Vol.0 (200x) No.0, 000–000

11institutetext: Faculty of Science & Technology, The ICFAI University
Tripura, Agartala, 799210, India<EMAIL_ADDRESS>
22institutetext: Department of Physics and Astronomy, University of
Southampton, Highfield, Southampton SO17 1BJ

# Effect of magnetic flux advection on the dynamics of shock in accretion flow
around a rotating black hole

Biplob Sarkar 11 Anjali Rao 22

(Received 2016 month day; accepted 20016 month day)

###### Abstract

We investigate the dynamical behaviour of a magnetized, dissipative, accretion
flow around a rapidly rotating black hole. We solve the magnetohydrodynamic
equations and calculate the transonic accretion solutions which may contain
discontinuous shock transitions. We investigate the effect of $\zeta-$
parameter (parametrizing the radial variation of the toroidal magnetic flux
advection rate) on the dynamical behaviour of shocks. For a rapidly rotating
black hole and for fixed injection parameters at the outer edge, we show that
stationary shocks are sustained in the global magnetized accretion solutions
for a wide range of $\zeta$ and accretion rate ($\dot{m}$). To investigate the
observational implications, we consider dissipative shocks and estimate the
maximum accessible energy from the post-shock corona (PSC) for nine stellar
mass black hole candidates. We compare this with the observed radio jet
kinetic power reported in the literature, whenever available. We find close
agreement between the estimated values from our model with those reported in
the literature.

###### keywords:

accretion, accretion discs – magnetohydrodynamics – stars: black holes – shock
waves – ISM: jets and outflows

##  1 Introduction

Magnetic fields have a significant effect in a variety of astrophysical
processes. Specifically in an accretion disc, there must be transport of
angular momentum outwards so that matter can fall in at the central object. In
this process, the utility of magnetic fields in the accretion disc was first
discussed by Shakura & Sunyaev (1973). By considering the magnetic field to be
turbulent, the authors integrated its role to the viscosity in the accretion
disc which is well-known as the $\alpha-$parameter these days. Moreover,
magnetic fields play a significant role in the origin of bi-directional jet
from the inner region of the accretion disc around a black hole (BH).
Blandford, & Znajek (1977) proposed that relativistic jet may be powered by
the rotational energy of the BH via torque exerted by magnetic field lines
threading the BH horizon. In an alternative mechanism, Blandford, & Payne
(1982) suggested that a spinning BH actually causes the ordered magnetic field
in the accretion disc to corotate with the accreting matter and this drives
quasi-relativistic jets via the release of gravitational potential energy.
Again, it was shown by Balbus, & Hawley (1991) that the Magneto-Rotational
Instability (MRI) may give rise to the disc viscosity, where even a weak
magnetic field can make the accretion flow go turbulent leading to the outward
transport of angular momentum. Recent 3D magnetohydrodynamic (MHD) simulations
assuming azimuthal magnetic fields have further demonstrated the formation of
a magnetically supported accretion disc around a BH (Machida et al., 2006;
Johansen & Levin, 2008). Optically thin accretion disc supported by magnetic
pressure can account for the “Bright/Hard” state in BH candidates which is
observed when the accretion disc undergoes transition from a Low/Hard state to
the High/Soft state (Oda et al., 2010, 2012).

In the model of magnetized accretion flow around BHs considered by Oda et al.
(2007, 2012); Sarkar & Das (2016); Das & Sarkar (2018), the parameter $\zeta$
parametrizes the radial variation of the toroidal magnetic flux advection rate
($\dot{\Phi}$). Oda et al. (2012) showed that the change in $\zeta$ results in
modification of $\dot{\Phi}$ which is responsible for the state transitions in
BH candidates (BHCs). When $\dot{\Phi}$ is relatively high and $\dot{M}$
surpasses a critical limit for the trigger of cooling instability, the
advection dominated accretion flow (ADAF) (Ichimaru, 1977; Narayan & Yi, 1994,
1995; Abramowicz et al., 1995) develops towards a magnetic pressure dominated
disc. In this situation, Oda et al. (2012) proposed that the transition of the
accretion disc would be bright hard-to-soft. While for a low $\dot{\Phi}$,
when $\dot{M}$ surpasses the limit, the ADAF advances towards an optically
thick disc. In this situation, Oda et al. (2012) suggested that the transition
would be dark hard-to-soft, which happens at under 0.1 $L_{\rm Edd}$ (Oda et
al., 2010).

Another significant aspect in the study of accretion flow around a BH is the
existence of global accretion solutions containing a shock, which have a
special importance. Such shocked accretion solutions occur when the supersonic
accretion flow approaching towards a BH experiences the centrifugal barrier in
the vicinity of the BH. Due to the centrifugal repulsion, the supersonic flow
may undergo a shock transition to become subsonic and the post-shock corona
(hereafter PSC) (Aktar et al., 2015; Sarkar & Das, 2016; Dihingia et al.,
2019) becomes hot, dense and puffed up (Fukue, 1987; Lu & Yuan, 1998; Das et
al., 2001; Gu & Lu, 2004; Das & Choi, 2009; Chattopadhyay & Chakrabarti, 2011;
Sarkar & Das, 2018; Dihingia et al., 2018a). Shocked accretion solutions
around BHs have the potential to explain many intriguing features of BH
candidates like the origin of outflows, jets (Das et al., 2014a, b, and
references therein) and the quasi-periodic oscillations (QPO’s) in the hard
X-ray spectrum of BHCs (Molteni et al., 1996; Chakrabarti et al., 2004).

In recent years, Sarkar & Das (2015, 2016); Sarkar et al. (2018); Sarkar & Das
(2018) have extensively investigated the implication of shocks in accretion
flow around non-rotating BHs, where the disc is threaded by toroidal magnetic
fields. Das & Sarkar (2018) extended the study to investigate the combined
effect of viscosity and toroidal magnetic fields on the dynamical structure of
the global accretion flow around rotating BHs. In these works, the authors
assumed $\zeta$ = 1 throughout, as a representative case. However, in general
it is expected that the variation of $\zeta$ will have significant effect on
the radial structure of the disc. In particular, Oda et al. (2007) showed that
the variation of $\zeta$ affects the temperature, plasma$-\beta$, accretion
rate and optical depth in the disc. Since $\zeta$ will affect the angular
momentum transport in the disc as well, the dynamics of shock is also likely
to be governed by the value of $\zeta$. In this paper, we aim to investigate
the effect of $\zeta$ on the dynamical behaviour of shocks in a viscous
axisymmetric magnetized accretion flow around a rapidly spinning BH. To mimic
the space-time geometry around a spinning BH, we use the pseudo Kerr potential
intoduced by Chakrabarti & Mondal (2006). The adopted pseudo potential in this
work represents a rotating black hole with spin parameter $a_{s}\lesssim 0.8$.
The relevant works related to other form of pseudo potentials commonly used by
the researchers are Chakrabarti, & Khanna (1992); Løvås (1998); Artemova et
al. (1996); Semerák, & Karas (1999); Mukhopadhyay (2002); Ivanov, & Prodanov
(2005); Ghosh, & Mukhopadhyay (2007); Ghosh et al. (2014); Karas, & Abramowicz
(2014). Very recently Dihingia et al. (2018b), suggested another effective
pseudo-potential which exactly describes the space-time geometry around a Kerr
black hole for the range of spin $0\leq a_{s}<1$. The focus of the present
work is to determine the critical value of $\zeta$ which would sustain
standing shocks in the magnetized accretion solutions around a rapidly
rotating BH. Also, we show that a wide range of $\zeta$ sustains standing
shocks in the accretion flow. Further for fixed flow parameters at the outer
edge, we show that for a flow with higher $\zeta$, the critical accretion rate
($\dot{m}^{\rm cri}$) that sustains shock is lower. Finally, we make use of
the present formalism to calculate the maximum shock luminosity
($\mathcal{L}_{\rm shock}^{\rm max}$) that corresponds to the maximum energy
dissipated across the shock. This is done for nine stellar mass BH sources,
where the mass, accretion rate and spin of the individual sources are taken
from the literature. We compare $\mathcal{L}_{\rm shock}^{\rm max}$ with the
reported values of jet kinetic power of the sources in the literature,
whenever available, and we find close agreement between the two.

In the following, we first present the model assumptions and governing
equations and the methodology to solve them (Section 2). In Section 3, we
present the shocked global accretion solutions, the properties of shock and
the critical values of $\zeta^{\rm cri}$ and $\dot{m}^{\rm cri}$ for shock.
Next, in Section 4, we compute the maximum shock luminosity from our model and
compare it for several stellar mass BH sources. Finally, we present the
summary of the work in Section 5.

##  2 Model for the accretion flow

The structure of magnetic fields in the accretion disc is considered to be the
same as described in Oda et al. (2012). On the basis of the results obtained
from 3D global as well as local MHD simulations of accretion flow around BHs,
we consider the magnetic fields in the accretion disc to be turbulent and
dominated by the toroidal component (Hirose et al., 2006; Machida et al.,
2006; Johansen & Levin, 2008). According to the results of these simulations,
the magnetic fields in the disc are decomposed into the mean fields, denoted
by ${\bf{B}}=(0,<B_{\phi}>,0)$, and the fluctuating fields,
$\delta{\bf{B}}=(\delta{B}_{r},\delta{B}_{\phi},\delta{B}_{z})$. Here, ‘$<>$’
expresses the azimuthal average of any quantity. The fluctuating components of
magnetic fields disappear on azimuthally averaging ($<\delta{\bf{B}}>=0$).
Moreover, the azimuthal component of the magnetic field is much more
significant as compared to the radial and vertical components,
$\mid<B_{\phi}>+\delta B_{\phi}\mid\gg\mid\delta B_{r}\mid~{}{\rm
and}\mid\delta B_{z}\mid$ (see Fig. 2 of Oda et al. (2012)). Eventually, the
azimuthally averaged form of the magnetic field is obtained as
$<{\bf{B}}>=<B_{\phi}>\hat{\phi}$ (Oda et al., 2007).

###  2.1 Governing Equations

In this work, we consider a steady, thin and axisymmetric magnetized accretion
flow around a rotating BH. Throughout the paper, we express the radial
coordinate ($x$) in units of $r_{g}=GM_{\rm BH}/c^{2}$, where $G$ is the
universal gravitational constant, $M_{\rm BH}$ is the mass of the BH and $c$
is the speed of light. Also, flow velocity is measured in units of $c$ and
time is measured in units of $GM_{\rm BH}/c^{3}$. We assume the ($x$, $\phi$,
$z$) coordinate system where the accretion flow is confined in the $x-\phi$
plane and the BH is located at the center of the coordinate system.

We denote the dynamical flow variables, namely, radial velocity, sound speed,
mass density, specific angular momentum, temperature, adiabatic index,
specific entropy, gas pressure and magnetic pressure of the flow by the
symbols $u$, $c_{s}$, $\rho$, $\lambda$, $T$, $\gamma$, $s$, $p_{\rm gas}$ and
$p_{\rm mag}$ respectively. Hence the mass flux conservation equation, radial
momentum equation, azimuthal momentum equation and the entropy generation
equation are, respectively,

$\dot{M}=2\pi u\Sigma x,$ $None$ ${u\frac{du}{dx}+\frac{1}{\rho}\frac{dp_{\rm
tot}}{dx}+\frac{d\Psi_{\rm eff}}{dx}+\frac{\left<B_{\phi}^{2}\right>}{4\pi
x\rho}=0},$ $None$ ${u\Sigma
x}\frac{d\lambda}{dx}+\frac{d}{dx}(x^{2}\mathbb{T}_{x\phi})=0,$ $None$
$uT\Sigma\frac{ds}{dx}=\frac{Hu}{1-\gamma}\left(\frac{\gamma p_{\rm
gas}}{\rho}\frac{d\rho}{dx}-\frac{dp_{\rm
gas}}{dx}\right)=\mathcal{Q}^{-}-\mathcal{Q}^{+}.$ $None$

Here, $\dot{M}$ represents the mass accretion rate and $\Sigma$ specifies the
mass density of the flow averaged in the vertical direction (Matsumoto et al.,
1984). It might be noticed that in the present work, we consistently regard
the direction of inward radial velocity as positive. Also, $p_{\rm tot}$ is
the total pressure of the flow which is considered to be $p_{\rm tot}=p_{\rm
gas}+p_{\rm mag}$. The gas pressure inside the disc is obtained as, $p_{\rm
gas}=R\rho T/\mu$ where $R$ is the gas constant and $\mu$ is the mean
molecular weight. We consider $\mu=0.5$ for a fully ionized hydrogen gas.
Also, the magnetic pressure of the flow is obtained as, $p_{\rm
mag}=<B_{\phi}^{2}>/8\pi$. We define the magnetic parameter of the flow as
$\beta=p_{\rm gas}/p_{\rm mag}$. Using the definition of magnetic parameter,
we obtain the total pressure of the flow as $p_{\rm tot}=p_{\rm
gas}(1+1/\beta)$. Moreover, $\mathbb{T}_{x\phi}$ denotes the $x\phi$ component
of the Maxwell stress and this component dominates the vertically integrated
total stress in the flow. When velocity in the radial direction is significant
in the accretion flow, $\mathbb{T}_{x\phi}$ is obtained as,

$\mathbb{T}_{x\phi}=\frac{<B_{x}B_{\phi}>}{4\pi}H=-\alpha_{T}(W+\Sigma
u^{2}),$ $None$

where $H$, $\alpha_{T}$ and $W$, respectively, denote the disc half-thickness,
constant of proportionality and the vertically integrated pressure of the flow
(Matsumoto et al., 1984). Along the same line as the seminal work of Shakura &
Sunyaev (1973), we consider $\alpha_{T}$ to remain constant everywhere along
the flow. In the event that $u$ is insignificant, such as for a Keplerian
flow, equation (5) restores to the ‘$\alpha$-model’ (Shakura & Sunyaev, 1973).

In left-hand side of equation (2), $\Psi_{\rm eff}$ is the effective potential
around a spinning BH suggested by Chakrabarti & Mondal (2006). The form of
$\Psi_{\rm eff}$ is given as,

$\Psi_{\rm
eff}=-\frac{{\mathcal{J}}+\sqrt{{\mathcal{J}^{2}}-4{\mathcal{I}}{\mathcal{K}}}}{2{\mathcal{I}}},$

where

${\mathcal{I}}=\frac{\varepsilon^{2}\lambda^{2}}{2x^{2}},$
${\mathcal{J}}=-1+\frac{\varepsilon^{2}\varphi\lambda
r^{2}}{x^{2}}+\frac{2\lambda a_{s}}{xr^{2}},$
${\mathcal{K}}=1-\frac{1}{r-r_{0}}+\frac{2a_{s}\varphi}{x}+\frac{\varepsilon^{2}\varphi^{2}r^{4}}{2x^{2}}.$

Here, radial distance in the cylindrical coordinates is represented by $x$ and
in spherical coordinates it is denoted by $r$. In addition, $\lambda$ is the
specific angular momentum of the flow. Further,
$r_{0}=0.085a_{s}^{2}+0.97a_{s}+0.04$,
$\varphi=2a_{s}/(2a_{s}^{2}+a_{s}^{2}x+x^{3})$ and
$\varepsilon^{2}=(a_{s}^{2}-2x+x^{2})/(2a_{s}^{2}/x+a_{s}^{2}+x^{2})$, where
$\varepsilon$ is the redshift factor and $a_{s}$ represents the spin of the
BH. The pseudo potential used in this work adequately describes the space time
geometry around a spinning BH for $a_{s}\lesssim 0.8$ (Chakrabarti & Mondal,
2006).

Assuming vertical hydrostatic equilibrium, we calculate the half-thickness of
the disc ($H$) as, $H(x)$ =
$c_{s}\sqrt{{x}/({\gamma{{\Psi_{r}^{{}^{\prime}}}}})}$ where
$\Psi_{r}^{{}^{\prime}}$ = $\left(\frac{\partial\Psi_{\rm eff}}{\partial
r}\right)$ z<<x, $z$ indicates vertical scale height in the cylindrical
coordinate system and $r=\sqrt{z^{2}+x^{2}}$ (Das et al., 2010; Aktar et al.,
2015). Here, the adiabatic sound speed is determined as $c_{s}=\sqrt{\gamma
p_{\rm tot}/\rho}$. In this work, we accept $\gamma$ to remain constant
throughout the flow and fix $\gamma=4/3$ (ultra-relativistic flow) for the
analysis that follows.

Equation (4) is the entropy generation equation, where the cooling rate and
heating rate of the flow are represented by $\mathcal{Q}^{-}$ and
$\mathcal{Q}^{+}$, respectively. 3D MHD simulations have revealed that the
dominant process contributing to the heating of the disc is the dissipation of
magnetic energy via the magnetic reconnection mechanism (Machida et al., 2006;
Hirose et al., 2006; Krolik et al., 2007). In view of this, the heating rate
is given by,

$\mathcal{Q}^{+}=\frac{<B_{x}B_{\phi}>}{4\pi}xH\frac{d\Omega}{dx}=-\alpha_{T}(W+\Sigma
u^{2})x\frac{d\Omega}{dx},$ $None$

where $\Omega$ represents the flow’s angular velocity.

Due to high temperature at the inner region of the accretion disc, it is
imperative that the plasma in this region be treated as two-temperature. In
such a case, the electrons in the flow can lose energy via synchrotron and
bremsstrahlung emission as well as through the inverse Comptonization effects.
However, in the present work, we have approximated the plasma to be single
temperature. Thus, the inverse Comptonization of the electrons has been
disregarded since inclusion of this process requires a two-temperature
analysis. Also, due to smaller size of the accretion disc in case of stellar
mass BH candidates, the accretion disc is strongly threaded by magnetic
fields. Hence, synchrotron emission by the electrons is expected to be
dominant over the bremsstrahlung process (Chattopadhyay & Chakrabarti, 2002;
Rajesh & Mukhopadhyay, 2010). Due to the above considerations, we consider
only synchrotron cooling to be active in the flow. Accordingly, the
synchrotron cooling rate is given as (Shapiro & Teukolsky, 1983),

$\mathcal{Q}^{-}=\frac{\widetilde{S}{c_{s}}^{5}\rho
H}{u}\sqrt{\frac{\Psi_{r}^{{}^{\prime}}}{x^{3}}}{\beta^{2}}{(\beta+1)^{-3}},$
$None$

with,

$\widetilde{S}=1.4827\times
10^{18}\frac{{\dot{m}}\mu^{2}e^{4}}{I_{n}m_{e}^{3}\gamma^{5/2}}G^{-1}M_{\odot}^{-1}c^{-3},$

where $e$ denotes the charge of the electron and $m_{e}$ represents the mass
of the electron. Also, $\dot{m}$ signifies the accretion rate in Eddington
units (${\dot{M}}_{\rm Edd}=1.39\times 10^{17}\times M_{\rm
BH}/M_{\odot}~{}{\rm g~{}s}^{-1}$). Here, we follow Rezzolla & Zanotti (2013);
Straub et al. (2014); Yuan & Narayan (2014) and use the efficiency factor of 1
to calculate the Eddington accretion rate. Moreover,
$I_{n}=(2^{n}n!)^{2}/(2n+1)!$ and $n(=1/(\gamma-1))$ indicates the polytropic
index of the flow. We ignore any coupling between electron and ion and
evaluate the temperature of electron in the flow using the relation
$T_{e}=(\sqrt{m_{e}/m_{i}})T_{i}$ (Chattopadhyay & Chakrabarti, 2002). Here,
$m_{i}$ indicates the mass of ion and $T_{i}$ refers to the ion temperature.
It is to be noted that although we assume a single temperature flow, the
temperature profile of ions and electrons is expected to become different at
least at the inner disc region. It was indicated by Mahadevan, & Quataert
(1997) that the exchange of energy between ions and electrons via Coulomb
collision is extremely ineffective in the accretion flow. Thus, the ions
cannot attain thermal equilibrium with the electrons. The ions in principle
retain the heat acquired via viscous dissipation in the disc. Again, since the
relativistic electrons are much lighter than the non-relativistic ions, the
electrons are able to radiate more effectively than the ions. Thus the
accretion flow develops a two-temperature structure which is mostly prominent
at the inner region of the disc.

Since we are dealing with hydromagnetic flows, as a consequence of induction
equation in steady-state (for the current analysis), we obtain,

${\bf\nabla}\times\left({\vec{u}}\times<B_{\phi}>\hat{\phi}-{\frac{4\pi}{c}}\eta{\vec{j}}\right)=0,$
$None$

where, $\vec{u}$ is the velocity vector, $\vec{j}$ is the current density and
$\eta$ is the resistivity of the flow. Equation (8) is now azimuthally
averaged. On account of significantly large length scales in accretion disc,
the Reynolds number achieves a very high value. As a result, we ignore the
magnetic diffusivity term. Further, we also disregard the dynamo term for the
present purpose. The resulting equation is then vertically integrated
considering the vanishing of the azimuthally averaged toroidal magnetic fields
at the surface of the disc. In view of these assumptions, the advection rate
of toroidal magnetic flux is determined as,

$\dot{\Phi}=-\sqrt{4\pi}u{B}_{0}(x)H,$ $None$

where,

$\displaystyle{B}_{0}(x)$
$\displaystyle=\langle{B}_{\phi}\rangle\left(x;z=0\right)$
$\displaystyle=\frac{2^{5/4}{\pi}^{1/4}(RT)^{1/2}{\Sigma}^{1/2}}{\mu^{1/2}H^{1/2}{\beta}^{1/2}}$

represents the mean azimuthal toroidal magnetic field in the disc equatorial
plane following Oda et al. (2007). In the accretion disc, it is generally
expected that $\dot{\Phi}$ will vary in the radial direction due to the
diffusion of magnetic fields and the dynamo action. However, to avoid the
complexity of determining the magnetic diffusivity term and dynamo term self
consistently from the local variables, we follow the prescription of Oda et
al. (2007) and adopt the parametric relation given by

$\dot{\Phi}\left(x;\zeta,\dot{M}\right)\equiv\dot{\Phi}_{\rm
inj}\left(\frac{x}{x_{\rm inj}}\right)^{-\zeta},$ $None$

where $\zeta$ represents a parameter describing the advection rate of magnetic
flux. Further, $\dot{\Phi}_{\rm inj}$ denotes the advection rate of the
toroidal magnetic field at the injection radius ($x_{\rm inj}$), equivalently
the outer edge of the accretion disc. When $\zeta$$=0$, magnetic flux remains
constant radially while, for $\zeta$$>0$, the magnetic flux increases with the
decrease of radius.

###  2.2 Transonic conditions and solution procedure

Due to the essential transonic nature of BH accretion flows, the infalling
matter is subsonic at the outer edge of the disc ($x_{\rm inj}$) and enters
the BH supersonically (Chakrabarti, 1990). The location in the radial
direction, where the flow behaviour changes smoothly from subsonic state to
supersonic state is usually termed as a critical point. In the following, we
solve the equations (1), (2), (3), (4), (9) and (10) simultaneously in order
to perform the critical point analysis (Sarkar & Das, 2016, and references
therein) and is given by,

$\frac{du}{dx}=\frac{\mathcal{N}}{\mathcal{D}},$ $None$

where the expressions for the numerator ($\mathcal{N}$) and denominator
($\mathcal{D}$) are provided in Das & Sarkar (2018). The expressions for the
radial gradients of $c_{s}$, $\lambda$ and $\beta$ can also be found in Das &
Sarkar (2018).

Now, the infalling matter must smoothly accrete onto the BH. This demands that
the flow variables must have finite values everywhere along the flow. Thus, if
the denominator $\mathcal{D}$ in equation (11) vanishes at any radial
location, the numerator $\mathcal{N}$ must also simultaneously vanish there
(${du}/{dx}={0}/{0}$), in order to maintain a smooth solution. Such a radial
location is known as a critical point ($x_{c}$). This gives us two critical
point conditions as $\mathcal{N}=0$ and $\mathcal{D}=0$. The expression of
Mach number ($M=u/c_{s}$) at $x_{c}$ is obtained by using the condition
$\mathcal{D}=0$, which is given as,

$M_{c}=\sqrt{\frac{-\varkappa_{2}-\sqrt{\varkappa^{2}_{2}-4\varkappa_{1}\varkappa_{3}}}{2\varkappa_{1}}},$
$None$

where

$\varkappa_{1}=2\alpha^{2}_{T}\gamma^{2}I_{n}(1-\gamma)(1-2g)(\beta_{c}+1)-\gamma^{2}[(1+\gamma)\beta_{c}+3],$
$\varkappa_{2}=2\gamma(\gamma\beta_{c}+2)+4\alpha^{2}_{T}\gamma
I_{n}g(1-g)(1-\gamma)(\beta_{c}+1),$
$\varkappa_{3}=2\alpha^{2}_{T}g^{2}I_{n}(1-\gamma)(\beta_{c}+1),$

where, the parameter $g$ is defined as $g=I_{n+1}/I_{n}$. To obtain the sound
speed ($c_{\rm sc}$) at $x_{c}$, we use the condition $\mathcal{N}=0$ to get a
cubic equation of the form,

${\mathcal{A}_{1}}c_{\rm sc}^{3}+{\mathcal{A}_{2}}c_{\rm
sc}^{2}+{\mathcal{A}_{3}}c_{\rm sc}+{\mathcal{A}_{4}}=0,$ $None$

where

${\mathcal{A}_{1}}=\widetilde{S}\sqrt{\frac{\Psi_{r}^{{}^{\prime}}}{x_{c}^{3}}}\frac{\beta_{c}^{2}}{(\beta_{c}+1)^{3}},$

${\mathcal{A}_{2}}=\frac{2\alpha^{2}_{T}I_{n}(\gamma
M_{c}^{2}+g)^{2}}{x_{c}\gamma^{2}}-\frac{M_{c}^{2}(4+2\gamma\beta_{c})}{2\gamma(1-\gamma)(\beta_{c}+1)}\left(\frac{d{\rm
ln}\Psi_{r}^{{}^{\prime}}}{dx}\right)$ $-\frac{2\alpha^{2}_{T}gI_{n}(\gamma
M_{c}^{2}+g)}{\gamma^{2}}\left(\frac{d{\rm
ln}\Psi_{r}^{{}^{\prime}}}{dx}\right)$
$-\frac{2\\{\beta_{c}(1+\gamma)+3\\}M_{c}^{2}}{\gamma(1-\gamma)(\beta_{c}+1)^{2}x_{c}}$
$+\frac{3(3+2\gamma\beta_{c})M_{c}^{2}}{2\gamma(1-\gamma)(\beta_{c}+1)x_{c}}+\frac{6\alpha^{2}_{T}gI_{n}(\gamma
M_{c}^{2}+g)}{x_{c}\gamma^{2}}$ $-\frac{8\alpha^{2}_{T}gI_{n}(\gamma
M_{c}^{2}+g)}{\gamma^{2}(\beta_{c}+1)x_{c}}-\frac{(1-4\zeta)M_{c}^{2}}{2\gamma(\beta_{c}+1)(1-\gamma)x_{c}},$

${\mathcal{A}_{3}}=-\frac{4{\lambda_{c}}\alpha_{T}I_{n}M_{c}(\gamma
M_{c}^{2}+g)}{x_{c}^{2}\gamma},$

${\mathcal{A}_{4}}=\left[\frac{\\{\beta_{c}(1+\gamma)+3\\}M_{c}^{2}}{(\gamma-1)(\beta_{c}+1)}-\frac{4\alpha^{2}_{T}I_{n}g(\gamma
M_{c}^{2}+g)}{\gamma}\right]\times\left(\frac{d\Psi_{r}^{{}^{\prime}}}{dx}\right).$

Here, we represent the values of the flow variables estimated at $x_{c}$ with
the subscript ‘c’.

Presently, we obtain the value of sound speed ($c_{\rm sc}$) at $x_{c}$ by
supplying the accretion flow parameters in equation (13). Then the value of
$u_{c}$ can be found using equation (12). The obtained values of $u_{c}$ and
$c_{\rm sc}$ are supplied in Eq. (11) to analyze the characteristics of the
critical points. At $x_{c}$, $du/dx$ possesses two particular values; one of
which holds for the accretion solution and the other is for wind. If both the
values of the velocity gradient at $x_{c}$ are real and of reverse sign,
$x_{c}$ is called a ‘saddle’ type critical point (Chakrabarti, 1990;
Chakrabarti & Das, 2004). Such a critical point is of unique significance
since a global accretion solution goes through it only. In this work, since
our objective is to explore the magnetized accretion flow around a BH, we
consider solely the accretion solutions in the subsequent investigation.

##  3 Results and discussion

In order to acquire a global accretion solution, we need to perform the
simultaneous solution of the coupled differential equations corresponding to
the radial gradient of flow velocity ($u$), sound speed ($c_{s}$), angular
momentum ($\lambda$) and plasma $\beta$. Additionally, the boundary values of
$u$, $c_{s}$, $\lambda$, $\beta$ and $\dot{m}$ must be known at a given radial
position ($x$). The Kerr parameter ($a_{s}$) and viscosity parameter
($\alpha_{T}$) also need to be specified. It is to be noted that, throughout
the paper we express angular momentum ($\lambda$) in units of the Keplerian
angular momentum $\lambda_{K}$(= $\sqrt{x^{3}/(x-2)^{2}}$). Owing to the
essential transonic nature of BH accretion solutions, flow must inevitably go
through a critical point. Thus, it is suitable to supply the boundary flow
variables at the critical point. With this, we carry out integration of the
equations corresponding to the radial gradient of $u$, $c_{s}$, $\lambda$ and
$\beta$ from the critical point once towards the inside up to exactly outside
the BH horizon and then outward up to the outer boundary of the disc. These
two sections can be joined to yield a full global transonic accretion
solution. As determined by the boundary parameters, the accretion flow has to
cross one critical point at the minimum and feasibly more (Sarkar & Das, 2013,
and references therein). A critical point which is located near the BH horizon
is called as inner critical point ($x_{\rm in}$) and that which is located far
from the BH horizon is known as outer critical point ($x_{\rm out}$).

Figure 1: Variation of Mach number as a function of logarithmic radial
distance. The flows are injected from the outer edge $x_{\rm inj}=1000$ with
specific energy ${{E}}_{\rm inj}$$=8.9\times 10^{-4}$, angular momentum
$\lambda_{\rm inj}$ = 0.11014$\lambda_{K}$, $\beta_{\rm inj}=10^{5}$, and
viscosity $\alpha_{T}=0.02$. The Kerr parameter is chosen as $a_{s}=0.8$.
Solid, dotted and dashed curves depict the results obtained for ($\zeta$,
$\dot{m}$) = (0.6, 0.05), (1.0, 0.05) and (1.0, 0.45), respectively. The
vertical arrows indicate the corresponding shock transitions positioned at
$x_{s}$ = 40.72 (solid), $x_{s}$ = 27.12 (dotted) and $x_{s}$ = 11.68
(dashed). See text for details.

To begin with, we explore the effect of the $\zeta$$-$parameter prescribing
the magnetic flux advection rate on the dynamical structure of the accretion
flow that contains shock wave. For this purpose, we fix the global parameters
as $\alpha_{T}=0.02$, $a_{s}=0.8$ and inject matter subsonically from the
outer edge of the disc at $x_{\rm inj}=1000$ with local flow variables as
$\lambda_{\rm inj}$ = 0.11014$\lambda_{K}$, $\beta_{\rm inj}=10^{5}$ and
${{E}}_{\rm inj}$$=8.9\times 10^{-4}$, respectively. The obtained results are
depicted in Fig. 1, where the variation of Mach number ($M$) is plotted
against the logarithmic radial distance. First we inject matter with accretion
rate $\dot{m}=0.05$ and $\zeta$$=0.6$. For these injection parameters,
subsonic flow at the outer edge approaches towards the BH and becomes
supersonic after crossing the outer critical point located at $x_{\rm
out}=654.52$. The supersonic flow proceeds towards the BH and sustains a
virtual centrifugally repulsive barrier in the neighbourhood of the BH. Due to
this obstruction, the accreting matter begins to pile up in the vicinity of
the BH leading to the increase of flow density profile. However, this
accumulation of matter cannot persist indefinitely and when the density scale
length achieves a critical limit, the flow may suffer a discontinuous shock
transition (Frank et al., 1992). As a result of this, the ordered kinetic
energy of the flow gets converted to disordered thermal energy. Whenever
dynamically feasible, such a discontinuous shock jump is preferred by the laws
of thermodynamics since the entropy of the flow increases after the shock
(Fukue, 1987; Becker & Kazanas, 2001; Fukumura & Tsuruta, 2004). The shock
transition in a magnetized accretion flow is governed by the conservation of:
(i) energy flux: ${{E}}_{+}$ = ${{E}}_{-}$
(ii) mass flux: $\dot{M}_{+}=\dot{M}_{-}$
(iii) pressure balance: $W_{+}+\Sigma_{+}u_{+}^{2}=W_{-}+\Sigma_{-}u_{-}^{2}$
(iv) magnetic flux advection rate: $\dot{\Phi}_{+}$ = $\dot{\Phi}_{-}$
where ‘$+$’ and ‘$-$’ represent the quantities evaluated immediately after and
before the shock transition ($x_{s}$) respectively (Landau & Lifshitz, 1959;
Sarkar & Das, 2016). In the psuedo-Kerr geometry, we follow Das & Sarkar
(2018) to calculate the local specific energy of the flow as ${{E}}(x)$
$=u^{2}/2+c_{s}^{2}/(\gamma-1)+$ $\Psi_{\rm eff}$
$+<B_{\phi}^{2}>/(4\pi\rho)$. For the present case, the shock is positioned at
$x_{s}=40.72$ as indicated with the solid vertical arrow. Soon after the shock
transition, flow gradually gains its radial velocity due to the gravitational
attraction and enters in to the BH supersonically after crossing the inner
critical point at $x_{\rm in}=2.52$. When the magnetic flux advection rate
parameter is increased further as $\zeta$$=1.0$ keeping all the remaining flow
parameters unaltered, we find that shock front moves inward and settles down
at $x_{s}=27.12$. In the figure, this is indicated using the dotted vertical
arrow. In reality, as $\zeta$ is increased, the magnetic fields carried with
the accreting matter are also enhanced that raise the synchrotron cooling
efficiency. Since the cooling is mostly effective at PSC, therefore post-shock
pressure reduces appreciably that eventually causes the shock front to move
further towards the BH in order to preserve the pressure balance across the
shock. Next, we choose $\zeta$$=1.0$ and $\dot{m}=0.45$ keeping rest of the
flow parameters same. Due to the increase of $\dot{m}$, the synchrotron
cooling efficiency in the flow is additionally augmented and the shock front
advances further towards the horizon. The shock now forms at $x_{s}=11.68$ as
shown with the dashed vertical arrow. In all the cases, arrows indicate the
overall direction of flow motion towards the BH. Based on the above analysis,
we consequently point out that the $\zeta$$-$parameter prescribing the
magnetic flux advection rate seems to play an important role in deciding the
accretion disc dynamics including shock waves, in addition to accretion rate
($\dot{m}$).

Figure 2: Variation of (a) radial velocity, (b) density in g/cm3, (c) the
toroidal component of magnetic field in Gauss, (d) the electron temperature in
Kelvin, (e) disc aspect ratio ($H/x$) and (f) effective optical depth as a
function of logarithmic radial coordinate. The solid, dotted and dashed curves
relate to the shocked accretion solutions portrayed in Fig. 1. The critical
points are indicated using filled circles where the closer one is the inner
critical point and the farthest represents the outer critical point. The
position of shock is displayed using vertical arrows. See text for details.

In Fig. 2, we show the radial disc structure corresponding to shock solutions
considered in Fig. 1. For this, we consider $M_{\rm BH}$ = 10 $M_{\odot}$ as a
standard reference value. Here in each panel we show the radial variation of a
flow variable where solid, dotted and dashed curves depict results
corresponding to ($\zeta$$=0.6$, $\dot{m}=0.05$), ($\zeta$$=1.0$,
$\dot{m}=0.05$) and ($\zeta$$=1.0$, $\dot{m}=0.45$), respectively. Moreover,
the filled dots represent the position of the critical points. In Fig. 2(a) we
show the variation of the radial velocity of the flow as a function of
distance from the BH. Since the accreting matter experiences greater
gravitational pull as it moves closer to the BH, the radial velocity profile
steadily increases towards the horizon. Thus, the subsonic flow starting from
the outer edge of the disc becomes supersonic as it crosses the outer critical
point. This supersonic flow experiences centrifugal barrier in the vicinity of
the BH and undergoes a shock jump to become subsonic as indicated by the
downward vertical arrow in each of the three cases. The subsonic flow again
becomes supersonic after passing through the inner critical point and enters
the BH horizon. In Fig. 2(b) we show the variation of density of the accretion
flow as a function of radial distance. The density of the flow rises as the
flow approaches the horizon and the density is boosted up across the shock
transition. The post-shock density rises due to the reduction of flow velocity
after the shock jump in order to conserve the mass flux ($\dot{M}$) across the
shock transition. This is evident from the sudden rise in density profile in
all the cases as indicated by the upward vertical arrow. Next, in Fig. 2(c) we
present the variation of the magnetic field strength $B_{\phi}$ as a function
of radial distance. For all the cases, we find that $B_{\phi}$ steadily
increases with the decrease of radial coordinate. This happens due to the
increase of magnetic flux advection rate ($\dot{\Phi}$) with deceasing radius.
Increase of $\zeta$ augments $\dot{\Phi}$ and leads to increase of $B_{\phi}$
profile as shown by the dotted curve. Also when $\dot{m}$ rises, there would
be an increase of magnetic field lines carried with the matter. So, $B_{\phi}$
profile would also increase as indicated using the dashed curve. Further,
$B_{\phi}$ also jumps across the shock transition due to the compression of
the flow at the shock location and the conservation of $\dot{\Phi}$ across the
shock. In Fig. 2(d), we show the variation of electron temperature ($T_{e}$)
in the flow as a function of radial distance. This is essential since $T_{e}$
is related to the nature of X-ray spectrum emitted from the disc. We observe a
rise in $T_{e}$ in the flow with decreasing radius for all the cases. $T_{e}$
is catastrophically augmented across the shock jump showing thereby that the
PSC is significantly hotter than the pre-shock region. When $\zeta$ value is
increased from $0.6$ to $1.0$ (keeping $\dot{m}$ fixed), there is increase of
magnetic field strength that increases the cooling of the electrons due to
synchrotron emission. Thus we observe a drop in $T_{e}$ profile in the inner
region (dotted curve). A further drop in the $T_{e}$ profile is also observed
when $\dot{m}$ is increased to $0.45$ retaining $\zeta$ fixed (dashed curve).
Augmented accretion rate additionally enhances the efficiency of synchrotron
emission by the electrons and also causes the drop in $T_{e}$ profile. In Fig.
2(e), we show the radial variation of the aspect ratio $(H/x)$ of the
magnetized accretion disc. We find that the aspect ratio remains below unity
throughout the disc for all the cases. Further, due to the high temperature
and density in the PSC of the disc, the PSC becomes puffed up and a sudden
rise in the disc aspect ratio is observed after the shock jump in all cases.
Lastly, in Fig. 2(f), we show the radial distribution of effective optical
depth in the vertical direction ($\tau_{\rm eff}$). In the disc, we calculate
$\tau_{\rm eff}$ using the relation $\tau_{\rm eff}=\sqrt{\tau_{\rm
syn}\tau_{\rm es}}$ (Rajesh & Mukhopadhyay, 2010) where, $\tau_{\rm syn}$
denotes the absorption effect arising due to thermal processes and is given by
$\tau_{\rm syn}=\left(H{\bar{q}}_{\rm
syn}/4\bar{a}T_{e}^{4}\right)\left(GM_{\rm BH}/c^{2}\right)$ (Rajesh &
Mukhopadhyay, 2010) where, ${\bar{q}}_{\rm syn}$ is the synchrotron emissivity
(Shapiro & Teukolsky, 1983) and $\bar{a}=5.6705\times 10^{-5}~{}{\rm g}~{}{\rm
s}^{-3}~{}{\rm K}^{-4}$ is the Stefan-Boltzmann constant. Here, $\tau_{\rm
es}$ denotes the scattering optical depth estimated as $\tau_{\rm
es}=\kappa_{\rm es}\rho H$ and where $\kappa_{\rm es}=0.38~{}{\rm cm}^{2}{\rm
g}^{-1}$ is the electron scattering opacity. Here, we observe that for all the
cases, $\tau_{\rm eff}$ remains well below unity. However, a rise in
$\tau_{\rm eff}$ is observed across the shock transition in every case due to
greater density at the PSC as observed in Fig. 2(b). When $\zeta$ is increased
retaining other parameters fixed, the magnetic activity in the disc is
augmented, leading to the increase in $\tau_{\rm syn}$ and the consequent
increase of $\tau_{\rm eff}$ profile (dotted curve). Also, when $\dot{m}$
rises to $0.45$ keeping rest of the parameters fixed, there is further rise in
$\tau_{\rm eff}$ profile due to increase of density in the accretion flow
(dashed curve).

Figure 3: Variation of (a) shock location $x_{s}$, (b) compression ratio $R$,
and (c) shock strength $\Theta$ as a function of $\zeta$ for flows injected
from $x_{\rm inj}=1000$ with $\beta_{\rm inj}=10^{5}$, $\alpha_{T}=0.02$,
${{E}}_{\rm inj}$$=8.9\times 10^{-4}$ and $\dot{m}=0.05$. The Kerr parameter
is chosen as $a_{s}=0.8$. Solid, dotted and dashed curves represent the
results corresponding to $\lambda_{\rm inj}$ = 0.10888$\lambda_{K}$,
$0.11014$$\lambda_{K}$ and $0.11141$$\lambda_{K}$ respectively. See text for
details.

One of the pertinent aspect in understanding the magnetically supported
accretion flow around the rotating BHs is to study the dependence of the shock
properties on the magnetic flux advection rate ($\dot{\Phi}$). For this
purpose in Fig. 3, we fix the outer edge of the disc at $x_{\rm inj}=1000$ and
inject matter to accrete with ${{E}}_{\rm inj}$ $=8.9\times 10^{-4}$,
$\beta_{\rm inj}=10^{5}$, $\alpha_{T}=0.02$ and $\dot{m}=0.05$, respectively.
Kerr parameter is held fixed at $a_{s}=0.8$. It has already been specified
that the $\zeta$$-$parameter decides the advection rate of toroidal magnetic
flux ($\dot{\Phi}$). Thus in Fig. 3(a), we present the variation of $x_{s}$
with $\zeta$ for three sets of specific angular momentum, $\lambda_{\rm inj}$
= 0.10888$\lambda_{K}$ (solid curve), $\lambda_{\rm inj}$ =
0.11014$\lambda_{K}$ (dotted curve) and $\lambda_{\rm inj}$ =
0.11141$\lambda_{K}$ (dashed curve). We observe that for a given $\lambda_{\rm
inj}$, shocks form for an ample range of $\zeta$ and when $\zeta$ is
increased, shock front moves towards the BH horizon. The increase of $\zeta$
leads to the increase of toroidal magnetic flux advection rate in the radial
direction as the flow proceeds towards the BH (Oda et al., 2007, 2012). With
the increase of magnetic activity in the disc, the flow gets efficiently
cooled. Since the cooling efficiency is greater in the PSC due to higher
temperature and density there, the post-shock pressure drops and the shock
front is pushed towards the BH in order to maintain pressure balance across
it. The increase of magnetic activity in the disc also enhances the angular
momentum transport towards the outer region of the disc. This weakens the
centrifugal repulsion against gravity and also contributes to shift the shock
towards the horizon. When $\zeta$$-$parameter is increased beyond a critical
value ($\zeta$cri), the shock conditions are no longer satisfied and standing
shock fails to form. Moreover, for a given $\zeta$, we find that when flow is
characterized with relatively large $\lambda_{\rm inj}$, shocks form over a
large range of distances and vice versa. This clearly indicates that shock
transition in BH accretion flows are centrifugally driven.

Now, corresponding to the shock solutions shown in Fig. 3(a), we present the
variation of relating shock compression ratio ($R$) and the shock strength
($\Theta$) with $\zeta$$-$parameter in Fig. 3(b) and Fig. 3(c), respectively.
Here, $R$ is defined as the ratio of the vertically integrated post-shock
density to pre-shock density as $R=\Sigma_{+}/\Sigma_{-}$. On the other hand,
$\Theta$ is defined as the ratio of pre-shock to post-shock Mach number
($\Theta=M_{-}/M_{+}$). Since stronger shocks are located closer to the BH
horizon, we find that $R$ steadily increases with $\zeta$ in Fig. 3(b). We
find that for shocks closest to the BH horizon, $R\sim 5$. Moreover, similar
variational characteristics of $\Theta$ is observed in Fig. 3(c) as in the
case of $R$. It is to be noticed that $R$ and $\Theta$ are almost independent
with the flux advection rate for the results shown using solid and dotted
curves. The reason for this behaviour is the weak angular momentum transport
rate for low value of $\zeta$. This claim is consistent with equation (5),
where, $\mathbb{T}_{x\phi}$ is proportional to the sum of vertically
integrated pressure of the flow ($W$) and the ram pressure ($\Sigma u^{2}$).
Here, $W$ has contribution from the gas pressure ($p_{\rm gas}$) as well as
magnetic pressure ($p_{\rm mag}$) in the flow. Now, from Fig. 2(c), we find
that when $\zeta$ value is increased, the value of $B_{\phi}$ increases at a
particular radial coordinate. Thus, $p_{\rm mag}(=<B_{\phi}>^{2}/8\pi)$ rises
with the increase of $\zeta$ for a given radial coordinate and consequently,
$\mathbb{T}_{x\phi}$ also augments at that location. So, $\zeta$ controls the
angular momentum transport rate in the flow. The major motivation in this
figure is to establish that the accretion flow admits shock for a wide range
of angular momentum as well as $\zeta$. The chosen values of $\lambda_{\rm
inj}$ for the solid and dotted curves are lower than that of the dashed curve.
So for the solid and dotted curves, the shocks form closer to the black hole
as compared to the dashed curve. Also when $\zeta$ is low ($\lesssim$ 0.8),
the rate of transport of angular momentum transport in the disc is expected to
be feeble. Due to the proximity of the shock location to the BH horizon and
meagre angular momentum transport rate, there is a small shift in shock
location when $\zeta$ is lowered and we do not observe a significant variation
in the compression ratio and shock strength for the results shown using solid
and dotted curves.

Figure 4: Shock location ($x_{s}$) variation as a function of $\dot{m}$. Flow
is injected from the outer edge $x_{\rm inj}=1000$ with $\beta_{\rm
inj}=10^{5}$, $\alpha_{T}=0.02$, ${{E}}_{\rm inj}$$=8.9\times 10^{-4}$ and
$\lambda_{\rm inj}$ = 0.11141$\lambda_{K}$. The spin of the BH is chosen as
$a_{s}=0.8$. Results plotted with solid, dotted and dashed curves are for
$\zeta$$=1.2$, $1.0$, and $0.8$, respectively. See the text for details.

In Fig. 4, we show the variation of shock location ($x_{s}$) as function of
$\dot{m}$. At the injection radius $x_{\rm inj}=1000$, the flow parameters are
chosen as ${{E}}_{\rm inj}$$=8.9\times 10^{-4}$, $\lambda_{\rm inj}$ =
0.11141$\lambda_{K}$, $\beta_{\rm inj}=10^{5}$ and $\alpha_{T}=0.02$,
respectively. The BH spin parameter is fixed as $a_{s}=0.8$. Here, solid,
dotted and dashed curves refer to the results corresponding to
$\zeta$$=1.2,1.0$ and $0.8$, respectively. We notice that for a given $\zeta$,
$x_{s}$ decreases with the increase of $\dot{m}$. It is already pointed out
that shock front advances towards the BH when $\dot{m}$ is increased (see Fig.
1) and the overall pressure in the PSC is expected to be reduced. Therefore,
for a given $\zeta$, we observe an anti-correlation between $x_{s}$ and
$\dot{m}$. When $\dot{m}$ rises beyond a critical limit ($\dot{m}^{\rm cri}$),
standing shocks do not from since the shock conditions are not satisfied
anymore. Again, $x_{s}$ is also found to decrease with the increase in
$\zeta$, when $\dot{m}$ is fixed. In reality, when $\zeta$ parameter is
augmented, synchrotron cooling efficiency in the flow is higher as there is
increased magnetic activity in the disc. Due to the efficient cooling at the
PSC, the post-shock pressure decreases and the shock location approaches
towards the BH in order to maintain the pressure balance across the shock
front. Thus, for a given $\dot{m}$, $x_{s}$ anti-correlates with $\zeta$.

##  4 Astrophysical Relevance

Table 1: Calculation of luminosity of shock. In Column 1, we make a list of
the names of the sources and in Column 2-4 we give the corresponding mass,
accretion rate and spin. In Column 5-6, we report the model parameters and in
Column 7-9, we indicate the maximum energy dissipated, the shock location and
the estimated maximum luminosity of shock obtained using equation (15). The
core radio luminosity values reported from observations are listed in column
10.

Object | $M_{\rm BH}$ | $\dot{m}$ | $a_{s}$ | ${E}_{\rm in}$ | $\lambda_{\rm in}$ | $\Delta{E}^{\rm max}$ | $x_{s}$ | $\mathcal{L}^{\rm max}_{\rm shock}$ | $\mathcal{L}^{\rm Obs}_{\rm jet}$
---|---|---|---|---|---|---|---|---|---
| $({\rm M}_{\odot})$ | $({\dot{M}}_{\rm Edd})$ |  | $(10^{-2})$ | $(\lambda_{K})$ |  ($10^{-2}c^{2}$) | $(r_{g})$ |  (erg s-1) |  (erg s-1)
A0620-00 |  $6.60$$~{}^{a}$ |  $5.0\times 10^{-4}$$~{}^{b}$ |  $0.12$$~{}^{c}$ | $-0.6517$ | $0.88972$ | $0.8754$ | $16.54$ | $3.61\times 10^{33}$ |  $1.0\times 10^{33}$$~{}^{\bigstar}$
LMC X-3 |  $6.98$$~{}^{d}$ |  $2.365$$~{}^{e}$ |  $0.25$$~{}^{f}$ | $-1.5036$ | $0.85117$ | $1.5642$ | $16.03$ | $3.23\times 10^{37}$ | $-$
H1743-322 |  $11.21$$~{}^{g}$ |  $0.127$$~{}^{h}$ |  $0.4$i | $-2.2296$ | $0.76128$ | $2.2488$ | $15.22$ | $4.0\times 10^{36}$ |  $3.60\times 10^{38}$$~{}^{\bigstar}$
XTE J1550-564 |  $9.10$$~{}^{j}$ |  $0.528$$~{}^{k}$ |  $0.49$$~{}^{k}$ | $-2.4323$ | $0.68529$ | $2.4216$ | $14.27$ | $1.45\times 10^{37}$ |  $3.41\times 10^{38}$$~{}^{l}$
GRO J1655-40 |  $5.31$$~{}^{m}$ |  $0.596$$~{}^{n}$ | $0.7^{o}$ | $-2.8775$ | $0.40495$ | $2.7427$ | $13.25$ | $1.08\times 10^{37}$ |  $1.95\times 10^{36}$$~{}^{p}$
XTE J1118+480† |  $7.50$$~{}^{q}$ |  $3.0\times 10^{-4}$$~{}^{b}$ |  $0.9$r | $-2.9702$ | $0.29669$ | $3.0866$ | $9.07$ | $8.68\times 10^{33}$ |  $4.0\times 10^{32}$$~{}^{\bigstar}$
LMC X-1† |  $10.91$$~{}^{s}$ |  $0.884$$~{}^{t}$ |  $0.92$$~{}^{t}$ | $-4.1158$ | $0.26757$ | $3.9125$ | $8.972$ | $4.71\times 10^{37}$ |  $2.0\times 10^{39}$$~{}^{u}$
Cyg X-1† |  $14.8$$~{}^{v}$ |  $0.061$$~{}^{w}$ |  $0.97$$~{}^{x}$ | $-3.3274$ | $0.35534$ | $3.4323$ | $8.02$ | $3.87\times 10^{36}$ |  $1.0\times 10^{37}$$~{}^{y}$
GRS1915+105† |  $10.1$$~{}^{z}$ |  $3.188$$~{}^{zx}$ |  $0.98$$~{}^{zy}$ | $-3.5997$ | $0.36401$ | $3.4312$ | $8.02$ | $1.38\times 10^{38}$ |  $1.0\times 10^{38}$$~{}^{zx}$

> References. aCantrell et al. (2010), bYang (2016), cGou et al. (2010),
> dOrosz et al. (2014), eKubota et al. (2010), fSteiner et al. (2014), gMolla
> et al. (2017), hBhattacharjee et al. (2017), iTursunov & Kološ (2018),
> jOrosz et al. (2011a), kSteiner et al. (2011), lFender et al. (2004), mMotta
> et al. (2014), nLuketic et al. (2010), oShafee et al. (2006), pMigliari et
> al. (2007), qKhargharia, et al. (2013), rChaty et al. (2003), sOrosz et al.
> (2009), tGou et al. (2009), uCooke et al. (2007), vOrosz et al. (2011b),
> wGou et al. (2011), xFabian et al. (2012), yGallo et al. (2005), zSteeghs et
> al. (2013), zxBelloni et al. (2000), zyMiller et al. (2013).
> ★ $L^{\rm Obs}_{\rm jet}=\dot{M}_{\rm out}c^{2}$ is used for the sources
> A0620-00, XTE J1118+480 and H 1743-322, where $\dot{M}_{\rm out}$ is the
> outflow rate. For the sources A0620-00 and XTE J1118+480, the outflow rate
> is adopted from Yang (2016). For H1743-322, $\dot{M}_{\rm out}$ is
> considered following Miller et al. (2006).

Thus far, we have dealt with non dissipative shocks where the specific energy
essentially remains conserved across the shock front (Chakrabarti, 1989).
However, as seen in Fig. 2(f), the sudden compression of the flow at the shock
causes the optical depth of the flow to rise significantly across the shock
front. Since the shock is thin and the optical depth there is very high,
thermal Comptonization of the photons is expected to be very important.
Chakrabarti & Titarchuk (1995) pointed out that due to the thermal
Comptonization process, the dissipation of energy of the accretion flow at the
shock is likely. This energy dissipation mechanism eventually reduces the
temperature of the flow in the PSC and the energy loss is proportional to the
difference in temperature in the post- and pre-shock flows. Following this,
the loss of energy ($\Delta{{E}}$) at the shock is assessed as (Das et al.,
2010),

$\Delta E=f_{d}n(c_{s+}^{2}-c_{s-}^{2}),$ $None$

where $c_{s-}$ and $c_{s+}$ denote the pre-shock and post-shock sound speeds,
respectively. Here, $f_{d}$ refers to the fraction of thermal energy
difference dissipated across the shock front, which is treated as a free
parameter in this work (Das et al., 2010; Singh & Chakrabarti, 2011; Sarkar &
Das, 2013; Kumar & Chattopadhyay, 2013; Sarkar et al., 2018; Das & Sarkar,
2018). The kinetic power lost from the disc can be evaluated by regarding the
shock luminosity as in Le & Becker (2004, 2005); Sarkar & Das (2016); Sarkar
et al. (2018),

$\mathcal{L}_{\rm total}=\mathcal{L}_{\rm shock}=\dot{M}\times\Delta E\times
c^{2}~{}~{}{\rm erg~{}s^{-1}},$ $None$

where, $\mathcal{L}_{\rm total}$, $\mathcal{L}_{\rm shock}$ and $\dot{M}$
denote the kinetic power lost by the disc, the luminosity of shock and the
accretion rate for a specified source, respectively. The energy loss at the
shock front due to Comptonization ($\Delta{{E}}$) is essentially a function of
the number of soft photons and the number density of electrons (Mondal et al.,
2014). In this paper, we fix $f_{d}=0.998$ all throughout, for the sake of
representation.

Subsequently, we compute the maximum luminosity of shock ($\mathcal{L}^{\rm
max}_{\rm shock}$) relating to maximum dissipation of energy at the shock.
Here, $\alpha_{B}=0.001$ and $\beta_{\rm in}=10^{5}$ are considered for all
cases. Also, in Oda et al. (2007), the authors have obtained radial structure
of the accretion disc considering $\zeta=1$, which remains optically thin
throughout. For $\zeta\sim 0$, the authors showed that steady global transonic
solutions with magnetic fields were not obtained beyond a certain critical
accretion rate. They further reported that for $\zeta\sim 1$, accretion
solutions connecting the outer boundary to the inner region can be obtained
for large accretion rates with low$-\beta$ region at the inner part of the
disc. Moreover, Oda et al. (2012) delineated that for small values of $\zeta$
(e.g., $\zeta\sim 0$), it is not possible to obtain the magnetically supported
low$-\beta$ disc solutions but the usual advection dominated accretion flow
(ADAF)/ radiatively inefficient accretion flow (RIAF) solutions can be
explained. Further in Table 1 of Oda et al. (2012), the authors showed the
formation of low$-\beta$ disc in global accretion solutions for
$0.5\leq\zeta\leq 1$, by suitably choosing the other disc parameters. For,
$\zeta>1$, the authors additionally showed that magnetic pressure would
dominate in the accretion flow even at low accretion rates. Considering all
these, we argue that $\zeta\geq 0.5$ is a reasonable approximation in the
accretion disc to obtain the low$-\beta$ solutions and this can support the
observational values. Thus, we choose $\zeta=1$ as a representative value in
the analysis that follows.

In Table 1, we present the physical parameters of the stellar mass BH sources
including model parameters and compute the maximum luminosity of shock.
Columns 1-4 of Table 1 show the names of sources, their mass ($M_{\rm BH}$),
accretion rate (${\dot{m}}$) and spin ($a_{s}$). In section 5-6, we note the
specific energy at the inner critical point (${{E}}_{\rm in}$) and the angular
momentum at the inner critical point ($\lambda_{\rm in}$). In column 7, we
note the maximum dissipated energy $\Delta{E}^{\rm max}$ and in column 8, we
indicate the position of shock ($x_{s}$). In column 10, the maximum luminosity
of shock $\mathcal{L}^{\rm max}_{\rm shock}$ is reported. Our primary
objective in this examination is to estimate the upper limit of energy that
can be removed from the PSC to power the outflowing matter from the disc as
Jets. Accordingly, we evaluate the maximum dissipated energy $\Delta{E}^{\rm
max}$ at the position of shock jump. For the stellar mass BH sources under
consideration, we find that the evaluated luminosities of shock are in
consensus with the core radio luminosity values $\mathcal{L}^{\rm Obs}_{\rm
jet}$ (in column 10) reported from observations (Fender et al., 2004; Cooke et
al., 2007; Gallo et al., 2007; Migliari et al., 2007; Belloni et al., 2000),
subject to availability. To estimate the observed jet luminosity ($L_{\rm
jet}^{\rm Obs}$) from the outflow rate ($\dot{M}_{\rm out}$), we have used the
relation $L_{\rm jet}^{\rm Obs}=\dot{M}_{\rm out}c^{2}$. Here, we assume 100%
energy conversion efficiency on the basis of the common definition $L_{\rm
Edd}=\dot{M}_{\rm Edd}c^{2}$ for the Eddington limit on the mass accretion
rate used by many authors (Rezzolla & Zanotti, 2013; Straub et al., 2014; Yuan
& Narayan, 2014). Although 100% energy conversion efficiency is unphysical,
however, the definition used would give us the maximum observed luminosity
achievable from the outflow rate available in the literature, which is
compared with the calculated maximum shock luminosity from our model. For the
BH sources marked with superscript ${\dagger}$, the reported values of spin
from the literature exceeds $0.8$. In such a case, it is inappropriate to use
the Chakrabarti & Mondal (2006) potential to calculate the shock luminosity
corresponding to these sources. Nevertheless, we carry out the analysis using
this potential in order to get a qualitative estimate of $\mathcal{L}^{\rm
max}_{\rm shock}$ for these sources.

##  5 Summary

We study the effects of the variation of magnetic flux advection rate
($\dot{\Phi}$) and accretion rate ($\dot{m}$) on the properties of shock in
accretion flow around a rapidly rotating BH ($a_{s}=0.8$). In the global
transonic accretion solutions, heating takes place due to the thermalization
of magnetic energy, while cooling occurs via synchrotron emission. In Fig. 1,
we present the combined effect of variation of $\dot{m}$ and
$\zeta$$-$parameter on the shock dynamics. For fixed injection parameters at
the outer edge, we find that the shock location shifts towards the horizon
with the increase of $\zeta$ (via the corresponding increase of $\dot{\Phi}$)
as well as the increase of $\dot{m}$. In Fig. 2, we have demonstrated the
radial variation of various flow variables corresponding to the global shocked
accretion solutions considered in Fig. 1. We observe that, the PSC is hotter,
denser, optically thick and also contains higher magnetic field strength as
compared to the pre-shock flow. In Fig. 3, we study the effect of the increase
of $\zeta$ on the shock dynamics. The increase of $\zeta$ leads to the
increase of toroidal magnetic flux advection rate in the radial direction as
the flow gets closer to the BH. With the increase of magnetic activity in the
disc, the flow is efficiently cooled. Since the cooling efficiency is greater
in the PSC due to higher temperature and density there, the post-shock
pressure drops and the shock front is pushed towards the BH in order to
maintain pressure balance across it. The increase of magnetic activity in the
disc (due to increase of $\zeta$) also enhances the angular momentum transport
towards the outer region of the disc. This further weakens the centrifugal
repulsion against gravity and also contributes to shift the shock towards the
horizon. As a consequence, the shock compression ratio ($R$) and the shock
strength ($\Theta$) also rise (Fig. 3 (b) & 3(c)). Furthermore, from Fig. 3 we
also observe that global accretion solutions with shock exist for an ample
range of $\zeta$$-$parameter and angular momentum. Next in Fig. 4, we explore
the effect of increase of $\dot{m}$ on the shock position. For a given set of
injection parameters at the outer edge, we observe that global shocked
accretion solutions exist for a wide range of $\dot{m}$. However, the range of
$\dot{m}$ for shock anti-correlates with the increase of $\zeta$. Finally in
Section 4, we consider the astrophysical application of the present formalism.
We estimate the maximum energy dissipated across the shock front via thermal
Comptonization phenomenon and the corresponding shock luminosity
($\mathcal{L}^{\rm max}_{\rm shock}$) for several stellar mass BHCs. We find
that the estimated $\mathcal{L}^{\rm max}_{\rm shock}$ is in well agreement
with the observed values of jet kinetic luminosity ($\mathcal{L}^{\rm
Obs}_{\rm jet}$). If the value of $\mathcal{L}^{\rm Obs}_{\rm jet}$ is
unavailable for a given source, the present model can predict the maximum
shock luminosity $\mathcal{L}^{\rm max}_{\rm shock}$ that may be observed.

In the end, we would like to mention the limitations of the present formalism.
For the sake of simplicity, we have adopted the Chakrabarti & Mondal (2006)
pseudo-Kerr potential to mimic the general relativistic effects around a
rotating BH. The potential gives satisfactory results till spin value
$a_{s}\leq 0.8$. Thus the accretion flow properties around an extremely
rotating BH cannot be explored satisfactorily in the present approach.
Although, we have attempted to calculate $\mathcal{L}_{\rm max}^{\rm shock}$
for four stellar mass BH candidates with $a_{s}>0.8$, this provides only a
qualitative estimate. Again, the magnetic fields are likely to cause the
generation of jets and outflows from the disc, which have not been considered
in the present work. Moreover, the adiabatic index is considered to be a
global constant in the paper, although it should be calculated self
consistently from the global accretion solution. Since the magnetized
accretion flow in this work remains optically thin throughout (Fig. 2f),
Comptonization of the soft synchrotron photons that are produced locally would
be important (Yuan & Narayan, 2014). However, to carry out a simplistic
treatment, we have excluded the cooling effect due to Comptonization in this
work. Ipser, & Price (1983) have indicated that the radiative cooling
efficiency in the accretion flow is enhanced due to the Comptonization
process. Again, recently Dihingia et al. (2018a) have studied two-temperature
shocked accretion flows around BHs including the effect of Comptonization. In
this work, the authors showed that due to the rise in density of the PSC as a
result of shock transition, all the radiative processes in the flow viz;
bremsstrahlung, synchrotorn and Comptonization due to synchrotorn photons are
very much efficient in the PSC. Thus for fixed outer boundary conditions, if
Comptonization due to synchrotorn photons is included in our present work, we
expect that the radiative cooling in the PSC would be enhanced and the shock
location would shift towards the horizon in order to maintain the pressure
balance across the shock. This will result in a quantitative change in the
related shock properties as reported in the present work. For example, the
shock compression ratio ($R$), shock strength ($\Theta$) and the critical
limits of $\zeta$ and $\dot{m}$, as found in Figs. (3) and (4), would be
quantitatively modified. However, the qualitative behaviour of the results is
expected to remain unaltered. We would like to address these issues in future.

##  6 Acknowledgements

BS acknowledges Dr. Ramiz Aktar and Mr. Indu Kalpa Dihingia for useful
discussions on specific topics of the paper. AR acknowledges a Commonwealth
Rutherford Fellowship. Authors would like to thank the referee for his
valuable comments and constructive suggestions, which helped to improve the
presentation of the paper.

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