11institutetext: Indian Institute of Astrophysics, II Block, Koramangala,
Bangalore 560034, India
11email<EMAIL_ADDRESS>22institutetext: Inter University Centre for
Astronomy and Astrophysics, Pune, India 33institutetext: Indian Centre for
Space Physics, 43 Chalantika, Garia Stn. Road, Kolkata 700084, India
44institutetext: Aryabhatta Research Institute of Observational Sciences
(ARIES), Nainital 263002, India

# Flux and spectral variability of Mrk 421 during its moderate activity state
using NuSTAR: Possible accretion disc contribution?

Santanu Mondal  Priyanka Rani 11 C. S. Stalin 2211 Sandip K. Chakrabarti 1133
Suvendu Rakshit 44

###### Abstract

Context. The X-ray emission in BL Lac objects is believed to be dominated by
synchrotron emission from their relativistic jets. However, when the jet
emission is not strong, one could expect signatures of X-ray emission from
inverse Compton scattering of accretion disc photons by hot and energetic
electrons in the corona. Moreover, the observed X-ray variability can also
originate in the disc, and gets propagated and amplified by the jet.

Aims. Here, we present results on the BL Lac object Mrk 421 using the Nuclear
Spectroscopic Telescope Array data acquired during 2017 when the source was in
a moderate X-ray brightness state. For comparison with high jet activity
state, we also considered one epoch data in April 2013 when the source was in
a very high X-ray brightness state. Our aim is to explore the possibility of
the signature of accretion disc emission in the overall X-ray emission from
Mrk 421 and also examine changes in accretion parameters considering their
contribution to spectral variations.

Methods. We divided each epoch of data into different segments in order to
find small scale variability. Data for all segments were fitted using simple
powerlaw model. We also fitted the full epoch data using the two component
advective flow model to extract the accretion flow parameters. Furthermore, we
estimated the X-ray flux coming from the different components of the flow
using the lowest normalization method and analysed the relations between them.
For consistency, we performed the spectral analysis using available models in
the literature.

Results. The simple powerlaw function does not fit the spectra well, and a
cutoff needs to be added. The spectral fitting of the data using the two
component advective flow model shows that the data can be explained with a
model where (a) the size of the dynamic corona at the base of the jet from
$\sim 28$ to $10\leavevmode\nobreak\ r_{s}$, (b) the disc mass accretion rate
from 0.021 to 0.051 $\dot{M}_{\rm Edd}$, (c) the halo mass accretion rate from
0.22 to 0.35 $\dot{M}_{\rm Edd}$, and (d) the viscosity parameter of the
Keplerian accretion disc from 0.18 $-$ 0.25. In the assumed model, the total
flux, disc and jet flux correlate with the radio flux observed during these
epochs.

Conclusions. From the spectral analysis, we conclude that the spectra of all
the epochs of Mrk 421 in 2017 are well described by the accretion disc based
two component accretion flow model. The estimated disc and jet flux relations
with radio flux show that accretion disc can contribute to the observed X-ray
emission, when X-ray data (that covers a small portion of the broad band
spectral energy distribution of Mrk 421) is considered in isolation. However,
the present disc based models are disfavoured with respect to the relativistic
jet models when considering the X-ray data in conjunction with data at other
wavelengths.

###### Key Words.:

galaxies: active – galaxies: jets – galaxies: nuclei – radiative transfer – BL
Lacertae objects: individual: Mrk 421

##  1 Introduction

Active galactic nuclei (AGN) harbour a supermassive black hole at their
center. They emit X-rays from their very central region and are believed to be
powered by accretion of matter onto supermassive black holes ($\sim
10^{6}-10^{10}M_{\odot}$) located at the centre. The process of accretion
leads to AGN emitting copious amount of radiation (luminosity $\sim
10^{42}-10^{48}$ erg s-1) as well as producing a variety of highly energetic
phenomena (Lynden-Bell, 1969; Shakura & Sunyaev, 1973; Rees, 1984). A majority
of AGN emit little to very low radio-emission and are called radio-quiet AGN,
while a minority emit ample radiation in the radio band and are called radio-
loud AGN (Kellermann et al., 1989). However, recently, it has been proposed
that AGN can be better divided based on their physical differences (Padovani,
2017) into non-jetted AGN (ones that lack strong relativistic jets) and jetted
AGN (ones with strong relativistic jets). Among the jetted AGN are blazars, a
peculiar category characterised by an extreme flux variations across the
entire electromagnetic spectrum over a range of timescales from minutes to
days (Wagner & Witzel, 1995; Ulrich et al., 1997), flat spectra in the radio
band, and high optical polarization and polarization variability (Angel &
Stockman, 1980; Andruchow et al., 2005; Rakshit et al., 2017). Such
characteristics are now attributed to their relativistic jets oriented close
to the line of sight to the observer and are subjected to Doppler boosting.

Blazars are divided into two categories, namely, flat spectrum radio quasars
(FSRQs) and BL Lac objects (BL Lacs). This division is historically based on
the presence or absence of emission lines in the spectra of those objects.
FSRQs have emission lines in their optical/infrared spectra, while BL Lacs
have either weak emission lines (equivalent width $<$ 5 Å) or featureless
spectra (Urry & Padovani, 1995). It is now believed that FSRQs are powerful
sources with strong relativistic jets and powered by the standard optically
thick and geometrically thin accretion disc. On the other hand, BL Lacs are
weak sources with radiatively inefficient and geometrically thin accretion
disc. In such discs, the reduced ultra-violet (UV) emission might not be able
to photo-ionise the broad line region (BLR) leading to weak or absent broad
emission lines in BL Lacs (Ghisellini, 2019).

Alternatively, the weakness or absence of emission lines in the spectra of BL
Lacs could be understood in the context of the continuum getting enhanced due
to Doppler boosting, and becoming dominant during the increased jet activity
of BL Lacs. It has been shown by Foschini (2012) that the faint lines detected
in BL Lacs could be due to the combined effect of the boosted jet emission and
a weak accretion disc. Removal of the contribution of the boosted jet emission
to the continuum leads to emission lines in BL Lacs having FWHM similar to
that of other AGN. This finding leads to hypothesize that during the low jet
activity or faint state of BL Lacs, the observed X-ray emission could have
significant contribution from the inverse Compton (IC) scattering of soft
photons from the accretion disc off the “hot corona”, in addition to the
synchrotron emission from the relativistic jet. According to Ghisellini et al.
(2011) a physical distinction between FSRQs and BL Lacs can be made based on
the ratio of the luminosity of the BLR to the Eddington luminosity ($L_{\rm
BLR}/L_{\rm Edd}$) with FSRQs having $L_{\rm BLR}/L_{\rm Edd}$ greater than 5
$\times$ $10^{-5}$ compared to BL Lacs.

The broad band spectral energy distribution (SED) of blazars, shows two
prominent peaks, one peaking around radio-optical-X-ray frequencies and the
other one peaking at X-ray to GeV $\gamma$-ray energies. In the leptonic
scenario of emission from blazar jets, the low energy peak is attributed to
synchrotron emission process, while the high energy peak is attributed to IC
process. Also, based on the location of the synchrotron peak in the SED,
blazars are further divided in low synchrotron peaked (LSP; $\nu_{\rm
peak}<10^{14}$ Hz), intermediate synchroton peaked (ISP; $\nu_{\rm
peak}=10^{14-15}$ Hz) and high synchrotorn peaked (HSP; $\nu_{\rm
peak}>10^{15}$ Hz) blazars. The observed X-ray emission in blazars is mostly
by synchrotron or synchrotron self Compton (SSC) processes, and thus jet
dominated. Therefore, the X-ray flux variations in them are believed to be due
to their relativistic jets. However, in the non-jetted category of AGN such as
Seyfert galaxies the observed X-ray emission is predominantly due to inverse
Compton scattering of optical/UV photons from the accretion disc by energetic
electrons in the corona and thus the X-ray flux variations in them is
attributed to the accretion disc (Haardt & Maraschi, 1993).

In the faint or low jet activity state of FSRQs, prominent accretion disc
emission is seen in the broad band SED (Raiteri et al., 2009) in many sources
and therefore in their faint state, their observed X-ray emission could also
have a contribution from the accretion disc/corona. Similarly, in the case of
BL Lacs too, continuum emission is dominated by non-thermal emission from
their relativistic jets. However, occasionally broad emission lines are seen
in their optical spectra which suggests that the BLR is photoionised by the
accretion disc emission. Emission lines (though weak) are seen in many BL Lac
objects (Raiteri et al., 2009; Vermeulen et al., 1995; Corbett et al., 2000;
Stocke et al., 2011). Also, broad emission lines are seen in many $\gamma$-ray
detected BL Lacs (Shaw et al., 2013), a large fraction of them belonging to
the LSP and ISP category. Recently, from X-ray observations carried out on the
BL Lac object Mrk 421 Chatterjee et al. (2018) found a break in the power
spectral density, which the authors attributed to the influence of the disc
onto the jet emission from Mrk 421. In the X-ray light curves of many blazars,
log normal (LN) flux distribution is observed similar to that seen in Seyfert
galaxies (Ackermann et al., 2015; Sinha et al., 2016) which again argues for
the contribution from the accretion disc to the observed X-ray emission of
blazars. According to Wandel & Urry (1991), the observed UV and X-ray emission
from BL Lacs can come from accretion disc. They were able to fit the observed
UV and soft X-ray spectrum of the BL Lac object PKS 2155$-$304 using accretion
disc spectra. Also Grandi & Palumbo (2004); Zdziarski & Grandi (2001) found
the signature of accretion disc emission in the X-ray spectra of the blazar 3C
273 and rthe adio galaxy 3C 120. However, this model of Wandel & Urry (1991)
was not supported by Smith & Sitko (1991), who from optical polarimetric
observations on the BL Lac PKS 2155$-$304, claimed to have found no
observational evidence for the contribution of the accretion disc to the
observed UV and optical continuum. A long term multi-wavelength lightcurve and
SEDs study showed that the data fits well with the LN model, inferred the
lognormality character might have originated from the disc and amplified by
the jet (Kapanadze et al., 2020; Acciari et al., 2021).

Mrk 421 (z = 0.03 de Vaucouleurs et al. 1991) is one of the closest BL Lac
objects hosted in an elliptical galaxy at a distance of 140 Mpc ($H_{0}$= 71
km $s^{-1}$ $Mpc^{-1}$, $\Omega_{m}$ =0.27, $\Omega_{\Lambda}$ = 0.73) and
hosting a 4$\times$ $10^{8}$ $M_{\odot}$ black hole (BH) at the center
(Wagner, 2008). The SED of Mrk 421 has the classical two-peak shape (Urry &
Padovani, 1995; Ulrich et al., 1997) and it has been studied extensively in a
broad energy band ranging from radio to very high energy $\gamma$-rays
(Fossati et al., 2008; Abdo et al., 2011; Shukla et al., 2012; Aleksić et al.,
2015). Most of the observed properties of Mrk 421 are understood to arise from
a relativistic jet at a small angle along our line of sight (Urry & Padovani,
1995). There are frequent multiwavelength (MWL) campaigns on this object to
understand its multiscale variability (Macomb et al., 1995; Gupta et al.,
2004; Acciari et al., 2009; Paliya et al., 2015; Pandey et al., 2017, and
references therein). It is an HSP BL Lac and is also the first extra-galactic
source detected in TeV $\gamma$-rays (Punch et al., 1992). As suggested by
several studies, the X-ray spectra of HBLs are curved and described well with
the log-parabola model (Massaro et al., 2004; Tramacere et al., 2007), where
the photon index is not a constant, rather varies with logarithmic energy. In
2006, the source was observed with a peak flux of $\sim 85$ mCrab in the
2$-$10 keV band, indicating that the first peak of SED occurred at an energy
beyond 10 keV (Tramacere et al., 2009; Ushio et al., 2009). Again, in 2013
April, Mrk 421 was observed to undergo a major X-ray outburst and was
extensively studied by multiple observational facilities, including the
Nuclear Spectroscopic Telescope Array (NuSTAR) and Swift satellites. Mrk421
showed very high optical emission during its historically low X-ray emission
during January-February 2013 and the presence of multiple compact regions
contributing to the broadband emission during low-activity states was
suggested by Baloković et al. (2016). On the contrary, low optical emission
during high X-ray emission was also reported during in February 2010 by
(Abeysekara et al., 2020). Therefore, the optical and X-ray emission are found
to show different behaviour (Carnerero et al., 2017). A MWL study has been
performed using the radio to TeV energy band data, very recently, where the
authors studied (anti)correlation between optical and hard X-ray energy bands
(MAGIC Collaboration et al., 2021). Also, it has been studied for its X-ray
and $\gamma$-ray flux variability characteristics (Kapanadze et al., 2016;
Rani et al., 2017, 2019; Rajput et al., 2020).

A huge amount of MWL data available on BL Lacs are well explained assuming the
centre is viewed along the jet. In such a view, though the jet emission
significantly gets Doppler boosted, one can also see the central region of BL
Lacs. Therefore, it is quite natural to expect contributions of both the
accretion disc in the UV/optical emission and corona in the observed X-rays.
When the jet emission dominates, it might not be possible to find the
signature of the accretion disc/corona in the continuum emission from BL Lacs.
However, when the jet is less active, emission from accretion disc/corona
could form a significant component of the observed continuum emission. Weak
(equivalent width $<$ 1 Å) and narrow FWHM = $300\pm 30$ km/sec) Ly$\alpha$
emission line has been seen in Mrk 421 by Stocke et al. (2011). The weakness
of the Ly$\alpha$ line in Mrk 421 could be due to the combined effects of (a)
the Doppler boosting of the continuum emission by the relativistic jet that is
closely aligned with the observer and (b) weak accretion disc. Taking Doppler
boosting into account, the intrinsic FWHM of Ly$\alpha$ line in Mrk 421 was
found to be similar to other AGN (Foschini, 2012). These observations clearly
demonstrate that during the faint state of BL Lacs, the accretion disc/corona
emission too can be an important contributor to the observed X-ray emission,
however, might not dominate the non-thermal emission from the jet. In a
detailed multiwavelength study by Acciari et al. (2021) showed that the flux
distributions of Mrk 421 in radio and soft X-rays are better described with a
Gaussian function, while in the optical, hard X-rays, high energy and VHE
$\gamma$-rays are preferably described with a LN distribution. Such
distribution of flux implies that the emission is being powered by a
multiplicative process rather than an additive one. In addition, Chakrabarti &
D’Silva (1994) mentioned that very high energy $\gamma$-rays from blazars
could be due to Fermi acceleration taking place at the surface of the
CENtrifugal pressure supported BOundary Layer (CENBOL) of a thick disc. Here,
CENBOL behaves like a dynamical corona also a base of the jet (see Mondal &
Chakrabarti, 2021). At this point, it is worth questing whether a static
corona or normal corona can explain the same behaviour? The answer is no, as
the normal corona cannot remain hot for ages unless heated by some underlying
heating mechanism. It has been observed that spectral index steepens with age
(Saikia et al., 2016, and references therein), because jets are not reheated
and only cools down with age. Therefore, it is essential to consider a
dynamical advective halo or dynamic corona to replenish the electrons.
Considering the above observed and theoretical constraints, in this work, we
aimed to fit the observed X-ray spectra of Mrk 421 during its moderately X-ray
brightness state with disc based models to derive various disc parameters
which could provide useful inputs into the accretion characteristics of Mrk
421. However, we note that in the literature, X-ray emission from Mrk 421 is
explained by processes that do not arise from the disc (MAGIC Collaboration et
al., 2021, and references therein). This paper is organized as follows: in
Section 2 we explain the observations and data reduction procedures, in
Section 3, we discuss the various analysis carried out on the data sets. The
conclusions are given in the final Section.

##  2 Observation and data reduction

Mrk 421 was in its moderate brightness state in the radio (a proxy for jet
emission) during 2017 and high brightness state during 2013. We show in Figure
1, its MWL light curves, that include the one month binned $\gamma$-ray light
curve for the energy range 100 MeV to 200 GeV 111
https://fermi.gsfc.nasa.gov/ssc/data/access/lat/4yr_catalog/ap_lcs.php from
the Fermi Gamma-ray Space Telescope, the optical V-band light curve from
Steward Observatory (Smith et al., 2009) and the 15 GHz radio light curve from
the Owens Valley Radio Observatory (OVRO; Richards et al. 2011). The V-band
light curve was directly taken from Steward archives and we have not done host
galaxy correction to the photometric points. Also shown in the same figure is
the degree of polarization in the optical V-band. From the soft X-ray light
curve from the Swift/XRT and the hard X-ray light curve from the Swift/BAT
(Arbet-Engels et al., 2021) it is evident that Mrk 421 was in a moderate X-ray
brightness state during the year 2017 compared to previous years. The vertical
dotted lines in Figure 1 show the epochs considered in the present paper.
Analysis of the X-ray data available during the low jet activity state of the
source through accretion disc based models fits to the observed X-rays could
be a way to derive various accretion disc parameters of the source. We
therefore searched the archives of the NuSTAR (Harrison et al. 2013) for the
availability of X-ray data and could find four epochs of both data available
during the same period. For comparison, we also selected one epoch during
2013, when the source was in a bright jet activity state that yielded high and
variable X-ray and $\gamma$-ray emission (Paliya et al., 2015; Acciari et al.,
2020). The details of the observations are given in Table 1. The flux of the
source in the optical, X-ray, and $\gamma$-ray bands and the optical
polarization during those four epochs are given in Table 2. From Table 2, it
is to be noted that though the source is detected in the $\gamma$-ray band
with good test statistics (TS) 222A TS of 25 roughly indicates detection at
the 5$\sigma$ level; Mattox et al. (1996) values, it is in a relatively
quiescent state (see Figure 1).

Considering the historic brightness state of the source, it is likely that the
source is in a relatively faint state in the optical in 2017. Carnerero et al.
(2017) presented the optical light curves during 2008 - 2015 in R-band. The
lowest brightness state was in 2008, when the R-band brightness was 13.5 mag.
The R-band brightness during the four epochs range between 13.09 and 13.32
mag, which suggest that during the four epochs considered in this work, the
R-band brightness is close to the historic faint state of Mrk 421. Low optical
emission can always be not a proxy for low X-ray emission as a flare in X-ray
can have no corresponding flare in the optical. In such cases, the optical and
X-ray emission may not be co-spatial (Carnerero et al., 2017). On the
contrary, there can be instances of an optical flare with no counterpart in
the X-ray band. Also, the multi-band flux variability characteristics of
blazars are highly complex, as revealed from recent observations Rajput et al.
(2019, 2020, 2021). However, inspection of Figure 1 indicate that the source
is in a relatively faint state across different bands of the electromagnetic
spectrum during 2017. Thus, during 2017, the contribution of relativistic jet
to the total emission in the optical/UV/X-ray may be relatively lower compared
to the high jet activity state in 2013.

The baseline gamma-ray flux (Figure 1) is $\sim 2\times 10^{-7}$ ph cm-2 s-1.
During epochs C and D, the gamma-ray activity state is also close to this
value. Considering the X-ray light curves from December 2012 - April 2018, the
minimum 0.3-2 keV X-ray flux was 5.71 ph s-1 and the maximum 0.3-2 keV X-ray
flux was 117.44 ph s-1. During January - December 2017, the minimum and
maximum flux values were 8.84 and 41.64 ph s-1 respectively. During the epochs
analysed in this work, the flux values ranged between 13.27 and 26.73 ph s-1
(Table 2). The source is thus not in the faintest X-ray state; however, the
X-ray flux values are much weaker than other times and hence the contribution
of jet emission during these four epochs is relatively lower compared to other
epochs. Also, during the period the $\gamma$-ray, optical and X-ray variations
are correlated, and they are also at the moderate brightness states. Between
epoch E (historic bright state; fluxes are given in subsection 3.4) and D, the
soft flux has decreased by 5.8 times, and the hard (2-10 keV) flux has
decreased by 22.6 times. The reduction of hard flux between E and D is much
larger, followed by soft X-ray. This points to relatively lower dominance of
jet emission in epoch D compared to E and emerging contribution of UV emission
from the accretion disc. Between epochs B and D, the hard X-ray flux has
decreased by a factor of $\sim 3.5$, while the UV flux has increased by a
factor of 1.2. The moderate brightness state in the optical, $\gamma$-ray and
radio bands along with low optical polarization in comparison to the
polarization at the high X-ray brightness state in 2013 (see Figure 1 and
Figure 2) indicates that the jet activity in Mrk 421 during 2017 is likely to
be lower. The $\gamma$-ray lightcurve in Figure 2 is one day binned. There are
also reports in the literature that point to varied correlation between
optical flux and degree of optical polarization (Gaur et al., 2012; Pandey et
al., 2021).

Models available in the literature on polarization in blazars point to shock
propagation through twisted fields inside the jet being the cause of the
changes in polarization (Blandford & Königl, 1979; Lyutikov et al., 2005;
Marscher et al., 2008; Itoh et al., 2016, and references therein). In the
present study, of the four epochs considered, in three epochs namely A to C,
the source was at a moderate X-ray brightness state, and larger than the
typical average X-ray brightness over a quiescent 4-5 months period (Abdo et
al., 2011). But during epoch D, the X-ray brightness of the source was lower
than the typical average quiescent brightness. Also, multi-band SED modelling
coupled with other analysis point to the presence of multiple emission zones
that give rise to the observed emission even during the quiescent state
(Baloković et al., 2016). Such multiple emission regions contributing to the
overall observed emission too will give rise to the low level of optical
polarisation typically observed in this source, except during instances that
are close to flaring episodes. Given the presence of multiple emission regions
contributing to the observed broad band emission during the quiescent state of
Mrk 421, we aim to explore the possibility that there is disc emission in the
observed X-ray spectrum and under this hypothesis deduce various accretion
parameters of the source. We therefore aim to use the Two Component Advective
Flow Two Component Advective Flow (tcaf; Chakrabarti & Titarchuk, 1995) model
on the four sets of data analysed in this work. However, we note that multi-
band SED fitting of the observations both in quiescent and flaring states
invoke SSC processes for the observed X-rays. Recently Jana et al. (2017)
estimated jet flux from tcaf model for systems where the lowest contribution
can be treated from the disc only (and that too not boosted by Doppler effect)
while the brightest part will have both the disc and the jet. Figure 3 shows
the optical spectra of the source taken from the Steward Observatory (Smith et
al., 2009) archives for the epochs considered in this present study. The
combined spectrum of the four epochs is shown in black. The positions of the
H$\alpha$ and H$\beta$ lines are marked with vertical dashed lines and zoomed
at the upper and lower corners. The emission lines are not visible in the
optical spectra taken during those four epochs using a moderate size optical
telescope. This indicates that even in the moderate X-ray brightness state,
the Doppler boosted jet emission dominates to swamp the emission lines.
Alternatively, high S/N observations with larger aperture telescopes, might
have detected faint emission lines. The NuSTAR data were extracted using the
standard NUSTARDAS v1.3.1
333https://heasarc.gsfc.nasa.gov/docs/nustar/analysis/ software. We ran
nupipeline task to produce cleaned event lists and nuproducts to generate the
spectra. We used a region of $30^{\prime\prime}$ for the source and
$50^{\prime\prime}$ for the background using ds9. The data were grouped by
grppha command, with a minimum of 30 counts rate per energy bin. The same
binning was used for all the observations.

As blazar variability timescales can range from hours to minutes depending on
the underlying physical processes (Aharonian et al., 2007; Paliya et al.,
2015), we split each epoch of observation into different segments, with each
segment containing $\sim 1.8$ ksec of data. This division of the data into
segments is to investigate spectral variations in hourly time scale. For that
purpose, first we made our own GTI files for each time range using the
gtibuild task in SAS444https://www.cosmos.esa.int/web/xmm-newton/sas-threads
environment and used those GTI files during the pipeline extraction (same as
in Mondal & Chakrabarti, 2019). As the data quality in each segment is highly
noisy above 20 keV, for the analysis of the data pertaining to each segment we
used the data in the energy range of 3$-$20 keV. However, considering each
epoch of observation as unique, the data is of good S/N up to 60 keV, and
therefore, for analysis of each epoch of observation we used the data in the
energy range of 3$-$60 keV. For spectral analysis of the data, we used
XSPEC555https://heasarc.gsfc.nasa.gov/xanadu/xspec/ (Arnaud, 1996) version
12.11.0. Each segment of the data was fitted using the simple power-law (pl)
model, while each epoch of observation was fitted using an accretion disc
based tcaf model and the thermal Comptonization model thcomp and the pl model.
We used the absorption model ${\it tbabs}$ (Wilms et al., 2000) with the
Galactic hydrogen column density fixed at 1.5$\times$ 1020 atoms cm-2 (Elvis
et al., 1989; Kalberla et al., 2005) throughout the analysis.

Table 1:  Log of observations. OBSID | Epoch | Date | MJDstart | MJDend | Exposure (s)
---|---|---|---|---|---
60202048002 | A | 03-01-2017 | 57756.99385 | 57757.52163 | 23691
60202048004 | B | 31-01-2017 | 57784.99038 | 57785.55288 | 21564
60202048006 | C | 28-02-2017 | 57812.92441 | 57813.49733 | 23906
60202048008 | D | 27-03-2017 | 57839.91052 | 57840.63969 | 31228
60002023031 | E | 14-04-2013 | 56396.90355 | 56397,29939 | 15605
Table 2:  Flux and polarization values of Mrk 421 during the epochs studied in this work. The $\gamma$-ray fluxes are in units of 10-7 ph cm-2 sec-1. The optical flux and polarization measurements during a and b belong to 27 February 2017 and 28 March 2017. The Soft and Hard represent the Swift/XRT 0.3-2 keV and 2-10 keV band fluxes. Epoch | Date |  $\gamma$-ray | TS | Radio | V-band | R-band | Soft | Hard | UV | Hard/Soft
---|---|---|---|---|---|---|---|---|---|---
|  | flux |  | mJy | (mag) | pol. (%) | (mag) |  ph$s^{-1}$ |  ph$s^{-1}$ |  ph$s^{-1}$ |
A | 03 Jan. 2017 | 3.4 | 97 | $0.46\pm 0.02$ | $13.67\pm 0.03$ | $2.98\pm 0.06$ | $13.32\pm 0.03$ | $26.68\pm 0.26$ | $6.42\pm 0.07$ | $--$ | 0.24
B | 31 Jan. 2017 | 2.1 | 29 | $0.50\pm 0.01$ | $13.61\pm 0.02$ | $1.31\pm 0.06$ | $13.26\pm 0.02$ | $26.73\pm 1.78$ | $6.46\pm 0.77$ | $7.05\pm 0.11$ | 0.24
C | 28 Feb. 2017 | 6.9 | 19 | $0.49\pm 0.01$ | $13.55\pm 0.02^{a}$ | $2.52\pm 0.06^{a}$ | $13.19\pm 0.03$ | $25.48\pm 0.85$ | $6.46\pm 0.42$ | $--$ | 0.25
D | 27 Mar. 2017 | 2.7 | 11 | $0.48\pm 0.01$ | $13.42\pm 0.02^{b}$ | $1.43\pm 0.05^{b}$ | $13.09\pm 0.02$ | $13.27\pm 0.44$ | $1.88\pm 0.26$ | $8.71\pm 0.21$ | 0.14
Figure 1: Long term flux and polarization variation of the source. From top,
we show the flux variation in the $\gamma$-ray, optical and radio bands. The
bottom panel shows the variation in the degree of polarization (%) in the
optical V-band.The blue dashed lines show the epochs of NuSTAR observations
studied here. The green dotted line correspond to data taken during April
2013. The bottom three panels show the fluxes from the Swift observation. The
X-ray and UV light curves are taken from Arbet-Engels et al. (2021).  Figure
2: Zoomed version of Figure 1, depicting clearly the brightness states of the
four epochs analysed in this work. Figure 3: Optical spectra taken from
Steward Observatory during the same observation period as considered in NuSTAR
analysis. Four different colors correspond to four epochs considered for X-ray
analysis and the black curve shows the combined spectrum. Zoomed regions
around H$\alpha$ and H$\beta$ lines are also shown at the top right and bottom
left.

##  3 Spectral analysis

###  3.1 Model fits to the spectra

First, we carried out simple pl model fits to each segment of the data in all
the epochs. The tbabs*pl model fits the data satisfactorily, with reduced
$\chi_{r}^{2}$ between 0.5$-$1.5. The model parameters and the reduced
$\chi_{r}^{2}$ for all the segments are given in Table 3. Secondly, for all
the data in each epoch of observations, we extracted the spectra in the 3$-$60
keV band and fitted them with both accretion disc and thermal Comptonization
models.

For the data belonging to each epoch of NuSTAR, fitting the spectra with the
phenomenological pl model (tbabs*pl), was found not to be satisfactory with
$\chi_{r}^{2}>1.2$ (see Table 4), except for epoch D, where it was unity. We
further fit the NuSTAR data using cutoffpl model, which returns a good fit
with $\chi_{r}^{2}\sim 1$. We hypothesize this as a signature of a
multiprocess system where disc and jet both can contribute. It is worth
mentioning that considering the MWL observation up to TeV band requires
synchrotron process based models to fit and infer the data and variability,
which are believed to be originated from the jet (see MAGIC Collaboration et
al., 2021, for a very recent study). Also, in the literature, the X-ray
spectrum in blazars are known to derive from the conventional power law form
with a tendency of show curvature, which is generally attributed to
synchrotron cooling break (see Gaur, 2020, and references therein).

In the literature about blazars, curved X-ray spectra are common and normally
used whenever the X-ray instrumental resolution allows to detect such
curvature(see e.g., Massaro et al., 2004). Here we provide a different view
based on the discussions in the literature (see earlier sections) that
variability originated in the disc can be amplified by the jet. Hence,
following our hypothesis, where the accretion disc-based model can be applied
to Mrk 421 data during its low or moderate jet activity state. We therefore
used the thermal Comptonization model thcomp (Zdziarski et al., 2020) along
with partial ionization zxipcf (Reeves et al., 2008) and diskbb as
tbabs*zxipcf*thcomp*diskbb to fit the NuSTAR data, which returned good fits.
For proper representation and estimation of parameters of the thcomp model, we
used “energies” command to extend the energy range from 0.01 to 1000 keV.
During the fitting we kept the ionization rate (log $\xi$) to 0.0 for the
zxipcf model and the scattering fraction ($f_{\rm sc}$) was kept fixed at 1
for all epochs for the thcomp model. Addition of diskbb provided the
temperature of the disc ($T_{\rm in}$) which was found to be reasonable (see
Table 5). For epoch D, the partial covering factor ($C_{f}$) was the lowest
$0.07$ and the temperature of the corona was the lowest among the four epochs.
This implies increased generation of disc photons, which on getting inverse
Compton scattered by the hot electrons in the corona cooled the corona down.
This is also consistent with the lowest-flux state of the source at epoch D.
We found temperature of the corona with values of 144$\pm$46, 55$\pm$8,
51$\pm$9, and 27$\pm$5 keV for epochs A, B, C, and D respectively, thus
decreasing. We thus found the temperature of the corona to decrease from epoch
A to D. The results of the best fitted model parameters are given in Table 5.
From the diskbb model fitted norm we estimated $R_{\rm in}$/$r_{\rm s}$, which
comes out to be 57, 25, 17, and 2 for epochs A to D. The estimated values are
significant and as per our hypothesis, this could be a signature of the
presence of possibly a truncated disc component.

Further, to study the accretion flow parameters and geometrical variation of
the flow and its contribution to the spectrum, we fitted the four observations
in the 3$-$60 keV band using the physical tcaf model. For spectral fitting, we
ran the source code directly in XSPEC as a local
model666https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node101.html. The
tcaf model777Currently the model is not available in XSPEC as a local/table
model. (Chakrabarti & Titarchuk, 1995) can successfully fit the X-ray data on
low mass X-ray binaries (Debnath et al. 2014, and references therein; Mondal
et al. 2014, and references therein; Iyer et al. 2015, and references therein;
Debnath et al. 2015, and references therein) and AGN (Mandal & Chakrabarti,
2008; Nandi et al., 2019; Mondal & Stalin, 2021), and has been used to infer
the outburst behaviour as well as the variability properties of compact
objects. This model requires five parameters, namely, (i) mass of the black
hole ($M_{\rm BH}$), (ii) disc accretion rate ($\dot{m}_{d}$), (iii) halo
accretion rate ($\dot{m}_{h}$), (iv) shock location, which is the boundary of
the CENBOL (Chakrabarti (1989), $X_{s}$ in unit of $r_{s}$, where $r_{s}$ is
the Schwarzschild radius, $2GM_{\rm BH}/c^{2}$), and (v) shock compression
ratio ($R$). For tcaf model fits, we froze the mass of the BH at $4\times
10^{8}$ M⊙ (Wagner, 2008), which gives $r_{s}=1.2\times 10^{14}\text{cm}$. The
quality of fits for all epochs appear to be good with the reduced
$\chi_{r}^{2}$ ranging from 0.9 to 1.1. It can be seen that the accretion rate
of the disc component varies in the range from 0.02$-$0.05$\dot{M}_{\rm Edd}$
and the halo accretion rate varies from 0.22$-$0.35$\dot{M}_{\rm Edd}$.
Increase of the disc accretion rate leads to increase in the number of soft
photons. This cools the corona by inverse Compton scattering, thus causing the
inner edge of the disc to move inward from $\sim 20$ to $10r_{s}$. The results
of the tbabs*zxipcf*tcaf model fits are given in Figure 4. From the parameters
obtained from tbabs*zxipcf*tcaf fits to the data we derived post-facto values
of $kT_{e}$ of 189, 55, 38, and 20 keV for epochs A, B, C, and D respectively.
We are not using any simple equation to calculate $kT_{e}$ in the source code,
which depends on all input parameters. Therefore, estimating error is beyond
the scope of this paper.

Thus, from the fitting of four epochs of data with both thcomp and tcaf models
we infer the following (i) tcaf model alone is a good representation of the
observations in the full energy band, proving different accretion flow
parameters, (ii) the thermal Comptonization thcomp model along with diskbb
well fits the data providing values of the temperature the corona, photon
index, and the temperature of the disc, (iii) the nature of variation of
$kT_{e}$ is in agreement with the variation in mass accretion rate, (iv) the
highest accretion rate obtained for the epoch D from tcaf is in agreement with
the lowest flux, and shock compression ratio observed, (v) the adequacy of
non-magnetic disc models to fit the data, can be the signature of low magnetic
field.

Figure 4: The 3$-$60 keV spectra fitted with tbabs*zxipcf*tcaf model along
with the residuals. The panels (a)-(d) correspond to the data acquired at
epochs A, B, C and D respectively.

###  3.2 Photon index and spectral flux

On hourly time scales, the photon index ($\Gamma_{\rm PL}$) is found to vary
between $2.2$ and $3.0$ ($\Delta\Gamma=0.8$) within a span of 83 days from
epoch A to D. We note that $\Gamma_{\rm PL}$ was always high, which implies
that the source was in a high/soft spectral state. The very low flux spectrum
with high $\Gamma_{\rm PL}$ belongs to epoch D. Figure 5, left panel shows the
variation of $\Gamma_{\rm PL}$ with segments of every epoch of observation.
Significant change in $\Gamma_{\rm PL}$ was observed during epoch D. On each
epoch of observation, the source exhibited small scale fluctuations in
$\Gamma_{\rm PL}$. The green shaded band shows the error bar and black
vertical lines show transition in each epoch. To check the behaviour of
$\Gamma_{\rm PL}$ of all the segments with the total flux in the 3$-$20 keV
band, we plotted the variation of $\Gamma_{\rm PL}$ with the total flux
$F_{3-20\leavevmode\nobreak\ keV}$ in Figure 5. We fitted the observed data
points in the $\Gamma$ vs. flux diagram using a linear function of the form
$\Gamma=a\times F_{3-20keV}+b$, taking into account the errors in both
$\Gamma$ and flux following Press et al. (1992). This fitting gave a
significant negative correlation between $\Gamma$ and flux with a correlation
co-efficient of $-$0.860, i.e. the value of photon index decreases with
increasing flux indicating a ‘harder when brighter’ trend in the 3$-$20 keV
band. The same behaviour has also been observed using the same data by MAGIC
Collaboration et al. (2021). Such behaviour is more often seen in the HSP
category of blazars (Giommi et al., 1990; Pian et al., 1998). Mrk 421 is an
HSP blazar and the harder when brighter trend could be most likely due to
change in the power-law component of the relativistic jet (Rani et al., 2017).
However, such harder when brighter trend in the X-ray band can also be due to
processes related to the accretion disc /corona as discussed towards the end
of the next section.

Table 3:  Results of tbabs*pl model fits to each segment from epoch A to D. The fluxes are in units of $10^{-10}\text{erg}\text{cm}^{-2}\text{s}^{-1}$. Here, $\Gamma_{\rm PL}$ is the powerlaw photon index. Epoch | Seg. No. | MJDstart | MJDend | $\Gamma_{\rm PL}$ | $F_{3-10\text{keV}}$ | $F_{10-20\text{keV}}$ | $F_{3-20\text{keV}}$ | HR | $\chi_{r}^{2}/dof$
---|---|---|---|---|---|---|---|---|---
| 1 | 57756.99385 | 57757.03783 | $2.413\pm 0.029$ | $2.423\pm 0.003$ | $0.937\pm 0.001$ | $3.360\pm 0.006$ | $0.387\pm 0.001$ | 1.0/158
| 2 | 57757.03783 | 57757.08181 | $2.473\pm 0.038$ | $2.472\pm 0.005$ | $0.901\pm 0.001$ | $3.374\pm 0.012$ | $0.364\pm 0.001$ | 0.8/99
| 3 | 57757.08181 | 57757.12580 | $2.375\pm 0.022$ | $2.623\pm 0.001$ | $1.053\pm 0.014$ | $3.676\pm 0.005$ | $0.401\pm 0.005$ | 1.2/208
| 4 | 57757.12580 | 57757.16978 | $2.373\pm 0.031$ | $2.805\pm 0.003$ | $1.128\pm 0.008$ | $3.933\pm 0.005$ | $0.402\pm 0.003$ | 0.9/154
| 5 | 57757.16978 | 57757.21376 | $2.391\pm 0.028$ | $2.829\pm 0.001$ | $1.118\pm 0.001$ | $3.947\pm 0.005$ | $0.395\pm 0.001$ | 0.9/161
A | 6 | 57757.21376 | 57757.25774 | $2.405\pm 0.022$ | $2.793\pm 0.008$ | $1.089\pm 0.004$ | $3.883\pm 0.005$ | $0.390\pm 0.002$ | 1.0/207
| 7 | 57757.25774 | 57757.30172 | $2.392\pm 0.034$ | $2.889\pm 0.009$ | $1.140\pm 0.005$ | $4.029\pm 0.002$ | $0.395\pm 0.002$ | 1.0/123
| 8 | 57757.30172 | 57757.34570 | $2.404\pm 0.022$ | $3.075\pm 0.008$ | $1.200\pm 0.007$ | $4.275\pm 0.009$ | $0.390\pm 0.002$ | 1.0/202
| 9 | 57757.34570 | 57757.38968 | $2.401\pm 0.024$ | $3.131\pm 0.002$ | $1.224\pm 0.006$ | $4.355\pm 0.011$ | $0.391\pm 0.002$ | 0.9/186
| 10 | 57757.38968 | 57757.43367 | $2.446\pm 0.036$ | $2.977\pm 0.010$ | $1.114\pm 0.013$ | $4.091\pm 0.002$ | $0.374\pm 0.005$ | 0.8/123
| 11 | 57757.43367 | 57757.47765 | $2.479\pm 0.021$ | $2.992\pm 0.007$ | $1.084\pm 0.006$ | $4.077\pm 0.011$ | $0.362\pm 0.002$ | 1.2/213
| 12 | 57757.47765 | 57757.52163 | $2.411\pm 0.022$ | $2.883\pm 0.009$ | $1.117\pm 0.004$ | $4.000\pm 0.005$ | $0.387\pm 0.002$ | 1.0/209
| 13 | 57784.99038 | 57785.03725 | $2.368\pm 0.027$ | $3.186\pm 0.002$ | $1.287\pm 0.005$ | $4.473\pm 0.004$ | $0.404\pm 0.002$ | 1.2/181
| 14 | 57785.03725 | 57785.08413 | $2.306\pm 0.030$ | $3.439\pm 0.015$ | $1.477\pm 0.004$ | $4.916\pm 0.004$ | $0.429\pm 0.002$ | 0.9/146
| 15 | 57785.08413 | 57785.13100 | $2.341\pm 0.022$ | $3.408\pm 0.005$ | $1.415\pm 0.005$ | $4.824\pm 0.002$ | $0.415\pm 0.002$ | 1.0/199
| 16 | 57785.13100 | 57785.17788 | $2.341\pm 0.036$ | $3.403\pm 0.008$ | $1.411\pm 0.009$ | $4.814\pm 0.003$ | $0.415\pm 0.003$ | 0.9/106
| 17 | 57785.17788 | 57785.22475 | $2.256\pm 0.026$ | $3.440\pm 0.007$ | $1.549\pm 0.004$ | $5.000\pm 0.004$ | $0.450\pm 0.001$ | 1.1/173
| 18 | 57785.22475 | 57785.27163 | $2.337\pm 0.022$ | $3.619\pm 0.004$ | $1.507\pm 0.014$ | $5.126\pm 0.010$ | $0.416\pm 0.004$ | 1.1/208
B | 19 | 57785.27163 | 57785.31850 | $2.314\pm 0.030$ | $3.537\pm 0.009$ | $1.507\pm 0.006$ | $5.045\pm 0.005$ | $0.426\pm 0.002$ | 0.8/154
| 20 | 57785.31850 | 57785.36538 | $2.299\pm 0.026$ | $3.449\pm 0.005$ | $1.491\pm 0.005$ | $4.939\pm 0.004$ | $0.432\pm 0.002$ | 1.0/177
| 21 | 57785.36538 | 57785.41225 | $2.432\pm 0.022$ | $3.259\pm 0.010$ | $1.237\pm 0.006$ | $4.496\pm 0.001$ | $0.380\pm 0.002$ | 1.0/199
| 22 | 57785.41225 | 57785.45913 | $2.470\pm 0.045$ | $3.025\pm 0.012$ | $1.105\pm 0.010$ | $4.130\pm 0.023$ | $0.365\pm 0.004$ | 0.8/75
| 23 | 57785.45913 | 57785.50600 | $2.520\pm 0.022$ | $2.542\pm 0.004$ | $0.884\pm 0.003$ | $3.426\pm 0.018$ | $0.348\pm 0.001$ | 1.1/200
| 24 | 57785.50600 | 57785.55288 | $2.509\pm 0.020$ | $2.127\pm 0.002$ | $0.747\pm 0.002$ | $2.874\pm 0.008$ | $0.351\pm 0.001$ | 1.1/241
| 25 | 57812.92441 | 57812.96848 | $2.566\pm 0.022$ | $2.510\pm 0.005$ | $0.834\pm 0.003$ | $3.345\pm 0.011$ | $0.332\pm 0.001$ | 1.0/189
| 26 | 57812.96848 | 57813.01255 | $2.443\pm 0.040$ | $2.565\pm 0.003$ | $0.963\pm 0.007$ | $3.528\pm 0.014$ | $0.375\pm 0.003$ | 0.6/86
| 27 | 57813.01255 | 57813.05662 | $2.548\pm 0.028$ | $2.301\pm 0.001$ | $0.775\pm 0.005$ | $3.080\pm 0.006$ | $0.337\pm 0.002$ | 0.7/162
| 28 | 57813.05662 | 57813.10069 | $2.462\pm 0.026$ | $2.577\pm 0.003$ | $0.950\pm 0.001$ | $3.526\pm 0.004$ | $0.369\pm 0.001$ | 0.9/171
| 29 | 57813.10069 | 57813.14476 | $2.476\pm 0.039$ | $2.773\pm 0.018$ | $1.008\pm 0.002$ | $3.781\pm 0.022$ | $0.364\pm 0.002$ | 0.9/98
| 30 | 57813.14476 | 57813.18883 | $2.396\pm 0.021$ | $2.968\pm 0.008$ | $1.166\pm 0.005$ | $4.134\pm 0.014$ | $0.393\pm 0.002$ | 1.2/215
| 31 | 57813.18883 | 57813.23291 | $2.387\pm 0.030$ | $2.951\pm 0.003$ | $1.170\pm 0.002$ | $4.121\pm 0.009$ | $0.396\pm 0.001$ | 1.0/148
C | 32 | 57813.23291 | 57813.27698 | $2.418\pm 0.032$ | $2.978\pm 0.006$ | $1.146\pm 0.002$ | $4.124\pm 0.002$ | $0.385\pm 0.001$ | 0.9/136
| 33 | 57813.27698 | 57813.32105 | $2.448\pm 0.023$ | $3.389\pm 0.002$ | $1.265\pm 0.003$ | $4.654\pm 0.008$ | $0.373\pm 0.001$ | 1.1/198
| 34 | 57813.32105 | 57813.36512 | $2.440\pm 0.035$ | $3.515\pm 0.006$ | $1.323\pm 0.002$ | $4.839\pm 0.031$ | $0.376\pm 0.001$ | 1.1/118
| 35 | 57813.36512 | 57813.40919 | $2.489\pm 0.025$ | $3.432\pm 0.013$ | $1.232\pm 0.001$ | $4.664\pm 0.021$ | $0.359\pm 0.001$ | 1.1/183
| 36 | 57813.40919 | 57813.45326 | $2.472\pm 0.023$ | $3.568\pm 0.009$ | $1.301\pm 0.003$ | $4.869\pm 0.010$ | $0.365\pm 0.001$ | 0.9/191
| 37 | 57813.45326 | 57813.49733 | $2.406\pm 0.017$ | $3.919\pm 0.006$ | $1.525\pm 0.005$ | $5.444\pm 0.017$ | $0.389\pm 0.001$ | 1.2/244
| 38 | 57839.91052 | 57839.95341 | $2.938\pm 0.067$ | $0.576\pm 0.001$ | $0.131\pm 0.003$ | $0.707\pm 0.002$ | $0.227\pm 0.005$ | 0.9/39
| 39 | 57839.95341 | 57839.99630 | $2.985\pm 0.106$ | $0.589\pm 0.001$ | $0.130\pm 0.005$ | $0.717\pm 0.002$ | $0.221\pm 0.008$ | 1.5/19
| 40 | 57839.99630 | 57840.03920 | $2.789\pm 0.069$ | $0.612\pm 0.004$ | $0.163\pm 0.001$ | $0.767\pm 0.008$ | $0.266\pm 0.002$ | 1.0/46
| 41 | 57840.03920 | 57840.08209 | $2.778\pm 0.086$ | $0.601\pm 0.009$ | $0.161\pm 0.003$ | $0.763\pm 0.001$ | $0.268\pm 0.006$ | 0.8/28
| 42 | 57840.08209 | 57840.12498 | $2.688\pm 0.073$ | $0.640\pm 0.001$ | $0.188\pm 0.002$ | $0.829\pm 0.001$ | $0.294\pm 0.003$ | 0.8/30
| 43 | 57840.12498 | 57840.16787 | $2.731\pm 0.048$ | $0.697\pm 0.004$ | $0.196\pm 0.002$ | $0.886\pm 0.005$ | $0.281\pm 0.003$ | 0.9/72
| 44 | 57840.16787 | 57840.21077 | $2.521\pm 0.069$ | $0.750\pm 0.002$ | $0.261\pm 0.002$ | $1.011\pm 0.008$ | $0.348\pm 0.003$ | 1.1/37
| 45 | 57840.21077 | 57840.25366 | $2.522\pm 0.052$ | $0.785\pm 0.001$ | $0.272\pm 0.001$ | $1.057\pm 0.003$ | $0.346\pm 0.001$ | 1.1/61
D | 46 | 57840.25366 | 57840.29655 | $2.609\pm 0.048$ | $0.765\pm 0.006$ | $0.244\pm 0.002$ | $1.009\pm 0.005$ | $0.319\pm 0.004$ | 1.0/73
| 47 | 57840.29655 | 57840.33944 | $2.869\pm 0.074$ | $0.836\pm 0.003$ | $0.205\pm 0.002$ | $1.040\pm 0.009$ | $0.245\pm 0.003$ | 0.5/129
| 48 | 57840.33944 | 57840.38234 | $2.673\pm 0.040$ | $0.951\pm 0.004$ | $0.284\pm 0.003$ | $1.235\pm 0.002$ | $0.299\pm 0.003$ | 0.9/104
| 49 | 57840.38234 | 57840.42523 | $2.523\pm 0.045$ | $1.093\pm 0.005$ | $0.380\pm 0.003$ | $1.472\pm 0.003$ | $0.348\pm 0.003$ | 1.2/77
| 50 | 57840.42523 | 57840.46812 | $2.667\pm 0.052$ | $1.141\pm 0.001$ | $0.343\pm 0.003$ | $1.484\pm 0.002$ | $0.301\pm 0.003$ | 0.9/59
| 51 | 57840.46812 | 57840.51101 | $2.633\pm 0.035$ | $1.138\pm 0.001$ | $0.353\pm 0.004$ | $1.491\pm 0.007$ | $0.310\pm 0.004$ | 1.2/121
| 52 | 57840.51101 | 57840.55391 | $2.604\pm 0.059$ | $1.109\pm 0.001$ | $0.355\pm 0.003$ | $1.464\pm 0.007$ | $0.320\pm 0.003$ | 0.9/47
| 53 | 57840.55391 | 57840.59680 | $2.590\pm 0.041$ | $1.175\pm 0.001$ | $0.381\pm 0.001$ | $1.556\pm 0.394$ | $0.324\pm 0.001$ | 0.9/88
| 54 | 57840.59680 | 57840.63969 | $2.588\pm 0.033$ | $1.186\pm 0.004$ | $0.386\pm 0.002$ | $1.572\pm 0.006$ | $0.325\pm 0.002$ | 1.0/133

Figure 5: Variation of $\Gamma_{\rm PL}$ with segment number (left panel). The green shaded band is the one $\sigma$ error region and the black lines show the transition from one observation to the other. The right panel shows the $\Gamma_{\rm PL}$ variation with total flux. The red solid line is the weighted linear least square fit to the data. Both panels show a significant change in $\Gamma_{\rm PL}$ from 2.2 to 3.0 in about 83 days. Table 4: Best fitting parameters and fluxes for the models tbabs*pl to NuSTAR. Here, flux is in $10^{-10}$ erg cm2sec-1 unit. Epoch | MJDstart | MJDend | $\Gamma_{\rm PL}$ | $F_{\rm 3-60keV}$ | $\chi_{r}^{2}/dof$
---|---|---|---|---|---
A | 57756.99385 | 57757.52163 | $2.436\pm 0.006$ | $5.166\pm 0.017$ | 1.2/456
B | 57784.99038 | 57785.55288 | $2.421\pm 0.006$ | $5.547\pm 0.006$ | 1.4/452
C | 57812.92441 | 57813.49733 | $2.488\pm 0.006$ | $5.335\pm 0.022$ | 1.5/451
D | 57839.91052 | 57840.63969 | $2.661\pm 0.011$ | $1.430\pm 0.001$ | 1.0/316
Table 5:  Parameters of tbabs*zxipcf*thcomp*diskbb model fits to the spectra. Here, $C_{\rm f}$ is the partial covering fraction, $kT_{e}$, and $\Gamma$ are the temperature of the corona and photon index of the spectrum respectively. $T_{\rm in}$ and Ndbb are the temperature and normalization of the diskbb model component. Epoch | MJDstart | MJDend | $C_{\rm f}$ | $kTe$ | $\Gamma$ | $T_{\rm in}$ |  Ndbb | $\chi_{r}^{2}/dof$
---|---|---|---|---|---|---|---|---
|  |  |  | [keV] |  | [keV] | $\times 10^{13}$ |
A | 57756.99385 | 57757.52163 | $0.26\pm 0.02$ | $144\pm 46$ | $2.49\pm 0.06$ | $0.07\pm 0.02$ | $3.13\pm 1.02$ | 1.0/451
B | 57784.99038 | 57785.55288 | $0.33\pm 0.07$ | $55\pm 8$ | $2.53\pm 0.09$ | $0.08\pm 0.02$ | $0.61\pm 0.17$ | 0.9/448
C | 57812.92441 | 57813.49733 | $0.3\pm 0.03$ | $51\pm 9$ | $2.6\pm 0.1$ | $0.08\pm 0.02$ | $0.26\pm 0.05$ | 1.1/446
D | 57839.91052 | 57840.63969 | $0.07\pm 0.01$ | $27\pm 5$ | $2.58\pm 0.07$ | $0.09\pm 0.02$ |  2.9e$-$3$\pm$3e$-$4 | 1.0/312
Table 6:  Results of tbabs*zxipcf*tcaf model fits to the spectra. $\dot{m}_{\rm d}$, and $\dot{m}_{\rm h}$ represent tcaf model fitted sub-Keplerian (halo) and Keplerian (disc) rates in Eddington rate unit respectively. $X_{s}$ (in Schwarzchild radius $r_{s}$), and $R$ are the model fitted shock location and shock compression ratio values respectively. $N_{t}$ is the model normalization. Fluxes are in $10^{-10}$ ergs cm-2 s-1. MJDstart | MJDend | $C_{\rm f}$ | $\dot{m}_{\rm d}$ | $\dot{m}_{\rm h}$ | $X_{\rm s}$ | R | $N_{t}\times 10^{4}$ | $F_{\rm tot}$ | $F_{\rm disk}$ | $F_{\rm jet}$ | $\chi_{r}^{2}$
---|---|---|---|---|---|---|---|---|---|---|---
57756.99385 | 57757.52163 | $0.31\pm 0.03$ | $0.021\pm 0.002$ | $0.221\pm 0.011$ | $19.77\pm 3.79$ | $1.91\pm 0.31$ | $1.36\pm 0.18$ | $5.09\pm 0.02$ | $1.61\pm 0.01$ | $3.48\pm 0.02$ | 1.0/450
57784.99038 | 57785.55288 | $0.36\pm 0.04$ | $0.028\pm 0.002$ | $0.232\pm 0.012$ | $9.80\pm 1.61$ | $2.42\pm 0.53$ | $2.02\pm 0.31$ | $5.30\pm 0.01$ | $1.25\pm 0.01$ | $4.05\pm 0.01$ | 0.9/447
57812.92441 | 57813.49733 | $0.24\pm 0.03$ | $0.035\pm 0.003$ | $0.278\pm 0.045$ | $10.81\pm 2.02$ | $2.23\pm 0.32$ | $1.35\pm 0.22$ | $5.16\pm 0.02$ | $1.71\pm 0.01$ | $3.44\pm 0.03$ | 1.1/446
57839.91052 | 57840.63969 | $0.06\pm 0.01$ | $0.051\pm 0.013$ | $0.346\pm 0.031$ | $9.90\pm 1.68$ | $1.72\pm 0.13$ | $0.42\pm 0.03$ | $1.41\pm 0.01$ | $1.41\pm 0.01$ | $-$ | 1.0/311

###  3.3 X-ray variability

To quantify the X-ray flux variation, we used the excess variance or the
fractional variability amplitude $F_{\rm{var}}$ (Edelson et al., 1996; Nandra
et al., 1997; Vaughan et al., 2003). $F_{\rm{var}}$ is the variance after
subtracting the contribution expected from measurement errors and is defined
as

$\centering{}F_{\rm{var}}=\sqrt{{S^{2}-{\bar{\sigma^{2}}_{\rm{err}}}\over\bar{x}^{2}}}\@add@centering$
(1)

where ${\bar{x}^{2}}=\sum_{i=1}^{N}{x_{i}/N}$ is the arithmetic mean of
$x_{i}$. ${S^{2}}$ and $\overline{\sigma_{\rm err}^{2}}$ are the sample
variance of the light curve and mean square error associated with the measured
fluxes $x_{i}$ respectively.

$\centering
S^{2}=\frac{1}{N-1}\sum_{i=1}^{N}(x_{i}-\bar{x})^{2},\@add@centering$ (2)
$\overline{\sigma_{\rm
err}^{2}}=\frac{1}{N}\sum_{i=1}^{N}\sigma^{2}_{\rm{err,i}}$ (3)

Using Eq. (1), we calculated the fractional variability amplitude for each
segment of observations and the measurement uncertainties of $F_{\rm{var}}$
were estimated following Vaughan et al. (2003)

$\centering{}\text{err}(F_{\rm{var}})=\sqrt{\Bigg{(}\sqrt{1\over{2N}}{\overline{\sigma_{\rm
err}^{2}}\over{{\bar{x}}^{2}}{F_{\rm{var}}}}\Bigg{)}^{2}+\Bigg{(}\sqrt{\overline{\sigma_{\rm
err}^{2}}\over{N}}{1\over\bar{x}}\Bigg{)}^{2}}\@add@centering$ (4)

The results of the variability analysis are listed in Table 7. We found the
amplitude of the flux variations to increase from epoch A to epoch D and the
highest flux variations were seen in epoch D.

Table 7:  Flux variability characteristics for each epoch of Mrk 421. OBS ID | Epoch | $F_{\rm{var}}\pm\text{err}(F_{\rm{var}})$
---|---|---
|  |  3$-$10 keV |  10$-$20 keV |  3$-$20 keV
60202048002 | A |  0.079$\pm$0.001 |  0.085$\pm$0.002 |  0.080$\pm$0.001
60202048004 | B |  0.139$\pm$0.001 |  0.203$\pm$0.002 |  0.156$\pm$0.001
60202048006 | C |  0.163$\pm$0.001 |  0.187$\pm$0.001 |  0.168$\pm$0.001
60202048008 | D |  0.272$\pm$0.001 |  0.353$\pm$0.002 |  0.276$\pm$0.021

In the observed period presented here, the hardness ratio (HR) was calculated
for each segment from the pl model fitted spectra for the energy range 3$-$20
keV. In our case $HR=\frac{F_{10-20keV}}{F_{3-10keV}}$, where, $F_{10-20keV}$
and $F_{3-10keV}$ are the flux in hard and soft energy bands. In Figure 6, we
show the HR variation with total observed flux. In $\sim$83 days, HR changed
by a factor of more than 2, and during the fourth epoch (2017 March 27), the
source had the lowest hard flux. We performed the weighted least square
fitting to the observed data points in the HR v/s flux diagram. We found a
strong positive correlation between the HR and total flux. We obtained a
correlation coefficient of 0.874 which also indicates that our spectrum become
harder as the flux increases, confirming the results which are interpreted
from photon index vs. flux diagram (Figure 5). It can also be seen from Table
3 that there is a noticeable transition in flux in all energy bands. Between
epoch C and epoch D the brightness in the soft band reduced by a factor of 7
in magnitude, whereas in the hard band the brightness dropped by a factor of
12. This large jump in flux between epoch C and D also implies that the corona
which reprocesses soft photons from the Keplerian disc and scatters them as
hard radiation has significantly cooled down, therefore can not produce enough
hard radiation, and the source moved to lower flux state. This abrupt change
in the flux is a signature of (a) sudden change in flow configuration, which
means viscosity has changed suddenly (Mondal et al., 2017) or (b) appearance
of some absorbers. The significant change in CENBOL geometry consistent with
the flux variation can also be seen from the tcaf model fitted parameters,
where the size of the CENBOL has changed by a factor of $\sim 2$. Also, this
is consistent with the estimated decrease in the temperature of the corona
from epochs A to D. From Table 7, it is evident that hard X-rays are more
variable than the soft X-rays. This might be indicative of more complex
dependence of variability on energy and other physical mechanisms. The
difference between soft and hard X-ray variability could be, for example,
related to the size of the emitting region, or the corona and thus the cooling
effect. There are arguments in the literature on the similar observational
flux variability characteristics noticed here. If the high-energy electron
cloud is located at the inner edge of the accretion disc, which is the
Comptonizing corona in our case or above its inner part (Zdziarski et al.,
1999), and the low-energy emitting region is associated to some outflow from
the disc or a second cloud with lower temperature (Petrucci et al., 2013),
then the hard flux could be more variable than the soft flux. This could
explain both the harder when brighter trend in the spectra and the hotter when
brighter trend in the corona of Mrk 421.

Figure 6: Variation of the soft (3$-$10 keV), hard (10$-$20 keV) and total
(3$-$20 keV) fluxes and hardness ratio (HR).

###  3.4 Contribution of the jet to spectral flux

In this section, we compare the spectral fitting with the accretion disc model
to estimate the contribution from both disc and jet. To investigate the
relative importance of jet emission over the accretion disc emission in Mrk
421 as well as to compare the parameters obtained from spectral fits to the
data acquired during 2017, we identified one NuSTAR data (OBSID = 60002023031,
hereinafter Epoch E) observed on 14 April 2013. The source was in a very
active state in $\gamma$-rays, optical and X-rays during epoch E, and the
X-ray emission during this epoch is expected to be dominated by the emission
from the relativistic jet. The R-band, the Swift/XRT soft and hard bands, and
UV fluxes are $11.90\pm 0.03$ (mag), $77.41\pm 0.26$, $50.62\pm 0.18$, and
$30.97\pm 0.74$ in ph s-1 unit respectively. The hardness ratio (hard/soft
flux) was 0.65 during this epoch. We fitted the spectra obtained during Epoch
E, with pl, thcomp and tcaf. The pl was not found to be useful, and indeed,
returned a poor fit with $\chi_{r}^{2}$ of 3.7 during epoch E, while tcaf fits
well ($\chi_{r}^{2}$ of 1.18) the data with $\dot{m}_{d}=0.43$,
$\dot{m}_{h}=0.304$, $X_{s}=27.47$, and $R=2.82$. The normalization from tcaf
fit was found to increase by a factor of about 60 times, compared to the
normalization obtained during epoch D, when the source was in the lowest flux
state in X-rays among all the epochs analysed here. The total X-ray flux was
also high with the value $27.57\times 10^{-10}$ ergs cm-2 s-1. In Figure 7 we
show the tcaf model fitted spectrum when the normalization is free (in panel
a) and for an average normalization (in panel b) obtained from the fit of
other epochs. The panel (b) does not fit the data very well when the
normalization for the fit is frozen to the average value obtained from the
relatively low flux data (epochs A-D) fitting. Moreover, it shows that the
spectrum is softening gradually, this can be due to the cooling effect in jet
and can be well explained by the logpar model, often used for this source.
Therefore, our comparative fitting of the less and very active jet epochs
shows that the disc could be present, however, might be weak and subdominant.
We also point out that multi-band data from UV/optical, X-ray, and $\gamma$
gathered during the same epochs analysed in this work was equally fit with
synchrotron and SCC models (MAGIC Collaboration et al., 2021). It is to be
noted that for moderately emitting X-ray jets, tcaf is enough to fit, since
tcaf uses cylindrical geometry for interception flux calculations. For
computation of average optical depth, tcaf code does not use a slab geometry
given in Sunyaev & Titarchuk (1980) rather it uses a spherical geometry. In
both the cases, the centrifugal force supported torus and matter inside the
funnel of the torus which constitutes the pre-jet matter till the sonic
surface) were taken care of while computing the intercepted photons. For
consistency, we checked that the thcomp also fits the data well with $kT_{\rm
e}=24\pm 6$ keV and $\Gamma=2.41\pm 0.05$.

As the normalization is a scale factor which depends on the source distance
and its inclination, therefore it is constant between epochs for a particular
source. If its value changes drastically, this could point to an additional
emission process contributing to the observed X-ray, for example, emission
from the jet. Therefore, the normalization can also be a probe to estimate the
flux contributions from both the disc and jet. This has been successfully
applied to black hole binary systems (Jana et al., 2017, and others). As Mrk
421 is a jetted candidate, we applied that finding for this candidate as a
case study. Considering that in the lowest normalization epoch, the flux
contribution is mainly from the disc, and increasing norm implies an excess
contribution from the jet in addition to the disc contribution. Therefore, the
total flux $F_{\rm total}$ can be written as the sum of disc and jet, as
below:

$F_{\rm total}=F_{\rm disk}+F_{\rm jet}.$ (5)

Here, $F_{\rm disk}$ can be obtained by using the lowest norm in epoch D to
all other epochs and the jet contribution can be estimated by subtracting the
lowest flux contribution from the disc from the total flux in Table 6 (in
Column 8 when the norm is free). Therefore, for this epoch, Fjet is zero. This
is also reflected in the third panel of Figure 8.

For all the epochs, we show in Figure 8 the relation ($ax^{b}$) between the
radio flux (a proxy for jet emission) against the X-ray emission coming from
different components of the accretion-jet system. The three panels in Figure 8
show the variation of the radio flux with the total flux, disc flux from tcaf
fits and the jet flux obtained from Equation 5. In this figure, the red
circles are the observed/estimated flux values, while the solid black curve is
the fitted function. We find that the radio flux is anti-correlated with the
X-ray flux from the accretion disc, while it is correlated with the jet X-ray
emission. The normalization in tcaf fits increases from Epochs D $\rightarrow$
C $\rightarrow$ A $\rightarrow$ B $\rightarrow$ E. This indicates that the
relative contribution of the jet emission over the disc emission increases in
the sequence D$\rightarrow$C$\rightarrow$A$\rightarrow$B$\rightarrow$E, with
the X-ray jet emission being maximum during epoch E and minimum during epoch
D.

Figure 7: The 3$-$60 keV spectra fitted with tbabs*zxipcf*tcaf model along
with the residuals for epoch E. The panels (a) shows the fit when tcaf model
norm is free and (b) corresponds to data fitted when norm is frozen to an
average low flux state norm around 1.3e4. Figure 8: Radio flux variation with
different components of the flux obtained from tcaf model fit. The red solid
circles are the observed data points and the black solid line is the
functional dependence between fluxes.

###  3.5 Joint fit to the Swift/XRT and NuSTAR data

For estimating various accretion parameters as discussed in the previous
section, we used only data from NuSTAR covering the energy range from 3-60
keV. It is likely, that inclusion of X-ray data with energies lower than 3 keV
could have an impact on the derived accretion parameters. To test this, we did
a joint spectral analysis of Swift/XRT and NuSTAR data with thcomp and tcaf
models.

For Swift/XRT data, the spectrum files were generated using the online product
generator (Evans et al., 2009)888https://www.swift.ac.uk/user_objects/. The
data were binned to 20 counts rate per energy bin. For the joint spectral
analysis, we used data in the energy range from 0.4 to 60 keV. For epochs A to
D, Swift/XRT data covering the energy range of 0.4 to 8 keV was used except
for epoch C, where the data from energy range 0.4 to 5 keV was used, owing to
poor S/N beyond 5 keV. The results of the thcomp model fit to the joint
Swift/XRT and NuSTAR data are given in Table 8. A comparison of Table 8 with
that of Table 5 indicates the parameters obtained from combined Swift/XRT \+
NuSTAR data and NuSTAR data alone agree within errors. Similarly, we show in
Figure 9, the tcaf model fits to the combined spectra, and the derived
parameters are given in Table 9. In tcaf model fits too, the parameters
obtained from the joint Swift/XRT and NuSTAR data agree within errors to the
parameters obtained using only NuSTAR data. In model fits to the combined data
set, an extra Gaussian component was required to fit the excess flux in the
0.5 - 0.7 keV range. We note that the disc parameters (that are needed for the
estimation of viscosity parameter) obtained from joint fitting are in
agreement to that obtained using NuSTAR data alone within errors.

Figure 9: The 0.4$-$60 keV combined Swift/XRT and NuSTAR spectra fitted with tbabs*zxipcf*(tcaf+gauss) model along with the residuals. The panels (a)-(d) correspond to the data acquired at epochs A, B, C and D respectively. Table 8:  Results of tbabs*zxipcf*thcomp*(diskbb+gauss) model fits to the joint Swift/XRT and NuSTAR spectra. The gauss component is required at $\sim 0.5-0.7$ keV energy range to fit the excess around that energy. The other parameters are the same as in Table 5. Epoch | MJDstart | MJDend | $C_{\rm f}$ | $kTe$ | $\Gamma$ | $T_{\rm in}$ |  Ndbb | $\chi_{r}^{2}/dof$
---|---|---|---|---|---|---|---|---
|  |  |  | [keV] |  | [keV] | $\times 10^{6}$ |
A | 57756.99385 | 57757.52163 | $0.57\pm 0.11$ | $138\pm 12$ | $2.40\pm 0.01$ | $0.07\pm 0.01$ | $4.07\pm 0.39$ | 1.0/750
B | 57784.99038 | 57785.55288 | $0.63\pm 0.03$ | $35\pm 4$ | $2.43\pm 0.02$ | $0.07\pm 0.02$ | $4.62\pm 0.25$ | 1.0/676
C | 57812.92441 | 57813.49733 | $0.65\pm 0.08$ | $38\pm 5$ | $2.48\pm 0.03$ | $0.07\pm 0.02$ | $3.91\pm 0.40$ | 1.1/489
D | 57839.91052 | 57840.63969 | $0.12\pm 0.01$ | $21\pm 3$ | $2.54\pm 0.04$ | $0.07\pm 0.01$ | $1.32\pm 0.11$ | 1.1/553
Table 9:  Results of tbabs*zxipcf*(tcaf+gauss) model fits to the joint Swift/XRT and NuSTAR spectra. The gauss component is required at $\sim 0.5-0.7$ keV energy range to fit the excess around that energy. The other parameters are the same as in Table 6. MJDstart | MJDend | $C_{\rm f}$ | $\dot{m}_{\rm d}$ | $\dot{m}_{\rm h}$ | $X_{\rm s}$ | R | $N_{t}\times 10^{4}$ | $F_{\rm tot}$ | $F_{\rm disk}$ | $F_{\rm jet}$ | $\chi_{r}^{2}/dof$
---|---|---|---|---|---|---|---|---|---|---|---
57756.99385 | 57757.52163 | $0.48\pm 0.03$ | $0.022\pm 0.003$ | $0.222\pm 0.010$ | $20.09\pm 2.48$ | $2.41\pm 0.36$ | $0.99\pm 0.13$ | $14.23\pm 0.04$ | $5.42\pm 0.02$ | $8.81\pm 0.05$ | 1.49/749
57784.99038 | 57785.55288 | $0.77\pm 0.13$ | $0.027\pm 0.001$ | $0.243\pm 0.011$ | $9.91\pm 1.23$ | $2.38\pm 0.34$ | $0.93\pm 0.15$ | $14.23\pm 0.04$ | $7.05\pm 0.05$ | $7.19\pm 0.06$ | 1.0/675
57812.92441 | 57813.49733 | $0.78\pm 0.16$ | $0.034\pm 0.004$ | $0.292\pm 0.011$ | $10.98\pm 2.59$ | $2.87\pm 0.21$ | $1.05\pm 0.24$ | $16.72\pm 0.42$ | $5.00\pm 0.01$ | $11.73\pm 0.30$ | 1.15/488
57839.91052 | 57840.63969 | $0.46\pm 0.02$ | $0.051\pm 0.008$ | $0.348\pm 0.021$ | $9.93\pm 1.63$ | $1.73\pm 0.19$ | $0.26\pm 0.04$ | $5.95\pm 0.08$ | $5.95\pm 0.08$ | – | 1.0/552

###  3.6 Estimating accretion disc viscosity parameter

Flux variability of different time scales (hours to years) in AGN can be
explained by several theoretical models (a) relativistic jet based models
(Marscher & Gear, 1985; Gopal-Krishna & Wiita, 1992; Calafut & Wiita, 2015)
where the variability time scale is large and (b) the accretion disc based
models where the variability can be in small scales. In case of the later
models, it is believed that the observed X-ray variability in both black hole
X-ray binaries and Seyfert galaxies mainly comes from two key components,
accretion disc and its dynamic corona where the X-rays are predominantly
produced close to the BH by IC scattering of the optical/UV seed photons from
the accretion disc. On the other hand, the geometrically thin, cold, and
optically thick accretion disc reprocesses the variable X-ray emission coming
from the innermost optically thin, hot corona in a complex way, therefore
playing mostly a passive role in the variability at the shortest timescales.
Furthermore, it has been interpreted by studying different timescales in
accretion disc of blazars that the observed variations are solely due to jet
or by the variations in the disc carried out, and amplified by jets (Wiita,
2006). Hence, the X-ray variability can be used to probe the properties of
accretion on to the BH. The physical scenarios that describe the large scale
variability in different accretion disc based models are the formation and
fragmentation of spiral shocks in accretion disc (Wiita et al., 1992;
Chakrabarti & Wiita, 1993), the asymmetries and geometric effects in accretion
disc (Mangalam & Wiita, 1993), or by the fluctuations that propagate from the
outer radii of the accretion disc towards the center and couple with the
emission from the inner region of the disc (Lyubarskii, 1997; Arévalo &
Uttley, 2006). It has been shown for black hole X-ray binaries that the
viscosity and inverse Compton cooling can be responsible for small scale
variability and different flux states (Mondal et al., 2017) during the
outburst phase of the BHs. The timescales of these variations can be scaled up
for AGN through the BH mass (McHardy et al., 2006).

Studying the variability using any physical accretion disc model is important
to understand the underlying dynamics behind these. X-ray flux variations in
blazars in the low flux state can possibly be explained by disc accretion.
Very recently, Chatterjee et al. (2018) discussed on the accretion disc origin
of variability of Mrk 421 using AstroSat data. Here, we use the results of the
spectral analysis of the X-ray observations of Mrk 421 by NuSTAR during 2017
using tcaf model to understand the flux variations possibly being caused by
accretion disc through the accretion rate variation, which also represents the
viscosity ($\alpha$) fluctuation.

It is believed that $\alpha$ parameterization provides the basis for
developing the accretion disc theory and related observations. There are
several methods to estimate $\alpha$ parameter. For instance, in the case of
observations, if we assume that the optical variability of the AGN is caused
by disc instabilities, then comparing the thermal timescales of $\alpha$-disc
(Shakura & Sunyaev, 1973, hereafter SS73) models with the observed variability
timescales, one can put constraints on the viscosity parameter for different
candidates (Siemiginowska & Czerny, 1989; Starling et al., 2004). In the
numerical simulation, Balbus & Hawley (1991) showed that $\alpha$-parameter
can be obtained from the outward transport of angular momentum in weakly
magnetized disc by magnetohydrodynamic turbulence. Pessah et al. (2007)
inferred observationally that in order to make MRI-driven turbulence, the
angular momentum transport is required for large values of the effective
$\alpha$ viscosity ($\alpha\geq 0.1$), the disc must be threaded by a
significant vertical magnetic field and the turbulent magnetic energy must be
in near equipartition with the thermal energy. On the other hand, in an
advection-dominated disc, Narayan & McClintock (2008) argued that the required
viscosity is 0.1$-$0.3. All these estimates seem to show a very broad range of
the $\alpha$ parameter, 0.001$-$0.6 (see also Hawley et al., 1995). However, a
narrow range of $\alpha$ parameter for blazars has been estimated by Xie et
al. (2009). In this work, we made an attempt to estimate the viscosity
parameter for Mrk 421.

To estimate the viscosity parameter we used the disc accretion rate returned
by tcaf model fits to the spectra. We assumed that the disc radiates locally
as a black body with an effective temperature,

$T=\left(\frac{3GM_{\rm BH}\dot{M}}{8\pi\sigma}\right)^{1/4}f^{1/4}r^{-3/4},$

where $f=[1-(r_{\rm in}/r)^{1/2}]\sim 1$, $r$ is the radial distance of the
disc. We considered the outer boundary of the disc as 500 $r_{s}$. The
kinematic viscosity $(\nu)$ can be written as

$\nu=\frac{\dot{M}}{3\pi\Sigma},$

where, $\Sigma$ is the surface density of the disc and can be obtained from
the density ($\rho$ in gm cm-3) and scale height ($H$ in cm) of the disc.
According to SS73 prescription, $\nu$ can be written as $\alpha c_{s}H$, where
$c_{s}$ is the isothermal sound speed, corresponding to the effective disc
temperature. After combining these equations and performing a few steps of
algebra, we obtained the $\alpha$-parameter value. Our estimated values of
$\alpha$ ranges between 0.18$-$0.25 in the Keplerian component of the two
component disc. We also estimated different time scales of the flow, for
example the dynamical ($t_{\rm dyn}=\sqrt{GM_{\rm BH}/r^{3}}$) and thermal
time scales ($t_{\rm th}=\alpha^{-1}t_{\rm dyn}$) are $\sim$ 2 years and
8$-$10 years respectively, whereas the viscous timescale ($t_{\rm vis}=t_{\rm
th}(H/r)^{2}$) is $\sim$ 30$-$36 Myr.

##  4 Conclusions

In this paper, we studied the flux and spectral variability of Mrk 421 using
NuSTAR data obtained in 2017 and in 2013. During the 2017 epoch, we found the
pl photon index to change from 2.2 to 3.0. We split each epoch of observation
into several segments to study the flux and spectral variability on short time
scales. We noticed that while the hardness ratio (Table 3) is almost doubled,
the average count rate of the source varied by a factor of 4 in $\sim 83$
days. We found that each epoch of observation in the energy band 3$-$60 keV
could be well fitted with phenomenological thcomp and physical tcaf models.
Each segment of data could be well fitted with the simple pl model. From the
obtained $F_{\rm var}$ (in Table 7), we conclude that the source was
significantly variable in all the epochs. We also found a strong correlation
between the pl index and the brightness of the source in the total energy band
with a “harder when brighter” behaviour.

Mrk 421 is a HSP blazar which generally lacks emission lines in the optical
spectra. Under such circumstances, the origin of the observed X-ray flux
variations lie in their jets (Abdo et al., 2011). Alternatively, during the
less active state, one may expect the contribution of accretion disc to the
observed emission, however, unlikely to dominate over jet contribution.
Recently, Chatterjee et al. (2018) noticed a break in the power spectral
density from X-ray observations of Mrk 421 at around the same time as that of
the NuSTAR data which is analysed here. The observations from AstroSat used by
Chatterjee et al. (2018) were also taken during the moderate brightness state
of Mrk 421. Such breaks, that are normally seen in the X-ray power spectral
density of Seyfert galaxies, are believed to be due to the X-ray being
produced in the accretion disc. We therefore, hypothesise that in the low or
moderate X-ray activity of Mrk 421, disc emission could manifest itself in the
observed X-rays and thus fit each epoch of data using the disc-based tcaf
model to investigate if the observed flux variations can be explained with
changes in the accretion rate or changes in the geometry of the accretion
disc. The derived parameters from the model fits show that the disc mass
accretion rate has varied by a factor of $\sim$3 from $0.02$ to $0.051$
$\dot{M}_{\rm Edd}$ and the size of the CENBOL as envisaged in tcaf model also
changed significantly. This size varied from 20 to 10 $r_{s}$ between epochs A
and D. In epoch D, the inner edge of the disc moved significantly inward with
the lowest shock compression ratio and the shrinking of the CENBOL was
maximum, and the accretion rate was the highest. This implies a transition of
the source to a lower flux state with a significant flux change. Also, the
temperature of the CENBOL was found to change from epoch A to D, using both
physical tcaf and phenomenological thcomp model fits. The CENBOL was found to
be hotter with increasing brightness of the source. Therefore, the shrinking
of the CENBOL, increase of the mass accretion rates, and the decrease in the
flux values follow the correlation with $\Gamma_{\rm PL}$ variation. Even
though both the model temperatures are showing the same profile, the static
corona (used in thcomp) cannot behave in the same way as the dynamic
corona/CENBOL (used in tcaf) does because the static corona would cool quickly
if there is no underlying heating mechanism. Therefore, the temperature
estimated from tcaf model is more realistic.

We note that the accretion disc-based model like tcaf can fit the spectra
without adding any other component. This indicates that even though Mrk 421 is
a HSP blazar, in the low/moderate activity state during the period of our
observation, the disc could contribute significantly to the total observed
X-ray emission, however, not dominating over the jet contribution. The
difference in the flux variations between the hard and soft bands and
subsequently the hardness ratio could also be due to changes in the size of
the emission region or the CENBOL. Thus, the hard emission might have its
origin from the changing corona region at the inner edge of the disc. On the
other hand, the presence of jet and its contribution to the total spectra
would imply that the gravitational potential energy of the infalling matter
not only gets transformed into radiation, but can also amplify magnetic field,
that allows the field to retrieve large store of rotational energy and
transform a part of it to jet power.

Thus, based on the modelling of the five epochs of NuSTAR data of Mrk 421 for
the energy band 3-60 keV from the relatively less and very high jet activity,
we conclude that (i) simple pl alone is not a good representation of the
observed X-ray rather cutoffpl fits the data well, (ii) both tcaf and thcomp
fit the observed X-ray emission well to give the accretion and spectral
properties though tcaf gives the physical parameters of the flow, (iii) non-
magnetic accretion disc models are found to be adequate to fit the
low/moderate X-ray activity state data of Mrk 421. However, the
X-ray/$\gamma$-ray correlation is one of the important characteristics in the
broadband emission of Mrk 421, that occurs during high and low activity, as
reported multiple times in the literature over the last years, and
particularly for the data obtained in the year 2017, as reported in MAGIC
Collaboration et al. (2021). The existence of this correlation implies a
substantial emission from the jet, probably related to SSC models, that occurs
also during the very low activity. And hence that, even during the low blazar
activity, the present disc-base models can not dominate the X-ray emission of
Mrk 421.

###### Acknowledgements.

We thank the referee for the critical comments on our manuscript. SM
acknowledges Andrzej Zdziarski for helpful discussions and helping in model
fitting. SM thanks Keith A. Arnaud for helping in model implementation in
XSPEC package. SM acknowledges funding from Ramanujan Fellowship grant (#
RJF/2020/000113) by SERB-DST, Govt. of India. This research has made use of
the NuSTAR Data Analysis Software (nustardas) jointly developed by the ASI
Science Data Center (ASDC), Italy and the California Institute of Technology
(Caltech), USA. This research has also made use of data obtained through the
High Energy Astrophysics Science Archive Research Center Online Service,
provided by NASA/Goddard Space Flight Center.

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