1 Department of Physics, Indian Institute of Technology, Kanpur 208016, India
2 DAA (retd), Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai
400005, India
3 Inter-University Centre for Astronomy & Astrophysics, Ganeshkhind,
Pune-411007, India
4 School of Physics and Astronomy, University of Southampton, Highfield
Campus, Southampton SO17 1BJ, UK
5 Department of Physics, Indian Institute of Technology, Hyderabad 502285,
India
6 Centre for Astrophysics, University of Southern Queensland, QLD 4300,
Australia

# LAXPC instrument onboard AstroSat: Five exciting years of new scientific
results specially on X-ray Binaries

J. S. Yadav1 P. C. Agrawal2 Ranjeev Misra3 Jayashree Roy3 Mayukh Pahari4,5 and
R. K Manchanda6

###### Abstract

With its large effective area at hard X-rays, high time resolution and having
co-aligned other instruments, AstroSat/LAXPC was designed to usher in a new
era in rapid variability studies and wide spectral band measurements of the
X-ray binaries. Over the last five years, the instrument has successfully
achieved to a significant extent these Science goals. In the coming years, it
is poised to make more important discoveries. This paper highlights the
primary achievements of AstroSat/LAXPC in unraveling the behavior of black
hole and neutron star systems and discusses the exciting possibility of the
instrument’s contribution to future science.

###### keywords:

Black hole—Neutron star—accretion—X-ray—radio jet—Space instrument.

<EMAIL_ADDRESS>
November 7, 2020January 6, 2021

12.3456/s78910-011-012-3 #### 000 0000 1– 1

##  1 Introduction

X-ray astronomy was born in 1962 with the chance discovery of a bright X-ray
source Scorpius X-1 (Sco X-1) in a rocket flight experiment. It is still the
brightest X-ray source in the sky radiating energy in X-rays at a prodigious
rate of $\sim$ $10^{36}$ ergs /sec. What process generates X-rays at such high
rate, remained a mystery for many years. In 1933 F.Zwicky and W.Baade made a
prophetic suggestion that when cores of high mass stars at the end of their
life collapse, it becomes a Neutron Star in a massive explosion called
supernova. The equation of state of these dense stars were investigated by J.
R. Oppenheimer and G. M. Volkoff in 1939. The Neutron stars remained a
theoretical curiosity till 1967 when A. Hewish and Jocelynn-Bell discovered a
radio source now called PSR B1919+21. producing periodic signal every 1.3373
sec. These pulsating objects came to be known as Pulsars, were interpreted by
T. Gold as spinning Neutron stars. About a few thousand Pulsars are now known.
X-rays with 33 msec pulsation period of were discovered from a source (named
as PSR B0531+21) in the Crab Nebula in 1969 in a rocket experiment. The Crab
pulsar was found to be spinning down in all the wavebands monotonically. This
suggests that these objects are rotation powered pulsars. However, these
discoveries did not contribute significantly to explain the nature of Sco X-1.
Soon after the launch of the first X-ray satellite UHURU in 1970, it
discovered a bright X-ray source Cen X-3, whose intensity was oscillating with
4.8 sec period. The object was found to be a Binary since its light curve
showed, periodic X-ray eclipses every 2.09 days. This discovery led to the
conclusion that the bright X-ray sources are compact objects, either a neutron
star or a black hole, in a binary with a companion star. The strong gravity of
the compact objects pulls matter from the companion star. This matter forms an
accretion disk from which the matter accretes to the compact object.

If the compact object is a neutron star, matter accretes either on the surface
of the neutron star in case of the neutron star with low magnetic field
($\leqslant$ 109 gauss), or accretes at the magnetic poles of a highly
magnetized neutron star. If the magnetic axis is offset from the rotation
axis, the X-ray intensity oscillates as the neutron star spins. Several
hundred X-ray binaries have been discovered in the Galaxy, majority of them
with neutron star as the X-ray source. The in falling matter at the poles
under strong gravity of the neutron star gets heated up as it settles down on
the surface of the neutron star or at its magnetic poles. The X-ray luminosity
of the binary is due to release of gravitational energy and depends on the
accretion rate.

###  1.1 Classification of X-ray Binaries and their Salient Features :

The X-ray binaries are classified in two categories; the Low Mass X-ray
Binaries (LMXBs) and the High Mass X-ray Binaries (HMXBs). The LMXBs usually
have a low mass ($\leq$ 2 Solar mass) companion star while the HMXBs usually
have a massive companion star ($\geqslant$ 5-30 Solar Mass). In the X-ray
binaries, the compact objects may be either a neutron star (NSXBs) or a black
hole (BHXBs). Among all the wavebands, the most rapid and violent brightness
changes are observed in the X-ray emitting sources. X-ray binaries (NSXBs or
BHXBs) exhibit intensity variations over a wide range of time scales ranging
from millisecond to months and years.

In a majority of the neutron star LMXBs, the magnetic field is weak (
$\leqslant$ $10^{9}$ gauss) but a small number of the neutron star LMXBs have
higher magnetic fields $\geqslant$ $10^{9}$ Gauss ($\sim$ $10^{12}$ Gauss).
The well known binary Her X-1 is an example of the LMXBs in which the neutron
star has a high magnetic field as inferred from the presence of the cyclotron
feature in its energy spectrum. These systems also exhibit regular X-ray
pulsations similar to those in the HMXBs. The LMXBs with low magnetic field
neutron stars, are thought to have evolved from high magnetic field neutron
star binaries in which the accretion has spun up the neutron stars over a long
period during which the magnetic field has significantly decayed. The
accreting millisec pulsars (spin periods $\sim$ msecs) are an example of the
spun up neutron stars from their progenitor LMXBs. The LMXBs with weak
magnetic field, are characterised by short orbital periods and sporadic or
regular short X-ray bursts (strictly not periodic), that have a typical
duration of about a few seconds to a minute or in rare cases longer.

The matter accretes from the companion star on the neutron star via Roche Lobe
overflow leading to the formation of an accretion disk around the neutron
star. From the inner part of the accretion disk, the matter falls on the
surface of the neutron star. The accreted matter piles up on the surface of
the neutron star forming a layer with increasing thickness. As the accretion
rate is variable, thickness of the layer in not uniform. When the accumulating
matter on the neutron star surface develops density and temperature conditions
in a localized region that ignite thermonuclear reactions a powerful X-ray
Burst occurs. These are called thermonuclear bursts. Initially the flame is
localized in a spot but it spreads quickly over the neutron star surface. Due
to uneven spread of the nuclear burning, the X-ray in-tensity is modulated due
to rapid spin of the neutron star. The intensity oscillations with frequency
of several hundred to about a kHz are detected in the LMXB bursts. This is one
of the ways to measure the spin periods of the neutron stars in the LMXBs.

Using the RXTE/PCA data, kHz QPOs were discovered from about a dozen LMXBs
(usually occurring in pairs) (van der Klis, 2001). The precise origin of kHz
QPOS is still not well understood. The LMXBs have multi-component continuum
energy spectra consisting of a disk blackbody component and a Thermal Compton
power-law. The thermal Compton part of the spectrum is most likely due to the
Compton up scattering of X-rays by hot electrons in a halo of accreted
material surrounding the disk.

A precise modeling of the continuum is essential to detect weak absorption
features known as Cyclotron Resonant Scattering Features (CRSFs) or commonly
referred as Cyclotron lines, that originate in the accretion column above the
magnetic poles. The CRSFs are detected in the strong magnetic field HMXBs as
well as in the spectra of the strong field LMXBs. In the neutron stars with
strong magnetic fields, the accreting matter is guided from the disk by the
magnetic field lines of the neutron star to its magnetic poles. The X-ray
spectrum emitted by the plasma in the accretion column is affected by the
magnetic field. This interaction gives rise to the CRSFs. The energy of
Cyclotron lines provides a direct measure of the dipole magnetic field of the
neutron star in the accretion column region.

The HMXBs usually have a pulsating X-ray source. The pulsation periods have a
range from $\sim$ 1 sec to hours. The slowest pulsar 2S 0114+65 has a spin
period of 2.7 hours. The spin rate of the pulsars is variable and depends on
the accretion rate. A study of the evolution of the pulsation periods of many
pulsars, shows that most pulsars exhibit both spin up as well as spin down
episodes. Depending on the accretion rate, a pulsar may make transition from
spin up to spin down and vice versa. In a class of X-ray Binaries, the optical
star is a Be star with a shell or disk of matter ejected by the companion.
Whenever the neutron star crosses the shell or disk during the course of its
orbital motion, the accretion rate shoots up resulting in dramatic increase in
brightness known as X-ray outbursts that occur once or twice in an orbital
period. Some times there is a sudden catastrophic release of matter from the
disk or shell of the Be star. This may be due to thermal instability that has
still not been well understood. This leads to a massive spike in the accretion
rate resulting in a giant outburst which is non-periodic and unpredictable.
This phenomenon also manifests in many X-ray transients that remain in a
dormant state with an exceedingly low accretion rate rendering them invisible.
This hibernation may last for years and even decades. Then due to reasons
still not understood, they spring back to life with giant X-ray outbursts
lasting for tens of days to months. QPOs of $\sim$ mHz to tens of Hz have been
detected in many HMXBs. Study of these QPOs provides insight into the
radiation environment closest to the neutron star.

In black hole X-ray binaries (BHXBs), accretion disk and relativistic radio
jets are integral part on all black hole mass scales and they provide simple
scaling of time and length with the mass of the black hole from supermassive
black holes in active galactic nuclei (AGNs) to stellar mass black holes in
BHXBs in our Galaxy. When accretion rate from the companion star is not
sufficient to support continuous accretion flow to the black hole, matter
fills the outer disk until a critical surface density is reached and an
outburst is triggered. An outburst in the BHXBs may last from $\sim$ 20 days
to several months. There are over 20 confirmed BHXBs and many more candidates
(Remillard & McClintock, 2006). Out of 20 BHXBs, 17 are transient X-ray
sources (mostly LMXBs) and the rest three are persistent bright X-ray sources
which have a massive O/B type stars as their companion (HMXBs) (Remillard &
McClintock, 2006). Many new BHXBs have been discovered with time (Mudambi
$et~{}al$., 2020a; Bhargava $et~{}al$., 2019), and MAXI shows now lightcurves
of around 40 BHXBs.

The BHXBs show different X-ray states at different stages of an outburst.
Initially, luminosity was the sole criteria to determine X-ray states (van der
Klis, 1994; Tanaka $et~{}al$., 1995). With the availability of larger data
sample during RXTE/PCA era, it became clear that X-ray states are not simple
function of the luminosity (Lewin & Van der Klis, 2006; Remillard &
McClintock, 2006). It is suggested that beside the mass accretion rate, there
must be at least one additional parameter which drives X-ray state transition.
The hardness-intensity diagram (HID) which shows a q-shaped track, has been
used to study the evolution of outbursts and associated X-ray states in the
BHXBs systems. The four distinct X-ray states identified in the HID are 1. Low
Hard state (LS), 2. High Soft state (HS), 3. Hard Intermediate State (HIMS),
and 4. Soft Intermediate state (SIMS) (Belloni, 2006). The LS state is
associated with relatively low accretion rate (lower than other bright X-rays
states) but can be observed at all luminosity. It is a radio loud state. The
energy spectrum is dominated by a hard power-law while power spectrum shows a
strong band-limited noise. The HS state is strongly dominated by thermal disk
component. Radio quenching is seen during this state. The intermediate states
(HIMS $\&$ SIMS) have significant contributions from both the hard power law
and the thermal disk components and show the most complex variability
characteristics including most of the quasi-periodic oscillations(QPOs).
Transient radio jets are seen during HIMS to SIMS transition. Remillard &
McClintock (2006) provided alternative scheme of X-rays states in BHXBs in
terms of Hard State (HS), Thermal State (TS) and Steep Power Law (SPL) state.
Here major difference is the SPL state which has significant contributions
from Thermal as well as non-thermal emissions.

###  1.2 LAXPC Instrument and its features to study Temporal and Spectral
properties :

Figure 1: Astrosat observations during observing cycles A2-A9 showing
percentage of time devoted to different instruments as primary instrument.

Large Area X-ray Proportional counter (LAXPC) instrument is one of the major
instruments onboard Astrosat which is India‘s first space science observatory
(Agrawal, 2006). There are three X-ray instruments on-board AstroSat which
cover a wide energy band. These X-ray instruments are: (i) LAXPC instrument
(Yadav $et~{}al$., 2016b; Agrawal $et~{}al$., 2017; Yadav $et~{}al$., 2017)
covering 3-80 keV region, (ii) a Cadmium-Zinc-Telluride Imager (CZTI; Bhalerao
$et~{}al$. (2017)) array covering 30-100 keV, and (iii) a Soft X-ray Imaging
Telescope (SXT; Singh $et~{}al$. (2016)) covering 0.3 - 8 keV. Beside these
X-ray instruments, there is an Ultra-Violet Imaging telescope (UVIT: Tandon
$et~{}al$. (2017)). All the above instruments are co-aligned to provide
simultaneous multi wavelength observations from optical to hard X-ray. There
is also a Scanning Sky Monitor (SSM) onboard Astrosat. The LAXPC instrument
uses three co-aligned identical LAXPC detectors (LAXPC10, LAXPC20 and LAXPC30)
to achieve large area. The performance of LAXPC detectors are discussed by
Antia $et~{}al$. (2020) in the same issue. A high detection efficiency in the
entire 3-80 keV energy region is achieved by having a 15 cm deep detection
volume divided in 5 identical Anode Layers and filling the Xe+CH4 gas at two
atmosphere pressure (1520 torr).

The LAXPC instrument was designed to meet the objectives of studying the
intensity variations with time, have a high detection efficiency in 3-80 keV
band, provide a large effective area over the broad energy range to be able to
study weak ($\sim$ a few milliCrab) sources and have a good spectral
resolution to measure the X-ray continuum spectra of X-ray binaries with
precision for deciphering the presence of weak cyclotron lines in the NSXBs. A
unique feature of the LAXPC is that every detected photon is time tagged to an
accuracy of 10 $\mu$sec to enable investigation of rapid intensity variations
over even sub-millisec scale. This feature provides LAXPC with the capability
to measure high frequency QPOs of even a few kHz. Good energy resolution (FWHM
$\sim$ 15% at 60 keV) was achieved by using an onboard gas purifier system and
continuously monitoring of the resolution of the Veto layer by shining 60 keV
X-rays on it from an Am241 source. We have done extensive lab as well as in
orbit calibration of LAXPC detectors (Antia $et~{}al$., 2017). Further
improvements and updates on LAXPC detector calibration and background are done
(Antia $et~{}al$., 2020; Misra $et~{}al$., 2020b) (this issue).

Since October 2016, AstroSat observatory has operated on the basis of targets
selection through scientific proposals. during the observing cycles A02-A09 in
the last 5 years. A comparison chart for the observations with different
instruments as primary instrument is shown in Figure 1. Primary instrument is
defined in the Astrosat science proposals (as science proposal requirement).
It may be noted that simultaneous data from all the instruments will be
available for a given source pointing if all the instruments are on during the
observation.

##  2 Highlight of Results on Neutron Star X-ray Binaries (NSXBs):

Large number of X-ray binaries with an accreting Neutron star as the X-ray
source, have been observed with the LAXPC instrument to investigate their
timing and spectral characteristics. Due to various reasons, briefly discussed
in Antia et al( 2020), less than 50% data of the sources have been analyzed so
far. Nevertheless, several new and interesting results have emerged from these
studies. Here we summarize highlights of some of these results.

Figure 2: Evolution of the Pulsar period of 4U 1909+07 during 2001-17.

###  2.1 Timing Properties : Spin Periods and their evolution, Low and High
Frequency QPOs.

The spin periods of the neutron stars are affected by the accretion rate and
the resulting accretion torque that varies with times. Usually the low and
high frequency QPOs are found in the LMXB pulsars while the low frequency QPOs
are more common in the HMXB pulsars. Astrosat has studied several neutron star
binaries to investigate their spin rates and their evolution with time. LAXPC
has observed a large number of HMXBs to investigate their periodic and
aperiodic variations like determination of the spin periods and their time
evolution, QPOs and their origin We summarize some of the significant results
derived from Astrosat.

The LMXB 3A 1822-371 has an orbital period of 5.2 hours and pulsates with 0.59
sec period (Jonker & van der Klis, 2001). The spin period from 1996 and 1998
data leads to a spin-up rate of -2.85$\times$ -12 s s-1. Subsequently
measurements of the spin period from several satellite observations have shown
that the pulsar is monotonically spinning up implying that the accretion rate
has been constant over $\sim$ 20 years. Astrosat LAXPC observations of 3A
1822-371 in September 2016 yielded a barycenter corrected pulsar spin period
to be Pspin =0.5914907$\pm$0.0000003 s confirming that the pulsar is still
spinning up (Amin $et~{}al$., 2020) Making a linear fit to all the measured
spin periods, gives a spin up rate=(-2.62$\pm$0.02)$\times$10-12 s s-1 giving
a spin up timescale (P/P)=7169 yr in agreement with the previous results
reported by Jain $et~{}al$. (2010).

Astrosat also studied accretion powered 2.26 millisecond pulsar SAX
J1748.9-2021, when a short and faint outburst occurred in it in 2017, with the
SXT and LAXPC instruments. From the spectral and timing analysis of the LAXPC
data,,the best-fitting orbital solution for the 2017 outburst was derived.
Using this an average local spin frequency of 442.361098(3) Hz was obtained.
The pulse profile obtained in 3-7 keV and 7-20 keV gave a constant fractional
amplitude $\sim$ 0.5% in contrast to earlier reported energy dependent
profile. The combined SXT+LAXPC spectrum in 1-50 keV showed the source to be
in a hard spectral state. This spectrum is best fitted with a single
temperature blackbody and thermal Comptonization model. Time resolved analysis
of the bursts revealed complex evolution in emission radius of blackbody for
the second burst suggestive of a mild expansion of photospheric radius.

The HMXB Pulsar 4U 1909+07 studied with the Astrosat-LAXPC observation of 2017
July, showed presence of 604 sec X-ray pulsations. Using the spin period
measurements obtained earlier with the various X-ray satellites and the period
deduced from the present work, a plot of Spin Period versus Time is shown in
Fig 2 (Jaisawal $et~{}al$., 2020b). Despite a brief episodes of spin-down,
there is a clear long term trend of spin-up. The pulse period changed from
604.7 s to 603.6 s between 2001-2017 resulting in an average spin-up rate of
1.71$\pm$10-9 ss-1. The pulse profiles show strong energy dependence evolving
from a complex broad structure in soft X-rays into a profile with a narrow
emission peak followed by a plateau in energy ranges above 20 keV. The change
in the pulse profile suggests a change in the beaming pattern.

Another HMXB 4U 1907+09 with a spin period of $\sim$ 404 sec was studied by by
using the LAXPC observations of 4th and 5th June (2020). Timing analysis of
the LAXPC data yielded a period of 442.33$\pm$0.07 sec. Two peaks of the pulse
profile of 4U 1907+09 shown in Fig 3 (left panel), exhibited different energy
dependence, the pulsed fraction of the main peak increased till about 40 keV
and decreased after that while the secondary peak disappeared at energy above
about 20 keV. Energy resolved pulse profiles created from combined data of the
three LAXPCs are shown in Fig 3 (right panel). The pulsar spin-down was found
to continue (Varun $et~{}al$., 2019b).

Figure 3: The pulse profile of 4U 1907+09 has double peak shape shown in the
left panel. The energy resolved pulse profiles created from combined data of
the three LAXPCs are shown in the right panel. Figure 4: The background
subtracted light curves from the 3 LAXPCs showing clear intensity oscillations
of $\sim$ 1 mHz and $\sim$ 1.7 mHz in the X-rays from the HMXB Pulsar 4U
0115+63

The Be binary with 194 sec pulsar GRO 12058+42 underwent a massive outburst in
April 2018 and was observed by Astrosat to investigate its timing and spectral
features. The SXT and LAXPC showed strong pulsations with a period of
194.2201$\pm$0.0016 s, and a spin-up rate of (1.65$\pm$0.06)$\times$10-11 Hz
s-1 Mukerjee $et~{}al$. (2020). Pulse profiles in 3-80 keV were found to be
energy dependent. The Power Density Spectrum (PDS) of the source revealed a
0.090 Hz QPO and its higher harmonics. Another HMXB pulsar OAO 1657-415 with
10.4 day orbital period, was studied by Jaisawal $et~{}al$. (2020a) with two
LAXPC observations in March and July 2019. The observations covered orbital
phases of 0.681-0.818 and 0.808-0.968. Despite being outside the eclipsing
regime, the PDS from the first data did not show any clear signature of
pulsation or quasi-periodic oscillations. However, during the second
observation, the X-ray pulsations at a period of 37.0375 s. were clearly
detected in the orbital phase range 0.808-0.92. The pulse profile of the
pulsar from the second observation consisted of a broad single peak with a
dip-like structure in the middle all across the observed energy range. The
evolution of the emission geometry was probed by constructing the pulse
profile in narrow time and energy segments. The energy spectrum of OAO
1657-415 is approximated by an absorbed power-law model with an iron
fluorescent emission line. These results are explained in the frame of stellar
wind accretion model.

Another HMXB X-ray binary pulsar 3A 0726-260 (4U 0728-25) was observed on 2016
May 6-7 with the LAXPC and SXT, after a gap of almost 20 years. The light
curves of the binary show Strong X-ray pulsations with a period of
103.144$\pm$0.001 seconds in 0.3-7 keV with the SXT and in 3-40 keV with the
LAXPC. The pulse profiles are energy dependent, and there is an indication
that the pulse shape changes from a broad single pulse to a double pulse at
higher energy. At energies above 20 keV pulsations with a period
103.145$\pm$0.001 seconds are detected for the first time and a double peaked
pulse profile is observed from the source. The energy spectrum of the source
is derived from the combined analysis of the SXT and LAXPC spectral data in
0.4-20 keV. The best spectral fit is obtained by a power law model with a
photon index (1.7$\pm$0.03) with high energy spectral cut-off at 12.9 $\pm$
0.7 keV and a broad Iron line at $\sim$ 6.3 keV (Roy $et~{}al$., 2020).

Detection of $\sim$ 1 mHz and $\sim$ 1.7 QPO in the Be X-ray Binary 4U 0115+
63 were reported by Roy $et~{}al$. (2019) using the Astrosat-LAXPC observation
of the HMXB. The QPOs were detected on 2015 October 24 during the peak of a
giant type II outburst. Prominent intensity oscillations seen at $\sim$ 1
and$\sim$ 1.7 mHz are shown in Fig 4. Cyclotron resonant absorption feature at
11 keV and its 3 harmonics are detected in the energy spectra. Possible models
to explain the origin of the $\sim$ mHz oscillations are examined. These
oscillations bear resemblance to the intensity oscillations observed from some
other neutron star and black hole sources and may have a common origin.
Current models to explain the instability in the inner accretion disk causing
the intense oscillations were examined and found to be inadequate.

###  2.2 Energy Spectra, Detection and Study of CRSFs:

LAXPC has observed a large number of HMXBs to investigate their periodic and
aperiodic variations and measure their continuum spectra and their
characterization to reveal the presence of Cyclotron lines (CRSFs). Thus far
Astrosat-LAXPC has discovered Cyclotron lines in a few pulsars in which
earlier there was no report of the presence of CRSF. Study of cyclotron line
energy and its profile enable determination of the magnetic field of the
neutron star and at the site of their origin in the accretion columns from the
following relation

$E_{c}=11.6(B/10^{12}G)(1+z)^{-1}keV$ (1)

E = Energy of the Cyclotron line, B = Magnetic field of the neutron star in
unit of 1012 gauss and z is gravitational red shift due to neutron star.

There are reports of shift of the line energy with time and correlation of
this with the X-ray luminosity. The results on this are conflicting and
reality of variation of line energy is still an open question.

Her X-1 was the first source in which Truemper $et~{}al$. (1978) discovered a
CRSF usually termed as Cyclotron line at $\sim$ 40 keV. The CRSF energy was
found to vary with pulse phase, X-ray luminosity, the phase of 35-day
precession cycle and with time (Staubert $et~{}al$., 2014). Using data
acquired from several X-ray satellites over a 20 year period, the CRSF energy
was inferred to decline by 4.5 keV from 1996 till 2012 (Staubert $et~{}al$.,
2014) as seen in Fig 5 from Bala $et~{}al$. (2020). After 2015 the trend was
reversed and the CRSF energy began to recover reaching a value of 37.4 keV
(Staubert $et~{}al$., 2019). Recent analysis of Her X-1 observations in 2018
with the LAXPC shows that since then the CRSF energy appears to be constant
around 37.5 keV (Bala $et~{}al$., 2020).

Figure 5: Shift in the energy of the Cyclotron line in Her X-1. The 2018
observation with the LAXPC gave a value of 37.5 keV.

Amin $et~{}al$. (2020) constructed the energy spectrum of the pulsar 3A
1822-371 using LAXPC10 and LAXPC20 data and fitted it using 3 different
continuum models. In all the cases an absorption feature at 23 keV was present
which is suggested to be a CRSF and if correct this implies a magnetic field
strength of B = (2.7-3.4) $\times$ $10^{12}$ Gauss for the neutron star. The
energy spectrum of HMXB 4U 1907+09 (spin period of $\sim$ 404 sec) was studied
by by (Varun $et~{}al$., 2019b) using the LAXPC observations of 4th and 5th
June (2020). A CRSF was detected in the spectrum at 18.5$\pm$0.2 keV . Pulse
phase-resolved spectroscopy was carried out with 10 independent phase bins and
variations of the CRSF parameters with pulse phase were found to be consistent
with previous studies.

The pulsar GRO J2058+42 was observed with the LAXPC. Its energy spectrum
extracted from the LAXPC data revealed presence of 3 CRSFs (Mukerjee
$et~{}al$., 2020). The cyclotron absorption features were detected in
(9.7-14.4) keV, (19.3-23.8) keV and (37.8-43.1) keV, one of which is the
fundamental line and the other 2 are harmonics The pulse phase resolved
spectroscopy of the source showed phase dependent variation in line energy and
relative strength of these features. An important finding from Astrosat is the
discovery of a Cyclotron line at $\sim$22 keV in the LAXPC energy spectrum by
of the HMXB pulsar 4U 1538-52 which has an orbital period of 3.75 days and the
spin period is currently $\sim$ 527 sec Varun $et~{}al$. (2019a).The Pulse
profile is double peaked at low energy and has a single peak in high energy
range, the transition taking place around the cyclotron line energy of the
source.

The CRSF is detected with a very high significance in the phase averaged
spectrum shown in Fig 6. It is one of the highest signal to noise ratio
detection of CRSF for this source. A detailed pulse phase resolved spectral
analysis with 10 independent phase bins was performed and parameters of the
continuum spectrum and CRSF parameters were derived. Theses show pulse phase
dependence over the entire phase with a CRSF energy variation of $\sim$13% in
agreement with previous studies. An increase in the centroid energy of the
CRSF observed between the 1996-2004 (RXTE) and the 2012 (Suzaku) observations,
is confirmed affirming that the increase in the line energy was a long-term
change.

AstroSat observed the Be/X-ray binary pulsar SXP 15.3 in the Small Magellanic
Cloud (SMC) during its outburst in late 2017, when the source reached a
luminosity level of $\sim$ $10^{38}$ erg s-1. Timing and spectral analysis
between 3 and 80 keV lead to the pulse profile that exhibits a significant
energy dependence. The pulse shape changes from a double peaked profile to a
single broad pulse at energies $>$ 15 keV. The energy spectrum suggests
presence of a Cyclotron Resonance Scattering Feature (CRSF) at $\sim$5 keV and
is independent of the choice of the continuum model. This feature is also
detected in the spectrum obtained by the NuStar. Till now CRSF has been
reported in about 36 X-ray pulsars in binaries (Maitra, 2017; Staubert
$et~{}al$., 2019). To the best of our knowledge this is the lowest energy
Cyclotron line confirmed in any pulsar. This indicates a magnetic field
strength of 6$\times$ $10^{11}$ G for the neutron star (Maitra $et~{}al$.,
2018).

Figure 6: The energy spectrum of the pulsar 4U 1538-52 derived from the LAXPC
data. Top panel shows the observed spectrum by the Black points and the best
fitted spectrum by a Red line. The middle panel shows Residuals when one fits
only the continuum spectrum and the bottom panel shows fit with continuum and
a Cyclotron line at 22 keV.

###  2.3 Thermonuclear Bursts and QPOs in LMXBs:

The thermonuclear bursts (or commonly called as Type-1 bursts) occur in the
LMXBs having a weakly magnetized ($<$ 109 G) neutron star. Detailed timing and
spectral studies of these bursts provides information about the spin period,
temperature of the thermonuclear burning surface and radius of the neutron
star. The event mode data in which each photon is time tagged to an accuracy
of about 10 microsec, enables studies of high frequency phenomenon like kHz
oscillations, Some significant results on the Type-1 bursts observed with the
LAXPC are summarized below.

LAXPC capability for high time resolution studies of phenomenon like detection
of Coherent Oscillations , kHz QPOs etc has been demonstrated by Verdhan
Chauhan $et~{}al$. (2017). They analyzed the LAXPC observations of 8th March
2016 for the LMXB source 4U 1728-24. A 3 ks data stretch revealed occurrence
of a typical Type-1 burst of about 20 sec duration. Dynamical power spectrum
of the data in the 3-20 keV band, shows presence of Coherent burst oscillation
whose frequency increased from 361.5 to 363.5Hz consistent with an earlier
result showing the same spin frequency of the neutron star. Our knowledge of
the spin periods of the neutron stars in LMXBs can also be derived from
detection of the Coherent Burst Oscillations in the initial phase of the
bursts. A kHz QPO, whose frequency drifted from $\sim$ 815 Hz to $\sim$ 850
Hz, was detected just before the burst. The QPO was detected below 10 keV as
well as in 10-20 keV band, which was not possible with the RXTE. Even for a
short observation with a drifting QPO frequency, the time-lag between the 5-10
and 10-20 keV bands can be constrained to be less than 100 microseconds.

Beri $et~{}al$. (2019) studied the LMXB 4U 1636-536 using 65 ks LAXPC
observation over 2 days and detected seven thermonuclear X-ray bursts
including a rare triplet of X-ray bursts. Time resolved spectroscopy performed
during these seven X-ray bursts suggested the presence of Photospheric Radius
Expansion in three of these X-ray bursts. A transient QPO at 5 Hz was also
detected. No evidence of kilo-hertz QPOs or coherent burst oscillations, was
found in the bursts and may perhaps be due to the hard spectral state of the
source.

Effects of Thermonuclear X-ray burst on non-burst emissions in the soft state
of the LMXB 4U 1728-34 was investigated by Bhattacharyya $et~{}al$. (2018) to
understand if a significant fraction of the burst emission, which is
reprocessed, contributes to the changes in the persistent emission during the
burst. This is important since it can introduce significant systematics in the
neutron star radius measurement using burst spectra. Analyzing the bursts data
for 4U 1728-34 in the soft state it was concluded that the burst emission is
not significantly reprocessed by a corona covering the neutron star.

Devasia $et~{}al$. (2021) detected 5 Type-1 thermonuclear X-ray bursts and one
burst-like event in the neutron star LMXB source Cygnus X-2 using LAXPC X-ray
data. An energy resolved burst profile analysis and time resolved spectral
analysis for each of the bursts was performed to characterize the burst
properties. An evolution of the blackbody temperature and radius is also
observed during each burst. A search for Coherent Burst Oscillations gave only
upper limits. A study of the hardness-intensity and color-color diagrams show
that during the 2016 LAXPC observation, Cygnus X-2 was in the early Flaring
Branch (FB).

The LMXB 4U 1323-62 is an interesting source from which periodic X-ray dips
were first detected by EXOSAT van der Klis & Jansen (1985); Parmar $et~{}al$.
(1989). It was extensively studied by PCA/ RXTE and a study of 40 Type 1
bursts showed a recurrence time of 2.45-2.59 hr. which is probably the orbital
period of the binary. The LMXB 4U 1323-62 was observed with the LAXPC and
detected 6 Type-1 thermonuclear X-ray bursts in $\sim$ 50 ks exposure. The
time gap between the successive thermonuclear bursts was found to be
consistent with the orbital period. Using the linear and orbital quadratic
ephemerides, the value of the Orbital period is estimated to be 2.65 to 2.70
h. The gap observed between the bursts in two case, is nearly double the wait
time of consecutive bursts as 2 bursts were missed in the data gaps. The
nearly fixed time gap is the time required to accumulate the accreted matter
to reach the level at which the thermonuclear reactions set in producing a
burst. The light curve of 4U 1323-62 also revealed the presence of two dips. A
known low Frequency QPO (LFQPO) was detected at $\sim$1 Hz, from the source.
However, no evidence of kHz QPO was found. The radius of the blackbody is
found to be consistent with the blackbody temperature and the black-body flux
of the bursts (Bhulla $et~{}al$., 2020).

The spectral and timing properties of the atoll source 4U 1705-44 were studied
by Agrawal $et~{}al$. (2018) using 100 ks data from the LAXPC instrument. The
source was in the high-soft state during the LAXPC observations and traced out
a banana track in the Hardness Intensity Diagram (HID). From the Power Density
Spectra (PDS) a broad Lorentzian feature centered at 1-13 Hz and a very low
frequency noise (VLFN) is detected. The energy spectra are well described by
sum of a thermal Comptonized component, a power-law and a broad (FWHM $\sim$2
keV) Iron line having equivalent width (EW $\sim$ 369-512 eV). Only
relativistic smearing in the accretion disc can not explain the origin of this
feature. A systematic evolution of the spectral parameters eg optical depth,
electron temperature etc. is seen as the source moves along the HID. Search of
correlation between frequency of the Broad Lorentzian and the spectral
parameters seems to suggest that the frequency varies with the strength of the
corona.

The structure of Corona of the well known Z source GX 17+2 was studied by Malu
$et~{}al$. (2020) in 0.8-50 keV using the Soft X-ray Telescope (SXT) and the
LAXPC data. For the first time, Cross-correlation studies were performed using
SXT soft and LAXPC hard light curves and they exhibited correlated and anti-
correlated lags of the order of a hundred seconds. Spectral modeling gave disk
radius of $\sim$ 12-16 km indicating that disk is close to the ISCO and a
similar value of disk radius was deduced based on the reflection model. Corona
size was inferred to be 27-46 km and 138-231 km depending on the model used.
Size of the X-ray emitting Boundary Layer (BL) was determined to be 57-71 km.
The observed lags and no movement of the inner disk front strongly indicates
that the varying corona structure is causing the X-ray variation in the Normal
Branch (NB) of Z source GX 17+2.

Bhulla $et~{}al$. (2020) investigated evolution of timing and spectral
properties of the bright Z-source GX 5-1 using February, 2017 observations
with the SXT and LAXPC instruments. The 0.8-20 keV spectra from simultaneous
SXT and LAXPC data at different locations of the hardness-intensity plot is
well described by a disk emission and a thermal Comptonized component. The
disk flux ratio (ratio of the disk flux to the total flux) increases
monotonically along the horizontal branch to the normal one. Thus in the
normal branch, the disk dominates the flux while in the horizontal branch the
Comptonized component dominates. The disk flux scales with the inner disk
temperature as T5.5 and not as T4 suggesting that either the inner radii
changes dramatically or that the disk is irradiated by the thermal component
changing its hardness factor. The PDS reveal a QPO whose frequency changes
from $\sim$ 30 Hz to 50 Hz and which is found to correlate well with the disk
flux ratio. In the 3-20 keV LAXPC band the r.m.s of the QPO increases with
energy (r.m.s / E0.8), while the harder X-ray seems to lag the soft ones with
a time-delay of a milliseconds. The results suggest that the spectral
properties of the source are characterized by the disk flux ratio and that the
QPO has its origin in the corona producing the thermal Comptonized component.

The results inferred, mainly from the LAXPC data, on various timing and
spectral properties of LMXBs summarized here, demonstrate the capability of
the LAXPC instrument for probing the bright as well as faint sources It is
hoped that in the coming years many more LMXBs will be studied and new results
will emerge from these studies.

###  2.4 Studies of New Ultraluminous X-ray (ULX) Pulsars with Astrosat:

The Ultraluminous X-ray (ULXs) sources detected in the Galaxy and other
galaxies were believed for a long time to be Intermediate Mass Black Holes
accreting matter from the companion star. This changed when Pulsations with
1.37 sec period were discovered from the ULX in M82 . Since then more ULX
Pulsars have been discovered and at present 8 ULX Pulsars have been discovered
and their properties are summarized in Table 1 taken from Chandra $et~{}al$.
(2020). Astrosat discovered ULX Pulsar nature of a transient X-ray source RX
J0209.6-7427 located in the Magellan Bridge when it became active after a gap
of 26 years in 2019 and was studied by Chandra $et~{}al$. (2020) with the SXT
and LAXPC instruments on Astrosat. The transient ﬁrst detected by ROSAT during
its 1993 outburst, went into a deep hibernation for 26 years and suddenly
sprang back to life in 2019 with a giant outburst in 2019.

Table 1: Summary of the 8 known ultraluminous X-ray pulsars. Name of ULX | Host Galaxy | Spin period (s) | Orbital period (days) | Spin-up/down |  $L_{X}~{}(10^{39}$ ergs s${}^{-1})$
---|---|---|---|---|---
M82 X-2 | M82 | 1.37 | $\sim 2.5$ | Spin-up | 4.9
NGC 7793 P13 | NGC 7793 | $\sim 0.42$ | 64 | Spin-up | $\sim 10$
NGC 5907 ULX | NGC 5907 | $\sim 1.13$ | 5.3 | Spin-up | $\sim 100$
NGC 300 ULX1 | NGC 300 | $\sim 31.6$ | - | Spin-up | 4.7
Swift J0243.6+6124 | Milky way | $\sim 9.86$ | $\sim 27.6$ | Spin-up | $\sim 2$
M51 ULX-7 | M51 | $\sim 2.8$ | $\sim 2$ | Spin-up | $\sim 10$
NGC 1313 X-2 | NGC 1313 | $\sim 1.5$ | - | Spin-up | $\sim 20$
RX J0209.6-7427 | SMC | $\sim 9$ | - | Spin-up | $\sim 1.6$
Figure 7: Evolution of the Pulsar period during the outburst of RX
J0209.6-7427 in 2019 is shown in this plot. Spin-up of the pulsar during the
outburst in clearly seen.

Using the SXT and LAXPC observations, Chandra $et~{}al$. (2020) detected
strong pulsations in RX J0209.6-7427 with 9.29 s periodicity over a broad
energy band covering 0.3-80 keV (first reported by the NICER mission in the
0.2-10 keV energy band). The pulsar exhibited a rapid spin-up during the
outburst as can be seen from Fig 7. Energy resolved folded pulse profiles were
generated in several energy bands in 3-80 keV. This is the first report of the
timing and spectral characteristics of this Be binary pulsar in hard X-rays.
There is suggestion of evolution of the pulse profile with energy. The energy
spectrum of the pulsar is determined and from the best-fitting spectral
values, the X-ray luminosity of RX J0209.6-7427 is inferred to be
1.6$\times$1039 erg s-1. These timing and spectral studies suggest that this
source has features of an ultraluminous X-ray pulsar in the Magellan Bridge.

Figure 8: The pulse profiles derived from SXT (0.5-7.0 keV), LAXPC (3-80 keV)
and CZTI (30-150 keV) are shown in the figure for the ULX Pulsar Swift
J0243.6+6124.

A second ULX Pulsar Swift J0243.6+6124, the first Galactic ultra-luminous
X-ray pulsar, was observed during its 2017-2018 outburst with AstroSat at both
sub- and super-Eddington levels of accretion with X-ray luminosities of Lx
$\sim$ 7$\times$1037 and 6$\times$1038 erg s-1. A detailed Broadband timing
and spectral study by Beri $et~{}al$. (2020) show that X-ray pulsations at
$\sim$ 9.85 s have been detected up to 150 keV when the source was accreting
at the super-Eddington level. The background subtracted pulse profiles for the
the SXT, LAXPC and CZTI instruments are shown in Fig 8 :

The pulse profiles are a strong function of both energy and source luminosity,
showing a double-peaked profile with pulse fraction increasing from 10% at
1.65 keV to 40-80 % at 70 keV. The continuum X-ray spectra are well-modeled
with a high energy cut-off power law ($\alpha$ $\sim$ 0.6-0.7) and one or two
blackbody components with temperatures of $\sim$ 0.35 keV and 1.2 keV,
depending on the accretion level. No iron line emission is observed at sub-
Eddington level, while a broad emission feature at around 6.9 keV is observed
at the super-Eddington level, along with a blackbody radius (121-142 km) that
indicates the presence of optically thick outflows.

Results on Neutron Star Binaries summarized above establish firmly the
capability of the LAXPC instrument for high resolution timing and spectral
studies. Large effective area enables studies of relatively fainter sources
like ULX Pulsars.

##  3  Study of Black Hole X-ray Binaries (BHXBs) with LAXPC/Astrosat

As has been mentioned earlier, the advantages that AstroSat/LAXPC has over
RXTE are (i) an higher effective area at energies $>$ 30 keV, (ii) event mode
data allowing for temporal analysis in arbitrary user defined energy bins and
(iii) simultaneous observations from other instruments on board. These have
primarily defined and driven the LAXPC study of Black hole binaries.

It has been known that BHXBs are variable on different time-scales and exhibit
both broad band continuum noise and narrow features termed as Quasi-periodic
oscillations. While there have been several models to explain the dynamic
origin of these variabilties, a consensus and conclusive explanation for them
remains illusive. One of the possible reasons for this impasse is that the
radiative processes that are involved in these variations have not been
clearly identified. The time-averaged spectra of black hole binaries are
usually represented by physically motivated radiative components such as disk
emission, thermal comptonization, and refection components.

The broad band spectrum from AstroSat allows for fitting of complex models
such as General Relativistic disk emission and blurred reflection components,
which can be used to constrain physical parameters such as the spin of the
black hole and the accretion rate. It is also imperative to identify which of
these components are responsible for the variability and in particular to
identify the spectral parameters which may be changing that cause the observed
variability. It is likely that more than one of these spectral parameters are
involved and estimating any time difference between the variations of these
parameters can provide critical information regarding the illusive dynamic
origin of phenomena. AstroSat observations of BHXBs can provide these
estimates paving the way for an emerging field where variability is quantified
and understood in terms of the active radiative processes of the system. Here
we present some of the scientific results obtained so far. It should be
emphasised that these results were possible due to the unique capabilities of
LAXPC (wide band spectroscopy, higher efficiency in the hard X-rays and fine
time resolution) and the simultaneous spectra available from SXT.

After decommissioning of RXTE in 2012, AstroSat/LAXPC is the only X-ray timing
instrument best suited for studying spectral-temporal characteristics of black
hole X-ray binaries (BHXBs). During last five years, LAXPC successfully
observed many BHXBs and very interesting results have come out. Here we
discuss some of BHXBs results which were observed on several occasions with
LAXPC instrument and in-depth analysis were performed. In §3.1-3.5 we present
the primary results from LAXPC grouped on the basis of broad topics which
highlight the advantage of the instrument. In §3.6, we enumerate these results
in a different degree of detail, for the individual black hole sources that
have been observed by LAXPC.

###  3.1 Energy dependent temporal properties:

Typically the variability is quantified by the power spectrum which is the
square of the Fourier transform of a lightcurve obtained over a certain energy
range. If normalized in a certain way, the integration of the power spectrum
(over some frequency range) gives the variance of the variation in that energy
and frequency range. Another useful quantity is the fractional root mean
square, (frms) which is the square root of the variance normalized the mean of
the lightcurve. It is insightful to consider this as a Data Cube where the
power is provided as a function of frequency and energy. Collapsing the Data
Cube over energy (or a given range of energy) provides the power spectrum in
that energy range. On the other hand collapsing the Data Cube over a given
range frequency, provides the frms as a function of energy for that frequency
range. Moreover, cross-frequency analysis between energy bands provides time-
lag as a function of energy.

LAXPC can efficiently provide frms and time-lag as a function of energy for
different frequency bands as shown in the early analysis of the black hole
systems GRS 1915+105 (Yadav $et~{}al$., 2016a) and Cygnus X-1 (Misra
$et~{}al$., 2017) Encouraged by these unprecedented results, attempts were
made to develop formalisms which could obtain physically meaningful radiative
parameter variations and time-lag between them. In the hard state, the spectra
of Cygnus X-1 above 3 keV, is dominated by thermal Comptonization. Maqbool
$et~{}al$. (2019). developed a model to predict the variability of such a
spectrum when the seed photon flux and the heating rate of the corona varies
with a time-lag between them. Fitting the predictions to the observed time-lag
and frms as a function of energy, led them to quantify the variability in the
seed photon flux and the coronal heating rate as a function of frequency.
Mudambi $et~{}al$. (2020a) used the same analysis to constrain the variability
of these quantities for the bright transient X-ray binary, MAXI J1820+070 in
its hard state observation by AstroSat/LAXPC allowing for a comparison with
Cygnus X-1. While these analysis are for the continuum variability, Jithesh
$et~{}al$. (2019) applied the model to a QPO observed by AstroSat LAXPC in the
hard state of the black hole system, SWIFT J1658.2-4242.

The model used in these works mentioned above, is only applicable to the
thermal Comptonization component, while a thermal component is often observed
in the spectra of black hole systems which are not in a pure hard state. A
generalized method which can predict energy dependent variability properties
for arbitrary spectral components is challenging since (i) the response of the
spectrum to parameter variation has to be done numerically and (ii) the
spectral parameters have to be cast into physical ones rather than empirical
ones (for e.g. heating rate of corona instead of say the asymptotic power-law
index for Comptonization models). An initial step in this direction has been
taken by Garg $et~{}al$. (2020) who have fitted the energy dependent frms and
time-lag of the QPO observed in GRS 1915+105 by AstroSat/LAXPC.

###  3.2 Variations of timing and spectral properties and their correlation :

An important and promising use of AstroSat/LAXPC data is the study of the
variation of timing features and their correlation with spectral parameters.
The sensitivity of LAXPC to detect small variation in rapid timing properties
was proved by the first time detection of the small $\sim 7$% variation of the
high frequency ($\sim 70$ Hz) QPO in GRS 1915+105 (Belloni $et~{}al$., 2019),
which although small has the potential to differentiate between generic
models. The QPO has also been reported by Sreehari $et~{}al$. (2019) who
showed that its presence depends on the strength of the high energy spectral
component.

The correlation with spectral parameters was demonstrated by Bhargava
$et~{}al$. (2019) where a long monitoring of the black hole transient, MAXI
J1535-571, showed a tight correlation between the QPO frequency with the high
energy spectral index rather than the flux. An important expected correlation
is that between QPO frequency and the inner radius of a truncated disk, since
a physical characteristic time-scale associated with the radius maybe
responsible for the QPO. Using LAXPC and SXT observations of the black hole
system GRS 1915+105, Misra $et~{}al$. (2020a) showed that there is such a
correlation and identified the QPO frequency with the dynamical time-scale
corrected by General Relativity as predicted decades ago. This important
result was possible because of the broad band spectral coverage by SXT and
LAXPC and the simultaneous rapid timing information provided by LAXPC.

###  3.3 Long term variability :

Long observations of BHXBs by LAXPC, which could be continuous exposure for
more than a day, or monitoring of sources on weeks/months time-scale have
provided extensive information regarding the long term behaviour of these
sources. A good example is the enigmatic black hole system GRS 1915+105 and
AstroSat was fortunate to observe it during a transition from a non-variable
class to a structured large amplitude one (Rawat $et~{}al$., 2019). This
allowed for tracking of the QPO and its energy dependent property rms and
time-lag as the source made the transition. The transient Swift J1658.2-4242
shows ’flip-flop’ state transitions on time-scale of minutes, which is
reflected both in the spectra and power density spectrum was studied by LAXPC
and other instruments (Bogensberger $et~{}al$., 2020).

Monitoring of an outburst from a black hole binary can be used to track the
spectral and timing evolution as has been done for two outbursts of 4U
1630-472 (Baby $et~{}al$., 2020). The study inferred the appearance of the
standard disk after a few hours of the burst and its persistence as the source
evolved to the soft state. Monitoring observations of Cygnus X-3 by LAXPC has
provided important clues on the formation of the radio jet and its connection
to the accretion disk (Pahari $et~{}al$., 2018b). The observations of Cyg X-3
provides a measurement of the orbital period and the discovery of low
frequency mHz QPO whose energy dependent rms and time-lag could be quantified
(Pahari $et~{}al$., 2017).

###  3.4 Broadband Spectral Fitting :

Although LAXPC spectral resolution is lower than that of other instruments
such as Nustar, however it does provide a wide energy band especially when
combined with SXT data. Hence AstroSat observations have been used to fit
complex spectral models to constrain system parameters. For example, after
fitting standard models to the AstroSat spectra of the black hole transient
MAXI J1535-571, Sreehari $et~{}al$. (2019), fitted the two component accretion
flow model to the data and found that the accretion rate to be nearly at the
Eddington value and could further constrain the mass of the black to be around
$\sim 6$ solar masses. For the same system, Sridhar $et~{}al$. (2019), used
the relativistic disk and blurred reflection model to constrain the spin,
distance to the source and mass of the black hole to be $\sim 10$ solar
masses. The different mass estimate are due to differences in the model and
other assumptions, but what is perhaps important is the ability of these
different models to make quantitative estimates given AstroSat data. Black
hole mass has also been estimated using two component flow for the black hole
binary 4U 1630-472, where the utility of long term monitoring of a source as
has been demonstrated. The extra galactic black hole systems LMC X-1 has a
well constrained distance and black hole mass and hence AstroSat data has
proved useful to constrain the black hole spin (Mudambi $et~{}al$., 2020b).
Relativistic disk fitting of well studied sources like GRS 1915+105, can be
used to confirm the black hole mass and to constrain the spin (Sreehari
$et~{}al$., 2020).

###  3.5 Synergy with other observatories:

Perhaps the most promising use of AstroSat data is when they are analyzed in
conjunction with data from other instruments of observatories. The
complementary nature of the different instruments provides a multi-faceted and
unprecedented view of the systems. Since black hole binaries are variable
simultaneous (or quasi-simultaneous) data would be optimal.

A good example of using the different capabilities of instruments was the
study of the black hole system 4U 1630-47 using the high energy grating
spectra from Chandra along with AstroSat’s LAXPC and SXT data (Pahari
$et~{}al$., 2018a). While the latter provided evidence for a highly ionized
wind, relativistic disk model fit to the continuum using AstroSat data showed
that the black hole is highly spinning, thus making an interesting connection
between wind outflow and black hole spin. Another example is when the
combination of high resolution Nustar data (3-80 keV) with AstroSat (SXT,
LAXPC and CZTI) (0.7-200 keV) of MAXI J1820+070 allowed for the use of more
complex and sophisticated models as compared to only AstroSat data (Mudambi
$et~{}al$., 2020a) or just Nustar data.

Even when the observations are not simultaneous, the use of multiple
instruments has proved extremely worthwhile as in the spectral and timing
study of the flip-flop state transitions of Swift J1658.2-4242 using a host of
instruments, XMM-Newton, NuSTAR, Swift, Insight-HXMT, INTEGRAL, ATCA and
AstroSat (Bogensberger $et~{}al$., 2020).

It is fortunate that at present along with AstroSat, there are two other
observatories capable of high time resolution operating namely NICER and
INSIGHT-HXMT. The three observatories together can provide unprecedented
timing information in an energy range (0.2-200 keV) which is about three order
of magnitude The improvement in our understanding that would be obtained by
such analysis is illustrated by Xiao $et~{}al$. (2019) where they performed
timing analysis of Swift J1658.2-4242 outburst in 2018 with Insight-HXMT,
NICER and AstroSat. They found a range of QPO activities detected by the
different instruments and quantified their dependence. The potential of such
analysis is clear, especially when one notes that the observations used by
them were often not strictly simultaneous. It is expected that soon, joint
analysis of AstroSat data with one or both of these instruments, will be
providing deeper insights into the nature of Black hole systems. The
importance having coordinated observations with all three observatories cannot
be over emphasised.

###  3.6 Important results of some of BHXBs with LAXPC instrument:

####  3.6.1 GRS 1915+105 :

GRS 1915+105 is a highly variable BHXBs (Yadav $et~{}al$., 1999; Belloni
$et~{}al$., 2000) with frequent radio emission (often referred as Microquasar)
(Fender $et~{}al$., 2004; Yadav, 2006). We have monitored this source
regularly. Yadav $et~{}al$. (2016a) have studied spectra and timing properties
of GRS 1915+105 when the source had the characteristics of being in Radio-
quiet during March 5-7, 2016.The energy dependent power spectra reveal a
strong low frequency (2 – 7 Hz) Quasi-periodic oscillation (LFQPO) and its
harmonic along with broad band noise. The QPO frequency changes rapidly with
flux. At the QPO frequencies, the time-lag as a function of energy has a non-
monotonic behavior such that the lags decrease with energy till about 15–20
keV and then increase for higher energies.

Figure 9: Color-color diagram during fast transition among three X-ray states.
Figure 10: Color-color diagram during fast transition among three X-ray
states.

Rawat $et~{}al$. (2019) have studied GRS 1915+105 during February and March,
2017 when the source underwent a major transition from a non-variable state
(close to $\chi$ class) to a periodic flaring state (similar to $\rho$ class).
It is shown that such transition takes place via an intermediate state when
the large-amplitude, irregular variability of the order of thousands of
seconds turned into a 100-150 sec nearly periodic flares similar to $\rho$
class/heartbeat oscillations. Figure 9 shows the color-color diagram during
this fast transition. It is interesting to note that HR2 remains same when
source transits to the intermediate state but when source finally attains the
flaring state, it regains the same HR1 but HR2 has changed. Belloni $et~{}al$.
(2019) have analysed 92 ks of data obtained with the LAXPC instrument and they
have detected around seven percent variation in High Frequency QPOs in GRS
1915+105. Banerjee $et~{}al$. (2020) have studied temporal and spectral
properties of GRS 1915+105 during $\theta$ class (Belloni $et~{}al$., 2000).
Sreehari $et~{}al$. (2020) have studied this source during the soft X-ray
state and have detected HFQPs in the range of 67.96 – 70.62 Hz with rms $\sim$
0.83 – 1.90 percent.

Misra $et~{}al$. (2020a) showed that there is a correlation between QPO
frequency and disk radii, and identified the QPO frequency with the dynamical
time-scale corrected by General Relativity as predicted decades ago, using
LAXPC and SXT simultaneous data of GRS 1915+105. Results of their analysis is
shown in Figure 10 which puts a tight limit on the spin of GRS 1915+105.

####  3.6.2 Cygnus X-3 and relativistic large radio jets :

Cygnus X-3 is an extraordinary HMXBs (a close binary with $\sim$ 4.8 hrs
orbital period) which produces brightest radio jets in our Galaxy. Although
the mass of the central object has not been confirmed using dynamical
measurements, the similarity of its spectro-temporal properties with known
black hole systems like GRS 1915+105 and XTE J1550-564 favours the binary
harbouring a black hole. The source is bright and persistent in X-rays, and
the X-ray emission originates as a result of wind accretion from a Wolf-Rayet
companion star onto the central compact object. In contrast to the canonical
BHXBs, six different spectral states have been reported in Cygnus X-3 by
Koljonen $et~{}al$. (2010) using simultaneous X-ray and radio observations.
Spectral states of Cygnus X-3, as observed by LAXPC are shown in the bottom
panel of Figure 11 where unfolded, best-fit spectra from SXT and LAXPC joint
fitting are shown. Details of spectral modelling are discussed in Koljonen
$et~{}al$. (2010). With LAXPC, we observed four different spectral states:
soft, hard, hypersoft and very high state. The presence of the very high state
is anticipated before but observed for the first time with LAXPC. Dramatic
changes in flux in both soft and hard bands in different states are visible.
Very high energetic photons, of the order of GeV, have been detected during
the hypersoft state when the radio emission is entirely quenched.

Figure 11: Top panel shows a typical 15-50 keV Swift/BAT lightcurve of Cygnus
X-3 over a timescale of 1.5 year. AstroSat/LAXPC observations during this
period are shown by stars and filled circles. Bottom panel shows unfolded
spectra using SXT+LAXPC joint fitting in the energy range 0.5-60 keV. Data
sets used for spectral fiting are shown with filled circle in respective
colours.

Astrosat/LAXPC performed over 20 observations of Cygnus X-3 during last five
years. The top panel of the Figure 11 shows 15-50 keV swift/BAT lightcurve
over 1.5 years and LAXPC pointing observations during this period are shown by
stars and filled circles. Using three long observations during 2015-2016
spanning over several orbits, Pahari $et~{}al$. (2017) determined the binary
orbital period of 17253.56 $\pm$ 0.19 sec, which is consistent with earlier
measurements. However, using one-year-long observation, a slow, low-amplitude
variability was observed with a periodicity of $\sim$35.8 days which may be
due to the orbital precession. During the peak of binary orbital phase, weak
but significant quasi-periodic oscillations have been observed at $\sim$5-8
mHz, $\sim$12-14 mHz and $\sim$18-24 mHz. Such detection is significant since
no QPO was detected with RXTE/PCA despite its most extensive archival data.
Interestingly, 7-15 mHz QPOs from Cygnus X-3 was reported with Exosat/ME
observations (van der Klis & Jansen, 1985). Detailed analysis with LAXPC
observations showed that QPOs were observed only during the flaring hard X-ray
state and when the source is brightest (the peak phase of the binary orbital
period). It is explained in terms of increased mass accretion rate when the
compact object passes through the denser wind section. An enhanced supply of
material may temporarily boost the temporal variability features.

Until now, investigating the connection between accretion disk and the radio
jets in Cygnus X-3 has been partially successful because of the lack of truly
simultaneous X-ray and radio data. X-ray and radio monitoring program of
Cygnus X-3 with LAXPC provided a rare opportunity to explore the radio/X-ray
connection in this X-ray binary.

Using long-term X-ray/radio monitoring campaign with Swift/BAT and 11.2 GHz
RATAN-600 telescope, it has been observed that Cygnus X-3 used to move into
the hard X-ray quenched state where hard X-ray flux decreased by order of
magnitude, and subsequently it showed major radio flare ejection event when
the radio flux density increases dramatically from few tens of mJy up to 20
Jy. Such conjunction was detected with LAXPC and RATAN-600 telescopes on 1-2
April 2017. Using detailed X-ray analysis, Pahari $et~{}al$. (2018b) found
that Cygnus X-3 undergo spectral state transition from the hypersoft state
(HPS) to a harder, more luminous state which was never observed before. We
term it as the very high state (VHS). Such a transition occurred within a few
hours when the radio flux density increases from $\sim$100 mJy to $\sim$478
mJy. Using SXT+LAXPC joint spectral analysis, they observed no hard X-rays
above 17 keV during the HPS state. Within hours timescale, a flat power law
appeared in the spectra with the power-law index of 1.49${}^{+0.04}_{-0.03}$
and extended up to 70 keV. Such an observation provided direct evidence of
synchrotron emission that originates in the radio-emitting blob which was
caught in the act of decoupling from the accretion disk. Such a detailed
radio/X-ray coupling event was observed for the first time and made possible
due to the LAXPC’s capabilities of higher efficiency for hard X-ray above 30
keV and high time resolution.

####  3.6.3 Cygnus X-1 :

Figure 12: The frequency times the power spectra of Cygnus X-1 for two energy
bands; 3-10 keV band (black points) and 20-40 keV band (red points).

Probably one of the best studied black hole systems, the bright black hole
system, Cygnus X-1, presents an excellent example to underline LAXPC
capabilities and the new understanding that LAXPC data can bring. The higher
effective area at high energies has allowed LAXPC to study in detail the
energy dependent rapid variability of Cygnus X-1 in the hard state. LAXPC data
analysis by Misra $et~{}al$. (2017), revealed the rms and time-lag variation
as a function of Fourier frequencies in the broad energy range of 4 – 80 keV,
extending the earlier results which were limited to 30 keV. They also showed
that the event mode data of LAXPC allows for flux resolved spectroscopy in
rapid ($\sim 1$) second time-scales, which for Cygnus X-1 revealed a
correlation between photon index and flux. A more extensive analysis of six
observations of Cygnus X-1 by Maqbool $et~{}al$. (2019), showed that the
energy and frequency dependent rapid variability is different for observations
which are in the hard state. More importantly, they could now quantitatively
fit and explain this variability in terms of a single zone stochastic
fluctuation model.

####  3.6.4 MAXI J1535-571 :

The bright outburst of this recently discovered black hole binary has been
observed by a large number of observatories including AstroSat and has
provided one of the most comprehensive view of any such systems. LAXPC
detected prominent QPOs at $\sim 2$ Hz in a wide energy range of 3-50 keV
(Sreehari $et~{}al$., 2019) when the source was in the hard intermediate
state. Spectral modelling using standard Comptonization models and physically
motivated ones like the two-component accretion flow revealed that source was
nearly at Eddington limit and its black hole mass is $\sim 6M_{\odot}$. A
detailed study of the broad band spectrum from SXT and LAXPC using
relativistic reflection models provided an estimate of the black hole spin
parameter ($a\sim 0.7$) and a mass of $\sim 10M_{\odot}$ (Sridhar $et~{}al$.,
2019). Correlation between the QPO frequency and spectral parameters was
studied by Bhargava $et~{}al$. (2019) who found that the frequency tightly
correlates with the photon index rather than the flux.

####  3.6.5 4U 1630-472 :

Among recurrent black hole X-ray transients, 4U 1630$-$472 is unique as it
shows frequent outbursts at about 600 days interval. The source was also at
the centre of focus due to its ‘superoutburst’ which typically lasts for two
years. Because no dynamical mass measurements have been performed, the source
has been categorised as the black hole candidate due to its similarity of
X-ray spectra-timing characteristics with other confirmed black hole X-ray
binaries. A dust scattering halo analysis placed the source between 4.7-11 kpc
Kalemci $et~{}al$. (2018) while the spectral index vs mass accretion rate
correlation was used to estimate the black hole mass of 10 $\pm$ 0.1 M$\odot$
(Seifina $et~{}al$., 2014).

Two major monitoring campaigns with AstroSat/LAXPC took place between 27
August $-$ 2 October 2016 and 4 August $-$ 17 September 2018 to monitor ‘mini’
outbursts from 4U 1630$-$472 which typically lasts 5-6 months. Using both
outburst observations, Baby $et~{}al$. (2020) performed detailed spectro-
timing analysis. Using broadband (0.7-20.0 keV) spectral modelling, they
detected no disk component during the onset of the 2016 outburst while later a
geometrically thin accretion disc with the temperature of $\sim$1.3 keV was
observed. Such behaviour was also accompanied by other observations like the
steepening of the powerlaw index from 1.8 to 2.6 and the decrease of rms
variability by $\sim$5%. However, all observations with LAXPC during 2018
outburst showed thermal disk emission dominated state with the disc
temperature vary between 1.26 $\pm$ 0.01 keV and 1.38 $\pm$ 0.01 keV.
Depending upon spectral characteristics, they observed three distinct spectral
states: low hard, intermediate and high soft states. Three states can be
distinguished in the ‘C’-shaped hardness intensity diagram track where the
hardness is the ratio between the counts in 6-20 keV and 2-6 keV and the
intensity is 2-20 keV X-ray flux calculated from LAXPC spectral modelling.
Such distinction was also apparent in the flux-rms plot where the flux was
calculated from the best-fit spectra while the rms was calculated using
0.005-10 Hz power density spectra. The transition among three states is in
agreement with the accretion geometry inferred from the two-component
accretion flow model (Chakrabarti and Titarchuk 1995).

Due to its high energy coverage, LAXPC spectra of 4U 1630-472 were also found
useful in measuring fundamental properties of the accreting compact object
when combined with concurrent spectra from other soft X-ray instruments like
SXT and Chandra. For example, the spin of a black hole was measured using
luminous high soft state observation with AstroSat and Chandra. Using
continuum spectral modeling method, Pahari $et~{}al$. (2018a) determined the
presence of a fast-spinning black hole in 4U 1630-472. Using relativistic
continuum spectral modeling on three independent measurements with Chandra,
SXT+LAXPC and Chandra+SXT+LAXPC and applying Markov Chain Monte Carlo
simulations on fitted spectral parameters, they constrained the spin of the
black hole to be 0.92 $\pm$ 0.04 within 99.7% confidence limit. Such a
measurement is essential for explaining accretion inflow and outflow
properties very close to the accreting black hole.

####  3.6.6 MAXI J1820+070 :

At a distance of 3.46${}^{+2.18}_{-1.03}$ kpc (Gandhi $et~{}al$., 2020), MAXI
J1820+070 was discovered as one of the closest and the brightest Galactic
BHXBs known till date (Corral-Santana $et~{}al$., 2016). With the Swift/BAT
peak flux of $\sim$4 Crab in 15-50 keV, the source showed back and forth
transition between hard and soft states. The source was proposed as a black
hole candidate depending upon the large amplitude hard X-ray variability
observed in the power density spectrum (Uttley $et~{}al$., 2018). However,
with the MAXI data, ‘q’ shaped track is clearly visible in the HID
(Chakraborty $et~{}al$., 2020) strengthening the fact that the source harbours
a black hole. AstroSat observed the source on two occasions during the peak of
the outburst.

Using the two days long observation with LAXPC on 30 March 2018, Mudambi
$et~{}al$. (2020a) observed 47.7 mHz QPO in the power density spectra which is
similar to what observed from two other BHXBs: GRS 1915+105 and IGR
J17091-3624. Using an SXT+LAXPC joint spectral analysis in the energy range
0.7-30 keV, they observed a flat spectral index of $\sim$1.61 and evidence for
cool disk at the temperature of $\sim$0.22 keV. They noted that the time lag
(where the reference band is 4.15-5.37 keV) increasing linearly with the
photon energy at 0.1, 1 and 10 Hz. However, the fractional rms decreases with
the energy at 0.1 and 1 Hz and independent of the photon energy at 10 Hz. The
observed temporal properties, e.g., typical 50-100 msec time lag, can be
explained qualitatively by the stochastic propagation fluctuation model
(Maqbool $et~{}al$., 2019). Such a detailed temporal analysis brings out the
importance of LAXPC observations.

Chakraborty $et~{}al$. (2020) studied the broadband spectra which include SXT,
LAXPC and CZTI in the energy range of 0.3$-$120 keV and the best fit spectral
modelling shows the presence of a soft excess component in the form of thermal
disk blackbody which is significantly below three keV, a thermal
comptonisation model with a very flat spectral index close to 1.4 and two
relativistic reflection models to fit broad and narrow iron line complex and a
Compton hump. Two relativistic models differ significantly in their coronal
electron temperature by a factor of $\sim$7\. Interestingly, the relativistic
reflection component dominates the broadband spectra. Comparing with the
detailed NuSTAR spectral study by Buisson $et~{}al$. (2019), they inferred
that the corona in MAXI J1820+070 is inhomogeneous and residing close to the
black hole ($<$ 3 Rg).

##  4 Active Galactic Nuclei

AstroSat/LAXPC has the potential to monitor the variability of Active Galactic
Nuclei (AGN) and in conjunction with SXT and UVIT to study the their broad
band spectra. Blazars are jet dominated AGN which show high amplitude
variability. Using AstroSat LAXPC and SXT observations Banerjee $et~{}al$.
(2020) constructed the power spectrum for the lighcurve of the blazar, Mrk
421, and detected a break or a characteristic time-scale. Such characteristic
time-scale have been detected before for X-ray binaries and regular AGN and
are believed to originate in the accretion disk. Hence this result suggests
that the jet variability also has an disk origin. For the blazar, RGB J0710 +
591, Goswami $et~{}al$. (2020) reported a significant deviation from a power-
law shape for the X-ray spectrum obtained from LAXPC and SXT. Such deviation
or curvature reflect the underlying shape of the particle energy spectrum that
produces the emission.

The AstroSat results suggested that particle distribution has a maximum energy
cutoff as predicted by models where the particles are shock accelerated and
radiatively cooled. Goswami $et~{}al$. (2020) studied the broad band spectral
energy distribution for the blazar 4C +21.35 using multi-wavelength data from
AstroSat and other observatories. They modeled the spectrum during flaring and
quiescent states with two compact regions which originated at different times
and moved away from the central region. For the regular AGN, RE J1034+396,
Chaudhury $et~{}al$. (2018) fitted the LAXPC and SXT spectra to reveal the
presence of a soft excess, consistent with earlier results.

Studies of faint sources like AGN by LAXPC is limited by the accuracy of the
instrument’s background estimation. It should be noted that most of the
results of AGN data analysis using the RXTE/PCA observations were undertaken
in the later times of the satellite operations when the response and
background were better characterised. With improvements and different
estimation techniques for LAXPC background (Antia $et~{}al$., 2020; Misra
$et~{}al$., 2020b) this issue, it is expected that in the near future there
will be significant spectral and variability studies of AGN using LAXPC data.

##  5 Summary

LAXPC instrument has carried on the legacy of RXTE PCA and HEXTE by being one
of the primary instruments to study rapid variability of X-ray systems. As
compared to RXTE/PCA, it has enhanced features which are (i) an higher
efficiency at hard X-rays (energies $\geq$ 30 keV), (ii) event mode data
allowing for temporal analysis in arbitrary user defined energy bins and (iii)
simultaneous SXT data at the soft X-ray band (0.5–8 keV). These advantages
have been demonstrated by a number of publications reporting important results
discussed in this paper for more than 30 sources. Moreover, the increasing
reservoir of archived and future LAXPC data is expected to provide an
unprecedented view of known sources as well as those that are yet to be
discovered. Improvements in the calibration, especially in the background
estimation, would lead to more detailed spectral and timing analysis of the
bright X-ray sources and will enable analysis of fainter sources such as a
larger number of active galactic nuclei.

It is fortunate that at present there are several X-ray missions in operation,
which have complimentary capabilities to the instruments on-board AstroSat.
Several works have already demonstrated the dramatic advantage of having
simultaneous (or even quasi-simultaneous) observations of a source with
multiple X-ray missions and with those in other wavebands. There is now an
unprecedented opportunity of detailed timing studies for a wide energy range
using simultaneous observations of AstroSat/LAXPC and NASA’s Nicer. Despite
the challenges of organizing and administrating such coordinated observations
by multiple missions (Nicer, Nustar, chandra and other), it is imperative for
break through results, that a large number of them should be undertaken.

## Acknowledgements

We acknowledge the strong support from Indian Space Research Organization
(ISRO) in various aspects of instrument development, space qualification,
software development, mission operation and data dissemination. We specially
acknowledge ISAC support for electronics development and during space
qualification tests. We thank IISU tean for providing us bellow pump along
with space qualified pump driver. We acknowl- edge support of the scientific
and technical staff of the LAXPC instrument team for their excellent team work
as well as staff of the TIFR Workshop who helped us at various level of LAXPC
instrument development.

## References

  * Agrawal (2006) Agrawal, P. C. 2006, Advances in Space Research, 38, 2989
  * Agrawal $et~{}al$. (2017) Agrawal, P. C., Yadav, J. S., Antia, H. M., $et~{}al$. 2017, Journal of Astrophysics and Astronomy, 38, 30
  * Agrawal $et~{}al$. (2018) Agrawal, V. K., Nandi, A., Girish, V., & Ramadevi, M. C. 2018, MNRAS, 477, 5437
  * Amin $et~{}al$. (2020) Amin, N., Roy, J., Chakroborty, S., $et~{}al$. 2020, private Communication
  * Antia $et~{}al$. (2020) Antia, H. M., Agrawal, P. C., Dedhia, D., $et~{}al$. 2020, JApA, this volume
  * Antia $et~{}al$. (2017) Antia, H. M., Yadav, J. S., Agrawal, P. C., $et~{}al$. 2017, ApJ Supp, 231, 10
  * Baby $et~{}al$. (2020) Baby, B. E., Agrawal, V. K., Ramadevi, M. C., $et~{}al$. 2020, MNRAS, 497, 1197
  * Bala $et~{}al$. (2020) Bala, S., Bhattacharya, D., Staubert, R., & Maitra, C. 2020, MNRAS, 497, 1029
  * Banerjee $et~{}al$. (2020) Banerjee, A., Bhattacharjee, A., Chatterjee, D., $et~{}al$. 2020, arXiv e-prints, arXiv:2007.05273
  * Belloni (2006) Belloni, T. 2006, Advances in Space Research, 38, 2801
  * Belloni $et~{}al$. (2000) Belloni, T., Klein-Wolt, M., Méndez, M., Van Der Klis, M., & Van Paradijs, J. 2000, arXiv preprint astro-ph/0001103
  * Belloni $et~{}al$. (2019) Belloni, T. M., Bhattacharya, D., Caccese, P., $et~{}al$. 2019, MNRAS, 489, 1037
  * Beri $et~{}al$. (2019) Beri, A., Paul, B., Yadav, J. S., $et~{}al$. 2019, MNRAS, 482, 4397
  * Beri $et~{}al$. (2020) Beri, A., Naik, S., Singh, K. P., $et~{}al$. 2020, MNRAS, arXiv:2010.08334
  * Bhalerao $et~{}al$. (2017) Bhalerao, V., Bhattacharya, D., Vibhute, A., $et~{}al$. 2017, Journal of Astrophysics and Astronomy, 38, 31
  * Bhargava $et~{}al$. (2019) Bhargava, Y., Belloni, T., Bhattacharya, D., & Misra, R. 2019, MNRAS, 488, 720
  * Bhattacharyya $et~{}al$. (2018) Bhattacharyya, S., Yadav, J. S., Sridhar, N., $et~{}al$. 2018, ApJ, 860, 88
  * Bhulla $et~{}al$. (2020) Bhulla, Y., Roy, J., & Jaaffrey, S. N. A. 2020, Research in Astronomy and Astrophysics, 20, 098
  * Bogensberger $et~{}al$. (2020) Bogensberger, D., Ponti, G., Jin, C., $et~{}al$. 2020, A&A, 641, A101
  * Buisson $et~{}al$. (2019) Buisson, D. J. K., Fabian, A. C., Barret, D., $et~{}al$. 2019, MNRAS, 490, 1350
  * Chakraborty $et~{}al$. (2020) Chakraborty, S., Navale, N., Ratheesh, A., & Bhattacharyya, S. 2020, MNRAS, 498, 5873
  * Chandra $et~{}al$. (2020) Chandra, A. D., Roy, J., Agrawal, P. C., & Choudhury, M. 2020, MNRAS, 495, 2664
  * Chaudhury $et~{}al$. (2018) Chaudhury, K., Chitnis, V. R., Rao, A. R., $et~{}al$. 2018, MNRAS, 478, 4830
  * Corral-Santana $et~{}al$. (2016) Corral-Santana, J. M., Casares, J., Muñoz-Darias, T., $et~{}al$. 2016, A&A, 587, A61
  * Devasia $et~{}al$. (2021) Devasia, J., Raman, G., & Paul, B. 2021, NA, 83, 101479
  * Fender $et~{}al$. (2004) Fender, R. P., Belloni, T. M., & Gallo, E. 2004, Monthly Notices of the Royal Astronomical Society, 355, 1105
  * Gandhi $et~{}al$. (2020) Gandhi, P., Rao, A., Charles, P. A., $et~{}al$. 2020, MNRAS, 496, L22
  * Garg $et~{}al$. (2020) Garg, A., Misra, R., & Sen, S. 2020, MNRAS, 498, 2757
  * Goswami $et~{}al$. (2020) Goswami, P., Sinha, A., Chandra, S., $et~{}al$. 2020, MNRAS, 492, 796
  * Jain $et~{}al$. (2010) Jain, C., Paul, B., & Dutta, A. 2010, MNRAS, 409, 755
  * Jaisawal $et~{}al$. (2020a) Jaisawal, G. K., Naik, S., Epili, P. R., $et~{}al$. 2020a, JApA, this volume
  * Jaisawal $et~{}al$. (2020b) Jaisawal, G. K., Naik, S., Ho, W. C. G., $et~{}al$. 2020b, MNRAS, 498, 4830
  * Jithesh $et~{}al$. (2019) Jithesh, V., Maqbool, B., Misra, R., $et~{}al$. 2019, ApJ, 887, 101
  * Jonker & van der Klis (2001) Jonker, P. G., & van der Klis, M. 2001, ApJL, 553, L43
  * Kalemci $et~{}al$. (2018) Kalemci, E., Maccarone, T. J., & Tomsick, J. A. 2018, ApJ, 859, 88
  * Koljonen $et~{}al$. (2010) Koljonen, K. I. I., Hannikainen, D. C., McCollough, M. L., Pooley, G. G., & Trushkin, S. A. 2010, MNRAS, 406, 307
  * Lewin & Van der Klis (2006) Lewin, W., & Van der Klis, M. 2006, Compact stellar X-ray sources, Vol. 39 (Cambridge University Press)
  * Maitra (2017) Maitra, C. 2017, Journal of Astrophysics and Astronomy, 38, 50
  * Maitra $et~{}al$. (2018) Maitra, C., Paul, B., Haberl, F., & Vasilopoulos, G. 2018, MNRAS, 480, L136
  * Malu $et~{}al$. (2020) Malu, S., Sriram, K., & Agrawal, V. K. 2020, MNRAS, doi:10.1093/mnras/staa2939
  * Maqbool $et~{}al$. (2019) Maqbool, B., Mudambi, S. P., Misra, R., $et~{}al$. 2019, MNRAS, 486, 2964
  * Misra $et~{}al$. (2020a) Misra, R., Rawat, D., Yadav, J. S., & Jain, P. 2020a, ApJL, 889, L36
  * Misra $et~{}al$. (2020b) Misra, R., Roy, J., & Yadav, J. S. 2020b, JApA, this volume
  * Misra $et~{}al$. (2017) Misra, R., Yadav, J. S., Verdhan Chauhan, J., $et~{}al$. 2017, ApJ, 835, 195
  * Mudambi $et~{}al$. (2020a) Mudambi, S. P., Maqbool, B., Misra, R., $et~{}al$. 2020a, ApJL, 889, L17
  * Mudambi $et~{}al$. (2020b) Mudambi, S. P., Rao, A., Gudennavar, S. B., Misra, R., & Bubbly, S. G. 2020b, MNRAS, 498, 4404
  * Mukerjee $et~{}al$. (2020) Mukerjee, K., Antia, H. M., & Katoch, T. 2020, ApJ, 897, 73
  * Pahari $et~{}al$. (2017) Pahari, M., Antia, H. M., Yadav, J. S., $et~{}al$. 2017, ApJ, 849, 16
  * Pahari $et~{}al$. (2018a) Pahari, M., Bhattacharyya, S., Rao, A. R., $et~{}al$. 2018a, ApJ, 867, 86
  * Pahari $et~{}al$. (2018b) Pahari, M., Yadav, J. S., Verdhan Chauhan, J., $et~{}al$. 2018b, ApJL, 853, L11
  * Parmar $et~{}al$. (1989) Parmar, A. N., Gottwald, M., van der Klis, M., & van Paradijs, J. 1989, ApJ, 338, 1024
  * Rawat $et~{}al$. (2019) Rawat, D., Pahari, M., Yadav, J. S., $et~{}al$. 2019, ApJ, 870, 4
  * Remillard & McClintock (2006) Remillard, R. A., & McClintock, J. E. 2006, Annu. Rev. Astron. Astrophys., 44, 49
  * Roy $et~{}al$. (2020) Roy, J., Agrawal, P. C., Singari, B., & Misra, R. 2020, Research in Astronomy and Astrophysics, 20, 155
  * Roy $et~{}al$. (2019) Roy, J., Agrawal, P. C., Iyer, N. K., $et~{}al$. 2019, ApJ, 872, 33
  * Seifina $et~{}al$. (2014) Seifina, E., Titarchuk, L., & Shaposhnikov, N. 2014, ApJ, 789, 57
  * Singh $et~{}al$. (2016) Singh, K. P., Stewart, G. C., Chandra, S., $et~{}al$. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9905, Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, ed. J.-W. A. den Herder, T. Takahashi, & M. Bautz, 99051E
  * Sreehari $et~{}al$. (2020) Sreehari, H., Nandi, A., Das, S., $et~{}al$. 2020, arXiv e-prints, arXiv
  * Sreehari $et~{}al$. (2019) Sreehari, H., Ravishankar, B. T., Iyer, N., $et~{}al$. 2019, MNRAS, 487, 928
  * Sridhar $et~{}al$. (2019) Sridhar, N., Bhattacharyya, S., Chandra, S., & Antia, H. M. 2019, MNRAS, 487, 4221
  * Staubert $et~{}al$. (2014) Staubert, R., Klochkov, D., Wilms, J., $et~{}al$. 2014, A&A, 572, A119
  * Staubert $et~{}al$. (2019) Staubert, R., Trümper, J., Kendziorra, E., $et~{}al$. 2019, A&A, 622, A61
  * Tanaka $et~{}al$. (1995) Tanaka, Y., Lewin, W., Van Paradijs, J., & Van Den Heuvel, E. 1995, Cambridge, UK: Cam, 126
  * Tandon $et~{}al$. (2017) Tandon, S. N., Hutchings, J. B., Ghosh, S. K., $et~{}al$. 2017, Journal of Astrophysics and Astronomy, 38, 28
  * Truemper $et~{}al$. (1978) Truemper, J., Pietsch, W., Reppin, C., $et~{}al$. 1978, ApJL, 219, L105
  * Uttley $et~{}al$. (2018) Uttley, P., Gendreau, K., Markwardt, C., $et~{}al$. 2018, The Astronomer’s Telegram, 11423, 1
  * van der Klis (1994) van der Klis, M. 1994, SSRv, 69, 208
  * van der Klis (2001) van der Klis, M. 2001, in American Institute of Physics Conference Series, Vol. 599, X-ray Astronomy: Stellar Endpoints, AGN, and the Diffuse X-ray Background, ed. N. E. White, G. Malaguti, & G. G. C. Palumbo, 406–415
  * van der Klis & Jansen (1985) van der Klis, M., & Jansen, F. A. 1985, NATURE, 313, 768
  * Varun $et~{}al$. (2019a) Varun, Maitra, C., Pragati, P., Harsha, R., & Biswajit, P. 2019a, MNRAS, 484, L1
  * Varun $et~{}al$. (2019b) Varun, Pradhan, P., Maitra, C., Raichur, H., & Paul, B. 2019b, ApJ, 880, 61
  * Verdhan Chauhan $et~{}al$. (2017) Verdhan Chauhan, J., Yadav, J. S., Misra, R., $et~{}al$. 2017, ApJ, 841, 41
  * Xiao $et~{}al$. (2019) Xiao, G. C., Lu, Y., Ma, X., $et~{}al$. 2019, Journal of High Energy Astrophysics, 24, 30
  * Yadav (2006) Yadav, J. 2006, The Astrophysical Journal, 646, 385
  * Yadav $et~{}al$. (1999) Yadav, J., Rao, A., Agrawal, P., $et~{}al$. 1999, The Astrophysical Journal, 517, 935
  * Yadav $et~{}al$. (2017) Yadav, J. S., Agrawal, P. C., Antia, H. M., $et~{}al$. 2017, Current Science, 113, 591
  * Yadav $et~{}al$. (2016a) Yadav, J. S., Misra, R., Verdhan Chauhan, J., $et~{}al$. 2016a, ApJ, 833, 27
  * Yadav $et~{}al$. (2016b) Yadav, J. S., Agrawal, P. C., Antia, H. M., $et~{}al$. 2016b, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9905, Space Telescopes and Instrumentation 2016: Ultraviolet to Gamma Ray, ed. J.-W. A. den Herder, T. Takahashi, & M. Bautz, 99051D


$\displaystyle={\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color<EMAIL_ADDRESS>E_{{\mathsf{E}}}+\beta_{{\mathsf{E}}}p_{{\mathsf{E}}}\nabla^{\text{g}}_{\mu}u^{\mu}+\beta_{{\mathsf{E}}}\lambda_{{\mathsf{E}}}Q^{\mu\nu}h_{\mu\lambda}\nabla^{\text{g}}_{\nu}u^{\lambda}-\beta_{{\mathsf{E}}}a_{{\mathsf{E}}}\Gamma_{Q}^{\mu\nu\rho\sigma}Q_{\mu\nu}\Big{(}\mathsf{H}_{\rho\sigma}-\aleph
a_{{\mathsf{E}}}Q_{\rho\sigma}\Big{)}}\leavevmode\nobreak\ ,$ $\displaystyle
r_{{\mathsf{F}}}$ $\displaystyle=\ell\gamma_{{\mathsf{F}}}\Delta
E\leavevmode\nobreak\ ,$ (182)

where we have used the identities in eq. 175 and defined

$\displaystyle\mathcal{X}^{[\rho\mu]\nu}$
$\displaystyle=2\frac{\partial\mathcal{F}}{\partial\nabla^{\text{g}}_{[\mu}Q_{\rho]\sigma}}Q_{\sigma}{}^{\nu}+2Q_{\sigma}{}^{[\mu}\frac{\partial\mathcal{F}}{\partial\nabla^{\text{g}}_{\rho]}Q_{\sigma\nu}}+2\frac{\partial\mathcal{F}}{\partial\nabla^{\text{g}}_{\nu}Q_{\sigma[\rho}}Q_{\sigma}{}^{\mu]}\leavevmode\nobreak\
.$ (183)

One may verify that the stress tensor $\tau^{\mu\nu}$ in section B.4 is
symmetric, but we have cast it in the canonical form by isolating a total-
derivative improvement term
$\nabla^{\prime\text{g}}_{\rho}\mathcal{X}^{[\rho\mu]\nu}$ that identically
drops out from the conservation equations in flat spacetime. The remaining
“canonical” stress tensor is asymmetric [8, 108]. Similarly, the last term in
the energy current $\epsilon^{\mu}$ drops out from the conservation equations
in flat spacetime.

Around a uniaxial nematic state, we can parametrise the fluctuations of the
order parameter as

$\displaystyle
Q^{\mu\nu}=Q_{\mathsf{S}}\left(p^{\mu}p^{\nu}-\frac{1}{d}h^{\mu\nu}\right)+Q^{\mu\nu}_{\mathsf{T}}\leavevmode\nobreak\
,$ (184)

where $\vec{p}^{2}\equiv p_{\mu}p^{\mu}=1$,
$p_{\mu}Q^{\mu\nu}_{\perp}=h_{\mu\nu}Q^{\mu\nu}_{\perp}=0$. The
$Q_{\mathsf{S}}$ and $Q_{\mathsf{T}}^{\mu\nu}$ components of the order
parameter are gapped and can be integrated out setting them to $Q_{0}$ and $0$
respectively. This leaves us with an equation of motion for the director
$p_{\mu}$ that takes the form

$u^{\lambda}\nabla^{\text{g}}_{\lambda}p^{\mu}=-\frac{1}{\sigma_{p}}\frac{\delta\mathcal{F}}{\delta
p_{\mu}}+\left(\bar{p}^{\mu[\sigma}p^{\rho]}+\gamma\bar{p}^{\mu(\sigma}p^{\rho)}\right)h_{\sigma\lambda}\nabla^{\text{g}}_{\rho}u^{\lambda}\leavevmode\nobreak\
,$ (185)

where $\bar{p}^{\mu\nu}=h^{\mu\nu}-p^{\mu}p^{\nu}$, and we have identified

$\displaystyle\sigma_{p}=\frac{4Q_{0}^{2}}{d-1}p_{\mu}p_{\rho}\bar{p}_{\nu\sigma}\sigma_{Q}^{\mu\nu\rho\sigma}=2Q_{0}^{2}\sigma_{Q}+2Q_{0}^{3}\frac{d-2}{d}\sigma_{Q,1}\leavevmode\nobreak\
,\qquad\gamma=\frac{2/Q_{0}}{d-1}p_{\mu}p_{\rho}\bar{p}_{\nu\sigma}\gamma^{\mu\nu\rho\sigma}=\frac{1}{Q_{0}}\gamma_{1}+\frac{d-2}{d}\gamma_{2}\leavevmode\nobreak\
.$ (186)

#### Director theory:

Following cue from our superfluid discussion, we can also write down a SK
description for the nematic director directly. To this end, we introduce the
doubled spatial director fields ${\mathbb{p}}_{1,2\alpha}$, satisfying
$\bbbeta^{\alpha}{\mathbb{p}}_{1,2\alpha}=0$, transforming under the KMS
symmetry as

$\displaystyle{\mathbb{p}}_{1\alpha}(\sigma)$
$\displaystyle\xrightarrow{\mathrm{{KMS}}}\Theta{\mathbb{p}}_{1\alpha}(\sigma),$
$\displaystyle\qquad{\mathbb{p}}_{2\alpha}(\sigma)$
$\displaystyle\xrightarrow{\mathrm{{KMS}}}\Theta{\mathbb{p}}_{2\alpha}(\sigma+i\hbar\,\Theta\bbbeta)\leavevmode\nobreak\
,$ (187)

with time-reversal eigenvalue $+1$. The director and its noise partner on the
physical spacetime are defined as
$p_{r,a\mu}=\frac{\partial\sigma^{\alpha}}{\partial
x^{\mu}}{\mathbb{p}}_{r,a\alpha}$, such that $\beta^{\mu}p_{r,a\mu}=0$, with
the KMS transformation in the classical limit

$p_{r\mu}\xrightarrow{\mathrm{{KMS}}}\Theta p_{r\mu}+\mathcal{O}(\hbar),\qquad
p_{a\mu}\xrightarrow{\mathrm{{KMS}}}\Theta\hat{p}_{a\mu}\equiv\Theta\left(p_{a\mu}+i\pounds_{\beta}p_{r\mu}\right)+\mathcal{O}(\hbar)\leavevmode\nobreak\
.$ (188)

Furthermore, it is useful to define the shifted noise director given by

$\displaystyle{\mathpzc p}_{a\mu}$
$\displaystyle=p_{a\mu}-\frac{1}{2}\mathcal{H}_{a\mu\nu}p^{\nu}+\frac{1}{2}n_{\mu}u^{\sigma}\mathcal{H}_{a\sigma\nu}p^{\nu}\leavevmode\nobreak\
,\qquad\beta^{\mu}{\mathpzc p}_{a\mu}=0\leavevmode\nobreak\ ,$ (189)

To impose the normalisation of $p_{\mu}$, we can introduce the Lagrange
multiplier terms similar to eq. 173, i.e.

$\displaystyle\mathcal{L}_{\text{norm}}=(p^{\mu}p_{\mu}-1)(p^{\nu}{\mathpzc
p}_{a\nu})+\frac{1}{4}(p^{\mu}p_{\mu}-1)^{2}\left(\frac{1}{2}h^{\mu\nu}\mathcal{H}_{a\mu\nu}+u^{\mu}N_{a\mu}\right)\leavevmode\nobreak\
,$ (190)

which are KMS-invariant, and set $p^{\mu}p_{\mu}=1$ and $p^{\mu}{\mathpzc
p}_{a\mu}=0$ onshell.

An argument similar to that employed for section B.4 can be used to show

$\displaystyle-\frac{1}{\sqrt{\gamma}}\pounds_{\beta}\Big{(}\sqrt{\gamma}\,\mathcal{F}\Big{)}$
$\displaystyle=-\varepsilon\,u^{\mu}\pounds_{\beta}n_{\mu}+\left(\rho\,u^{\mu}u^{\nu}-\mathcal{F}\,h^{\mu\nu}\right)\left(\frac{1}{2}\pounds_{\beta}h_{\mu\nu}-\pounds_{\beta}n_{(\mu}\vec{u}_{\nu)}\right)$
$\displaystyle\qquad+\frac{\partial\mathcal{F}}{\partial\nabla^{\text{g}}_{\mu}p_{\lambda}}\left(u^{\rho}\nabla^{\text{g}}_{\rho}p_{\lambda}\pounds_{\beta}n_{\mu}+\nabla_{\text{g}}^{\nu}p_{\lambda}\left(\frac{1}{2}\pounds_{\beta}h_{\mu\nu}-\pounds_{\beta}n_{(\mu}\vec{u}_{\nu)}\right)\right)$
$\displaystyle\qquad+\frac{\partial\mathcal{F}}{\partial\nabla^{\text{g}}_{\mu}p_{[\rho}}p^{\nu]}\left(\nabla^{\prime\text{g}}_{\rho}\left(\pounds_{\beta}h_{\mu\nu}-2\pounds_{\beta}n_{(\mu}\vec{u}_{\nu)}\right)+h_{\lambda\nu}\nabla^{\text{g}}_{\rho}u^{\lambda}\pounds_{\beta}n_{\mu}+2h_{\lambda(\rho}\nabla^{\text{g}}_{\mu)}u^{\lambda}\pounds_{\beta}n_{\nu}\right)$
$\displaystyle\qquad-\left(\frac{\partial\mathcal{F}}{\partial
p_{\mu}}+\frac{\partial\mathcal{F}}{\partial\nabla^{\text{g}}_{\lambda}p_{\mu}}\nabla^{\text{g}}_{\lambda}\right)\left(\pounds_{\beta}p_{\mu}-\frac{1}{2}\pounds_{\beta}h_{\mu\rho}p^{\rho}+\pounds_{\beta}n_{(\mu}\vec{u}_{\rho)}p^{\rho}\right)\leavevmode\nobreak\
,$ (191)

for a free energy density $\mathcal{F}$ that only depends on $T$, $\varpi$,
$p_{\mu}$, $\nabla^{\text{g}}_{\lambda}p_{\mu}$, and $h^{\mu\nu}$. This allows
one to immediately write down the effective action

$\displaystyle\mathcal{L}$
$\displaystyle=-\varepsilon\,u^{\mu}N_{a\mu}+\frac{1}{2}\left(\rho\,u^{\mu}u^{\nu}-\mathcal{F}\,h^{\mu\nu}\right)\mathcal{H}_{a\mu\nu}-\left(\frac{\partial\mathcal{F}}{\partial
p_{\mu}}+\frac{\partial\mathcal{F}}{\partial\nabla^{\text{g}}_{\lambda}p_{\mu}}\nabla^{\text{g}}_{\lambda}\right){\mathpzc
p}_{a\mu}$
$\displaystyle\qquad+\frac{\partial\mathcal{F}}{\partial\nabla^{\text{g}}_{\mu}p_{\lambda}}\left(u^{\rho}\nabla^{\text{g}}_{\rho}p_{\lambda}N_{a\mu}+\frac{1}{2}\nabla_{\text{g}}^{\nu}p_{\lambda}\mathcal{H}_{a\mu\nu}\right)+\frac{\partial\mathcal{F}}{\partial\nabla^{\text{g}}_{\mu}p_{[\rho}}p^{\nu]}\left(\nabla^{\text{g}}_{\rho}\mathcal{H}_{a\mu\nu}+h_{\lambda\nu}\nabla^{\text{g}}_{\rho}u^{\lambda}N_{a\mu}+2h_{\lambda(\rho}\nabla^{\text{g}}_{\mu)}u^{\lambda}N_{a\nu}\right)$
$\displaystyle\qquad+i{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T\begin{pmatrix}-N_{a\mu}\\\ \frac{1}{2}\mathcal{H}_{a\mu\nu}\\\
{\mathpzc
p}_{a\mu}-\frac{1}{2}\gamma\bar{p}_{\mu}^{\lambda}p^{\tau}\mathcal{H}_{a\tau\lambda}\\\
{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\beta_{{\mathsf{E}}}\Pi^{{\mathsf{E}}}_{a}}\end{pmatrix}^{\\!\\!\intercal}\\!\\!\begin{pmatrix}T\kappa^{\mu\rho}&0&0&{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color<EMAIL_ADDRESS>0&\eta^{\mu\nu\rho\sigma}&0&{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color<EMAIL_ADDRESS>0&0&\sigma_{p}\bar{p}^{\mu\rho}&{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color<EMAIL_ADDRESS>{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}0}&{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}-p_{{\mathsf{E}}}h^{\rho\sigma}-\lambda_{{\mathsf{E}}}p^{\langle\rho}p^{\sigma\rangle}}&{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}0}&{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}\gamma_{{\mathsf{E}}}/\beta_{{\mathsf{E}}}}\end{pmatrix}\\!\\!\begin{pmatrix}-N_{a\rho}\\\
\frac{1}{2}\hat{\mathcal{H}}_{a\rho\sigma}\\\
\\!\\!\\!\\!\hat{\leavevmode\nobreak\ \leavevmode\nobreak\ {\mathpzc
p}_{a\rho}}-\frac{1}{2}\gamma\bar{p}_{\rho}^{\lambda}p^{\tau}\hat{\mathcal{H}}_{a\tau\lambda}\\\
{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color<EMAIL_ADDRESS>$\displaystyle\qquad+{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color@cmyk@fill{1}{0.50}{0}{0}i{\text{k}}_{\resizebox{}{}{\scalebox{0.8}[1.0]{{\leavevmode\hbox{\set@color{\scriptsize
B}}}}}}T\gamma_{{\mathsf{F}}}\Pi^{{\mathsf{F}}}_{a}\hat{\Pi}^{{\mathsf{F}}}_{a}}\leavevmode\nobreak\
,$ (192)

yielding section 4.3 in the main text.