# Emission lines from X-ray illuminated accretion disc in black hole binaries

Santanu Mondal,1,2 Tek P. Adhikari,3 Chandra B. Singh4
1Indian Institute of Astrophysics, II Block, Koramangala, Bengaluru 560034,
India
2Physics Department, Ben-Gurion University of the Negev, Be’er-Sheva 84105,
Israel
3Inter-University Center for Astronomy and Astrophysics (IUCAA), Pune 411007,
India
4South-Western Institute for Astronomy Research, Yunnan University, University
Town, Chenggong District, Kunming 650500, P.R. China
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

X-ray flux from the inner hot region around central compact object in a binary
system illuminates the upper surface of an accretion disc and it behaves like
a corona. This region can be photoionised by the illuminating radiation, thus
can emit different emission lines. We study those line spectra in black hole
X-ray binaries for different accretion flow parameters including its geometry.
The varying range of model parameters captures maximum possible observational
features. We also put light on the routinely observed Fe line emission
properties based on different model parameters, ionization rate, and Fe
abundances. We find that the Fe line equivalent width $W_{\rm E}$ decreases
with increasing disc accretion rate and increases with the column density of
the illuminated gas. Our estimated line properties are in agreement with
observational signatures.

###### keywords:

accretion, accretion disc – X-ray binaries:black holes – atomic processes –
line:formation – radiative transfer

††pubyear: 2020††pagerange: Emission lines from X-ray illuminated accretion
disc in black hole binaries–References

##  1 Introduction

X-ray observations show prominent Fe K emission lines at 6-7 keV and Compton
hump at $\sim$20-30 keV in the spectra of black hole binaries which indicates
that the material above the disc is being illuminated by the hard X-ray
source. The Fe emission lines with various shapes have been routinely observed
in many black hole X-ray binaries (Tomsick et al., 2014; Plant et al., 2014;
Mondal et al., 2014b; Debnath et al., 2015; Xu et al., 2020; Dong et al.,
2020a, and references therein) using high-resolution X-ray spectroscopy.
Variations of these features with time imply that they are produced in the
vicinity of the X-ray sources themselves. Several theoretical models in the
literature have been proposed to explain those observed features of the
emission lines. Detailed calculations of the radiative transfer of X-rays in
an optically thick medium were carried out by Ross et al. (1978) and Ross
(1979), solving the transfer of the continuum photons using the Fokker–Planck
diffusion equation, including a modified Kompaneets operator for more
realistic Compton scattering, while the transfer of lines is calculated using
the escape probabilities approximation.

In the case of black hole binaries, there are several ways to illuminate the
disc by the X-ray radiation, for example, at the outer boundary of the disc,
relatively less gravity, increases the height of the disc, intercepts more
radiation or the winds from the corona or from the disc itself scatters the
X-rays down to the disc. The effects of light bending, specifically the return
of some fraction of the disc radiation to the disc itself, can become
significant (Cunningham, 1976) to enhance the strength of the emission. The
emission lines generated in low mass X-ray binaries (LMXRBs) in the X-ray
illuminated accretion disc was studied by Kallman & White (1989). Authors
showed that the Fe K line broadening is dominated by rotation or by
Comptonization through higher optical depth, rather than from an accretion
disc corona. Intensity estimation of the reprocessed emission from an
irradiated slab of gas has shown that the Fe K$\alpha$ line is the strongest
line in the reflection spectrum (George & Fabian, 1991; Matt et al., 1991). It
can be by virtue of its relatively high cosmic abundance and large fluorescent
yield. If the strong lines from other abundant elements such as carbon,
nitrogen, and oxygen at lower energies can be detected, it would provide a lot
of information on the ionisation state of the accretion disc. Several models
of reflection from photoionized discs have shown that the soft X-ray features
are very sensitive to the ionisation parameter of the disc material (Ross &
Fabian, 1993).

Observations of Fe line emission have evidenced to be important for the
determination of the spin parameter, one of the intrinsic parameters that
describe black holes (Laor, 1991; Dabrowski et al., 1997; Brenneman &
Reynolds, 2006). It was pointed out by Fabian et al. (1989), that, if the
reflected emission comes from the accretion disc, then relativistic and
Doppler effects would broaden the emission lines, particularly in high spin
black hole cases, where if reflection occurs close to the black hole (Fabian
et al., 2000; Reynolds & Nowak, 2003; Miller et al., 2008; Steiner et al.,
2011; Dauser et al., 2012). Recent high-resolution X-ray observation indeed
detected this line broadening features and was used to estimate the spin
parameter of black holes (Iwasawa et al., 1996; Mondal et al., 2016; Dong et
al., 2020b, and references therein).

Several models have been introduced in the literature to explain the emission
lines from the disc from time to time. For example, the scattering of photons
by cold electrons using Green’s functions approach was first derived by
Lightman & Rybicki (1980) and Lightman et al. (1981), and their implications
for AGN observations discussed in Lightman & White (1988). Guilbert & Rees
(1988) proposed a theoretical model to study reflected emission assuming that
irradiation on the surface of the disc was weak enough so the gas remains
neutral, but yet would reprocess the radiation-producing observable spectral
features. Matt et al. (1993) studied emission properties based on mass
accretion variation. Zycki et al. (1994) carried out similar calculations
including ionisation balance and thermal balance in the medium along with the
distribution of X-ray intensity with optical depth, yet neglecting the
intrinsic emission inside the gas. Later, Magdziarz & Zdziarski (1995)
calculated the cold reflection using Green’s function approach, highly
dependent on viewing angle, however, their calculations did not include line
production. Apart from the above mentioned models, there are many notable
models such as, reflionx by Ross & Fabian (1993), TITAN code by Dumont et al.
(2000), which was later extended by Różańska et al. (2002) to treat cases of
Compton thick media. Różańska et al. (2002) demonstrated that the use of
hydrostatic equilibrium is of crucial importance to study the disc
illumination by the hard X-ray. All other earlier studies assumed constant
density in the material. It has been argued that a plane parallel slab under
hydrostatic equilibrium could represent the surface of an accretion disc more
accurately (Nayakshin et al., 2000), and that its reflected spectrum is, in
fact, different from the one predicted by constant density models (see also,
Rozanska & Czerny, 1996; Nayakshin & Kallman, 2001; Różańska & Madej, 2008;
Różańska et al., 2011). Recently, in the last couple of years, Garcia and his
collaborators proposed a model, xillver, (García et al., 2013) where authors
have updated the above models adding more physical processes and atomic data,
to make it more broadly applicable. However, all these models used the
incident spectra from the phenomenological power law model, did not take into
account the detailed accretion solution and the effects of flow parameters and
its geometry.

Despite the existence of a large number of models in the literature, there is
still a lack of understanding of the origin of the corona, its optical depth,
and temperature profile to generate physical emission from that region, which
is responsible for illuminating the disc. There are models which successfully
explain the scenario of scattering of soft photons by the corona at different
location of the disc (Sunyaev & Titarchuk, 1980; Haardt & Maraschi, 1993;
Zdziarski et al., 2003), however, without taking into account the physical
origin of the corona. Therefore, a self-consistent modeling of accretion, as
well as emission lines from reflected disc component, remains to be done,
which motivates us to study line emissions using a physical accretion disc
model. For our purpose, we use two-component solution, TCAF (Chakrabarti &
Titarchuk, 1995; Chakrabarti, 1997, hereafter CT95), generated theoretical
spectra. According to this model, the so-called truncation of the disc is the
location of the shock (Chakrabarti, 1989), formed by low angular momentum,
hot, sub-Keplerian halo, satisfying Rankine-Hugoniot conditions. Numerical
simulations have shown that the two-component flow forms and is stable even in
presence of spatially- and temporally- varying viscosity parameters and
cooling processes (Giri & Chakrabarti, 2013; Giri et al., 2015; Roy &
Chakrabarti, 2017). Beyond the shocked region, both disc and halo accretion
pile up to decide the optical depth and temperature of that region. The same
region also upscatters the incoming soft radiation from the disc via inverse
Comptonization. The model has indeed a reflection mechanism, where disc to
shock and vice-versa interceptions are taken into account in an iterative way.
The disc inclination effect has not been taken into account in the current
model. Later, this model was modified to see the effects of cooling and mass
loss coupled with hydrodynamics (Mondal & Chakrabarti, 2013). In the last
couple of years, the present model is also implemented in XSPEC to study both
LMXRBs (Debnath et al., 2014; Mondal et al., 2014b) and AGNs (Nandi et al.,
2019; Mondal & Stalin, 2021) data successfully. The observational studies
using the current model can explain different spectral states, the evolution
of quasi-periodic oscillations (QPOs), estimation of mass and spin of the
black hole which covers current research topics in the field. To study the
emission line we use publicly available photoionization code cloudy C17.01
version (Ferland et al., 2017). Earlier, Bianchi et al. (2006) used cloudy to
study the emission lines from 8 nearby Seyfert 2 galaxies and revealed that
the soft X-ray emission of all the objects is likely to be dominated by the
photoionized gas. Recently, Adhikari et al. (2015, 2016, 2019) used cloudy in
detail to study emission spectra in the AGN environment and its application in
different geometry and thermodynamic conditions.

The goal of the present manuscript is (i) to use TCAF model generated spectra
to illuminate the disc and to produce emission lines, (ii) to study how
different model parameters (e.g., disc, halo accretion rates, and the disc
geometry) affect line shapes and its intensity, (iii) to study the effect of
Fe abundances ($A_{\rm Fe}$) and ionisation parameter ($\xi$) on the line
properties in different spectral states, (iv) to estimate the profile of the
equivalent width only of the Fe lines with disc mass accretion rate, (v) to
see the temperature structure of the gas cloud above the disc, and (vi) to use
the simulated Fe lines to compare with observed Fe line features of the black
hole binaries. In this work, we are not going into the details explaining the
origin of emission lines from other species than iron, which will be studied
in follow-up works.

The paper is organized as follows: in section 2, we discuss the details of the
accretion disc model, which is used to generate the illuminating spectra and
how it is used to form emission lines from cloudy. In section 3, we describe
emission lines variation with different model parameters for instance,
accretion rates (as the model uses two accretion components), the geometry of
the flow, ionisation parameter, and Fe abundance. The equivalent width profile
with disc mass accretion rate and ionisation parameter and the temperature
structure of the gas cloud are also discussed. Our simulated results are also
compared with observations. In section 4, the change in shape of Fe lines with
model parameters and its intensity variation is discussed. Finally, we draw
our concluding remarks in section 5.

##  2 Modelling

In order to generate emission lines, we use theoretical spectra from
Chakrabarti (1997) model. In Figure 1, we present a cartoon diagram of the
two-component model where a cold, high angular momentum Keplerian disc resides
at the equatorial plane and it is flanked by the sub-Keplerian flow which we
call halo, which is hot and low angular momentum flow. The optically thin pre-
shock halo does not radiate efficiently therefore energy and entropy are
advected with the flow. At the center of the co-ordinate, a black hole of mass
$M_{\rm BH}$ is located. For the supply of soft (seed) photons for thermal
Comptonization, we assume a Keplerian disc on the equatorial plane, truncated
at the shock radius. This disc emits a flux of radiation the same as that
produced by a Shakura & Sunyaev (1973) disc. The soft photons emerging out
from the Keplerian disc are reprocessed via Compton or inverse Compton
scattering within the corona. Injected photons may undergo a single, multiple
or no scattering at all with the hot electrons in between its emergence from
the Keplerian disc and its escape from the halo. As the radiation passes
through the corona, the probability of repeated scattering by the same photon
decreases exponentially, however, the gain in energy is exponentially higher.
A balance of these two processes gives a power-law distribution of the energy
density. For the temperature estimation of the corona, we solve the thermally
decoupled two-temperature equations for electrons and protons. The location of
the shock can be found after solving flow equations by providing initial
specific energy and specific angular momentum value to the flow at the outer
boundary. However, as we are not solving the transonic flow coupled with the
radiative transfer, we use corona size or shock location as a free input
parameter of the model.

In the present model, we have computed the model spectra due to thermal
Comptonization only. As the innermost region of the disc is rapidly falling in
and is, in fact, supersonic (Chakrabarti, 1989; Chakrabarti & Titarchuk, 1995;
Mondal & Chakrabarti, 2013). At high accretion rates, when the flow becomes
cold and the thermal Comptonization can be ignored, the bulk motion takes the
role of energy shifting of seed photons. This so-called bulk motion
Comptonization (BMC) spectrum is decided by the upper limit of the velocity of
the infalling matter and thus the spectral slope saturates to around 2.0 even
when the accretion rates are varied. The theoretical works (Chakrabarti &
Titarchuk, 1995; Titarchuk & Zannias, 1998), Monte Carlo simulations (Laurent
& Titarchuk, 1999), and the observational results (Shaposhnikov & Titarchuk,
2009; Titarchuk & Seifina, 2009) all point to these saturation effects. Since
this property is solely due to the unique properties of a black hole
accretion, this is not affected by our analysis of thermal Comptonization,
valid for relatively lower accretion rates of the Keplerian component.

Figure 1: A cartoon diagram of the geometry of the flow (CT95). The scattering
of the soft photons from the Keplerian disc are the Zigzag trajectories. These
photons are Comptonized by the CENtrifugal pressure supported BOundary Layer
(post-shock region of the sub-Keplerian flow or corona) and are radiated as
hard X-rays.

The theoretical TCAF model uses five parameters namely, (i) mass of black hole
(MBH in $M_{\odot}$ unit), (ii) disc mass accretion rate ($\dot{m}_{\rm d}$),
(iii) halo mass accretion rate ($\dot{m}_{\rm h}$). The accretion rates are in
the Eddington accretion rate ($\dot{M}_{\rm Edd}$) unit, (iv) location of the
shock or boundary layer of the corona ($X_{\rm s}$ in $r_{\rm s}$ =$2GM_{\rm
BH}$/$c^{2}$ unit, where, $G$ and $c$ are the gravitational constant and speed
of light respectively). Here, $X_{\rm s}$ is the boundary layer of the corona
and we are solving steady state equations, therefore at any instant standing
shock will form at a particular location, and (v) shock compression ratio
($R=\rho_{-}/\rho_{+}$, where, $\rho$ is the density of the pre (-) and post
(+) shock flow). Here, we briefly discuss how model parameters change the
shape of the emitted spectra. Parameters (ii) to (v) collectively defines the
properties of the disc and corona of the flow, for instance, the electron
density and temperature, photon spectrum, and density. The fractional
interception of soft photons from the disc by the corona and vice-versa are
taken into account. During the scattering of soft photons by the corona,
cooling also takes place, which is also responsible for changing spectral
shapes (Done et al., 2007; Mondal & Chakrabarti, 2013; Yuan & Narayan, 2014).
All of these depend on the mass of the black hole, thus mass estimation can be
done satisfactorily (Molla et al., 2017) by analyzing full outburst data of
different X-ray missions, by keeping the mass of the BH as a free parameter or
by keeping the normalization fixed for a particular outburst for a particular
BH, details are explained in the paper. According to Chakrabarti (1997), if
one increases the halo rate keeping other parameters unchanged, the model will
produce a hard spectrum. Similarly, increasing the disc rate leaving other
parameters frozen will produce a soft spectrum. Figure 2 shows the typical
model spectra for varying $\dot{m}_{\rm d}$ when other parameters are fixed.
Increasing $\dot{m}_{\rm d}$ generates softer spectra. The parameter values
for each spectrum are given in the figure. The solid lines show the set of
spectra with cutoff at higher energy $\sim 1$ MeV whereas the dotted lines
show cutoff around 100 keV. When the location of the shock is increased
keeping other parameters fixed, the spectrum will become harder. A similar
effect can be also seen for the compression ratio (R). The effects are
discussed later in subsection 3.3. In general, in outbursts of black hole
binaries, during rising and declining phases with different spectral states,
all the parameters change smoothly in a multidimensional space (Mondal et al.,
2014b; Debnath et al., 2015; Chatterjee et al., 2016; Jana et al., 2016, and
others). The accretion rate ratio (ARR, the ratio of disc rate with halo
accretion rate) is also an indicator of spectral states change (Mondal et al.,
2014b), where the spectra move from hard (HS) to soft state (SS) through
intermediate states when ARR changes from low to high.

To generate the model spectra we use the outer boundary of the disc at 500
$r_{\rm s}$, extend up to $X_{\rm s}$ and corona starts from $X_{\rm s}$, and
extend up to 3 $r_{\rm s}$. The illuminating radiation contains different
components, including blackbody, Comptonization, and reflection. Photo-
ionisation processes are simulated with numerical spectral synthesis code
cloudy version C17.01 (Ferland et al., 2017). The cloudy solves different
radiation steps to generate emission lines (for detail code description and
most recent updates see Ferland et al., 1995, 2017).

In cloudy, a slab of gas is divided into a large number of thin zones which
results a plane parallel open geometry. In this geometry, the photon once
escaped will be lost to infinity without further interactions. A spherical
closed geometry can be implemented to include the multiple interactions of an
escaped photon as well. However, for this case of studying illumination of
disc by the photons from corona, the assumption of plane parallel geometry is
more realistic. Here, we note that the radiative transfer is solved in 1
dimension only. The radiation field from the accretion disc is normally
incident on the slab of gas and its reprocessing is initiated. We do not take
into account the incidence at different angles. The gas density which has been
used as the input parameter is taken from the mass accretion rate, and the
column density ($N_{\rm H}$) is kept constant to $1.28\times 10^{21}$ cm-2.
However, the effect of $N_{\rm H}$ has been verified in our study in a range.
For the detail discussions on the different possible geometries of the
radiation matter interactions and their implementation in various
astrophysical environments, we refer the reader to the Hazy1111www.nublado.org
documentation of cloudy. Moreover, Adhikari et al. (2015); Adhikari (2019)
have extensively discussed the use of open and closed geometries in cloudy for
various cases of AGN absorption and emission regions.

The next important assumption we employ in cloudy modelling is the use of
constant density case. This means that the gas number density is kept constant
to a given value across all the zones of the given gas cloud. Here we note
that, an alternative situation of constant pressure can arise when a gas cloud
is illuminated by the radiation energy where the gas number density is
stratified across the zones of cloud. This phenomenon of radiation pressure
confinement is discussed and implemented in various photoionized environments
of different astrophysical systems in the literatures (Różańska et al., 2006;
Stern et al., 2014; Baskin et al., 2014; Adhikari et al., 2015, 2019;
Adhikari, 2019). However, Adhikari et al. (2018) have shown that when there is
high gas number density and hence the high gas pressure, the radiation
pressure confinement is very weak, and the model structures are very similar
between constant density and constant pressure assumptions. Since the gas
number density in the accretion disc of black hole X-ray binaries are quite
high, we choose to use the constant density assumption for simulating the
photoionisation process.

The level of ionisation is determined by balancing all ionisation and
recombination processes. Ionisation processes include photo, Auger,
collisional ionisation, and charge transfer. Recombination processes include
radiative, low-temperature dielectronic, high temperature dielectronic, three-
body recombination, and charge transfer. The free electrons are assumed to
have a predominantly Maxwellian velocity distribution with a kinetic
temperature determined by the balance between heating (photoelectric,
mechanical, cosmic-ray, etc.) and cooling (predominantly inelastic collisions
between electrons and other particles) processes. The line emission and
continuum radiative transfer processes are solved simultaneously.

The incident radiation illuminates the disc and ionizes it. The ionisation
parameter is calculated using the equation:

$\xi=\frac{4\pi F_{\rm inc}}{n_{\rm H}}\text{erg cm s${}^{-1}$},$ (1)

where, $F_{\rm inc}$ is the Hydrogen ionizing flux integrated in the range
1-1000 Rydbergs and $n_{\rm H}$ is Hydrogen number density in cm-3 unit. The
chemical composition of the disc is set to default Solar values in cloudy and
it is mentioned when the Fe abundances ($A_{\rm Fe}$) is varied. Solar
abundances in cloudy are adopted from Grevesse & Sauval (1998). The electron
number density at some radius r of the flow is calculated from $\dot{m}_{\rm
h}$ above the disc, given by, $n_{\rm e}=\frac{\dot{m}_{\rm h}}{4\pi
r^{2}m_{\rm p}\varv}$ in $\text{cm}^{-3}$ unit, where $m_{\rm p}$ and $\varv$
are the mass of the proton and velocity of the inflow, which can be written as
$1/r^{1/2}$, when the disc radius is away from the black hole. Thus the
typical value of $n_{e}$ varies in a range from $1.4\times 10^{10}$-$1.4\times
10^{13}\text{cm}^{-3}$ for the accretion rate range 0.001-1 $\dot{M}_{\rm
Edd}$ and $r$=500$r_{\rm s}$. For the purpose of this work, the electron
density self consistently derived from our model can be used as the hydrogen
number density $n_{\rm H}$ in the cloudy computations. The model parameters
used to construct the spectral radiation shape or spectral energy distribution
(SEDs), which are used in the cloudy modelling are discussed in each figure.

Figure 2: Incident radiation spectra derived from the model for various values
of mass accretion rates, $\dot{m_{\rm d}}$, when the other model parameters
are fixed at $\dot{m_{\rm h}}=1.0$, $X_{\rm s}$=30, and $R$=2.5. Two clusters
of SEDs: a)first cluster with cut off around 100 keV shown by dotted colored
lines, and b) second cluster which with cut off around 1000 keV shown by solid
colored lines. The detail parameters for other SEDs are mentioned with the
corresponding figures.

##  3 Results

For the gas cloud above the disc which is illuminated by the central
radiation, we have calculated model reflection spectra covering a wide range
of disc parameters, with disc rate ($\dot{m}_{\rm d}$=0.001 - 2.0), the halo
rate ($\dot{m}_{\rm h}$=0.001, 0.01, 0.1, and 1.5), the shock location
($X_{\rm s}$=10, 30, 50, 80, and 100), the shock compression ratio ($R$=1.5,
2.5, 3.5, and 4.0), the ionisation parameter ($\log\xi$=1, 2, 3, and 4), and
the iron abundance ($A_{\rm Fe}$ = 0.5, 1.0, 2.0, and 5.0) relative to its
Solar value. The mass of the black hole is fixed at $7M_{\odot}$ throughout
the paper. These range of parameter values take into account for most of the
observed spectral states and features of black hole binaries, and all
parameters have significant effect on emission lines. For simplicity,
abundances of all other elements considered are kept fixed at Solar values.

###  3.1 The effect of varying $\dot{m}_{\rm d}$

Figure 3 shows reflected spectra for four different values of $\dot{m}_{\rm
d}$, mentioned in the figure. In all cases, the incident spectra have
$\dot{m}_{\rm h}$=1.0, $X_{\rm s}=30$, $R=2.5$, $\log\xi=3$, and the iron has
Solar abundance ($A_{\rm Fe}$ = 1.0). Increasing $\dot{m}_{\rm d}$ increases
the number of soft photons, thus the intensity of emission lines at the high
energy regime of the spectrum decreases, might be due to the presence of less
number of high energy photons, however, the possibility of forming more lines
is observed. It is clear from the emission spectra that both illuminating flux
and the spectral shape of the ionizing radiation incident on the surface of
the disc have a significant impact on the ionisation balance of the gas, and
thus on the reflected spectrum.

Figure 3: Reflected spectra for four different values of the $\dot{m}_{\rm
d}$, with $\dot{m}_{\rm d}$ = 0.001, 0.005, 0.05, and 0.1. The incident
spectrum has $\log\xi$ = 3, and iron has Solar abundance, the other
theoretical model parameters are $\dot{m}_{\rm h}$ =1.0, $M_{\rm BH}$=7.0 M⊙,
$X_{\rm s}$=30, and $R=2.5$. Successive spectra have been offset by factors of
1000 for clearer visibility.

The equivalent width $W_{\rm E}$ of an emission line is calculated by using an
expression,

$W_{E}=\int_{5.5\text{keV}}^{7.2\text{keV}}\frac{F_{E}^{C}-F_{E}^{t}}{F_{E}^{C}}dE,$
(2)

where $F_{E}^{C}$ and $F_{E}^{t}$ are the fluxes in the reflected continuum
and around the line energy centroid respectively. For the purpose of studying
the emission from highly ionized Fe, we sum up the $W_{\rm E}$ of Fe-emission
lines in the energy band $5.5-7.2$ keV and define it as the Fe-line $W_{\rm
E}$. In the computations of the $W_{\rm E}$ in this energy band, the
contribution of the ions Fe ii to Fe xxvi are considered.

Figure 4 shows that the general trend of $W_{\rm E}$ is decreasing with
increasing $\dot{m}_{\rm d}$. This is expected as at low accretion rate most
of the photons are concentrated at the high energy part of the spectrum, where
the Compton scattering becomes important, therefore hard spectra are more
efficient in heating the illuminated layer, which emits more lines. As the
accretion rate increases, the incident spectrum becomes softer thus less
number of hard photons participate in line emission. It also shows that
increase in column density increases $W_{\rm E}$, which satisfies that $W_{\rm
E}$ $\propto$ $N_{\rm H}$. The red, blue, green, and black lines correspond to
$N_{\rm H}$ values $10^{21}$, $10^{23}$, $10^{24}$, and $10^{25}$
respectively. The $W_{\rm E}$ is estimated for the lines Feii-Fexxvi. Our
estimated variation of $W_{\rm E}$ is in the same line of Zycki & Czerny
(1994); García et al. (2013), where authors showed that $W_{\rm E}$ decreases
with increasing photon index ($\Gamma$), which is the same in our model as
increase in $\dot{m}_{\rm d}$ makes the spectrum softer thus the higher value
of ($\Gamma$). A recent analysis of MAXI J1820+070 (Xu et al., 2020), showed a
similar agreement. However, opposite behavior can also be observed if the
lines originate from the inner edge of the disc, where the disc moves inward
with accretion rate, which increases rotational velocity, therefore increases
line broadening, thus the $W_{\rm E}$ (Tomsick et al., 2009; Debnath et al.,
2015). The $W_{\rm E}$ is also an important measurable quantity to diagnose
the evolution of QPOs in the accretion disc, reported in Galactic black hole
GRS 1915+105 (Miller & Homan, 2005). Earlier the current model was also used
to show the evolution of QPOs with disc accretion rate for the outbursting
candidates (Mondal et al., 2014b; Chakrabarti et al., 2015), where the QPO
frequency increases with accretion rate. Thus combining these two effects in
the future we will be able to study the QPOs evolution consistently from the
Fe line fitting. Apart from the variation of the Fe line width, our results
also include emission lines at low energies e.g., N and O to a range between
0.4 to 0.8 keV, which have been observed in high-resolution XMM-Newton
observation of black hole candidate Swift J1753.5-0127 (Mostafa et al., 2013).
The range of our simulated $W_{\rm E}$ agrees with estimations from the model
fitted observed data (Titarchuk & Seifina, 2009). However, it should be noted
that as the flow comes closer to the black hole with increasing disc accretion
rate, BMC becomes effective, and also the gravitational effects take a
significant role, that can skew the line properties (Titarchuk & Seifina,
2009).

Figure 4: $W_{\rm E}$ of Fe line integrated in the energy range $5.5-7.2$ keV
as a function of $\dot{m}_{d}$. We extend $\dot{m}_{d}$ values to super-
Eddington rate to see the trend of behavior. Various colours represent the
$W_{\rm E}$ computed for various values of $N_{\rm H}$. Other theoretical
parameters used in these computations are: $\dot{m}_{\rm h}$ =1.0, $M_{\rm
BH}$=7.0 M⊙, $X_{\rm s}=30$ and R = 2.5 respectively. In the cloudy modelling,
$\log\xi$ = 3.0 and Solar values for Fe abundances are used. Various colors in
the plot depicts the varying $N_{\rm H}$ employed.

###  3.2 The effect of varying $\dot{m}_{\rm h}$

In Figure 5, we carry out the same task for four different values of
$\dot{m}_{\rm h}$, when $\dot{m}_{\rm d}$=0.02 and keeping all other
parameters the same as in Figure 3. As we discussed for increasing
$\dot{m}_{\rm h}$ spectra become harder thus increases the high energy photons
at high energies which affect the emission features, as the harder
illuminating spectra have greater ionizing efficiency, change the shape of the
reflection spectra significantly. For $\dot{m}_{\rm h}$=0.1 and 1.5 (HS), the
illuminated gas is more highly ionized than for $\dot{m}_{\rm h}$=0.001 (SS)
and 0.01 (intermediate state). One can observe that more emission lines are
visible in the high energy regime and less lines are observed at low energy
regime which is due to the outer layer of the illuminated cloud gets fully
ionized for higher $\dot{m}_{\rm h}$. However, an opposite feature is observed
for the low $\dot{m}_{\rm h}$ spectra. In the intermediate state (red curve),
throughout the energy band emission lines are clearly visible with a
relatively high intensities.

Figure 5: Reflected spectra for four values of the $\dot{m}_{\rm h}$, with
$\dot{m}_{\rm h}$ = 0.001, 0.01, 0.1, and 1.5. The incident spectrum has
$\log\xi$ = 3, and iron has Solar abundance, the other theoretical model
parameters are $\dot{m}_{\rm d}$ =0.02, $M_{\rm BH}$ = 7.0 M⊙, $X_{\rm s}$ =
30.0, and R = 2.5. Successive spectra have been offset by factors of 1000 for
clearer visibility.

###  3.3 The effect of varying geometry

Figure 6 shows emission lines due to variation in accretion geometry e.g.
varying size of the corona and its height (hs). Here, we choose four different
values of corona size i.e. $X_{\rm s}$, with $X_{\rm s}$=10, 30, 50, and 100,
and four different values of R, with $R$=1.5, 2.5, 3.0, and 4.0, from SS to HS
through intermediate states. Both parameters change hs and also the optical
depth of the corona, thus the emitted spectra. The change in $X_{\rm s}$ and
$R$, changes the hs as $\sim$ $\sqrt{(R-1)}X_{\rm s}/R$ and the total
accretion rate at the post-shock region after pilling up of matter can be
written as $R\dot{m}_{\rm h}$+$\dot{m}_{\rm d}$ (Mondal et al., 2014a),
indicates that the physical quantities (temperature and optical depth etc.) of
the corona depends on $R$. Furthermore, varying $R$ during the outburst phase
also means variable mass outflow, as the TCAF model proposed that the corona
is the base of the mass outflow/jet, which extracts thermal energy from the
inner hot region and makes the spectrum softer (Chakrabarti, 1999; Singh &
Chakrabarti, 2011; Mondal et al., 2014a), therefore change in emission line
properties. However, in the present modeling we do not include jet effects.
Figure 6a shows the lines for different $X_{\rm s}$, at the intermediate
$X_{\rm s}$, the intensity of a few lines is higher than low and high $X_{\rm
s}$, some extra lines are visible around 0.3 keV. Figure 6b, shows the lines
for different values of R, with increasing R, spectrum moves from SS to HS,
thus high energy photon intensity increases, which increases the ionisation
rate. Therefore more lines are visible with increasing R, especially below
0.01 keV and between 0.2-0.4 keV.

Figure 6: Reflected spectra (a) for four values of the $X_{\rm s}$, with
$X_{\rm s}$ = 10, 30, 50, and 100, when $R$=2.5 and (b) for four values of R,
with $R$= 1.5, 2.5, 3.0, and 4.0, when $X_{\rm s}$=30. The all incident
spectra have $\log\xi$ = 3, and iron has Solar abundance, the other
theoretical model parameters are $\dot{m}_{\rm d}$ =0.005, $\dot{m}_{\rm h}$ =
1.0, and $M_{\rm BH}$=7.0 M⊙.

###  3.4 The effect of varying $\xi$

The flux of the continuum that hits the Cloudy gas is fixed by the gas density
and the ionization parameter as described in Eq. 1. The open geometry (a plane
parallel slab with several thin zones) of the emitting gas in Cloudy is
assumed which is different from the geometry of how TCAF is formulated. For a
given gas cloud, the ionization parameter at the illuminated surface is
defined. However, in the inner zones of the slab, it can change according to
Eq. 1. Nevertheless, with our assumption of constant density gas (with thin
slab) this change is negligible. For a closed (spherical) geometry, the change
in ionization parameter can be significant and should be represented by a
distribution (Adhikari et al., 2015). Also, if one assumes a constant pressure
gas (instead of constant density), then, there can be several orders of
magnitude difference in ionization parameter between the illuminated side and
the back side of the slab (Adhikari et al., 2016; Adhikari et al., 2018;
Adhikari et al., 2019, and references therein). Therefore, we examine the
effect of $\xi$ on the emission lines for different values of $\xi$ applied to
two different values of ARR, keeping the other TCAF model parameters fixed. In
both panels of Figure 7, each curve corresponds to a particular value of
log$\xi$=1, 2, 3, and 4. The other model parameters are $M_{\rm BH}$=7.0 M⊙,
$X_{\rm s}$ = 30, R = 2.5, and the iron has Solar abundance. In each panel,
the plotted spectra have been rescaled for visual clarity. The scaling factors
are, from bottom to top, 1, 103, 106, and 109. The left panel in Figure 7a
shows emission spectra when ARR=0.001, i.e. a very hard state of the
illuminating spectrum. On the other hand, Figure 7b, when ARR=20, i.e. a
softer state of the illuminating spectrum, and the effects are indeed visible.
For the same ARR, increasing $\xi$ means a higher ionisation rate, which
raises the temperature of the illuminated region of the slab thus ionize the
gas at a larger optical depth. Hence ions from lower atomic number (Z)
elements are completely stripped from all their elements, while heavy Z
elements get partially ionized. Thus the emission spectra lack lines from most
of the low-Z elements, are either absent or the intensity of the lines went
down abruptly when $\log\xi$ increases and progressively showing for heavy-Z
elements. This specific trend is observed for $\log\xi$=4. The Fe K$\alpha$
line is peaking at 7keV. However, an opposite scenario is observed in Figure
7b when both ARR and $\xi$ are high. All lines are visible even for higher
$\xi$ in comparison of low ARR (hard state) in the left panel. This infers
that in the hard state, the partly ionized disc can emit more lines as more
hard photons take part in line emission. However, for a fully ionized disc
with harder illumination, it emits fewer lines. Whereas in the soft state, the
disc emits more lines even for a higher ionisation parameter ($\xi$) as the
disc does not get completely ionised. However, the left panel with $\log\xi$=4
can show up emission lines at low energies if the column density of the gas
cloud is increased. If the slab thickness is less and the radiation is hard,
it will pass through the slab with little scattering or without interacting
with the medium.

Figure 7: Reflected spectra for four values of the $\log\xi$, with $\log\xi$ =
1, 2, 3, and 4. The iron has Solar abundance, $M_{\rm BH}$=7.0 M⊙, $X_{\rm s}$
= 30 and R = 2.5. The other disc parameters are (a) $\dot{m}_{\rm d}$ =0.001,
$\dot{m}_{\rm h}$ = 1.0, and (b) $\dot{m}_{\rm d}$ =0.02, $\dot{m}_{\rm h}$ =
0.001. Successive spectra have been offset by factors of 1000 for clearer
visibility.

Figure 8 shows the plot of $W_{\rm E}$ versus ionisation parameter, where
$W_{\rm E}$ tends to decrease with increase in ionisation parameter until low
ionisation region, however, it sharply increases in higher ionisation regime,
$\xi>10^{3}$. After some value of ionisation, $\xi\sim 10^{4}$, again the
$W_{\rm E}$ decrease which is because of a wider part of the disc becomes
fully ionized, these typical behaviors can also be seen mainly in case of
Seyfert 1 galaxies atmosphere (Zycki & Czerny, 1994). As evidenced from Zycki
& Czerny (1994) $W_{\rm E}$ becomes maximum around $\xi=3\times 10^{8}$ for
Seyfert 1 galaxies and decreases for further increase in $\xi$. The difference
of $\sim$ 4 orders of magnitude in the $\xi$ value at which $W_{\rm E}$ peaks
in X-ray binaries and Seyfert galaxies is due to the considerable difference
in their luminosity. Considering this point, the results from Figure 8 of this
paper corroborate with the results of Zycki & Czerny (1994) (Figure 19 of
their paper). The black, pink, and green curves corresponds to different
values of $N_{\rm H}$. The peak value of $W_{\rm E}$ is highest for the lowest
value of $N_{\rm H}=10^{22}$. This is due to the fact that, the larger
fraction of the slab of gas with low column density is at high temperature,
and thus more ionized, as compared to that with higher column density. This
fact can also be realized by looking at the Figure 10, where the temperature
for the lowest column density stays high even at the backside of the slab.
However, for the case with highest column density, the temperature falls off
rapidly with the depth of the slab and unable to ionize the Fe inside. Here,
we study the dependence of our result on the ionization parameter defined at
the surface of the gas cloud. From observations, $\xi$ is always constrained
to have a value in the range. This is why we are exploring the range of
values. Similar studies were also carried out by Zycki & Czerny (1994) and
García et al. (2013).

Figure 8: Variation of equivalent widths as a function of ionisation parameter
$\xi$. The incident SED used for this case is parametrized by: $\dot{m}_{\rm
d}$ =1.2, $\dot{m}_{\rm h}$ =1.0, $M_{\rm BH}$=7.0 M⊙, $X_{\rm s}$ = 30 and R
= 2.5 respectively. In the cloudy modelling, Solar abundances are used and
various colors in the plot depicts the varying $N_{\rm H}$ employed.

###  3.5 The effect of varying $A_{\rm Fe}$

The amount of a particular element present in the gas changes the continuum
opacity, which in turn affects the photoionisation heating rate. Therefore the
abundances for each element are taken into account in a photoionisation effect
can greatly affect the ionisation balance, the observable spectral line
features in the reprocessed radiation. At the same time, the abundance of a
particular element influences the strength of the emission lines. Considering
the relevant effect of Fe line emission in the analysis of the X-ray spectra
for accreting binary candidates, we have carried out generating emission
spectra in which the Fe abundance is varied between sub-Solar, Solar, and
super-Solar values. All other elements considered in these calculations are
set to their Solar values. The effects of varying abundance of iron are shown
in Figure 9(a-b). The illuminating radiation has $\log\xi$ = 3, and the other
model parameters are the same as in Figure 7 for two different spectral
states, one is a very hard state when ARR=0.001 (left panel) and the other one
is a softer state when ARR=20 (right panel), are shown in Figure 9a and b
respectively. In each panel, the plotted spectra have been re-scaled for
visual clarity. The scaling factors are, from bottom to top, 1, $10^{3}$,
$10^{6}$, and $10^{9}$. The effect of $A_{\rm Fe}$ substantially changes the
Fe emission features, evident in the emission spectra. For higher $A_{\rm
Fe}$, increases the strength of overall emission lines including Fe emission.
A significant difference at the low energy emission can be seen, where for
higher ARR more lines are visible with higher intensity compared to low ARR.
The reason behind this is that harder radiation (low ARR) fully ionised the
disc therefore less lines are generated.

Figure 9: Reflected spectra for different values of $A_{\rm Fe}$, with $A_{\rm
Fe}$ = 0.5, 1.0, 2.0 and 5.0 when $\log\xi$=3. The TCAF model parameters which
are fixed for both cases are, $M_{\rm BH}$=7.0 M⊙, $X_{\rm s}$ = 30 and R =
2.5. The accretion rates for the two cases of spectral states are: (a)
$\dot{m}_{\rm d}$ =0.001, $\dot{m}_{\rm h}$ = 1.0, and (b) $\dot{m}_{\rm d}$
=0.02, $\dot{m}_{\rm h}$ = 0.001. Successive spectra have been offset by
factors of 1000 for clearer visibility.

###  3.6 Temperature profile and the effect of varying disc column density
$N_{\rm H}$

We studied the dependence of our results on the thickness of the cloud by
varying the column density of the cloud, $N_{\rm H}$. Figure 10 shows that
increase in optical depth decreases the temperature of the illuminated medium
for the disc parameter $\dot{m_{d}}=1.2$, $\dot{m_{h}}=1.0$, $X_{\rm s}$=30,
and $R$=2.5. This behaviour is due to the increased reprocessing of the X-ray
photons by the larger surface of the gas cloud. The column density and optical
depth considered here can potentially affect the temperature structure of the
gas cloud by varying heating and cooling efficiency, thus the strength of the
emission. At the surface of the gas layer where illuminating radiation hits is
hot and Compton heating and cooling dominates. The temperature at the surface
remains at Compton temperature. In this region radiation field thermalises,
thus the temperature of the gas remains constant $\sim 4\times 10^{5}$ K. As
the optical depth increases, cold region exists and the temperature decreases,
after a certain optical depth ($\sim 0.05$), the temperature falls sharply by
orders in magnitude. If the column density is typically low, the cooling rate
is also low, thus the illuminating radiation thermalises the layers of the
cloud, thus the temperature remains constant to a fixed value does not fall as
we see in high column density case. The blue, orange, green, and red curves
show temperature profile for the values of $N_{\rm H}$ = $10^{21}$, $10^{23}$,
$10^{24}$, and $10^{25}$ respectively. However, the scenarios may change if a
different set of input SEDs and ionisation parameter are considered.

Figure 10: Temperature as a function of the optical depth for models with
variation in column densities. The structure presented here is for the
incident SED generated using model parameters $\dot{m_{d}}=1.2$,
$\dot{m_{h}}=1.0$, $X_{\rm s}$=30, and $R$=2.5. $\log\xi$ = 3.0 and Solar
values for Fe abundances are used in cloudy.

Figure 11 shows the effect of $N_{\rm H}$ on reflection spectra of Fe line
between 5.5-7.2 keV, where line intensity increases with increasing column
density. It also shows that when intensity increases, line width increases as
evident from Figure 4. The change in line shape and the line to continuum
contrast on increasing the column density are related to the behaviour of the
curve of growth (COG) (i.e., the plot of line $W_{E}$ against the ionic column
density). The curve of growth generally has three parts : linear part (where
$W_{E}$ increases linearly with ionic column density), exponential part
($W_{E}$ increases exponentially), and saturated part (lines are saturated).
Such a behaviour can be seen in Adhikari (2019). For the lowest column
density, the emission lines are produced corresponding to the ionic column
density at the linear part of the COG. Once the column density increases, the
lines are produced from an exponential part of the COG, hence, having higher
$W_{E}$ and broader lines. Due to the higher ionic column density, the
collisional broadening becomes more significant and the line shape becomes
complex. If we keep increasing the column density further, it should remain
unchanged as we approach the saturated part of COG. The Fe XXVI (6.96 keV)
line will show up for the cases where $\dot{m}_{\rm d}$ is low i.e. more hard
photons can contribute and also the ionization rate is high. Therefore Figure
15a shows the Fe XXVI line up, whereas the panel b does not show up as the
$\dot{m}_{\rm d}$ is high and $\dot{m}_{\rm h}$ is low (typical soft-state).
Since Figure 11 is shown for the $\dot{m}_{\rm d}=1.2$ (which lacks hard X-ray
photons), hence the Fe XXVI line is not present there. Furthermore, the 5.6
keV features are not Fe-line features (these are Cr and Ti line features not
relevant at present). In addition the 6.1 (6.07-6.11) keV features for Mn, Ti,
Cr, are not relevant at present as well. We put them for the sake of
completeness and in the future the high-resolution observations may detect
these lines.

Figure 11: Fe-line spectra for various values of $N_{\rm H}$ and for the same
model parameters as in Fig. 10

##  4 Shape of Fe Line

In Figure 12, we show Fe line emission for varying $\dot{m}_{\rm d}$, same as
in Figure 3. Increasing disc mass accretion rate, the observed 6.6 keV and 6.7
keV double lines intensity increases. In all four cases, Fe line $\sim 6.6$
keV splits into two and many other lines from low Z species are produced
between 5-8 keV energy band, some of them are mentioned in Figure 11. In this
case, ARR value increases refers increase in disc rate i.e. no of soft photon
increases keeping the hot flow component (halo rate) fixed. Therefore spectra
become softer. As the ARR value increases the second peak intensity increases
and the first peak decreases. For high ARR, the outer layer of the gas cloud
falls quickly from very hot phase to cold phase at negligibly small Thomson
optical depth, therefore it can ionize more layer in the cloud to emit lines.

However, Figure 13 shows a different line properties when halo rate increases
from sub-Eddington to Eddington rate, Fe line $\sim 6.5$ KeV starts splitting
and the component at $\sim 6.7$ keV has higher intensity than the lower energy
peak. It is also noticeable that many complex lines are disappeared at low
halo rate, and they appeared at high halo rate, this implies that when
spectral states move from soft to hard, the intensity of illuminating spectrum
at high energy increases, thus the ionisation rate, producing more lines.
Here, increase in hot flow rate refers decrease in ARR which generate more
hard photons to illuminate the disc surface. Therefore, at high ARR ( black
and red lines) the illuminating radiation has not enough high energy photons
to trigger thermal instability as the temperature of the gas cloud decrease,
therefore less no of Fe species contributed and showed a single sharp peak. As
the ARR goes down ($<$1, green and blue lines), the temperature of the gas
cloud increases, the Fe line broadens.

Figure 12: Same as Figure 3, but for the reflected spectra of the Fe lines. We
use offset 100 for all Fe line spectra for visual clarity. Figure 13: Same as
Figure 5, but for the reflected spectra of Fe lines.

Figure 14a-b show the effect of varying $X_{\rm s}$ and R on the reflected Fe
line spectra. In both cases, Fe lines showed a double peak $\sim$6.5 keV.
Figure 14a shows that increasing $X_{\rm s}$, the low energy peak intensity of
Fe lines increases, however, in low and high $X_{\rm s}$, lines showed
opposite nature compared to the intermediate corona size. Also, an extra
emission line $\sim$7 keV is observed in low and high $X_{\rm s}$ with the
increase in intensity for high $X_{\rm s}$, these can be understood from the
hardening of the illuminating spectra, and more number of hard photons are
contributing in ionizing the disc. Figure 14b, showed an opposite behavior in
the double peak Fe line, where the intensity of the low energy peak $\sim$6.6
keV increases with R, and the high energy peak intensity did not change much,
which showed flipping behavior for $R$=4, the strong shock condition. It
should be noted that the double peaks which are observed are not due to any
gravitational broadening effects nor due to photon bending effects, rather two
different transitions of Fe lines at different energies. The low energy iron
lines are not shown up, which can be due to completely stripped off of the
electrons from the atomic levels.

Figure 14: Same as Figure 6, but for the reflected spectra of Fe lines.

Figure 15 a-b show the effect of varying $\log\xi$ on the reflected Fe line
spectra for the same sets of model parameters in Figure 7 for HS and SS. At
the pure HS with low $\log\xi$, a single and broad Fe line is observed at
$\sim 6.5$ keV. It becomes narrow when $\log\xi$ increases and the line splits
when $\log\xi$ is 3, at very high ionisation value the line intensity goes
down and an another sharp-peaked line at 6.96 keV with high intensity is
observed. This evidences that the gas becomes so ionized that H like iron (Fe
XXVI) is the dominant species. However, for the same $\log\xi$, in the SS
Figure 15b, only one line is observed at 6.5 keV and it is narrower in low and
high $\log\xi$. The appearance of the single peak line in the SS follows the
similar discussion of temperature structure change of the gas cloud with the
hardness of the illuminating radiation. As the ARR is high ($\gg 1$), incident
radiation has not enough hard photons to trigger the thermal instability,
therefore not many Fe species contributed in line emission.

Figure 15: Same as Figure 7. Reflected spectra for Fe lines.

Figure 16a-b show the effect of varying $A_{\rm Fe}$ (increases from black to
blue line) on the reflected Fe line spectra for the same sets of model
parameters in HS and SS as in Figure 9. Figure 16a shows the double peak
nature of Fe lines with increasing intensity in low energy line, whereas high
energy line intensity decreases, this is due to the hard radiation increases
ionisation rate with increasing Fe abundances. Thus more complex lines are
formed. However, Figure 16b shows a single line with increasing intensity for
increasing $A_{\rm Fe}$, indeed shows a significant effect of Fe abundances.
Also, the illuminating spectra have less effect on the temperature structure
of the gas cloud in the SS (ARR $>$ 1) as less number of hard photons
contributed to trigger the thermal instability. However, from observation, we
see that the line broadens and breaks into a double peak during the outburst
phase, and the intensity increases as the spectrum moves to the soft state
(Mondal et al., 2016) when the inner edge of the disc comes much closer to the
BH. These two combined results infer that gravitational broadening might be
more dominating over photoionisation in the soft state, when the Fe lines are
originated at the inner edge of the disc.

In the current modeling, we did not consider any gravitational effect as in
Fabian et al. (1989) or Laor (1991). The lines are generated due to photo-
ionization processes or due to atomic transitions in Cloudy. It is also worth
mentioning that Cloudy does not have the gravitational broadening effect, but
the broadening due to the thermal motion of the gas is incorporated.
Furthermore, our location of the shock is not close to the innermost stable
circular orbit, rather farther away, where the line broadening effect may not
be effective enough. Moreover, general relativistic or photon bending effects
are important to get those double-peaked lines, which is beyond the scope of
the paper.

Figure 16: Same as Figure 9. Reflected spectra for Fe lines.

##  5 Conclusions

In this paper, we have presented reprocessed spectra from an accretion disc,
which is emitted as a reflected spectra. The incident illuminating spectra are
generated using TCAF model. Each model is characterized by the disc and halo
mass accretion rates, location of shock/size of the corona and the shock
compression ratio of the flow. Next, the reprocessing of this incident
radiation is simulated by using the photoionisation code cloudy, where a disc
is parametrized by: the ionisation parameter at the surface, the hydrogen
number density, the column density and the Fe abundance with respect to its
Solar value. The range of parameters we use here, covers the observed spectral
states for accreting black hole binaries. The parameter ranges are following:
$0.001\leq\dot{m}_{\rm d}\leq 2.0$, $0.001\leq\dot{m}_{h}\leq 1.5$, $10\leq
X_{s}\leq 100$, $1.5\leq R\leq 4.0$, $1\leq\log\xi\leq 4$, and $0.5\leq A_{\rm
Fe}\leq 5$. The illuminating spectra consist disc blackbody and the
Comptonization component of the spectra taking into account the reflection,
covering the energy range from 0.001 to 1000 keV.

In comparison to other line emission models in the literature, our
illuminating spectra vary in a multidimensional parameter space including the
dynamics of the flow, which is indeed observed for outbursting black hole
candidates. The key findings of the present model are as follows:

  1. 1.

Increasing $\dot{m}_{\rm d}$, the spectrum becomes softer, the intensity of
the emission lines at the high energy part of the spectrum decreases, but more
lines are observed at low energies. Varying $\dot{m}_{\rm d}$ splits Fe line
into double peak $\sim 6.65$ keV (Fe XXV) with higher intensity at low energy
line and many complex lines are formed (see Figure 12). The Fe line intensity
correlates with $\dot{m}_{\rm d}$.

  2. 2.

Changing $\dot{m}_{\rm h}$, covering from hard to soft followed by
intermediate, shows that in the soft state more lines are observed throughout
the energy band, however, in the intermediate state (red line in Figure 5)
emission lines are clearly visible throughout the energy band. In hard state,
more/less lines are visible at high/low energies. In the pure soft state and
intermediate state (black and red lines in Figure 13), Fe lines have a single
peak $\sim 6.65$ keV with higher equivalent line width in the soft state and a
sharp peak in the intermediate state. The Fe line splits into a double peak
when it enters into hard state with opposite behavior in peak intensity.

  3. 3.

Increasing $X_{\rm s}$, low energy peak intensity of Fe line increases,
whereas the peak intensity of high energy decreases. But both peaks are
comparable for $X_{\rm s}=100$ case (see Figure 14a) and a sharp peaked H like
iron (Fe XXVI) line at $6.96$ keV has been observed.

  4. 4.

Increasing R, low energy peak intensity is higher than high energy peak.
However, when shock gets stronger, the peak intensity nature becomes opposite.
This can be due to, at strong shock spectrum becomes harder thus efficiency of
illuminating the disc by high energy photons is more, which emits lines with
higher intensity (see Figure 14b).

  5. 5.

In the soft spectral state, a single peaked Fe line is produced for different
ionisation parameters, where the line width is less for the lowest and highest
ionisation parameters compared to intermediate ionisation rate. In contrast,
in the pure hard state, line splits at higher ionisation values, and at the
highest ionisation value a high intensity, sharp peaked line at $6.96$ keV is
observed. This implies that the gas become so ionized that H like iron (Fe
XXVI) is the dominant species. This ion has a large fluorescent yield
resulting in a strong line.

  6. 6.

A similar feature is observed when Fe abundance is varied. In soft spectral
state, the intensity and width of the single peaked Fe line increase with Fe
abundance, however, in pure hard state, the lines split into two and the
intensity of the low energy peaks increase with increasing Fe abundance.

  7. 7.

The measurable quantity which relates the observation is the $W_{\rm E}$ of
the Fe lines, which decrease with increasing disc accretion rate, and
increases with the column density of the gas cloud. The equivalent width also
decreases with ionisation parameter ($\xi$) up to a certain value
($\xi<10^{3}$) and then again increases sharply with a peak $\sim 10^{4}$ and
then falls sharply.

  8. 8.

The temperature profile of the gas cloud above the disc changes by orders in
magnitude with increasing optical depth, depending on the column density of
the medium.

In the present paper, we did not fit the simulated Fe lines directly with the
observed lines. In the future we aim to directly fit the observed data using
the current model as a table model or as a local model. The current model also
does not include the inclination effect in the theoretical model generation,
which will be published elsewhere.

## Acknowledgements

We thank the anonymous referee for making critical comments that improved the
quality of the manuscript. We thank Agata Różańska for insightful comments
that improved the presentation of the manuscript. SM acknowledges Ramanujan
Fellowship (file #RJF/2020/000113) by SERB-DST, Govt. of India and Kreitman
Fellowship by Kreitman School of Advanced Graduate Studies at the Ben-Gurion
University of the Negev, Israel, during which the work was started. T. P. A
gratefully acknowledges the Inter-University Center for Astronomy and
Astrophysics (IUCAA), Pune, India and the Nicolaus Copernicus Astronomical
Center of the Polish Academy of Sciences, Poland for providing the access to
the Computational Cluster, where the numerical simulations used in this paper
are performed. C.B.S. is supported by the National Natural Science Foundation
of China under grant No. 12073021.

## Data Availability

Data information may not be applicable for this article. No new data has been
analyzed as it is mostly a theory-based article. The numerically created data
that support the findings of this study are available from the corresponding
author, upon reasonable request.

## References

  * Adhikari (2019) Adhikari T. P., 2019, Photoionization Modelling as a Density Diagnostic of Line Emitting/Absorbing Regions in Active Galactic Nuclei, doi:10.1007/978-3-030-22737-1.
  * Adhikari et al. (2015) Adhikari T. P., Różańska A., Sobolewska M., Czerny B., 2015, ApJ, 815, 83
  * Adhikari et al. (2016) Adhikari T. P., Różańska A., Czerny B., Hryniewicz K., Ferland G. J., 2016, ApJ, 831, 68
  * Adhikari et al. (2018) Adhikari T. P., Hryniewicz K., Różańska A., Czerny B., Ferland G. J., 2018, ApJ, 856, 78
  * Adhikari et al. (2019) Adhikari T. P., Różańska A., Hryniewicz K., Czerny B., Behar E., 2019, ApJ, 881, 78
  * Baskin et al. (2014) Baskin A., Laor A., Stern J., 2014, MNRAS, 438, 604
  * Bianchi et al. (2006) Bianchi S., Guainazzi M., Chiaberge M., 2006, A&A, 448, 499
  * Brenneman & Reynolds (2006) Brenneman L. W., Reynolds C. S., 2006, ApJ, 652, 1028
  * Chakrabarti (1989) Chakrabarti S. K., 1989, ApJ, 347, 365
  * Chakrabarti (1997) Chakrabarti S. K., 1997, ApJ, 484, 313
  * Chakrabarti (1999) Chakrabarti S. K., 1999, A&A, 351, 185
  * Chakrabarti & Titarchuk (1995) Chakrabarti S., Titarchuk L. G., 1995, ApJ, 455, 623
  * Chakrabarti et al. (2015) Chakrabarti S. K., Mondal S., Debnath D., 2015, MNRAS, 452, 3451
  * Chatterjee et al. (2016) Chatterjee D., Debnath D., Chakrabarti S. K., Mondal S., Jana A., 2016, ApJ, 827, 88
  * Cunningham (1976) Cunningham C., 1976, ApJ, 208, 534
  * Dabrowski et al. (1997) Dabrowski Y., Fabian A. C., Iwasawa K., Lasenby A. N., Reynolds C. S., 1997, MNRAS, 288, L11
  * Dauser et al. (2012) Dauser T., et al., 2012, MNRAS, 422, 1914
  * Debnath et al. (2014) Debnath D., Chakrabarti S. K., Mondal S., 2014, MNRAS, 440, L121
  * Debnath et al. (2015) Debnath D., Mondal S., Chakrabarti S. K., 2015, MNRAS, 447, 1984
  * Done et al. (2007) Done C., Gierliński M., Kubota A., 2007, A&ARv, 15, 1
  * Dong et al. (2020a) Dong Y., García J. A., Liu Z., Zhao X., Zheng X., Gou L., 2020a, MNRAS, 493, 2178
  * Dong et al. (2020b) Dong Y., García J. A., Steiner J. F., Gou L., 2020b, MNRAS, 493, 4409
  * Dumont et al. (2000) Dumont A. M., Abrassart A., Collin S., 2000, A&A, 357, 823
  * Fabian et al. (1989) Fabian A. C., Rees M. J., Stella L., White N. E., 1989, MNRAS, 238, 729
  * Fabian et al. (2000) Fabian A. C., Iwasawa K., Reynolds C. S., Young A. J., 2000, PASP, 112, 1145
  * Ferland et al. (1995) Ferland G. J., et al., 1995, in American Astronomical Society Meeting Abstracts. p. 108.02
  * Ferland et al. (2017) Ferland G. J., et al., 2017, Rev. Mex. Astron. Astrofis., 53, 385
  * García et al. (2013) García J., Dauser T., Reynolds C. S., Kallman T. R., McClintock J. E., Wilms J., Eikmann W., 2013, ApJ, 768, 146
  * George & Fabian (1991) George I. M., Fabian A. C., 1991, MNRAS, 249, 352
  * Giri & Chakrabarti (2013) Giri K., Chakrabarti S. K., 2013, MNRAS, 430, 2836
  * Giri et al. (2015) Giri K., Garain S. K., Chakrabarti S. K., 2015, MNRAS, 448, 3221
  * Grevesse & Sauval (1998) Grevesse N., Sauval A. J., 1998, Space Sci. Rev., 85, 161
  * Guilbert & Rees (1988) Guilbert P. W., Rees M. J., 1988, MNRAS, 233, 475
  * Haardt & Maraschi (1993) Haardt F., Maraschi L., 1993, ApJ, 413, 507
  * Iwasawa et al. (1996) Iwasawa K., et al., 1996, MNRAS, 282, 1038
  * Jana et al. (2016) Jana A., Debnath D., Chakrabarti S. K., Mondal S., Molla A. A., 2016, ApJ, 819, 107
  * Kallman & White (1989) Kallman T., White N. E., 1989, ApJ, 341, 955
  * Laor (1991) Laor A., 1991, ApJ, 376, 90
  * Laurent & Titarchuk (1999) Laurent P., Titarchuk L., 1999, Astrophysical Letters and Communications, 38, 173
  * Lightman & Rybicki (1980) Lightman A. P., Rybicki G. B., 1980, ApJ, 236, 928
  * Lightman & White (1988) Lightman A. P., White T. R., 1988, ApJ, 335, 57
  * Lightman et al. (1981) Lightman A. P., Lamb D. Q., Rybicki G. B., 1981, ApJ, 248, 738
  * Magdziarz & Zdziarski (1995) Magdziarz P., Zdziarski A. A., 1995, MNRAS, 273, 837
  * Matt et al. (1991) Matt G., Perola G. C., Piro L., 1991, A&A, 247, 25
  * Matt et al. (1993) Matt G., Fabian A. C., Ross R. R., 1993, MNRAS, 262, 179
  * Miller & Homan (2005) Miller J. M., Homan J., 2005, ApJ, 618, L107
  * Miller et al. (2008) Miller J. M., et al., 2008, ApJ, 679, L113
  * Molla et al. (2017) Molla A. A., Chakrabarti S. K., Debnath D., Mondal S., 2017, ApJ, 834, 88
  * Mondal & Chakrabarti (2013) Mondal S., Chakrabarti S. K., 2013, MNRAS, 431, 2716
  * Mondal & Stalin (2021) Mondal S., Stalin C. S., 2021, Galaxies, 9, 21
  * Mondal et al. (2014a) Mondal S., Chakrabarti S. K., Debnath D., 2014a, Ap&SS, 353, 223
  * Mondal et al. (2014b) Mondal S., Debnath D., Chakrabarti S. K., 2014b, ApJ, 786, 4
  * Mondal et al. (2016) Mondal S., Chakrabarti S. K., Debnath D., 2016, Ap&SS, 361, 309
  * Mostafa et al. (2013) Mostafa R., Mendez M., Hiemstra B., Soleri P., Belloni T., Ibrahim A. I., Yasein M. N., 2013, MNRAS, 431, 2341
  * Nandi et al. (2019) Nandi P., Chakrabarti S. K., Mondal S., 2019, ApJ, 877, 65
  * Nayakshin & Kallman (2001) Nayakshin S., Kallman T. R., 2001, ApJ, 546, 406
  * Nayakshin et al. (2000) Nayakshin S., Kazanas D., Kallman T. R., 2000, ApJ, 537, 833
  * Plant et al. (2014) Plant D. S., Fender R. P., Ponti G., Muñoz-Darias T., Coriat M., 2014, MNRAS, 442, 1767
  * Reynolds & Nowak (2003) Reynolds C. S., Nowak M. A., 2003, Phys. Rep., 377, 389
  * Ross (1979) Ross R. R., 1979, ApJ, 233, 334
  * Ross & Fabian (1993) Ross R. R., Fabian A. C., 1993, MNRAS, 261, 74
  * Ross et al. (1978) Ross R. R., Weaver R., McCray R., 1978, ApJ, 219, 292
  * Roy & Chakrabarti (2017) Roy A., Chakrabarti S. K., 2017, MNRAS, 472, 4689
  * Rozanska & Czerny (1996) Rozanska A., Czerny B., 1996, Acta Astron., 46, 233
  * Różańska & Madej (2008) Różańska A., Madej J., 2008, MNRAS, 386, 1872
  * Różańska et al. (2002) Różańska A., Dumont A. M., Czerny B., Collin S., 2002, MNRAS, 332, 799
  * Różańska et al. (2006) Różańska A., Goosmann R., Dumont A. M., Czerny B., 2006, A&A, 452, 1
  * Różańska et al. (2011) Różańska A., Madej J., Konorski P., Sadowski A., 2011, A&A, 527, A47
  * Shakura & Sunyaev (1973) Shakura N. I., Sunyaev R. A., 1973, A&A, 500, 33
  * Shaposhnikov & Titarchuk (2009) Shaposhnikov N., Titarchuk L., 2009, ApJ, 699, 453
  * Singh & Chakrabarti (2011) Singh C. B., Chakrabarti S. K., 2011, MNRAS, 410, 2414
  * Steiner et al. (2011) Steiner J. F., et al., 2011, MNRAS, 416, 941
  * Stern et al. (2014) Stern J., Laor A., Baskin A., 2014, MNRAS, 438, 901
  * Sunyaev & Titarchuk (1980) Sunyaev R. A., Titarchuk L. G., 1980, A&A, 500, 167
  * Titarchuk & Seifina (2009) Titarchuk L., Seifina E., 2009, ApJ, 706, 1463
  * Titarchuk & Zannias (1998) Titarchuk L., Zannias T., 1998, ApJ, 493, 863
  * Tomsick et al. (2009) Tomsick J. A., Yamaoka K., Corbel S., Kaaret P., Kalemci E., Migliari S., 2009, ApJ, 707, L87
  * Tomsick et al. (2014) Tomsick J. A., et al., 2014, ApJ, 780, 78
  * Xu et al. (2020) Xu Y., Harrison F. A., Tomsick J. A., Walton D. J., Barret D., García J. A., Hare J., Parker M. L., 2020, ApJ, 893, 30
  * Yuan & Narayan (2014) Yuan F., Narayan R., 2014, ARA&A, 52, 529
  * Zdziarski et al. (2003) Zdziarski A. A., Lubiński P., Gilfanov M., Revnivtsev M., 2003, MNRAS, 342, 355
  * Zycki & Czerny (1994) Zycki P. T., Czerny B., 1994, MNRAS, 266, 653
  * Zycki et al. (1994) Zycki P. T., Krolik J. H., Zdziarski A. A., Kallman T. R., 1994, ApJ, 437, 597