# Spectral and Temporal Studies of Swift J1658.2–4242 using AstroSat
Observations with JeTCAF Model

Santanu Mondal1 and V. Jithesh2
1Indian Institute of Astrophysics, II Block Koramangala, Bengaluru 560034,
India
2Department of Physics and Electronics, CHRIST (Deemed to be University),
Hosur Main Road, Bengaluru - 560029, India
E-mail<EMAIL_ADDRESS>(SM)E-mail<EMAIL_ADDRESS>(VJ)Equal First Author

(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present the X-ray spectral and temporal analysis of the black hole X-ray
transient Swift J1658.2–4242 observed by AstroSat. Three epochs of data have
been analysed using the JeTCAF model to estimate the mass accretion rates and
to understand the geometry of the flow. The best-fit disc mass accretion rate
($\dot{m}_{d}$) varies between $0.90^{+0.02}_{-0.01}$ to
$1.09^{+0.04}_{-0.03}$ $\dot{M}_{\rm Edd}$ in these observations, while the
halo mass accretion rate changes from $0.15^{+0.01}_{-0.01}$ to
$0.25^{+0.02}_{-0.01}$ $\dot{M}_{\rm Edd}$. We estimate the size of the
dynamic corona, that varies substantially from $64.9^{+3.9}_{-3.1}$ to
$34.5^{+2.0}_{-1.5}$ $r_{g}$ and a moderately high jet/outflow collimation
factor stipulates isotropic outflow. The inferred high disc mass accretion
rate and bigger corona size indicate that the source might be in the
intermediate to soft spectral state of black hole X-ray binaries. The mass of
the black hole estimated from different model combinations is $\sim
14M_{\odot}$. In addition, we compute the quasi-periodic oscillation (QPO)
frequencies from the model-fitted parameters, which match the observed QPOs.
We further calculate the binary parameters of the system from the decay
profile of the light curve and the spectral parameters. The estimated orbital
period of the system is $4.0\pm 0.4$ hr by assuming the companion as a mid or
late K-type star. Our analysis using the JeTCAF model sheds light on the
physical origin of the spectro-temporal behaviour of the source, and the
observed properties are mainly due to the change in both the mass accretion
rates and absorbing column density.

###### keywords:

accretion, accretion discs $-$ black hole physics $-$ X-rays: binaries $-$
ISM: jets and outflows $-$ shock waves $-$ X-rays: individual: Swift
J1658.2$-$4242

††pubyear: 2023††pagerange: Spectral and Temporal Studies of Swift
J1658.2–4242 using AstroSat Observations with JeTCAF Model–A

##  1 Introduction

Accretion onto black hole (BH) in the X-ray binaries produces electromagnetic
radiation at different energy bands. In the X-ray band, almost all black hole
X-ray binaries (BHXRBs) show spectral variability with different spectral
states from hard (HS) to soft (SS) through intermediate states (Homan et al.,
2001; Belloni et al., 2005; Remillard & McClintock, 2006; Nandi et al., 2012;
Debnath et al., 2015; Shui et al., 2021; Baby et al., 2021, and references
therein). These spectral states change can occur due to several reasons, e.g.,
change in mass accretion rate or corona (Chakrabarti & Titarchuk, 1995;
Narayan et al., 1996; Laurent & Titarchuk, 2001; Remillard & McClintock, 2006;
Mondal et al., 2014; Chatterjee et al., 2016), accretion disc instabilities,
and/or hysteresis behaviour of the cooling and viscosity (Janiuk et al., 2002;
Meyer-Hofmeister et al., 2005; Mondal et al., 2017) during the outburst phase.
In the long term outbursts or between outbursts, flaring and re-flaring events
are observed, which can also be explained from the change in viscosity of
accreting matter (Chakrabarti et al., 2019).

In the temporal domain too, the sources show variability mainly in the form of
different types of quasi-periodic oscillations (QPOs), whose frequency ranges
from mHz (low frequency; LF) to kHz (high frequency; HF). The LFQPOs can be
divided into three types (type A, B, and C) depending on their root mean
square (rms) and Q values (Casella et al., 2005). Type C QPOs are mainly
identified in the hard-intermediate state (HIMS), while type A and B QPOs are
detected in the soft-intermediate state (SIMS; Wijnands et al., 1999; Casella
et al., 2005; Motta et al., 2011). Several models have been proposed to
explain the origin and evolution of LFQPOs in X-ray binaries. The proposed
models are generally based on three different mechanisms: oscillation of the
shock or dynamic corona (Molteni et al., 1996; Chakrabarti et al., 2015),
Lense-Thirring precession (Stella & Vietri, 1998; Stella et al., 1999; Ingram
et al., 2009), and instabilities (e.g. Tagger & Pellat, 1999; Titarchuk &
Fiorito, 2004; Cabanac et al., 2010). LFQPOs are often detected in transient
BHXRBs and their origin provides essential information regarding the accretion
properties and accretion flow geometry around the BH. The appearance and
disappearance of these QPOs are also interlinked with the spectral states,
indicating that both spectral and temporal behaviours originated from the same
system and geometrical flow configuration. Therefore, it is worth exploring
the models which can explain both spectral and timing signatures
simultaneously.

AstroSat (Singh et al., 2014; Agrawal et al., 2017) is India’s first multi-
wavelength astronomical observatory, which contains five instruments onboard:
Soft X-ray Telescope (SXT), Large Area X$-$ray Proportional Counter (LAXPC),
Cadmium Zinc Telluride Imager (CZTI), Scanning Sky Monitor (SSM) and an Ultra-
Violet Imaging Telescope (UVIT). Among them, LAXPC is ideal for the rapid time
variability studies of X-ray binaries with moderate spectral capabilities. The
high sensitivity and medium resolution spectral capability of SXT in the 0.3–8
keV energy band are useful for broadband spectral studies simultaneously with
LAXPC. These capabilities provided a unique opportunity to investigate the
spectro-timing properties of several X-ray binaries (including new
transients), particularly BHXRBs using AstroSat. The analysis of the first
AstroSat data sets of Cygnus X–1 identified the source in the canonical hard
state characterised by a prominent thermal Comptonisation component having a
photon index of $\sim 1.8$ and the coronal temperature ($\rm kT_{e}$) of $>60$
keV along with signatures of weak reflection. The temporal analysis revealed
two broad peaks in the power density spectrum. For both components, the
fractional rms exhibited a decreasing trend with energy, while the time lag
showed an increasing trend with energy (Misra et al., 2017). A more detailed
study with six AstroSat observations, Maqbool et al. (2019) estimated the
energy-dependent fractional rms and time lag at different frequencies. In
addition, by invoking a fluctuation propagation model with simple truncated
disk geometry, they explained the mechanism responsible for the observed
timing properties of the source. AstroSat observations of MAXI J1535$-$571
identified the source in the HIMS of BHXRBs. Using the broadband spectrum with
reflection models, Sridhar et al. (2019) estimated the spin, BH mass and
distance to the source, and the values are 0.67, 10.4 $M_{\odot}$ and 5.4 kpc,
respectively. From the timing analysis, Bhargava et al. (2019) reported the
presence of strong QPOs, which vary in the range of 1.7–3.0 Hz. A tight
correlation between the QPO frequency and the power-law spectral index has
been observed, while the QPO frequency is not correlated with the flux. From
the strong correlation, the authors argue that the QPOs are closely related to
the Comptonising region rather than the accretion disk.

The independent observations of MAXI J1820+070 with AstroSat and
contemporaneous NuSTAR revealed an inhomogeneous corona characterised by high
coronal temperature (Chakraborty et al., 2020). Moreover, the broadband
coverage using the three simultaneously observing X-ray instruments (SXT,
LAXPC and CZTI) onboard AstroSat helps to estimate the BH mass of the source,
which is in the range of 6.7–13.9 $M_{\odot}$. AstroSat observed GRS 1915+105
multiple times and identified the source in different variability classes
(Yadav et al., 2016b; Banerjee et al., 2021; Majumder et al., 2022). In
addition, Sreehari et al. (2020) studied the broadband spectrum of the source
to estimate the mass and the spin of the system, and they concluded that GRS
1915+105 hosts a maximally rotating, comparatively higher mass BH and the
X-ray binary source accreting at Super-Eddington rate. Spectro-timing
properties of a new black hole transient MAXI J1348-630 have been investigated
using the simultaneous/quasi-simultaneous AstroSat and NICER observations. The
analysis revealed several interesting characteristics of the source, including
the flip-flop behaviour, different types of QPOs and energy-dependent
behaviour of variability components (Jithesh et al., 2021). The detailed
broadband spectral study using the reflection models constrained the spin and
inclination of the system and the values are $>0.97$ and $\sim 33^{\circ}$,
respectively (Mall et al., 2023). The studies mentioned above showcase the
spectro-timing capabilities of AstroSat in broadband energy.

Swift J1658.2$-$4242 is a low-mass black hole X$-$ray transient in the
Galactic plane and its detection was reported by Swift/BAT on 2018 February 16
(Barthelmy et al., 2018). INTEGRAL also detected the source during the
observations of the Galactic center field on 2018 February 13 (Grebenev et
al., 2018). The spectral hardness was typical for BHXRBs in the hard state
during that period (Grebenev et al., 2018; Grinberg et al., 2018). The radio
observations from ATCA imply that the source is a black hole X-ray binary
located at a distance $>3$ kpc (Russell et al., 2018). LFQPO frequencies were
detected by several satellites (NuSTAR, Swift, XMM-Newton; Xu et al. (2018,
2019), AstroSat; Beri & Altamirano (2018); Jithesh et al. (2019); Xiao et al.
(2019), Insight-HXMT and NICER; Xiao et al. (2019)). The X-ray spectra showed
high absorption column density (Lien et al., 2018; Jithesh et al., 2019, and
references therein). Furthermore, several dips were observed in the NuSTAR
light curve (Xu et al., 2018), which can be a signature of high inclination
($i>65^{\circ}$). The joint observations from NuSTAR, Swift and XMM-Newton
identified the source in the intermediate state of BHXRBs and observed a rapid
flux variability in 40 s accompanied by a transient QPO with a frequency of
6–7 Hz (Xu et al., 2019). The authors proposed that the accretion disc
instabilities triggered at a large disk radius may be the reason for the
unusual spectro-timing properties. A detailed study using the Insight-HXMT,
NICER and AstroSat observations (Xiao et al., 2019) identified different
timing and spectral states of the source through the hardness-intensity
diagram and hardness-rms diagram. In addition, the simultaneous observations
by three satellites identified the source in HIMS and detected a QPO at $\sim
1.5$ Hz frequency. The detected QPO characteristics are remarkably similar in
the three energy bands considered. Extreme varieties of ‘flip-flop’ behaviours
have been observed in the source from the XMM-Newton, NuSTAR, AstroSat, Swift,
Insight-HXMT and INTEGRAL observations (Bogensberger et al., 2020). During the
flip-flop events, the flux was varied by $\sim 77\%$ and the flip-flop
transitions occurred at random integer multiples of different fundamental
periods in the two intervals of the outburst. In addition, the source
exhibited changes in the power density spectrum simultaneous with the sharp
changes in the flux and showed a transition between different types of QPOs
(particularly type C and A) during the flip-flop. The spectral modelling of
the flip-flop states revealed that the change in the inner accretion disc
temperature leads to the major flux variation during the flip-flop events.

In this work, we revisited the broadband X-ray spectral and temporal
properties of Swift J1658.2–4242 from AstroSat observations to understand the
accretion flow behaviour of the system using accretion-ejection based physical
models. For this, we analyzed the simultaneous SXT and LAXPC data of Swift
J1658.2$-$4242 and modelled the spectra with the physical model JeTCAF. In the
next section, we describe the observations and the data reduction procedures
used in this work. The spectral and temporal analysis results are presented in
Section §3. In Section §3.4, we estimate the binary parameters from the decay
of the light curve and the spectral modelling. Finally, we discuss the results
and draw brief conclusions, which are presented in Section §4.

##  2 Observation and Data Reduction

For this work, we used three AstroSat observations of Swift J1658.2–4242 and
the observation details are given in Table 1.

Table 1: Observation Log. (1) Data; (2) Observation ID; (3) date of observation; (4) modified Julian day; (5) exposure time. Data | ObsID | Date | MJD | Exp.
---|---|---|---|---
|  | (in 2018) |  | (ks)
Data1 | T02_004T01_9000001910 | February 20–21 | 58170.5 | 20
Data2 | T02_011T01_9000001940 | March 03–04 | 58180.5 | 30
Data3 | T02_020T01_9000001990 | March 28–29 | 58205.5 | 30

###  2.1 Soft X-ray Telescope

SXT (Singh et al., 2016, 2017) is an imaging telescope onboard AstroSat
operating in the soft X-ray energy range, 0.3–8 keV. The Level 1 Photon
Counting (PC) mode data is reprocessed to obtain the cleaned Level 2 data from
all orbits in the observation using the SXT pipeline software (version:
AS1SXTLevel2-1.4a). The different orbit event files were merged using the SXT
event merger script (based on Julia) and obtained a merged, exposure-corrected
event file for the further extraction of science products. The source spectrum
is extracted using the tools available in xselect, and we used a circular
region of radius 16 arcmin to extract source events from the merged event
file. We generated the SXT off-axis auxiliary response file (ARF) by the
sxtARFModule tool using the on-axis ARF of version 20190608 provided by the
SXT instrument team. A blank sky background spectrum provided by the SXT
instrument team is used as the background spectrum in the spectral modelling.
While fitting the SXT spectrum, we modified the gain of the response file
using the xspec command gain. The tool gain has two parameters: the slope and
offset. The offset is free to vary, while the slope is fixed at 1. The
spectrum in the 0.7–6 keV energy band is used for the spectral modelling.

###  2.2 Large Area X-ray Proportional Counter

LAXPC has three units, LAXPC10, LAXPC20 and LAXPC30 operate in the 3–80 keV
energy band. The three units have a total effective area of $\sim 8000~{}\rm
cm^{2}$ and record the X-ray events with a time resolution of $10\,\rm\mu s$
(Yadav et al., 2016b, a; Antia et al., 2017; Agrawal et al., 2017). We used
the Event Analysis (EA) mode data of the source and reduced the Level 1 data
using the LAXPC software (LaxpcSoft) to generate the Level 2 cleaned event
file. Among the three units, LAXPC10 and LAXPC30 showed abnormal behaviours
(gain change, low gain and gas leakage). In addition, the LAXPC 30 has no
longer been operational since 2018 March 8. Thus, we used only the LAXPC20
spectrum for the spectral analysis. We modelled the LAXPC20 spectrum in the
5-25 keV energy range. We restrict our spectral analysis below 25 keV because
the background dominates above 25 keV.

##  3 Analysis and Results

The spectro-temporal studies of Swift J1658.2-4242 have been done by several
authors using different models. The authors concluded that the source is
highly variable, highly absorbed with a high inclination angle (Xu et al.,
2018). In addition, the source showed a unique feature during the outburst
(particularly in the intermediate state), namely the rapid disappearance of
QPO in the high flux state (transient QPO) but in the low flux state, a
significantly higher rms reported at the QPO frequency (Xu et al., 2019;
Jithesh et al., 2019; Bogensberger et al., 2020). Though several models have
been employed to understand the spectral state and QPOs separately in the
past, none of the models have taken into account the accretion behavior, how
the mass accretion rate changed, and what was the geometry of the corona.
Therefore, it is worth performing the analysis using an accretion disc-jet
based model to shed further light on the aspects of the observed features.
This motivates us to apply the JeTCAF model to this source, which is discussed
in the next section.

###  3.1 JeTCAF model

The spectral properties of BHXRBs can be well explained using the combined
disc-corona models (Galeev et al., 1979; Zdziarski et al., 2003; Gilfanov et
al., 2003). Alternatively, an accretion disc based two-component advective
flow (TCAF; Chakrabarti & Titarchuk, 1995) model is also available, which
describes the accretion processes, the physical origin of the corona and
temporal properties of the flow from the model parameters. However, this model
does not take into account the signatures of jet from the system. Only
recently, JeTCAF (Jet in TCAF; Mondal & Chakrabarti, 2021) model has been
proposed, which includes the effect of both mass accretion and outflows in the
emitted spectrum. Throughout this paper, we employed the JeTCAF model to fit
the AstroSat spectra.

Matter moving inwards towards the BH becomes supersonic and forms a shock in
accretion (Fukue, 1987; Chakrabarti, 1989). Since then, many numerical
simulations have been performed to verify the formation of the shock in sub-
Keplerian flow (Chakrabarti & Molteni, 1993; Ryu et al., 1997; Giri &
Chakrabarti, 2013; Garain et al., 2014; Kim et al., 2017, and references
therein). From this shocked region to the inner sonic point is known as post-
shock region. As the infalling matter gets compressed due to the geometrical
effect, its optical depth and temperature also change. In this region,
electrons and protons are thermally decoupled due to their different cooling
timescales, which is related to their mass difference. Therefore, for this
region, we solve two temperature equations, and the electron temperature is
obtained as a function of optical depth (cf. sec 2.4 in Chakrabarti &
Titarchuk, 1995). The average temperature of the local plasma is estimated
after deriving an accurate analytical form considering a spherical geometry of
the flow. The typical value of the temperature of this region varies from a
few 100 keV (in the hard spectral state) to a few keV (in the soft state).
However, the temperature may become higher depending on the input parameters,
which vary in multi-dimensional space. A large set of spectra were generated
using a range of parameter space in Debnath et al. (2014) to generate a model
fits file. It takes into account the maximum possible range of electron
temperature and energy spectral index.

The JeTCAF model has three components: first, the Keplerian disc is lying at
the equatorial plane which produces multicolour disc blackbody spectrum and
supplies soft photons. Second, the sub-Keplerian flows above and below the
Keplerian flow, which forms the dynamic corona or post-shock region. This
region upscatters the soft photons from the Keplerian component as in TCAF
model (Chakrabarti & Titarchuk, 1995). As the repeated scattering of photons
gains energy from the corona, the corona cools down. This gain of energy is
taken care of by an enhancement factor. The third component is the mass
outflow/jet implemented in JeTCAF. Considering the temperature of the post-
shock region as the base, the jet temperature in the JeTCAF model falls down
inverse to the distance away from the base of the jet. The temperature of this
component varies between the other two components (the Keplerian disc and
post-shock region). Therefore, soft photons from the standard disc can also
get upscattered by the jet component and produce hard radiation at the
shoulder of the multicolour disc blackbody peak. At the same time, the hard
radiation from the hot inner region gets down scattered by the bulk motion of
the jet when it passes through the jet, which is called bulk motion
Comptonisation by the jet (Titarchuk & Shrader, 2005; Mondal & Chakrabarti,
2021). The second component from the jet produces an excess in the spectrum
above 10 keV, similar to the so-called reflection hump. The present version of
the JeTCAF model does not consider the Iron line, which will be incorporated
in the future. However, in the present analysis, the data above 6 keV is not
taken into account due to calibration issues, therefore the iron line is not
required.

In addition to the spectral properties, the JeTCAF model also explains the
origin of temporal properties, e.g., the origin of QPOs, similar to the TCAF
model. As the matter comes closer to the BH, due to compression, it heats up,
at the same time, the soft photons from the Keplerian disc get upscattered by
the hot electrons, therefore cooling increases. However, both heating and
cooling processes have different timescales. Once the cooling timescale
matches with the heating timescale, resonance occurs and QPOs originate
(Molteni et al., 1996; Chakrabarti et al., 2015; Garain et al., 2014). Thus,
the model is capable of simultaneously explaining the spectral and temporal
properties of BHXRBs.

The JeTCAF model has six parameters. These parameters are namely, (1) mass of
the BH ($M_{\rm BH}$), if the mass is unknown, (2) the Keplerian or disc mass
accretion rate ($\dot{m}_{d}$), (3) sub-Keplerian or halo mass accretion rate
($\dot{m}_{h}$), (4) size of the dynamic corona or the location of the shock
($X_{s}$) in Schwarzschild radius ($r_{g}=2GM_{\rm BH}/c^{2}$) unit, where, G
and $c$ are the gravitational constant and the speed of light, respectively,
(5) shock compression ratio (R=post-shock density/pre-shock density), and (6)
jet collimation factor ($f_{\rm col}$=solid angle subtended by the
outflow/inflow). The mass accretion rates are in the unit of Eddington rate
$\dot{M}_{\rm Edd}$. As the matter falls in and goes out as jet/outflows,
there should have some connections between accretion and ejection, which are
derived in Chakrabarti (1999). It shows that the ratio of the solid angles
between the outflow and inflow automatically comes to the relation, which we
denoted as the jet collimation parameter ($f_{\rm col}$) in the present
version of JeTCAF. However, it can be derived from the geometry of the
accretion-ejection flows, which we plan to implement in the future. If the jet
collimation parameter increases, the mass outflow rate, optical depth and the
number of scattering increase. This makes the spectrum harder. Apart from the
above parameters, the model requires a scale factor, the normalization (norm),
to match the theoretical spectra to the observed one. Therefore, it should not
vary drastically from observation to observation for a single source in a
single outburst. Several works (Molla et al., 2016, 2017, and others) have
discussed this issue and estimated the mass of the central BH using the
constant normalization method. In this model, the norm, which for simplicity
could be written as $R_{z}^{2}/(4\pi D^{2})\sin{i}$, where, ‘$R_{z}$’ is the
effective height of the Keplerian component in km at the pre-shock region, ‘D’
is source distance (in 10 kpc unit) and ‘$i$’ is disc inclination angle with
the line of sight is also a variable. However, since the value of ‘$R_{z}$’ is
unknown, we leave it as a free parameter to be determined from the best-
fitting value. This is very similar to the estimation of Rin, the inner edge
of the disk, obtained from the normalization of the DISKBB model fit.

The JeTCAF model is implemented as a local additive
model111https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node100.html in
xspec (version 12.12.0; Arnaud, 1996) for the spectral fitting. Along with the
JeTCAF model, we have used TBABS (Wilms et al., 2000), partial covering
absorption (PCFABS), and partial covering absorption by partially ionized
plasma (ZXIPCF; Reeves et al., 2008). Using these model components, we fitted
the broadband AstroSat spectra in the 0.7–25 keV energy band.

Figure 1:  AstroSat LAXPC and SXT light curves of Swift J1658.2–4242 from
Data1 (left), Data2 (middle) and Data3 (right). The top and bottom panels in
each plot represent the LAXPC (3–50 keV) and SXT (0.3–8 keV) light curves,
respectively. The time bin size for LAXPC and SXT light curves is 50 s. We
used LAXPC10 and LAXPC20 for the light curve extraction in Data1 and Data2,
while for Data3, LAXPC20 alone was used. The black dashed horizontal lines in
Data2 represent the demarcation of the two flux levels.

###  3.2 Spectral fitting

As mentioned in §2, we used three AstroSat observations of Swift J1658.2–4242
for this study and the LAXPC and SXT light curves are given in Figure 1. In
all observations, the source intensity varies in both SXT and LAXPC.
Particularly in Data2, the source exhibits a significant flux variability,
where the source intensity gradually increases in the initial period of the
observation, then it changes abruptly to a high intensity level. After
spending a few kilo seconds in the high intensity level, the source comes back
to low intensity state and goes into the high intensity quickly once again. In
the last part of the observation, the source makes a transition from the high
intensity to the lower intensity level (see the middle panel of Figure 1).
This variability behaviour is similar to flip-flop behaviour observed in
BHXRBs (Miyamoto et al., 1991; Takizawa et al., 1997; Sriram et al., 2012,
2016). In addition, the source exhibited different variability features in the
AstroSat LAXPC data. In particular, a low-frequency QPO at $\sim 1.56$ Hz
drifted to $\sim 1.74$ Hz during the course of the first observation (Data1),
while a transient QPO associated with the flip-flop behaviour has been
observed in the Data2. In Data3, a QPO detected at $\sim 3.98$ Hz (see Jithesh
et al., 2019). The main motivation of Jithesh et al. (2019) was to understand
the energy-dependent variability properties and the spectral state. The
single-zone stochastic propagation model successfully explained the energy-
dependent variability properties of the source and shed light on the mechanism
responsible for the variability. In the broadband spectral analysis, they used
the phenomenological model to understand the spectral state mainly. This
motivates us to do detailed spectral modelling with a complex physical model
like JeTCAF to understand the observed properties of the source.

Figure 2: The SXT (black points) and LAXPC (red points) unfolded spectra from
Data1 (top panel), Data2 (middle panel), Data3 (bottom panel) are fitted with
the model M2.

We fitted the broadband spectra of these AstroSat observations using the
JeTCAF model along with different absorption models available in the
literature. Our fitting uses different model combinations to estimate the
physical accretion parameters and the significance of absorption, as the
source is highly absorbed. First, we fit the data using model
TBABS*PCFABS*JeTCAF (M1), where the Galactic absorption column density
($N_{\rm HG}$) was fixed to its known value of $1.83\times 10^{22}$ cm-2
(Kalberla et al., 2005). This gives a poor fit with reduced $\chi^{2}$
($\chi_{r}^{2}$) $\gtrsim$2\. Therefore, for this model fit, we could not
estimate errors for model parameters. The parameter values are given in the
top panel of Table 2. In the next step of the analysis, we let the $N_{\rm
HG}$ parameter vary, and obtained an improved fit ($\chi_{r}^{2}\sim 0.9-1.1$
for higher $N_{\rm HG}$ values. However, the Galactic absorption neither
varies significantly from observation to observation in the same outburst nor
can be much higher than (by a factor of few) its known line of sight value
between the source and the observer. Nonetheless, this model fit gives an
indication that some intrinsic absorption is there, which needs to be taken
care of by using some physically motivated absorption models. A similar
conclusion was drawn by Xu et al. (2019), however, by considering a single
component absorption model with $N_{\rm H}$ value an order of magnitude higher
than known $N_{\rm HG}$ (Kalberla et al., 2005).

Moreover, the JeTCAF model parameters show that the disc mass accretion rate
($\dot{m}_{\rm d}$) of the source is close to the Eddington accretion rate and
the mass outflow parameter ($f_{\rm col}$) was also moderately high. For such
high accretion state source, it might be possible that soft photons from the
disc may get Comptonised by some warm and optically thick corona which can be
originated from the disc wind or outflows (Done et al., 2018, and references
therein). Therefore, to consider such effects, we added a partially ionised
plasma (pip) model (ZXIPCF) to M1 and the model TBABS*PCFABS*ZXIPCF*JeTCAF is
designated as M2. During fitting with M2, we kept NHG to the known Galactic
value. The best fit is achieved using model M2 with $\chi_{r}^{2}$ values of
$\sim 1.0$ for all data sets. These results and fit statistics are given in
the second panel of Table 2. It can be seen that the column density of PCFABS
model (NHpcf) varies between $8.5-12.7\times 10^{22}$ cm-2 and the partial
covering factor (C${}_{f}^{pcf}$) was fluctuating between 0.58 and 0.95. The
column density of ZXIPCF model (NHpip) varies noticeably between
$3.9-8.3\times 10^{22}$ cm-2 with partial covering factor (C${}_{f}^{pip}$)
$>0.9$ and pegged to its upper limit. The ionisation parameter ($\xi$) is
varied between 2.5-25 erg cm s-1 and is relatively low. The above model
parameters and their covering coefficients imply that the source is highly
absorbed, and significant intrinsic absorption is there, which can be due to
wind/outflow from the highly accreting disc. The model M2 is fitted to the
data and is shown in Figure 2. Panels A, B, and C in Figure 2 denote the
Data1, 2, and 3, respectively. The JeTCAF model continuum and different
absorption models are adequate to fit the present data sets.

During the fitting with different model combinations, the mass of the black
hole (MBH) in JeTCAF model is used as a free parameter, since we do not know
its value beforehand. As expected, the mass should not change from epoch to
epoch, but rather remain constant. This is consistent with the yielded average
mass value from X-ray spectral fitting using M2, which is $\sim 14$ M⊙ (see
Table 2). The disc mass accretion rate ($\dot{m}_{d}$) in JeTCAF model (in M2)
varies between 0.9 to 1.04 $\dot{M}_{\rm Edd}$, whereas the halo accretion
rate ($\dot{m}_{h}$) is relatively low and varies in the range of 0.15–0.24
$\dot{M}_{\rm Edd}$. We have noticed a significant change in the size of
dynamic corona ($X_{\rm s}$) from $\sim 34~{}r_{g}$ to $\sim 65~{}r_{g}$
during the outburst period, and anti-correlates with $\dot{m}_{d}$. The shock
compression ratio (R) variation was significant, from 5.6 to 6.6 in Data1, 2,
and 3. The jet collimation factor ($f_{\rm col}$) was low ($\sim 0.01$) during
the first observation and increased to $\sim 0.4$ in the second and third
observations. This is also reflected in both the column densities, that have
increased in Data2 and 3 compared to Data1. This implies that jet/mass outflow
from the inner region of the disc was present, changed the shape of the
continuum. We note that the JeTCAF model norm varies between 8–11 for all
observations. For Data1, the norm is 3. As we have only three observations, we
can not apply the constant normalization technique as used in Molla et al.
(2016, 2017). However, naturally, a consistent BH mass is getting from the
best fit. The best-fit spectral parameters are provided in Table 2. All errors
are estimated in the 90% confidence level using
err222https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node79.html
https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node28.html command.
Interestingly, in both model combinations, JeTCAF model parameters show a
similar trend of parameter variations and the parameter values are quite
similar. This concludes that the JeTCAF model parameter estimation is robust.
We have computed the best-fit JeTCAF model spectra without absorption using
the source code for the best-fitted model M2 parameters, which is shown in
Figure 5.

The source exhibited flip-flop behaviour in Data2. To understand this
variability behaviour, we further extracted the flux-resolved spectra from
Data2 and performed the spectral modelling using all three models discussed
above. During this fitting, we fixed the mass of the black hole to the values
obtained from the Data2 and freed the other parameters. The best-fit spectral
parameters from the flux-resolved spectral analysis using model M2 are given
in Table 2 (as Seg 1 and 2). The two segments are also fitted well with model
M2, with $\chi_{r}^{2}\sim$ 1\. The spectral fits are shown in Figure 3. In
this analysis, both mass accretion rates ($\dot{m}_{\rm d}$ and $\dot{m}_{\rm
h}$) increase as the flux increases from low to high flux states, while the
size of the corona and shock compression ratio decrease during the transition.
In addition, we notice that $\dot{m}_{d}$ has increased on the same day, which
has changed the size of the dynamic corona from $\sim 40~{}r_{g}$ to $\sim
37~{}r_{g}$. The $f_{\rm col}$ parameter is more or less the same, while the R
parameter increases its value from 5.7 to 6.1. The norm of JeTCAF model is
$\sim 10$ for both segments. The NHpcf value has increased from 6 to 8 $\times
10^{22}$ cm-2 and the NHpip has decreased from 14 to 8 $\times 10^{22}$ cm-2
along with a decrease in $\xi$ from 63 to 13 erg cm s-1. In the next section,
we estimate the QPO frequency using the best-fit shock parameters of the
JeTCAF model in M2.

Figure 3: The flux-resolved spectral analysis of Data2 with the model M2. The
top and bottom panels represent the spectral fit for the low and high flux
levels, respectively. Table 2: The broadband spectral parameters of Swift
J1658.2$-$4242 when fitted with models M1, and M2. Here, NHpcf, C${}_{f}^{\rm
pcf}$, NHpip, and C${}_{\rm f}^{\rm pip}$ are hydrogen column densities and
dimensionless covering fractions for PCFABS and ZXIPCF models, respectively
and $\xi$ is the ionization parameter for the ZXIPCF model. The C${}_{\rm
f}^{\rm pip}$ parameter obtained from the fit is high $>0.9$ and pegged at the
upper limit for all observations. The $\dot{m}_{d}$, $\dot{m}_{h}$, $X_{s}$,
$R$, and $f_{\rm col}$ are the disc and halo mass accretion rates, location of
the shock or size of the corona, shock compression ratio, and jet collimation
factor respectively. f denotes the values are frozen. The Galactic absorption
(NHG) value is frozen to $1.83\times 10^{22}$ cm-2 for all observations. For
poor fits with $\chi_{r}^{2}\gtrsim 2$, errors of the model M1 parameters are
not estimated.

Model | Data | Seg | $N_{\rm Hpcf}$ | C${}_{f}^{\rm pcf}$ | $N_{\rm Hpip}$ | $\log(\xi_{\rm pip})$ | MBH | $\dot{m}_{\rm d}$ | $\dot{m}_{\rm h}$ | $X_{\rm s}$ | R | fcol | $\chi^{2}_{r}/dof$ |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  |  |  $(10^{22}$ cm${}^{-2})$ |  |  $(10^{22}$ cm${}^{-2})$ | (erg cm s-1) | (M⊙) | $(\dot{M}_{\rm Edd})$ | $(\dot{M}_{\rm Edd})$ | $(r_{\rm g})$ |  |  |  |  |
| 1 | - | $8.1$ | $0.95$ | $-$ | $-$ | $13.0$ | $0.91$ | $0.14$ | $65.0$ | $5.1$ | $<0.01$ | 2.4/235 |  |
| 2 | - | $7.3$ | $0.95$ | $-$ | $-$ | $13.8$ | $0.92$ | $0.20$ | $34.2$ | $5.6$ | $0.40$ | 3.4/482 |  |
M1 |  | 1 | $7.6$ | $0.95$ | $-$ | $-$ | ${}^{f}13.8$ | $1.05$ | $0.24$ | $38.4$ | $6.0$ | $0.39$ | 1.8/434 |  |
|  | 2 | $10.1$ | $1.0$ | $-$ | $-$ | ${}^{f}13.8$ | $1.16$ | $0.21$ | $41.2$ | $5.3$ | $0.32$ | 3.3/382 |  |
| 3 | - | $7.9$ | $0.95$ | $-$ | $-$ | $14.0$ | $1.05$ | $0.27$ | $32.1$ | $6.2$ | $0.33$ | 1.9/490 |  |
| 1 | - | $8.5^{+0.9}_{-0.7}$ | $0.95^{+0.07}_{-0.04}$ | $3.9^{+0.2}_{-0.1}$ | $0.4^{+0.1}_{-0.1}$ | $13.5^{+0.5}_{-0.4}$ | $0.90^{+0.02}_{-0.01}$ | $0.15^{+0.01}_{-0.01}$ | $64.9^{+3.9}_{-3.1}$ | $5.6^{+0.2}_{-0.1}$ | $<0.01$ | 1.0/232 |  |
| 2 | - | $9.2^{+1.0}_{-0.9}$ | $0.79^{+0.05}_{-0.04}$ | $8.3^{+0.5}_{-0.4}$ | $1.4^{+0.2}_{-0.1}$ | $13.6^{+0.6}_{-0.5}$ | $0.93^{+0.04}_{-0.04}$ | $0.20^{+0.01}_{-0.01}$ | $34.5^{+2.0}_{-1.5}$ | $5.7^{+0.3}_{-0.2}$ | $0.39^{+0.05}_{-0.03}$ | 1.0/479 |  |
M2 |  | 1 | $6.0^{+0.4}_{-0.3}$ | $0.72^{+0.06}_{-0.06}$ | $14.4^{+1.1}_{-0.9}$ | $1.8^{+0.3}_{-0.1}$ | ${}^{f}13.6$ | $1.07^{+0.03}_{-0.02}$ | $0.22^{+0.02}_{-0.01}$ | $40.1^{+2.2}_{-1.9}$ | $5.7^{+0.4}_{-0.4}$ | $0.33^{+0.04}_{-0.03}$ | 1.0/431 |  |
|  | 2 | $7.8^{+0.9}_{-0.9}$ | $0.79^{+0.06}_{-0.04}$ | $7.8^{+1.0}_{-0.7}$ | $1.1^{+0.2}_{-0.2}$ | ${}^{f}13.6$ | $1.09^{+0.04}_{-0.03}$ | $0.25^{+0.02}_{-0.01}$ | $37.4^{+2.1}_{-2.1}$ | $6.1^{+0.5}_{-0.4}$ | $0.32^{+0.02}_{-0.01}$ | 1.0/379 |  |
| 3 | - | $12.7^{+1.0}_{-0.8}$ | $0.58^{+0.10}_{-0.07}$ | $7.3^{+0.9}_{-0.7}$ | $0.7^{+0.1}_{-0.1}$ | $14.2^{+0.7}_{-0.5}$ | $1.04^{+0.03}_{-0.03}$ | $0.24^{+0.01}_{-0.01}$ | $41.1^{+2.8}_{-2.4}$ | $6.6^{+0.4}_{-0.2}$ | $0.35^{+0.03}_{-0.03}$ | 1.0/487 |  |

###  3.3 QPO frequency from model parameters

The QPO frequencies can be estimated using the shock oscillation model
(Molteni et al., 1996; Chakrabarti et al., 2008) as below,

$\nu_{\rm QPO}=\frac{c}{r_{\rm g}\;M_{\rm BH}\;RX_{\rm s}^{1.5}}.$ (1)

For the three observations, the computed QPO frequencies are 2.53, 6.37, and
4.04 Hz, respectively. The estimated QPO frequencies, the observed QPO
frequencies and the Q-factor of the QPOs are given in Table 3. The estimated
QPO values are consistent with the observed QPOs. The $\nu_{\rm QPO}$ values
correlate with the $\dot{m}_{d}$, where the highest and lowest QPOs are
observed during the highest and lowest accretion rate periods. Apart from the
QPOs, the model can explain the broadband noise variability. In the case of
QPOs, we considered that the shock is axisymmetric. However, it is not
necessarily always, in a complicated disc, shock can be non-axisymmetric due
to some perturbations (see Chakrabarti & Wiita, 1993; Garain & Kim, 2022, for
recent simulation). Therefore, the spiral shock arms may deform and reform
again, which can produce broad and multi-peaked power density spectra. Thus,
it is also possible to explain broadband noise in the light of JeTCAF model
after implementing non-axisymmetric shock.

Table 3: Estimated QPO frequencies using M2 model parameters against those reported by Jithesh et al. (2019) Data |  Estimated $\nu_{\rm QPO}$ |  Observed $\nu_{\rm QPO}$ | Q
---|---|---|---
| (Hz) | (Hz) |
1 | $2.53^{+0.25}_{-0.19}$ | $1.57^{+0.01}_{-0.01}$ | 8.60
| $--$ | $1.71^{+0.04}_{-0.04}$ | 4.39
2 | $6.37^{+0.62}_{-0.46}$ | $6.57^{+0.05}_{-0.05}$ | 3.23
3 | $4.04^{+0.43}_{-0.34}$ | $3.97^{+0.04}_{-0.02}$ | 8.30

###  3.4 Estimating Binary Parameters

Figure 4: The 1-day binned MAXI light curve of Swift J1658.2–4242 in the 2–20
keV energy band over the period 2018 January 14 to October 9. The day “0"
represents MJD 58132.5. The vertical dashed, dot-dashed, and solid blue lines
correspond to three AstroSat observations considered in this work. The solid
green line represents the exponential decay function fitted to the light
curve.

Decay of X-ray light curve is common in BHXRBs with irradiated or unirradiated
discs and they generally show three decay phases in the long-term light curve
during the outburst evolution (King & Ritter, 1998; King, 1998; Dubus et al.,
1999; Frank et al., 2002, and references therein). The first two decay
profiles follow the exponential, while the third one has a linear profile
(King & Ritter, 1998). It should be noted that the third phase of decay might
not always be observed as it appears much later in the outburst. Later, Mondal
(2020) revisited these decay profiles by parameterizing $\alpha$ and combined
the mass and orbital period of the companion star to understand the effect of
the binary parameters. It is also noted that the binary parameters can be
estimated from the same solution. Figure 4 shows the 1-day binned MAXI light
curve of Swift J1658.2–4242 in the 2–20 keV energy band over the period 2018
January 14 to October 9. The light curve clearly shows two decays, the first
one starts between Observation 2 and 3, and the second decay starts later
after Observation 3. Both decay profiles more or less follow the exponential
function as explained in King & Ritter (1998) for a given viscosity parameter
($\alpha$) using disc irradiation. The first decay of the source, though it
shows a sharp change, does not have sufficient data points at the slope of the
decay. Hence, we did not fit this phase. As expected, this phase can provide a
different decay (viscosity) timescale as the two decay phases have different
physical origins. The third phase of decay was not observed in the light
curve. Thus, we considered the second decay phase for the fitting, which is
much clearer and lasted for a longer time. The solid green line in Figure 4
represents the best-fit exponential decay function, $f(t)=Aexp(-t/t_{v}$)
fitted to the light curve. Here, $t_{v}$ is the light curve decay timescale or
the viscous timescale and A is the normalization. The fitted function gives A
= $0.51\pm 0.05$ photons cm-2 s-1 and $\tau=76.7\pm 4.9$ day with reduced
$\chi^{2}=4.96$ for 131 degrees of freedom. After getting the $t_{v}$, we
estimate the binary parameters as discussed below.

The amount of matter accreting from the companion star falling into a black
hole may depend on the companion properties and binary separation or orbital
period of the companion star. Therefore, by fitting the spectrum and the decay
profile of the light curve, it might be possible to put some light on the
binary parameters. To do that, we considered the unirradiated disc mass
accretion rate and the decay profile of the light curve. The effective disc
temperature can be written as (Shakura & Sunyaev, 1973),

$T=\left(\frac{3GM_{\rm BH}\dot{M}_{d}}{8\pi\sigma r_{\rm
out}^{3}}\right)^{1/4},$ (2)

where G, $\dot{M}_{d}$, $\sigma$, and $r_{\rm out}$ are the gravitational
constant, disc mass accretion rate in g/s unit, Stefan-Boltzman constant, and
outer radius of the disc, respectively. This gives the sound speed at the
radius for a given temperature (T) as

$c_{s}=\left(\frac{\gamma kT}{\mu m_{p}}\right)^{1/2}.$ (3)

Here, $\gamma$ (= 5/3), $k$, $\mu$ (= 0.6), and $m_{p}$ are the adiabatic
index of the flow, Boltzman constant, mean molecular weight, and the mass of
the proton, respectively. We can write the height of the disc as,

$H=c_{s}\left(\frac{r_{\rm out}^{3}}{GM_{\rm BH}}\right)^{1/2}.$ (4)

Once we have all these quantities, the viscous timescale can be written from
Shakura & Sunyaev (1973),

$t_{v}=\frac{r_{\rm out}^{2}}{\alpha c_{s}H},$ (5)

here, $\alpha$ is the viscosity parameter, for which we have chosen a typical
value $\sim 0.1$ (King et al., 2007) for the rest of the analysis.

After doing some simple algebra, $r_{\rm out}$ can be written as,

$r_{\rm out}=6.15\times 10^{8}~{}\text{cm}~{}~{}\dot{M}_{d}^{1/5}M_{\rm
BH}^{-1/5}t_{v}^{4/5}\alpha^{4/5}.$ (6)

Considering $M_{\rm BH}$, $\dot{M}_{d}$ (in Table 2), and $t_{v}$ from
spectral and light curve fitting, the $r_{\rm out}$ value comes out to be
$2.8\times 10^{10}$ cm. The standard assumption regarding the outer accretion
disc radius is that the disc is tidally truncated at about 80% (see Cannizzo,
1994) the Roche Lobe radius ($R_{L}$) of the primary. This gives $R_{L}\sim
3.5\times 10^{10}$ cm. Now, the binary separation ($a$) can be estimated using
the relation (Eggleton, 1983),

$R_{L}=\frac{0.49aq^{2/3}}{0.6q^{2/3}+\ln[1+q^{1/3}]},$ (7)

here, $q$ is the binary mass ratio (donor mass/accretor mass; $M_{d}/M_{\rm
BH}$). The optical observation of this binary system has inferred that the
companion might be a mid or late K-type star (Bahramian et al., 2018) or an
evolved star. Therefore, the mass of the donor can lie between 0.5 and 0.8
$M_{\odot}$. Considering the lower and upper $M_{d}$ values, the upper and
lower limit of $a$ come out to be $a_{u}=2.3\times 10^{11}$ cm and
$a_{l}=2.0\times 10^{11}$ cm.

The orbital period ($P_{\rm orb}$) of the donor star can be estimated using
Kepler’s third law,

$P_{\rm orb}(hr)=6.83a^{3/2}\left(1+q\right)^{-1/2}\left(\frac{M_{\rm
BH}}{M_{\odot}}\right)^{-1/2}.$ (8)

The $a_{u}$ and $a_{l}$ values give the upper and lower limits of $P_{\rm
orb}$ come out as 4.4 hr and 3.6 hr. Therefore, our estimated possible $P_{\rm
orb}$ value is $4.0\pm 0.4$ for a reasonable $\alpha$-parameter value of 0.1.

##  4 Discussion and Conclusion

In this paper, we have presented the spectral and temporal analysis of the
black hole transient Swift J1658.2–4242 using the SXT and LAXPC onboard
AstroSat. The data is fitted using the JeTCAF model along with different
absorption model components to understand the spectral and temporal properties
of the source. The estimated absorption column densities for different
absorption models are high and in the range of $\sim 6-13\times 10^{22}$ cm-2
for PCFABS and $\sim 4-14\times 10^{22}$ cm-2 for ZXIPCF models, while the NH
for TBABS model was fixed to the Galactic value. The higher value of $N_{H}$
suggests that the neutral absorption is probably intrinsic to the source,
which might be originated from the significant wind or outflow from the disc.
It is believed that mass accretion close to or higher than Eddington rate may
launch winds (Lipunova, 1999; Done et al., 2018, and references therein),
which might be the case for this source. It is further supported by the
estimated disc mass accretion rate.

The JeTCAF model provides the mass accretion rate and the accretion geometry
of the system directly. The best-fit disc mass accretion rate is high and
close to the Eddington rate in these observations. The jet/outflow collimation
factor for Data 2 and 3 is also moderately high $\sim 0.4$, which has modified
the shape of the continuum spectra. However, the low or negligibly small value
in Data1 says that outflow was mild or absent during this epoch. The shock
location or corona size reduced from 65 to 35 $r_{g}$ during the observation
period due to the change in mass accretion rates. The smaller corona size for
a higher accretion rate and vice-versa are consistent with the cooling rate
due to the scattering of soft photons by hot electrons in the corona.
Moreover, the disc mass accretion rate is always higher than the hot flow or
halo mass accretion rate along with a bigger size of the corona implies that
the source belongs to the intermediate to soft spectral state of BHXRBs.

Several works in the literature, which studied months or weeks of continuous
observations of several black hole candidates in the framework of TCAF model
(Mondal et al., 2014; Debnath et al., 2015; Molla et al., 2016, 2017, and
references therein), concluded that, during the rising phase of the outburst,
the halo rate is high and slowly goes down, and the disc rate goes up. At the
same time, shock moves inward. However, in the declining phase, the scenario
is the opposite. As we do not have continuous monitoring observations for
weeks or months, we can not verify those profiles. However, the basic
relations between parameters in our study agree with previous studies. In
addition, different model combinations fit to the data show that JeTCAF model
parameters do not vary from one to the other. This suggests that the accretion
flow parameters estimated using JeTCAF model are robust.

The QPO frequencies from all three epochs are estimated by considering the
shock oscillation model and using the model fitted shock location and
compression ratio parameters. The computed QPO frequencies are consistent with
the observed frequencies (see Table 3). The $\nu_{\rm QPO}$ values are also
correlated with the $X_{s}$, where the highest frequency QPO ($\sim 6.6$ Hz)
observed at the lowest shock location values and vice-versa, indicating that
the cooling may take a role in this change. The increase in $\dot{m}_{d}$
generates more soft photons in the disc, which are upscattered by the hot
corona and gain energy, therefore cooling rate increases (Mondal et al., 2017,
and references therein). Thus, the corona size ($X_{s}$) decreases and
$\nu_{QPO}$ increases. The Q-factor of the QPO correlates with the geometry of
the corona, but anti-correlates with the disc mass accretion rate. As pointed
out earlier, the size of the corona increases or decreases depending on the
interplay between the disc and halo mass accretion rates. Therefore, all these
changes in corona geometry, spectral state, QPO frequency, and its Q-factor
are interlinked through the fundamental flow parameters of accretion, which
are the mass accretion rate.

The source exhibited ‘flip-flop’ behaviour in one of the observations,
however, the physical reason is unclear. There are several explanations for
this e.g., accretion disc instability, ejection of jets, fluctuation in
viscosity etc (Miyamoto et al., 1991; Takizawa et al., 1997; Lyubarskii, 1997;
Lasota, 2001; Fender et al., 2004; Sriram et al., 2012, 2016). We note that,
during the AstroSat observation period, the detection of jet ejection or radio
flare was not reported (Bogensberger et al., 2020). However, the JeTCAF model
fitted parameters do show the signature of outflows, from the observation 2,
when the ‘flip-flop’ behaviour was triggered. The higher value of $f_{\rm
col}$ may indicate that the outflow was not likely to be a collimated jet,
rather wind type. In addition, from the JeTCAF model fits, we can see that the
accretion rate changed significantly, which is sufficient to change the flux
behaviour of the source (see Figure 5, the dashed and dash-dotted lines for
segments 1 and 2). Therefore, we argue that the ‘flip-flop’ may originate
either due to the change in accretion rates, possibly when the disc accretion
rate dominates over the halo accretion rate or due to ejection of mass
outflows, which significantly obscured the central continuum emission.
Furthermore, all the characteristics mentioned above might be the consequences
of the same underlying mechanism, which is the change in the mass accretion
rates.

Modelling the broadband X-ray spectra with the JeTCAF model helps us estimate
the binary parameters for the system. We used the disc mass accretion rate
from the JeTCAF model and fitted the long-term MAXI light curve of the source
to derive the binary parameters. The decay phase of the light curve is
considered for this analysis, and the estimated decay timescale (viscous
decay) is $\sim 77$ day for the source. As the viscous timescale depends on
the outer radius of the disc, it connects both the orbital separation and the
period of the system for a given binary mass ratio. Our estimate of binary
separation ($a$) varies between $2.0-2.3\times 10^{11}$ cm, for the companion
mass of 0.5 – 0.8 $M_{\odot}$, by assuming the companion as a mid or late
K-type star (Bahramian et al., 2018). The orbital period of the companion
comes out to be between 3.6 to 4.4 hr. The estimated short orbital period of
Swift J1658.2–4242 is consistent with other BHXRB systems (3.2–6.3 hr; Zurita
et al., 2008; Russell et al., 2014; Jaisawal et al., 2015; Corral-Santana et
al., 2016, and references therein). Detailed optical photometric studies using
large telescopes are required to verify the estimated orbital period of the
source.

## Acknowledgements

We thank the anonymous referee for making insightful comments and suggestions
that improved the paper. SM acknowledges Ramanujan Fellowship
(#RJF/2020/000113) by SERB-DST, Govt. of India, for this research. VJ would
like to acknowledge the Centre for Research, CHRIST (Deemed to be University),
for the financial support in the form of a Seed Money Grant (SMSS-2217). The
research is based on the results obtained from the AstroSat mission of the
Indian Space Research Organization (ISRO), archived at the Indian Space
Science Data Centre (ISSDC). This work has used the data from the LAXPC and
SXT instruments. We thank the LAXPC Payload Operation Center (POC) and the SXT
POC at TIFR, Mumbai, for providing the data via the ISSDC data archive and the
necessary software tools.

## Data Availability

The data used in this article are available in the ISRO’s Science Data Archive
for AstroSat Mission
(https://astrobrowse.issdc.gov.in/astro_archive/archive/Home.jsp). The model
used in the paper can be shared on reasonable request to the corresponding
author.

## References

  * Agrawal et al. (2017) Agrawal P. C., et al., 2017, Journal of Astrophysics and Astronomy, 38, 30
  * Antia et al. (2017) Antia H. M., et al., 2017, ApJS, 231, 10
  * Arnaud (1996) Arnaud K. A., 1996, in Jacoby G. H., Barnes J., eds, Astronomical Society of the Pacific Conference Series Vol. 101, Astronomical Data Analysis Software and Systems V. p. 17
  * Baby et al. (2021) Baby B. E., Bhuvana G. R., Radhika D., Katoch T., Mandal S., Nandi A., 2021, MNRAS, 508, 2447
  * Bahramian et al. (2018) Bahramian A., Britt C., Strader J., 2018, The Astronomer’s Telegram, 11342, 1
  * Banerjee et al. (2021) Banerjee A., Bhattacharjee A., Chatterjee D., Debnath D., Chakrabarti S. K., Katoch T., Antia H. M., 2021, ApJ, 916, 68
  * Barthelmy et al. (2018) Barthelmy S. D., D’Avanzo P., Deich A., Gronwall C., Melandri A., Page K. L., Palmer D. M., 2018, GRB Coordinates Network, 22416, 1
  * Belloni et al. (2005) Belloni T., Homan J., Casella P., van der Klis M., Nespoli E., Lewin W. H. G., Miller J. M., Méndez M., 2005, A&A, 440, 207
  * Beri & Altamirano (2018) Beri A., Altamirano D., 2018, The Astronomer’s Telegram, 12072, 1
  * Bhargava et al. (2019) Bhargava Y., Belloni T., Bhattacharya D., Misra R., 2019, MNRAS, 488, 720
  * Bogensberger et al. (2020) Bogensberger D., et al., 2020, A&A, 641, A101
  * Cabanac et al. (2010) Cabanac C., Henri G., Petrucci P. O., Malzac J., Ferreira J., Belloni T. M., 2010, MNRAS, 404, 738
  * Cannizzo (1994) Cannizzo J. K., 1994, ApJ, 435, 389
  * Casella et al. (2005) Casella P., Belloni T., Stella L., 2005, ApJ, 629, 403
  * Chakrabarti (1989) Chakrabarti S. K., 1989, ApJ, 347, 365
  * Chakrabarti (1999) Chakrabarti S. K., 1999, A&A, 351, 185
  * Chakrabarti & Molteni (1993) Chakrabarti S. K., Molteni D., 1993, ApJ, 417, 671
  * Chakrabarti & Titarchuk (1995) Chakrabarti S., Titarchuk L. G., 1995, ApJ, 455, 623
  * Chakrabarti & Wiita (1993) Chakrabarti S. K., Wiita P. J., 1993, A&A, 271, 216
  * Chakrabarti et al. (2008) Chakrabarti S. K., Debnath D., Nandi A., Pal P. S., 2008, A&A, 489, L41
  * Chakrabarti et al. (2015) Chakrabarti S. K., Mondal S., Debnath D., 2015, MNRAS, 452, 3451
  * Chakrabarti et al. (2019) Chakrabarti S. K., Debnath D., Nagarkoti S., 2019, Advances in Space Research, 63, 3749
  * Chakraborty et al. (2020) Chakraborty S., Navale N., Ratheesh A., Bhattacharyya S., 2020, MNRAS, 498, 5873
  * Chatterjee et al. (2016) Chatterjee D., Debnath D., Chakrabarti S. K., Mondal S., Jana A., 2016, ApJ, 827, 88
  * Corral-Santana et al. (2016) Corral-Santana J. M., Casares J., Muñoz-Darias T., Bauer F. E., Martínez-Pais I. G., Russell D. M., 2016, A&A, 587, A61
  * Debnath et al. (2014) Debnath D., Chakrabarti S. K., Mondal S., 2014, MNRAS, 440, L121
  * Debnath et al. (2015) Debnath D., Mondal S., Chakrabarti S. K., 2015, MNRAS, 447, 1984
  * Done et al. (2018) Done C., Tomaru R., Takahashi T., 2018, MNRAS, 473, 838
  * Dubus et al. (1999) Dubus G., Lasota J.-P., Hameury J.-M., Charles P., 1999, MNRAS, 303, 139
  * Eggleton (1983) Eggleton P. P., 1983, ApJ, 268, 368
  * Fender et al. (2004) Fender R. P., Belloni T. M., Gallo E., 2004, MNRAS, 355, 1105
  * Frank et al. (2002) Frank J., King A., Raine D., 2002, Accretion Power in Astrophysics, 3 edn. Cambridge University Press, doi:10.1017/CBO9781139164245
  * Fukue (1987) Fukue J., 1987, PASJ, 39, 309
  * Galeev et al. (1979) Galeev A. A., Rosner R., Vaiana G. S., 1979, ApJ, 229, 318
  * Garain & Kim (2022) Garain S. K., Kim J., 2022, arXiv e-prints, p. arXiv:2212.08310
  * Garain et al. (2014) Garain S. K., Ghosh H., Chakrabarti S. K., 2014, MNRAS, 437, 1329
  * Gilfanov et al. (2003) Gilfanov M., Revnivtsev M., Molkov S., 2003, A&A, 410, 217
  * Giri & Chakrabarti (2013) Giri K., Chakrabarti S. K., 2013, MNRAS, 430, 2836
  * Grebenev et al. (2018) Grebenev S. A., Mereminskiy I. A., Prosvetov A. V., Ducci L., Bozzo E., Savchenko V., Ferrigno C., 2018, The Astronomer’s Telegram, 11306, 1
  * Grinberg et al. (2018) Grinberg V., Eikmann W., Kreykenbohm I., Wilms J., 2018, The Astronomer’s Telegram, 11318, 1
  * Homan et al. (2001) Homan J., Wijnands R., van der Klis M., Belloni T., van Paradijs J., Klein-Wolt M., Fender R., Méndez M., 2001, ApJS, 132, 377
  * Ingram et al. (2009) Ingram A., Done C., Fragile P. C., 2009, MNRAS, 397, L101
  * Jaisawal et al. (2015) Jaisawal G. K., Homan J., Naik S., Jonker P., 2015, The Astronomer’s Telegram, 7361, 1
  * Janiuk et al. (2002) Janiuk A., Czerny B., Siemiginowska A., 2002, ApJ, 576, 908
  * Jithesh et al. (2019) Jithesh V., Maqbool B., Misra R., T A. R., Mall G., James M., 2019, ApJ, 887, 101
  * Jithesh et al. (2021) Jithesh V., Misra R., Maqbool B., Mall G., 2021, MNRAS, 505, 713
  * Kalberla et al. (2005) Kalberla P. M. W., Burton W. B., Hartmann D., Arnal E. M., Bajaja E., Morras R., Pöppel W. G. L., 2005, A&A, 440, 775
  * Kim et al. (2017) Kim J., Garain S. K., Balsara D. S., Chakrabarti S. K., 2017, MNRAS, 472, 542
  * King (1998) King A. R., 1998, MNRAS, 296, L45
  * King & Ritter (1998) King A. R., Ritter H., 1998, MNRAS, 293, L42
  * King et al. (2007) King A. R., Pringle J. E., Livio M., 2007, MNRAS, 376, 1740
  * Lasota (2001) Lasota J.-P., 2001, New Astron. Rev., 45, 449
  * Laurent & Titarchuk (2001) Laurent P., Titarchuk L., 2001, ApJ, 562, L67
  * Lien et al. (2018) Lien A. Y., et al., 2018, GRB Coordinates Network, 22419, 1
  * Lipunova (1999) Lipunova G. V., 1999, Astronomy Letters, 25, 508
  * Lyubarskii (1997) Lyubarskii Y. E., 1997, MNRAS, 292, 679
  * Majumder et al. (2022) Majumder S., Sreehari H., Aftab N., Katoch T., Das S., Nandi A., 2022, MNRAS, 512, 2508
  * Mall et al. (2023) Mall G., Vadakkumthani J., Misra R., 2023, Research in Astronomy and Astrophysics, 23, 015015
  * Maqbool et al. (2019) Maqbool B., et al., 2019, MNRAS, 486, 2964
  * Meyer-Hofmeister et al. (2005) Meyer-Hofmeister E., Liu B. F., Meyer F., 2005, A&A, 432, 181
  * Misra et al. (2017) Misra R., et al., 2017, ApJ, 835, 195
  * Miyamoto et al. (1991) Miyamoto S., Kimura K., Kitamoto S., Dotani T., Ebisawa K., 1991, ApJ, 383, 784
  * Molla et al. (2016) Molla A. A., Debnath D., Chakrabarti S. K., Mondal S., Jana A., 2016, MNRAS, 460, 3163
  * Molla et al. (2017) Molla A. A., Chakrabarti S. K., Debnath D., Mondal S., 2017, ApJ, 834, 88
  * Molteni et al. (1996) Molteni D., Sponholz H., Chakrabarti S. K., 1996, ApJ, 457, 805
  * Mondal (2020) Mondal S., 2020, Advances in Space Research, 65, 693
  * Mondal & Chakrabarti (2021) Mondal S., Chakrabarti S. K., 2021, ApJ, 920, 41
  * Mondal et al. (2014) Mondal S., Debnath D., Chakrabarti S. K., 2014, ApJ, 786, 4
  * Mondal et al. (2017) Mondal S., Chakrabarti S. K., Nagarkoti S., Arévalo P., 2017, ApJ, 850, 47
  * Motta et al. (2011) Motta S., Muñoz-Darias T., Casella P., Belloni T., Homan J., 2011, MNRAS, 418, 2292
  * Nandi et al. (2012) Nandi A., Debnath D., Mandal S., Chakrabarti S. K., 2012, A&A, 542, A56
  * Narayan et al. (1996) Narayan R., McClintock J. E., Yi I., 1996, ApJ, 457, 821
  * Reeves et al. (2008) Reeves J., Done C., Pounds K., Terashima Y., Hayashida K., Anabuki N., Uchino M., Turner M., 2008, MNRAS, 385, L108
  * Remillard & McClintock (2006) Remillard R. A., McClintock J. E., 2006, ARA&A, 44, 49
  * Russell et al. (2014) Russell T. D., Soria R., Motch C., Pakull M. W., Torres M. A. P., Curran P. A., Jonker P. G., Miller-Jones J. C. A., 2014, MNRAS, 439, 1381
  * Russell et al. (2018) Russell T. D., Miller-Jones J. C. A., Sivakoff G. R., Tetarenko A. J., 2018, The Astronomer’s Telegram, 11322, 1
  * Ryu et al. (1997) Ryu D., Chakrabarti S. K., Molteni D., 1997, ApJ, 474, 378
  * Shakura & Sunyaev (1973) Shakura N. I., Sunyaev R. A., 1973, A&A, 24, 337
  * Shui et al. (2021) Shui Q. C., et al., 2021, MNRAS, 508, 287
  * Singh et al. (2014) Singh K. P., et al., 2014, in Takahashi T., den Herder J.-W. A., Bautz M., eds, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series Vol. 9144, Space Telescopes and Instrumentation 2014: Ultraviolet to Gamma Ray. p. 91441S, doi:10.1117/12.2062667
  * Singh et al. (2016) Singh K. P., et al., 2016, in Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray. p. 99051E, doi:10.1117/12.2235309
  * Singh et al. (2017) Singh K. P., et al., 2017, Journal of Astrophysics and Astronomy, 38, 29
  * Sreehari et al. (2020) Sreehari H., Nandi A., Das S., Agrawal V. K., Mandal S., Ramadevi M. C., Katoch T., 2020, MNRAS, 499, 5891
  * Sridhar et al. (2019) Sridhar N., Bhattacharyya S., Chandra S., Antia H. M., 2019, MNRAS, 487, 4221
  * Sriram et al. (2012) Sriram K., Rao A. R., Choi C. S., 2012, A&A, 541, A6
  * Sriram et al. (2016) Sriram K., Rao A. R., Choi C. S., 2016, ApJ, 823, 67
  * Stella & Vietri (1998) Stella L., Vietri M., 1998, ApJ, 492, L59
  * Stella et al. (1999) Stella L., Vietri M., Morsink S. M., 1999, ApJ, 524, L63
  * Tagger & Pellat (1999) Tagger M., Pellat R., 1999, A&A, 349, 1003
  * Takizawa et al. (1997) Takizawa M., et al., 1997, ApJ, 489, 272
  * Titarchuk & Fiorito (2004) Titarchuk L., Fiorito R., 2004, ApJ, 612, 988
  * Titarchuk & Shrader (2005) Titarchuk L., Shrader C., 2005, ApJ, 623, 362
  * Wijnands et al. (1999) Wijnands R., Homan J., van der Klis M., 1999, ApJ, 526, L33
  * Wilms et al. (2000) Wilms J., Allen A., McCray R., 2000, ApJ, 542, 914
  * Xiao et al. (2019) Xiao G. C., et al., 2019, Journal of High Energy Astrophysics, 24, 30
  * Xu et al. (2018) Xu Y., et al., 2018, ApJ, 865, 18
  * Xu et al. (2019) Xu Y., et al., 2019, ApJ, 879, 93
  * Yadav et al. (2016a) Yadav J. S., et al., 2016a, ApJ, 833, 27
  * Yadav et al. (2016b) Yadav J. S., et al., 2016b, in Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray. p. 99051D, doi:10.1117/12.2231857
  * Zdziarski et al. (2003) Zdziarski A. A., Lubiński P., Gilfanov M., Revnivtsev M., 2003, MNRAS, 342, 355
  * Zurita et al. (2008) Zurita C., Durant M., Torres M. A. P., Shahbaz T., Casares J., Steeghs D., 2008, ApJ, 681, 1458

##  Appendix A Best-fit model spectra without absorption

We have computed the best-fit M2 model spectra for all observations and
segments (bottom panel of Table 2) to see the shape of the continuum spectra
without any absorption. During extracting the model spectra, we set all NH,
covering fraction, and ionization parameter values to zero and the “constant"
scale-factor for the second spectrum (which is LAXPC in our case) is set to 1
to bring the both model components in the same line. All spectra are shown in
Figure 5, where different colours correspond to different observations and
segments. From the figure, it is visible that the significant fraction of the
bolometric flux comes from the AstroSat energy band.

Figure 5: The best-fitted M2 model spectra are shown for all observations and
segments. During extracting the model, we set all NH values to zero and the
“constant" scale-factor for the second spectrum (LAXPC) in XSPEC is set to 1
to keep the model continuous.