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On the origin of core radio emissions from black hole sources in the realm of relativistic shocked accretion flow

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We study the relativistic, inviscid, advective accretion flow around the black holes and investigate a key feature of the accretion flow, namely the shock waves. We observe that the shock-induced accretion solutions are prevalent and such solutions are commonly obtained for a wide range of the flow parameters, such as energy (${\cal E}$) and angular momentum ($\lambda$), around the black holes of spin value $0\le a_{\rm k} < 1$. When the shock is dissipative in nature, a part of the accretion energy is released through the upper and lower surfaces of the disc at the location of the shock transition. We find that the maximum accretion energies that can be extracted at the dissipative shock ($\Delta{\cal E}^{\rm max}$) are $\sim 1\%$ and $\sim 4.4\%$ for Schwarzschild black holes ($a_{\rm k}\rightarrow 0$) and Kerr black holes ($a_{\rm k}\rightarrow 1$), respectively. Using $\Delta{\cal E}^{\rm max}$, we compute the loss of kinetic power (equivalently shock luminosity, $L_{\rm shock}$) that is enabled to comply with the energy budget for generating jets/outflows from the jet base ($i.e.$, post-shock flow). We compare $L_{\rm shock}$ with the observed core radio luminosity ($L_R$) of black hole sources for a wide mass range spanning $10$ orders of magnitude with sub-Eddington accretion rate and perceive that the present formalism seems to be potentially viable to account $L_R$ of $16$ Galactic black hole X-ray binaries (BH-XRBs) and $2176$ active galactic nuclei (AGNs). We further aim to address the core radio luminosity of intermediate-mass black hole (IMBH) sources and indicate that the present model formalism perhaps adequate to explain core radio emission of IMBH sources in the sub-Eddington accretion limit.
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arXiv:2205.07737v1 [astro-ph.HE] 16 May 2022
MNRAS 000,115 (0000) Preprint 17 May 2022 Compiled using MNRAS L
X style file v3.0
On the origin of core radio emissions from black hole sources in
the realm of relativistic shocked accretion flow
Santabrata Das1Anuj Nandi2, C. S. Stalin3, Suvendu Rakshit4,
Indu Kalpa Dihingia5, Swapnil Singh2, Ramiz Aktar6, Samik Mitra1
1Indian Institute of Technology Guwahati, Guwahati, 781039, Assam, India
2Space Astronomy Group, ISITE Campus, U. R. Rao Satellite Center, Outer Ring Road, Marathahalli, Bangalore 560037, India
3Indian Institute of Astrophysics, Koramangala, Bangalore 560034, India
4Aryabhatta Research Institute of Observational Sciences, Manora Peak, Nainital 263002, India
5Discipline of Astronomy, Astrophysics and Space Engineering, Indian Institute of Technology Indore, Indore 453552, India
6Department of Astronomy, Xiamen University, Xiamen, Fujian 361005, People’s Republic of China
Accepted XXX. Received YYY; in original form ZZZ
We study the relativistic, inviscid, advective accretion flow around the black holes
and investigate a key feature of the accretion flow, namely the shock waves. We ob-
serve that the shock-induced accretion solutions are prevalent and such solutions are
commonly obtained for a wide range of the flow parameters, such as energy (E) and
angular momentum (λ), around the black holes of spin value 0 ak<1. When the
shock is dissipative in nature, a part of the accretion energy is released through the
upper and lower surfaces of the disc at the location of the shock transition. We find
that the maximum accretion energies that can be extracted at the dissipative shock
(∆Emax) are 1% and 4.4% for Schwarzschild black holes (ak0) and Kerr
black holes (ak1), respectively. Using ∆Emax, we compute the loss of kinetic power
(equivalently shock luminosity, Lshock ) that is enabled to comply with the energy bud-
get for generating jets/outflows from the jet base (i.e., post-shock flow). We compare
Lshock with the observed core radio luminosity (LR) of black hole sources for a wide
mass range spanning 10 orders of magnitude with sub-Eddington accretion rate and
perceive that the present formalism seems to be potentially viable to account LRof 16
Galactic black hole X-ray binaries (BH-XRBs) and 2176 active galactic nuclei (AGNs).
We further aim to address the core radio luminosity of intermediate-mass black hole
(IMBH) sources and indicate that the present model formalism perhaps adequate to
explain core radio emission of IMBH sources in the sub-Eddington accretion limit.
Key words: accretion, accretion disc - black hole physics - X-rays: binaries - galaxies:
active - radio continuum: general.
The observational evidence of the ejections of mat-
ter from the BH-XRBs (Rodriguez, Mirabel, & Marti
1992;Mirabel & Rodr´ıguez 1994) and AGNs
(Jennison & Das Gupta 1953;Junor, Biretta, & Livio
1999) strongly suggests that there possibly exists a viable
coupling between the accreting and the outflowing matters
(Feroci et al. 1999;Willott et al. 1999;Ho & Peng 2001;
Pahari et al. 2018;Russell et al. 2019a;de Haas et al.
E-mail: (SD)
E-mail: (AN)
2021). Since the ejected matters are in general colli-
mated, they are likely to be originated from the inner
region of the accretion disc and therefore, they may
reveal the underlying physical processes those are ac-
tive surrounding the black holes. Further, observational
studies indicate that there is a close nexus between the
jet launching and the spectral states of the associated
black holes (Vadawale et al. 2001;Chakrabarti et al. 2002;
Gallo, Fender, & Pooley 2003;Fender, Homan, & Belloni
2009;Radhika et al. 2016;Blandford, Meier, & Readhead
2019). All these findings suggest that the jet generation
mechanism seems to be strongly connected with the ac-
cretion process around the black holes of different mass
©0000 The Authors
2Das et al.
irrespective to be either BH-XRBs or AGNs. Meanwhile, nu-
merous efforts were made both in theoretical (Chakrabarti
1999;Das & Chakrabarti 1999;Blandford & Begelman
1999;Das, Chattopadhyay, & Chakrabarti 2001;
McKinney & Blandford 2009;Das et al. 2014;Ressler et al.
2017;Aktar, Nandi, & Das 2019;Okuda et al. 2019) as well
as observational fronts to explain the disc-jet symbiosis
(Feroci et al. 1999;Brinkmann et al. 2000;Nandi et al.
2001;Fender, Belloni, & Gallo 2004;Miller-Jones et al.
2012;Miller et al. 2012;Sbarrato, Padovani, & Ghisellini
2014;Radhika et al. 2016;Svoboda, Guainazzi, & Merloni
2017;Blandford, Meier, & Readhead 2019).
The first ever attempt to examine the correlation be-
tween the X-ray (LX) and radio (LR) luminosities for black
hole candidate GX 339-4 during its hard states was carried
out by Hannikainen et al. (1998), where it was found that
LRscales with LXfollowing a power-law. Soon after, Fender
(2001) reported that the compact radio emissions are asso-
ciated with the Low/Hard State (LHS) of several black hole
binaries. Similar trend was seen to follow by several such
BH-XRBs (Corbel et al. 2003;Gallo, Fender, & Pooley
2003). Later, Merloni, Heinz, & di Matteo (2003) revis-
ited this correlation including the low-luminosity AGNs
(LLAGNs) and found tight constraints on the correlation
described as the Fundamental Plane of the black hole activity
in a three-dimensional plane of (LR, LX, MBH ), where MBH
denotes the mass of the black hole. Needless to mention that
the above correlation study was conducted considering the
core radio emissions at 5 GHz in all mass scales ranging from
stellar mass (10 M) to Supermassive (10610 M)
black holes. To explain the correlation, Heinz & Sunyaev
(2003) envisaged a non-linear dependence between the
mass of the central black hole and the observed flux
considering core dominated radio emissions. Subsequently,
several group of authors further carried out the similar
works to reveal the rigor of various physical processes re-
sponsible for such correlation (Falcke, K¨
ording, & Markoff
ording, Falcke, & Corbel 2006;Merloni et al.
2006;Wang, Wu, & Kong 2006;Panessa et al. 2007;
ultekin et al. 2009;Plotkin et al. 2012;Corbel et al. 2013;
Dong & Wu 2015;Panessa et al. 2015;Nisbet & Best 2016;
ultekin et al. 2019).
In the quest of the disc-jet symbiosis, many au-
thors pointed out that the accretion-ejection phe-
nomenon is strongly coupled and advective accreting disc
plays an important role in powering the jets/outflows
(Das & Chakrabarti 1999;Blandford & Begelman
1999;Chattopadhyay, Das, & Chakrabarti 2004;
Aktar, Nandi, & Das 2019, and references therein). In
reality, an advective accretion flow around the black holes
is necessarily transonic because of the fact that the infalling
matter must satisfy the inner boundary conditions imposed
by the event horizon. During accretion, rotating matter
experiences centrifugal repulsion against gravity that yields
a virtual barrier in the vicinity of the black hole. Eventually,
such a barrier triggers the discontinuous transition of the
flow variables to form shock waves (Landau & Lifshitz
1959;Frank et al. 2002). In reality, the downstream flow
is compressed and heated up across the shock front that
eventually generates additional entropy all the way up to
the horizon. Hence, accretion solutions harboring shock
waves are naturally preferred according to the 2nd law
of thermodynamics (Becker & Kazanas 2001). Previous
studies corroborate the presence of hydrodynamic shocks
(Fukue 1987;Chakrabarti 1989;Nobuta & Hanawa
1994;Lu et al. 1999;Fukumura & Tsuruta 2004;
Chakrabarti & Das 2004;Mo´scibrodzka, Das, & Czerny
2006;Das & Czerny 2011;Aktar, Das, & Nandi 2015;
Dihingia et al. 2019), and magnetohydrodynamic (MHD)
shocks (Koide, Shibata, & Kudoh 1998;Takahashi et al.
2002;Das & Chakrabarti 2007;Fukumura & Kazanas
2007;Takahashi & Takahashi 2010;Sarkar & Das 2016;
Fukumura et al. 2016;Okuda et al. 2019;Dihingia et al.
2020) in both BH-XRB and AGN environments.
Extensive numerical simulations of the accretion disc
independently confirm the formation of shocks as well
(Ryu, Chakrabarti, & Molteni 1997;Fragile & Blaes 2008;
Das et al. 2014;Generozov et al. 2014;Okuda & Das
2015;Sukov´a & Janiuk 2015;Okuda et al. 2019;
Palit, Janiuk, & Czerny 2020). Due to the shock com-
pression, the post-shock flow becomes hot and dense that
results in a puffed up torus like structure which acts as the
effective boundary layer of the black hole and is commonly
called as post-shock corona (hereafter PSC). In general,
PSC is hot enough (T&109K) to deflect outflows which
may be further accelerated by the radiative processes
active in the disc (Chattopadhyay, Das, & Chakrabarti
2004). Hence, the outflows/jets are expected to carry a
fraction of the available energy (equivalently core emis-
sion) at the PSC, which in general considered as the
base of the outflows/jets (Chakrabarti 1999;Das et al.
2001;Chattopadhyay & Das 2007;Das & Chattopadhyay
2008;Singh & Chakrabarti 2011;Sarkar & Das 2016).
Becker and his collaborators showed that the energy
extracted from the accretion flow via isothermal shock can
be utilized to power the relativistic particles emanating
from the disc (Le & Becker 2005;Becker, Das, & Le 2008;
Das, Becker, & Le 2009;Lee & Becker 2020). Moreover,
magnetohydrodynamical study of the accretion flows around
the black holes also accounts for possible role of shock as
the source of high energy radiation (Nishikawa et al. 2005;
Takahashi et al. 2006;Hardee, Mizuno, & Nishikawa 2007;
Takahashi & Takahashi 2010).
An important generic feature of shock wave is that
it is likely to be radiatively efficient. For that, shocks be-
come dissipative in nature where an amount of accreting
energy is escaped at the shock location through the disc
surface resulting the overall reduction of downstream flow
energy all the way down to the horizon. This energy loss
is mainly regulated by a plausible mechanism known as the
thermal Comptonization process (Chakrabarti & Titarchuk
1995;Das, Chakrabarti, & Mondal 2010, and references
therein). Assuming the energy loss to be proportional to
the difference of temperatures across the shock front, the
amount of energy dissipation at the shock can be estimated
(Das, Chakrabarti, & Mondal 2010), which is same as the
accessible energy at the PSC. A fraction of this energy could
be utilized to produce and power outflows/jets as they are
likely to originate from the PSC around the black holes
(Chakrabarti 1999;Das, Chattopadhyay, & Chakrabarti
2001;Aktar, Das, & Nandi 2015;Okuda et al. 2019).
Being motivated with this appealing energy extraction
mechanism, in this paper, we intend to study the stationary,
axisymmetric, relativistic, advective accretion flow around
MNRAS 000,115 (0000)
Core radio emissions from black hole sources 3
the black holes in the realm of general relativity and self-
consistently obtain the global accretion solutions containing
dissipative shock waves. Such dissipative shock solution has
not yet been explored in the literature for maximally rotat-
ing black holes having spin ak1. We quantitatively esti-
mate the amount of the energy released through the upper
and lower surface of the disc at the shock location and show
how the liberated energy affects the shock dynamics. We also
compute the maximum available energy dissipated at the
shock for 0 ak<1. Utilizing the usable energy available
at the PSC, we estimate the loss of kinetic power (which is
equivalent to shock luminosity) from the disc (Lshock) which
drives the jets/outflows. It may be noted that the kinetic
power associated with the base of the outflows/jets is inter-
preted as the core radio emission. Further, we investigate
the observed correlation between radio luminosities and the
black hole masses, spanning over ten orders of magnitude
in mass for BH-XRBs as well as AGNs. We show that the
radio luminosities in both BH-XRBs and AGNs are in gen-
eral much lower as compared to the possible energy loss at
the PSC and therefore, we argue that the dissipative shocks
seem to be potentially viable to account the energy bud-
get associated with the core radio luminosities in all mass
scales. Considering this, we aim to reveal the missing link
between the BH-XRBs and AGNs in connection related to
the jets/outflows. Employing our model formalism, we es-
timate the core radio luminosity of the intermediate mass
black hole (IMBH) sources in terms of the central mass.
The article is organized as follows: In Section 2, we de-
scribe our model and mention the governing equations. We
present the solution methodology in Section 3. In Section
4, we discuss our results in detail. In Section 5, we discuss
the observational implications of our formalism to explain
the core radio emissions from black holes in all mass scales.
Finally, we present the conclusion in Section 6.
We consider a steady, geometrically thin, axisymmetric,
relativistic, advective accretion disc around a black hole.
Throughout the study, we use a unit system as G=MBH =
c= 1, where MBH ,Gand care the mass of the black hole,
gravitational constant and speed of light, respectively. In this
unit system, length and angular momentum are expressed in
terms of GMBH/c2and GMBH /c. Since we have considered
MBH = 1, the present analysis is applicable for black holes
of all mass scales.
In this work, we investigate the accretion flow around
a Kerr black hole and hence, we consider Kerr metric in
Boyer-Lindquist coordinates (Boyer & Lindquist 1967) as,
ds2=gµν dxµdxν,
=gttdt2+ 2g dtdφ +grr dr2+gθ θ 2+gφφ 2,(1)
where xµ(t, r, θ, φ) denote coordinates and gtt =(1
2r/Σ), g=2akrsin2θ/Σ, grr = Σ/∆, gθθ = Σ and
gφφ =Asin2θ/Σ are the non-zero metric components. Here,
A= (r2+a2
ksin2θ, Σ = r2+a2
kcos2θ, ∆ = r2
k, and akis the black hole spin. In this work, we follow
a convention where the four velocities satisfy uµuµ=1.
Following Dihingia, Das, & Nandi (2019), we obtain the
governing equations that describe the accretion flow for a
geometrically thin accretion disc which are given by,
(a) the radial momentum equation:
,r +1
2grr gtt,r
+grr grr,r
+uφutgrr g
gtt,r gtφ,r+1
2uφuφgrr gφφgtt,r
+(grr +urur)
e+pp,r = 0.
(b) the continuity equation:
M=4πrurρH, (3)
where eis the energy density, pis the local gas pressure,
Mis the accretion rate treated as global constant, and r
stands for radial coordinate. Moreover, Hrefers the local
half-thickness of the disc and is given by (Riffert & Herold
1995;Peitz & Appl 1997;Dihingia, Das, & Nandi 2019),
; with F=γ2
k)2+ 2∆a2
where γ2
φ= 1/(1 v2
φ) is the bulk azimuthal Lorentz factor
and v2
φ=uφuφ/(utut). We define the radial three velocity
in the co-rotating frame as v2=γ2
rand thus, we have
the bulk radial Lorentz factor γ2
v= 1/(1 v2), where v2
In order to solve equations (2-3), a closure equation in
the form of Equation of State (EoS) describing the rela-
tion among the thermodynamical quantities, namely den-
sity (ρ), pressure (p) and energy density (e) is needed. For
that we adopt an EoS for relativistic fluid which is given by
(Chattopadhyay & Ryu 2009),
f=1 + Θ 9Θ + 3
+ 2 +mp
+ Θ me+ 3mp
me+ 2mp,
where Θ (= kBT /mec2) is the dimensionless temperature,
meis the mass of electron, and mpis the mass of ion,
respectively. According to the relativistic EoS, we express
the speed of sound as as=p2ΓΘ/(f+ 2Θ), where Γ =
(1 + N)/N is the adiabatic index, and N= (1/2)(df/dΘ)
is the polytropic index of the flow (Dihingia, Das, & Nandi
In this work, we use a stationary metric gµν which has
axial symmetry and this enables us to construct two Killing
vector fields tand φthat provide two conserved quantities
for the fluid motion in this gravitational field and are given
huφ= constant; hut= constant = E,(4)
where h[= (e+p)] is the specific enthalpy of the fluid, Eis
the relativistic Bernoulli constant (i.e., the specific energy
of the flow). Here, ut=γvγφ/pλggtt , where λ(=
uφ/ut) denotes the conserved specific angular momentum.
MNRAS 000,115 (0000)
4Das et al.
We simplify equations (2) and (3) to obtain the wind equa-
tion in the co-rotating frame as,
dr =N
where the numerator Nis given by,
r(r2) +γ2
r2+ 2akγ2
φr2(r3) 2a2
Γ + 1 ra2
dr ,
and the denominator Dis given by,
v(Γ + 1) ,(7)
where, Ω = uφ/utis the angular velocity of the flow.
Following Dihingia, Das, & Nandi (2019), we obtain the
temperature gradient as,
dr =
2N+ 1 ra2
dr +5
dr .(8)
In order to obtain the accretion solution around the
black hole, we solve equations (5-8) following the methodol-
ogy described in Dihingia, Das, & Nandi (2019). While do-
ing this, we specifically confine ourselves to those accre-
tion solutions that harbor standing shocks (Fukue 1987;
Chakrabarti 1989;Yang & Kafatos 1995;Lu et al. 1999;
Chakrabarti & Das 2004;Fukumura & Tsuruta 2004;Das
2007;Chattopadhyay & Kumar 2016;Sarkar & Das 2016;
Dihingia et al. 2019). In general, during the course of ac-
cretion, the rotating infalling matter experiences centrifu-
gal barrier at the vicinity of the black hole. Because of
this, matter slows down and piles up causing the accumu-
lation of matter around the black hole. This process con-
tinues until the local density of matter attains its criti-
cal value and once it is crossed, centrifugal barrier trig-
gers the transition of the flow variables in the form of
shock waves. In reality, shock induced global accretion so-
lutions are potentially favored over the shock free solu-
tions as the entropy content of the former type solution
is always higher (Das, Chattopadhyay, & Chakrabarti 2001;
Becker & Kazanas 2001). At the shock, the kinetic energy
of the supersonic pre-shock flow is converted into ther-
mal energy and hence, post-shock flow becomes hot and
behaves like a Compton corona (Chakrabarti & Titarchuk
1995;Iyer, Nandi, & Mandal 2015;Nandi et al. 2018;
Aktar, Nandi, & Das 2019). As there exists a temperature
gradient across the shock front, it enables a fraction of
the available thermal energy to dissipate away through the
disc surface. Evidently, the energy accessible at the post-
shock flow is same as the available energy dissipated at
the shock. A part of this energy is utilized in the form
of high energy radiations, namely the gamma ray and the
X-ray emissions, and the rest is used for the jet/outflow
generation as they are expected to be launched from the
post-shock region (Chakrabarti 1999;Becker, Das, & Le
2008;Das, Becker, & Le 2009;Becker, Das, & Le 2011;
Sarkar & Das 2016). These jets/outflows further consume
some energy simultaneously for their thermodynamical ex-
pansion and for the work done against gravity. The remain-
ing energy is then utilized to power the jets/outflows.
It may be noted that for radiatively inefficient adiabatic
accretion flow, the specific energy in the pre-shock as well
as post-shock flows remains conserved. In reality, the energy
flux across the shock front becomes uniform when the shock
width is considered to be very thin and the shock is non-
dissipative (Chakrabarti 1989;Frank et al. 2002) in nature.
However, in this study, we focus on the dissipative shocks
where a part of the accreting energy is released vertically at
the shock causing a reduction of specific energy in the post-
shock flow. The mechanism by which the accreting energy
could be dissipated at the shock is primarily governed by the
thermal Comptonization process (Chakrabarti & Titarchuk
1995) and because of this, the temperature in the post-
shock region is decreased. Considering the above scenario,
we model the loss of energy (∆E) to be proportional
to the temperature difference across the shock front and
Eis estimated as (Das, Chakrabarti, & Mondal 2010;
Singh & Chakrabarti 2011;Sarkar & Das 2016, and refer-
ences therein),
where βis the proportionality constant that accounts the
fraction of the accessible thermal energy across the shock
front. Here, the quantities expressed using the subscripts
’ and ‘+’ refer their immediate pre-shock and post-shock
values, respectively. Needless to mention that because of the
energy dissipation at the shock, the post-shock flow energy
(E+) can be expressed as E+=EE, where Edenotes
the energy of the pre-shock flow. In this work, we treat E
and E+as free parameters and applying them, we calculate
Efrom the shocked accretion solutions. Needless to men-
tion that the post-shock flow may become bound due to the
energy dissipation (∆E>0) across the shock front, however,
all the solutions under consideration are chosen as unbound
in the pre-shock domain. With this, we calculate βusing
equation (9) for shocked accretion solutions that lies in the
range 0 < β < 1.
It is noteworthy that in this work, the global accretion
solutions containing shocks are independent of the accretion
rate as radiative cooling processes are not taken into ac-
count for simplicity. This eventually imposes limitations in
explaining the physical states of the accretion flow although
the model solutions are suffice to characterize the accretion
flow kinematics in terms of the conserved quantities, namely
energy and angular momentum of the flow.
Now, based on the above insight on the energy bud-
get, the total usable energy available in the post-shock flow
is ∆E. Keeping this in mind, we calculate the loss of ki-
netic power by the disc corresponding to ∆Ein terms of
the observable quantities and obtain the shock luminosity
(Le & Becker 2004,2005) as,
Lshock =˙
M×E × c2erg s1,(10)
where Lshock is the shock luminosity and ˙
Mis the accretion
rate. With this, we compute Lshock considering the dissipa-
tive shock mechanism and compare it with core radio lu-
minosity observed from the black hole sources. Indeed, it is
clear from equation (10) that Lshock may be degenerate due
MNRAS 000,115 (0000)
Core radio emissions from black hole sources 5
Figure 1. Plot of Mach number (M=v/as) as function of
radial coordinate (r). Here, the flow parameters are chosen as
E= 1.002 and λ= 2.01, and black hole spin is considered as
ak= 0.99. Results depicted with solid (purple), dashed (orange)
and dotted (green) curves are obtained for ∆E= 0,0.0025, and
0.0167, respectively. At the inset, inner critical points (rin ) are
zoomed which are shown using open circle, open triangle and a
cross whereas outer critical point (rout ) is shown using filled cir-
cle. Vertical arrows represent the locations of the shock transition
(rs) and the arrows indicate the overall direction of flow motion
towards the black hole. See text for details.
to the different combinations of ˙
Mand ∆E. In this work,
we choose the spin value of the black hole in the range
0ak0.99. Moreover, in order to represent the LHS
of the black hole sources (as ‘compact’ jets are commonly
observed in the LHS (Fender, Belloni, & Gallo 2004)), we
consider the value of accretion rate in the range ˙m=
M/ ˙
MEdd = 1051.0 (Wu & Liu 2004;Athulya M. et al.
2021), where ˙
MEdd is the Eddington mass accretion rate and
is given by ˙
MEdd = 1.39 ×1017 (MBH/M) g sec1. Fur-
thermore, in order to examine the robustness of our model
formalism, we vary the mass of the central black hole in a
wide range starting form stellar mass to Supermassive scale,
and finally compare the results with observations.
In Fig. 1, we depict the typical accretion solutions around
a rotating black hole of spin ak= 0.99. In the figure, we
plot the variation of Mach number (M=v/a) as func-
tion of radial coordinate (r). Here, the flow starts its jour-
ney from the outer edge of the disc at redge = 5000 sub-
sonically with energy E= 1.002 and angular momentum
λ= 2.01. As the flow moves inward, it gains radial veloc-
ity due to the influence of black hole gravity and smoothly
makes sonic state transition while crossing the outer crit-
ical point at rout = 117.5285. At the supersonic regime,
rotating flow experiences centrifugal barrier against gravity
that causes the accumulation of matter in the vicinity of
the black hole. Because of this, matter locally piles up re-
sulting the increase of density. Undoubtedly, this process is
not continued indefinitely due to the fact that at the critical
limit of density, centrifugal barrier triggers the discontin-
uous transition in the flow variables in the form of shock
waves (Fukue 1987;Frank et al. 2002). At the shock, super-
sonic flow jumps into the subsonic branch where all the pre-
shock kinetic energy of the flow is converted into thermal
energy. In this case, the flow experiences shock transition
at rs= 50.47. Just after the shock transition, post-shock
flow momentarily slows down, however gradually picks up
its velocity and ultimately enters into the black hole su-
personically after crossing the inner critical point smoothly
at rin = 1.4031. This global shocked accretion solution is
plotted using solid (purple) curve where arrows indicate the
direction of the flow motion and the vertical arrow indi-
cates the location of the shock transition. Next, when a
part of the flow energy (∆E) is radiated away through the
disc surface at the shock, the post-shock thermal pressure
is reduced and the shock front is being pushed further to-
wards the horizon. Evidently, the shock settles down at a
smaller radius in order to maintain the pressure balance
across the shock front. Following this, when ∆E= 0.0025
is chosen, we obtain rs= 24.47 and rin = 1.4047, and
the corresponding solution is plotted using the dashed curve
(orange). When the energy dissipation is monotonically in-
creased, for the same set of flow parameters, we find the clos-
est standing shock location at rs= 8.22 for ∆E= 0.0167.
This solution is presented using dotted curve (green) where
rin = 1.4147. For the purpose of clarity, in the inset, we
zoom the inner critical point locations as they are closely
separated. In the figure, critical points and the energy dissi-
pation parameters are marked. What is more is that follow-
ing Chakrabarti & Molteni (1993); Yang & Kafatos (1995);
Lu et al. (1997); Fukumura & Kazanas (2007), the stabil-
ity of the standing shock is examined, where we vary the
shock front radially by an infinitesimally small amount in
order to perturb the radial momentum flux density (Trr ,
Dihingia, Das, & Nandi (2019)). When shock is dynamically
stable, it must come back to its original position and the cri-
teria for stable shock is given by, κ(rs) = dT rr
dr dT rr
dr <
0 (Fukumura & Kazanas 2007). Invoking this criteria, we
ascertain that all the standing shocks presented in Fig.
1are stable. For the same shocked accretion solutions,
we compute the various shock properties (see Das 2007;
Das, Becker, & Le 2009), namely, shock location (rs), com-
pression ratio (R), shock strength (S), scale height ratio
(H+/H), and present them in Table 1. In reality, as ∆E
increases, shock settles down at the lower radii (Fig. 1) and
hence, the temperature of PSC increases due to enhanced
shock compression. Moreover, since the disk thickness is
largely depends on the local temperature, the scale height
ratio increases with the increase of ∆Eyielding the PSC
to be more puffed up for stronger shock. Accordingly, we
infer that geometrically thick PSC seems to render higher
energy dissipation (equivalently Lshock) that possibly leads
to produce higher core radio luminosity.
We examine the entire range of E+and λthat pro-
vides the global transonic shocked accretion solution around
MNRAS 000,115 (0000)
6Das et al.
Table 1. Various shock properties computed for solutions pre-
sented in Fig. 1, where λ= 2.01, E= 1.002 are chosen. See
the text for details.
Erout rin rsR S H+/H
0.0 117.5285 1.4031 50.47 1.58 1.75 1.11
0.0025 117.5285 1.4047 24.47 2.43 2.86 1.20
0.0167 117.5285 1.4147 8.22 3.67 4.35 1.26
Note: ∆Eis the energy loss, rout is the outer critical point, rin
is the inner critical point, rsis the shock location, Ris the com-
pression ratio, Sis the shock strength, and H+/Hrefers scale
height ratio.
Figure 2. Plot of parameter space in λ− E+plane that admitted
shock induced global accretion solutions around the black holes.
For fixed ak= 0.99, we obtain shocked accretion solution passing
through the inner critical point and having the minimum energy
+. The maximum amount of energy is lost by the flow via the
disc surface at the shock for Emin
+. See text for details.
a rapidly rotating black hole of spin value ak= 0.99. The
obtained results are presented in Fig. 2, where the effective
region bounded by the solid curve (in red) in λ− E+plane
provides the shock solutions for ∆E= 0. Since energy dis-
sipation at shock is restricted, the energy across the shock
front remains same that yields E=E+. When energy dis-
sipation at shock is allowed (i.e., E>0), we have E+<E
irrespective to the choice of λvalues. We examine all possi-
ble range of ∆Ethat admits shock solution and separate the
domain of the parameter space in λ− E+plane using dashed
curve (in blue). Further, we vary λfreely and calculate the
minimum flow energy with which flow enters into the black
hole after the shock transition. In absence of any energy dis-
sipation between the shock radius (rs) and horizon (rh), i.e.,
in the range rh< r < rs, this minimum energy is identical
to the minimum energy of the post-shock flow (E+) and we
denote it as Emin
+. Needless to mention that Emin
depends on the spin of the black hole (ak) marked in the
Figure 3. Plot of maximum available energy across the shock front
(∆Emax) as function of the black hole spin (ak). Obtained results
are depicted by the filled circles in orange color which are joined
by the green lines. See text for details.
figure. It is obvious that for a given ak, the maximum en-
ergy that can be dissipated at the shock is calculated as
Emax =E− Emin
Subsequently, we freely vary all the input flow param-
eters, namely Eand λ, and calculate ∆Emax for a given
ak. The obtained results are presented in Fig. 3, where we
depict the variation of ∆Emax as function ak. We find that
around 1% of the flow energy can be extracted at the dissi-
pative shock for Schwarszchild black hole (weakly rotating,
ak0) and about 4.4% of the flow energy can be extracted
for Kerr black hole (maximally rotating, ak1).
In the next section, we use equation (10) to estimate the
shock luminosity (Lshock) (equivalent to the kinetic power re-
leased by the disc) for black hole sources that include both
BH-XRBs and AGNs. While doing this, the jets/outflows
are considered to be compact as well as core dominated sur-
rounding the central black holes. Further, we compare Lshock
with the observed core radio luminosity (LR) of both BH-
XRBs and AGNs.
In this work, we focus on the core radio emission at 5
GHz from the black hole sources in all mass scales starting
from BH-XRBs to AGNs. We compile the mass, distance,
and core radio emission data of the large number of sample
sources from the literature.
5.1 Source Selection: BH-XRBs
We consider 16 BH-XRBs whose mass and distance are well
constrained, and the radio observations of these sources in
LHS are readily available (see Table 2). The accretion in
MNRAS 000,115 (0000)
Core radio emissions from black hole sources 7
Table 2. Physical and observable parameters of BH-XRBs. Core radio luminosities (LR) are complied from the literature for several
sources, if available. For the rest, LRis calculated using source distance (D), observation frequency (ν) and core radio flux (F5) values
using LR= 4πνF5D2, where Drefers source distance.
Source Name Mass Distance Spin νRadio Flux Core radio luminosity References
(MBH) (D) (ak) (F5) at 5 GHz (LR)
(in M) (in kpc) (in GHz) (in mJy) (in 1030 erg s1)
4U 1543-47 9.42 ±0.97 7.5±1.00.85 4.8 3.18 4.00 1.03 1.29 1,2,3,44
Cyg X-1 14.8±1.0 1.86 ±0.12 >0.99 15 6.00 19.60 0.124 0.406 4,5,3,45ab
GRO J1655-40 6.3±0.25 3.2±0.20.98 4.86 1.46 2.01 0.087 0.120 6,7,3,46
GRS 1915+105 12.4+2.0
1.60.99 5.0 25.75 198.77 11.396 87.967 8,8,3,47
XTE J1118+480 7.1±1.3 1.8±0.6 — 15 6.27.5 0.069 0.084 9,9,3
XTE J1550-564 9.1±0.6 4.4±0.50.78 4.8 0.88 7.45 0.098 0.829 10,10,3,48
Cyg X-3 2.4+2.1
1.17.4±1.1 — 4.36 269.15 11,12,13
GX 339-4 10.08+1.81
1.80 8.4±0.9>0.97 — 0.00178 0.8128 14,15,13,49
XTE J1859+226 6.55 ±1.35 6 11 0.6 — 0.151 0.199 16,17ab, 13,50
H 1743-322 11.21+1.65
1.96 8.5±0.8<0.7 4.8 0.12 2.37 0.05 0.984 18,19,20,19
IGR J17091-3624 10.612.3 11 17 <0.27 5.5 0.17 2.41 0.18 2.52 21,22,22,51
4U 1630-472 10.0±0.1 11.5±0.30.98 4.86 1.4±0.3 1.08 1.98 23,24,25,52ab
4.80 2.6±0.3
MAXI J1535-571 6.47+1.36
1.33 4.1+0.6
0.50.99 5.5 0.18 377.20 0.02 41.74 26,27,28,53
MAXI J1348-630 11 ±2 2.2+0.5
0.6— 5.5 3.4±0.2 0.108 29,30,31
MAXI J1820+070 5.73 8.34 2.96 ±0.33 0.2 4.7 62 ±4 3.06 32,33,34,54
V404 Cyg 9.0+0.2
0.62.39 ±0.14 >0.92 4.98 0.141 0.680 0.005 0.023 35,36,37,55
Swift J1357.2-0933 >9.3 2.36.3 — 5.5 — 0.0043 0.033 38,39ab, 40
MAXI J0637-430 8.010.0 — 5.5 0.066 ±0.015 0.043 41,42,43
References: 1: Orosz (2003), 2: Park et al. (2004), 3: G¨
ultekin et al. (2019), 4: Orosz et al. (2011a), 5: Reid et al. (2011), 6:
Greene, Bailyn, & Orosz (2001), 7: Jonker & Nelemans (2004), 8: Reid et al. (2014), 9: McClintock et al. (2001), 10: Orosz et al. (2011b),
11: Zdziarski, Mikolajewska, & Belczynski (2013), 12: McCollough, Corrales, & Dunham (2016), 13: Merloni, Heinz, & di Matteo (2003),
14: Sreehari et al. (2019), 15: Parker et al. (2016), 16: Nandi et al. (2018), 17a:Hynes et al. (2002), 17b:Zurita et al. (2002),
18: Molla et al. (2017), 19: Steiner, McClintock, & Reid (2012), 20: Corbel et al. (2005), 21: Iyer, Nandi, & Mandal (2015), 22:
Rodriguez et al. (2011), 23: Seifina, Titarchuk, & Shaposhnikov (2014), 24: Kalemci, Maccarone, & Tomsick (2018), 25: Hjellming et al.
(1999), 26: Sreehari et al. (2019), 27: Chauhan et al. (2019), 28: Russell et al. (2019a), 29: Lamer et al. (2020), 30: Chauhan et al. (2021),
31: Russell et al. (2019b), 32: Torres et al. (2020), 33: Atri et al. (2020), 34: Trushkin et al. (2018), 35: Khargharia, Froning, & Robinson
(2010), 36: Miller-Jones et al. (2009), 37: Plotkin et al. (2019), 38: Corral-Santana et al. (2016), 39a:Mata S´anchez et al. (2015), 39b:
Shahbaz et al. (2013), 40: Paice et al. (2019), 41: Baby et al. (2021), 42: Tetarenko et al. (2021), 43: Russell et al. (2019), 44: Shafee et al.
(2006), 45a:Zhao et al. (2021), 45b:Kushwaha, Agrawal, & Nandi (2021), 46: Stuchl´ık & Koloˇs (2016), 47: Sreehari et al. (2020), 48:
Miller et al. (2009), 49: Ludlam, Miller, & Cackett (2015), 50: Steiner, McClintock, & Narayan (2013), 51: Wang et al. (2018), 52a:
King et al. (2014), 52b:Pahari et al. (2018), 53: Miller et al. (2018), 54: Guan et al. (2021), 55: Walton et al. (2017)
: Mass estimate of these sources are uncertain, till date. : VLA observation; : ATCA observation; : VLBA observation in 2014.
Note: References for black hole mass (MBH), distance (D), Fνor LR, and spin (ak) are given in column 8 in sequential order. Data are
complied based on the recent findings (see also Merloni, Heinz, & di Matteo (2003); G¨
ultekin et al. (2019)).
LHS (Belloni et al. 2005;Nandi et al. 2012) is generally cou-
pled with the core radio emission (Fender, Belloni, & Gallo
2004) from the sources. Because of this, we include the ob-
servation of compact radio emission at 5 GHz to calcu-
late the radio luminosity while excluding the transient ra-
dio emissions (i.e., relativistic jets) commonly observed in
soft-intermediate state (SIMS) (see Fender, Belloni, & Gallo
2004;Fender, Homan, & Belloni 2009;Radhika & Nandi
2014;Radhika et al. 2016, and references therein). It may
be noted that the core radio luminosity of some of these
sources are observed at different frequency bands (such as
15 GHz). For Cyg X-1, 15 GHz radio luminosity was con-
verted to 5 GHz radio luminosity assuming a flat spectrum
(Fender et al. 2000), whereas for XTE J1118+480, we con-
vert the 15 GHz radio luminosity to 5 GHz radio luminos-
ity using a radio spectral index of α= +0.5 considering
Fν=να(Fender et al. 2001). For these sources, we calcu-
late 5 GHz radio luminosity using the relation LRνLν=
MNRAS 000,115 (0000)
8Das et al.
4πνF5D2(see G¨
ultekin et al. 2019), where ν5 GHz, F5
are the 5 GHz flux, and Dis the distance of the source,
respectively. It may be noted that our BH-XRB source
samples differ from Merloni, Heinz, & di Matteo (2003) and
ultekin et al. (2019) because of the fact that we use most
recent and refined estimates of mass and distance of the
sources under consideration, and accordingly we calculate
their radio luminosity. Further, we exclude the source LS
5039 from Table 2as it is recently identified as NS-Plusar
source (Yoneda et al. 2020). In Table 2, we summarize the
details of the selected sources, where columns 1 8 rep-
resent source name, mass, distance, spin, observation fre-
quency (ν), radio flux (F5), core radio luminosity (LR) and
relevant references, respectively.
5.2 Source Selection: SMBH in AGN
We consider a group of AGN sources following
ultekin et al. (2019) (hereafter G19) that includes both
Seyferts and LLAGNs. For these sources, G¨
ultekin et al.
(2019) carried out the image analysis to extract the core
radio flux (Fν) that eventually renders their core radio
luminosity (LR). Here, we adopt a source selection criteria
as (a) MBH >105Mand (b) source observations at
radio frequency ν5 GHz, that all together yields 61
source samples. Subsequently, we calculate the core radio
luminosity of these sources as LR= 4πνF5D2, where F5
denote the core radio luminosity at ν= 5 GHz frequency
and obtain LR= 1032.51040.8erg s1.
Next, we use the catalog of
Rakshit, Stalin, & Kotilainen (2020) (hereafter R20)
to include Supermassive black holes (SMBHs) in our sample
sources. The R20 catalog contains spectral properties of
500,000 quasars up to redshift factor (z)5 covering a
wide range of black hole masses 1071010M. The mass
of the SMBHs in the catalog is obtained by employing the
Virial relation where the size of the broad line region can be
estimated from the AGN luminosity and the velocity of the
cloud can be calculated using the width of the emission line.
Accordingly, the corresponding relation for the estimation
of SMBH mass is given by (Kaspi et al. 2000),
log MBH
M=a+blog λLλ
1044erg s1+ 2 log ∆V
km s1,
where Lλis the monochromatic continuum luminosity at
wavelength λand ∆Vis the FWHM of the emission
line. The coefficients aand bare empirically calibrated
based on the size-luminosity relation either from the re-
verberation mapping observations (Kaspi et al. 2000) or in-
ternally calibrated based on the different emission lines
(Vestergaard & Peterson 2006). Depending on the redshift,
various combinations of emission line (Hβ, Mg II, C IV)
and continuum luminosity (L5100,L3000,L1350 ) are used.
A detailed description of the mass measurement method is
described in R20.
The majority of AGN in R20 sample have MBH >
108M. As the low-luminosity AGNs (LLAGNs) with mass
MBH <107Mare not included in R20 sample, we explore
the low-luminosity AGN catalog of Liu et al. (2019) (here-
after L19). It may be noted that in L19, the black hole mass
is estimated by taking the average of the two masses ob-
tained independently from the Hαand Hβlines.
In order to find the radio-counterpart and to estimate
the associated radio luminosity, we cross-match both cat-
alogs (i.e., L19 and R20) with 1.4 GHz FIRST survey
(White et al. 1997) within a search radius of 2 arc sec. The
radio-detection fraction is 3.4% for R20 and 11.7% for L19
AGN samples. We note that the present analysis deals with
core radio emissions of black hole sources and many AGNs
show powerful relativistic jets which could be launched
due to Blandford-Znajek (BZ) process (Blandford & Znajek
1977) instead of accretion flow. Meanwhile, Rusinek et al.
(2020) reported that the jet production efficiency of radio
loud AGNs (RL-AGNs) is 10% of the accretion disc radia-
tive efficiency, while this is only 0.02% in the case of radio
quiet AGNs (RQ-AGNs) suggesting that the collimated, rel-
ativistic jets ought to be produced by the BZ mechanism
rather than the accretion flow. Subsequently, we calculate
the radio-loudness parameter (R, defined by the ratio of
FIRST 1.4 GHz to optical g-band flux) and restrict our
source samples for radio-quiet (R < 19; see Komossa et al.
2006) AGNs. As some radio sources are present in both cat-
alogs (i.e., L19 and R20), we exclude common sources from
R20. With this, we find 1207 and 911 radio-quiet AGNs in
the R20 and L19 AGN sample, respectively. Accordingly,
the final sample contains 2118 AGNs with black hole mass
in the range 105.1<(MBH/M)<1010.3.
The FIRST catalog provides 1.4 GHz integrated radio
flux (F1.4), which is further converted to the luminosity L1.4
(in watt/Hz) at 1.4 GHz using the following equation as,
L1.4= 4π×107×D2
(1 + z)(1+α)×F1.4,(12a)
where we set the spectral index α=0.8 considering Fν=
να(Condon 1992) and DLrefers the luminosity distance.
Thereafter, we obtain the core radio luminosity LRat 5 GHz
adopting the relation (Yuan et al. 2018) given by,
log LR= (20.9±2.1) + (0.77 ±0.08) log L1.4.(12b)
where LRis expressed in erg s1. The radio luminosity at 5
GHz of our AGN sample has a range of LR= 1036.21041.2
erg s1.
Following Rusinek et al. (2020), we further calculate the
mean jet production efficiency of our sample and it is found
to be only 0.02% compared to the disc radiative effi-
ciency. Such a low jet production efficiency suggests that
the production of the jets in our sample is possibly due to
accretion flow rather than the BZ process. Moreover, we
calculate the 0.212 keV X-ray luminosity (Lx) from the
XMM-Newton data (Rosen et al. 2016, 3XMM-DR7) for 119
AGNs having both X-ray and radio flux measurements. The
Lxranges from 1 ×1041 2×1046 erg s1with a median
of 1044 erg s1. The ratio of X-ray (0.212 KeV) lumi-
nosity to radio luminosity (LRat 1.4 GHz) has a range of
Lx/LR1.5×1026.6×105with a median of 2.6×104.
5.3 Comparison of Lshock with Observed Core Radio
Emission (LR) of BH-XRBs and AGNs
In Fig. 4, we compare the shock luminosity (equivalently loss
of kinetic power) obtained due to the energy dissipation at
MNRAS 000,115 (0000)
Core radio emissions from black hole sources 9
Figure 4. Plot of kinetic power Lshock (in erg s1) released through the upper and lower surface of the disc due to the energy dissipation
at the accretion shock as function of the central black hole mass (MBH). The same is compared with the observed core radio emission
(LR) of BH-XRBs and AGNs source samples. Shaded region (light-green) represents the model estimate of Lshock obtained for accretion
rates 105.˙m.1 and 0 ak<1. Open circles denote BH-XRBs, whereas open diamonds, red dots and blue dots represent the
AGN samples taken from G¨
ultekin et al. (2019), Liu et al. (2019) and Rakshit, Stalin, & Kotilainen (2020), respectively. Open squares
and open triangles illustrate LRfor IMBH sources. Solid, dotted, dot-dashed and dashed lines indicate the results obtained from liner
regression for AGNs (L19), AGNs (R20), AGNs (G19), and BH-XRBs, respectively. See text for details.
the shock with the observed core radio luminosities of cen-
tral black hole sources of masses in the range 31010 M.
The chosen source samples contain several BH-XRBs and a
large number of AGNs. In the figure, the black hole mass
(in units of M) is varied along the x-axis, observed core
radio luminosity (LR) is varied along y-axis (left side) and
shock luminosity (Lshock) is varied along the y-axis (right
side), respectively. We use ∆Emax calculated for black holes
having spin range 0 ak0.99 (see Fig. 3), to compute the
shock luminosity Lshock which is analogous to the core radio
luminosity (LR) of the central black hole sources. Here, the
radio core is assumed to remain confined around the disk
equatorial plane (θπ/2) in the region rrs. We vary
the accretion rate in the range 105˙m1 to include
both gas-pressure and radiation pressure dominated disc
(Kadowaki, de Gouveia Dal Pino, & Singh 2015, and refer-
ences therein) and obtain the kinetic power Lshock that is de-
picted using light-green color shade in Fig. 4. The open green
circles correspond to the core radio emission from the 16
BH-XRBs while the dots and diamonds represent the same
for AGNs. The black diamonds represent 61 AGN source
samples adopted from G¨
ultekin et al. (2019). The red dots
(908 samples) denote the low-luminosity AGNs (LLAGNs)
(Liu et al. 2019) and the blue dots (1207 samples) represent
the quasars (Rakshit, Stalin, & Kotilainen 2020). At the in-
set, these three sets of AGN source samples are marked as
AGNs (G19), AGNs (L19) and AGNs (R20), respectively. It
is to be noted that we exclude Cyg X-3 from this analysis
due to the uncertainty of its mass estimate and in the fig-
ure, we mark this source using red asterisk inside open circle.
We carry out the linear regression analysis for (a) BH-XRBs,
(b) AGNs (G19), (c) AGNs (L19), and (d) AGNs (R20) and
estimate the correlation between the mass (MBH) and the
core radio luminosity (LR) of the black hole sources. We find
that LRM1.5
BH for BH-XRBs (dashed line), LRM0.98
BH for
AGNs (G19) (dot-dashed line), LRM0.38
BH for AGNs (L19)
(solid line), and LRM0.54
BH for AGNs (R20) (dotted line),
respectively. Fig. 4clearly indicates that the kinetic power
released because of the energy dissipation at the shock seems
to be capable of explaining the core radio emission from the
MNRAS 000,115 (0000)
10 Das et al.
central black holes. In particular, the results obtained from
the present formalism suggest that for ˙m.1, only a fraction
of the released kinetic power at the shock perhaps viable to
cater the energy budget required to account the core radio
emission for supermassive black holes although LRfor stellar
mass black holes coarsely follows shock luminosity (Lshock).
It is noteworthy to mention that the radio luminosity of
AGNs from G19 are in general lower compared to the same
for sources from R20 and L19 catalogs. In reality, AGNs
from L19 and R20 are mostly distant unresolved sources
where it remains challenging to separate the core radio flux
from the lobe regions. Hence, a fraction of the lobe contri-
bution is likely to be present in the estimation of their LR
values even for radio quiet AGNs. Nonetheless, we infer that
the inclusion of the L19 and R20 sources will not alter the
present findings of our analysis at least qualitatively.
5.4 LRfor Intermediate Mass Black Holes
The recent discovery by the LIGO collaboration resolves the
long pending uncertainty of the possible existence of the in-
termediate mass black holes (IMBHs) (Abbott et al. 2020).
They reported the detection of IMBH of mass 142 Mwhich
is formed through the merger of two smaller mass black
holes. This remarkable discovery establishes the missing link
between the stellar mass black holes (MBH .20M) and
the Supermassive black holes (MBH &106M). Due to lim-
ited radio observations of the IMBH sources, model com-
parison with observation becomes unfeasible. Knowing this
constrain, however, there remains a scope to predict the ra-
dio flux for these sources by knowing the disc X-ray lumi-
nosity (LX), source distance (D), and possible range of the
source mass (MBH). Following Merloni, Heinz, & di Matteo
(2003), we obtain the radio flux (F5) at 5 GHz using the
relation given by,
F5=10 ×LX
3×1031 erg s10.6
10 kpc 2
Thereafter, using equation (13), we calculate LR= 4πνF5D2
(see Table 3). As a case study, we choose two IMBH sources
whose LXand Dare known from the literature and ex-
amine the variation of LRin terms of the source mass
(MBH). Since the mass of IC 342 X-1 source possibly lie
in the range of 50 .MBH/M.103(Cseh et al. 2012;
Agrawal & Nandi 2015), we obtain the corresponding LR
values which is depicted by the open squares joined with
straight line in Fig. 4. Similarly, we estimate LRfor M82
X-1 source by varying MBH in the range 250 500 M
(Pasham, Strohmayer, & Mushotzky 2014) and the results
are presented by open triangles joined with straight line in
Fig. 4. Needless to mention that the predicted LRfor these
sources reside below the model estimates. With this, we ar-
gue that the present model formalism is perhaps adequate to
explain the energetics of the core radio emissions of IMBH
In this paper, we study the relativistic, inviscid, advective,
accretion flow around the black holes and address the im-
plication of the dissipative accretion shock in explaining the
core radio emissions from the central engines. We observe
that with the appropriate choice of the set of flow param-
eters, namely energy (E) and angular momentum (λ), the
global transonic accretion solutions pass through the shock
discontinuity (rs) around the black holes. When the shocks
are considered to be dissipative (i.e., accretion energy is be-
ing dissipated across the shock front) in nature, it reduces
the local temperature that eventually decreases the post-
shock pressure causing the shock front to settle down at
smaller radii (see Fig. 1). Hence, the size of the PSC (rs)
decreases as the level of energy dissipation (∆E) is increased.
We further point out that the shock induced global accre-
tion solutions are generic solutions and such solutions are
possible for wide range of Eand λaround weakly as well
as rapidly rotating black holes (see Fig. 2). Subsequently,
we calculate the maximum amount of accreting energy that
can be extracted at the shock and find that ∆E 1% for
Schwarszchild black hole (ak0) and ∆E 4.4% for Kerr
black hole (ak1) (see Fig. 3).
We implement our model formalism to explain the ob-
served core radio emissions emanated from the vicinity of
the black holes, in particularly when the compact core is yet
to be separated from the central region. While doing this,
we explore the entire range of the black hole masses starting
from stellar mass to Suppermassive scale. We find that for
105.˙m.1, the dissipative shock model formalism is ca-
pable to account the energy budget associated with the core
radio luminosity of large number of the central black hole
sources particularly with 2176 AGNs although 16 BH-XRBs
are also seen to comply sparsely. It appears that the present
model estimate suffers overestimation from the radio lumi-
nosity of BH-XRB sources that perhaps causes the reticence
of adopted model formalism. We also emphasize that one
would get the degenerate Lshock due to the suitable combi-
nation of ˙
Mand ∆Eas delineated in equation (10), which
remains in broad agreement with LR(see Fig. 4). In reality,
Lshock is corroborated the core radio emissions from the re-
gion which is still not decoupled from the accretion disk to
form jets and hence, Lshock > LRseems to be not unreal-
istic as only a part of Lshock contributes to radio emission
(other parts will be exhausted for (a) thermodynamical ex-
pansion and (b) against gravity). We further attempt to fill
the missing link between the BH-XRBs and AGNs includ-
ing two IMBH sources and predict LRvalues as function of
source mass (MBH) as their mass uncertainty is yet to be set-
tled. We find that LR(as function of MBH) for these IMBH
sources resides inside the domain of the model estimates (see
Fig. 4) and therefore, we indicate that the plausible explana-
tion of the core radio emission of these IMBH sources could
be understood from this model formalism.
It is noteworthy to refer that there exists alternative
scenarios involving magnetic fields where the rotational en-
ergy of the black hole is imparted to power the launching
jets (Blandford & Znajek 1977). On the contrary, a recent
study indicates that the jet driving mechanism in all astro-
physical objects possibly uses energy directly from the accre-
tion disc, rather than black hole spin (King & Pringle 2021).
MNRAS 000,115 (0000)
Core radio emissions from black hole sources 11
Table 3. Physical and observational parameters of IMBH sources.
Source Name Distance (D) X-ray luminosity (LX) Mass Range (MBH) Predicted Radio Luminosity(LR) References
(in Mpc) (in erg s1) (in M) (in erg s1)
IC342 X-1 3.93 5.34 ×1039 50 1000 3.09 ×1032 3.20 ×1033 1,2,3
M82 X-1 3.90 2.00 ×1040 300 500 2.77 ×1033 4.12 ×1033 4,5,6
References: 1: Tikhonov & Galazutdinova (2010), 2: Agrawal & Nandi (2015), 3: Cseh et al. (2012), 4: Sakai & Madore (1999), 5:
Feng & Kaaret (2009), 6: Pasham, Strohmayer, & Mushotzky (2014)
: Mass estimate of this sources is uncertain, till date.
Note: References for source distance (D), X-ray luminosity (LX) and mass are given in column 6 in sequential order.
Moreover, it is inferred that jets from BH-XRBs are linked
with the accretion states (Fender, Homan, & Belloni 2009;
Radhika et al. 2016, and references therein) indicating the
launching of jets possibly happens from the accretion disc
In addition, the accretion disk geometry is generally
depends on ˙mwhen radiative cooling processes are ac-
tive inside the disk. However, in this work, we focus in
examining the non-dissipative accretion flow and being
transonic, flow must satisfy the regularity conditions. Be-
cause of these extra conditions, out of the three constants
of the motions, namely, E,λ, and ˙m, only two are suf-
ficient (i.e.,Eand λ) to obtain the accretion solutions
(Das, Chattopadhyay, & Chakrabarti 2001) and therefore,
the half-thickness (H) of the disk remains independent on
We further indicate that the shock induced global
accretion solutions are potentially promising in explaining
the spectral state transitions of BH-XRBs (Nandi et al.
2018;Radhika et al. 2018;Aneesha, Mandal, & Sreehari
2019;Baby et al. 2020;Aneesha & Mandal 2020),
when the two-component accretion flow configu-
ration is espoused (Chakrabarti & Titarchuk 1995;
Smith et al. 2001;Smith, Heindl, & Swank 2002;
Mandal & Chakrabarti 2007;Iyer, Nandi, & Mandal
2015). With this, we further infer that the shock tran-
sition radius perhaps be visualized as the inner edge of
the truncated disc (Esin, McClintock, & Narayan 1997;
Done, Gierli´nski, & Kubota 2007;Kylafis et al. 2012).
Finally, we point out the limitations of the present
model formalism. In our analysis, we ignore viscosity, mag-
netic fields and various radiative processes to avoid com-
plexity, although these physical processes are likely to
be relevant in the context of accretion physics. In ad-
dition, the accreting matter may dissipate both angular
momentum and accretion rate from the post-shock region
when jets/outflows are present (Fukumura & Kazanas 2007;
Takahashi & Takahashi 2010). However, the mechanisms re-
sponsible for the mass loss and angular momentum loss from
the disc still remain inconclusive. In addition, the estima-
tion of mass loss generally depends on the outflow geom-
etry which is again largely model dependent (Chakrabarti
1999;Das et al. 2001;Aktar, Das, & Nandi 2015, and refer-
ences therein). Hence, in this work, we only focus in studying
the accretion flow for simplicity and therefore, the obtained
shocked-induced global accretion solutions remain indepen-
dent on accretion rate ( ˙m) as radiative cooling processes are
neglected, although ˙meventually regulates the estimation of
Lshock. Because of this, it remains theoretically infeasible to
describe the physical states of accretion flow as well as its
geometrical morphology around black holes while explain-
ing the core radio luminosity by means of energy dissipation
across the shock front. Considering all these issues, we argue
that the overall findings of the present paper are expected
to remain unaltered at least qualitatively. In addition, we
point out that we restrict ourselves while carrying out the
LRMBH correlation analysis without invoking disc X-ray
luminosity (LX) due to the lack of X-ray observations for
large number of chosen source samples. We plan to take up
these issues for our future works and will be reported else-
The data underlying this article are available in the pub-
lished literature.
We thank the anonymous reviewers for constructive com-
ments and useful suggestions that help to improve the
quality of the paper. SD thanks Science and Engineering
Research Board (SERB), India for support under grant
MTR/2020/000331. SD also thank Department of Physics,
IIT Guwahati for providing the infrastructural support to
carry out this work. AN and SS thanks GH, SAG; DD,
PDMSA and Director, URSC for encouragement and con-
tinuous support to carry out this research. IKD thanks the
financial support from Max Planck partner group award at
Indian Institute Technology of Indore (MPG-01).
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