# Understanding the X-ray spectral curvature of Mkn 421 using broadband
AstroSat observations

Jyotishree Hota,1 Zahir Shah,2,3 Rukaiya Khatoon,4 Ranjeev Misra,2 Ananta C.
Pradhan,1 and Rupjyoti Gogoi,4
1 Department of Physics and Astronomy, National Institute of Technology
Rourkela, Odisha ,India
2 Inter-University Center for Astronomy and Astrophysics, Post Bag 4,
Ganeshkhind, Pune - 411007, India
3 Department of Physics, Central University of Kashmir, India
4 Tezpur University,Napaam-784028, Assam, India E-mail:
hotajyoti<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We present a time-resolved X-ray spectral study of the high energy peaked
blazar Mkn 421 using simultaneous broadband observations from the LAXPC and
SXT instruments on-board AstroSat. The $\sim 400$ ksec long observation taken
during 3–8 January, 2017 was divided into segments of 10 ksecs. Each segment
was fitted using synchrotron emission from particles whose energy distribution
was represented by a log-parabola model. We also considered particle energy
distribution models where (i) the radiative cooling leads to a maximum energy
($\xi_{max}$ model), (ii) the system has energy dependent diffusion (EDD) and
(iii) has energy dependent acceleration (EDA). We found that all these models
describe the spectra, although the EDD and EDA models were marginally better.
Time resolved spectral analysis allowed for studying the correlation between
the spectral parameters for different models. In the simplest and direct
approach, the observed correlations are not compatible with the predictions of
the $\xi_{max}$ model. While the EDD and EDA models do predict the
correlations, the values of the inferred physical parameters are not
compatible with the model assumptions. Thus, we show that spectrally
degenerate models, can be distinguished based on spectral parameter
correlations (especially those between the model normalization and spectral
shape ones) making time-resolved spectroscopy a powerful tool to probe the
nature of these systems.

###### keywords:

galaxies: active – BL Lacertae objects: general – BL Lacertae objects:
individual: Mkn 421 – acceleration of particles – diffusion – X-rays: galaxies

††pubyear: 2021††pagerange: Understanding the X-ray spectral curvature of Mkn
421 using broadband AstroSat observations–Understanding the X-ray spectral
curvature of Mkn 421 using broadband AstroSat observations

##  1 Introduction

Blazars belong to a subclass of radio loud active galactic nuclei (AGN) with a
relativistic jet directed towards the observer (Urry & Padovani, 1995;
Blandford et al., 2019). They are characterized by non-thermal broadband
spectra, rapid variability, strong radio and optical polarization (Sambruna,
2000; Fan et al., 2008). The spectral energy distribution (SED) of a blazar
extends from radio to $\gamma$-ray energies (Fossati et al., 1998) and is
generally characterized by a double hump structure with the low energy hump
peaking in the optical to X-ray energies and the high energy hump with peak
located in $\gamma$-ray band. The high-energy emission can be produced either
through leptonic models or through hadronic models. Based on leptonic model,
there are two types of emission processes, that have been used to explain the
broadband SED viz. synchrotron process and the inverse-Compton (IC) process.
The low energy hump is produced by relativistic electrons interacting with
ambient jet magnetic field through synchrotron process, where as the high
energy hump is mainly explained by IC scattering of low-energy photons (Urry &
Mushotzky, 1982; Ghisellini et al., 1985; Begelman et al., 1987; Blandford &
Levinson, 1995; Bloom & Marscher, 1996; Sokolov et al., 2004). The seed
photons for IC scattering in leptonic model can be either synchrotron photons
(synchrotron self-Compton, (SSC): Jones et al. (1974); Maraschi et al. (1992);
Ghisellini et al. (1993)) or photons external to jet (external Compton (EC):
Dermer et al. (1992); Sikora et al. (1994); Błażejowski et al. (2000); Shah et
al. (2017)). Alternatively, in the hadronic processes, the the higher
frequency emission can also be initiated by relativistic protons such as the
proton synchrotron process and pion production. (Mannheim & Biermann, 1992;
Mücke & Protheroe, 2001).

Blazars are broadly classified into Bl Lacs and FSRQs. Additionally, based on
the location of the peak of synchrotron component, BL Lacs are further
classified into low-energy peaked BL Lacs (LBL), ($\nu_{p}$ < $10^{14}$ Hz),
Intermediate peaked BL lacs (IBL) ($10^{14}$ < $\nu_{p}$ < $10^{15}$) and
high-energy peaked BL Lacs (HBL) ($\nu_{p}$ > $10^{15}$) (Padovani & Giommi,
1995).

The Blazar Mkn 421 is a well-known nearby HBL source, which is located at a
redshift of z = 0.031 (Punch et al., 1992). It shows rapid variability in a
time scale of few minutes to days. Mkn 421 has been observed across the
electromagnetic spectrum by various advanced observatories in order to explore
its broadband SED characteristics and underlying physical processes associated
(Krawczynski et al., 2001; Brinkmann et al., 2005; Albert et al., 2007; Lichti
et al., 2008; Fossati et al., 2008; Acciari et al., 2009; Tramacere et al.,
2009; Horan et al., 2009; Ushio et al., 2009, 2010; Aleksić et al., 2010;
Isobe et al., 2010; Aleksić et al., 2012; Shukla, A. et al., 2012; Bartoli et
al., 2016; Sinha et al., 2016).

Mkn 421 is an extremely bright source and displays a strong variability in all
wavelengths from radio to very high energy gamma rays on multiple time scales
(Wagner & Witzel, 1995; Falomo et al., 2014; Goyal, 2020). Its long-term
multi-wavelength study gives clues on the emission mechanisms, structure and
physical parameters inside the relativistic jets. Several previous studies
have reported correlated variability between the X-ray and TeV $\gamma$-ray
emission from Mkn 421 during flaring period (Macomb et al., 1995; Buckley et
al., 1996; Fossati et al., 2004; Albert et al., 2007; Fossati et al., 2008;
Donnarumma et al., 2009; Abdo et al., 2011; Acciari et al., 2011; Cao & Wang,
2013; Aleksić et al., 2015b) as well as during low activity period (Aleksić et
al., 2015a; Baloković et al., 2016). Where as no significant correlation
between optical/UV and X-ray/Tev emission has been reported (Macomb et al.,
1995; Albert et al., 2007; Cao & Wang, 2013).

In its flaring state the X-ray spectra of Mkn 421 shows significant curvature
(Fossati et al., 2000; Massaro et al., 2004a), and the log-parabola model has
been shown to provide the satisfactorily fit to such spectral shapes (Massaro
et al., 2004a, b, 2006; Donnarumma et al., 2009; Sinha et al., 2015). The
broad band X-ray spectrum of Mkn 421 was also analyzed by Ushio et al. (2010)
with the parametric forward-fitting SYNCHROTRON model incorporated in XSPEC.
The authors showed that the synchrotron emission from exponentially moderated
power-law electron distribution provided a good fit to the X-ray spectrum
(Ushio et al., 2010). Detailed correlation study among the best fit model
parameters has been carried out in different studies (Tramacere et al., 2007a;
Massaro et al., 2008; Tramacere et al., 2009, 2011; Sinha et al., 2015).
Further, Goswami et al. (2020) showed that an analytical model obtained by
considering the energy dependent electron escape time scale, provides a good
fit to the hard curved hard X-ray spectra of Mkn 421.

Recently, Chatterjee et al. (2018) have analyzed the X-ray light curves of Mkn
421 by using the observations made with SXT and LAXPC telescopes on board the
AstroSat satellite. They showed that the X-ray power spectral density (PSD) is
a piece-wise power-law with a break. The strong variability observed during
this long observation also makes it an ideal one to study spectral variability
as a function of time. In this work, we undertake a time resolved spectral
study using different particle energy distribution models. The motivation is
to find out how well these models represent the observed spectra, obtain
correlations between spectral parameters and use these correlations to check
for self consistency of the model.

The paper is organised as follows: in section 2, we give details of the
AstroSat observation and describe the data analysis procedure. In section 3,
we present details of the spectral analysis of the observed data. In section
section 4 the results are summarized and their implications are discussed.

##  2 AstroSat X-RAY OBSERVATIONS AND ANALYSIS

AstroSat is India’s first multi-wavelength space astronomy satellite. It has
five scientific instruments onboard which carry observation of a cosmic source
over a wide range of energies from UV to hard X-ray (Agrawal, 2006; Singh et
al., 2014; Rao et al., 2016). The onboard instruments like Ultra-Violet
Imaging Telescope (UVIT (130–300 nm ); Tandon et al. (2017a, b)), Soft X-ray
focusing Telescope (SXT (0.3–8.0 keV); Singh et al. (2017b, 2016)),

Large Area X-ray Proportional Counter (LAXPC (3 – 80 keV); Yadav et al.
(2016b)) and Cadmium Zinc Telluride Imager (CZTI (10 – 100 keV); Rao et al.
(2017)), allow for simultaneous observations in UV, soft X-rays and hard
X-rays. AstroSat has the unique capability of probing the strictly
simultaneous spectral curvature over broad energy ranges (X-ray and UV).
During 3 – 8 January, 2017, AstroSat carried a ToO observation of the flaring
activity of Mkn 421 for a total observation time of 409 ksec (Chatterjee et
al., 2018). The data of this observation is available on the AstroSat data
archive astrobrowse
111https://astrobrowse.issdc.gov.in/astroarchive/archive/Home.jsp. The
information of instrument-specific software/tools for processing the data is
provided in the ASSC website 222http://astrosat-ssc.iucaa.in/?q=data and
analysis. The details of the data reduction of SXT and LAXPC observations are
as follows.

###  2.1 SXT

SXT is a X-ray focusing telescope which operates in the 0.3 – 8.0 keV energy
band (Singh et al., 2017a). It is used for X-ray imaging and spectroscopy
purposes with angular resolution of 2′ and FOV of $\sim$ 40′ diameter. The SXT
observed Mkn 421 in the Photon Counting (PC) mode in 57 orbits during the
period 3 – 8 January, 2017. We processed the archived Level-1 data of each
orbit with the SXT pipeline (AS1SXTLevel2,version 1.4b
333http://www.tifr.res.in/∼astrosat$\\_$sxt/sxtpipeline.html) and obtained
cleaned Level-2 event file for each orbit. In order to avoid the time-
overlapping of events from the consecutive orbits, we merged the cleaned
Level-2 event files from all the orbits into a single cleaned event file by
using the standard julia script, developed by the instrument team. The science
products were generated from the merged event files using XSELECT (V2.4d)
package which is in-built in HEASOFT. A circular region of 15′ radius centered
at the source location was used to extract the source count rate, the 15′
source region was selected so that it includes more than 90$\%$ of the source
photons. The spectral analysis was carried out by using a background spectrum
“SkyBkg$\\_$comb$\\_$EL3p5$\\_$Cl$\\_$Rd16p0$\\_$v01.pha" provided by the SXT
POC team and the response matrix file (RMF)
(“sxt$\\_$pc$\\_$mat$\\_$g0to12.rmf") and an off-axis auxiliary response file
(ARF) generated using SXT ARF generation
tool444http://www.tifr.res.in/∼astrosat$\\_$sxt/dataanalysis.html. The grppha
tool was used to ensure a minimum of 40 counts per bin. Due to uncertainities
in the response at low energies, we restricted the spectra analysis to 0.5–7.0
keV. Further, we add a systematic uncertainty of 3% to the data in order to
account for uncertainties in response calibrations. Gain variation was taken
into account using the gain fit tool in Xspec with the slope fixed at one. The
best fit offset value for different segments was found to be typically $\sim
0.03$ keV.

###  2.2 LAXPC

The LAXPC is one of the primary instruments of AstroSat which consists of
three non-imaging proportional counters namely LAXPC10, LAXPC20, LAXPC30. Each
proportional counter are having an effective area of $\sim$ 2000 $cm^{2}$. The
LAXPC works in the energy range 3 – 80 keV with high time resolution ($\sim$
10 $\mu$s) (Yadav et al., 2016a; Antia et al., 2017; Agrawal et al., 2017;
Misra et al., 2017). We processed the Level-1 data of the source with the
LAXPCSOFT package which is based on Fortran codes and available at the
AstroSat Science Support Cell (ASSC) 555http://astrosat-ssc.iucaa.in. The
command laxpc_make_event were used to create the combined level-2 event file
and laxpc_make_stdgti were used to generate good time interval (gti) file in
which the intervals of South Atlantic Anomaly (SAA) and Earth occultation were
removed. Finally the tasks laxpc$\\_$make$\\_$lightcurve and
laxpc$\\_$make$\\_$spectra were used to generate the source lightcurve and
spectra respectively, from the gti file. The corrected light curve has not
been background subtracted. For background estimation we have used the faint
source method (Misra et al., 2021). The background subtraction has done
separately by using the standard FTool lcmath command. We limited the spectral
analysis to 4 - 18 keV as we found that the background completely dominated
the spectra at higher energies. Systematic uncertainty of 3% was included in
the spectral analysis. Only LAXPC 10 and LAXPC 20 were used for the analysis,
since LAXPC 30 had substantial gas leak by the time of the observation.

###  2.3 TIME VARIABILITY

Figure 1: Light curve of Mkn 421 in the energy range of 0.5–7.0 keV from SXT
(top panel) and 3.0-30.0 keV from LAXPC10 (middle panel) and LAXPC20 (bottom
panel).

The SXT light curve is generated for the time bin of 100 sec in the energy
range 0.5 – 7 keV (top panel of Figure 1) where as the LAXPC10 and LAXPC 20
light curve are obtained for the time bin of 100 sec in the energy range 3 –
30 keV (middle and bottom panels of Figure 1). The AstroSat observations of
Mkn 421 during 3 – 8 January, 2017 revealed that the source was in high flux
state and undergoing through strong variablity in soft and hard X-rays (see
Figure 1). The calculated variability amplitude ($F_{var}$) of the source for
SXT, LAXPC10, LAXPC20 using LCSTATS are 0.21772 $\pm$ 0.004, 0.30334 $\pm$
0.005, 0.37697 $\pm$ 0.009, respectively.

##  3  Time-resolved spectral analysis

The X-ray emission in Mkn 421 is known to be synchrotron emission produced by
the interaction of non-thermal relativistic electrons with an ambient magnetic
field. We assume that the emission arises from a spherical region of radius, R
which is filled with tangled magnetic field, B and relativistic isotropic
electron distribution, $n(\gamma)$. If we express the electron lorentz factor,
$\gamma$ in terms of $\xi$ such that $\xi=\gamma\sqrt{\mathbb{C}}$, where
$\mathbb{C}=1.36\times 10^{-11}\frac{\delta B}{1+z}$ with z being the redshift
of source and $\delta$ as jet Doppler factor, then the synchrotron flux
received by the observer at energy $\epsilon$ will be (Begelman et al., 1984),

$\displaystyle F_{\rm
syn}(\epsilon)=\frac{\delta^{3}(1+z)}{d_{L}^{2}}V\,\mathbb{A}\int_{\xi_{min}}^{\xi_{max}}f(\epsilon/\xi^{2})n(\xi)d\xi$
(1)

where, $d_{L}$ is the luminosity distance, V is the volume in the emission
region, $\mathbb{A}=\frac{\sqrt{3}\pi e^{3}B}{16m_{e}c^{2}\sqrt{\mathbb{C}}}$
and f(x) is the synchrotron emmisivity function (Rybicki & Lightman, 1986). We
solved Equation 1 numerically and included it as a local convolution model,
_$synconv\otimes n(\xi)$_ in XSPEC. This allows for the photon spectrum to be
modelled for any particle energy distribution $n(\xi)$ represented by any
XSPEC model. It is important to mention here that XSPEC “energy" variable in
the _$synconv\otimes n(\xi)$_ model should be interpreted as
$\xi=\gamma\sqrt{\mathbb{C}}$. Thus, any parameter of the model, such as the
pivot energy of the logparabola model $\xi_{piv}$, would be related to
$\gamma$ as $\gamma_{piv}=\xi_{piv}/\sqrt{\mathbb{C}}$. In order to account
for Galactic absorption, we used the TBabs model (Wilms et al., 2000)
available in the XSPEC with the equivalent hydrogen column density ($N_{H}$)
fixed at $1.92\times 10^{20}$cm-2 as obtained from the online
tool666https://heasarc.gsfc.nasa.gov/cgi-bin/Tools/w3nh/w3nh.pl which is
developed by LAB survey group (Kalberla et al., 2005).

The broadband X-ray spectrum of Mkn 421 is generally characterized by a smooth
broad curvature ((Fossati et al., 2000; Tanihata et al., 2004; Massaro et al.,
2004a; Tramacere et al., 2007a; Tramacere et al., 2007b; Tramacere et al.,
2009)), which can be obtained by considering the input particle density to be
curved. There are several possible curved models which may give acceptable
fits to the curved spectrum. Here, we fitted the observed X-ray spectrum in
each 10 ksec time segment by synchrotron emission from a particle energy
distribution described by empirical models viz power law (PL), logparabola
(LP), broken power law (BPL), and power law with exponential cutoff (CPL). We
have also fitted the observed spectrum with synchrotron emission from physical
models such as power-law with a maximum energy due to radiative cooling,
energy dependent diffusion (EDD), and energy dependent acceleration (EDA). Our
results show that the empirical models (LP, BPL, CPL) fit the observed
spectrum well (see top panel of Figure 2), while power law model gives a
poorer fit as compared to other models (bottom panel of Figure 2). The
motivation of our work is to obtain a more consistent physical scenario
responsible for the emission, hence, we have discussed only physical models in
detail.

Figure 2:  Comparison of reduced-$\chi^{2}$ of synchrotron convolved
logparabola model with the reduced $\chi^{2}$ of synchrotron convolved broken
power law model (top left panel), the reduced $\chi^{2}$ of synchrotron
convolved Power law with an exponential cutoff model (top right panel) and the
reduced $\chi^{2}$ of synchrotron convolved PL model (bottom panel). In each
panel, the solid line is identity line.

####  3.0.1 Logparabola model

We first fitted the broadband X-ray spectra using _$synconv\otimes n(\xi)$_ ,
where $n(\xi)$ is given by a logparabola form i.e.,

$n(\xi)=K({{\xi}/{\xi_{r}}})^{-{\alpha}-{\beta}{\log({{\xi}/{\xi_{r}}})}}\\\ $
(2)

Here, ${\alpha}$ is the particle spectral index at the reference energy,
$\xi^{2}=\xi_{r}^{2}$, ${\beta}$ is the curvature parameter and K is the
normalization of the particle density at $\xi=\xi_{r}$. While performing the
spectral fit in each time bin with _$synconv\otimes n(\xi)$_ model (Equation
1), we have fixed the $\xi_{r}^{2}$ at 1 keV, and the three parameters, viz.
${\alpha}$, ${\beta}$ and norm $\mathbb{N}$ were kept free. Here $\mathbb{N}$
is defined as

$\mathbb{N}=\frac{\delta^{3}(1+z)}{d_{L}^{2}}V\,\mathbb{A}K$ (3)

The model provides reasonable reduced-$\chi^{2}$ for all the time segments, as
shown in top left panel of Figure 3 where the reduced $\chi^{2}$ and the best-
fit parameters are plotted as a function of time. The rest of the panels show
the correlation between the different parameters. In Table 1 (left), we
provide the best-fit parameters for all the time segment and the reduced
$\chi^{2}$ value using logparabola model. We note that $\beta$ and $\alpha$
are clearly inversely correlated. The value of $\alpha$ decreases slowly with
normalization till $\mathbb{N}\sim 2$ and then decreases more rapidly, while
for $\beta$ there is a slow increase followed by a rapid increase. A strong
anti-correlation is obtained between $\alpha$ and flux, $F_{0.5-18.0\ kev}$
with r-value and p-value as $\sim-0.89$ and $1.6\times 10^{-14}$,
respectively, which is hardening when brightening features commonly seen in
blazars (left bottom panel of Figure 3) . Whereas a positive correlation is
observed between $\beta$ and flux, $F_{0.5-18.0\ kev}$ with r-value and
p-value $\sim 0.64$ and $1.6\times 10^{-5}$, respectively (right bottom panel
of Figure 3).

Figure 3: Top left panel: Best fit results obtained by fitting the wideband
spectra in each time segment with synchrotron convolved logparabola model. Top
right panel: scatter plot between best fitted $\alpha$ and $\beta$ values,
middle left panel: scatter plot between best fitted $\mathbb{N}$ and $\alpha$
values, middle right panel: scatter plot between best fitted $\mathbb{N}$ and
$\beta$ values, bottom left panel: scatter plot between flux, $F_{0.5-18.0\
kev}$ and best fitted $\alpha$ values, bottom right panel: scatter plot
between flux, $F_{0.5-18.0\ kev}$ and best fitted $\beta$ values.

####  3.0.2 Particle distribution with a maximum energy

We next consider the case of particles getting accelerated by a shock and then
subsequently these accelerated particles lose energy by emitting radiation
through the synchrotron process. Since the synchrotron radiative loss is
proportional to energy squared, it is more significant for higher energies,
leading to a maximum energy for the distribution. The steady state particle
density for such a scenario is determined by

$\frac{\partial}{\partial\gamma}\left[\left(\frac{\gamma}{\tau_{acc}}-\beta_{s}\gamma^{2}\right)n_{a}\right]+\frac{n_{a}}{\tau_{esc}}=Q\delta(\gamma-\gamma_{0})$
(4)

where $\tau_{acc}$ and $\tau_{esc}$ are the acceleration and escape time-
scales respectively, and the cooling term contains
$\beta_{s}=\frac{4}{3}\frac{\sigma_{T}B^{2}}{8\pi m_{e}c}$, with B as the
magnetic field, $\sigma_{T}$ the Thomson cross section, and $m_{e}$ the
electron mass. It is assumed that there is mono-energetic injection of
particles at energy $\gamma_{0}$. The solution to the above equation (after
transforming $\gamma$ to $\xi=\sqrt{\mathbb{C}}\gamma$) is (Kardashev, 1962),

$n(\xi)=K\xi^{-p}\left(1-\frac{\xi}{\xi_{max}}\right)^{(p-2)}$ (5)

where $K=Q_{0}\tau_{a}\gamma_{0}^{p-1}\mathbb{C}^{p/2}$,
$p=\tau_{acc}/\tau_{esc}+1$ is the particle spectral index,
$\xi_{max}=\gamma_{max}\sqrt{\mathbb{C}}$ and
$\gamma_{max}=1/(\beta_{s}\tau_{acc})$.

Using Equation 5 as an input particle density, we again perform the
$\chi^{2}$-fit of the combined SXT, LAXPC10 and LAXPC20 spectrum with
_$synconv\otimes n(\xi)$_ model (here after $\xi_{max}$ model) in all the 39
time bins. In each time bin the fit was carried with three free parameters,
$\xi_{max}$, $p$, and the normalization $\mathbb{N}$ defined as

$\mathbb{N}=\frac{\delta^{3}(1+z)}{d_{L}^{2}}V\,\mathbb{A}Q_{0}\tau_{acc}{\gamma_{0}^{p-1}}\mathbb{C}^{p/2}$
(6)

We find that this model provides an acceptable fit to all time segments,
although the reduced $\chi^{2}$ is typically larger than that for the log-
parabola model. The reduced $\chi^{2}$ for the two models are plotted against
each other in the top left panel of Figure 4.

In Table 1 (right), we provide the best-fit parameters for all the time
segment and the reduced $\chi^{2}$ value using $\xi_{max}$ model. A strong
anti-correlation is obtained between $p$ and flux, $F_{0.5-18.0\ kev}$ with
r-value and p-value as $\sim-0.89$ and 9.62 $\times 10^{-15}$, respectively
(bottom left panel of Figure 5). Also a moderate anti-correlation is obtained
between $\xi_{max}$ and flux, $F_{0.5-18.0\ kev}$ with r-value and p-value as
$\sim-0.68$ and 1.4 $\times 10^{-6}$, respectively (right bottom panel of
Figure 5). The correlation plots between best fitted parameters are shown in
Figure 5. The correlation between the fitted parameters is similar to the
results obtained for the log-parabola model. $\xi_{max}$ and the index $p$ are
correlated to each other and show little variation with normalization till
$\mathbb{N}\sim 2$ and rapidly decrease for higher values. The tight
correlation between $p$ and $\xi_{max}$ implies that variation in the index is
perhaps the primary driver of the variability.

In the framework of this model, variation of the index, $p$ can only occur if
$\tau_{esc}$ or $\tau_{acc}$ change. Now, there is no direct dependence of
$\xi_{max}$ on $\tau_{esc}$, while $\xi_{max}$ is inversely proportional to
$\tau_{acc}$ which in turn is directly proportional to $p$. Thus, variation in
$\tau_{acc}$ should lead to $\xi_{max}$ being inversely proportional to $p$
which is completely contrary to the observed positive correlation between best
fit values of $p$ and $\xi_{max}$ (Figure 5). Thus, although the model
provides reasonable spectral fitting to the data, the correlation between $p$
and $\xi_{max}$ can be used to rule it out. Another issue is the variation of
the normalization with index,
$\mathbb{N}\propto(\gamma_{0}^{2}\mathbb{C})^{p/2}$ (Equation 6), which
implies that a change in index $\Delta p$ should result in a corresponding
change in normalization $\Delta\mathbb{N}/\mathbb{N}=log(\xi_{0}^{2})\Delta
p/2$, where $\xi_{o}^{2}\equiv\gamma_{0}^{2}\mathbb{C}$. Now since
$\xi_{o}^{2}$ corresponds to the observed energy of the photon (in keV)
produced by electrons with energy $\gamma_{0}$, it should be a significantly
small number since $\gamma_{o}$ is small compared to the $\gamma$ required to
produce X-ray photons. Thus, $|log(\xi_{0}^{2})|>>1$ and there should be a
significant variation of the normalization when the index changes. However,
the spectral fit parameters show that when $p$ changes from $2.2$ to $2.6$,
the normalization varies only by a few percentage (top right panel of Figure
5). As pointed out by Shah et al. (2021), the power-law solution to Equation 4
(for $\gamma<\gamma_{max}$), approximately pivots around $\gamma_{0}$ when $p$
is varied, leading to large variation in the number density at
$\gamma(>>\gamma_{0}$), which is typically not observed.

The model can be perhaps made compatible with the data if it is assumed that
the acceleration time-scale depends on the magnetic field such that
$\tau_{acc}\propto B^{-n}$ or $B\propto(p-1)^{-1/n}$. In that case,
$\xi_{max}=\sqrt{\mathbb{C}}/(\beta_{s}\tau_{acc})\propto
B^{n-3/2}\propto(p-1)^{-(n-3/2)/n}$. Thus, $\xi_{max}$ would be positively
correlated with $p$ if $n<3/2$ and will be linearly correlated if $n\sim 3/4$.
However, this will not address the issue of the normalization being
insensitive the variations of the spectral index.

Figure 4: Comparison of reduced-$\chi^{2}$ of synchrotron convolved
logparabola model with: top left panel: the reduced $\chi^{2}$ of $\xi_{max}$
model, top right panel: the reduced $\chi^{2}$ of synchrotron convolved EDD
model and bottom left panel: the reduced $\chi^{2}$ of synchrotron convolved
EDA model. Solid line in each panel represents identity line.

Figure 5: Top left panel: Best fit results obtained by fitting the wideband
spectra in each time segment with $\xi_{max}$ model. Top right panel: scatter
plot between best fitted $\mathbb{N}$ and $p$ values, middle left panel:
scatter plot between best fitted $\mathbb{N}$ and $\xi_{max}$ values and
middle right panel: scatter plot between best fitted $\xi_{max}$ and $p$
values, bottom left panel: scatter plot between flux, $F_{0.5-18.0\ kev}$ and
best fitted $p$ values, bottom right panel: scatter plot between flux,
$F_{0.5-18.0\ kev}$ and best fitted $\xi_{max}$ values.

####  3.0.3 Energy dependent diffusion (EDD) model

The curvature in the spectrum can also be fitted by considering energy
dependence of the escape time-scale $\tau_{esc}(\gamma)$. The situation can be
visualized by assuming that the electrons gain relativistic energy by crossing
a shock front and this acceleration ceases when they diffuse away from the
shock. If this diffusion is occuring within a region having a tangled magnetic
field, the diffusion coefficient may depend on the gyration radius which in
turn depends on the energy of the electron. Thus, the diffusion coefficient or
equivalently the escape time-scale $\tau_{esc}$ may be energy dependent. We
can parameterize this energy dependence by assuming that
$\tau_{esc}=\tau_{esc,R}\left(\frac{\gamma}{\gamma_{R}}\right)^{-\kappa}$.
Note that $\tau_{esc}$ cannot be larger than the free-streaming value and this
limit will be reached when the gyration radius is of the order of size of the
region. We can then identify $\gamma_{R}$ as the energy at which this limit is
reached, $\tau_{esc,R}$ as the free streaming value and restrict the validity
of the dependence to $\gamma<\gamma_{R}$. The solution to Equation 4, with
this dependence of the escape time-scale and neglecting the synchrotron energy
loss, becomes

$n(\xi)=Q_{o}\tau_{acc}\sqrt{\mathbb{C}}\xi^{-1}\rm{exp}\left[-\frac{\eta_{R}}{\kappa}\,\left(\left(\frac{\xi}{\xi_{R}}\right)^{\kappa}-\left(\frac{\xi_{0}}{\xi_{R}}\right)^{\kappa}\right)\right]$
(7)

where $\xi_{R}=\sqrt{\mathbb{C}}\gamma_{R}$,
$\xi_{0}=\sqrt{\mathbb{C}}\gamma_{0}$ and
$\eta_{R}\equiv\tau_{acc}/\tau_{esc,R}$. For small value of $\kappa<<1$, the
solution reduces to a log-parabola form

$n(\xi)\propto(\xi/\xi_{R})^{-\eta_{R}-1-\eta_{R}\kappa log(\xi/\xi_{R})}$ (8)

and for $\xi\rightarrow 0$, the distribution is identical to that given in
Equation 5 for $\gamma<<\gamma_{max}$ and $p=\eta_{R}+1$ i.e., when the escape
time-scale is energy independent and equal to the free-streaming value of
$\tau_{esc,R}$. The particle energy distribution given in Equation 7 depends
on a number of parameters which would be degenerate when it is used for
spectral fitting. Hence, we recast the distribution as

$n(\xi)=K^{\prime}\xi^{-1}\rm{exp}\left[-\frac{\psi}{\kappa}\,\xi^{\kappa}\right]$
(9)

where
$\psi=\eta_{R}\left(\mathbb{C}\gamma_{R}^{2}\right)^{-\kappa/2}=\eta_{R}\xi_{R}^{-\kappa}$,
and the normalization,

$K^{\prime}=Q_{0}\tau_{acc}\sqrt{\mathbb{C}}\rm{exp}\left[\frac{\eta_{R}}{\kappa}\,\left(\frac{\gamma_{0}}{\gamma_{R}}\right)^{\kappa}\right]$
(10)

We performed a $\chi^{2}$-fit to the combined SXT, LAXPC10 and LAXPC20 spectra
using _$synconv\otimes n(\xi)$_ model with input particle density given by
Equation 9 and we refer to this model as synchrotron convolved EDD model. The
free parameters for the model are $\psi$, $\kappa$, and $\mathbb{N}$ given by

$\mathbb{N}=\frac{\delta^{3}(1+z)}{d_{L}^{2}}V\mathbb{A}K^{\prime}$ (11)

The reduced-$\chi^{2}$ obtained in all the time-segments suggests that the
synchrotron convolved EDD model provides good fit to the broadband X-ray
spectrum. For comparison with the logparabola fit, we have plotted the
reduced-$\chi^{2}$ values obtained from the synchrotron convoluted log-
parabola against the reduced-$\chi^{2}$ of synchrotron convolved EDD model in
top right panel of Figure 4, the figure suggest that both log-parabola and EDD
model fits the X-ray spectrum well, while the EDD model provides comparatively
better fit to the spectra of some of the time segments. In Table 2 (right), we
provide the best-fit parameters for all the time segment and the reduced
$\chi^{2}$ values using EDD model. A strong anti-correlation is obtained
between $\psi$ and flux, $F_{0.5-18.0\ kev}$ with r-value and p-value as
$\sim-0.90$ and 3.9 $\times 10^{-15}$, respectively (bottom left panel of
Figure 6). On the other hand a strong correlation is obtained between $\kappa$
and flux, $F_{0.5-18.0\ kev}$ with r-value and p-value as $\sim 0.84$ and 1.5
$\times 10^{-11}$, respectively (bottom right panel of Figure 6). Since
$\psi=\eta_{R}\xi_{R}^{-\kappa}$ one expects log$\psi$ to be linearly and
inversely proportional to $\kappa$ and this is indeed the case. The left panel
of Figure 6 shows the variation of log${}_{10}\psi$ with $\kappa$ and the
straight line fit, log${}_{10}\psi=-0.38\kappa+0.42$, which implies that
$\eta_{R}\sim 2.6$ and $\xi_{R}\sim 2.4$. Moreover the normalization as given
by Equation 11 implies that
log$\mathbb{N}=\left[\frac{\eta_{R}}{\kappa}\,A^{\kappa}\right]+B$. This form
does represent the variation of the normalization as shown in the right panel
of Figure 6, where the solid line represents the form with $A=0.26$ and
$B=1.3$ and $\eta_{R}=2.6$. This provides an estimate of $\gamma_{0}\sim
0.26\gamma_{R}$. While correlations between the best fit spectral parameters
are consistent with the predictions of the model, the inferred physical values
are not quite as expected. The model is not consistent for electrons with
energy $>\gamma_{R}$. However, the observed photon energy corresponding to
$\gamma_{R}$ is estimated to be $\xi_{R}^{2}\sim 5.75$ keV, which is within
the energy range considered for spectral fitting. Moreover the injection
energy $\gamma_{0}$ is not significantly smaller than $\gamma_{R}$. These
issues will be discussed further in the last section.

Figure 6: Best fit results obtained by fitting the wideband spectra in each
time segment with synchrotron convolved EDD model. Top left panel : Plot
between best fitted $\kappa$ and $\log_{10}\psi$ values, Top right panel: plot
between $\ln\mathbb{N}$ and $\kappa$ values. Solid curve in each panel is best
fitted function as described in the text. Bottom left panel: scatter plot
between flux, $F_{0.5-18.0\ kev}$ and best fitted $\kappa$ values, bottom
right panel: scatter plot between flux, $F_{0.5-18.0\ kev}$ and best fitted
$\psi$ values.

####  3.0.4 Energy dependent acceleration model

We next consider a scenario in which the acceleration process is energy
dependent (Massaro et al., 2004a, 2006; Tramacere et al., 2007a). While
Massaro et al. (2004a) considered that the probability that an electron
accelerates, decreases with energy, here we invoke an energy dependent
acceleration time-scale i.e.
$\tau_{acc}=\tau_{acc,R}\left(\frac{\gamma}{\gamma_{R}}\right)^{\kappa}$. If
we assume that the increase in the time-scale with energy is related to the
increasing gyration radius of the particle, then we have to identify
$\gamma_{R}$ as the energy corresponding to a gyration radius equal to the
size of the system i.e., $\gamma_{R}$ would have the same meaning as it had
for the energy dependent diffusion case. If we neglect the synchrotron energy
loss, then the steady state solution of Equation 4 for such an energy
dependent $\tau_{acc}$ becomes

$n(\xi)=Q_{0}\tau_{acc,R}\sqrt{\mathbb{C}}\xi_{R}^{-\kappa}\xi^{\kappa-1}\exp\left[-\frac{\eta_{R}}{\kappa}\left(\left(\frac{\xi}{\xi_{R}}\right)^{\kappa}-\left(\frac{\xi_{0}}{\xi_{R}}\right)^{\kappa}\right)\right]$
(12)

where $\xi_{0}=\sqrt{\mathbb{C}}\gamma_{0}$,
$\xi_{R}=\sqrt{\mathbb{C}}\gamma_{R}$ and
$\eta_{R}\equiv\tau_{acc,R}/\tau_{esc}$. For small value of $\kappa<<1$, the
solution again reduces to a log-parabola form

$n(\xi)\propto(\xi/\xi_{R})^{-\eta_{R}+\kappa-1-\eta_{R}\kappa
log(\xi/\xi_{R})},$ (13)

and for $\kappa\rightarrow 0$ and $\gamma<<\gamma_{max}$, the solution is
identical to that obtained in Equation 5. Again, in order to remove the
degeneracy in the parameters of the particle energy distribution given in
Equation 12, we recast the distribution as

$n(\xi)=K^{\prime}\xi^{\kappa-1}\rm{exp}\left[-\frac{\psi}{\kappa}\,\xi^{\kappa}\right]$
(14)

where
$\psi=\eta_{R}\left(\mathbb{C}\gamma_{R}^{2}\right)^{-\kappa/2}=\eta_{R}\xi_{R}^{-\kappa}$,
and the normalization,

$K^{\prime}=Q_{0}\tau_{acc,R}\sqrt{\mathbb{C}}\xi_{R}^{-\kappa}\rm{exp}\left[\frac{\eta_{R}}{\kappa}\,\left(\frac{\xi_{0}}{\xi_{R}}\right)^{\kappa}\right]$
(15)

We refer to the _$synconv\otimes n(\xi)$_ model as the synchrotron convolved
EDA model. While fitting the broadband X-ray spectrum in each time bin with
the synchrotron convolved EDA model, the parameters $\kappa$, $\psi$ and the
normalization $\mathbb{N}$ were kept as free parameters. Here $\mathbb{N}$ is
defined as

$\mathbb{N}=\frac{\delta^{3}(1+z)}{d_{L}^{2}}V\,\mathbb{A}K^{\prime}$ (16)

The synchrotron convolved EDA model also provided slightly better fit to the
broadband X-ray spectra as compared to the synchrotron convolved logparabola
model for some of the time segments (see bottom panel Figure 4). In Table 2
(left), we provide the best-fit parameters for all the time segment and the
reduced $\chi^{2}$ values using EDA model. A strong anti-correlation is
obtained between $\psi$ and flux, $F_{0.5-18.0\ keV}$ with r-value and p-value
as $\sim$ \- 0.95 and 1.3 $\times 10^{-20}$, respectively (bottom left panel
of Figure 7). On the other hand a correlation is obtained between $\kappa$ and
flux, $F_{0.5-18.0\ kev}$ with r-value and p-value as $\sim 0.78$ and
3.96$\times 10^{-9}$, respectively (bottom right panel of Figure 7).

The relation $\psi=\eta_{R}\xi_{R}^{-\kappa}$ shows that log$\psi$ should be
linearly and inversely proportional to $\kappa$. The left panel of Figure 7
shows that this is indeed satisfied although the relation has significantly
larger scatter than the equivalent relation for the EDD model (Figure 6). The
Figure shows the variation of log$\psi$ with $\kappa$ and the straight line
fit, $\log_{10}\psi=-0.23\kappa+0.42$, which implies that $\eta_{R}\sim 2.63$
and $\xi_{R}\sim 1.70$. Moreover the variation of the normalization with
$\kappa$ as shown in the right panel of Figure 7 is in agreement with the
normalization as given by Equation 16,
log$\mathbb{N}=\frac{\eta_{R}}{\kappa}\,A^{\kappa}-\kappa\log\xi_{R}+B$. In
the Figure, the solid line represents the form with $A=0.19$, $B=2.70$,
$\xi_{R}=1.70$, and $\eta_{R}=2.63$ . This provides an estimate of
$\gamma_{0}\sim 0.19\gamma_{R}$. Thus the results obtained for the EDA model
is qualitatively similar to those obtained for the EDD one.

Figure 7: Best fit results obtained by fitting the wideband spectra in each
time segment with synchrotron convolved EDA model. left panel : Plot between
best fitted $\kappa$ and $\log_{10}\psi$ values, right panel: plot between
$\ln\mathbb{N}$ and $\kappa$ values. Solid curve in each panel is best fitted
function as described in the text. Bottom left panel: scatter plot between
best fitted average flux in the range 0.5 – 18.0 keV and $\kappa$ values,
bottom right panel: scatter plot between best fitted average flux in the range
0.5 – 18.0 keV and $\psi$ values.

##  4 Summary and Discussion

We have carried out time-resolved X-ray (0.5 – 18 keV) spectral study of
AstroSat observation of Mkn 421 during 3 – 8 January, 2017, when the source
exhibited large variations. The long 409 ksec observation was divided into
segments of length 10 ksec and each segment was fitted by a synchrotron
spectrum arising from a non-thermal distribution of electrons. Each spectra is
well described by emission from a log-parabola distribution of electrons
having variable non-zero curvatures.

We investigated the possibility that the curvature in the particle
distribution is due to synchrotron cooling, which causes a deviation from a
power-law form at high energies and has a maximum value of Lorentz factor
$\gamma_{max}$ and found that each of the spectra can be reasonably
represented by the model. The index of the particle distribution at low
energies was found to have a positive correlation with the maximum energy of
the particles. However, in the simplest picture, where the observed variation
is primarily due to changes in the acceleration time-scale, the correlation
should have been negative. A more complex model where the magnetic field is
correlated with the acceleration time-scale may produce the positive
correlation observed, but fails to explain the variation of normalization with
index.

We considered the possibility that the spectral curvature is because the
electron escape or diffusion time-scale is energy dependent i.e.,
$\tau_{esc}=\tau_{esc,R}\left(\frac{\gamma}{\gamma_{R}}\right)^{-\kappa}$ and
again found that each spectra can be well represented by this Energy Dependent
Diffusion (EDD) model. Moreover, two of the three parameters of the model, the
normalization and $\psi$ ($\psi$ being a combination of other physical
parameters), were found to be correlated with the third parameter $\kappa$ in
complete accordance with the predictions of the model. The matching of the
expected to the observed correlations allow for an estimation of the
characteristic synchrotron photon energy $\xi_{R}^{2}\sim 5.75$ keV arising
from electron with energy $\gamma_{R}$ and that the energy at which electrons
are injected into the acceleration region $\gamma_{0}\sim 0.25\gamma_{R}$.
Instead of energy dependent escape time-scale, we considered the possibility
that the acceleration time scale is energy dependent i.e.,
$\tau_{acc}=\tau_{acc,R}\left(\frac{\gamma}{\gamma_{R}}\right)^{\kappa}$ and
find that such an Energy Dependent Acceleration (EDA) model gives
qualitatively similar results to the EDD one. The spectra are well described
by the EDA model and the parameter correlations are roughly as predicted, with
$\xi_{R}^{2}\sim 2.89$ keV and $\gamma_{0}\sim 0.19\gamma_{R}$. Although the
EDD and EDA models successfully represent the spectra and the correlation
between fitted parameters are consistent with the models, there are
inconsistencies which are discussed below.

For the EDD and EDA models, the best fit values of $\kappa$ ranges from 0.3 –
1.0. This is somewhat surprising since one would assume that the value of
$\kappa$ is determined by micro-physics of the system and at different
physical regimes should have discrete values. For example, if the turbulence
is of Kolomogrov type its value should be $1/3$, if it is of Kraichnan type it
should be $1/2$ and for Bohm diffusion it should be $1$ (Asano et al., 2014;
Teraki & Asano, 2019; Goswami et al., 2020). One of the above situation should
have been applicable to the system giving rise to a constant $\kappa$ or even
if the system’s micro-physics changed, $\kappa$ should have changed discreetly
and perhaps not in a continuous manner as observed. In other words, the models
suggest that the turbulence scale index varies rapidly and continuously for
these systems. Another issue is the value of $\gamma_{R}$ as inferred from the
correlation between spectral parameters. If the energy dependence of the
acceleration or escape time-scale is related to gyration radius of the
electron, then $\gamma_{R}$ may be identified with the energy where the
gyration radius becomes as large as the size of the system and hence beyond
this energy, the time-scales should be the free streaming values and energy
independent. For the model to be consistent, the typical synchrotron photon
energy emitted by particles with $\gamma_{R}$, should be much larger than the
photon energy range being considered. However, the corresponding photon energy
($\xi^{2}$) turns out to be within the X-ray spectral range being fitted.

Perhaps a more generic issue with all such class of models is the variation of
the normalization with spectral shape. In general, for these models, there is
a continuous injection of electron at a low energy $\gamma_{0}$ and then the
processes of acceleration, diffusion and cooling, create a non-thermal
distribution of the electron energies in the steady state. Variation of the
injection rate, would lead to a simple up and down scaling of the distribution
without changes in its shape. On the other hand, variations in the
acceleration or diffusion time-scales would lead to changes in the
distribution shape, which in turn would lead to rather dramatic changes in the
density of particles at energies $>>\gamma_{0}$. In the simple picture, where
the distribution is a power-law, changes in the index would lead to large
variations in the value of the distribution at large $\gamma$ since the power-
law would be approximately “pegged” at the pivot point $\gamma_{0}$ (Shah et
al., 2021). This is generically true and one would expect large variations in
the inferred normalization at some large $\gamma$ with spectral shape
variations, which is not observed. For the EDD and EDA models considered here,
while the observed correlation between the normalization and $\kappa$ can be
explained in the framework of the model, the value of $\gamma_{0}$ turns out
to be close to those producing the X-rays ($\gamma_{0}\sim 0.2\gamma_{R}$),
which is physically inconsistent.

The models considered in this work are simple ones which have analytical
solutions, but in reality the physical situation may be more complex and a
more sophisticated technique to efficiently test non-analytical models need to
be developed and applied. For example, both the acceleration and diffusion
time-scales could be energy dependent, which would perhaps be more physical
(Massaro et al., 2004a, 2006; Tramacere et al., 2007a). The energy dependence
of these time-scale could be different than the power-law form used in this
work. Stochastic acceleration (which may be represented as an additional
second order differential term in Equation 4) may play an important part in
describing the particle distribution. The observed photon spectrum may be
arising from a distribution of particles which having escaped from an
acceleration region, and radiate in a surrounding cooling region. Perhaps, an
important assumption of these models, that the particle distribution is in
steady state may not be valid. In other words, it is implicitly assumed in
these models, that the acceleration and escape time-scales are significantly
smaller than the observed variability time-scale of the photon spectrum. Note
that this is not exactly the same as having a one-time injection of particles
into the system and studying their evolution but to solve the corresponding
time-dependent equation with some physical parameter varying continuously with
time. In a recent work Shah et al. (2021) have shown that such a model may
indeed solve the generic problem of large spectral shape variation with modest
changes in the normalization.

In this work, we have shown that time-resolved spectroscopy of AstroSat data
of Mkn 421 can discriminate and rule out simple models. Even though the models
considered can explain the spectra of each time segment and even if the
correlation between spectral parameters can be explained within the framework
of the model, the inferred estimates of physical values can be used to argue
against these models in their simplest forms. A primary ingredient to do this
is to also consider the variation of the normalization with spectral shape
parameters. This opens the need, and provides the motivation to develop more
sophisticated models and test them against data.

## Acknowledgements

We thank the anonymous referee for insightful comments and constructive
suggestions. The authors JH and ACP would like to acknowledge Inter University
Centre for Astronomy and Astrophysics (IUCAA), Pune, India for providing
facilities to carry out this work. ZS is supported by the Department of
Science and Technology, Govt. of India, under the INSPIRE Faculty grant
(DST/INSPIRE/04/2020/002319). This publication uses data from the Astrosat
mission of the Indian Space Research Organisation(ISRO), archived at the
Indian Space Science Data Centre(ISSDC). This work has used the data from the
Soft X-ray Telescope (SXT) developed at TIFR, Mumbai. LaxpcSoft software is
used for analysis of the LAXPC data and we acknowledge the LAXPC Payload
Operation Center (TIFR, Mumbai). This research has made use of data, software
and/or web tools obtained from the High Energy Astrophysics Science Archive
Research Center (HEASARC), a service of the Astrophysics Science Division at
NASA/GSFC and of the Smithsonian Astrophysical Observatory’s High Energy
Astrophysics Division.

## DATA AVAILABILITY

The data and the model used in this article will be shared on reasonable
request to the corresponding author, Jyotishree Hota (email:
<EMAIL_ADDRESS>or Zahir Shah (email: shahzahir4@gmail.com).

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Table 1: Best fit parameters of time resolved spectral analysis fitted with
the synchrotron convolved Log-parabola model and $\xi_{max}$ model.

Obs. | Log-parabola |  | $\xi_{max}$
---|---|---|---
| $\alpha$ | $\beta$ | norm | $log_{10}F$ | $\chi_{\rm red}^{2}$ |  | p | $\xi_{max}$ | norm | $log_{10}F$ | $\chi_{\rm red}^{2}$
1 | $2.84_{-0.04}^{+0.04}$ | $1.36_{-0.08}^{+0.08}$ | $1.29_{-0.01}^{+0.01}$ | $-9.076_{-0.004}^{+0.004}$ | 1.13 |  | $2.69_{-0.15}^{+0.18}$ | $4.80_{-0.15}^{+0.18}$ | $1.39_{-0.01}^{+0.01}$ | $-9.063_{-0.004}^{+0.004}$ | 1.15
2 | $2.98_{-0.03}^{+0.03}$ | $1.11_{-0.06}^{+0.06}$ | $1.33_{-0.01}^{+0.01}$ | $-9.060_{-0.003}^{+0.003}$ | 0.92 |  | $2.81_{-0.03}^{+0.03}$ | $5.54_{-0.19}^{+0.20}$ | $1.46_{-0.01}^{+0.01}$ | $-9.049_{-0.003}^{+0.003}$ | 0.93
3 | $2.98_{-0.04}^{+0.04}$ | $1.17_{-0.08}^{+0.08}$ | $1.26_{-0.01}^{+0.01}$ | $-9.089_{-0.004}^{+0.004}$ | 0.97 |  | $2.81_{-0.03}^{+0.03}$ | $5.33_{-0.21}^{+0.23}$ | $1.39_{-0.01}^{+0.01}$ | $-9.079_{-0.004}^{+0.004}$ | 1.02
4 | $3.00_{-0.08}^{+0.08}$ | $0.95_{-0.15}^{+0.15}$ | $1.23_{-0.02}^{+0.02}$ | $-9.083_{-0.008}^{+0.008}$ | 0.97 |  | $2.85_{-0.07}^{+0.08}$ | $6.04_{-0.56}^{+0.75}$ | $1.35_{-0.03}^{+0.03}$ | $-9.075_{-0.009}^{+0.009}$ | 0.98
5 | $2.92_{-0.05}^{+0.05}$ | $0.99_{-0.08}^{+0.08}$ | $1.31_{-0.01}^{+0.01}$ | $-9.048_{-0.005}^{+0.005}$ | 0.93 |  | $2.77_{-0.04}^{+0.04}$ | $5.83_{-0.28}^{+0.32}$ | $1.42_{-0.02}^{+0.02}$ | $-9.036_{-0.005}^{+0.005}$ | 0.92
6 | $2.82_{-0.04}^{+0.04}$ | $1.04_{-0.08}^{+0.08}$ | $1.37_{-0.01}^{+0.01}$ | $-9.014_{-0.004}^{+0.004}$ | 1.01 |  | $2.70_{-0.04}^{+0.04}$ | $5.66_{-0.25}^{+0.27}$ | $1.47_{-0.02}^{+0.02}$ | $-9.002_{-0.004}^{+0.004}$ | 1.02
7 | $2.86_{-0.04}^{+0.04}$ | $1.01_{-0.07}^{+0.07}$ | $1.49_{-0.01}^{+0.01}$ | $-8.982_{-0.004}^{+0.004}$ | 1.00 |  | $2.73_{-0.03}^{+0.03}$ | $5.77_{-0.24}^{+0.26}$ | $1.60_{-0.02}^{+0.02}$ | $-8.971_{-0.004}^{+0.004}$ | 1.01
8 | $2.81_{-0.04}^{+0.04}$ | $1.26_{-0.08}^{+0.08}$ | $1.54_{-0.01}^{+0.01}$ | $-8.984_{-0.004}^{+0.004}$ | 0.94 |  | $2.67_{-0.03}^{+0.03}$ | $4.95_{-0.17}^{+0.18}$ | $1.64_{-0.02}^{+0.02}$ | $-8.972_{-0.004}^{+0.004}$ | 0.92
9 | $2.87_{-0.05}^{+0.05}$ | $1.13_{-0.08}^{+0.09}$ | $1.44_{-0.01}^{+0.01}$ | $-9.010_{-0.004}^{+0.004}$ | 0.99 |  | $2.72_{-0.04}^{+0.04}$ | $5.34_{-0.23}^{+0.25}$ | $1.55_{-0.02}^{+0.02}$ | $-8.999_{-0.005}^{+0.005}$ | 0.99
10 | $2.92_{-0.03}^{+0.03}$ | $1.01_{-0.06}^{+0.06}$ | $1.43_{-0.01}^{+0.01}$ | $-9.009_{-0.003}^{+0.003}$ | 0.93 |  | $2.77_{-0.03}^{+0.03}$ | $5.72_{-0.22}^{+0.24}$ | $1.56_{-0.01}^{+0.01}$ | $-8.999_{-0.003}^{+0.003}$ | 0.93
11 | $2.89_{-0.04}^{-0.04}$ | $1.03_{-0.07}^{+0.07}$ | $1.48_{-0.01}^{+0.01}$ | $-8.992_{-0.004}^{+0.004}$ | 0.99 |  | $2.75_{-0.03}^{+0.03}$ | $5.69_{-0.24}^{+0.26}$ | $1.60_{-0.02}^{+0.02}$ | $-8.981_{-0.004}^{+0.004}$ | 0.99
12 | $2.96_{-0.04}^{+0.04}$ | $0.98_{-0.07}^{+0.07}$ | $1.49_{-0.01}^{+0.01}$ | $-8.997_{-0.004}^{+0.004}$ | 0.97 |  | $2.82_{-0.03}^{+0.03}$ | $5.97_{-0.26}^{+0.29}$ | $1.63_{-0.02}^{+0.02}$ | $-8.987_{-0.004}^{+0.004}$ | 1.00
13 | $2.91_{-0.04}^{+0.04}$ | $1.16_{-0.07}^{+0.08}$ | $1.57_{-0.01}^{+0.01}$ | $-8.983_{-0.004}^{+0.004}$ | 0.98 |  | $2.76_{-0.03}^{+0.03}$ | $5.33_{-0.21}^{+0.22}$ | $1.71_{-0.02}^{+0.02}$ | $-8.972_{-0.004}^{+0.004}$ | 0.99
14 | $2.85_{-0.05}^{+0.05}$ | $1.31_{-0.09}^{+0.09}$ | $1.60_{-0.02}^{+0.02}$ | $-8.977_{-0.005}^{+0.005}$ | 1.04 |  | $2.70_{-0.04}^{+0.04}$ | $4.94_{-0.19}^{+0.21}$ | $1.73_{-0.02}^{+0.02}$ | $-8.964_{-0.005}^{+0.005}$ | 1.02
15 | $2.98_{-0.12}^{+0.11}$ | $0.99_{-0.15}^{+0.16}$ | $1.68_{-0.03}^{+0.03}$ | $-8.950_{-0.011}^{+0.010}$ | 0.85 |  | $2.91_{-0.09}^{+0.10}$ | $6.44_{-0.59}^{+0.78}$ | $1.86_{-0.04}^{+0.04}$ | $-8.945_{-0.011}^{+0.011}$ | 0.88
16 | $2.85_{-0.04}^{+0.04}$ | $1.09_{-0.08}^{+0.09}$ | $1.83_{-0.02}^{+0.02}$ | $-8.894_{-0.004}^{+0.004}$ | 1.00 |  | $2.69_{-0.03}^{+0.03}$ | $5.25_{-0.23}^{+0.26}$ | $1.97_{-0.02}^{+0.02}$ | $-8.885_{-0.004}^{+0.004}$ | 1.00
17 | $2.79_{-0.04}^{+0.04}$ | $1.09_{-0.06}^{+0.06}$ | $2.00_{-0.01}^{+0.01}$ | $-8.846_{-0.004}^{+0.004}$ | 1.06 |  | $2.66_{-0.03}^{+0.04}$ | $5.45_{-0.26}^{+0.30}$ | $2.12_{-0.02}^{+0.02}$ | $-8.833_{-0.004}^{+0.004}$ | 1.06
18 | $2.81_{-0.03}^{+0.03}$ | $1.09_{-0.07}^{+0.08}$ | $2.06_{-0.02}^{+0.02}$ | $-8.838_{-0.004}^{+0.004}$ | 1.02 |  | $2.66_{-0.03}^{+0.03}$ | $5.22_{-0.20}^{+0.22}$ | $2.20_{-0.02}^{+0.02}$ | $-8.828_{-0.004}^{+0.004}$ | 1.00
19 | $2.79_{-0.03}^{+0.03}$ | $1.14_{-0.07}^{+0.07}$ | $2.04_{-0.01}^{+0.01}$ | $-8.841_{-0.003}^{+0.003}$ | 1.08 |  | $2.66_{-0.03}^{+0.02}$ | $5.29_{-0.19}^{+0.19}$ | $2.17_{-0.02}^{+0.02}$ | $-8.831_{-0.003}^{+0.003}$ | 1.15
20 | $2.67_{-0.03}^{+0.03}$ | $1.38_{-0.06}^{+0.06}$ | $2.22_{-0.01}^{+0.01}$ | $-8.808_{-0.003}^{+0.003}$ | 1.15 |  | $2.58_{-0.02}^{+0.02}$ | $4.80_{-0.10}^{+0.13}$ | $2.32_{-0.02}^{+0.02}$ | $-8.795_{-0.003}^{+0.003}$ | 1.15
21 | $2.61_{-0.04}^{+0.04}$ | $1.37_{-0.07}^{+0.07}$ | $2.23_{-0.02}^{+0.02}$ | $-8.795_{-0.004}^{+0.004}$ | 0.95 |  | $2.56_{-0.03}^{+0.03}$ | $4.90_{-0.16}^{+0.13}$ | $2.31_{-0.03}^{+0.03}$ | $-8.782_{-0.004}^{+0.004}$ | 0.99
22 | $2.61_{-0.04}^{+0.04}$ | $1.18_{-0.07}^{+0.07}$ | $2.16_{-0.02}^{+0.02}$ | $-8.787_{-0.004}^{+0.004}$ | 0.94 |  | $2.57_{-0.03}^{+0.03}$ | $5.37_{-0.18}^{+0.19}$ | $2.25_{-0.03}^{+0.02}$ | $-8.774_{-0.004}^{+0.004}$ | 1.01
23 | $2.54_{-0.03}^{+0.03}$ | $1.09_{-0.06}^{+0.06}$ | $2.22_{-0.02}^{+0.02}$ | $-8.748_{-0.003}^{+0.003}$ | 1.00 |  | $2.51_{-0.03}^{+0.02}$ | $5.50_{-0.19}^{+0.17}$ | $2.28_{-0.02}^{+0.02}$ | $-8.735_{-0.004}^{+0.003}$ | 1.02
24 | $2.47_{-0.04}^{+0.04}$ | $1.27_{-0.08}^{+0.08}$ | $2.33_{-0.02}^{+0.02}$ | $-8.729_{-0.004}^{+0.004}$ | 1.24 |  | $2.43_{-0.03}^{+0.02}$ | $4.84_{-0.17}^{+0.13}$ | $2.35_{-0.03}^{+0.03}$ | $-8.716_{-0.004}^{+0.004}$ | 1.23
25 | $2.55_{-0.03}^{+0.03}$ | $1.26_{-0.06}^{+0.07}$ | $2.26_{-0.02}^{+0.02}$ | $-8.760_{-0.003}^{+0.003}$ | 1.20 |  | $2.49_{-0.02}^{+0.02}$ | $4.86_{-0.15}^{+0.13}$ | $2.31_{-0.02}^{+0.02}$ | $-8.749_{-0.003}^{+0.003}$ | 1.21
26 | $2.47_{-0.03}^{+0.03}$ | $1.59_{-0.07}^{+0.07}$ | $2.25_{-0.02}^{+0.02}$ | $-8.781_{-0.003}^{+0.003}$ | 1.10 |  | $2.43_{-0.02}^{+0.02}$ | $4.27_{-0.13}^{+0.08}$ | $2.26_{-0.02}^{+0.02}$ | $-8.768_{-0.003}^{+0.003}$ | 1.08
27 | $2.42_{-0.04}^{+0.04}$ | $1.62_{-0.08}^{+0.08}$ | $2.22_{-0.02}^{+0.02}$ | $-8.780_{-0.003}^{+0.003}$ | 1.11 |  | $2.41_{-0.03}^{+0.02}$ | $4.30_{-0.15}^{+0.08}$ | $2.20_{-0.03}^{+0.03}$ | $-8.765_{-0.004}^{+0.004}$ | 1.13
28 | $2.41_{-0.04}^{+0.04}$ | $1.85_{-0.07}^{+0.07}$ | $2.28_{-0.02}^{+0.02}$ | $-8.790_{-0.003}^{+0.003}$ | 1.07 |  | $2.42_{-0.02}^{+0.02}$ | $4.13_{-0.03}^{+0.09}$ | $2.27_{-0.02}^{+0.03}$ | $-8.773_{-0.003}^{+0.003}$ | 1.11
29 | $2.35_{-0.05}^{+0.05}$ | $1.71_{-0.08}^{+0.08}$ | $2.29_{-0.02}^{+0.02}$ | $-8.762_{-0.004}^{+0.004}$ | 1.04 |  | $2.40_{-0.02}^{+0.03}$ | $4.33_{-0.07}^{+0.15}$ | $2.27_{-0.03}^{+0.03}$ | $-8.765_{-0.004}^{+0.004}$ | 1.09
30 | $2.30_{-0.06}^{+0.05}$ | $1.92_{-0.10}^{+0.10}$ | $2.38_{-0.02}^{+0.02}$ | $-8.754_{-0.004}^{+0.004}$ | 1.13 |  | $2.33_{-0.02}^{+0.04}$ | $3.92_{-0.04}^{+0.18}$ | $2.30_{-0.03}^{+0.04}$ | $-8.745_{-0.004}^{+0.003}$ | 1.12
31 | $2.34_{-0.07}^{+0.07}$ | $2.02_{-0.11}^{+0.11}$ | $2.39_{-0.03}^{+0.03}$ | $-8.770_{-0.005}^{+0.005}$ | 0.99 |  | $2.37_{-0.04}^{+0.04}$ | $3.91_{-0.11}^{+0.11}$ | $2.34_{-0.05}^{+0.05}$ | $-8.755_{-0.005}^{+0.006}$ | 1.05
32 | $2.26_{-0.04}^{+0.04}$ | $1.94_{-0.09}^{+0.09}$ | $2.28_{-0.02}^{+0.02}$ | $-8.767_{-0.003}^{+0.003}$ | 1.15 |  | $2.31_{-0.03}^{+0.02}$ | $3.90_{-0.12}^{+0.03}$ | $2.19_{-0.03}^{+0.03}$ | $-8.749_{-0.004}^{+0.002}$ | 1.15
33 | $2.14_{-0.04}^{+0.03}$ | $1.67_{-0.07}^{+0.07}$ | $2.33_{-0.02}^{+0.02}$ | $-8.698_{-0.003}^{+0.003}$ | 1.18 |  | $2.26_{-0.03}^{+0.01}$ | $4.26_{-0.16}^{+0.04}$ | $2.21_{-0.03}^{+0.02}$ | $-8.684_{-0.003}^{+0.004}$ | 1.19
34 | $2.15_{-0.04}^{+0.04}$ | $1.83_{-0.08}^{+0.08}$ | $2.44_{-0.02}^{+0.02}$ | $-8.700_{-0.003}^{+0.003}$ | 1.19 |  | $2.26_{-0.03}^{+0.01}$ | $4.09_{-0.18}^{+0.01}$ | $2.31_{-0.03}^{+0.03}$ | $-8.683_{-0.004}^{+0.003}$ | 1.18
35 | $2.16_{-0.04}^{+0.04}$ | $2.33_{-0.07}^{+0.07}$ | $2.42_{-0.02}^{+0.02}$ | $-8.757_{-0.003}^{+0.003}$ | 1.13 |  | $2.29_{-0.01}^{+0.01}$ | $3.70_{-0.05}^{+0.02}$ | $2.28_{-0.02}^{+0.02}$ | $-8.739_{-0.003}^{+0.002}$ | 1.19
36 | $2.28_{-0.04}^{+0.04}$ | $2.13_{-0.09}^{+0.09}$ | $2.38_{-0.02}^{+0.02}$ | $-8.773_{-0.003}^{+0.003}$ | 0.97 |  | $2.32_{-0.02}^{+0.02}$ | $3.71_{-0.01}^{+0.07}$ | $2.28_{-0.03}^{+0.03}$ | $-8.757_{-0.003}^{+0.003}$ | 0.96
37 | $2.45_{-0.09}^{+0.08}$ | $1.22_{-0.32}^{+0.36}$ | $2.20_{-0.06}^{+0.07}$ | $-8.745_{-0.013}^{+0.013}$ | 0.97 |  | $2.33_{-0.02}^{+0.03}$ | $3.54_{-0.03}^{+0.10}$ | $2.34_{-0.05}^{+0.04}$ | $-8.770_{-0.005}^{+0.005}$ | 1.09
38 | $2.51_{-0.09}^{+0.09}$ | $2.05_{-0.19}^{+0.20}$ | $2.29_{-0.04}^{+0.04}$ | $-8.826_{-0.007}^{+0.007}$ | 0.96 |  | $2.44_{-0.05}^{+0.05}$ | $3.73_{-0.17}^{+0.19}$ | $2.31_{-0.06}^{+0.06}$ | $-8.813_{-0.007}^{+0.007}$ | 0.98
39 | $2.48_{-0.04}^{+0.04}$ | $1.82_{-0.07}^{+0.08}$ | $2.28_{-0.02}^{+0.02}$ | $-8.801_{-0.003}^{+0.003}$ | 1.03 |  | $2.46_{-0.02}^{+0.02}$ | $4.13_{-0.06}^{+0.10}$ | $2.29_{-0.03}^{+0.03}$ | $-8.787_{-0.003}^{+0.004}$ | 1.08

Notes: $log_{10}F_{0.5-18\,keV}$ is logarithm of flux (ergs cm-2 s-1) in the
range 0.5–18.0 keV.

Table 2: Best fit parameters of time resolved spectral analysis fitted with
the synchrotron convolved energy dependent diffusion model and energy
dependent acceleration model.

Obs. | Energy dependent diffusion (EDD) model |  | Energy dependent acceleration (EDA) model
---|---|---|---
| $\psi$ | $\kappa$ | norm | lg10F | $\chi_{\rm red}^{2}$ |  | $\kappa$ | $\psi$ | norm | lg10F | $\chi_{\rm red}^{2}$
1 | $1.82_{-0.05}^{+0.03}$ | $0.50_{-0.02}^{+0.04}$ | $46.94_{-14.26}^{+13.13}$ | $-9.071_{-0.004}^{+0.004}$ | 1.13 |  | $0.44_{-0.03}^{+0.02}$ | $2.24_{-0.02}^{+0.03}$ | $214.30_{-43.70}^{+110.16}$ | $-9.072_{-0.004}^{+0.004}$ | 1.13
2 | $1.96_{-0.04}^{+0.02}$ | $0.40_{-0.02}^{+0.03}$ | $176.34_{-60.39}^{+61.58}$ | $-9.056_{-0.003}^{+0.003}$ | 0.92 |  | $0.35_{-0.02}^{+0.02}$ | $2.31_{-0.02}^{+0.02}$ | $891.18_{-245.51}^{+447.00}$ | $-9.057_{-0.003}^{+0.003}$ | 0.92
3 | $1.96_{-0.05}^{+0.03}$ | $0.42_{-0.02}^{+0.04}$ | $136.88_{-51.07}^{+53.04}$ | $-9.086_{-0.004}^{+0.004}$ | 0.98 |  | $0.37_{-0.02}^{+0.02}$ | $2.33_{-0.02}^{+0.02}$ | $707.28_{-235.06}^{+347.33}$ | $-9.086_{-0.004}^{+0.004}$ | 0.98
4 | $1.96_{-0.06}^{+0.10}$ | $0.37_{-0.08}^{+0.04}$ | $242.5_{-122.27}^{+122.27}$ | $-9.080_{-0.008}^{+0.008}$ | 0.97 |  | $0.33_{-0.06}^{+0.03}$ | $2.28_{-0.04}^{+0.05}$ | $1335.13_{-676.50}^{+676.50}$ | $-9.080_{-0.008}^{+0.008}$ | 0.97
5 | $1.90_{-0.05}^{+0.02}$ | $0.38_{-0.03}^{+0.04}$ | $201.07_{-86.05}^{+154.30}$ | $-9.043_{-0.005}^{+0.005}$ | 0.92 |  | $0.33_{-0.03}^{+0.02}$ | $2.23_{-0.02}^{+0.02}$ | $1042.25_{-404.43}^{+1021.79}$ | $-9.044_{-0.005}^{+0.005}$ | 0.92
6 | $1.81_{-0.05}^{+0.03}$ | $0.40_{-0.02}^{+0.04}$ | $127.04_{-53.44}^{+56.58}$ | $-9.010_{-0.004}^{+0.004}$ | 1.01 |  | $0.35_{-0.02}^{+0.03}$ | $2.16_{-0.02}^{+0.02}$ | $640.11_{-237.26}^{+372.11}$ | $-9.010_{-0.004}^{+0.004}$ | 1.01
7 | $1.84_{-0.04}^{+0.03}$ | $0.39_{-0.03}^{+0.03}$ | $163.54_{-58.98}^{+84.97}$ | $-8.978_{-0.004}^{+0.004}$ | 0.99 |  | $0.34_{-0.02}^{+0.02}$ | $2.18_{-0.02}^{+0.02}$ | $875.75_{-302.27}^{+519.84}$ | $-8.979_{-0.004}^{+0.004}$ | 0.99
8 | $1.79_{-0.05}^{+0.02}$ | $0.48_{-0.02}^{+0.04}$ | $63.28_{-19.40}^{+18.48}$ | $-8.980_{-0.004}^{+0.004}$ | 0.93 |  | $0.42_{-0.03}^{+0.02}$ | $2.19_{-0.02}^{+0.02}$ | $294.60_{-61.89}^{+153.88}$ | $-8.980_{-0.004}^{+0.004}$ | 0.93
9 | $1.85_{-0.05}^{+0.04}$ | $0.43_{-0.03}^{+0.04}$ | $103.14_{-36.55}^{+55.09}$ | $-9.006_{-0.004}^{+0.004}$ | 0.99 |  | $0.38_{-0.03}^{+0.02}$ | $2.21_{-0.02}^{+0.03}$ | $498.99_{-142.52}^{+398.01}$ | $-9.006_{-0.004}^{+0.004}$ | 0.99
10 | $1.90_{-0.03}^{+0.03}$ | $0.38_{-0.02}^{+0.03}$ | $204.21_{-66.56}^{+94.64}$ | $-9.006_{-0.003}^{+0.003}$ | 0.93 |  | $0.34_{-0.02}^{+0.02}$ | $2.24_{-0.02}^{+0.02}$ | $1085.63_{-318.21}^{+625.98}$ | $-9.006_{-0.003}^{+0.003}$ | 0.93
11 | $1.88_{-0.04}^{+0.04}$ | $0.39_{-0.03}^{+0.03}$ | $177.95_{-65.98}^{+98.79}$ | $-8.988_{-0.004}^{+0.004}$ | 0.98 |  | $0.34_{-0.02}^{+0.02}$ | $2.22_{-0.02}^{+0.02}$ | $939.96_{-321.74}^{+631.42}$ | $-8.988_{-0.004}^{+0.004}$ | 0.98
12 | $1.95_{-0.04}^{+0.03}$ | $0.36_{-0.03}^{+0.03}$ | $319.35_{-122.48}^{+204.53}$ | $-8.993_{-0.004}^{+0.004}$ | 0.97 |  | $0.32_{-0.02}^{+0.02}$ | $2.27_{-0.02}^{+0.02}$ | $1789.81_{-689.15}^{+1224.59}$ | $-8.994_{-0.004}^{+0.004}$ | 0.97
13 | $1.89_{-0.04}^{+0.04}$ | $0.43_{-0.03}^{+0.03}$ | $124.96_{-39.96}^{+59.58}$ | $-8.979_{-0.004}^{+0.004}$ | 0.98 |  | $0.38_{-0.03}^{+0.02}$ | $2.26_{-0.02}^{+0.02}$ | $619.33_{-161.62}^{+424.85}$ | $-8.980_{-0.004}^{+0.004}$ | 0.98
14 | $1.82_{-0.05}^{+0.05}$ | $0.49_{-0.04}^{+0.04}$ | $61.95_{-18.57}^{+29.53}$ | $-8.972_{-0.005}^{+0.005}$ | 1.03 |  | $0.42_{-0.03}^{+0.02}$ | $2.23_{-0.03}^{+0.03}$ | $319.51_{-91.67}^{+184.50}$ | $-8.973_{-0.005}^{+0.005}$ | 1.03
15 | $1.95_{-0.08}^{+0.11}$ | $0.38_{-0.08}^{+0.05}$ | $290.08_{-159.55}^{+159.55}$ | $-8.947_{-0.011}^{+0.011}$ | 0.85 |  | $0.33_{-0.07}^{+0.04}$ | $2.27_{-0.04}^{+0.06}$ | $1519.87_{-830.64}^{+830.64}$ | $-8.947_{-0.011}^{+0.011}$ | 0.85
16 | $1.82_{-0.06}^{+0.03}$ | $0.43_{-0.02}^{+0.05}$ | $128.74_{-53.95}^{+48.71}$ | $-8.892_{-0.004}^{+0.004}$ | 1.00 |  | $0.38_{-0.03}^{+0.02}$ | $2.18_{-0.02}^{+0.02}$ | $577.66_{-166.21}^{+399.66}$ | $-8.892_{-0.004}^{+0.004}$ | 1.00
17 | $1.77_{-0.05}^{+0.03}$ | $0.41_{-0.03}^{+0.05}$ | $140.72_{-59.26}^{+62.65}$ | $-8.840_{-0.004}^{+0.004}$ | 1.05 |  | $0.37_{-0.03}^{+0.02}$ | $2.12_{-0.02}^{+0.02}$ | $614.09_{-175.44}^{+514.86}$ | $-8.841_{-0.004}^{+0.004}$ | 1.07
18 | $1.78_{-0.05}^{+0.03}$ | $0.44_{-0.02}^{+0.04}$ | $119.11_{-44.37}^{+35.35}$ | $-8.835_{-0.004}^{+0.004}$ | 1.00 |  | $0.38_{-0.02}^{+0.02}$ | $2.15_{-0.02}^{+0.02}$ | $579.50_{-168.66}^{-259.54}$ | $-8.836_{-0.004}^{+0.004}$ | 1.01
19 | $1.76_{-0.03}^{+0.02}$ | $0.45_{-0.02}^{+0.03}$ | $100.68_{-28.84}^{+26.77}$ | $-8.838_{-0.003}^{+0.003}$ | 1.09 |  | $0.39_{-0.02}^{+0.02}$ | $2.14_{-0.02}^{+0.02}$ | $493.87_{-113.25}^{+189.08}$ | $-8.839_{-0.003}^{+0.003}$ | 1.09
20 | $1.65_{-0.03}^{+0.03}$ | $0.55_{-0.03}^{+0.03}$ | $43.84_{-7.74}^{+9.94}$ | $-8.804_{-0.003}^{+0.003}$ | 1.14 |  | $0.45_{-0.02}^{+0.02}$ | $2.10_{-0.02}^{+0.02}$ | $224.03_{-41.00}^{+55.94}$ | $-8.804_{-0.003}^{+0.003}$ | 1.14
21 | $1.58_{-0.03}^{+0.06}$ | $0.56_{-0.04}^{+0.02}$ | $36.29_{-5.09}^{+13.56}$ | $-8.790_{-0.004}^{+0.004}$ | 0.95 |  | $0.46_{-0.02}^{+0.02}$ | $2.05_{-0.01}^{+0.02}$ | $195.89_{-39.26}^{+55.69}$ | $-8.791_{-0.004}^{+0.004}$ | 0.95
22 | $1.60_{-0.04}^{+0.04}$ | $0.48_{-0.03}^{+0.03}$ | $58.37_{-14.23}^{+20.59}$ | $-8.782_{-0.004}^{+0.004}$ | 0.95 |  | $0.41_{-0.02}^{+0.02}$ | $2.01_{-0.02}^{+0.02}$ | $287.94_{-67.12}^{+125.10}$ | $-8.783_{-0.004}^{+0.004}$ | 0.95
23 | $1.53_{-0.03}^{+0.03}$ | $0.47_{-0.03}^{+0.03}$ | $55.11_{-11.75}^{+16.30}$ | $-8.743_{-0.003}^{+0.003}$ | 0.99 |  | $0.40_{-0.02}^{+0.02}$ | $1.92_{-0.02}^{+0.02}$ | $273.81_{-55.50}^{+101.95}$ | $-8.744_{-0.003}^{+0.003}$ | 0.99
24 | $1.43_{-0.04}^{+0.04}$ | $0.58_{-0.04}^{+0.04}$ | $27.10_{-5.30}^{+7.31}$ | $-8.725_{-0.004}^{+0.004}$ | 1.22 |  | $0.47_{-0.03}^{+0.02}$ | $1.89_{-0.02}^{+0.02}$ | $127.10_{-21.97}^{+47.54}$ | $-8.726_{-0.004}^{+0.004}$ | 1.23
25 | $1.51_{-0.03}^{+0.03}$ | $0.55_{-0.03}^{+0.03}$ | $34.16_{-5.79}^{+7.65}$ | $-8.757_{-0.003}^{+0.003}$ | 1.19 |  | $0.45_{-0.02}^{+0.02}$ | $1.96_{-0.01}^{+0.02}$ | $166.69_{-25.87}^{+48.02}$ | $-8.758_{-0.003}^{+0.003}$ | 1.19
26 | $1.43_{-0.03}^{+0.03}$ | $0.69_{-0.03}^{+0.04}$ | $17.18_{-2.31}^{+2.58}$ | $-8.777_{-0.003}^{+0.003}$ | 1.08 |  | $0.55_{-0.02}^{+0.02}$ | $1.97_{-0.02}^{+0.02}$ | $80.94_{-10.58}^{+15.46}$ | $-8.778_{-0.003}^{+0.003}$ | 1.08
27 | $1.39_{-0.05}^{+0.03}$ | $0.70_{-0.03}^{+0.04}$ | $15.46_{-2.51}^{+2.20}$ | $-8.775_{-0.003}^{+0.003}$ | 1.10 |  | $0.56_{-0.03}^{+0.02}$ | $1.93_{-0.02}^{+0.02}$ | $68.19_{-8.36}^{+16.21}$ | $-8.776_{-0.003}^{+0.003}$ | 1.10
28 | $1.39_{-0.04}^{+0.04}$ | $0.76_{-0.03}^{+0.04}$ | $13.70_{-1.70}^{+1.89}$ | $-8.783_{-0.003}^{+0.003}$ | 1.05 |  | $0.59_{-0.02}^{+0.02}$ | $1.98_{-0.02}^{+0.02}$ | $62.28_{-7.70}^{+10.53}$ | $-8.784_{-0.003}^{+0.003}$ | 1.05
29 | $1.35_{-0.05}^{+0.04}$ | $0.73_{-0.04}^{+0.04}$ | $14.19_{-2.25}^{+2.54}$ | $-8.756_{-0.004}^{+0.004}$ | 1.03 |  | $0.56_{-0.02}^{+0.03}$ | $1.92_{-0.03}^{+0.02}$ | $67.99_{-14.43}^{+10.59}$ | $-8.757_{-0.004}^{+0.004}$ | 1.03
30 | $1.27_{-0.06}^{+0.05}$ | $0.84_{-0.05}^{+0.05}$ | $10.16_{-1.45}^{+1.67}$ | $-8.748_{-0.004}^{+0.004}$ | 1.11 |  | $0.64_{-0.03}^{+0.03}$ | $1.90_{-0.03}^{+0.03}$ | $44.68_{-6.62}^{+9.24}$ | $-8.750_{-0.004}^{+0.004}$ | 1.11
31 | $1.32_{-0.07}^{+0.05}$ | $0.84_{-0.05}^{+0.07}$ | $10.92_{-2.02}^{+1.91}$ | $-8.764_{-0.005}^{+0.005}$ | 0.99 |  | $0.65_{-0.04}^{+0.03}$ | $1.95_{-0.03}^{+0.04}$ | $45.17_{-6.74}^{+13.14}$ | $-8.765_{-0.005}^{+0.005}$ | 1.00
32 | $1.23_{-0.04}^{+0.04}$ | $0.87_{-0.04}^{+0.05}$ | $8.86_{-0.97}^{+1.09}$ | $-8.762_{-0.003}^{+0.003}$ | 1.13 |  | $0.65_{-0.03}^{+0.03}$ | $1.88_{-0.02}^{+0.02}$ | $38.74_{-4.49}^{+5.89}$ | $-8.763_{-0.003}^{+0.003}$ | 1.13
33 | $1.12_{-0.04}^{+0.03}$ | $0.82_{-0.03}^{+0.04}$ | $8.82_{-0.90}^{+0.95}$ | $-8.693_{-0.003}^{+0.003}$ | 1.16 |  | $0.61_{-0.02}^{+0.02}$ | $1.72_{-0.02}^{+0.02}$ | $37.91_{-3.96}^{+5.23}$ | $-8.694_{-0.003}^{+0.003}$ | 1.18
34 | $1.13_{-0.04}^{+0.04}$ | $0.87_{-0.04}^{+0.05}$ | $8.51_{-0.92}^{+0.98}$ | $-8.695_{-0.003}^{+0.003}$ | 1.16 |  | $0.64_{-0.02}^{+0.03}$ | $1.77_{-0.02}^{+0.02}$ | $36.69_{-4.51}^{+4.97}$ | $-8.696_{-0.003}^{+0.003}$ | 1.17
35 | $1.16_{-0.03}^{+0.03}$ | $1.00_{-0.04}^{+0.04}$ | $7.24_{-0.53}^{+0.60}$ | $-8.751_{-0.003}^{+0.003}$ | 1.11 |  | $0.73_{-0.02}^{+0.02}$ | $1.88_{-0.02}^{+0.02}$ | $30.37_{-2.60}^{+2.94}$ | $-8.752_{-0.003}^{+0.003}$ | 1.11
36 | $1.26_{-0.05}^{+0.04}$ | $0.91_{-0.04}^{+0.04}$ | $9.01_{-0.92}^{+0.98}$ | $-8.767_{-0.003}^{+0.003}$ | 0.94 |  | $0.68_{-0.02}^{+0.03}$ | $1.94_{-0.02}^{+0.02}$ | $39.81_{-4.59}^{+4.97}$ | $-8.768_{-0.003}^{+0.003}$ | 0.94
37 | $1.24_{-0.06}^{+0.06}$ | $1.02_{-0.06}^{+0.06}$ | $7.74_{-0.94}^{+1.14}$ | $-8.780_{-0.005}^{+0.005}$ | 0.97 |  | $0.75_{-0.03}^{+0.03}$ | $1.98_{-0.03}^{+0.03}$ | $32.41_{-4.51}^{+5.62}$ | $-8.781_{-0.005}^{+0.005}$ | 0.96
38 | $1.49_{-0.10}^{+0.08}$ | $0.81_{-0.07}^{+0.10}$ | $13.88_{-3.73}^{+4.36}$ | $-8.821_{-0.007}^{+0.007}$ | 0.96 |  | $0.62_{-0.04}^{+0.07}$ | $2.12_{-0.05}^{+0.04}$ | $67.46_{-21.68}^{+21.47}$ | $-8.823_{-0.007}^{+0.007}$ | 0.96
39 | $1.47_{-0.05}^{+0.03}$ | $0.73_{-0.03}^{+0.04}$ | $16.42_{-2.64}^{+2.18}$ | $-8.797_{-0.003}^{+0.003}$ | 1.03 |  | $0.58_{-0.03}^{+0.02}$ | $2.03_{-0.02}^{+0.03}$ | $71.91_{-8.22}^{+17.04}$ | $-8.797_{-0.003}^{+0.003}$ | 1.03

Notes: $log_{10}F_{0.5-18\,keV}$ is logarithm of flux (ergs cm-2 s-1) in the
range 0.5–18.0 keV.