# The accretion properties of a low-mass Active Galactic Nucleus: UGC 6728

Prantik Nandi,1 Sachindra Naik,1 Arka Chatterjee,2 Sandip K. Chakrabarti,3
Samar Safi-Harb2 and Neeraj Kumari1
1Astronomy and Astrophysics Division, Physical Research Laboratory,
Ahmedabad-380009, India
2Department of Physics & Astronomy, University of Manitoba, Winnipeg, R3T 2N2,
Canada
3Indian Centre for Space Physics, Netaji Nagar, Kolkata, 700099, India
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a comprehensive analysis of approximately $15$ years ($2006-2021$)
of X-ray observations of UGC 6728, a low-mass bare AGN, for the first time.
Our study encompasses both spectral and temporal aspects of this source. The
spectral properties of this source are studied using various phenomenological
and physical models. We conclude that (a) the observed variability in X-ray
luminosity is not attributed to the Hydrogen column density ($N_{H}$) as UGC
6728 exhibits a bare nucleus, implying a negligible $N_{H}$ contribution along
the line of sight, and (b) the spectral slope in the X-ray band demonstrates a
systematic variation over time, indicating a transition from a relatively hard
state to a comparatively soft state. We propose that the underlying accretion
dynamics around the central object account for this behavior. By performing
X-ray spectral fitting, we estimate the mass of the central supermassive black
hole (SMBH) in UGC 6728 to be $M_{BH}=(7.13\pm 1.23)\times 10^{5}$ M⊙ with
spin $a=0.97^{+0.20}_{-0.27}$ and inclination angle $i=49.5\pm 14.5$ degree.
Based on our spectral and temporal analysis, we suggest that UGC 6728 lacks a
prominent Compton hump or exhibits a very subtle hump that remains
undetectable in our analysis. Furthermore, the high-energy X-ray photons in
this source are likely to originate from the low-energy X-ray photons through
inverse Compton scattering in a Compton cloud, highlighting a connection
between the emission in two energy ranges. We notice a strong soft excess
component in the initial part of our observations, which later reduced
substantially. This variation of soft excess is explained in view of accretion
dynamics.

###### keywords:

galaxies: active – galaxies: Seyfert – X-rays: galaxies – X-rays: individual:
UGC 6728.

††pubyear: 2023††pagerange: The accretion properties of a low-mass Active
Galactic Nucleus: UGC 6728–The accretion properties of a low-mass Active
Galactic Nucleus: UGC 6728

##  1 Introduction

Active galactic nuclei (AGNs) are among the most luminous sources which are
powered by the process of accretion of matter from the host galaxy (Storchi-
Bergmann & Schnorr-Müller, 2019) onto the supermassive black holes (SMBHs)
(Lynden-Bell, 1969; Rees, 1984) located at their center. The mass of an SMBH
ranges from $10^{5}$ to $10^{9}$ M⊙ (Kormendy & Richstone, 1995; Peterson et
al., 2004; Event Horizon Telescope Collaboration et al., 2019). The AGNs are
generally characterized by their complex spatial and temporal variabilities
(McHardy et al., 1999; Nandra, 2001; Webb & Malkan, 2000; Ballantyne et al.,
2014; Zoghbi et al., 2017) over the entire range of the electromagnetic
spectrum, starting from radio to very high energy $\gamma-$rays. The X-ray
photons are believed to be originated from a region closer to the central SMBH
(Fabian et al., 1989; Wilkins et al., 2021). Thus, X-ray spectroscopy is a
powerful tool to probe the innermost regions of the accretion disc around the
SMBHs. It allows us to determine the physical conditions of the plasma
surrounding the central X-ray source. By observing these X-ray photons, we can
characterize the fundamental parameters of the black hole, such as its mass,
spin, etc. These parameters play a crucial role in understanding the physical
nature of the central engine and its cosmological evolution (Fabian, 2012;
Kormendy & Ho, 2013). However, our knowledge of the physical processes in AGNs
is mostly based on the galaxies which harbour a SMBH of mass greater than
$10^{6}$ M⊙. However, the low mass AGNs $(M_{BH}<10^{6}M_{\odot})$ are yet to
be explored. One can explore the missing link to probe whether the accretion
mechanism is independent of the mass of the black hole.

According to standard accretion disc theory (Pringle et al., 1973; Shakura &
Sunyaev, 1973), X-ray photons are thought to be originated from the innermost
region of the accretion disc, which surrounds the central black hole.
Observationally, the X-ray spectra of AGNs are explained by the power-law
continuum (Bianchi et al., 2009; Sobolewska & Papadakis, 2009) with high-
energy cut-off (Fabian et al., 2015, 2017; Tortosa et al., 2018; Middei et
al., 2019; Baloković et al., 2020). The power-law continuum is thought to be
generated by inverse Comptonization (Sunyaev & Titarchuk, 1980a) of thermal
optical/UV photons in a hot corona or Compton cloud located close to the
accretion disc (Haardt & Maraschi, 1991). However, the geometry and location
of the Compton reprocessing region are still not well understood. This cloud
can be located above the disc (Haardt & Maraschi, 1991, 1993; Poutanen &
Svensson, 1996) or at the base of the jet (Chakrabarti & Titarchuk, 1995;
Fender et al., 1999; Markoff et al., 2005). This region is formed by the hot
($T\sim 10^{9}$ K) and radiatively inefficient plasma (Sunyaev & Titarchuk,
1980a), which behaves like a quasi-Bondi flow (Ichimaru, 1977).

In addition to the power-law continuum, the other spectral features, such as
absorption (Turner et al., 1997; Risaliti et al., 1999; Turner & Miller,
2009), reflection (Ross & Fabian, 1993, 2005; García et al., 2013, 2014), soft
X-ray excess emission (Halpern, 1984; Singh et al., 1985; Arnaud et al., 1985;
Miniutti & Fabian, 2004; Crummy et al., 2006; Walton et al., 2013; Liebmann et
al., 2018; Nandi et al., 2021, 2023), Fe lines (Ross & Fabian, 2005; Fabian et
al., 2009; García & Kallman, 2010) and a high-energy cut-off (Sunyaev &
Titarchuk, 1980a; Haardt & Maraschi, 1991, 1993) are also present in the X-ray
spectra of the AGNs. The absorption effect on the X-ray spectrum is produced
by the equivalent Hydrogen column density along the line of sight (Antonucci &
Miller, 1985; Netzer, 2013; Jana et al., 2020). We often observe an excess
emission in the soft X-ray (below $2$ keV) energy band, known as ‘soft-excess’
(Halpern, 1984; Singh et al., 1985; Arnaud et al., 1985; Magdziarz et al.,
1998; Jin et al., 2012; Done et al., 2012; Petrucci et al., 2013, 2018; Kubota
& Done, 2018; Middei et al., 2020; Nandi et al., 2021). This feature is mostly
observed in Seyfert 1 AGNs (Pravdo et al., 1981; Arnaud et al., 1985; Singh et
al., 1985; Turner & Pounds, 1989), a subclass of radio-quiet AGNs. The X-ray
spectra below $10$ keV are often associated with several fluorescent emission
lines. Among them, the most prominent is the Fe Kα line at $\sim 6.4$ keV
(Pounds et al., 1990; Ross & Fabian, 2005; Fabian et al., 2009; García &
Kallman, 2010). From the albedo calculation of the accretion disc, the
Comptonized photons produce the reflection component (Chakrabarti & Titarchuk,
1995). The signature of this component is visible in $15$ to $50$ keV range
with a peak at around $\sim 30$ keV, which is called the ‘Compton hump’
(Krolik, 1999).

In this scenario, X-ray observations play a crucial role in constraining the
geometry, such as the size and location, of the emitting region with a proper
understanding of its formation. A common characteristic observed in X-ray
studies of nearby bright Seyfert galaxies is the detection of X-ray
reverberation lags. This phenomenon suggests that the X-ray emitting region is
compact and lies within a few tens of the gravitational radii of the black
hole (Fabian et al., 2009; De Marco et al., 2013; Cackett et al., 2014;
Emmanoulopoulos et al., 2014; Uttley et al., 2014; Kara et al., 2016). The
spectral modelling of optical to X-ray spectra from AGNs also yields a similar
size of X-ray emitting region (Petrucci et al., 2013; Done et al., 2013; Kara
et al., 2013; Nandi et al., 2019; Nandi et al., 2021). From a physical
perspective, the two-component advective flow (TCAF) (Chakrabarti & Titarchuk,
1995) model may provide a self-consistent picture of the accretion dynamics
around the central SMBHs. This model is quantified by four flow parameters:
two types of accretion flow rates, namely, disc rate $(\dot{m}_{d})$ (Shakura
& Sunyaev, 1973) and halo rate $(\dot{m}_{h})$ (Chakrabarti, 1989, 1990, 1995)
in the unit of Eddington rate $(\dot{M}_{Edd})$, the dimension of the Compton
cloud or CENBOL through shock location $(X_{s})$ in the unit of Schwarzschild
radius $(R_{s}=2GM_{BH}/c^{2})$, and the density of this cloud by the shock
compression ratio $(R)$ (Chakrabarti, 1989), which is defined as the ratio of
the post-shock and the pre-shock flow densities. Along with these parameters,
we also need an intrinsic parameter, namely, the mass of the central black
hole (in the unit of M⊙) and an extensive parameter, called normalization,
which is used to match the observed spectrum with the theoretical one, and
essentially a function of the distance of the source and the collecting area
of the instrument. This model is used to constrain the geometry around the
black hole mass from few solar mass (Chakrabarti, 1997; Dutta & Chakrabarti,
2010; Debnath et al., 2014) to supermassive AGNs (Mandal & Chakrabarti, 2008;
Nandi et al., 2019; Nandi et al., 2021). However, for the reflection-dominated
AGN spectra, various geometries for the Compton cloud have been proposed for
the X-ray spectra fitting (Dauser et al., 2013; García et al., 2014;
Niedźwiecki et al., 2016; Ingram et al., 2019; Mastroserio et al., 2021;
Dovčiak et al., 2022). It is to be noted that the TCAF model includes the
reflection component in the very process of obtaining the solution
(Chakrabarti & Titarchuk, 1995).

UGC 6728 is a relatively less explored nearby
($z$=0.0065111https://ned.ipac.caltech.edu/byname?objname=UGC+6728&hconst=67.8&omegam=0.308&omegav=0.692&wmap=4&corr_z=1
(Kalberla et al., 2005)) Seyfert 1 AGN, which shows broad emission lines in
the optical band (Bentz et al., 2016). This AGN is a late-type galaxy and
contains an SMBH of mass M${}_{BH}=(7.1\pm 4.0)\times 10^{5}$ M⊙ (Bentz et
al., 2016). The X-ray observation of UGC 6728 reported that it has a ‘bare
nucleus,’ i.e., little or no obscuration along the line of sight to the
nucleus (Walton et al., 2013).

We organize the paper as follows. In Section 2, we describe the details of the
data selection and reduction techniques for different observatories. Then in
Section 3, we perform our analysis of X-ray data from various observations and
present the corresponding results. We further divide our analysis into two
sub-sections. In Section 3.2, we analyze and discuss the timing properties of
corresponding observations. Section 3.1 describes the various phenomenological
and physical models to extract different physical properties of this source.
Later, we discuss our findings from the X-ray data analysis of this source in
Section 4. Finally, we draw our conclusions based on our results in Section 5.

##  2 Observations and Data Reduction

In the present work, we use publicly available archival data of the low mass,
bare AGN UGC 6728 from XMM-Newton, Swift, and NuSTAR using
HEASARC222http://heasarc.gsfc.nasa.gov/. We reprocessed all the data using
HEAsoft v6.26.1 (Arnaud, 1996), which includes Xspec v12.10.1f.

###  2.1  NuSTAR

NuSTAR is a hard X-ray focusing telescope that operates in the energy range
from $3.0$ to $79.0$ keV using two identical moduli, FPMA and FPMB (Harrison
et al., 2013). This telescope observed UGC 6728 in two phases in 2016 and
2017. The details of the observations are listed in Table 1. We obtained the
observational data from NASA’s HEASARC
archive333https://heasarc.gsfc.nasa.gov/cgi-bin/W3Browse/w3browse.pl and used
standard data reduction
technique444https://heasarc.gsfc.nasa.gov/docs/nustar/analysis/ to reprocess
the data. The data were reprocessed by NuSTAR Data Analysis Software
NUSTARDASv1.4.1. We used standard filtering criteria to generate clean event
files using the nupipeline task. We used the latest calibration data
(20230918) from the NuSTAR
database555https://heasarc.gsfc.nasa.gov/FTP/caldb/data/nustar/fpm/ to
calibrate the data. The source and the background products were extracted by
considering circular regions of $60$ arcsec and $120$ arcsec radii,
respectively. We considered the centre of the source region to be at the
source coordinates. The background region was considered far from the source
region to avoid any contamination. Later, we used the nuproduct task to
extract the source spectra and rebinned them by ensuring at least 50 counts
per bin by using the GRPPHA task.

###  2.2  XMM-Newton

UGC 6728 was observed with XMM-Newton (Jansen et al., 2001) only once on 23
February 2006. The XMM-Newton observation of the source was carried out in
Full Window Mode. The details of the observation are given in Table 1. We used
the Science Analysis System (SAS v16.1.0666https://www.cosmos.esa.int/web/xmm-
newton/sas-thread-epatplot) to reprocess the raw data of EPIC-pn (Strüder et
al., 2001) with calibration data files dated 2 February 2018. To prevent the
edge of CCD and the bad pixels, we used only the unflagged events using FLAG
== 0. Along with this, we also used PATTERN $\leq$ $4$ for single and double
pixels. We exclude the photon flares by considering appropriate Good Time
Interval (GTI) files to acquire the maximum signal-to-noise ratio. For the
spectrum, initially, we used circular regions of $30$ arcsec and $60$ arcsec
radii for the source and background, respectively. Then we apply the SAS task
especget for spectrum extraction. After that, we inspected the spectrum for
photon pile-up using SAS task epaplot. To remove the pile-up effect, we
considered numerous values of the inner radii for the source and investigated
the presence of pile-up777https://www.cosmos.esa.int/web/xmm-newton/sas-
thread-epatplot. Finally, we found that in an annular region with inner and
outer radii of $10$ arcsec and $30$ arcsec, respectively, the source spectrum
became free from the pile-up effect. We use this annular region to extract the
final source spectrum and use it to fit various spectral models. The response
files (arf and rmf files) were generated with the SAS task arfgen and rmfgen,
respectively. Finally, we rebinned the data by ensuring at least $100$ counts
per bin using GRPPHA on the final $0.2$ to $10.0$ keV EPIC-pn spectrum.

###  2.3  Swift

Swift/XRT (Burrows et al., 2005) is an X-ray focusing telescope sensitive in
the energy range of $0.3$ to $10.0$ keV. UGC 6728 was monitored with this
telescope in three epochs from 2006 to 2021. Between them, this source was
observed simultaneously with NuSTAR in 2016 and 2017 (see Table 1 for
details). To generate X-ray spectra between $0.3$ to $10.0$ keV, we used
standard online tools provided by the UK Swift Science Data
Centre888https://www.swift.ac.uk/user_objects/ (Evans et al., 2009). In the
case of 2021 observations, we combined all the spectra into a single spectrum
to achieve enough exposure time. We also considered these observations
individually in our spectral fitting using a simple power-law model. However,
we did not observe any significant difference in the model parameters.
Finally, we used GRPPHA task to rebin the XRT spectra, ensuring a minimum of
10 counts per bin.

Table 1: Observation Log.

ID | Date | Obs. ID | Instrument | Exposures
---|---|---|---|---
| (yyyy-mm-dd) |  |  | (ks)
XMM | 2006-02-23 | 0312191601 |  XMM-Newron/epic-pn | 11.8
SU | 2009-06-06 | 704029010 |  Suzaku/XIS-HXD | 46 & 39†
XRT1 | 2016-07-10 | 00081098001 |  Swift/XRT | 06.9
N1 | 2016-07-10 | 60160450002 |  NuSTAR/FPMA-FPMB | 22.6
XRT2 | 2017-10-13 | 00088256001 |  Swift/XRT | 06.9
N2 | 2017-10-13 | 60376007002 |  NuSTAR/FPMA-FPMB | 58.1
XRT3 | 2020-09-05 | 00013662001 |  Swift/XRT | 01.8
XRT4 | 2021-04-13 | 00013662002 |  Swift/XRT | 02.2
| -2021-10-25 | 00096132008 |  |

† exposure time for XIS and HXD, respectively.

###  2.4  Suzaku

Suzaku observed UGC 6728 on 6 June 2009, with the X-ray imaging spectrometer
(XIS) (Koyama et al., 2007) and the Hard X-ray Detector (HXD) (Takahashi et
al., 2007) for effective exposures of approximately $\sim 46$ ks and $\sim 39$
ks, respectively. The XIS observations were carried out in STANDARD mode,
whereas the HXD observation was carried out in WELL mode. This observation was
previously used by Walton et al. (2013) in their study, where they reported
the presence of narrow and broad iron emission features. However, they
encountered challenges in constraining the reflection parameters and
consequently, fixed the inclination angle for this source at 45 degrees.

For the data reduction, we followed the standard procedures as described in
the Suzaku Data Reduction
Guide999https://heasarc.gsfc.nasa.gov/docs/suzaku/analysis/abc/ and applied
recommended screening criteria to generate the final spectra and light curves.
We used calibration files101010www.astro.isas.jaxa.jp/suzaku/caldb/ (dated
2016–16–06) in FTOOLS v6.25 to reprocess the event files. Circular regions of
radii 200 arcsecs and 250 arcsecs with center at the source coordinates were
considered for the source and background products, respectively. The final
spectrum and light curve were obtained by merging data from the two front-
illuminated detectors (XIS0 and XIS3). The response files were generated using
XISRESP task. In our analysis, we excluded the data in the 1.6–2.0 keV range
to avoid the effect of known Si K-edge. The XIS spectrum was subsequently
binned using the GRPPHA task, with a minimum of 100 counts per bin. For the
HXD/PIN data reduction, we applied hxdpinxbpi and hxdpinxblc tasks on the
event file to produce the spectrum and light curves. Furthermore, the HXD
spectrum was corrected for the non-X-ray background, cosmic X-ray background
and dead time.

##  3 Result

###  3.1 Spectral Analysis

We performed X-ray spectral analysis using data from Swift/XRT, XMM-
Newton/EPIC-pn, Suzaku/XIS-HXD and NuSTAR/FPMA-FPMB instruments, covering a
duration of approximately 15 years (2006-2021). The spectral analysis was
carried out using $\chi^{2}$ statistics onto the observational data in Xspec
v12.10.1f (Arnaud, 1996) software package. The details of the observations
from multiple observatories used in the current work are presented in Table 1.
For the Suzaku observation, data in the range of 0.5–30 keV are used in the
spectral analysis. In 2016 and 2017, the source was observed with Swift/XRT
and NuSATR simultaneously. For these observations, data in the energy ranges
of 0.3–10 keV and 3–79 keV are used from XRT and NuSTAR, respectively. Apart
from these simultaneous broad-band observations, UGC 6728 was also observed
with XMM-Newton in 2006 and Swift/XRT in 2020 and 2021 in the soft X-ray
energy band (below 10 keV).

Data from these observations are fitted using multiple spectral models, and
the error associated with each parameter of the model is determined at a 90%
confidence level (or $1.6\sigma$) using the
‘error’111111https://heasarc.gsfc.nasa.gov/xanadu/Xspec/manual/node79.html
task in Xspec. The unabsorbed luminosity is calculated using the
‘clumin’121212https://heasarc.gsfc.nasa.gov/xanadu/Xspec/manual/node286.html
task on the composite model. The Eddington luminosity for UGC 6728 is computed
as $L_{Edd}=8.91\times 10^{43}$ erg/s, based on a mass of $7.1\times 10^{5}$
$M_{\odot}$ (Bentz et al., 2016). In this work, we adopted the following
cosmological parameters:
$H_{0}=70\text{~{}km~{}s}^{-1}\text{~{}Mpc}^{-1};~{}\Lambda_{0}=0.73,\text{~{}and
}\Omega_{M}=0.27$ (Bennett et al., 2003).

####  3.1.1  Model Construction

As the X-ray spectral properties of UGC 6728 are not well understood to date,
our motivation was to characterize the X-ray spectrum of this source. For this
purpose, at first, we parameterize the X-ray spectrum across a broad energy
range, from 0.3 to 79.0 keV, using various simple models (Power law, Gaussian,
etc.). Then, we focus on the primary continuum within the energy range of 3.0
to 10.0 keV. According to current understanding, the X-ray continuum photons
are produced mainly by the inverse Compton scattering of thermal photons from
the accretion disc in a hot electron cloud. This process, being non-thermal,
often leads to a power-law-type spectrum with a high-energy cut-off.
Therefore, we begin our analysis by employing the power law
(powerlaw131313https://heasarc.gsfc.nasa.gov/xanadu/Xspec/manual/node216.html)
and cut-off power law
(cutoffpl141414https://heasarc.gsfc.nasa.gov/xanadu/Xspec/manual/node161.html)
models to characterize the primary continuum. The equivalent hydrogen column
density along the line of sight of the source modifies the shape of the power
law component. To account for this, we consider the Galactic hydrogen column
density as $N_{H,gal}$ at $4.42\times 10^{20}$
cm-2151515https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3nh/w3nh.pl using a
multiplicative model
TBabs161616https://heasarc.gsfc.nasa.gov/xanadu/Xspec/manual/node268.html
(Verner et al., 1996) in Xspec. Along with this, we introduce another
multiplicative model component zTBabs (Wilms et al., 2000) to incorporate the
extragalactic hydrogen column density ($N_{H}$) with redshift (z) $=0.0065$.
So, our base model to fit the X-ray continuum is expressed as
TBabs*zTBabs*constant*(powerlaw/cutoffpl), where constant is used for cross-
normalisation for different instruments.

We fit each spectrum by this baseline model, and the corresponding variation
of $\chi_{red}$ for the 2017 XRT2+N2 observation is shown in the top panel of
Figure 1. During the spectral fitting, a distinctive line-like feature emerged
in the energy range of 6-7 keV for the SU (MJD-54988), XRT1+N1 (MJD-57579),
and XRT2+N2 (MJD-58093) observations. To account this feature, we introduce a
Gaussian model,
zGauss171717https://heasarc.gsfc.nasa.gov/xanadu/Xspec/manual/node176.html
with z=0.0065 in Xspec and the corresponding variation of $\chi_{red}$ for the
2017 XRT2+N2 observation is shown in the second panel of Figure 1.

After successfully fitting the primary continuum with the Fe-line, we extend
the X-ray spectrum into the high-energy regime (above 10.0 keV) and observe no
significant deviations. This suggests that the reflection component in the
X-ray spectra above 10.0 keV is either absent or not significant within a 90%
confidence level. Next, we extend the X-ray spectra into the low energy regime
(below 3.0 keV) and observe a significant deviation of the model from the
observed spectra. Given that UGC 6728 is a Seyfert 1 AGN, it is expected to
exhibit a soft-excess component below 2.0 keV. We initially model this
component by powerlaw (Nandi et al., 2021, 2023). Subsequently, we fit the
broadband spectra (0.3–79.0 keV) using physical models, such as TCAF, relxill.
We present the variation of $\chi_{red}$ for the 2017 XRT2+N2 observation in
the last panel of Figure 1 using TCAF model fitting.

####  3.1.2  Power-law

We start the spectral fitting using the power-law model to characterize the
primary continuum within the energy range of 3.0-10.0 keV. This model,
represented in Xspec as tbabs*zTbabs*(powerlaw+zGauss), is used as the
baseline for our analysis. The zGauss component is used to compensate for the
Fe-line at 6.4 keV. Later, we substitute the power law model with a cut-off
power law model, represented as cutoffpl. The spectral fitting results are
presented in Table 2. It is to be noted that there are no significant
variations in the spectral index ($\Gamma$) observed in either case.

For the spectral analysis of XMM-Newton (MJD–53789) data in the 3.0-10.0 keV
range, we employ tbabs*tbabs*cutoffpl model in our fitting. This model
provided a satisfactory fit with a spectral index of
$\Gamma=1.39^{+0.10}_{-0.11}$ and cut-off energy of
$E_{c}=489.54^{+258.54}_{-254.62}$ keV, resulting in a $\chi^{2}$ value of
42.71 for 37 degrees of freedom (dof). The continuum luminosity in the
2.0–10.0 keV range is calculated as $\log(L_{x})=41.85\pm 0.28$. No Fe-line
was detected during this observation.

Applying a similar methodology, we analyze 0.5-30.0 keV data from the Suzaku
(MJD–54988) observation of the source. The estimated spectral index ($\Gamma$)
and cut-off energy ($E_{c}$) are determined to be $1.50^{+0.06}_{-0.07}$ and
$469.65^{+155.52}_{-271.71}$ keV, respectively. During this analysis, we also
detect an Fe-line at $6.54^{+0.10}_{-0.09}$ keV with an equivalent width
($EW$) of $188^{+15}_{-17}$ eV. The normalization ($N$) is calculated to be
$12.2^{+2.07}_{-1.88}\times 10^{-3}$ photons/keV/cm-2/s, resulting in a
reduced $\chi^{2}$ ($\chi^{2}_{red}$) of 1.02. Along with this, the 2.0-10.0
keV source luminosity is determined to be $\log(L_{x})=43.03\pm 0.45$.

We proceed to analyze the XRT1+N1 (MJD–57579) observation using the baseline
model. During this analysis, we observe changes in the spectral parameters.
The spectral index ($\Gamma$) changed from $1.39^{+0.10}_{-0.11}$ to
$1.64^{+0.04}_{-0.04}$, and the corresponding cut-off energy ($E_{c}$) varied
from $389.54^{+258.54}_{-254.62}$ to $376.45^{+287.82}_{-266.78}$ keV. The
normalization ($N$) is determined to be $1.30^{+0.16}_{-0.14}\times 10^{-3}$
photons/keV/cm-2/s, and the reduced $\chi^{2}$ ($\chi^{2}_{red}$) value for
this fitting is 1.08 for 162 $dof$. During the spectral fitting, we detect an
Fe-line at $6.40^{+0.02}_{-0.02}$ keV with an equivalent width ($EW$) of
$104^{+8}_{-9}$ eV. The result is listed in Table 2. The continuum luminosity
in the 2.0 to 10.0 keV range is estimated to be $\log(L_{x})=41.83\pm 0.35$.

In the subsequent analysis, we employ the combined observations from XRT2+N2
(MJD–58093) to conduct spectral analysis using the cut-off power-law model.
The results indicate a spectral index ($\Gamma$) of $1.69^{+0.03}_{-0.02}$ and
a cut-off energy ($E_{c}$) of $344.89^{+100.15}_{-157.83}$ keV. The goodness
of fit is evaluated by a $\chi^{2}$ value of 685.54 for 665 $dof$. During this
analysis, an Fe-line was detected at $6.41^{+0.01}_{-0.01}$ keV with
equivalent width $EW=80^{+7}_{-8}$ eV and the result is listed in Table 2. The
continuum luminosity in the energy range of 2.0–10.0 keV is estimated to be
$\log(L_{x})=42.14\pm 0.37$.

Using the same model for the remaining Swift/XRT observations, XRT3
(MJD–58097) and XRT4 (MJD–59414), we determine the photon indices as
$\Gamma=1.66^{+0.11}_{-0.10}$ and $\Gamma=1.85^{+0.06}_{-0.07}$, with cut-off
energies of $E_{c}=359.60^{+195.78}_{-168.74}$ keV and
$E_{c}=258.60^{+101.88}_{-129.26}$ keV, respectively. The calculated
luminosity within the 2.0-10.0 keV energy range is $\log(L_{x})=41.78\pm 0.13$
and $\log(L_{x})=41.90\pm 0.35$, respectively.

Table 2:  cutoffpl fitting result for all spectra. The unabsorbed luminosity
is calculated for the energy range from 2.0 to 10.0 keV.

ID | MJD | $N_{H}$ | $\Gamma$ | $E_{c}$ |  Fe $K_{\alpha}$ | EW | $N_{Fe}$ | $N$ | $\chi^{2}/dof$ | $\log(L)$
---|---|---|---|---|---|---|---|---|---|---
|  | $(10^{20})$ |  | (keV) |  |  | $(10^{-6})$ | $(10^{-3})$ |  |
XMM | 53789 | $0.10^{+0.15}_{-0.12}$ | $1.39^{+0.10}_{-0.11}$ | $489.54^{+287.82}_{-254.62}$ | – | – | – | $1.10^{+0.07}_{-0.08}$ | $42.71/37$ | $41.85\pm 0.28$
SU | 54988 | $0.15^{+0.11}_{-0.11}$ | $1.50^{+0.06}_{-0.07}$ | $469.65^{+155.52}_{-171.71}$ | $6.54^{+0.10}_{-0.11}$ | $188^{+15}_{-17}$ | $361.4^{+3.84}_{-4.57}$ | $12.2^{+2.07}_{-1.88}$ | $327.31/320$ | $43.03\pm 0.45$
XRT1+N1 | 57579 | $0.10^{+0.10}_{-0.10}$ | $1.64^{+0.04}_{-0.04}$ | $376.45^{+258.54}_{-266.78}$ | $6.40^{+0.02}_{-0.02}$ | $104^{+8}_{-9}$ | $6.51^{+0.15}_{-0.15}$ | $1.30^{+0.16}_{-0.14}$ | $175.68/162$ | $41.83\pm 0.35$
XRT2+N2 | 58093 | $0.10^{+0.12}_{-0.10}$ | $1.69^{+0.03}_{-0.02}$ | $344.89^{+100.15}_{-157.83}$ | $6.41^{+0.01}_{-0.01}$ | $80^{+7}_{-8}$ | $11.6^{+1.11}_{-1.10}$ | $3.31^{+0.22}_{-0.23}$ | $685.54/665$ | $42.14\pm 0.37$
XRT3 | 58097 | $0.09^{+0.13}_{-0.09}$ | $1.66^{+0.11}_{-0.10}$ | $359.60^{+195.78}_{-168.74}$ | – | – | – | $1.46^{+0.26}_{-0.24}$ | $14.31/17$ | $41.78\pm 0.33$
XRT4 | 59414 | $0.11^{+0.15}_{-0.12}$ | $1.85^{+0.06}_{-0.07}$ | $258.60^{+101.88}_{-129.26}$ | – | – | – | $2.70^{+0.27}_{-0.27}$ | $22.81/27$ | $41.90\pm 0.35$

####  3.1.3 Nthcomp

In order to understand the physical processes leading to the primary continuum
X-ray emission from UGC 6728, we fit the broadband energy spectrum with the
thermal Comptonization model,
Nthcomp181818https://heasarc.gsfc.nasa.gov/xanadu/Xspec/manual/node205.html
(Zdziarski et al., 1996; Życki et al., 1999). The model depends on the seed
photon energy $(kT_{bb})$, which we consider to be $200$ eV, corresponding to
the Wien’s temperature of the disc for this source. It is worth mentioning
that we tested a range of values of $(kT_{bb})$ from $150$ to $250$ eV and
found no significant deviations in the spectral fitting. For our analysis, we
opted $inp\\_type=1$, which is the case when the seed photons are of a
multicolour blackbody nature. The composite model we used in Xspec reads as

TBabs*zTBabs*constant*(Nthcomp+zGauss).

For the spectral fitting, we initially focus on the $3.0$ to $10.0$ keV range
spectra and apply the asymptotic power law model. Subsequently, we extend the
spectra above $10.0$ keV and below $3.0$ keV, keeping the photon index
constant. Then, we apply the Nthcomp model replacing the power-law component
in the broadband spectral fitting. Interestingly, we observe positive
residuals below $2.0$ keV, indicating the presence of soft-excess emission in
this source. Additionally, we provide annotations for the model components of
the broadband spectral fitting ($0.3-79.0$ keV range) during the 2017
(XRT2+N2) observation in the respective panels. We illustrate the variation of
$\chi^{2}$ for the Nthcomp model in Figure 1.

Figure 1: Variation of $\chi_{red}$ is shown for each model component while
fitting the broadband spectrum of UGC 6728 during the 2017 XRT2+N2
observation. Initially, we fitted the data above $3.0$ keV with the Nthcomp
model. Then we added zGaussian for the Fe-line at $\sim 6.41$ keV. Later, we
replaced Nthcomp by TCAF for the broadband spectrum ($0.3-79.0$ keV).

We begin the spectral fitting using spectra obtained from the 2006 XMM-Newton
(XMM) observation. For the continuum spectrum (above $3.0$ keV), we find the
spectral index $(\Gamma)=1.38\pm 0.14$. We replace the powerlaw model with
Nthcomp for the primary continuum by fixing $(\Gamma)$ at $1.38$. The
estimated electron temperature $(kT_{e})$ from the spectral fitting is
$475.89\pm 250.35$ keV, accompanied by a Nthcomp normalization of
$1.10^{+0.07}_{-0.08}\times 10^{-3}$ photons/keV/sec/cm2. The reduced
$\chi^{2}$ ($\chi^{2}_{red}=\chi^{2}/\text{dof}$) is found to be $1.08$ for 37
degrees of freedom (dof).

Using the baseline model, we proceed to fit data from the Suzaku observation
(MJD–54988), and the corresponding results are presented in Table 3. The
spectral fitting provides the photon index, $\Gamma$, of $1.50\pm 0.04$ and an
electron energy, $kT_{e}$, of $350.42\pm 151.74$ keV. Using the values of
$\Gamma$ and $kT_{e}$ in Equation 1, the value of the optical depth, $\tau$,
is determined to be $0.68^{+0.71}_{-0.34}$. A Fe-line at $6.54\pm 0.10$ with
an equivalent width (EW) of $188^{+15}_{-17}$ eV is detected in the source
spectrum obtained from this observation. The normalization is found to be
$7.65^{+1.07}_{-2.0}$ photons/keV/sec/cm2 with a reduced
$\chi^{2}/dof=277.31/319$.

We employ similar spectral fitting procedures for the 2016 (XRT1+N1) and 2017
(XRT2+N2) observations. The estimated values of $\Gamma$ are $1.64\pm 0.03$
and $1.69\pm 0.02$ for the 2016 and 2017 observations, respectively. The
electron temperatures estimated from the X-ray broadband ($0.3$-$79.0$ keV
range) spectral fitting remain at $256.40\pm 86.21$ keV and $249.18\pm 41.95$
keV, respectively (see Table 3). Additionally, We identify the presence of an
iron Kα line at $6.40\pm 0.02$ and $6.41\pm 0.01$ keV for XRT1+N1 and XRT2+N2
observations with equivalent widths (EW) of $104^{+3}_{-4}$ and
$80.9^{+3}_{-3}$ eV, respectively. The normalizations of the Nthcomp model
fitting of these broadband X-ray spectra are $1.27^{+0.16}_{-0.14}\times
10^{-3}$ and $3.23^{+0.22}_{-0.23}\times 10^{-3}$ photons/keV/sec/cm2,
respectively. The corresponding $\chi^{2}_{red}$ values are $1.08$ and $1.03$
with $161$ and $662$ dofs (see Table 3), respectively.

Subsequently, we carry out spectral analysis of the Swift/XRT observations
from 2020 (XRT3) and 2021 (XRT4) using Nthcomp model. No iron lines are
detected in the spectra from these observations. The power-law indices
$(\Gamma)$ from the spectral fitting are $1.66\pm 0.10$ and $1.85\pm 0.05$ for
XRT3 and XRT4, respectively and the corresponding electron temperatures
$(kT_{e})$ are $248.78\pm 124.36$ and $178.32\pm 126.32$ keV, respectively.

Through the comprehensive spectral fitting of X-ray broadband data from all
observations using the Nthcomp model, we determine that the hydrogen column
density $(N_{H})$ along the line of sight is negligible and remains relatively
constant throughout our observations. This parameter, $(N_{H})$ varies from
$0.09^{+0.13}_{-0.09}\times 10^{20}$ cm-2 to $0.12^{+0.15}_{-0.12}\times
10^{20}$ cm-2, where the Galactic hydrogen column density $(N_{H,gal})$ is
fixed at $4.42\times 10^{20}$ cm-2. Additionally, we observe an increase in
the photon index $(\Gamma)$ from $1.38\pm 0.4$ to $1.85\pm 0.5$, while the
electron temperature $(kT_{e})$ exhibited a decrease from $475.89\pm 250.35$
keV to $178.32\pm 126.32$ keV. Furthermore, we calculate the optical depth
$(\tau)$ for each observation using the formula (Zdziarski et al., 1996):

$\tau=\sqrt{\frac{9}{4}+\frac{3}{\theta_{e}(\Gamma+2)(\Gamma-1)}}-\frac{3}{2}.$
(1)

Here, $\theta_{e}=\frac{kT_{e}}{m_{e}c^{2}}$ is the thermal energy of
electrons with respect to the rest mass energy. The values of $\tau$ for each
observation are given in Table 3. The maximum possible error in $\tau$ is
calculated using the formula
$\delta\tau\sim(\frac{1}{2}\frac{\Delta\theta_{e}}{\theta_{e}}+\frac{\Delta\Gamma}{\Gamma})\times\tau$,
where $\Delta\theta_{e}$ and $\Delta\Gamma$ are taken from the estimated
errors presented in Table 3. The variation of $\tau$ is presented in Table 3
as well as in Figure 2 (second panel from top). We find the values of optical
depth to be always below $1.0$. This suggests that the Compton cloud of the
source was optically thin for the entire observation period.

Table 3:  Results obtained from the spectral fitting above 3.0 keV spectra
with Nthcomp model. The optical depth $\tau$ is calculated using Equation 1.

ID | MJD | $N_{H}$ | $\Gamma$ | $kT_{e}$ | $N$ | $\chi^{2}/dof$ | $\tau^{\dagger}$
---|---|---|---|---|---|---|---
|  | $(10^{20})$ |  | (keV) | $(10^{-3})$ |  |
XMM | 53789 | $0.12^{+0.15}_{-0.12}$ | $1.38^{+0.14}_{-0.14}$ | $475.89\pm 250.35$ | $1.10^{+0.07}_{-0.08}$ | $40.02/37$ | $0.68^{+0.69}_{-0.26}$
SU | 54988 | $0.15^{+0.11}_{-0.12}$ | $1.50^{+0.04}_{-0.04}$ | $350.42\pm 151.74$ | $7.65^{+1.07}_{-2.0}$ | $277.31/319$ | $0.68^{+0.71}_{-0.34}$
XRT1+N1 | 57579 | $0.10^{+0.10}_{-0.10}$ | $1.64^{+0.03}_{-0.03}$ | $256.40\pm 86.21$ | $1.27^{+0.16}_{-0.14}$ | $174.10/161$ | $0.69^{+0.62}_{-0.24}$
XRT2+N2 | 58093 | $0.10^{+0.12}_{-0.10}$ | $1.69^{+0.02}_{-0.02}$ | $249.18\pm 41.95$ | $3.23^{+0.22}_{-0.23}$ | $680.27/662$ | $0.66^{+0.52}_{-0.22}$
XRT3 | 59097 | $0.09^{+0.13}_{-0.09}$ | $1.66^{+0.10}_{-0.10}$ | $248.74\pm 124.36$ | $1.45^{+0.26}_{-0.24}$ | $14.20/14$ | $0.69^{+0.51}_{-0.28}$
XRT4 | 59414 | $0.12^{+0.15}_{-0.12}$ | $1.85^{+0.05}_{-0.05}$ | $178.32\pm 126.32$ | $2.72^{+0.27}_{-0.27}$ | $21.77/26$ | $0.71^{+0.55}_{-0.32}$

Figure 2: Variation of different parameters of powerlaw fitting with time for
all spectra. The corresponding parameters are given in Table 3.

####  3.1.4 CompTT

From the thermal Comptonization model (Nthcomp), we derive the temperature of
the Compton cloud and spectral index from the model fitting. We also estimate
the optical depth using Equation 1. However, it is important to note that for
most of the observations covering energies up to 10.0 keV, constraining the
electron temperature from spectral fitting is challenging, resulting in
relatively large errors in optical depth calculations. To validate the derived
values of these fundamental parameters, we employ another physical model
called
CompTT191919https://heasarc.gsfc.nasa.gov/xanadu/Xspec/manual/node158.html on
the X-ray spectra of UGC 6728. The CompTT (Sunyaev & Titarchuk, 1980b;
Titarchuk, 1994; Hua & Titarchuk, 1995; Titarchuk & Lyubarskij, 1995) model is
characterized by the input soft photons with a temperature ($T_{0}$)
determined from the Wien’s law. This model incorporates the plasma temperature
of the Compton cloud ($kT_{e}$), optical depth ($\tau$), and the assumed
geometry (disc, sphere, analytical approximation). In our analysis, we adopt a
spherical geometry. The default input value for the soft photon temperature is
set at 200 eV (0.2 keV) in the fitting. It is to be noted that the parameter
$T_{0}$ is treated as a free parameter in the fitting. The composite model we
used in Xspec reads as,

TBabs*zTBabs*constant(CompTT+zGauss).

We begin our spectral fitting of data above 3.0 keV from the XMM-Newton
observation in 2006 using this model, and the corresponding results are
presented in Table 4. For this observation, we find the soft photon
temperature $(T_{0})$ at $0.21\pm 0.14$ keV with Compton cloud temperature
$(kT_{e})$ at $470.54^{+145.62}_{-148.26}$ keV. The optical depth $(\tau)$ for
spherical cloud is estimated as $0.75^{+0.42}_{-0.35}$ and the model
normalization is found to be $8.95^{+2.85}_{-0.86}\times 10^{-6}$
photons/keV/sec/cm2. During the model fitting, we did not observe any
significant deviations of the observational data from the continuum model
around $\sim 6.4$ keV. Therefore, we discard the model component zGauss for
this particular spectral fitting of XMM-Newton data.

We follow a similar methodology for fitting the data from the Suzaku
observation (MJD–54988) within the energy range of 3.0 to 30.0 keV. As with
the Nthcomp model fitting (see 3.1.3), we initially consider the 3.0 keV to
10.0 keV spectrum for this observation when employing the CompTT model. Then,
we extend the observed spectrum to higher-energy regions and perform fitting
accordingly. However, we are unable to distinguish the reflection component
above 10.0 keV due to poor data quality. It is plausible that the reflection
was either absent or barely present during this observation. While conducting
the spectral fitting, we identify the presence of Fe-line at
$6.54^{+0.10}_{-0.09}$ keV with an equivalent width of $290^{+15}_{-17}$ eV.
From the continuum fitting, using the composite model described above, we
determine the following parameters: soft photon temperature
$T_{0}=0.20^{+0.15}_{-0.18}$ keV, temperature of the Compton cloud
$kT_{e}=386.87^{+103.08}_{-098.65}$ keV and optical depth
$\tau=0.61^{+0.32}_{-0.31}$.

We then attempt to fit two broadband X-ray spectra from 2016 and 2017
observations of UGC 6728 obtained from the simultaneous Swift/XRT and NuSTAR
observations, covering the energy band of 3.0 to 79.0 keV. In both the
observations, we detect a narrow Fe-line at $6.4\pm 0.02$ keV with an
equivalent width of $104^{+8}_{-9}$ eV and $80^{+7}_{-8}$ eV for XRT1+N1
(MJD–57579) and XRT2+N2 (MJD–58093) observations, respectively. The disc
temperature ($T_{0}$) is found to be consistent for both observations at
$0.18\pm 0.13$ keV. The temperature of the Compton cloud ($kT_{e}$) is found
to be $215.22^{+59.36}_{-58.74}$ keV for XRT1+N1 and
$251.41^{+77.41}_{-82.26}$ keV for XRT2+N2. The optical depth vary between
$0.87^{+0.21}_{-0.23}$ and $0.62^{+0.35}_{-0.41}$ for these two observations.
The model normalizations for these observations are estimated to be
$34.8^{+8.14}_{-9.54}\times 10^{-6}$ photons/keV/sec/cm2 for XRT1+N1 and
$16.0^{+3.76}_{-4.19}\times 10^{-6}$ photons/keV/sec/cm2 for XRT2+N2. Detailed
results are provided in Table 4.

In the subsequent analysis, we perform spectral fitting using Swift/XRT data.
The disc temperature ($T_{0}$) is found to be consistent at
$0.14^{+0.17}_{-0.19}$ for the observations XRT3 and XRT4. The temperature of
the Compton cloud ($kT_{e}$) is found to be $248.78^{+126.35}_{-146.24}$ keV
and $215.06^{+104.63}_{-151.21}$ keV for XRT3 and XRT4, respectively. The
optical depth is found to vary accordingly. The estimated values of $\tau$ are
found to be $0.74^{+0.32}_{-0.24}$ for XRT3 and $0.39^{+0.41}_{-0.28}$ for
XRT4. The Fe-line is found to be absent in these two observations, leading us
to exclude the zGauss component from the composite model. The normalizations
for the continuum of these observations using the CompTT model are estimated
to be $16.0^{+3.76}_{-4.19}\times 10^{-6}$ photons/keV/sec/cm2 for XRT3 and
$52.1^{+7.47}_{-8.15}\times 10^{-6}$ photons/keV/sec/cm2 for XRT4. The
parameters obtained from the CompTT model fitting are presented in Table 4.

Throughout the long-term (2006–2021) X-ray spectral fitting of the continuum
of UGC 6728 using CompTT model, we find that the extra-galactic hydrogen
column density $(N_{H})$ was nearly constant and insignificant. This is
attributed to the ‘bare’ nature of the nucleus of this AGN. The disc
temperature $(T_{0})$ vary between $0.14\pm 0.2$ to $0.21\pm 0.14$ keV. The
Compton cloud temperature $(kT_{e})$ and the optical depth $(\tau)$ also
exhibit variations between $215.06^{+104.63}_{-151.21}$ to
$470.54^{+145.62}_{-148.26}$ keV and $0.87^{+0.21}_{-0.23}$ to
$0.39^{+0.41}_{-0.28}$, respectively. These variations of $kT_{e}$ and $\tau$
closely resemble that observed from the thermal Comptonization model (Nthcomp)
fitting. The optical depth $\tau$ is found to be consistently below 1.0,
suggesting that the cloud was optically thin throughout the observational
period.

Table 4:  Results obtained from the spectral fitting of 3.0–10.0 keV data with
CompTT model.

ID | MJD | $N_{H}$ | $T_{0}$ | $kT_{e}$ | $\tau$ | $N$ | $\chi^{2}/dof$
---|---|---|---|---|---|---|---
|  | $(10^{20})$ | (keV) | (keV) |  | $(10^{-6})$ |
XMM | 53789 | $0.12^{+0.12}_{-0.13}$ | $0.21^{+0.14}_{-0.14}$ | $470.54^{+145.62}_{-148.26}$ | $0.75^{+0.42}_{-0.35}$ | $8.95^{+2.85}_{-0.86}$ | $39.71/32$
SU | 54988 | $0.15^{+0.22}_{-0.11}$ | $0.20^{+0.15}_{-0.18}$ | $386.87^{+103.08}_{-098.65}$ | $0.61^{+0.32}_{-0.31}$ | $44.1^{+6.51}_{-7.17}$ | $197.36/189$
XRT1+N1 | 57579 | $0.11^{+0.12}_{-0.12}$ | $0.18^{+0.12}_{-0.13}$ | $215.22^{+59.36}_{-58.74}$ | $0.87^{+0.21}_{-0.23}$ | $19.9^{+7.04}_{-10.2}$ | $173.31/159$
XRT2+N2 | 58093 | $0.10^{+0.11}_{-0.11}$ | $0.19^{+0.13}_{-0.15}$ | $251.41^{+77.41}_{-82.26}$ | $0.62^{+0.35}_{-0.41}$ | $34.8^{+8.14}_{-9.54}$ | $683.22/661$
XRT3 | 59097 | $0.09^{+0.12}_{-0.09}$ | $0.15^{+0.15}_{-0.14}$ | $248.78^{+126.35}_{-146.24}$ | $0.74^{+0.32}_{-0.24}$ | $16.0^{+3.76}_{-4.19}$ | $26.01/24$
XRT4 | 59414 | $0.12^{+0.10}_{-0.11}$ | $0.14^{+0.17}_{-0.19}$ | $215.06^{+104.63}_{-151.21}$ | $0.39^{+0.41}_{-0.28}$ | $52.1^{+7.47}_{-8.15}$ | $51.40/54$

####  3.1.5 TCAF

By conducting Nthcomp model fitting of X-ray spectra spanning over
approximately 15 years $(2006-2021)$, we derive several valuable physical
parameters related to the spectral hardness and electron temperature of the
Compton cloud or corona in UGC 6728. The calculated value of optical depth
consistently indicated an optically thin Compton cloud. Nevertheless,
fundamental properties such as the mass of the central object, accretion flow
rates, and size of the Compton cloud for this accreting supermassive black
hole are still not derived from the above analysis. Knowledge of these
parameters would contribute to a deeper comprehension of this massive
accreting system. To address this, we employ the Two Component Advective Flow
(TCAF) model (Chakrabarti & Titarchuk, 1995) to analyze the X-ray spectrum of
this object. For the spectral fitting, the model used is

TBabs*zTBabs*constant(TCAF+zGauss).

The TCAF model is based on four accretion flow parameters: (i) Keplerian disc
accretion rate ($\dot{m}_{d}$) in units of the Eddington rate
($\dot{m}_{Edd}$), (ii) Sub-Keplerian halo accretion rate ($\dot{m}_{h}$) in
units of the Eddington rate ($\dot{m}_{Edd}$), (iii) shock compression ratio
(R), and (iv) shock location ($X_{s}$) in units of the Schwarzschild radius
($r_{g}=2GM/c^{2}$) with an intrinsic parameter which is the mass of the
central black hole in units of the solar mass ($M_{\odot}$). The upper limit,
lower limit, and default values of each parameter are kept in a file called
lmodel.dat. These values are provided in Table 5. We use initpackage and lmod
task in Xspec to run this table to fit the X-ray spectrum of UGC 6728 with the
TCAF model. We run the source code about $\sim 10^{5}$ times and select the
best spectrum by minimizing $\chi$.

Table 5:  The TCAF parameter space is defined in the file lmod.dat. The two
columns for minima and maxima are provided for the range of iterations.
Between them, the 1st column indicates the soft bound and the 2nd column gives
the hardbound of the parameters.

Model parameters | Parameter units | Default value | Min. | Min. | Max. | Max. | Increment
---|---|---|---|---|---|---|---
$\rm M_{BH}$ | $\rm M_{\odot}$ | $\rm 1.0\times 10^{5}$ | $2\times 10^{4}$ | $2\times 10^{4}$ | $5.5\times 10^{8}$ | $5.5\times 10^{8}$ | $10.0$
$\dot{m}_{d}$ | $\rm\dot{m}_{Edd}$ | $\rm 0.001$ | $0.0001$ | $0.0001$ | $1.0$ | $2.0$ | $0.0001$
$\dot{m}_{h}$ | $\rm\dot{m}_{Edd}$ | $\rm 0.01$ | $0.0001$ | $0.0001$ | $2.0$ | $3.0$ | $0.0001$
$\rm X_{s}$ | $\rm r_{g}$ | $\rm 50.0$ | $8.0$ | $8.0$ | $1000.0$ | $1000.0$ | $1.0$
$\rm R$ |  | $\rm 2.5$ | $1.01$ | $1.01$ | $6.8$ | $6.8$ | $0.1$

We begin the spectral fitting process of X-ray data from the XMM-Newton
observation (2006) using the baseline model as described above. During model
fitting, any significant deviation in spectral fit near $\sim 6.4$ keV is not
seen. Therefore, the Gaussian component zGauss is excluded from the fitting
procedure. The fitted parameters obtained from the TCAF model for this
specific observation are presented in Table 6 and the model fitted spectrum
with the variation of $\chi$ is presented in Figure 3. For this observation,
we obtain $M_{BH}=7.28^{+1.09}_{-1.08}\times 10^{5}$ M⊙ for the central SMBH
with disc and halo mass accretion rates $(\dot{m}_{d})$ and $(\dot{m}_{h})$,
of $0.05^{+0.01}_{-0.01}$ $\dot{m}_{Edd}$ and $0.07^{+0.01}_{-0.01}$
$\dot{m}_{Edd}$, respectively. Furthermore, we determine the shock location or
the boundary of the Compton cloud $(X_{s})$ to be at $31.58^{+4.65}_{-3.56}$
$r_{g}$ with a shock strength of $(R)=1.92^{+0.99}_{-0.91}$ for this XMM-
Newton observation.

Using a similar approach, we fit the broadband ($0.5-30.0$ keV) X-ray spectra
from Suzaku (MJD–54988), and the corresponding result is shown in Table 5 and
the spectra are presented in Figure 3. The estimated values of model
parameters are : $M_{BH}=7.09^{+0.59}_{-0.57}\times 10^{5}$ M⊙,
$\dot{m}_{d}=0.05^{+0.01}_{-0.01}$ $\dot{m}_{Edd}$,
$\dot{m}_{h}=0.08^{+0.01}_{-0.01}$ $\dot{m}_{Edd}$,
$X_{s}=29.09^{+2.10}_{-2.09}$ $r_{g}$, and $R=1.40^{+0.07}_{-0.09}$. The
normalization of the continuum model is estimated as
$335.4^{+5.65}_{-8.62}\times 10^{-5}$ photons/keV/sec/cm2 and
$\chi^{2}/dof=277.31/320$. The X-ray luminosity for the energy band 0.5–10.0
keV is found to be $\log(L_{x})=43.06\pm 0.43$.

Next, we analyze the broadband ($0.3-79.0$ keV) X-ray spectra from XRT1+N1
(2016) and XRT2+N2 (2017) using the baseline model mentioned earlier. Our
analysis yielded the following estimation of the TCAF model parameters for
2016 observation as : $M_{BH}=7.11^{+0.39}_{-0.32}\times 10^{5}$ M⊙,
$\dot{m}_{d}=0.06^{+0.01}_{-0.01}$ $\dot{m}_{Edd}$,
$\dot{m}_{h}=0.09^{+0.01}_{-0.01}$ $\dot{m}_{Edd}$,
$X_{s}=23.62^{+2.55}_{-2.56}$ $r_{g}$, and $R=1.68^{+0.09}_{-0.08}$.
Similarly, for the 2017 observation, the corresponding parameter values are
estimated as $M_{BH}=7.14^{+0.22}_{-0.19}\times 10^{5}$ M⊙,
$\dot{m}_{d}=0.07^{+0.01}_{-0.01}$ $\dot{m}_{Edd}$,
$\dot{m}_{h}=0.11^{+0.01}_{-0.01}$ $\dot{m}_{Edd}$,
$X_{s}=21.13^{+2.07}_{-1.56}$ $r_{g}$, and $R=1.34^{+0.06}_{-0.05}$. We
observe that the normalization for XRT1+N1 (2016) observation is
$1.51^{+0.37}_{-0.38}\times 10^{-5}$ photons/keV/sec/cm2, which is increased
to $1.81^{+0.80}_{-0.78}\times 10^{-5}$ photons/keV/sec/cm2 during the XRT2+N2
(2017) observation. We observe a change in X-ray luminosity from $Lx\sim
6.0\times 10^{41}$ (XRT1+N1: 2016) to $Lx\sim 1.5\times 10^{42}$ erg/s
(XRT2+N2: 2017) in about 500 days. It is to be noted that this type of
spectral variability in a short timescale is commonly seen in AGNs (Ballantyne
et al., 2014; Zoghbi et al., 2017). The details of the parameter space are
presented in Table 6, and corresponding TCAF fitted spectra from these two
observations are shown in Figure 3 along with the variation of $\chi$. It is
to be noted that, for these broadband spectra, we find the Fe Kα line at
$6.4\pm 0.02$ and $6.41\pm 0.01$ keV for XRT1+N1 and XRT2+N2 observations with
equivalent width (EW) of $104^{+8}_{-9}$ and $80.9^{+7}_{-8}$ eV,
respectively.

We then proceed to fit the data from the last two Swift/XRT observations, XRT3
and XRT4, using the baseline model. No Fe-line is observed in these spectra.
From the TCAF model fitting, we estimate the mass of the central black hole to
be $M_{BH}=7.07^{+2.05}_{-2.10}\times 10^{5}$ and
$M_{BH}=7.03^{+1.15}_{-1.19}\times 10^{5}$ M⊙, disc accretion rates
$\dot{m}_{d}=0.07\pm 0.02$ and $0.08\pm 0.01$ $\dot{m}_{Edd}$, sub-Keplerian
halo accretion rates $\dot{m}_{h}=0.10^{+0.02}_{-0.02}$ and
$\dot{m}_{h}=0.09^{+0.01}_{-0.01}$ $\dot{m}_{Edd}$ for both the observations,
respectively. We find the shock locations at $18.38^{+5.60}_{-5.35}$ $r_{g}$
and $11.03\pm 4.65$ $r_{g}$ with shock strengths of $1.77^{+0.56}_{-0.50}$ and
$1.68^{+0.16}_{-0.19}$ for XRT3 and XRT4 observations, respectively. The
parameters obtained from the TCAF model fitting are presented in Table 6 and
the model fitted spectra with the variation of $\chi$ are presented in the
bottom middle and bottom right panels of Figure 3.

Through the comprehensive spectral fitting of the long-term ($2006-2021$)
X-ray observations of UGC 6728 using TCAF model, we encounter a negligible
amount of extra-galactic hydrogen column density $(N_{H})$. The estimated mass
of the central black hole is $7.13\pm 1.23~{}~{}\times~{}~{}10^{5}$ M$\odot$,
which closely aligns with the mass estimated from the reverberation mapping
method conducted by Bentz et al. (2016). Furthermore, this model provides
valuable insights into the dynamics of accretion flow surrounding the central
black hole.

We observe variations in the disc accretion rate $(\dot{m}_{d})$, ranging from
$0.05\pm 0.01$ $\dot{m}_{Edd}$ to $0.08\pm 0.01$ $\dot{m}_{Edd}$, which has an
increasing trend over time, reaching its maximum during the 2021 observation.
The shock location, representing the boundary of the Compton cloud, moves
accordingly. The Compton cloud is found to be the largest
($31.58^{+4.65}_{-3.56}$ $r_{g}$) during the 2006 XMM-Newton observation.
Subsequently, the location of the shock gradually shifts inward. The halo
accretion rate $(\dot{m}_{h})$ displays variations from $0.09\pm 0.01$
$\dot{m}_{Edd}$ to $0.11\pm 0.01$ $\dot{m}_{Edd}$, peaking during the 2017
(XRT2+N2) observation. Consequently, this observation exhibits a high X-ray
luminosity for that time of observation. We also detect a narrow Fe K$\alpha$
line at $\sim 6.41$ keV with an equivalent width (EW) of approximately 100 eV
in the XRT1+N1 (2016) and XRT2+N2 (2017) observations.

The variations of accretion flow parameters of TCAF model, namely
$\dot{m}_{d}$, $\dot{m}_{h}$, $X_{s}$, and $R$, are presented in Table 6 and
potted in Figure 2, with other parameters. The unabsorbed X-ray luminosity in
the $0.5-10.0$ keV range, determined using the ‘clumin’ task in Xspec for all
the observations, is provided in Table 6. It is observed that the source
remained in the sub-Eddington luminosity regime throughout the observation
period. The long-term variations of the X-ray spectral parameters are also
illustrated in Figure 2.

Figure 3:  TCAF model fitted X-ray spectra of UGC 6728 from various
observations from 2009-2021, along with the residuals obtained from the
spectral fitting. Table 6:  Parameters obtained from the TCAF model fitting
with all the spectra. The unabsorbed X-ray luminosity is calculated for the
energy range from $0.5$ to $10.0$ keV.

ID | MJD | $N_{H}$ | $M_{BH}$ | $\dot{m}_{d}$ | $\dot{m}_{h}$ | $X_{s}$ | $R$ | $N$ | $\chi^{2}/dof$ | $\log(L_{x})$
---|---|---|---|---|---|---|---|---|---|---
|  | $(10^{20})$ | $(10^{5}M_{\odot})$ | $(\dot{m}_{Edd})$ | $(\dot{m}_{Edd})$ | $(r_{g})$ |  | $(10^{-5})$ |  |
XMM | 53789 | $0.12^{+0.10}_{-0.10}$ | $7.28^{+1.09}_{-1.08}$ | $0.05^{+0.01}_{-0.01}$ | $0.07^{+0.01}_{-0.01}$ | $31.58^{+4.65}_{-3.56}$ | $1.92^{+0.99}_{-0.91}$ | $0.92^{+0.63}_{-0.65}$ | $214.12/209$ | $42.14\pm 0.25$
SU | 54988 | $0.15^{+0.11}_{-0.12}$ | $7.09^{+0.59}_{-0.57}$ | $0.05^{+0.01}_{-0.01}$ | $0.08^{+0.01}_{-0.01}$ | $29.09^{+2.10}_{-2.09}$ | $1.40^{+0.07}_{-0.09}$ | $7.40^{+0.95}_{-0.92}$ | $277.31/320$ | $43.06\pm 0.43$
XRT1+N1 | 57579 | $0.10^{+0.10}_{-0.10}$ | $7.11^{+0.39}_{-0.32}$ | $0.06^{+0.01}_{-0.01}$ | $0.09^{+0.01}_{-0.01}$ | $23.62^{+2.55}_{-2.56}$ | $1.68^{+0.09}_{-0.08}$ | $1.51^{+0.37}_{-0.38}$ | $252.10/232$ | $41.78\pm 0.22$
XRT2+N2 | 58093 | $0.10^{+0.10}_{-0.10}$ | $7.14^{+0.22}_{-0.19}$ | $0.07^{+0.01}_{-0.01}$ | $0.11^{+0.01}_{-0.01}$ | $21.13^{+2.07}_{-1.56}$ | $1.34^{+0.06}_{-0.05}$ | $1.81^{+0.80}_{-0.78}$ | $671.74/626$ | $42.16\pm 0.17$
XRT3 | 59097 | $0.09^{+0.10}_{-0.09}$ | $7.07^{+2.05}_{-2.10}$ | $0.07^{+0.02}_{-0.02}$ | $0.10^{+0.02}_{-0.02}$ | $18.38^{+5.60}_{-5.35}$ | $1.77^{+0.56}_{-0.50}$ | $0.63^{+0.80}_{-0.88}$ | $22.40/22$ | $42.15\pm 0.22$
XRT4 | 59414 | $0.12^{+0.10}_{-0.12}$ | $7.03^{+1.15}_{-1.19}$ | $0.08^{+0.01}_{-0.01}$ | $0.09^{+0.01}_{-0.01}$ | $11.03^{+4.65}_{-4.65}$ | $1.68^{+0.16}_{-0.19}$ | $0.98^{+0.80}_{-0.95}$ | $52.37/59$ | $42.14\pm 0.15$

####  3.1.6 relxillCp

From the power-law fitting above 10.0 keV, we observed that the high-energy
spectra of UGC 6728 lacked a reflection component in the high-energy domain
above 10 keV. However, the presence of Fe-line (see Table 2) suggests the
existence of a reflection component over the primary continuum above 10.0 keV.
Despite the considerable uncertainty in the temperature estimation, we also
observed a correlation between the equivalent width of the Fe-line and the
temperature of the Compton cloud. It is anticipated that the equivalent width
decreases as the temperature of the Compton cloud decreases unless the photons
from outflows excite the Fe atoms (Chakrabarti & Titarchuk, 1995). To
investigate the presence of reflection in the X-ray spectra of UGC 6728, we
employ the relativistic reflection model RELXILL (García et al., 2013, 2014;
Dauser et al., 2013; Dauser et al., 2014, 2016) on the broadband spectra. This
model assumes a broken power-law emission profile with the broken radius
$R_{br}$. The emissivity is expressed as $E(r)\sim r^{-q_{1}}$ for $r<R_{br}$
and $E(r)\sim r^{-q_{2}}$ for $r>R_{br}$, where $q_{1}$ and $q_{2}$ are the
inner and outer emissivity index, respectively. The model parameter $R_{refl}$
provides information about the ratio of intensity emitted towards the disc
compared to escaping to infinity (Dauser et al., 2016). Other parameters, such
as the ionization parameter $(\xi)$, iron abundance $(A_{Fe})$, inclination
angle $(i^{0})$, inner radius of the accretion disc $(R_{in})$, and the spin
of the black hole $(a)$, are allowed to vary during the fitting process.

For the broadband X-ray spectral fitting, we use relxillCp flavour with the
final model specified in Xspec as Tbabs*zTbabs*constant*relxillCp. The
relxillCp flavour does not assume any particular geometry, and the primary
continuum emission is given by the thermal Comptonized model depending on the
spectral index $(\Gamma)$ and cut-off energy $(E_{c})$. We do not use
additional Gaussian components to fit the Fe-line as this model incorporates
Fe abundance as a free parameter. We fix the values of the outer radius of the
accretion disc at $R_{out}=1000~{}r_{g}$ and redshift $z=0.0065$ and the
corresponding results are represented in Table 7.

We apply the relxillCp model to SU observation (MJD–54988), and this model
provided a satisfactory fit of the X-ray spectrum with
$\chi^{2}/dof=621.96/563$. The parameters for the primary continuum are
estimated as $\Gamma=1.50^{+0.04}_{-0.05}$ and $E_{c}=358.52^{+6.89}_{-19.5}$
keV. The inner and outer emissivity indices are found to be
$3.00^{+0.84}_{-0.97}$ and $5.23^{+0.59}_{-0.52}$ for
$R_{in}=3.94^{+1.60}_{-0.98}~{}r_{g}$ and
$R_{br}=15.49^{+8.64}_{-8.97}~{}r_{g}$, respectively. The Fe abundance for
this observation was estimated as $1.05^{+0.70}_{-0.74}~{}A_{\odot}$ with
ionisation parameter $\log(\xi)=3.71^{+0.32}_{-0.33}$ and reflection fraction
$R_{refl}=0.74^{+0.11}_{-0.10}$. The intrinsic parameters of this source were
estimated as the black hole spin $a=0.99^{+0.12}_{-0.34}$ and inclination
angle $i=48.20^{+14.2}_{-15.4}$ degrees. The spectrum is plotted on the left
panel of Figure 4.

We followed a similar procedure for XRT1+N1 (MJD–57579) and XRT2+N2
(MJD–58093) observations for the broadband, and the corresponding estimated
values from the X-ray spectral fitting is given in Table 7. For these
observations, the parameters for the primary continuum were obtained as
follows: the spectral index $\Gamma$ are $1.65^{+0.04}_{-0.03}$ and
$1.70^{+0.02}_{-0.03}$ with cut-off energy $E_{c}$ are
$273.68^{+29.56}_{-67.38}$ and $33.24^{+15.62}_{-56.24}$ keV, respectively.
The accretion disc extended up to $R_{in}=6.14^{+1.21}_{-1.11}~{}r_{g}$ and
$R_{in}=21.49^{+1.32}_{-1.33}~{}r_{g}$ with break radius
$R_{br}=39.61^{+10.22}_{-10.18}~{}r_{g}$ and $R_{br}=15.0~{}r_{g}$,
respectively. We fixed $R_{br}$ at its default value ($15.0~{}r_{g}$) for
XRT2+N2 observation as this parameter was not well constrained during fitting.
The emissivity indices are found at $q_{1}=2.85^{+0.47}_{-0.61}$ and
$q_{1}=3.00^{+0.63}_{-0.58}$ respectively for inner region where $r<R_{br}$
and $q_{2}=3.01^{+0.52}_{-0.51}$ and $q_{2}=3.00^{+0.50}_{-0.59}$ respectively
for outer region of these observations. The Fe abundance and the ionization
parameters are nearly same and the values are
$A_{Fe}=0.51^{+0.20}_{-0.21}~{}A_{\odot}$ and
$0.50^{+0.20}_{-0.23}~{}A_{\odot}$ with $\log(\xi)=3.45^{+0.63}_{-0.62}$ and
$3.37^{+0.22}_{-0.23}$, respectively. The reflection fraction is determined to
be $R_{refl}=0.11^{+0.05}_{-0.07}$ and $0.12^{+0.08}_{-0.05}$ for these
observations. It is expected that the intrinsic parameters would be constant
for all observations and the estimated values for black hole spin and
inclination angles are $a=0.98^{+0.20}_{-0.18}$ and $a=0.96^{+0.29}_{-0.30}$
with $i=53.59^{+15.7}_{-16.1}$ and $i=46.62^{+13.6}_{-12.3}$ degrees,
respectively. The spectra are plotted on the middle and right panels of Figure
4.

Figure 4: relxillCp model fitted X-ray spectra of UGC 6728 from various
observations from 2009–2021, along with the residuals obtained from the
spectral fitting. Table 7:  Parameters obtained from the relxillCp model
fitting with all the spectra.

Model Parameters | SU | XRT1+NU1 | XRT2+NU2
---|---|---|---
| (MJD–54988) | (MJD–57579) | (MJD–58093)
$N_{H}(10^{20}~{}cm^{-2})$ | $0.23^{+0.21}_{-0.20}$ | $0.75^{+0.20}_{-0.22}$ | $0.73^{+0.25}_{-0.23}$
$q_{1}$ | $3.00^{+0.84}_{-0.97}$ | $2.85^{+0.47}_{-0.61}$ | $3.00^{+0.63}_{-0.58}$
$q_{2}$ | $5.23^{+0.59}_{-0.52}$ | $3.01^{+0.52}_{-0.51}$ | $3.00^{+0.50}_{-0.59}$
$R_{br}(r_{g})$ | $15.49^{+8.64}_{-8.97}$ | $39.61^{+10.22}_{-10.18}$ | $15.00^{f}$
$R_{in}(r_{g})$ | $3.64^{+1.60}_{-0.98}$ | $6.14^{+1.21}_{-1.11}$ | $5.66^{+1.32}_{-1.33}$
$a$ | $0.99^{+0.12}_{-0.34}$ | $0.98^{+0.20}_{-0.18}$ | $0.96^{+0.29}_{-0.30}$
$i(^{0})$ | $48.20^{+14.2}_{-15.4}$ | $53.59^{+15.7}_{-16.1}$ | $46.62^{+13.6}_{-12.3}$
$\Gamma$ | $1.50^{+0.04}_{-0.05}$ | $1.65^{+0.04}_{-0.03}$ | $1.70^{+0.02}_{-0.03}$
$\log\xi$ | $3.71^{+0.32}_{-0.33}$ | $3.45^{+0.63}_{-0.62}$ | $3.37^{+0.22}_{-0.23}$
$A_{fe}(A_{\odot})$ | $1.05^{+0.70}_{-0.74}$ | $0.51^{+0.20}_{-0.21}$ | $0.50^{+0.20}_{-0.23}$
$kT_{e}(keV)$ | $358.52^{+6.89}_{-19.5}$ | $273.68^{+29.56}_{-67.38}$ | $233.24^{+15.62}_{-56.24}$
$R_{refl}$ | $0.74^{+0.11}_{-0.10}$ | $0.11^{+0.05}_{-0.07}$ | $0.12^{+0.08}_{-0.05}$
$N(10^{-5})$ | $12.2^{+1.6}_{-1.1}$ | $3.94^{+0.58}_{-0.58}$ | $9.01^{+0.35}_{-0.30}$
$\chi^{2}/dof$ | 621.96/563 | 223.48/224 | 800.86/802

f indicates the parameter that was fixed during X-ray spectrum fitting.

###  3.2 Timing Analysis

We conducted a comprehensive timing analysis of the light curves with time
binsize=$500$s obtained from the XMM-Newton, Swift, and NuSTAR observations of
UGC 6728 (see Table 1). To investigate the variability, we partitioned the
X-ray light curves from the XMM-Newton and Swift observations in the 0.3-10
keV energy range into two segments: the soft excess component (0.3-2.0 keV
range) and the primary continuum component (3.0-10.0 keV range). The light
curves in the 0.3-10 keV range are illustrated in the top panels of Figure 5.
The bottom panels in the figure show the light curves of the source in the
soft and hard X-ray ranges. The left panels represent the light curves from
the XMM-Newton observation, whereas the other panels represent the light
curves from the Swift/XRT observations of UGC 6728. In this figure, it becomes
apparent that the source count rates during the XMM-Newton observation
significantly exceed the count rates observed during the XRT observations.
This discrepancy may be attributed to the use of different instruments for the
respective observations.

To explore the variability in the high-energy regime, we considered light
curves from NuSTAR observations. For the correlation study in the broad energy
range, we considered the NuSTAR observations in 2016 (N1) and 2017 (N2) due to
their substantial exposure times (above 20 ks; see Table 1). The light curves
obtained from the NuSTAR observations in the $3.0-79.0$ keV energy range were
divided into two energy bands: $3.0-6.0$ keV and $7.0-79.0$ keV ranges and
were utilized for the correlation study (see Figure 6). Notably, we excluded
the light curve in the $6.0-7.0$ keV band as it predominantly comprises
photons associated with the Fe Kα emission line, while our primary interest
lies in investigating the continuum emission.

Figure 5: Variation of photon counts with respect to time from XMM-Newton and
Swift/XRT observations of UGC 6728 at different epochs (see Table 1). The
light curves in the $0.3-$10.0 keV range are shown in the top panels, whereas
the bottom panels show the light curves in the soft X-ray band (represented by
red dots/line) and hard X-ray band (represented by blue dots/line), plotted
together for comparison. The top and bottom left panels represent light curves
from the XMM-Newton observation, whereas the other panels are from the
Swift/XRT observations of UGC 6728.

####  3.2.1 Variability

X-ray variability is a powerful tool to probe the nearby regions of the
central supermassive black bole of an AGN. Since UGC 6728 has a ‘bare-nucleus’
(see Table 3 and Table 6), the X-ray photons originated at the Compton cloud
are not intercepted by any other medium such as BLR, NLR or molecular torus.
As a result, the variability in the X-ray emission of this source is very
likely produced by the Compton cloud itself or the very inner edge of the
accretion disc. To study the temporal variability of UGC 6728, we estimated
different variability parameters for all the X-ray observations used in the
present work. The results are presented in Table 1. The fractional variability
$F_{var}$ (Edelson et al., 1996; Nandra et al., 1997; Rodríguez-Pascual et
al., 1997; Edelson et al., 2001; Vaughan et al., 2003; Edelson & Malkan, 2012)
for a light curve of $x_{i}$ counts/s with error $\sigma_{i}$ for $N$ number
of data points, with a mean count rate $\mu$ and standard deviation $\sigma$
is given by,

$F_{var}=\sqrt{\frac{\sigma^{2}_{XS}}{\mu^{2}}}$ (2)

where, $\sigma_{XS}$ is the excess variance (Nandra et al., 1997; Edelson et
al., 2002) which is defined as,

$\sigma^{2}_{XS}=\sigma^{2}-\frac{1}{N}\sum_{i=1}^{N}\sigma^{2}_{i}.$ (3)

We defined the normalized excess variance as
$\sigma^{2}_{NXS}=\sigma^{2}_{XS}/\mu^{2}$ and the uncertainties on
corresponding parameters $(\sigma^{2}_{NXS},\text{and}F_{var})$ are calculated
as described by (Vaughan et al., 2003; Edelson & Malkan, 2012). To investigate
the variability in the X-ray light curves, we calculated peak-to-peak
amplitude as $R=x_{max}/x_{min}$, where $x_{max}$ and $x_{min}$ are the
maximum and minimum flux, respectively.

The X-ray light curves from different energy bands exhibit different
magnitudes of variability. The results from the variability analysis are
presented in Table 8. During 2006 XMM-Newton observation, the highest count
rates were observed in all energy bands ranging from $0.3$ to $10.0$ keV. For
this entire energy range ($0.3$-$10.0$ keV), the maximum count rate
$(x_{max})$, the minimum count rate $(x_{min})$ and the mean count rate
$(\mu)$ were $8.44$ counts/s, $3.47$ counts/s, and $5.60$ counts/s,
respectively. For this observation, the soft band (0.3-3.0 keV range) mean
count rate $(\mu)$ was determined to be $2.58$ counts/s, with a maximum count
rate $(x_{max})$ and minimum count rate $(x_{min})$ of $3.95$ counts/s and
$1.55$ counts/s, respectively. Along with this, the primary continuum
(3.0-10.0 keV) exhibited a similar trend with $\mu$, $x_{max}$ and $x_{min}$
of $3.02$ counts/s, $4.50$ counts/s, and $1.92$ counts/s, respectively. The
variability parameters such as the peak-to-peak amplitude $(R)$, normalized
excess variance $\sigma^{2}_{NXS}$, and fractional variability $(F_{var})$ are
found to be comparable for the light curves in the low ($0.3-2.0$ keV), high
($3.0-10.0$ keV) and total ($0.3-10.0$ keV) energy ranges of this observation.
We found that $R$ varies in the range of $2.32$ to $2.56$, whereas
$\sigma^{2}_{NXS}$ and $F_{var}$ are approximately the same at $0.04\pm 0.004$
and $\sim 20\%$, respectively, for the low-energy, high energy, and the entire
energy bands. The detailed results are given in Table 8.

In the case of Swift/XRT observations, the exposure time is comparatively
lower with respect to other observations (see Table 1). As a result, we have a
smaller number of data points for these observations (see Table 8 and Figure
5). In the soft energy band ($0.3-2.0$ keV), we found that the maximum count
rates ranged from $0.35$ count/s to $1.05$ count/s, with corresponding minimum
count rates varying from $0.09$ count/s to $0.17$ count/s. The mean count
rates ranged from $0.16$ count/s to $0.48$ count/s. We observe a similar trend
in the high energy band ($3.0-10.0$ keV), with maximum count rates ranging
from $0.40$ count/s to $0.61$ count/s, minimum count rates varying from $0.01$
count/s to $0.18$ count/s, and mean count rates ranging from $0.19$ count/s to
$0.30$ count/s. For the entire energy range ($0.3-10.0$ keV) of the Swift/XRT
observations, the maximum count rates varied from $0.22$ count/s to $1.16$
count/s, minimum count rates ranged from $0.1$ count/s to $0.23$ count/s, and
the mean count rates varied from $0.20$ count/s to $0.58$ count/s. In all the
Swift/XRT observations, we observed that XRT2 had comparatively higher counts
compared to other observations, as shown in Figure 5 and presented in Table 8.
Due to the low count rate and high error associated with each data point, we
encountered negative values for normalized excess variance
($\sigma^{2}_{NXS}$), resulting in imaginary fractional variability
($F_{var}$) for most of the XRT observations. However, for observations with
positive $\sigma^{2}_{NXS}$, we observed fractional variability ($F_{var}$)
ranging from $35.4\%$ to $43.5\%$, with an average of $39.5\%$ for the low
energy band, and from $30.1\%$ to $38.8\%$, with a mean of $35.0\%$ for the
entire energy band. In the energy range from 3.0 to 10.0 keV, only XRT3 showed
a positive $\sigma^{2}_{NXS}=(0.35\pm 0.07)$ with $F_{var}=18.7\pm 2.1\%$. The
detailed results are presented in Table 8. The soft X-ray variability studies
could be improved with future missions, such as Colibrí (Heyl et al., 2019;
Caiazzo et al., 2019; Heyl et al., 2020).

We have only two NuSTAR observations of UGC 6728 in 2016 (N1) and 2017 (N2).
The entire energy band ($3.0-79.0$ keV) is divided into several sub-energy
bands, namely, $3.0-6.0$ keV for the primary continuum, $6.0-7.0$ keV for the
Fe Kα line and $7.0-79.0$ keV for the high energy counter-part. Figure 6
displays the temporal variation of the X-ray photons in the energy bands of
$3.0-6.0$ keV and $7.0-79.0$ keV for the 2016 (N1) and 2017 (N2) observations.
The results of the variability analysis for different energy bands of these
observations can be found in Table 8. For the N1 observation, we observed a
maximum count rate of $0.20$ count/s and a minimum count rate of $0.04$
count/s in the $3.0-6.0$ keV energy band. The average count rate ($\mu$) in
this band was $0.10$ count/s. A similar count rate pattern was observed in the
high-energy counterpart ($7.0-79.0$ keV), with maximum, minimum and average
count rates of $0.35$ count/s, $0.09$ count/s, and $0.18$ count/s,
respectively. The Fe K$\alpha$ energy band ($6.0-7.0$ keV) exhibited lower
count rates, with maximum count rates of $0.08$ count/s, minimum count rates
of $0.01$ count/s, and an average count rate of $0.04$ count/s. For the entire
X-ray energy band ($3.0-79.0$ keV), the count rates ranged from a maximum
value of $0.53$ count/s to a minimum value of $0.17$ count/s, with a mean
value of $0.32$ count/s. The variability parameter ($F{var}$) ranged from
$11.6\pm 5.2\%$ to $31.6\pm 2.6\%$, indicating higher variability in the lower
energy band compared to the higher energy band for this observation. In the
entire energy range, the average variability was $F_{var}=22.9\pm 5.6\%$.
Please refer to Table 8 for detailed results.

In the case of the 2017 NuSTAR observation (N2), we observed a similar trend
to N1. For the lower energy band ($3.0-6.0$ keV), the maximum, minimum and
average count rates were $0.37$ count/s, $0.07$ count/s, and 0.22 count/s,
respectively. In the Fe K$\alpha$ energy band ($6.0-7.0$ keV), the photon
counts varied from a maximum value of $0.12$ count/s to a minimum value of
$0.02$ count/s, with an average count rate of $\mu=0.07$ count/s. However, for
the high-energy counterpart ($7.0-79.0$ keV), the errors associated with each
photon count were higher than the standard deviation of the light curve.
Consequently, the fractional variability $F{var}$ became imaginary, and those
values are not presented in Table 8. For this light curve, we found that the
$x_{max}=0.05$ count/s and $x_{min}=0.02$ count/s with $\mu=0.03$ count/s. For
the entire energy range ($3.0-79.0$ keV), the maximum count rate was $1.17$
count/s, the minimum count rate was $0.24$ count/s, and the average count rate
was $\mu=0.62$ count/s. We found that the variability parameter $F_{var}$
remained nearly constant for all energy bands as well as for the entire energy
range, ranging from $21.8\pm 4.1\%$ to $26.5\pm 2.0\%$. The detailed results
are presented in Table 8.

Table 8: The table presents the variability statistics in different energy ranges for various observations for 500s time bins. It is important to note that in some cases, the average error of the observational data exceeds the $1\sigma$ limit, leading to negative excess variance. Consequently, these cases have imaginary values for $F_{var}$, which are not included in the table. ID | Energy band | $N$ | $x_{max}$ | $x_{min}$ | $\mu$ | $R=\frac{x_{max}}{x_{min}}$ | $\sigma^{2}_{NXS}$ | $F_{var}$
---|---|---|---|---|---|---|---|---
| keV |  | Count/s | Count/s | Count/s |  |  | $(\%)$
XMM | 0.3-2.0 | 82 | 3.95 | 1.55 | 2.58 | 2.56 | $0.04\pm 0.004$ | $21.2\pm 1.25$
XMM | 3-10.0 | 82 | 4.50 | 1.92 | 3.02 | 2.32 | $0.04\pm 0.003$ | $19.1\pm 2.11$
XMM | 0.3-10.0 | 82 | 8.44 | 3.47 | 5.60 | 2.44 | $0.04\pm 0.003$ | $20.1\pm 1.10$
XRT1 | 0.3-2.0 | 17 | 0.35 | 0.09 | 0.16 | 3.92 | $0.18\pm 0.036$ | $43.5\pm 2.35$
XRT1 | 3-10.0 | 17 | 0.54 | 0.18 | 0.30 | 3.03 | $-0.03$ | $--$
XRT1 | 0.3-10.0 | 17 | 0.41 | 0.12 | 0.21 | 3.60 | $0.14\pm 0.038$ | $38.0\pm 2.01$
XRT2 | 0.3-2.0 | 21 | 1.05 | 0.17 | 0.48 | 6.01 | $0.13\pm 0.032$ | $35.4\pm 2.14$
XRT2 | 3-10.0 | 21 | 0.40 | 0.11 | 0.23 | 9.71 | $-0.021$ | $--$
XRT2 | 0.3-10.0 | 21 | 1.16 | 0.23 | 0.58 | 5.10 | $0.09\pm 0.03$ | $30.1\pm 2.00$
XRT3 | 0.3-2.0 | 05 | 0.18 | 0.13 | 0.16 | 1.40 | $-0.24$ | $--$
XRT3 | 3-10.0 | 05 | 0.48 | 0.17 | 0.27 | 2.92 | $0.35\pm 0.07$ | $18.7\pm 2.1$
XRT3 | 0.3-10.0 | 05 | 0.22 | 0.15 | 0.20 | 1.43 | $-0.39$ | $--$
XRT4 | 0.3-2.0 | 08 | 0.51 | 0.10 | 0.29 | 4.91 | $0.16\pm 0.061$ | $39.5\pm 2.20$
XRT4 | 3-10.0 | 08 | 0.61 | 0.01 | 0.19 | 60.1 | $-0.05$ | $--$
XRT4 | 0.3-10.0 | 08 | 0.61 | 0.10 | 0.35 | 5.78 | $0.150\pm 0.061$ | $38.8\pm 2.3$
N1 | 3.0-6.0 | 51 | 0.20 | 0.04 | 0.10 | 5.45 | $0.101\pm 0.019$ | $31.6\pm 2.6$
N1 | 6.0-7.0 | 51 | 0.08 | 0.01 | 0.04 | 7.30 | $0.019\pm 0.015$ | $13.7\pm 2.4$
N1 | 7.0-79.0 | 51 | 0.35 | 0.09 | 0.18 | 3.80 | $0.014\pm 0.010$ | $11.6\pm 5.2$
N1 | 3.0-79.0 | 51 | 0.53 | 0.17 | 0.32 | 3.13 | $0.052\pm 0.012$ | $22.9\pm 5.6$
N2 | 3.0-6.0 | 131 | 0.37 | 0.07 | 0.22 | 5.61 | $0.070\pm 0.006$ | $26.5\pm 2.0$
N2 | 6.0-7.0 | 131 | 0.12 | 0.02 | 0.07 | 4.77 | $0.048\pm 0.009$ | $21.8\pm 4.1$
N2 | 7.0-79.0 | 129 | 0.05 | 0.02 | 0.03 | 2.10 | $-0.98$ | $--$
N2 | 3.0-79.0 | 131 | 1.17 | 0.24 | 0.62 | 4.91 | $0.062\pm 0.003$ | $25.0\pm 02.0$

####  3.2.2 Correlation

For the temporal analysis of the long-term X-ray observation of UGC 6728, we
concentrated on two epochs of NuSTAR observations: 2016 and 2017 which were
accompanied by low energy ($0.3-10.0$ keV) observations from Swift/XRT. To
study the correlation between the light curves, we have chosen the
simultaneous observations from NuSTAR (N1 and N2) in two energy bands: (i)
$3.0-6.0$ keV and (ii) $7.0-79.0$ keV. We excluded the light curve in
$6.0-7.0$ keV band as it contains primarily photons from the Fe Kα line, which
appears at $\sim 6.4$ keV (see Section 3.1).

The $\zeta-$discreate cross-correlation function
(ZDCF202020www.weizmann.ac.il/particle/tal/research-activities/software,
(Alexander, 1997)) was utilized for this analysis. The likelihood of the
correlation function was calculated for the same DCF function using $15000$
simulated points in the ZDCF code for the light curves obtained from the
NuSTAR observations. We fitted a Gaussian function to estimate the peak and
the error on the peak is obtained using the formula provided by Gaskell &
Peterson (1987). Our approach followed a similar procedure as described in
Nandi et al. (2021). Initially, we fitted all the light curves with a straight
line and found the $\chi^{2}_{red}<1.5$ with different slopes and intercepts.
However, these data points are not suitable for linear fitting. Next, we
performed the correlation analysis of these light curves to ensure the
simultaneity of their observations (see the upper panel of Figure 6).

The ZDCF obtained from both NuSTAR observations (N1 and N2) show a similar
type of delay pattern. In the case of 2016 NuSTAR observation (N1), a delay of
$66.8\pm 500$s is observed. We fitted a Gaussian function (indicated by the
dotted black line in the lower panel of Figure 6). The uncertainty of the
position of the peak is obtained as $\Delta\tau_{d}=162.5$s, which is smaller
than the time bin size (500s) of the light curves. Hence, we considered $500$s
as the error of the position of the peak. The detailed results are given in
Table 9. We calculated the likelihood function, represented by deep-green bars
in the lower panel of Figure 6, to accurately determine the position of the
peak. A similar procedure was followed for the 2017 observation (N2). In this
case, we found that the delay between the two given energy bands is
$\tau_{d}^{zdcf}=66.8$s. The uncertainty of the position of the peak, obtained
from the prescription provided by (Gaskell & Peterson, 1987) is
$\Delta\tau_{d}=46.5$s, is much smaller than the time bin size. As with the N1
observation, we considered the time bin size ($500$s) as the uncertainty for
the parameter $\tau_{d}^{zdcf}$. The results are summarized in Table 9.

Figure 6: Top panel: The light curves of UGC 6728 in the energy bands of $3.0$ to $6.0$ keV (red) and $7.0$ to $79.0$ keV (blue) are presented for two epochs of NuSTAR observations in 2016 and 2017. The high energy band have higher count rates in comparison to the low energy band for these two observations. Lower Panel: $\zeta$-discrete cross-correlation functions (light-green) are plotted for the light curves in $3.0-6.0$ keV and $7.0-79.0$ keV ranges. The likelihood functions (dark-green), simulated using $15000$ points, are plotted along with the ZDCF. Table 9: The parameters used for estimating the time delay are presented. The peak value of the correlation function is denoted by $\epsilon^{z}_{\tau}$, and the corresponding time delay is represented by $\tau^{zdcf}_{d}$. The uncertainty of the position of the peak of the correlation function is calculated using the procedure described in Gaskell & Peterson (1987) and is given by $\Delta\tau_{d}$. To ensure the reliability of the results, we compared this error with the time bin size and selected the larger one for our analysis. Further details can be found in Section 3.2.2. Id | Epochs | Bin size | $\Delta\tau_{d}$ | $\epsilon^{z}_{\tau}$ | $\tau^{zdcf}_{d}$
---|---|---|---|---|---
| Year | (s) | (s) |  | (s)
N1 | 2016 | 500 | 162.5 | $0.72\pm 0.10$ | $66.8\pm 500$
N2 | 2017 | 500 | 46.5 | $0.78\pm 0.06$ | $-50.9\pm 500$

##  4 discussions

In this study, we investigated the X-ray observations of UGC 6728 utilizing
data from XMM-Newton, NuSTAR and Swift/XRT spanning from 2006 (MJD-53789) to
2021 (MJD-59414) (see Table 1). We observed that UGC 6728 is a highly spinning
$(a=0.98\pm 0.3)$ black hole with an inclination angle around $\sim 50$
degrees. The results of our spectral and temporal analysis are presented in
Section 3. In this section, we will discuss the key findings derived from our
data analysis.

###  4.1 Evolution of Primary Continuum

The ‘bare-nucleus’ (Walton et al., 2013) of UGC 6728 was observed over the
period of 2006 to 2021 by various X-ray observatories, revealing both spectral
and temporal variabilities. Throughout the long-term X-ray observations, the
X-ray luminosity of UGC 6728 is found to vary in the range of $\sim
10^{41.8}-10^{44.2}$ erg/s. The spectral index of the primary continuum
($3.0-10.0$ keV) also displayed variations ranging from $1.38\pm 0.14$ to
$1.85\pm 0.05$, during this observational period. Correspondingly, the
electron temperature of the Compton cloud varied from $475.89\pm 250.35$ keV
to $178.32\pm 126.32$ keV. The hardest spectrum was observed in 2006 XMM-
Newton observation, with a spectral index of $\Gamma=1.38\pm 0.14$ and an
electron temperature of $475.9\pm 250.4$ keV according to the Nthcomp model.
This is the highest temperature of the electron cloud obtained from all the
observations. It is worth noting that the spectral index $\Gamma$ is lower
than that of other sources mentioned in the literature. This could be
attributed to the relatively poor data quality, as the exposure time was
limited for this observation. If we take into consideration the uncertainty of
this parameter, its value falls within the range of $1.14$ to $1.52$, which is
consistent with the observed $\Gamma$ values reported in the literature (Kriss
et al., 1980). The corresponding temperature of the electron also appears to
be high, with a large uncertainty for this observation. It is important to
mention that the parameter, $kT_{e}$, is estimated using various models such
as cut-off power-law, Nthcomp, and CompTT. The significant uncertainties in
the estimation of $kT_{e}$ might be due to the absence of high-energy data
beyond 10.0 keV. The model TCAF provides insights into the accretion flow
dynamics around the central object for this observation. By fitting the $0.2$
to $10.0$ keV spectra with the TCAF model, we found that the sub-Keplerian
halo accretion rate ($\dot{m}_{h}=0.07\pm 0.01$) was higher than the disc
accretion rate ($\dot{m}_{d}=0.05\pm 0.01$) during that time. This suggests
that the flow was sub-Keplerian-dominated for that observational period. This
behavior is commonly observed in AGNs (Nandi et al., 2019; Nandi et al.,
2021). We found that the boundary of the Compton cloud or the shock location
$(X_{s})$ was estimated to be $\sim 31.6$ $r_{g}$ with a relatively weak shock
strength of $R\sim 1.92$ based on spectral fitting. This is the highest shock
strength when all the X-ray observations of this source are considered. A
higher value of $R$ indicates the higher temperature of the Compton cloud,
which is consistent with the results obtained from the Nthcomp model fitting,
implying an outward push of the boundary of the Compton cloud.

In the next Suzaku observation in 2009, we observe that the X-ray luminosity
of UGC 6728 is maximum. This increase in luminosity can be attributed to the
high normalization of the primary continuum during this observation. It is
noted that the spectral index changed from $1.38\pm 0.14$ to $1.50\pm 0.07$,
and the corresponding electron temperature decreased from $475.9\pm 250.4$ to
$350.42\pm 151.74$ keV. This variation in spectral index and electron
temperature could be explained by the variation in accretion dynamics. During
this observation, we found that the accretion rates remained constant with
those of the previous observation. However, the shock strength decreased from
$1.92$ to $1.40$. As a result, the electrons in the Compton cloud could not
gain energy from the shock, leading to a decrease in temperature. Moreover, we
observed a prominent Fe-line at $6.54\pm 0.1$ keV with an equivalent width of
$188\pm 17$ eV. The presence of the Fe-line suggests the existence of either
reflection or a high abundance of iron in the accreting medium. To distinguish
between these possibilities, we applied the relxillCp model to this
observation and found that the high abundance $(A_{Fe}=1.05\pm 0.72)$ of iron
with a relatively small amount of reflection $(R_{refl}=0.74\pm 0.11)$.

We conducted a broadband ($0.3-79.0$ keV) spectral analysis of UGC 6728 using
combined data from the 2016 Swift/XRT (XRT1) and NuSTAR (N1) observations. For
this particular observation, we observed a relatively softer spectrum
($\Gamma=1.64\pm 0.03$) for the primary continuum ($3.0-79.0$ keV) compared to
the 2006 (XMM) observation. The corresponding temperature of the electron
cloud in the Compton cloud also changed from $475.9\pm 250.35$ keV to
$254.4\pm 86.21$ keV. We determined the optical depth of the Compton cloud to
be $0.69\pm 0.02$, indicating an optically thin cloud. The disc accretion rate
$(\dot{m}_{d})$ and the halo accretion rate $(\dot{m}_{h})$, changed from
$0.05\pm 0.01$ $\dot{M}_{Edd}$ to $0.06\pm 0.01$ $\dot{M}_{Edd}$ and $0.07\pm
0.01$ $\dot{M}_{Edd}$ to $0.09\pm 0.01$ $\dot{M}_{Edd}$, respectively, with
respect to the XMM observation. The increase in $\dot{m}_{d}$ implies an
increased supply of soft photons, which interact with the Compton cloud or
corona, resulting in a relatively softer spectrum. Consequently, the shock
location moved inward, with $X_{s}$ shifting from $31.58^{+4.65}_{-3.56}$
$r_{g}$ to $23.62\pm 2.55$ $r_{g}$. The strength of the shock also increased
from $1.40\pm 0.08$ to $1.68\pm 0.09$. These changes contributed to the
observation of a relatively softer spectrum during this particular
observation. By fitting a single Gaussian, we detected a narrow Fe Kα line at
$6.40\pm 0.02$ keV with an equivalent width (EW) of $104\pm 9$ eV. The
detection of Fe-line suggests the presence of reprocessed emission. To
investigate this, we applied the relxillCp model on the broadband spectra. The
model output indicated a reflection fraction of $R_{refl}=0.11\pm 0.06$ along
with an Fe abundance of $A_{Fe}=0.51\pm 0.20$ $A_{\odot}$.

In the 2017 observation, which was carried out by Swift/XRT (XRT2) and NuSTAR
(N2) simultaneously, we obtained another broadband spectrum
$(0.3-79.0~{}\text{keV})$ with a higher X-ray luminosity compared to the
previous broadband observation in 2016 (see the first panel of Figure 5). By
fitting the primary continuum ($3.0-10.0$ keV) of this broadband spectrum with
the Nthcomp model, we found a marginal increase in the spectral index from
$1.64$ to $1.69$, along with a decrease in the electron temperature $(kT_{e})$
of the Compton cloud from $256.40\pm 86.21$ keV to $249.18\pm 41.95$ keV, with
large uncertainties in $kT_{e}$. We also found the presence of an optically
thin Compton cloud with optical depth $0.66\pm 0.02$ during this observation.
Additionally, we detected a narrow Fe line near 6.41 keV with an equivalent
width (EW) of $80.9\pm 8$ eV. To investigate the presence of reflection
component beyond 10.0 keV, we used relxillCp model to fit the broadband
spectra and found that $A_{Fe}=0.50\pm 0.2$ $A_{\odot}$ and $R_{refl}=0.11\pm
0.6$. The low values of these parameters indicate the high energy spectrum is
barely dominated by reflection above 10.0 keV.

The increase in X-ray luminosity suggests a change in the accretion flow,
which can be quantified through TCAF model fitting. We observed the accretion
rates, $\dot{m}_{d}$ and $\dot{m}_{h}$, increase from $0.06\pm 0.01$
$\dot{M}_{Edd}$ to $0.07\pm 0.01$ $\dot{M}_{Edd}$ and $0.09\pm 0.01$
$\dot{M}_{Edd}$ to $0.11\pm 0.01$ $\dot{M}_{Edd}$, respectively. As a result,
the luminosity of UGC 6728 increased by $\sim 2.4$ times from the previous
(2016) observation. Moreover, the shock location shifted inward from $23.62\pm
2.56$ $r_{g}$ to $21.23\pm 2.07$ $r_{g}$, accompanied by a corresponding
decrease in the shock strength from $1.68\pm 0.038$ to $1.34\pm 0.80$.

In the 2020 observation carried out by Swift/XRT (XRT3), we encountered a
spectrum similar to that of the previous observation in 2017. The flow
parameters, as well as the spectral index ($\Gamma$) and electron temperature
($kT_{e}$), remained unchanged compared to the parameters obtained from the
2017 observation. Specifically, we found $\Gamma=1.66$ with $kT_{e}=249.18$
keV and corresponding optical depth $(\tau)$ is $0.69\pm 0.02$. We also
estimated the X-ray luminosity in $0.5-10.0$ keV range and found that this is
nearly the same as in the case of the 2017 observation (see Table 3).

The accretion dynamics exhibited minimal changes in this observation. The TCAF
fitting suggests the accretion rates for 2020 observation to be
$\dot{m}_{d}=0.07\pm 0.02$ $\dot{M}_{Edd}$ and $\dot{m}_{h}=0.11\pm 0.01$
$\dot{M}_{Edd}$ which are similar to those during the 2017 observation. The
shock location is located near $18.38\pm 5.6$ $r_{g}$ with a shock strength of
$1.77\pm 0.56$ for this observation.

In the 2021 observation from Swift/XRT (XRT4), we found the softest X-ray
spectrum for UGC 6728. The spectral index was determined to be $\Gamma=1.85\pm
0.05$, with an electron temperature of $kT_{e}=178.32\pm 126.32$ keV. The
corresponding optical depth was measured to be $\tau=0.71\pm 0.02$. During
this observation, we estimated the maximum disc accretion rate as
$\dot{m}_{d}=0.08\pm 0.01$ $\dot{M}_{Edd}$. Additionally, the halo accretion
rate decreased from $0.109\pm 0.01$ $\dot{M}_{Edd}$ to $0.09\pm 0.01$
$\dot{M}_{Edd}$.

The increase in the disc accretion rate leads to the cooling of the Compton
cloud due to the presence of soft photons generated in the disc. As a result,
the Compton cloud cannot maintain its size and gradually shrinks. This causes
the shock location to move inward, eventually settling at $11.03\pm 4.65$
$r_{g}$, with a shock strength of $R=1.68\pm 0.2$. The reduction in the size
of the Compton cloud affects the production of hard photons, resulting in the
observed soft spectrum.

Throughout the X-ray observation period (2006-2021), we found that the Compton
cloud remained optically thin. However, the spectral index $(\Gamma)$
exhibited systematic variations. The hardest spectra were observed in the 2006
(XMM) observation, followed by a transition to a softer state. The softest
state was observed in the 2021 (XRT4) observation. The variations in $\Gamma$
are summarized in Table 3 and visualized in Figure 2.

As the source transitioned from one state to another state, the other spectral
parameters, such as the accretion rates, the size of the Compton cloud, the
temperature of the Compton cloud etc, changed accordingly. We found that the
disc accretion rate $(\dot{m}_{d})$ increased from $0.051\pm 0.01$
$\dot{M}_{Edd}$ (in Table 6) to $0.08\pm 0.01$ $\dot{m}_{Edd}$ as the spectra
became softer. On the other hand, the halo accretion rate $(\dot{m}_{h})$
increased from $0.07\pm 0.01$ $\dot{m}_{Edd}$ to $0.09\pm 0.01$
$\dot{m}_{Edd}$ before reaching a relatively constant value of $\sim 0.1\pm
0.01$ $\dot{m}_{Edd}$.

The change in $\dot{m}_{d}$ also influenced the position of the shock location
$(X_{s})$. As the disc rate increased and the source transitioned from a
harder to a softer state, the shock location moved inward. This resulted in a
decrease in the size of the Compton cloud, a behavior commonly observed in
AGNs and Galactic black holes (Chakrabarti & Titarchuk, 1995; Nandi et al.,
2019; Nandi et al., 2021). The shock strength $(R)$ showed relatively minor
variations, ranging from $1.34\pm 0.06$ to $1.92\pm 0.99$, with an average
value of $1.69\pm 0.59$.

In addition, We also estimated the hydrogen column densities $(N_{H})$ along
the line of sight, while assuming a constant galactic hydrogen column density.
The results showed that the extragalactic hydrogen column densities varied
within a range of approximately $(10^{19}-10^{20})$/cm2. This result supports
that the source has a ‘bare-nucleus’, as previously reported by Walton et al.
(2013). Furthermore, we conducted an estimation of the mass of the central
supermassive black hole (SMBH) using TCAF fitting. The obtained value for the
SMBH mass was $(7.13\pm 1.23)\times 10^{5}$ M⊙. This measurement aligns with
the results obtained by (Bentz et al., 2016) ) using an independent method,
further confirming the consistency of our findings.

###  4.2 Evolution of Soft Excess

The soft excess (Arnaud et al., 1985; Singh et al., 1985; Fabian et al., 2002;
Gierliński & Done, 2004), characterized by an excess emission below 2.0 keV,
is an extraordinary feature in the X-ray spectra of most Seyfert 1 AGNs.
Despite its common occurrence, the origin of this excess emission remains
unclear. However, multiple explanations are proposed to explain this feature
in literature (Miniutti & Fabian, 2004; Crummy et al., 2006; Sobolewska &
Done, 2007; Done et al., 2007; Mehdipour et al., 2011; Done et al., 2012;
Lohfink et al., 2012; Walton et al., 2013; Liebmann et al., 2018). In this
context, a long-term X-ray observation of UGC 6728 showed a strong correlation
between the primary continuum and the soft excess (Nandi et al., 2023). In
this study, we also observe the presence of the soft excess. However, the
strength of this component exhibited variability throughout our analysis.

The presence of a strong soft excess is observed in 2006 (XMM) and 2009 (SU)
observations as presented in the top two panels of Figure 7. During these
observations, the spectra appear relatively harder with
$(\Gamma=1.39^{+0.10}_{-0.11}~{}\text{and}~{}\Gamma=1.50^{+0.06}_{-0.07})$
compared to observations performed in later epochs. The electron cloud
exhibited temperatures of $kT_{e}=470.54^{+145.62}_{-148.26}$ and
$kT_{e}=368.87^{+103.08}_{-98.65}$ keV and optical depth
$\tau=0.75^{+0.42}_{-0.35}$ and $\tau=0.61^{+0.32}_{-0.31}$, respectively.
Given that the source was in a soft state during these observations, a larger
Compton cloud was observed, with radii of $X_{s}=31.58^{+4.65}_{-3.56}$
$r_{g}$ and $X_{s}=29.09^{+2.10}_{-2.09}$ $r_{g}$, respectively. The disc and
halo accretion rates are estimated to be $0.05\pm$ and $0.07\pm 0.01$
$\dot{m}_{Edd}$, respectively, for these two observations. The increased size
of the Compton cloud could produce a relatively cooler electron region near
the outer boundary of CENBOL or Compton cloud, leading to the enhanced
production of photons in the soft energy band (below 3.0 keV), as described in
Nandi et al. (2021). As a result, a strong presence of soft excess could be
observed during the initial epochs.

The soft excess component exhibited a diminished presence in the 2016
(XRT1+N1) and 2017 (XRT2+N2) observations, as illustrated in the bottom panels
of Figure 7. In comparison to the preceding XMM and SU observations, these
spectra appeared softer. From the spectral fitting by the TCAF model, we
estimated the size of the Compton cloud for these observations were
$X_{s}=23.62^{+2.55}_{-2.56}$ $r_{g}$ and $X_{s}=21.13^{+2.07}_{-1.56}$
$r_{g}$ respectively. The reduction in the size of the Compton cloud led to a
decreased number of photons being generated in the primary continuum as well
as a weaker soft excess component in these observations. On the other hand, we
also observed weaker reflection fraction ($R_{refl}$) for weaker soft-excess
(Fabian et al., 2002). It would be interesting to observe UGC 6728 in the
future to see whether the soft excess diminishes or re-brightens, preserving
the primary continuum Mehdipour et al. (2023).

Figure 7:  The variation of the strength of soft excess with respect to the
primary continuum is plotted for the observations from 2006 to 2017. The trend
indicates a decline in the strength of the soft excess as the source moved
from a relatively softer state to a comparatively harder state.

###  4.3 Correlation between Spectral Parameters

Table 10: Correlation among different spectral parameters estimated from the spectral fitting of long term $(\sim 15\text{ years})$ X-ray observations with various satellites. The correlation analysis involved Parameter-1 and Parameter-2, which are displayed in the first and second columns, respectively. The third column represents the Pearson Correlation Coefficient (PCC), while the corresponding $p$-values are presented in the final column. A positive PCC value indicates a positive correlation between the two parameters, whereas a negative value indicates an anti-correlation. Parameter-1 | Parameter-2 | PCC | $p-value$
---|---|---|---
$\Gamma$ | $kT_{e}$ | $-0.97$ | $<0.05$
$\Gamma$ | $\dot{m}_{d}$ | $+0.88$ | $<0.05$
$\Gamma$ | $\dot{m}_{h}$ | $+0.66$ | $0.15$
$\Gamma$ | $\dot{m}$ | $+0.87$ | $<0.05$
$\Gamma$ | $X_{s}$ | $-0.96$ | $<0.05$
$\Gamma$ | $R$ | $-0.24$ | $0.64$
$\dot{m}_{d}$ | $X_{s}$ | $-0.94$ | $<0.05$
$\dot{m}_{d}$ | $\dot{m}_{h}$ | $-0.59$ | $0.22$
$\dot{m}_{d}$ | $R$ | $+0.06$ | $0.91$
$\dot{m}_{h}$ | $X_{s}$ | $-0.52$ | $0.29$
$\dot{m}_{h}$ | $R$ | $-0.54$ | $0.28$

The spectral analysis using Nthcomp and TCAF models reveals the nature of the
region where the X-ray photons are generated. For UGC 6728, we found that the
Compton cloud remains optically thin throughout the entire observation period
(see Figure 3). However, we observe a transition of the source from a
comparatively hard state ($\Gamma=1.38$) to a comparatively soft state
($\Gamma=1.85$) between 2006 and 2021. The X-ray luminosities ($0.5-10.0$ keV)
remain relatively constant, around $L_{x}\sim 10^{42.15}$ erg/s.

We examined the correlation among various spectral parameters and the
corresponding result is presented in Table 10, and selected correlations are
plotted in Figure 8. The Pearson Correlation Coefficient (PCC) is utilized to
examine the relationships between
parameters212121https://www.socscistatistics.com/tests/pearson/default2.aspx.
We observe a strong anti-correlation between the spectral index ($\Gamma$) and
the temperature of the Compton cloud ($kT_{e}$) with a PCC value of $-0.97$.
This anti-correlation is commonly observed in AGNs (Nandi et al., 2021), and
it is displayed in the upper left panel of Figure 8 with linear interpolation.
However, due to the inability to constrain $kT_{e}$ accurately with the
available spectra up to $79.0$ keV, we obtained high $p$-values during the
calculation of the correlation coefficient. Consequently, correlations
involving $kT_{e}$ are not presented in Table 10.

Next, we found a strong correlation between $\Gamma$ and the disc accretion
rate ($\dot{m}_{d}$) with a PCC of $0.88$ ($p$-value = $0.051$). As the disc
rate increases, more disc (seed) photons are generated, leading to
interactions with the Compton cloud and a subsequent decrease in cloud
temperature. Consequently, a smaller number of hard photons are produced,
resulting in the transition of the source from a hard to a soft state, as
indicated by an increase in $\Gamma$. We also plotted the variation of
$\Gamma$ and $\dot{m}_{d}$ with linear interpolation in the upper left panel
of Figure 8. The parameter $\Gamma$ is also loosely correlated with the halo
accretion rate $\dot{m}_{h}$ with PCC=$0.66$. However, the high $p$-value
($p$-value = $0.15$) makes it challenging to draw any definitive conclusions
based on this correlation coefficient.

Furthermore, we calculated the correlation between $\Gamma$ and the total
accretion rate $\dot{m}$, and the corresponding PCC is estimated to be $0.88$
with $p$-value=$<0.05$. This correlation is well-established and commonly
observed in AGNs and Galactic black holes. We observed a strong global anti-
correlation trend between $\Gamma$ and $X_{s}$, with a Pearson Correlation
Coefficient (PCC) value of $-0.96$ and corresponding $p$-value of $<0.05$.
This finding suggests that as the shock location or the boundary of the
Compton cloud ($X_{s}$) decreases, the spectrum tends to soften, aligning with
the concept described by Chakrabarti & Titarchuk (1995). In line with this
understanding, we can also anticipate an anti-correlation between $X_{s}$ and
$\tau$, as illustrated in the lower left panel of Figure 8.

In our investigation of the accretion flow parameters estimated from the
spectral fitting using TCAF model, we explored correlations among
$\dot{m}_{d}$, $\dot{m}_{h}$, $X_{s}$, and $R$. It is generally anticipated
that the shock location moves inward as the disc accretion rate increases
(Chakrabarti & Titarchuk, 1995; Chakrabarti, 1995). We discovered a similar
correlation between $\dot{m}_{d}$ and $X_{s}$, exhibiting an anti-correlation
with a Pearson Correlation Coefficient (PCC) of $-0.94$ and a corresponding
$p$-value of $<0.05$. The relationship between $\dot{m}_{d}$ and $X_{s}$ is
depicted with linear interpolation in the right panel of Figure 8. As for the
other parameters, we observed weak correlations or anti-correlations among
them. This outcome is expected since these parameters are intrinsic to TCAF
and act independently during X-ray spectral fitting.

Figure 8: Correlation between different spectral parameters estimated from the
spectral fitting of X-ray spectra.

##  5 Conclusions

We conducted a comprehensive analysis of X-ray observations spanning
approximately 15 (2006-2021) years for UGC 6728, a low-mass Seyfert 1 AGN
which has a highly spinning black hole at the centre with the inclination
angle nearly $\sim 50$ degrees. Throughout this period, we observed
significant variations in the X-ray luminosity of the source. The X-ray
luminosity exhibited fluctuations on a yearly timescale, with a factor of
approximately 2.4 variations observed in the 2016-2017 observations. Our
investigation focused on the broadband X-ray spectra covering the energy range
of 0.3-79.0 keV. We find this source is retaining its ‘bare’-nature throughout
the observational period. In this section, we present a summary of our major
findings.

  1. 1.

Based on our spectral analysis of the long-term X-ray data of UGC 6728, we
determined that the variations in luminosity are not dependent on the hydrogen
column density ($N_{H}$). This outcome is consistent with the expectation that
this source possesses a negligible $N_{H}$ value ($N_{H}\leq 1.0\times
10^{19},\text{cm}^{-2}$) along the line-of-sight.

  2. 2.

Throughout the extended observation period of approximately 15 years
(2006-2021), the spectral slopes of UGC 6728 exhibit significant variations.
Notably, this undergoes a systematic transition from a comparatively harder
state $(\Gamma\sim 1.38)$ to a comparatively softer state $(\Gamma\sim 1.85)$
from 2006 to 2021.

  3. 3.

The observed variation of $\Gamma$ can be attributed to the accretion flow
dynamics surrounding the central supermassive black hole and the
characteristics of the Compton cloud. Our analysis reveals that the disc rate
$(\dot{m}_{d})$ increases from $0.05\pm 0.01$ $\dot{M}_{Edd}$ to $0.08\pm
0.01$ $\dot{M}_{Edd}$, while $\Gamma$ changes from $1.38\pm 0.14$ to
$1.85\pm.0.05$, indicating a transition of this source from a hard state to a
soft state. Correspondingly, the shock location $(X_{s})$ or the size of the
Compton cloud and compression ratio $(R)$ exhibit variations. Notably, the
Compton cloud is optically thin $(\tau<1.0)$ throughout the entire observation
period.

  4. 4.

The intensity of the soft excess component varied over the duration of our
observational period. The 2006 and 2009 observations displayed profound soft
excess, compared with the diminished presence of soft excess observed in 2016
and 2017 epochs. The variation in the strength of soft excess could be
explained by the variation of the accretion dynamics.

  5. 5.

In our timing study, we observed a similar order of variability in the high
energy band (above $3.0$ keV). Furthermore, correlation analysis between two
high energy bands, specifically $3.0-6.0$ keV and $7.0-79.0$ keV, does not
reveal any time delays, suggesting that the photons in both energy ranges may
have originated from a similar mechanism. Since the high energy spectra (above
$3.0$ keV) of UGC 6728 can be adequately described by a single power-law
model, we can infer that this source exhibits either no reflection hump or a
very minimal hump in relation to the primary continuum above $10.0$ to $15.0$
keV.

  6. 6.

By performing X-ray spectral fitting using TCAF model, we determined the mass
of the central supermassive black hole (SMBH) to be $M_{BH}=(7.13\pm
1.23)\times 10^{5}$ M⊙ with a high spin at $a=0.97^{+0.20}_{-0.27}$ and
inclination angle of $49.5\pm 14.5$ degree.

## Acknowledgements

We would like to express our sincere gratitude to the reviewers for their
valuable insights and comprehensive evaluation of our work, greatly
contributing to its improved quality and clarity. This research work at PRL is
funded by the Department of Space, Government of India. AC and SSH are
supported by the Canadian Space Agency (CSA) and the Natural Sciences and
Engineering Research Council of Canada (NSERC) through the Discovery Grants
and the Canada Research Chairs programs. 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 and the High Energy Astrophysics Division of the
Smithsonian Astrophysical Observatory. This work has made use of data obtained
from the NuSTAR mission, a project led by Caltech, funded by NASA and managed
by NASA/JPL, and has utilized the NuSTARDAS software package, jointly
developed by the ASDC, Italy and Caltech, USA. This research has made use of
observations obtained with XMM-Newton, an ESA science mission with instruments
and contributions directly funded by ESA Member States and NASA. This work
made use of Swift/XRT data supplied by the UK Swift Science Data Centre at the
University of Leicester, UK.

## Data Availability

We used archival data of Swift/XRT and NuSTAR observatories for this work.
These data are publicly available on their corresponding websites. Appropriate
links are given in the text.

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