# Accretion Properties and Estimation of Spin of Galactic Black Hole Candidate
Swift J1728.9–3613 with NuSTAR during its 2019 outburst

Skye R. Heiland,1 Arka Chatterjee,1 Samar Safi-Harb,1 Arghajit Jana,2 and
Jeremy Heyl3
1Department of Physics and Astronomy, University of Manitoba, Winnipeg,
Manitoba, R3T 2N2, Canada
2Institute of Astronomy, National Tsing Hua University, Hsinchu, 300044,
Taiwan
3Department of Physics and Astronomy, University of British Columbia,
Vancouver, V6T 1Z1, Canada E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Black hole X-ray binaries (BHXRBs) play a crucial role in understanding the
accretion of matter onto a black hole. Here, we focus on exploring the
transient BHXRB Swift J1728.9–3613 discovered by Swift/BAT and MAXI/GSC during
its January 2019 outburst. We present measurements on its accretion
properties, long time-scale variability, and spin. To probe these properties
we make use of several NICER observations and an unexplored data set from
NuSTAR, as well as long term light curves from MAXI/GSC. In our timing
analysis we provide estimates of the cross-correlation functions between light
curves in various energy bands. In our spectral analysis we employ numerous
phenomenological models to constrain the parameters of the system, including
flavours of the relativistic reflection model Relxill to model the Fe
K$\alpha$ line and the $>15$ keV reflection hump. Our analysis reveals that:
(i) Over the course of the outburst the total energy released was $\sim
5.2\times 10^{44}$ ergs, corresponding to roughly 90% the mass of Mars being
devoured. (ii) We find a continuum lag of $8.4\pm 1.9$ days between light
curves in the $2-4$ keV and $10-20$ keV bands which could be related to the
viscous inflow time-scale of matter in the standard disc. (iii) Spectral
analysis reveals a spin parameter of $\sim 0.6-0.7$ with an inclination angle
of $\sim 45^{\circ}-70^{\circ}$, and an accretion rate during the NuSTAR
observation of $\sim 17\%~{}L_{\rm Edd}$.

###### keywords:

accretion, accretion discs – black hole physics – relativistic processes –
X-rays: individual: Swift J1728.9–3613

††pubyear: 2023††pagerange: Accretion Properties and Estimation of Spin of
Galactic Black Hole Candidate Swift J1728.9–3613 with NuSTAR during its 2019
outburst–Accretion Properties and Estimation of Spin of Galactic Black Hole
Candidate Swift J1728.9–3613 with NuSTAR during its 2019 outburst

##  1 Introduction

Black hole X-ray binaries (BHXRBs) are a class of astrophysical objects in
which a black hole and a main sequence star are gravitationally bound. These
systems make ideal laboratories for studying the physics of accretion and
matter in the presence of an extreme gravitational well, in addition to
testing the predictions of general relativity. BHXRBs are further categorized
into transient and persistent classes. The transient class remains dormant for
most of its lifetime, except for occasional outbursts where the luminosities
($L_{\rm X}$) reach beyond $10^{35}$ erg s-1(e.g., Tetarenko et al., 2016).
These outbursts and their spectral evolution are traced through the so-called
‘q’ or hardness–intensity diagram (e.g., Homan et al., 2001; Remillard &
McClintock, 2006). A full outburst starts with a low/hard state followed by an
intermediate state reaching to the high/soft state while rising to the peak
luminosity. During the declining phase, the path reverses ending with the hard
state. ‘Failed’ or hard/low state only outbursts are those where high/soft
state remains absent (see Alabarta et al., 2021, and references therein).
Energy dependent variations in count rates are a universal feature of
outbursting BHXRBs, which are usually attributed to their accretion
properties. A few major physical drivers of accretion are viscosity (e.g.,
Shakura & Sunyaev, 1973; Smith et al., 2002; Chatterjee et al., 2020),
electron cloud temperature and optical depth (Sunyaev & Titarchuk, 1980,
1985), the Compton cooling rate of the cloud (e.g., Chakrabarti & Titarchuk,
1995; Garain et al., 2012, 2014). In general, measurements of such parameters
are performed using long term light-curve variations and detailed spectral
studies using various phenomenological or physical models.

Apart from accretion properties, outbursting BHXRBs also provide a chance to
measure the intrinsic properties of the black hole itself, e.g. mass ($M_{\rm
BH}$) and spin ($a^{*}$). The dynamical method, wherein mass is obtained using
radial velocity measured from absorption lines in the spectrum from the
companion star, provides the most accurate measure of the mass (Casares &
Jonker, 2014). One can also use dips and eclipses in the light curve to
predict the mass ratio and inclination of the binary system (Horne, 1985).
However, only around 20 BHXRBs are measured this way (e.g., Corral-Santana et
al., 2016). Apart from the dynamical method, one can measure the mass using
spectral fitting (e.g., Kreidberg et al., 2012; Torres et al., 2019; Kubota et
al., 1998; Shaposhnikov & Titarchuk, 2007; Jana et al., 2022) or the X-ray
reverberation method (Mastroserio et al., 2019). Direct measurement of the
spin could be carried out by imaging the shape of the photon sphere or radius
of the innermost stable circular orbit (ISCO) (Chan et al., 2013). Using radio
interferometry, the EHT-collaboration directly captured images of the
supermassive black holes M 87* and our own Sgr A* (see Event Horizon Telescope
Collaboration et al. (2019, 2022) and references therein). For galactic
BHXRBs, spin is usually measured by X-ray spectroscopy (e.g Draghis et al.
(2022)), though in principle X-ray interferometry could be used to produce
EHT-style observations that directly image the shape of the BH shadow in some
Seyfert galaxies (Uttley et al.,, 2020; den Hartog et al., 2020).

Within X-ray spectroscopy there are two primary methods that have proved
successful in measuring black hole spin, namely continuum fitting (e.g., Zhang
et al., 1997; McClintock et al., 2006; Steiner et al., 2014) and relativistic
reflection (e.g., Fabian et al., 1989; Miller et al., 2012; Reynolds, 2021).
In continuum fitting, the inner edge of the accretion disc or ISCO ($r_{\rm
in}$) is measured through the inner disc temperature. The optically thick and
geometrically thin accretion disc considers the relativistic form prescribed
by (Novikov & Thorne, 1973) where the spin influences $r_{\rm in}$. For
extreme prograde motion (i.e. $a\rightarrow 1$), $r_{\rm in}$, marginally
stable and bound orbits and, the photon sphere merge into the surface of the
event horizon. As the inner radius becomes smaller, the excess binding energy
loss, spent into heating the accretion disc, becomes increasingly larger while
approaching $r_{\rm in}$. The corresponding temperature $T_{\rm in}$ can then
be measured using relativistic models like Kerrbb (Zhang et al., 1997; Li et
al., 2005). This method was employed to measure the spins of LMC X–1 (Gou et
al., 2009; Mudambi et al., 2020), H 1743-322 (Steiner et al., 2009), GRS
1915+105 (Sreehari et al., 2020), LMC X–1 (Jana et al., 2021a), LMC X–3
(Bhuvana et al., 2021), and MAXI J1820+070 (Zhao et al., 2021).

In contrast to continuum fitting, the relativistic reflection method instead
models the spectral shape using the blurred iron K$\alpha$ line at 6.4 keV and
associated reflection hump above 15 keV (Fabian et al., 1989; George & Fabian,
1991). This fluorescent line distorts due to the gravitational redshift
experienced by the reflected spectra of a spinning black hole, with maximal
distortion occurring for a maximally spinning prograde orbit. The ‘reflection
hump’ arises due to the hard X-ray reflection from the accretion disc. Often
the irradiation of the disc shows a profile steeper than $r^{-5}$, invoking a
compact corona along the rotation axis of the black hole. This is the so-
called ‘lamp-post’ model of the coronal geometry (Miniutti & Fabian, 2004),
and the height and compactness of this corona change alongside other spectral
properties (Fabian et al., 2015). As the Doppler effect significantly modifies
the observed spectrum with changing inclination (Luminet, 1979; Viergutz,
1993), these models calculate the parameters of the system and modify the iron
line profile accordingly. A central assumption invoked in this method is that
the inner radius of the accretion disk extends all the way to the inner most
stable circular orbit or ISCO. This need not be the case, especially in the
low/hard state (e.g. Zdziarski & Marco, 2020; Done et al., 2007). The
spectroscopic reflection model Relxill (García et al., 2013; Dauser et al.,
2014; García et al., 2014) provides a measure of the iron abundance ($A_{\rm
Fe}$) within the disc, and various flavours of disc/emissivity profiles can be
assumed. Within the past few years, Relxill has successfully measured the spin
of many galactic black holes, such as MAXI J1535–571 (Miller et al., 2018),
Cygnus X–1 (Tomsick et al., 2018), XTE J1752–223 (García et al., 2018), GX
339–4 (García et al., 2019), MAXI J1631–479 (Xu et al., 2020), XTE J1908–094
(Draghis et al., 2021), MAXI J1813–095 (Jana et al., 2021b). Most recently,
Draghis et al. (2022) reported spins of ten new Galactic black hole candidate
using the Relxill model. Apart from these two, one can measure spin using
timing properties where low-frequency Quasi Periodic Oscillations (QPOs) are
assumed to originate from  Lense-Thirring precession (Ingram et al., 2009).

Swift J1728.9–3613 or MAXI J1728–36 went into outburst on 17 January 2019 and
continued until June 2019. The source was reported by Swift/BAT on 28 January
2019 (Barthelmy et al., 2019). However, it was initially discovered by
MAXI/GSC on 26 January 2019 (Negoro et al., 2019). On 31 January 2019, MeerKAT
detected the radio counterpart of the source with a magnitude of $11.2\pm 0.6$
mJy at 1.28 GHz (Bright et al., 2019). Integral observed the source on 19
February 2019 and detected the source flux as $103\pm 4$ mCrab in the $20-40$
keV range (Ducci et al., 2019). Following the MAXI and Swift detection, NICER
has been monitoring the source since 29 January 2019 (Enoto et al., 2019),
producing a total of 103 observations. Out of these, $\sim 70$ of them lie
within the outbursting period. An optical counterpart with a magnitude of 16.7
was observed on 28 January 2019 by the 60 cm BOOTES-3/YA robotic telescope (Hu
et al., 2019). Recently, Saha et al. (2022) analyzed NICER spectra of Swift
J1728.9–3613 during the outburst and obtained a lower mass limit for the
compact object of $4.6~{}M_{\odot}$, obtained assuming $a^{*}=0$ from the
inner radius of the disk and a distance of $\sim 10$ kpc. This estimate
further strengthens Swift J1728.9–3613’s candidacy as a BHXRB. NuSTAR observed
Swift J1728.9–3613 on 3 February 2019 but has thus far remained unexplored.
This is the observation we utilized to estimate the spin and accretion
properties.

In this paper, we concentrate on the MAXI/GSC light curve to explore the long
time scale variability. We explored the first two observations of NICER to
understand the beginning of the outburst. Later, we systematically examine the
NuSTAR data to extract accretion disc properties and spin using numerous
models such as Diskbb, Cutoffpl, Nthcomp, and Relxill. The paper is structured
as follows. The data analysis process is briefly described in Section 2.
Results obtained from MAXI/GSC light curve are presented in Section 3.1.
Spectral analysis with NICER is presented in Section 3.2. Spectral properties
are analyzed and constrained in Section 3.3, while Relxill is employed in
Section 3.3.3. Finally, we draw our conclusions and discuss our results in
Section 4. All errors associated with model parameters are quoted at the 90%
confidence level unless otherwise stated.

Table 1: Log of Observations of Swift J1728.9–3613 Instrument | Date (UT) | Obs ID | Exposure (ks) | Count s-1
---|---|---|---|---
NICER/XTI | 2019-01-29 | 1200550101 | 1.86 | $581.7\pm 0.7$
NICER/XTI | 2019-01-30 | 1200550102 | 13.7 | $992.0\pm 0.4$
NuSTAR | 2019-02-03 | 90501303002 | 55.9 | $144.7\pm 0.02$

##  2 Observations and Data Reduction

NuSTAR observed Swift J1728.9–3613 on 3 February 2019 for a total exposure of
55.9 ks (see Table 1). NuSTAR is a hard X-ray focusing telescope, consisting
of two identical modules: FPMA and FPMB (Harrison et al., 2013). The raw data
were reprocessed with the NuSTAR Data Analysis Software (NuSTARDAS, version
1.4.1). Cleaned event files were generated and calibrated by using the
standard filtering criteria in the nupipeline task and the latest calibration
data files available in the NuSTAR calibration database
(CALDB)111http://heasarc.gsfc.nasa.gov/FTP/caldb/data/nustar/fpm/. The source
and background products were extracted by considering circular regions with
radii 60 arcsec and 90 arcsec, at the source coordinates and away from the
source, respectively. The spectra and light curves were extracted using the
nuproduct task. We re-binned the spectra with 25 counts per bin by using the
grppha task. Additionally, we divided the light-curves in two segments using
xselect and run nuproduct using user defined gti files.

Following the discovery of Swift J1728.9–3613, NICER began to monitor the
source. We analysed the first two of the 70 NICER observations. NICER is a
soft X-ray telescope whose primary X-ray Timing Instrument (XTI) consists of
56 identical FPMs (50 of which are functional) that record the energies of
arriving photons, as well as their time of arrival to within 300 ns of UTC
(Gendreau et al., 2012). Raw data were processed with the NICER Data Analysis
Software (NICERDAS, version
9)222https://heasarc.gsfc.nasa.gov/docs/nicer/nicer_analysis.html. Cleaned
event files were generated with standard filtering criteria in the nicerl2
task, while background files were generated with the nibackgen3C50 task.
Response files were generated with the nicerarf and nicerrmf tasks. Spectra
and light curves were extracted with xselect and spectra were rebinned with 25
counts per bin using the grppha task. We have not applied any systematic on
NuSTAR or NICER data.

##  3 Results and Discussion

###  3.1 Long Term Delay

MAXI/GSC (Matsuoka et al., 2009) has monitored the source daily and provided
one day binned light curves at various energy ranges. From these curves, as
presented in the top panel of Fig. 1, we observed that the peak of the
outburst occurred within 10-12 days of the beginning. Comparable to other
canonical outbursts, rate variations were observed in various energy ranges.
During the peak luminosity in the high/soft state, the hard 10-20 keV counts
were substantially less than the softer 2-4 keV counts. In total, we found the
outburst lasted for about 150 days starting from MJD 58500 to 58150.

Figure 1: Top panel: MAXI/GSC light curves of Swift J1728.9–3613 during the
outburst in three different energy bands, binned at 1 day. The blue vertical
line represents the date of the primary NuSTAR observation analyzed in Section
3.3, while the orange lines show when the two NICER observations analyzed in
Section 3.2 were taken. Bottom panels: Estimations of the cross–correlation
functions between light curves of the indicated energy bands and likelihood
distributions of the true time lags. Estimation was performed with the
$\zeta$-transformed discrete correlation function ($\zeta$-DCF) and its
associated likelihood (PLIKE) algorithm from Alexander (2014). $\tilde{\tau}$
and $\sigma$ are the mean and standard deviation of the Gaussian fits, while
$\rho$ is the maximum DCF value.

The MAXI/GSC light curves can be used to obtain an estimate for the amount of
energy released by the outburst in each energy band as

$E_{i}=4\pi D^{2}\tilde{E}_{i}\int_{{\rm MJD}_{\rm start}}^{{\rm MJD}_{\rm
stop}}L_{i}(t)dt$

where $D$ is the distance to the source, $\tilde{E}_{i}$ is the average photon
energy in the $i^{\rm th}$ band, and $L_{i}$ is the corresponding MAXI/GSC
light curve. Assuming a distance of 10 kpc and an average photon energy of 3.0
keV, 7.0 keV, and 15 keV in each energy band respectively, we calculate an
energy output of $\sim 2.1\times 10^{44}$ ergs in the $2-4$ keV band, $\sim
2.2\times 10^{44}$ ergs in the $4-10$ keV band, and $\sim 9.0\times 10^{43}$
ergs in the $10-20$ keV band, all over the course of the entire outburst.
Considering all energy bands, the total energy released is $\sim 5.2\times
10^{44}$ ergs, which translates to $5.8\times 10^{26}$ g of mass assuming a
mass-to-radiation conversion efficiency of 0.1 (Reynolds, 2021). The converted
matter is equivalent to $2.9\times 10^{-7}M_{\odot}$ or 91% of the mass of
Mars. However, the bolometric luminosity should be higher than what we have
calculated from X-rays. Thus, it is expected that our estimation of the matter
conversion is moderately reserved.

We inspected the time delays among the energy bands using the
$\zeta$-DCF333https://www.weizmann.ac.il/particle/tal/research-
activities/software algorithm (Alexander, 2014), presented in the bottom
panels of Fig. 1. A positive delay refers to the softer component arriving
later, while the reverse indicates delayed arrival of the harder component.
The $2-4$ keV light curve correlates with both $4-10$ and $10-20$ keV light
curves, having maximum correlation coefficients ($\rho$) of $0.95$ and $0.72$
respectively. The peak delays ($\tilde{\tau}$) between the corresponding light
curves were found to be $(3.7\pm 0.8)$ and $(8.4\pm 1.9)$ days, with standard
deviations $11.1$ and $9.9$ days respectively. A similar correlated pattern
was also observed between $4-10$ and $10-20$ keV light curves with maximum
coefficient, delay, and width $\rho=0.77$, $\tilde{\tau}=(4.7\pm 1.1)$ days,
$\sigma=10.0$ days respectively. This delay is consistent with what we would
expect examining the other two light curves, i.e. $8.4\;{\rm days}-3.7\;{\rm
days}=4.7\;{\rm days}$. We should note that $\sigma$ here refers to the
estimated width of the cross-correlation function itself, whereas the errors
associated with the true delays are estimated from the width of the
corresponding likelihood distribution. These distributions were calculated
with the PLIKE algorithm, also presented by Alexander (2014). Our analysis
showed that harder ($10-20$ keV) radiation as a whole arrived before its
softer counterpart. The magnitude of the delay increased with increasing
differences in X-ray energy, exhibiting a maximum delay between the $2-4$ and
$10-20$ keV photons. An accreting BHXRB spectrum is typically approximated by
the sum of a hard power-law and soft blackbody disk component, and from the
light curve analysis we can infer that the harder spectral component dominated
the rising phase before giving way to more prominent soft disk emission. This
scenario is similar to what was observed earlier in the outburst profiles of
GX 339–4, XTE J1650–500 (Smith et al., 2002; Chatterjee et al., 2020). A
possible reason for this delayed peak in soft emission could be the viscous
delay with which the standard disc spirals inward before heating enough to
glow in the X-ray. Influenced by the viscous delay, the disc gradually
modifies the spectral and temporal properties, triggering the outburst profile
of the BHXRB (Smith et al., 2002). The left panel of Fig. 2 shows the
variation in hardness ratio (HR) vs. intensity (in terms of the accretion rate
$\dot{M}=L/L_{\rm Edd}$) over the full outburst as observed by MAXI/GSC, and
illustrates a typical ‘q’ diagram for an outbursting BHXRB.

Figure 2: Left Panel (a): Hardness–accretion rate diagram of Swift
J1728.9–3613 over the course of its outburst, produced with MAXI/GSC
observations. The MAXI hardness ratio (HR) is defined as the ratio of count
rates in the $4-20$ keV and $2-4$ keV bands. Colour represents progression of
the outburst through time (MJD-58500 days) as indicated at the top. The
maximum disc accretion rate was $\dot{M}_{\rm max}=0.28~{}L_{\rm Edd}$ during
peak luminosity. The primary NuSTAR observation is marked with $\odot$. Right
Panel (b): Hardness–intensity diagram (HID) for the first NuSTAR observation
of the Swift J1728.9–3613 outburst. Points are binned at 100s with the NuSTAR
HR defined as the ratio of count rates in the $6-78$ keV and $3-6$ keV bands.
Colour represents progression through time as indicated at the top, in
seconds. The dashed line through the data is the best fit having a slope and
intercept of $278.2\pm 10.5$ and $65.2\pm 6.0$ counts/s, respectively.

###  3.2 NICER

As mentioned previously, the X-ray spectrum of a BHXRB is typically
approximated with the sum of a multi-colour disc blackbody (MCD) and power-law
component. Reprocessed or reflected emission may also be observed, such as a
Fe K$\alpha$ line at $\sim 6.4$ keV, a reflection hump at $\sim 15-40$ keV
(see bottom panel of Fig. 4), and in our case, an apparent Ni emission line at
$\sim 8$ keV (Corliss & Sugar, 1981; Molendi et al., 2003; Medvedev et al.,
2018). Spectral analysis was carried out in HEASEARC’s spectral analysis
package XSPEC version 12.12.1 (Arnaud, 1996). We used the Tbabs model to
account for interstellar absorption with the WILM abundance (Wilms et al.,
2000) and the cross-section of Verner et al. (1996). The MCD component was
handled with the Diskbb model, and is included in all spectral models
presented in this paper.

The first two NICER observations of Swift J1728.9–3613 were taken on MJD 58512
and MJD 58513 respectively, which correspond to the beginning of the rising
phase of the outburst – or the low/hard state. We fit the data for the first
two observations in the $0.3-10.0$ keV range with an absorbed MCD and power
law model [Tbabs*(Diskbb + Powerlaw)]. This returned an inner disc temperature
of $T_{\rm in}=1.05_{-0.02}^{+0.03}$ keV for the first observation and $T_{\rm
in}=1.229_{-0.006}^{+0.006}$ keV for the second. We find a reduction in photon
index from $\Gamma=2.02_{-0.07}^{+0.06}$ to $1.72_{-0.08}^{+0.08}$, with
normalization varying from $\sim 1.7$ to $\sim 0.9~{}{\rm
ph~{}keV^{-1}~{}cm^{-2}}$; hydrogen column density stayed approximately
constant ($n_{\rm H}=4.67_{-0.07}^{+0.07}\times 10^{-22}{\rm cm^{-2}}$ to
$4.58_{-0.04}^{+0.04}\times 10^{-22}{\rm cm^{-2}}$). Table 2 contains detailed
results for each fit, along with their $\chi^{2}$ value per degree of freedom
(dof) and model flux. Removing the MCD component from the model substantially
worsens both fits, giving an F–statistic of $\sim 200$ and $\sim 3000$ for the
first two observations respectively. This indicates the presence of a rapidly
growing standard accretion disc at the beginning of the outburst.

Table 2: NICER Spectral Analysis Results: Diskbb+Powerlaw

Observation ID |  $n_{\rm H}$ ($10^{-22}{\rm cm^{-2}}$) |  $T_{\rm in}$ (keV) | normdiskbb | $\Gamma$ | normPL |  $\chi^{2}/$dof |  $F_{2-10}$ ($10^{-9}$ ergs cm-2)
---|---|---|---|---|---|---|---
1200550101 | $4.67_{-0.07}^{+0.07}$ | $1.05_{-0.02}^{+0.03}$ | $109_{-20}^{+20}$ | $2.02_{-0.07}^{+0.06}$ | $1.7_{-0.3}^{+0.3}$ | 891/902 | $4.498_{-0.010}^{+0.010}$
1200550102 | $4.58_{-0.04}^{+0.04}$ | $1.229_{-0.006}^{+0.006}$ | $215_{-8}^{+8}$ | $1.72_{-0.08}^{+0.08}$ | $0.9_{-0.2}^{+0.2}$ | 1008/944 | $6.177_{-0.014}^{+0.014}$

$F_{2-10}$ is the total model flux in the $2-10$ keV range.

###  3.3 NuSTAR

We performed the spectral analysis of the NuSTAR data in the $3-78$ keV energy
range. The right panel of Fig. 2 shows the variation in HR vs. count rate
throughout the primary NuSTAR observation, with points binned at 100s. We
observe that during the observation HR and intensity were positively
correlated, lying along the line with slope, intercept, correlation
coefficient, and null hypothesis probability $(278.2\pm 10.5)$, $(65.2\pm
6.0)$ counts/s, $r=0.877$, and $p<10^{-5}$ respectively. Correlation between
these two parameters is typical of BHXRBs during intermediate spectral states
where disc-corona coupling becomes stronger (Churazov et al., 2001; Taylor et
al., 2003). In Section 3.3.1 we resolve the spectrum between the two regions
in the HR diagram with unfolded spectra from both regions presented in Fig. 3.
Section 3.3.2 is dedicated to examining the full time-averaged spectrum of the
observation using several non-relativistic models, and in Section 3.3.3 we
explore the same spectrum with several flavours of the Relxill model family.
Gaussian components are included to model the Fe and Ni emission lines where
necessary in all non-relativistic models. Fig. 4 shows the full spectrum of
the NuSTAR observation fit with a simple MCD and power-law model, which
disregards all reflected emission. The bottom panel of Fig. 4 displays the
residuals of that fit and showcases the reflected emission, i.e. the 6.4 keV
Fe K$\alpha$ line and reflection hump above $\sim 15$ keV.

####  3.3.1 Hardness-Resolved Spectra

The ‘gap’ between data points that occurs at $\sim 45,000$s shows a sudden
softening of Swift J1728.9–3613’s spectrum, alongside a reduction in intensity
during the observation. We extracted spectra for both regions of the HID
(denoting the earlier region i and the later region ii) and fit them with an
absorbed MCD and cutoff power law model Cutoffpl, which very roughly
approximates the continuum emission of a thermally Comptonized medium. Fig. 3
shows the resulting spectral fits for both regions.

Between regions i and ii the inner disc temperature, photon index, and cutoff
energy did not appreciably vary, returning values of $T_{\rm
in}=1.195_{-0.011}^{+0.010}$ keV, $\Gamma=2.27_{-0.07}^{+0.07}$, and $E_{\rm
cut}=94_{-34}^{+78}$ keV in region i; and $T_{\rm in}=1.216_{-0.005}^{+0.005}$
keV, $\Gamma=2.13_{-0.03}^{+0.03}$, and $E_{\rm cut}=79_{-18}^{+28}$ keV in
region ii. In region i we also detected signatures of both the Fe K$\alpha$
line at $E_{\rm Fe}=6.34_{-0.14}^{+0.11}$ keV and the Ni XXVIII line at
$E_{\rm Ni}=8.06_{-0.28}^{+0.10}$ keV. In region ii we only detected the Fe
K$\alpha$ line (with greater uncertainty) at $E_{\rm Fe}=6.2_{-0.3}^{+0.4}$
keV, and the apparent hydrogen column density changed from $n_{\rm
H}=4.19_{-0.12}^{+0.25}\times 10^{-22}{\rm cm^{-2}}$ to a more uncertain
$n_{\rm H}=3.14_{-0.7}^{+0.7}\times 10^{-22}{\rm cm^{-2}}$. Between regions we
found a reduction in the harder power law component, with normalization
varying from $\sim 3.3$ to $\sim 1.5~{}{\rm ph~{}keV^{-1}~{}cm^{-2}}$
(illustrated in Fig. 3). This explains the softening of the spectrum as well
as the disappearance of the reprocessed Ni emission line. Variations on this
timescale ($\sim 100$s) being primarily influenced by the power-law component
are consistent with the findings of Churazov et al. (2001) in the soft state
of Cyg X-1. Detailed results of both fits are presented in Table 3.

Figure 3: Models (both total and their individual components) for both spectra extracted from region i and region ii of the HID presented in the right panel of Fig. 2. Table 3: NuSTAR Hardness Resolved Spectral Results | i | ii
---|---|---
$n_{\rm H}$ ($10^{22}{\rm cm^{-2}}$) | $4.19_{-0.12}^{+0.25}$ | $3.8_{-0.7}^{+0.7}$
$T_{\rm in}$ (keV) | $1.195_{-0.011}^{+0.010}$ | $1.216_{-0.005}^{+0.005}$
normdiskbb | $313_{-12}^{+13}$ | $303_{-6}^{+7}$
$\Gamma$ | $2.27_{-0.07}^{+0.07}$ | $2.13_{-0.03}^{+0.03}$
$kT_{e}$ (keV) | $94_{-34}^{+78}$ | $79_{-18}^{+28}$
normPL (${\rm ph~{}keV^{-1}cm^{-2}}$) | $3.3_{-0.5}^{+0.6}$ | $1.54_{-0.07}^{+0.07}$
$E_{\rm Fe}$ (keV) | $6.34_{-0.14}^{+0.11}$ | $6.2_{-0.3}^{+0.4}$
$\sigma_{\rm Fe}$ (keV) | $0.80_{-0.07}^{+0.08}$ | $0.3_{-0.3}^{+0.5}$
normFe ($10^{-3}$) | $6.5_{-1.4}^{+0.5}$ | $1.1_{-0.9}^{+1.0}$
$E_{\rm Ni}$ (keV) | $8.06_{-0.28}^{+0.10}$ | -
$\sigma_{\rm Ni}$ (keV) | $0.37_{-0.13}^{+0.19}$ | -
normNi ($10^{-3}$) | $1.7_{-0.3}^{+1.2}$ | -
$\chi^{2}$/dof | 869/887 | 604/599
$F_{2-10}$ ($10^{-9}$ ergs cm-2) | $9.952_{-0.012}^{+0.012}$ | $8.862_{-0.022}^{+0.022}$
Figure 4: Top panel: Time-averaged unfolded spectrum from FPMB (green) fit
with an absorbed MCD and power law model, with no Gaussian components. Both
the individual model components as well as their sum are presented. Bottom
Panel: Residuals in terms of (data$-$model)/error obtained from spectral
analysis for the same model. Figure 5: Residuals in terms of (data-
model)/error obtained from spectral analysis for the Powerlaw, Cutoffpl,
Nthcomp, RelxillCp, RelxillLp, and RelxillLpCp models. Reduced $\chi^{2}$
values ($\tilde{\chi}^{2}$) for each model are quoted in each panel. The first
three models (i.e., the non-relativistic ones) include two Gaussian components
for the Fe K$\alpha$ and Ni emission lines.

####  3.3.2 Time averaged spectrum: Phenomenological Models

Examining the entire spectrum of the observation, we first fit the data with
an absorbed power–law and MCD model. This model returned a reasonable fit with
$\chi^{2}=950$ for 956 degrees of freedom (dof) with estimates of the inner
disc temperature $T_{\rm in}=1.185_{-0.010}^{+0.015}$ keV, a photon index of
$\Gamma=2.46_{-0.02}^{+0.02}$, and a column density of $n_{\rm
H}=4.63_{-0.28}^{+0.28}\times 10^{-22}\mathrm{cm^{-2}}$. We find signatures of
the Fe and Ni emission lines at $E_{\rm Fe}=6.48_{-0.02}^{+0.02}$ keV and
$E_{\rm Ni}=7.99_{-0.03}^{+0.04}$ keV. The reflection hump was clearly visible
in the residuals at energies above $\sim 15$ keV (see Fig. 5). This was
repeated with an exponential cutoff power-law component Cutoffpl. This
provided no significant improvement to the fit, returning an inner disc
temperature of $T_{\rm in}=1.16_{-0.03}^{+0.03}$ keV, a slightly softer photon
index of $\Gamma=2.24_{-0.08}^{+0.08}$, cutoff energy $kT_{e}=94_{-39}^{+93}$
keV, and column density $n_{\rm H}=4.32_{-0.23}^{+0.27}\times
10^{-22}\mathrm{cm^{-2}}$; we again find signatures of the Fe and Ni emission
lines at $E_{\rm Fe}=6.27_{-0.13}^{+0.16}$ keV and $E_{\rm
Ni}=8.0_{-0.2}^{+0.3}$ keV. The reflection hump was still clearly visible in
the residuals.

A much better description of continuum emission due to thermal Comptonization
is given by the model Nthcomp (Zdziarski et al., 1996), which attempts to
simulate the upscattering of photons through the corona from a seed spectrum
parameterized by the inner temperature of the accretion disc. The high energy
cutoff is parameterized by the electron temperature of the medium. Nthcomp is
not a power law, but can still be parameterized with an asymptotic photon
index (Życki et al., 1999). Using the inner disc temperature returned by
Diskbb, $T_{\rm in}=1.149_{-0.009}^{+0.006}$ keV, as the low energy rollover
in Nthcomp, we estimate the hot electron temperature to be
$kT_{e}=132_{-123}^{+476}$ keV with photon index
$\Gamma=2.36_{-0.01}^{+0.02}$. This model also returned the lowest hydrogen
column density of all of the presented models with $n_{\rm
H}=3.27_{-0.35}^{+0.91}\times 10^{-22}\mathrm{cm^{-2}}$. For completeness, we
may also calculate the optical depth ($\tau$) of the corona as given by
Zdziarski et al. (1996),

$\tau=\sqrt{\frac{9}{4}+\frac{m_{e}c^{2}}{kT_{e}}\frac{3}{(\Gamma-1)(\Gamma+2)}}-\frac{3}{2},$

returning a value of $\tau=0.55_{-0.42}^{+3.54}$ for the Nthcomp model. The
optically thin corona was also proposed for Cyg X-1 (Churazov et al., 2001)
during its high/soft state.

We also attempted to fit the spectrum with the non-relativistic reflected
power-law model Pexrav from Magdziarz & Zdziarski (1995), alongside a regular
power-law component to account for direct coronal emission. Since this model
accounts for the reflection and reprocessing of X-rays from the corona by the
accretion disc, it can be used to estimate the reflection fraction ($R_{\rm
refl}$), defined as the fraction of X-rays reprocessed by the disc vs. those
received via direct emission. It also estimates abundances of elements heavier
than He, including iron ($A_{\mathrm{Fe}}$), as well as the inclination angle
$i$. This model returned a close fit with the reflection hump reduced in the
residuals and a $\chi^{2}=916$ for 949 dof. However, most parameters were
unable to be constrained. Hence, we opted for relativistic model to estimate
those parameters. One of the few parameters that remained well-behaved was the
reflection fraction, $R_{\rm refl}=0.48_{-0.06}^{+0.06}$, and we should expect
Relxill to estimate reflection fractions of the same order.

Finally, we should note explicitly that in all the aforementioned models, two
Gaussian model components were used to model to the Fe K$\alpha$ and Ni
emission lines. We observed two more emission lines at 27.9 keV and 30.8 keV
having widths of 0.07 keV and 0.06 keV respectively; the normalization of both
lines was around $10^{-4}$ ph cm-2 s-1. The origin of these emission lines
could be attributed to the radioactive isotope 241Am, as reported by García et
al. (2018); Connors et al. (2022). Similar lines were also observed in
AstroSat/LAXPC spectra, which are an instrumental feature (Sreehari et al.,
2019). To fit those lines, we added two additional Gaussian components to each
model and froze the line energy, width, and normalization as obtained from the
Powerlaw model. Detailed results of all spectral fits for this section are
presented in the first three columns of Table 4, and the residuals for each
model are presented in Fig. 5.

####  3.3.3 Time averaged spectrum: Relxill

We used different flavors of the relativistic reflection model Relxill (García
et al., 2013, 2014; Dauser et al., 2014, 2016) to probe the reprocessed
emission more accurately. In this model, the reflection fraction ($R_{\rm
refl}$) is given by the ratio between the Comptonized emission to the disc and
to the infinity. A broken power-law emission profile is assumed with $E(r)\sim
R^{-q_{\rm in}}$ for $r>R_{\rm br}$ and $E(r)\sim R^{-q_{\rm out}}$ for
$r<R_{\rm br}$, where E(r), $q_{\rm in}$, $q_{\rm out}$ and $R_{\rm br}$ are
the emissivity, inner emissivity index, outer emissivity index, and break
radius respectively. The other free parameters in this model are the
ionization parameter ($\xi$), iron abundance ($A_{\rm Fe}$) and inclination
angle ($i$).

We started our analysis with the RelxillCp model, with the final model reading
in XSPEC as TBabs*(diksbb+RelxillCp). The primary emission for this model is
given by the Comptonized model Nthcomp (Zdziarski et al., 1996; Życki et al.,
1999). To verify that Relxill does indeed present an advantage over our non-
relativistic models, we compared Nthcomp with no additional Gaussian
components to model the Fe/Ni emission lines to the RelxillCp model through an
F-test, returning an F-statistic of $\sim 15$ (probability $3.1\times
10^{-25}$). This confirms a statistical need to include modeling of reflected
emission in the NuSTAR spectrum. RelxillCp directly estimates the coronal
properties in terms of the $\Gamma$ and $kT_{\rm e}$. During fitting, we
linked the seed photon temperature ($T_{\rm S}$) with the inner disc
temperature ($T_{\rm in}$). We fixed the outer radius of the disc at $R_{\rm
out}=1000$ $r_{g}$.

Analysis with RelxillCp returned a good fit with $\chi^{2}=916$ for 952 dof.
We obtained an inner temperature of $T_{\rm in}=1.2358_{-0.0007}^{+0.0007}$
keV, photon index $\Gamma=2.381_{-0.003}^{+0.003}$, iron abundance $A_{\rm
Fe}=0.47_{-0.06}^{+0.06}~{}A_{\odot}$, and ionization parameter
$\log(\xi)=3.39_{-0.04}^{+0.04}$. The ISCO was extended up to $R_{\rm
in}=2.2_{-0.6}^{+0.4}$ $R_{\mathrm{ISCO}}$, with the break radius extending
out to $R_{\rm br}=14.3_{-0.6}^{+0.6}$ $r_{g}$. The inner emissivity index
appears steep with $q_{1}=8.6_{-0.4}^{+0.4}$, and we find a much flatter outer
emissivity index of $q_{2}=0.35_{-0.11}^{+0.10}$. We estimate a reflection
fraction of $R_{\rm refl}=0.67_{-0.06}^{+0.06}$ and a BH spin parameter of
$a^{*}=0.65_{-0.06}^{+0.04}$, with inclination angle $i=69.8_{-0.5}^{+0.5}$
degrees.

While the RelxillCp model does not assume any particular coronal geometry, the
RelxillLp and RelxillLpCp flavors of the Relxill family assume a lamp-post
geometry where the corona is assumed to be a compact source located above the
BH (García & Kallman, 2010; Dauser et al., 2016). The incident primary
emission is given by either Cutoffpl (RelxillLp) or Nthcomp (RelxillLpCp). The
height of corona ($h$) is an input parameter in this model.

Analysis with the RelxillLp and RelxillLpCp models both returned good fits
with $\chi^{2}/{\rm dof}=940/954$ and $\chi^{2}/{\rm dof}=933/954$
respectively. Both returned similar inner temperatures of $T_{\rm
in}=1.220_{-0.004}^{+0.004}$ keV and $T_{\rm in}=1.211_{-0.005}^{+0.005}$ keV
respectively. RelxillLp returned a photon index of
$\Gamma=2.421_{-0.003}^{+0.003}$ while RelxillLpCp returned a slightly harder
photon index of $\Gamma=2.244_{-0.002}^{+0.002}$. RelxillLp also returned a
reflection fraction of $R_{\rm refl}=0.76_{-0.04}^{+0.04}$, almost two times
more than the $R_{\rm refl}=0.470_{-0.014}^{+0.014}$ given by RelxillLpCp.
Both models returned similar values of the coronal height, with
$h=3.59_{-0.11}^{+0.15}~{}r_{g}$ for RelxillLp and
$h=5.484_{-0.009}^{+0.009}~{}r_{g}$ for RelxillLpCp. Otherwise, RelxillLp
returned an iron abundance $A_{\rm Fe}=1.50_{-0.12}^{+0.12}~{}A_{\odot}$,
ionization parameter $\log(\xi)=4.48_{-0.05}^{+0.05}$, $R_{\rm
in}=2.5_{-0.5}^{+0.5}~{}R_{\mathrm{ISCO}}$, inclination angle $i=58_{-2}^{+2}$
degrees, and spin parameter $a^{*}=0.70_{-0.31}^{+0.15}$. RelxillLpCp returned
an iron abundance $A_{\rm Fe}=0.44_{-0.04}^{+0.04}~{}A_{\odot}$, ionization
parameter $\log(\xi)=3.01_{-0.02}^{+0.02}$, $R_{\rm
in}=1.7_{-0.5}^{+1.4}~{}R_{\mathrm{ISCO}}$, inclination angle
$i=37_{-3}^{+3}~{}^{\circ}$, and spin parameter $a^{*}=0.63_{-0.07}^{+0.07}$.
Detailed results of all spectral fits for this section are presented in the
last three columns of Table 4.

###  3.4 Error Estimation

While fitting the NuSTAR spectrum, we found that some of the parameters were
degenerate. To remove this degeneracy and find the global minimum, we employed
Markov Chain Monte Carlo (MCMC) in
XSPEC444https://heasarc.gsfc.nasa.gov/xanadu/xspec/manual/node43.html, which
constrains the uncertainty range of the parameters. We used the RelxillLp
model to estimate the errors using MCMC, as the model provides a physical
picture of the corona. As a caveat, it should be noted that the coronal
geometry influences the emissivity profile of the reflection spectrum. Thus,
the parameters could modify if coronal geometry varies. We used 1,000,000
steps with 8 walkers using the Goodman-Weare algorithm. We chose to discard
the first 10,000 steps, i.e. the ‘transient’ or ‘burn-in’ phase. The posterior
distribution of the RelxillLp model fitted parameters are plotted in Figure 6,
with errors quoted at the 1 $\sigma$.

Figure 6: Posterior distribution of the spectral parameters obtained from the
MCMC analysis with the RelxillLp model. Plotting was performed using corner
(Foreman-Mackey, 2016). Central dashed lines correspond to the peak values
whereas $1\sigma$ confidence levels are represented by dashed lines on either
side. Table 4: NuSTAR Spectral Analysis Results

| Powerlaw | Cutoffpl | Nthcomp | RelxillCp | RelxillLp | RelxillLpCp
---|---|---|---|---|---|---
$n_{\mathrm{H}}$ ($10^{-22}\mathrm{cm^{-2}}$) | $4.63_{-0.28}^{+0.28}$ | $4.32_{-0.23}^{+0.27}$ | $3.27_{-0.35}^{+0.91}$ | $4.05_{-0.03}^{+0.03}$ | $4.12_{-0.09}^{+0.09}$ | $4.32_{-0.11}^{+0.11}$
$T_{\mathrm{in}}$ (keV) | $1.185_{-0.010}^{+0.015}$ | $1.16_{-0.03}^{+0.03}$ | $1.149_{-0.009}^{+0.006}$ | $1.2358_{-0.0007}^{+0.0007}$ | $1.220_{-0.004}^{+0.004}$ | $1.211_{-0.005}^{+0.005}$
$\Gamma$ | $2.46_{-0.02}^{+0.02}$ | $2.24_{-0.08}^{+0.08}$ | $2.36_{-0.01}^{+0.02}$ | $2.381_{-0.003}^{+0.003}$ | $2.421_{-0.004}^{+0.004}$ | $2.244_{-0.002}^{+0.002}$
$E_{\mathrm{Fe}}$ (keV) | $6.48_{-0.02}^{+0.02}$ | $6.27_{-0.13}^{+0.16}$ | $6.40_{-0.02}^{+0.12}$ | - | - | -
$\sigma_{\mathrm{Fe}}$ (keV) | $0.50_{-0.05}^{+0.05}$ | $0.6_{-0.2}^{+0.3}$ | $0.62_{-0.07}^{+0.10}$ | - | - | -
normFe ($10^{-3}$) | $3.5_{-0.8}^{+0.4}$ | $4.6_{-0.2}^{+5.3}$ | $6.0_{-0.8}^{+1.2}$ | - | - | -
$E_{\mathrm{Ni}}$ (keV) | $7.99_{-0.03}^{+0.04}$ | $8.0_{-0.2}^{+0.3}$ | $8.1_{-0.1}^{+0.2}$ | - | - | -
$\sigma_{\mathrm{Ni}}$ (keV) | $0.56_{-0.06}^{+0.07}$ | $0.5_{-0.2}^{+0.3}$ | $0.48_{-0.18}^{+0.12}$ | - | - | -
normNi ($10^{-3}$) | $1.7_{-0.6}^{+0.4}$ | $1.3_{-1.3}^{+0.9}$ | $1.7_{-1.1}^{+1.0}$ | - | - | -
$kT_{e}$ (keV) | - | $94_{-39}^{+93}$ | $132_{-123}^{+476}$ | $400^{f}$ | $400^{f}$ | $400^{f}$
$R_{\mathrm{refl}}$ | - | - | - | $0.67_{-0.06}^{+0.06}$ | $0.76_{-0.04}^{+0.04}$ | $0.470_{-0.014}^{+0.014}$
$A_{\mathrm{Fe}}$ ($A_{\sun}$) | - | - | - | $0.47_{-0.06}^{+0.06}$ | $1.50_{-0.12}^{+0.12}$ | $0.44_{-0.04}^{+0.04}$
$a^{*}$ | - | - | - | $0.65_{-0.06}^{+0.04}$ | $0.70_{-0.31}^{+0.15}$ | $0.63_{-0.07}^{+0.07}$
$i$ (∘) | - | - | - | $69.8_{-0.5}^{+0.5}$ | $58_{-2}^{+2}$ | $37_{-3}^{+3}$
$R_{\mathrm{in}}$ ($R_{\mathrm{ISCO}}$) | - | - | - | $2.2_{-0.6}^{+0.4}$ | $2.5_{-0.5}^{+0.5}$ | $1.7_{-0.5}^{+1.4}$
$R_{\mathrm{br}}/h$ ($r_{g}$) | - | - | - | $14.3_{-0.6}^{+0.6}$ | $3.59_{-0.11}^{+0.15}$ | $5.484_{-0.009}^{+0.009}$
$\log\xi$ | - | - | - | $3.39_{-0.04}^{+0.04}$ | $4.48_{-0.05}^{+0.05}$ | $3.01_{-0.02}^{+0.02}$
$q_{1}$ | - | - | - | $8.61_{-0.41}^{+0.37}$ | - | -
$q_{2}$ | - | - | - | $0.35_{-0.11}^{+0.10}$ | - | -
normdiskbb | $305_{-21}^{+14}$ | $379_{-31}^{+33}$ | $464_{-16}^{+21}$ | $276_{-6}^{+6}$ | $265_{-6}^{+6}$ | $299_{-8}^{+8}$
norm | $4.5_{-0.2}^{+0.2}$ | $2.9_{-0.5}^{+0.7}$ | $0.130_{-0.001}^{+0.002}$ | $0.0315_{-0.0008}^{+0.0008}$ | $0.0415_{-0.0002}^{+0.0003}$ | $0.04694_{-0.00008}^{+0.00008}$
$\chi^{2}$/dof | 950/956 | 907/955 | 908/955 | 916/952 | 940/954 | 933/954
$\tilde{\chi}^{2}$ | 0.9936 | 0.9502 | 0.9504 | 0.9618 | 0.9854 | 0.9775
$F_{2-10}$ ($10^{-9}$ ergs cm-2 s-1) | $9.656_{-0.01}^{+0.01}$ | $9.652_{-0.010}^{+0.010}$ | $9.804_{-0.010}^{+0.010}$ | $9.824_{-0.010}^{+0.010}$ | $9.832_{-0.010}^{+0.010}$ | $9.807_{-0.010}^{+0.010}$

f indicates parameters that were fixed during analysis.

‘norm’ refers to the normalization of the primary model component.

##  4 Concluding Remarks

We analyzed the Swift J1728.9–3613 data obtained from MAXI/GSC in the form of
one-day binned lightcurves, two NICER/XTI observations taken during the rising
phase of the outburst, and one NuSTAR/FPMB observation taken near the peak of
the outburst. Spectral analysis was performed on the NICER observations in the
$0.3-10$ keV range in order to understand the parameters of the system at the
beginning of the outburst. We then examined the NuSTAR observation in the
$3-78$ keV range to more closely probe the accretion process and to measure
parameters such as spin, inner disc radius, and inclination angle.

The long term lightcurve analysis revealed that, in total, Swift J1728.9–3613
devoured roughly 90% of the mass of Mars and released $\sim 5.2\times 10^{44}$
ergs of energy in $2-20$ keV energy band. We also find that lightcurves in
various energy bands correlate with each other, where softer photons are
delayed compared to their harder counterparts. The maximum delay was observed
between the $2-4$ and $10-20$ keV photons with a mean lag of
$\tilde{\tau}=(8.4\pm 1.9)$ days. Production of such a delay is likely related
to the viscous delay of matter spiraling through the accretion disc during the
outburst. This is supported by the presence of a strong multi-color disc
component in the NICER spectra during the rising phase, with an increase
normalization between the first and second observations.

Various models were used to understand the key parameters involved in the
accretion process. In resolving the spectrum of Swift J1728.9–3613 during the
primary NuSTAR observation we revealed a sudden decrease in hard emission,
with power-law normalization decreasing from $3.3_{-0.5}^{+0.6}$ to
$1.54_{-0.07}^{+0.07}$ ph keV-1 cm-2. Such a reduction could have have
resulted from large amounts of ejecta being carried away from the corona in a
relatively short amount of time, so-called blobby jets. These ejections can
appear to move at superluminal speeds, and have been definitively reported on
in several other outbursting black holes (e.g. Fender et al., 1998; Hjellming
& Rupen, 1995). Turning to the time-averaged spectrum, we obtained a hydrogen
column density of $4.63^{+0.28}_{-0.28}\times 10^{-22}$ cm-2 using the WILM
(Wilms et al., 2000) abundance from the Tbabs*{Diskbb+Powerlaw} model. Using
the ANGR (Anders & Grevesse, 1989) abundance, we found a lower
($2.99^{+0.11}_{-0.11}\times 10^{-22}$ cm-2) column density for the same
model. The disc temperature varied from 1.15 to 1.23 keV across all the
models, with Powerlaw yielding an inner temperature of
$T_{in}=1.18^{+0.015}_{-0.010}$ keV. From the Nthcomp model, we estimated a
coronal electron temperature of $kT_{e}=132^{+476}_{-123}$ keV. The coronal
temperature remains unconstrained from our time-averaged spectral analysis.
However, when we examined the hardness-resolved spectra using the Cutoffpl
model, we found the cutoff temperatures were better constrained, having values
$E_{\rm cut}=94_{-34}^{+78}$ keV in region i and $E_{\rm cut}=79_{-18}^{+28}$
keV in region ii. We found signatures of the Fe K$\alpha$ line around
$(6.48\pm 0.02)$ keV with an equivalent width of $53$ eV. Apart from the Fe
K$\alpha$ line, we also observed an Ni emission line around $(8.0\pm 0.3)$ keV
with an equivalent width of $60$ eV. This Ni line has previously been observed
in several BHXRBs and AGNs (Corliss & Sugar, 1981; Molendi et al., 2003;
Fukazawa et al., 2016; Medvedev et al., 2018). It is possible that the
$8.06_{-0.28}^{+0.10}$ keV line is observed due to the blue-shifted wing of
the Fe K$\alpha$. However, if this were the case, we would also expect to
observe a corresponding redshifted wing at roughly 6.4 - (8 - 6.4) = 4.8 keV
(Cui et al., 2000). Given that we detect strong signatures of the emission
line at 6.4 keV in the first place and the 8 keV Ni line has been previously
identified, we believe that Ni emission line could be the more plausible
explanation. Future generation satellites, such as Colibrì, (Heyl et al.,
2019; Caiazzo et al., 2019) would be capable of resolving these lines more
accurately and locating line emitting regions.

We applied phenomenological models, like Cutoffpl to fit the NuSTAR data and
estimate the luminosity of the accretion disc during that observation. Using
$L_{2-10}=4\pi D^{2}F_{2-10}$, assuming a distance of 10 kpc, we find
$L_{2-10}=1.09\times 10^{38}~{}\rm{erg~{}s^{-1}}$. Assuming a mass-to-
radiation conversion efficiency of $\eta=0.1$ based on our measured spin value
(Reynolds, 2021), we find a mass accretion rate of $\dot{M}=L/\eta
c^{2}=1.2\times 10^{18}~{}\rm{g~{}s^{-1}}$. Using the lower mass limit of
Swift J1728.9–3613 reported by Saha et al. (2022), $\sim 5M_{\odot}$
accounting for spin, we estimate $L/L_{\rm Edd}\sim 0.17$ during the
observation. Combining this mass with the MAXI/GSC light curves, we find a
maximum accretion rate of $L/L_{\rm Edd}\sim 0.28$ during the peak of the
outburst.

Three models from the Relxill family were selected for analysis with the
NuSTAR data in order to estimate the more complex properties of Swift
J1728.9–3613, namely RelxillCp, RelxillLp, and RelxillLpCp. Since RelxillCp
assumes that the irradiation of the disc behaves like a broken power-law, this
model estimates a steep inner emissivity index of $q_{1}=8.61_{-0.41}^{+0.37}$
that becomes much flatter ($q_{2}=0.35_{-0.11}^{+0.10}$) at the break radius
$R_{\rm br}=14.3_{-0.6}^{+0.6}~{}r_{g}$. Otherwise, the three models estimate
spin parameters of $0.65_{-0.06}^{+0.04}$, $0.70_{-0.31}^{+0.15}$, and
$0.63_{-0.07}^{+0.07}$; inner disc radii of $2.2_{-0.6}^{+0.4}$
$R_{\mathrm{ISCO}}$, $2.5_{-0.5}^{+0.5}~{}R_{\mathrm{ISCO}}$, and
$1.7_{-0.5}^{+1.4}~{}R_{\mathrm{ISCO}}$; and inclination angles of
$69.8_{-0.5}^{+0.5}$ degrees, $58_{-2}^{+2}$ degrees, and $37_{-3}^{+3}$
degrees, respectively.

## Acknowledgements

We acknowledge the anonymous reviewer for the helpful comments and suggestions
which improved the paper. SH, AC, SSH and JH are supported by the Canadian
Space Agency (CSA) and the Natural Sciences and Engineering Research Council
of Canada (NSERC) through the Discovery Grants and the Canada Research Chairs
programs. AJ acknowledge the support of the grant from the Ministry of Science
and Technology of Taiwan with the grand number MOST 110-2811-M-007-500 and
MOST 111-2811-M-007-002. This research made use of the NuSTAR Data Analysis
Software (NuSTARDAS) jointly developed by the ASI Space Science Data Center
(ASSDC, Italy) and the California Institute of Technology (Caltech, USA).

## Data Availability

We used publicly available archival data of MAXI, NICER and NuSTAR
observatories for this work. All the models used in this work, are publicly
available. Appropriate links are provided in the text.

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