 11: The unfolded spectra along with the data to model ratio for
const*TBabs(xillverCp) fit to epoch A (left panel) and epoch B (right panel)
observations of MR 2251$-$178\. Here blue points are for FPMA and yellow
points are for FPMB.

Figure 12: The unfolded spectra along with the data to model ratio for
const*TBabs(xillverCp) fit to epoch C (left panel), epoch D (middle panel) and
epoch E (right panel) observations of MR 2251$-$178\. Here blue points are for
FPMA and yellow points are for FPMB. In epoch E the zTBabs component was added
to the model.

###  4.1 NGC 3227

NuSTAR observed this source nine times between November 2016 and December
2019. Of these, results on seven epochs were reported by Kang et al. (2021),
wherein they were able to constrain $\rm{E_{cut}}$ in three epochs using
phenomenological fits and $\rm{kT_{e}}$ in two epochs using physical model
fits. Here, we report results for two additional epochs and for all the nine
epochs, we used both phenomenological and physical model fits to model the
spectra.

Ratios of the model const*TBabs(zpo + zgauss), const*TBabs(pexrav + zgauss)
and const*TBabs(xillverCP) fits to the observed FPMA and FPMB spectra carried
out on the OBSID 60202002002, the one with the longest exposure time are shown
in Fig. 2. Weak FeK$\alpha$ line is present in all the OBSIDs and therefore in
the const*TBabs(zpo) and const*TBabs(pexrav) model fits we included a zgauss
component to account for the line. All the above models could fit the spectra
of all epochs of NGC 3227 reasonably well except for epochs G, H and I, where
the $\chi^{2}/dof$ was greater than 1.2 and there is evidence for the presence
of significant absorption at the low energy end (see Fig. 5). Addition of an
extra absorption component zTBabs that takes into account the effects of host
galaxy absorption to all the models improved the fit significantly (see left,
middle and right panel of Fig. 8). The const*TBabs(pexrav + zgauss) model fit
to the FPMA/FPMB spectra showed evidence for variation in $\rm{E_{cut}}$. To
analyze the variation of $\rm{E_{cut}}$ with time as well as to find the
relation between $\rm{E_{cut}}$ and $\rm{kT_{e}}$ we used the $\rm{E_{cut}}$
values obtained from xillver model fits. The lowest value of $\rm{E_{cut}}$ of
92${}^{+10}_{-8}$ keV was obtained in epoch B and the highest value of
$\rm{E_{cut}}$ was observed in epoch E (378${}^{+152}_{-91}$ keV) while
intermediate values of $\rm{E_{cut}}$ were obtained during other epochs. These
observed variation of $\rm{E_{cut}}$ indicates that the coronal temperature of
NGC 3227 must be changing with time. This is also evident in Fig. 6 (upper
panel) where the values of $\rm{E_{cut}}$ are plotted against epochs. To
quantify the significance of the variation of $\rm{E_{cut}}$ we fitted a
constant (mean of all the $\rm{E_{cut}}$ measurements) to the variation of
$\rm{E_{cut}}$ with epoch and calculated the $\chi^{2}$ and the null
hypothesis probability (p) that $\rm{E_{cut}}$ does not change with time
(shown as a dashed line in the top panel of Fig. 6). We found $\chi^{2}$/dof
$>>$ 10, and a $p$ value of 0.0 that the $\rm{E_{cut}}$ does not change with
time. The values of $\rm{E_{cut}}$ obtained during the first two epochs (A and
B) are similar to the value of $\rm{E_{cut}}$ = 90$\pm$20 keV reported by
Markowitz et al. (2009). Vasudevan et al. (2013a) from an analysis of XMM and
BAT spectra pointed $\rm{E_{cut}}$ to lie outside the BAT range at $>$ 636
keV. Recently Kang et al. (2021) too reported $\rm{E_{cut}}$ values for epoch
A, B and G, and lower limits for epochs C,D,E and F from pexrav model fits.
The values of $\rm{E_{cut}}$ obtained here using xillver for epochs A, B and G
are in agreement with that of Kang et al. (2021) from using pexrav model.
Also, our model fits could constrain $\rm{E_{cut}}$ during epochs E and F
using xillver.

Comptonization model fits using xillverCP provided values of $\rm{kT_{e}}$
which too was found to vary between epochs. We could constrain $\rm{kT_{e}}$
for epochs A, B, C and G, and obtain lower limits for epochs D, E, F, H and I.
Kang et al. (2021) using the same model used here could constrain
$\rm{kT_{e}}$ for only epochs A and B. Our results for epochs A and B are in
agreement with that of Kang et al. (2021). The variation of $\rm{kT_{e}}$ is
shown in the lower panel of Fig. 6. The trend seen in the variation of
$\rm{kT_{e}}$ with epochs is similar to the variation of $\rm{E_{cut}}$. From
Comptonization model fits we found the lowest value of $\rm{kT_{e}}$ =
28${}^{+9}_{-6}$ keV during epoch B while the highest value of $\rm{kT_{e}}$ =
56${}^{+131}_{-18}$ keV was obtained for epoch E. We obtained lower limits of
$\rm{kT_{e}}$ of 80, 45, 47, 85 and 36 keV during epochs D, E, F, H and I
respectively. Spectral fits using xillverCP along with the data to model ratio
for all the epochs are shown in Figures 7 and 8. The results of the model fits
are given in Table 2. To quantify the significance of the variation of
$\rm{kT_{e}}$ we fitted a constant (mean of all the $\rm{kT_{e}}$
measurements) to the variation of $\rm{kT_{e}}$ with epoch and calculated the
$\chi^{2}$ and the null hypothesis probability (p) that $\rm{kT_{e}}$ does not
change with time (shown as a dashed line in Fig. 6). We found $\chi^{2}$/dof
of 12.86, and a $p$ value of 0.005 that $\rm{kT_{e}}$ does not change with
time. The contour plots between $\Gamma$ against $\rm{kT_{e}}$ and R against
$\rm{kT_{e}}$ for NGC 3227 for all the epochs are shown in Fig. 14 and 15
(left panels).From the contour plots it is evident that the coronal
temperature variation is prominent in NGC 3227. We therefore, conclude that we
found variation in the temperature of the corona in NGC 3227. According to
Petrucci et al. (2001) for an optically thick corona ($\tau$ $>$ 1)
$\rm{E_{cut}}$ = 3$\rm{kT_{e}}$. However, the relation between $\rm{E_{cut}}$
and $\rm{kT_{e}}$ can be complicated in the case of a non-static corona such
as the one with outflows (Liu et al. 2014). Also, according to Middei et al.
(2019), the relation of $\rm{E_{cut}}$ = 2$-$3 $\rm{kT_{e}}$ is valid only for
low values of $\tau$ and $\rm{kT_{e}}$. For NGC 3227 using all the five epochs
in which we could constrain both $\rm{E_{cut}}$ and $\rm{kT_{e}}$ we found
$\rm{E_{cut}}$ = 3.94$\pm$0.62 $\rm{kT_{e}}$ which is similar to that of
Petrucci et al. (2001) and Middei et al. (2019)

Table 5: Results of the correlation analysis between different parameters for NGC 3227. Provided are the slope (m), intercept (c), Pearson’s correlation coefficient (r) and the probability (p) for null hypothesis (no correlation) from OLS fit and the least squares fit from simulated points. See text in Section 4.4. Parameter | OLS | Simulated
---|---|---
| m | c | r | p | m | c | r | p
$\Gamma$/Flux | -0.07$\pm$0.03 | 1.90$\pm$0.03 | -0.63 | 0.07 | -0.07$\pm$0.03 | 1.91$\pm$0.04 | -0.64 | 0.06
$\rm{kT_{e}}$/Flux | 18$\pm$28 | 22$\pm$31 | 0.42 | 0.58 | 18$\pm$150 | 13$\pm$169 | 0.08 | 0.52
$\Gamma$/$\rm{kT_{e}}$ | 0.001$\pm$0.0006 | 1.75$\pm$0.03 | 0.86 | 0.14 | 0.0005$\pm$0.0002 | 1.78$\pm$0.02 | 0.80 | 0.20
$R$/$\rm{kT_{e}}$ | -0.005$\pm$0.005 | 1.03$\pm$0.22 | -0.57 | 0.43 | -0.001$\pm$0.002 | 0.90$\pm$0.17 | -0.36 | 0.60
$R$/Flux | -0.51$\pm$0.13 | 1.31$\pm$0.13 | -0.84 | 0.01 | -0.04$\pm$0.15 | 0.78$\pm$0.16 | -0.09 | 0.68
$\tau$/Flux | -0.29$\pm$0.62 | 2.35$\pm$0.65 | -0.17 | 0.66 | -0.98$\pm$1.88 | 3.15$\pm$2.12 | -0.35 | 0.65
$y$/Flux | -0.15$\pm$0.70 | 13.06$\pm$0.79 | -0.15 | 0.85 | -0.53$\pm$1.50 | 14.01$\pm$1.71 | -0.21 | 0.66
$\tau$/$\rm{kT_{e}}$ | -0.043$\pm$0.002 | 4.19$\pm$0.09 | -0.99 | 0.00 | -0.02$\pm$0.00 | 3.44$\pm$0.28 | -0.97 | 0.03

###  4.2 MR 2251$-$178

This source has five epochs of observations that are public and having
exposure $>$ 20 ks. In this work we analyzed all of them. Simple power law
fits to the FPMA spectra of all the epochs is shown in the right panel of Fig.
1. From this figure, noticeable change in the spectra could not be
ascertained. The ratio of the model fits const*TBabs(zpo), const*TBabs(pexrav)
and const*TBabs(xillverCP) to the observed FPMA and FPMB spectra on OBSID
90601637002, the one with the longest exposure time are shown in Fig. 4. From
two sets of observations from Einstein separated by about a year Halpern
(1984) found evidence of variable X-ray absorption in MR 2251$-$178, with the
column density changing from $<$ 5 $\times$ 1021 cm-2 to 2 $\times$ 1022 cm-2,
suggested to be due to the presence of a warm absorber. EXOSAT and Ginga
observations revealed strong correspondence between the absorbing column
density and the flux of the source with the low energy absorption decreasing
with the increasing flux of the source. These observations were explained by
variable absorption in photo-ionized gas along the line of sight (Mineo &
Stewart 1993; Pan et al. 1990). However, Walter & Courvoisier (1992) from an
analysis of the EXOSAT data argued that the variability seen in the source can
be explained without invoking the presence of a warm absorber. From the ratio
of the observed data to the model fit we did not find any signature of
absorption that could affect the source spectra in all observations but in
epoch E and to take care of this we added the zTBabs component to all the four
models in epoch E(see Fig 11 and 12). Model fits to the four sets of
observations that span about five years using const*TBabs(zpo) do not reveal
the presence of FeK$\alpha$ line in the spectra, the reflection bump was also
found to be either negligible or weak (see top panel of Fig. 4), more likely
due to poor S/N. However, the ratio plot for the model const*TBabs(zpo) to the
epoch D spectra revealed the presence of the FeK$\alpha$ line at around 6.4
keV. We therefore added a zgauss component with the model and found the energy
of the line at 6.49${}^{+0.37}_{-0.27}$ keV with a fixed width of 0.1 keV,
letting the parameter free did not significantly improve the fit. For this
source previously the FeK$\alpha$ line was reported to be present in the Ginga
observations with an equivalent width of 125${}^{+100}_{-105}$ eV (Mineo &
Stewart 1993). Relatively strong FeK$\alpha$ was also reported to be present
in the BeppoSaX observation (Orr et al. 2001) and a narrow FeK$\alpha$ line
was present in the Chandra observations (Gibson et al. 2005). From BeppoSAX
observations in the 0.1$-$200 keV band, Orr et al. (2001) found a
$\rm{E_{cut}}$ value of around 100 keV which is similar to that obtained here.

Figure 13: Correlation of $\Gamma$ and $\rm{kT_{e}}$ with the X-ray brightness
of NGC 3227. The orange dashed line is the ordinary linear least squares (OLS)
fit to the data points. The black dashed line is the linear line drawn using
the median values of the simulated parameters. The grey shaded region
indicates the upper and lower errors in the fit parameters for OLS and the
green region indicates the errors in the fit parameter obtained from the
simulation. In the least square analysis, epochs in which we were unable to
constrain $\rm{kT_{e}}$ were dropped (indicated with red point).

To find evidence for the change in $\rm{kT_{e}}$ if any, we carried out
fitting of the observations with the physical model, const*TBabs(xillverCP).
Examination of the results of the fit (Table 3) shows that the $\rm{kT_{e}}$
obtained during all the epochs agree within errors. Though we could not find
any signature of $\rm{kT_{e}}$ variation with epochs from the xillverCP fit,
the $\rm{E_{cut}}$ values obtained from the xillver model fit during epochs A,
C and D agree within errors except that of epoch B. This could be due to the
quality of the data in epoch B, as the values of $\rm{E_{cut}}$ and
$\rm{kT_{e}}$ obtained during that epoch also have large error bars. To test
for the $\rm{kT_{e}}$ variation if any, we plotted the 90 per cent contours
between $\rm{kT_{e}}$ and $\Gamma$. The 90 percent contours of $\rm{kT_{e}}$
against $\Gamma$ overlap (see Fig. 14) and we conclude that in MR 2251$-$178
we did not find any variation of $\rm{kT_{e}}$ with time. The 90 per cent
contours of R against $\rm{kT_{e}}$ are also shown in the bottom panel of Fig.
15. Considering all the four epochs where $\rm{E_{cut}}$ and $\rm{kT_{e}}$
could be constrained we found $\rm{E_{cut}}$ = 4.84$\pm$0.11 $\rm{kT_{e}}$.
This is deviant from the generally adopted $\rm{E_{cut}}$ = 2$-$3
$\rm{kT_{e}}$ (Petrucci et al. 2001). It is likely the relation between
$\rm{E_{cut}}$ and $\rm{kT_{e}}$ is complex and may depend on other physical
properties of the sources. Homogeneous analysis of a large number of sources
are needed to establish the relation between $\rm{E_{cut}}$ and $\rm{kT_{e}}$
as well as on its dependence on other physical properties. Spectral fits using
xillverCP along with the residuals for all the epochs are shown in Fig. 11 and
12.

Figure 14: The 90 per cent confidence level contours between $\rm{kT_{e}}$ and
$\Gamma$ for the xillverCP model fit to the FPMA/FPMB spectra of NGC
3227(left), MR 2251$-$178(middle) and NGC 5548(right). The colours indicate
different OBSIDs. Here for MR 2251$-$178 the limits of $\rm{kT_{e}}$ do not
have a correspondence with the values given in Table 3. This is because the
contours were generated by freezing the $R$ parameter so as to put all the
five OBSIDs together.

Figure 15: The 90 per cent confidence level contours between R and
$\rm{kT_{e}}$ for the xillverCP model fit to the FPMA/FPMB spectra of NGC
3227(left), MR 2251$-$178(middle) and NGC 5548(right). Figure 16: Correlation
between $\Gamma$ and $R$ with $\rm{kT_{e}}$ of NGC 3227. The dashed lines and
shaded regions have the same meaning as in Fig. 13. Epochs in which
$\rm{kT_{e}}$ could not be constrained (shown as red points) were not included
in the linear least square analysis.

###  4.3 NGC 5548

NGC 5548 was observed by NuSTAR six times between July 2013 to January 2021.
Of these, Zhang et al. (2018) have reported results for five epochs. In this
work we carried out both phenomenological and physical model fits for all the
six epochs. We fit the spectra using const*TBabs(zpo+zgauss) and
const*TBabs(pexrav+zgauss) to model the primary power law emission and the
reflection component along with the FeK$\alpha$ line with the width of the
line being frozen to the value of 0.1 keV. From the ratio of the observed data
to the model fit using TBabs we found the presence of absorption component in
the spectra. So unlike the other sources, for NGC 5548, we added an extra
component zTBabs in each model to account for the absorption by the host
galaxy. Moreover, as xillver self consistently models the reflected spectrum
and its associated FeK$\alpha$ line, we also fit the spectra using xillver and
$\rm{E_{cut}}$ values obtained using xillver model were used to find variation
in $\rm{E_{cut}}$. The $\rm{E_{cut}}$ obtained from xillver model fits are
$>487$ keV, $>480$ keV, 129${}^{+15}_{-13}$ keV, 179${}^{+40}_{-23}$ keV,
133${}^{+15}_{-13}$ and $>$394 keV for epochs A, B, C, D, E and F
respectively. The residuals of the fit to the spectra for the epoch D that has
the maximum exposure time, for various models are given in Fig. 3 and the
results of the fit are given in Table 4. From xillver model fits, we could
constrain $\rm{E_{cut}}$ only for epochs C, D and E, while Zhang et al. (2018)
could constrain $\rm{E_{cut}}$ in the four epochs (A, C, D and E). For the
epochs in common, though our values of $\rm{E_{cut}}$ are in agreement with
Zhang et al. (2018) the results do not match for epoch E. Though Zhang et al.
(2018) claims to have detected $\rm{E_{cut}}$ variation in NGC 5548, our
analysis could not confirm changes in $\rm{E_{cut}}$. This could be due to
differences in the choice of binning and the energy ranges used in both the
work. Ursini et al. (2015) via the joint fitting of XMM-Newton, Chandra,
NuSTAR, and INTEGRAL data obtained lower limits for $\rm{E_{cut}}$ in all the
epochs except for epoch D. Using the simultaneous XMM-Newton and NuSTAR data
from 2013 campaign Cappi et al. (2016) also fitted the 4$-$79 keV epoch A/B ,
epoch D and epoch E spectra using cut-off power-law and pexmon. Their values
of $\rm{E_{cut}}$ are in agreement with the $\rm{E_{cut}}$ values obtained in
this work.

Fitting the physical Comptonization model xillverCP to ascertain the change in
coronal temperature, we got the highest $\rm{kT_{e}}$ value of
65${}^{+147}_{-24}$ keV during epoch D, for epochs A, B and F we obtained
lower limits of 53, 54 and 71 keV, while for the remaining two epochs C and E,
we obtained similar values of 39${}^{+14}_{-10}$ keV and 38${}^{+12}_{-9}$ keV
respectively. The values of $\rm{kT_{e}}$ between epochs agree to each other
within error bars. The model fits to the data along with the data to model
ratio for all the epochs of observations are shown in Fig. 9 and 10. The
$\rm{E_{cut}}$ values obtained here points to variability. However, the
derived $\rm{kT_{e}}$ values are consistent with each other within error bars.
The contour plots between $\rm{kT_{e}}$ against $\Gamma$ as well as R against
$\rm{kT_{e}}$ are shown in the right most panels of Fig. 14 and 15. From these
plots, the change in $\rm{kT_{e}}$ is not evident. We thus conclude that we
have not found evidence for variation in $\rm{kT_{e}}$ in NGC 5548.
Considering all the epochs in which we could constrain both $\rm{E_{cut}}$ and
$\rm{kT_{e}}$, we found $\rm{E_{cut}}$ = 3.19$\pm$0.32 $\rm{kT_{e}}$ which is
in agreement with the empirical relation of Petrucci et al. (2001).

###  4.4 Correlation between different parameters

We discuss below the correlation between various parameters. This is
restricted to the source NGC 3227 as the temperature of its corona is found to
vary in this work. Since the errors in the measured $\rm{E_{cut}}$ and
$\rm{kT_{e}}$ values are not symmetric and there is no conventional way to
take care of such errors during correlation study we adopted two procedures to
find the correlation between various parameters.

In first case, we neglected the corresponding errors and considered only the
best fit values of the parameters and performed the ordinary linear least
square (OLS) fit between them. The Pearson’s correlation coefficient (r) and
the null hypothesis probability (p) for no correlation were also estimated
using the best fit values.

In the second case, to take into account the non-symmetric errors we simulated
$10^{5}$ points from each rectangular box around the best fit values (x,y)
with x and y boundaries of ($\rm{x_{low}}$,$\rm{x_{high}}$) and
($\rm{y_{low}}$,$\rm{y_{high}}$) respectively. Here $\rm{x_{low}}$,
$\rm{y_{low}}$ and $\rm{x_{high}}$, $\rm{y_{high}}$ refer to the lower and
upper errors in x and y values. Linear least squares fit was done for $10^{5}$
times yielding distribution of the slope (m), the intercept (c), the Pearson’s
rank correlation coefficient and the probability of no correlation. The median
values of the distributions were taken to represent the best fit values of the
correlation. All the values and the errors for the unweighted as well as for
the simulated correlation are given in Table 5.

####  4.4.1 $\Gamma$ v/s Flux

In Seyfert galaxies, the X-ray spectra are generally found to be softer with
increasing X-ray flux (Markowitz et al. 2003). We show in Fig. 13, the
correlation between $\Gamma$ and the brightness of NGC 3227. For the source,
each point in the Figure corresponds to $\Gamma$ and flux obtained by
xillverCP model fits to each epoch of spectra. Ordinary linear least squares
fit to the data are shown by orange dashed lines. The black dashed lines show
the linear least square fit using the median values of the simulated points
using the lower and upper errors in $\Gamma$ and flux values. The grey and
green shaded regions are the area bounded by the errors in the least square
fit parameters. For NGC 3227, we found anti correlation between $\Gamma$ and
flux (see Fig. 13 and Table 5) between epochs of observations significant at
the greater than 90 per cent level.

####  4.4.2 $\rm{kT_{e}}$ v/s Flux

Correlation between the changes in the temperature of the corona with other
physical parameters of the sources such as its apparent brightness as well as
its spectral shape can provide important constraints in enhancing our
understanding of AGN corona. The correlation between $\rm{kT_{e}}$ and flux is
shown in Fig. 13. Also, shown in the figure are the ordinary and simulated
linear least squares fit to the data. No correlation is found between
$\rm{kT_{e}}$ and flux in NGC 3227.

####  4.4.3 $\Gamma$ v/s $\rm{kT_{e}}$

In Fig. 16 the correlation between $\Gamma$ and $\rm{kT_{e}}$ is shown. The
orange and black dashed lines represent the ordinary and simulated least
square fit to the data. The correlation between $\Gamma$ and $\rm{kT_{e}}$ is
found to be not significant.

####  4.4.4 $R$ v/s $\rm{kT_{e}}$

The correlation between the distant reflection fraction $R$ and $\rm{kT_{e}}$
is presented in the bottom panel of Fig. 16. We did not find any correlation
between these two parameters.

###  4.5 Nature of corona in AGN

The primary X-ray emission from thermal Comptonization depends on the optical
depth $\tau$ and $\rm{kT_{e}}$ as (Zdziarski et al. 1996; Życki et al. 1999),

$\tau=\sqrt{\frac{9}{4}+\frac{3}{\theta\Big{[}\Big{(}\Gamma+\frac{1}{2}\Big{)}^{2}-\frac{9}{4}\Big{]}}}-\frac{3}{2}\\\
$ (7)

where, $\theta={kT_{e}}/{m_{e}c^{2}}$. We show in Fig. 18 the variation of the
reflection fraction, the optical depth and the Compton $y$ parameter with
flux. The Compton $y$ parameter is defined as (Petrucci et al. 2001),

$y\simeq
4\Bigg{(}\frac{4kT_{e}}{mc^{2}}\Bigg{)}\Bigg{[}1+\Bigg{(}\frac{4kT_{e}}{mc^{2}}\Bigg{)}\Bigg{]}\tau(1+\tau)$
(8)

According to Stern et al. (1995), a Comptonized corona must have a constant
$y$ in equilibrium. We too found no correlation of $y$ with the flux of the
source. The parameter $\tau$ is found not to show any statistically
significant variation with flux. $R$ is found to be not correlated with
flux(see Fig. 17).

Figure 17: Distant reflection fraction $R$, the optical depth $\tau$ and
Compton-y parameter as a function of flux for the sources NGC 3227. The dashed
lines and the shaded regions have the same meaning as in Fig. 13. Points shown
as red in the Figure were not included in the linear least squares fit.

Multiple epochs of observations on few AGN available today point to increased
$\rm{E_{cut}}$ with the flux of the sources. For example, from analysis of
NuSTAR and XMM-Newton observations of NGC 5548, Zhang et al. (2018) found
$\rm{E_{cut}}$ to be positively correlated with the flux of the source.
Similarly in Mrk 335 too, Keek & Ballantyne (2016) through joint fitting of
Suzaku and NuSTAR found $\rm{E_{cut}}$ to be positively correlated with flux.
Recently from flux resolved spectroscopy of Ark 564, Barua et al. (2020) found
the temperature of the corona to decrease with increasing flux. Also, for the
source ESO 103$-$035 Barua et al. (2021) found a positive correlation between
$\rm{kT_{e}}$ and flux. Scenarios that could cause the change in the
temperature of the corona or $\rm{E_{cut}}$ in AGN are (a) Compton cooling and
(b) expanding corona. In the Compton cooling scenario, as the source
brightens, there will be increased seed UV photons from the disk for
Comptonization leading to a cooling of the corona. This leads to a cooler when
brighter behaviour. In the expanding corona scenario, the increase in
$\rm{kT_{e}}$ during high flux states of the AGN could be due to changes in
the geometry of the corona. This has been invoked to explain the
$\rm{E_{cut}}$ changes in Mrk 335 (Keek & Ballantyne 2016) and NGC 5548 (Zhang
et al. 2018) . According to the expanding corona model, at low flux state, the
corona is warm, optically thick, compact and close to the black hole. This
causes more illumination of the disk leading to larger reflection fraction. As
the source increases in brightness, the corona expands, the optical depth
drops and the temperature rises. A reduced reflection fraction during this
period argues that the corona expands vertically from the disk. In NGC 3227
too, the corona temperature is found to vary with time. We found a negative
correlation between the optical depth $\tau$ and $\rm{kT_{e}}$, with $\tau$
decreasing towards higher temperatures (see Fig. 18). We calculated $\tau$
using Equation 7 and according to Zdziarski et al. (1996) $\tau$ is geometry
dependent and equals the radial optical depth in an uniform sphere. The
negative correlation between $\tau$ and $\rm{kT_{e}}$ argues for a change in
the geometry of the corona (Ballantyne et al. 2014). Tortosa et al. (2018) too
from the analysis of a sample of AGN found a negative correlation between
$\tau$ and $\rm{kT_{e}}$. According to the authors this negative correlation
could not be explained with a fixed disk-corona configuration in radiative
balance. The possible explanation for this kind of behaviours could be (a) due
to the change in the geometry and position of the corona and/or (b) variation
in the fraction of the intrinsic disk emission to the total disk emission. We
note that broad band spectral energy distribution fits to simultaneous UV to
hard X-ray data alone will be able to provide strong constrain on $\tau$. This
in turn can put constrains on the role of accretion disk emission to
$\rm{kT_{e}}$ changes. In NGC 3227 we found no statistically significant
correlation between $\rm{kT_{e}}$ and flux, $R$ and flux, $\Gamma$ and flux as
well as $\Gamma$ and $\rm{kT_{e}}$.

##  5 Summary

In this work, we carried out spectral analysis of the NuSTAR data for three
Seyfert type AGN namely NGC 3227, NGC 5548 and MR 2251$-$178 including a few
OBSIDs not analyzed yet in comparison to previous works. We summarize our
results below

  1. 1.

All the sources were found to show moderate variations in their average
brightness during the epochs analyzed in this work.

  2. 2.

In NGC 3227, we found unambiguous evidence for the change in the temperature
of the corona. This change in $\rm{kT_{e}}$ is also reflected in the variation
in $\rm{E_{cut}}$. For NGC 5548 and MR 2251$-$178 we found no evidence for the
variation in the temperature of the corona.

Our knowledge on the variation in the temperature of the corona is known only
for less than half a dozen sources. Details on such coronal temperature
variation in more AGN are needed to pin point the reasons for the temperature
of the corona to vary and its effect on the other physical properties of the
sources.

##  6 Acknowledgements

We thank the anonymous referee for her/his useful comments and suggestions
which improved the quality and the clarity of the paper. We thank the NuSTAR
Operations, Software and Calibration teams for support with the execution and
analysis of these observations. This research has made use of the NuSTAR Data
Analysis Software (NuSTARDAS) jointly developed by the ASI Science Data Center
(ASDC, Italy) and the California Institute of Technology (USA). This research
has made use of data and/or software provided by the High Energy Astrophysics
Science Archive Research Center (HEASARC), which is a service of the
Astrophysics Science Division at NASA/GSFC.

Figure 18: Correlation between $\tau$ and $\rm{kT_{e}}$ of NGC 3227. The
dashed lines and the shaded regions have the same meaning as in Fig. 13.
Points shown as red in the Figure were not considered for the linear least
squares fit.

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