# Accretion disk around Reissner-Nordström black hole coupled with a nonlinear
electrodynamics field

G. Abbas a,b<EMAIL_ADDRESS>H. Rehman a<EMAIL_ADDRESS>Tao
Zhu c,d<EMAIL_ADDRESS>Qiang Wu c,d<EMAIL_ADDRESS>G. Mustafa e,f
<EMAIL_ADDRESS>a National Astronomical Observatories, Chinese Academy
of Sciences, Beijing 100101, China
b Department of Mathematics, The Islamia University of Bahawalpur, Bahawalpur
Pakistan
c Institute for Theoretical Physics and Cosmology, Zhejiang University of
Technology, Hangzhou, 310032, China
d United Center for Gravitational Wave Physics (UCGWP), Zhejiang University of
Technology, Hangzhou, 310032, China e United Center for Gravitational Wave
Physics (UCGWP), Zhejiang University of Technology, Hangzhou, 310023, China
f New Uzbekistan University, Mustaqillik Ave. 54, Tashkent 100007, Uzbekistan

###### Abstract

The phenomenon by which matter accumulates in the vicinity of a huge celestial
object is known as accretion. The gravitational energy is excreted as a
consequence of infalling matter onto compact objects. The accretion procedure
around celestial bodies like neutron stars, white dwarfs, and black holes has
considerable importance because of its ability to transform gravitational
energy into radiation. This study investigates the particle’s geodesic motion
and accretion around the spherically symmetric Reissner-Nordström black hole
coupled with a nonlinear electrodynamics field utilizing isothermal fluid. The
formation of the disc-like structure in the accretion process arises from the
geodesic motion exhibited by particles near the black hole. The circular
orbits, radiant flux energy, radioactive efficiency, and radiant temperature,
can be determined. Our study focuses on the examination of particles
exhibiting stable circular orbits within the equatorial plane. We analyze the
perturbations experienced by particles throughout employing restoring forces
and the oscillatory behavior of the particles around a compact object. We
conduct an analysis of the fluid’s critical flow and maximum accretion rate.
Our results show how the black hole parameter $\zeta$ and charge $q$ affect
the circular geodesic of particles and the maximum accretion rate of the
Reissner-Nordström black hole coupled with nonlinear electrodynamics.

##  I Introduction

The theory of general relativity (GR) speculates on the presence of black
holes (BHs) as mysterious objects. The extremely strong gravitational field in
the universe is considered to have originated from BH. Additionally, it is
believed that BHs possess strong magnetic fields and spin. In light of these
characteristics, BHs are the ideal astrophysical laboratory for studying the
nature of gravity and the matter around it. Based on the examination of
observational evidence, empirical data has recently confirmed the existence of
BH. The first accomplishment represented the discovery of gravitational waves
arising due to the collision of two BHs in a binary system, as observed
through the collaborative work of LIGO and Virgo a1 . Another significant role
of the Event Horizon Telescope is its utilization of baseline Interferometry
to capture the first images of the BH shadow of M87 a2 ; a3 , as well as
revealed image of Sgr A∗ a4 .

It is believed that cosmic entities, such as BHs, undergo mass accumulation
via the phenomenon of accretion. They might also serve to analyze modified
theories of gravity. The existence of an accretion disc is an essential
element in pursuing the higher accretion rate encompassing these compact
objects. Diffuse matter creates the accretion disc and emits energy by slowly
spiraling into a centrally condensed object. Accretion is the process by which
a fluid nearby attracts particles to a compact object like a BH. Whenever the
fluid velocity is identical to the sound speed, these particles must pass
through the critical point. The fluid is projected onto its central mass at
supersonic speeds. The BH mass needs to be raised as a result of this event.
a5 . It is fascinating to analyze numerous usual radii as a consequence of
examining the particles’ geodesic structure near the BH, such as the innermost
stable circular orbit (ISCO) and marginally bound orbit ($r_{mb}$). In the
examination of BH accretion discs, the considered radii are the significant
factors.

The ISCO is associated with the inner boundary of the accretion disc around a
BH, and their radii can be used to compute the energy emission efficiency,
which is a measure of how quickly energy from the rest mass turns into
radiation. The locations of unstable or stable circular orbits correspond to
the greatest or lowest value of the effective potential, accordingly.
According to Newtonian theory, it is believed that the ISCO does not have a
minimum radius. This is supported by the observation that the ISCO can assume
any radius once the effective potential reaches its smallest value for all
possible values of angular momentum a8 . For any minimum or maximum value of
the angular momentum, the effective potential in GR and particles rotating
near the Schwarzschild BH comprises two extremes, so it corresponds with the
two points. One can explored ISCO at $r=3r_{g}$ a8 ; a9 , where $r_{g}$
denotes the Schwarzschild radius. In b2  and b3 , researchers studied the
effects of ISCO in the vicinity of Kerr BH and introduced these
characteristics in GR.

Thorne and Novikov b4  determined the Kerr and Schwarzschild BH accretion
discs efficiency. In b6  Johannsen created the accretion discs around such
BHs, while Johannsen and Psaltis b5  presented the Kerr-like metric. The
geodesic structure and spherical orbits of charged particles near revolving,
weakly magnetic BHs have been identified by Tursunov et al. b7 . Since the
particles in the accretion disc revolve in stable orbits, oscillations in the
radial, as well as vertical directions with epicyclic frequencies, arise if
the particles are perturbed.

Because of this, understanding orbital and epicyclic frequencies is important
for understanding the mechanisms of the accretion discs that surround BHs.
Moreover, the accretion disc and geodesic structure have been analyzed in the
literature for various BHs in b8 -ca7 . The universal implications of non-
linear electrodynamics (NED) theory, with the aim of examining the problem of
universal evolution, as suggested by the Born-Infeld theory a63a -a63c . The
study has emphasized the significance of NED in the field of cosmology,
particularly regarding the time transition that both microscopic and
macroscopic regions experience. In the last few years, there has been a
significant amount of interest in cosmological models that incorporate NLED,
as evidenced by the attention obtained in various studies a63d -a63f . The
study of the NED phenomenon in celestial objects has experienced a notable
expansion as a result of noteworthy findings a63g -a63k . The remarkable
characteristics of Einstein’s gravitational solutions and the NED field are
revealed when examining their implications within the framework of the Big
Bang cosmological model. The potential significance of NED fields in the
cosmos cannot be understated. To have a comprehensive understanding of these
solutions, it is essential to recognize the relationship between powerful NED
fields. Previous research has investigated BHs characterized by many horizons
within the framework of NED fields a63l -a63p . Recently, the nonlinear BH
(RN-BH coupled with the NED parameter $\zeta$) solution has been computed in
the framework of the NED field given in Ref. ac1 . Also, the first law of
thermodynamics Smarr formula, and the physical characteristics of this BH are
investigated in ac1 .

With the above motivations, this paper aims to investigate the properties of
the circular geodesic and accretion disc surrounding RN-BH coupled with the
NED parameter $\zeta$. To be conservative, we restrict our analysis to the
polar coordinate system and the equatorial plane and circular orbits and
calculate in detail the effects of the NED parameter $\zeta$ on $r_{ph}$,
$r_{mb}$, and $r_{isco}$. Moreover, the critical accretion is calculated using
certain dynamic isothermal fluid parameters. The paper will be completed in
the following manner. In Sec. II, we present a brief review of the RN-BH
space-time coupled with the NED parameter $\zeta$. Sec. III is devoted to
discussions on the general formulation for particle movement in the given
subsections such as flux radiant energy, circular motion, oscillations, and
stable circular orbits. In Sec. IV and its subsections, we determine the
generic formulas for numerous dynamical parameters, critical flow speed,
accretion for an isothermal fluid, and accretion rate. In Sec V, we examined
the solution of the RN-BH coupled with the NED parameter $\zeta$ and a
circular geodesic in the equatorial plane. In Sec 6 we discuss the summary of
this article.

##  II Black hole spacetime in the NED model

The action of the NED model is given by ac1

$S=\int d^{4}x\sqrt{-g}\left(\frac{R}{16\pi G}+\mathcal{L(F)}\right),$ (1)

where $G$ is the gravitational constant, $R$ indicates the Ricci scalar of the
spacetime, and ${\cal L}(F)$ is the NED Lagrangian which is defined as

$\displaystyle{\cal L}({\cal F})=-{\cal F}-\zeta\sqrt{-{\cal F}},$ (2)

with

$\displaystyle{\cal F}\equiv\frac{1}{4}{F}_{\mu\nu}{F}^{\mu\nu}$ (3)

being the Maxwell invariant and $\zeta$ the coupling constant. For later
convenience, hereafter we set $G=1$. To obtain the spherical symmetric BH
solution from Eq. (1), one can start with the following metric ansatz,

$ds^{2}=-f(r)dt^{2}+\frac{dr^{2}}{f(r)}+r^{2}d\Omega^{2},$ (4)

where

$\displaystyle d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\varphi^{2},$ (5)

and $\psi(r)$ is the metric function. The Einstein field equation can be
written as

$G^{\nu}_{\mu}=8\pi T^{\nu}_{\mu},$ (6)

where $G^{\nu}_{\mu}$ is the Einstein tensor and $T^{\nu}_{\mu}$ is the
energy-momentum tensor of the nonlinear electromagnetic field which is of the
form ac1

$T^{\nu}_{\mu}=\frac{1}{4\pi}(\mathcal{L}\delta^{\nu}_{\mu}-\mathcal{L_{F}}F_{\mu\gamma}F^{\nu\gamma}),$
(7)

where $\mathcal{L_{F}}=\frac{\partial\mathcal{L}}{\partial\mathcal{F}}$. The
four components of Einstein’s field equations are consistent and practically
reduce to $G^{0}_{0}=8\pi T^{0}_{0}$ which can be explicitly written as

$\frac{rf^{\prime}(r)+f(r)-1}{r^{2}}=-\frac{q^{2}}{r^{4}}-\frac{4\zeta
q\sqrt{q}}{3r}\ln(r),$ (8)

from (8) after some manipulation. By solving this equation one can obtain the
following metric

$f(r)=1-\frac{2M}{r}+\frac{q^{2}}{r^{2}}-\frac{4q\sqrt{q}\zeta}{3r}\ln(r).$
(9)

##  III General formulation for the geodesic motion of test particles

This section establishes the general formulation for the geodesic motion of
the massive test particles by examining the RN-BH coupled NED parameter
$\zeta$, which follows timelike geodesics. We assume $\xi_{t}=\partial_{t}$
and $\xi_{\phi}=\partial_{\phi}$ are killing vectors associated with
fundamental constants, indicated as $E$ and $L$ (conserved energy and angular
momentum) associated with the specified trajectory

$E=-g_{\mu\nu}\xi_{t}^{\mu}u^{\nu}\equiv-u_{t},$ (10)

and

$L=g_{\mu\nu}\xi_{\phi}^{\mu}u^{\nu}\equiv u_{\phi},$ (11)

where $u^{\mu}=\frac{dx^{\mu}}{d\tau}=(u^{t},u^{r},u^{\theta},u^{\phi})$ is
the four-velocity vector with $\tau$ being the affine parameter of the
timelike geodesics. For timelike geodesics, the four-velocity vector $u^{\mu}$
fulfills the normalization condition $u^{\mu}u_{\mu}=1$ and then one obtains

$[g_{rr}(u^{r})^{2}+g_{\theta\theta}(u^{\theta})^{2}]=[1-g^{tt}(u_{t})^{2}-g^{\phi\phi}(u_{\phi})^{2}].$
(12)

From Eqs. (10), (11) and (12), in the equatorial plane (i.e.
$\theta=\frac{\pi}{2}$), we have

$u^{t}=-\frac{E}{f(r)},$ (13) $u^{\theta}=0,$ (14)
$u^{\phi}=-\frac{L}{r^{2}},$ (15)

and

$u^{r}=\sqrt{-f(r)\left(1-\frac{E^{2}}{f(r)}+\frac{L^{2}}{r^{2}}\right)}.$
(16)

Then Eq. (16) gives

$(u^{r})^{2}+V_{eff}=E^{2},$ (17)

where

$V_{eff}=f(r)\left[1+\frac{L^{2}}{r^{2}}\right],$ (18)

is the effective potential. From the above result, it is obvious that the
effective potential relies on the radial distribution, angular momentum, and
space-time parameter $f(r)$. The effective potential is incredibly helpful in
the geodesic motion of the particles due to its ability to identify the
position of ISCO by analyzing the local extrema of the effective potential.

###  III.1 Circular Motion of Test Particles

Let us first consider the circular motion of the massive particles in the
equatorial plane $(\theta=\pi/2)$. For circular motion, one has the conditions

$\displaystyle u^{r}=\frac{dr}{d\tau}=0\;\;{\rm
and}\;\;\dot{u}^{r}=\frac{d^{2}r}{d\tau^{2}}=0.$ (19)

With these two conditions and by using Eq. (17), one has

$\displaystyle V_{eff}=E^{2}\;\;{\rm and}\;\;\frac{d}{dr}V_{eff}=0.$ (20)

Then one can obtain the angular velocity $\Omega_{\phi}$, specific energy $E$,
and the specific angular momentum $L$ associated with the test particle that
are provided respectively by

$\displaystyle\Omega_{\phi}^{2}=\frac{1}{2r}f^{\prime}(r),$ (21)
$\displaystyle E^{2}=\frac{2f^{2}(r)}{2f(r)-rf^{\prime}(r)},$ (22)

and

$\displaystyle L^{2}=\frac{r^{3}f^{\prime}(r)}{2f(r)-rf^{\prime}(r)}.$ (23)

Frome Eqs. (22) and (23), $E$ and $L$ should be real if

$\displaystyle 2f(r)-rf^{\prime}(r)>0.$ (24)

From the above expression, the specific area of the circular orbit can be
investigated. So, the inequality (24), is essential for the existence of
circular orbits. For bound orbits, one requires $E^{2}<1$ while the marginally
bound orbit corresponds to $E^{2}=1$. Thus the marginally bound orbit
satisfies

$\displaystyle 2[f(r)-1]f(r)+rf^{\prime}(r)=0.$ (25)

This equation determines the radius of the marginally bound orbit. On the
other hand, it is easy to see that Eqs. (22) and (23) diverge if

$\displaystyle-rf^{\prime}(r)+2f(r)=0.$ (26)

With this equation, one can determine the photon sphere radius which is
essential for the investigation of gravitational lensing.

###  III.2 Radiant Energy Flux and Circular Orbits

The presence of stable circular orbits is dependent upon the local minima of
the effective potential, which is attained when
$\frac{d^{2}V_{eff}}{dr^{2}}>0$. From Eq. (18), we have

$\displaystyle\frac{d^{2}V_{eff}}{dr^{2}}=\left(1+\frac{L^{2}}{r^{2}}\right)f^{\prime\prime}(r)-\frac{4L^{2}}{r^{3}}f^{\prime}(r)+\frac{6L^{2}}{r^{4}}f(r).$
(27)

By employing the requirements $V_{eff}=0$, $\frac{dV_{eff}}{dr}=0$, and
$\frac{d^{2}V_{eff}}{dr^{2}}=0$, one is able to calculate the radius of ISCO,
i.e.,$r_{isco}$. Furthermore, the process of accretion is possible when
$r<r_{isco}$. When particles fall from a state of rest to an infinite
distance, accreting onto compact objects, they emit gravitational energy,
which is converted into radiation. In h2 , the expression for the energy flux
radiating across the accretion disc is formulated on the basis of the angular
velocity $\Omega_{\phi}$, the specific energy $E$, and the specific angular
momentum $L$ as

$\displaystyle
K=-\frac{\dot{M}\Omega_{\phi},_{r}}{{4\pi\sqrt{-g}(E-L\Omega_{\phi})^{2}}}\int^{r}_{r_{isco}}(E-L\Omega_{\phi})L,_{r}dr,$
(28)

where radiant flux is represented by $K$, mass accretion rate is $\dot{M}$,
$\Omega_{\phi},_{r}\equiv\frac{d\Omega_{\phi}}{dr}$ and $g$ is the determinant
of the metric tensor $g_{\mu\nu}$. $g$ is given by

$\displaystyle g=\det(g_{\mu\nu})=-r^{4}\sin^{2}\theta,$ (29)

In order to properly analyze our findings within the equatorial plane, we
establish the relationship $\sin\theta=\sin\frac{\pi}{2}=1$. By applying Eqs.
(21-23), we obtain

$\displaystyle K(r)$ $\displaystyle=$ $\displaystyle\frac{-\dot{M}}{4\pi
r^{4}}\sqrt{\frac{r}{2f^{\prime}(r)}}$
$\displaystyle\times\frac{[rf^{\prime}(r)-2f(r)][rf^{\prime\prime}(r)-f^{\prime}(r)]}{[2f(r)+rf^{\prime}(r)]^{2}}\int^{r}_{r_{isco}}F(r)dr,$

where the function $F(r)$ is given by

$\displaystyle F(r)$ $\displaystyle=$
$\displaystyle\sqrt{\frac{r}{2f^{\prime}(r)}}\Big{[}rf^{\prime}(r)+2f(r)\Big{]}$
(31) $\displaystyle\times\frac{rf(r)f^{\prime\prime}(r)-2rf^{\prime
2}(r)+3f(r)f^{\prime}(r)}{[2f(r)-rf^{\prime}(r)]^{2}}.$

We assume that the accretion disc is in a state of thermal equilibrium, thus
according to the Stefan-Boltzmann law, one has $K(r)=\sigma T^{4}(r)$ with
$\sigma$ being the Stefan-Boltzmann constant, which is the relation between
energy flux and temperature. Consequently, the radiation produced from the
accretion disc is presumed to possess characteristics similar to those of
black body radiation. By considering the thermal black body radiation, it is
simple to determine the temperature distribution of the accretion disc by
utilizing the given equation, from which one can figure out disc luminosity
$L(\nu)$ where $\nu$ is the frequency of the radiated photons. The disc
luminosity is determined by R3

$\displaystyle
L(\nu)=\frac{8\cos\gamma}{\pi}\int^{r_{f}}_{r_{i}}\int^{2\pi}_{0}\frac{\nu_{e}^{3}r}{e^{\frac{\nu_{e}}{T}}-1}d\phi
dr.$ (32)

where $\gamma$ is the inclination angle of the accretion disc. From the
preceding result, we can see that the flux energy is denoted by
$I(\nu)=L(\nu)/(4\pi)$. The maximum efficiency $\eta^{*}$ can be obtained by

$\displaystyle\eta^{*}=1-E_{isco}.$ (33)

Here, $E_{isco}$ is the energy of the particles at ISCO. This proceeding
relation holds when all emitted photons have the ability to escape infinity.
When a fluid element experiences a perturbation, the resulting particles move,
which corresponds to a circular orbit within the plane $\theta=\frac{\pi}{2}$.

###  III.3 Oscillations

In accretion processes, numerous types of oscillatory motion are seen as a
result of restoring forces. The oscillatory motion in both the horizontal and
vertical directions arises from the influence of restoring forces acting upon
perturbations within the accretion discs. In an accretion disc, the number of
restoring forces arises from the rotational motion of the disc in the presence
of a vertical gravitational field.

When a fluid element moves radially, it returns to its state of equilibrium
through the rotational motion of the fluid by virtue of the presence of a
restoring force. It is worth noting that the gravitational force within
accretion discs serves as a counterbalance to the centrifugal force due to
central objects. The fluid element is dragged outside or inside and returned
towards the original radius utilizing $\Omega_{r}$ epicyclic frequency,
depending on whether the latter exceeds the former or vice versa. When the
fluid element encounters a vertical perturbation within the plane
$\frac{\pi}{2}$, the field of gravitation pinches the elements that are
perturbed, causing it to return to its original equilibrium state. The element
of fluid exhibits harmonic oscillations within the equatorial plane due to the
presence of a restoring force, characterized by a vertical epicyclic frequency
$\Omega_{\theta}$. Three different types of motion, harmonic vertical motion
with a vertical frequency, circular motion with an orbital frequency, and
radial motion with a radial frequency, are responsible for the behavior of
particles within the accretion disc. Consequently, within the equatorial
plane, we can examine the redial motion and vertical motion in the vicinity of
circular orbits.

Now, let us investigate the radial and vertical motions that are represented
by $\frac{1}{2}\left(\frac{dr}{dt}\right)^{2}=V_{eff}^{(r)}$ and
$\frac{1}{2}(\frac{d\theta}{dt})^{2}=V_{eff}^{(\theta)}$, exhibited by the
particles under consideration, where using Eq. (12) for explain the radial and
vertical motions we consider $u^{r}=0$, $u^{\theta}=0$, respectively.
Considering $u^{r}=\frac{dr}{d\tau}=\frac{dr}{dt}u^{t}$ and
$u^{\theta}=\frac{d\theta}{d\tau}=\frac{d\theta}{dt}u^{t}$, we can deduce the
following equations,

$\displaystyle\frac{1}{2}\left(\frac{dr}{dt}\right)^{2}=-\frac{1}{2}\frac{f^{3}(r)}{E^{2}}\left[1-\frac{E^{2}}{f(r)}+\frac{L^{2}}{r^{2}\sin^{2}(\theta)}\right]=V_{eff}^{(r)}$
(34)

and

$\displaystyle\frac{1}{2}\left(\frac{d\theta}{dt}\right)^{2}=-\frac{1}{2}\frac{f^{2}(r)}{E^{2}r^{2}}\left[1-\frac{E^{2}}{f(r)}+\frac{L^{2}}{r^{2}\sin^{2}\theta}\right]=V_{eff}^{(\theta)}.$
(35)

In the equatorial plane, we shall investigate the vertical and radial
epicyclic frequencies in the vicinity of circular orbits by considering small
perturbations denoted as $\delta r$ and $\delta\theta$. Differentiating Eq.
(34) and Eq. (36) with respect to time $t$, one obtains

$\displaystyle\frac{d^{2}r}{dt^{2}}=\frac{dV_{eff}^{(r)}}{dr}.$ (36)

For a particle having a perturbation in its original radius at $r=r_{0}$,
characterized by a deviation $\delta r=r-r_{0}$, the resulting equation can be
expressed as follows

$\displaystyle\frac{d^{2}}{dt^{2}}(\delta
r)=\frac{d^{2}V_{eff}^{(r)}}{dr^{2}}(\delta
r)\Rightarrow(\delta\ddot{r})+\Omega_{r}^{2}(\delta r)=0,$ (37)

where $\Omega_{r}^{2}\equiv-\frac{d^{2}}{dr^{2}}V_{eff}^{(r)}$ and dots
denotes derivatives with respect to $t$. Through an analogous methodology,
when considering a perturbation in the vertical direction
$\delta\theta=\theta-\theta_{0}$, we arrive at the following result

$\displaystyle\frac{d^{2}(\delta\theta)}{dt^{2}}=\frac{{d^{2}V_{eff}}^{(\theta)}}{dr^{2}}(\delta\theta)\Rightarrow(\delta\ddot{\theta})+\Omega_{\theta}^{2}(\delta\theta)=0,$
(38)

where $\Omega_{\theta}^{2}\equiv-\frac{d^{2}}{d\theta^{2}}V_{eff}^{(\theta)}$.
In the Equatorial plane, Eqs. (34) and (36) lead to

$\displaystyle\Omega_{r}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{2E^{2}r^{4}}\Big{\\{}\big{[}(r^{2}+L^{2})3f(r)-2E^{2}r^{2}\big{]}r^{2}f(r)f^{\prime\prime}(r)$
(39)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+2r^{2}\big{[}(r^{2}+L^{2})3f(r)-E^{2}r^{2}\big{]}f^{\prime
2}(r)$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-6L^{2}f^{2}(r)\big{[}2rf^{\prime}(r)-f(r)\big{]}\Big{\\}},$

and

$\displaystyle\Omega_{\theta}^{2}=\frac{f^{2}(r)L^{2}}{E^{2}r^{4}}.$ (40)

The prime notation in Eq. (39) denotes differentiation with respect to the
radial coordinate $r$. The subsequent section presents a comprehensive
investigation of the fundamental dynamical equations governing RN-BH coupled
with NED.

##  IV Basic Dynamical Equations

In this section, we performed an analysis of the fundamental formalism for the
accretion process around RN-BH coupled with NED. For this purpose, we follow
the fundamental formalism that was established by Babichev et al. R4 ; R5 .
Let us start by considering an ideal fluid that is characterized by its
energy-momentum tensor

$\displaystyle T^{\mu\nu}=(\rho+p)u^{\nu}u^{\mu}+g^{\mu\nu}p.$ (41)

The quantities denoted as $p$ and $\rho$ correspond to the pressure and energy
density of the fluid, respectively. In the equatorial plane, the four-velocity
$u^{\mu}$ can be described as follows

$\displaystyle u^{\mu}=\frac{dx^{\mu}}{d\tau}=(u^{t},u^{r},0,0),$ (42)

where $\tau$ is the proper time. By combining the above equation with the
normalization condition $(u^{\mu}u_{\mu}=1)$, we obtain

$\displaystyle u^{t}=\frac{\sqrt{f(r)+(u^{r})^{2}}}{f(r)}.$ (43)

The requirement $u^{t}>0$ indicates flow is forward, whereas for accretion
(flow is inward), the assumption $u^{r}<0$ holds. In order to analyze the
accretion process, it is necessary to calculate the conservation equation of
energy-momentum and conservation equation of particle-number. The conservation
of energy-momentum tensor reads $T_{;\mu}^{\mu\nu}=0$, which leads to

$\displaystyle
T_{;\mu}^{\mu\nu}=\frac{1}{\sqrt{-g}}(\sqrt{-g}T^{\mu\nu})_{,\mu}+\Gamma_{\alpha\mu}^{\nu}T^{\alpha\mu}=0,$
(44)

where $\sqrt{-g}=r^{2}\sin\theta$, $\Gamma$ is Christoffel symbol’s of 2nd
kind and $(;)$ is the covariant derivative associated with the metric
$g_{\mu\nu}$. By utilizing the BH metric, Eq. (44) can be transformed into the
form of

$\displaystyle
T_{,r}^{10}+\frac{1}{\sqrt{-g}}T^{10}(\sqrt{-g})_{,r}+2\Gamma_{01}^{0}T^{10}=0.$
(45)

By solving Eq. (45), one has

$\displaystyle\frac{d}{dr}[(\rho+p)u^{r}r^{2}{\sqrt{f(r)+(u^{r})^{2}}}]=0.$
(46)

By performing the integration on the above equation, we are able to derive the
resulting expression

$\displaystyle(\rho+p)u^{r}r^{2}{\sqrt{f(r)+(u^{r})^{2}}}=C_{0}.$ (47)

In Eq. (47), $C_{0}$ represents the integration constant. Using the principle
of conservation coupled with the four-velocity, as expressed by the equation
$u_{\mu}T_{;\nu}^{\mu \nu}=0$, we attain

$\displaystyle(\rho+p)u_{;\nu}^{\mu}{u_{\mu}u^{\nu}}+(\rho+p)_{,\nu}u_{\mu}{u^{\mu}u^{\nu}}$
$\displaystyle+(\rho+p){u_{\mu}u^{\mu}}u_{;\nu}^{\nu}+p_{,\nu}g^{\mu\nu}u_{\mu}+p{u_{\mu}}g_{;\nu}^{\mu\nu}=0.$
(48)

Since $g_{;\nu}^{\mu\nu}=0$ and utilizing $u^{\mu}u_{\mu}=1$, one gets

$\displaystyle(p+\rho){u_{;\nu}^{\nu}}+u^{\nu}{\rho,_{\nu}}=0,$ (49)

and since $A_{;a}^{b}=\partial_{a}A^{b}+\Gamma_{ac}^{b}A^{c}$, we acquire

$\displaystyle
u^{r}\rho_{,r}+[\Gamma_{0c}^{0}u^{c}+u_{,r}^{r}+\Gamma_{1c}^{1}u^{c}+\Gamma_{2c}^{2}u^{c}+\Gamma_{3c}^{3}u^{c}](\rho+p)=0.$
(50)

From Eq. (50), we attain

$\displaystyle\frac{\rho^{\prime}}{(\rho+p)}+\frac{u^{\prime}}{u}+\frac{2}{r}=0,$
(51)

and solving it yields

$\displaystyle r^{2}u^{r}\exp\left(\int{\frac{d\rho}{p+\rho}}\right)=-C_{1},$
(52)

where $C_{1}$ is the integration constant. Assuming $u^{r}<0$, it follows that
the above expression also has a negative sign and we calculate

$\displaystyle(p+\rho)\sqrt{(u^{r})^{2}+f(r)}\exp\left(-\int{\frac{d\rho}{p+\rho}}\right)=C_{2}.$
(53)

The constant of integration, indicated as $C_{2}$ and the flux mass equation
is

$\displaystyle(\rho u^{\mu})_{;\mu}\equiv\frac{1}{\sqrt{-g}}(\sqrt{-g}\rho
u^{\mu})_{,\mu}=0.$ (54)

Then utilizing Eq. (54), we get

$\displaystyle\frac{1}{\sqrt{-g}}(\sqrt{-g}\rho
u^{\mu})_{,r}+\frac{1}{\sqrt{-g}}(\sqrt{-g}\rho u^{\theta})_{,\theta}=0.$ (55)

Note that the term $\frac{1}{\sqrt{-g}}(\sqrt{-g}\rho u^{\theta})_{,\theta}$
in Eq. (55) can be omitted since we only focus on the equatorial plane.
Therefore, the expression $\sqrt{-g}\rho u^{\mu}$ is considered to be
constant, i.e.,

$\displaystyle\rho u^{r}r^{2}=C_{3},$ (56)

where $C_{3}$ represents the integration constant.

###  IV.1 Dynamical Parameters

Figure 1: The behaviors of $u$ as a function of $r$ for different values of
electric charge $q$ and the NED paramater $\zeta$. The left panel corresponds
to a fixed $q=1$ various values of $\zeta$ and the right panel corresponds to
a fixed $\zeta=-1.5$ various values of electric charge $q$.

Let us examine the accretion of the isothermal fluids characterized by the
equation of state $p=k\rho$, where $k$ is the state function. For isothermal
fluid $p\propto\rho$ it is necessary that the speed of sound remain constant
in the accretion procedure. Moreover, from Eqs. (52), (53) and (56), one can
determine

$\displaystyle\frac{\rho+p}{\rho}\sqrt{f(r)+(u^{r})^{2}}=C_{4},$ (57)

where $C_{4}$ represents the integration constant. Considering $p=k\rho$ and
substitute it into Eq. (57), one has

$\displaystyle
u=\frac{\sqrt{A_{4}^{2}-\frac{(k+1)^{2}\left[3\left(-2Mr+q^{2}+r^{2}\right)-4\zeta
q^{3/2}r\ln r\right]}{3r^{2}}}}{k+1}.$ (58)

The graphs presented in Fig. 1 illustrate the relationship between radial
velocity $u$ and $r$. In the left panel of Fig. 1, we analyze that initially,
radial velocity increases to its maximum for a small BH radius with increasing
the coupling parameter $\zeta$ and then moves toward a decreasing trend with
increasing the BH radius. It is worthwhile to note that the maximum radial
velocity attains at $\zeta=-0.5$. It is important to observe the impact of the
coupling parameter, as radial velocity increases with increasing the coupling
parameter for the given considered domain and show the stability of the
system. In the right panel of Fig. 1, we observe that the radial velocity
attain maximum value at $q=0.1$ and decline gradually towards equilibrium
position.

Figure 2:  The behaviors of the fluid density $\rho$ as a function of $r$ for
different values of electric charge $q$ and the NED paramater $\zeta$. The
left panel corresponds to a fixed $q=1$ various values of $\zeta$ and the
right panel corresponds to a fixed $\zeta=-1.5$ various values of electric
charge $q$.

Now we can determine the density of fluid from Eq. (56), which is given by

$\displaystyle\rho=\frac{A_{3}(k+1)}{r^{2}\sqrt{A_{4}^{2}-\frac{(k+1)^{2}\left(3\left(-2Mr+q^{2}+r^{2}\right)-4\zeta
q^{3/2}r\ln(r)\right)}{3r^{2}}}}.$

The behavior of $\rho$ as a function of $r$ for different values of the
electric charge and the NED parameter $\zeta$ are presented in Fig. 2. From
the left panel of Fig. 2, it is noted that initially fluid density declines
for small BH radius and then grows rapidly toward maximum. Notably, the fluid
density converges more rapidly for the smallest value of the BH parameter
$\zeta$. Also in the right panel of Fig. 2, observed opposite behavior as in
Fig. 2.

###  IV.2 Mass Evolution

Based on astronomical investigation, it is proposed that the mass of BH
gradually varies over time due to various phenomena such as the emission of
Hawking radiation and mass accreting around the BH. The mass accretion rate of
RN-BH coupled with the NED parameter $\zeta$ can be computed as
$\dot{M}\equiv\frac{dM}{dt}=-\int T_{t}^{r}ds$ in which $ds=\sqrt{-g}d\theta
d\phi$ and also $T_{t}^{r}=(p+\rho)u_{t}u^{r}$. As a consequence, accretion
rate $\dot{M}$ is acquired by

$\displaystyle\dot{M}=-4\pi r^{2}u(p+\rho)\sqrt{u^{2}+f(r)}\equiv-4\pi C_{0}.$
(60)

By asssuming $C_{0}=-C_{1}C_{2}$ and
$C_{2}=(p_{\infty}+\rho_{\infty})\sqrt{f(r_{\infty})}$, we have

$\displaystyle\dot{M}=4\pi
C_{1}(p_{\infty}+\rho_{\infty})\sqrt{f(r_{\infty})}M^{2}.$ (61)

The evolution of time and BH mass can be obtained by considering $M_{i}$ is
the initial mass and utilizing the Eq. (61), we obtained

$\displaystyle\frac{dM}{M^{2}}=\mathcal{F}t,$ (62)

where $\mathcal{F}\equiv 4\pi C_{1}(p+\rho)\sqrt{f(r_{\infty})}$. From
Eq.(62), we have

$\displaystyle
M_{t}=\frac{M_{i}}{1-\mathcal{F}M_{i}t}\equiv\frac{M_{i}}{1-\frac{t}{t_{cr}}},$
(63)

the expression for the time accretion $t_{cr}$ is calculated by using the
formula $t_{cr}=[4\pi C_{1}(p+\rho)\sqrt{f(r_{\infty})}M_{i}]^{-1}$. According
to Eq. (63), it is evident that at $t=t_{cr}$, the BH mass increases up to
infinity within a finite time. So, the mass accretion rate of a BH is

$\displaystyle\dot{M}=4\pi C_{1}(p+\rho)M^{2}.$ (64)

Figure 3 depicts the relationship between the mass accretion rate and the
variable $r$. Under the influence of altered values of the parameter $\zeta$,
it is observed that the accretion rate will initially decline along the BH
radius $r$ and then increase to its maximum. Also, we note that the accretion
rate $\dot{M}$ grow more rapidly for the smallest value of the BH parameter
$\zeta$. Moreover, in the right panel of Fig. 3, we can see that the accretion
rate gives the same behavior along BH radius $r$ as that in the left panel of
Fig. 3, but the impact of BH charge $q$ is totally reversal to the case in the
left panel.

Figure 3: The mass accretion rate $\dot{M}$ as a function of $r$ for different
values of electric charge $q$ and the NED paramater $\zeta$. The left panel
corresponds to a fixed $q=1$ various values of $\zeta$ and the right panel
corresponds to a fixed $\zeta=-1.5$ various values of electric charge $q$.

###  IV.3 Critical Accretion

The flow of fluid is static farthest from the BH, but it starts to move and
accelerates inward due to the gravitational field exerted by the BH. When
fluid flows inward, it reaches a sonic point where the velocity of the fluid
is equivalent to the speed of sound. By utilizing Eqs. (56) and (57), we have

$\displaystyle\frac{\rho^{\prime}}{\rho}+\frac{u^{\prime}}{u}+\frac{2}{r}=0,$
(65)

and

$\displaystyle\frac{\rho^{\prime}}{\rho}\left[\frac{d\ln(p+\rho)}{d\ln\rho}-1\right]+\frac{uu^{\prime}}{u^{2}+f(r)}+\frac{1}{2}\frac{f^{\prime}(r)}{u^{2}+f(r)}=0.$
(66)

From Eq. (66), we attain

$\displaystyle\frac{d\ln u}{d\ln r}=\frac{D_{1}}{D_{2}},$ (67)

where

$\displaystyle D_{1}=\frac{rf^{\prime}(r)}{2(u^{2}+f(r))}-2V^{2},$ (68)

and

$\displaystyle D_{2}=V^{2}-\frac{u^{2}}{u^{2}+f(r)}.$ (69)

From Eqs. (67)-(69), we attain

$\displaystyle V^{2}=\frac{d\ln(p+\rho)}{d\ln\rho}-1.$ (70)

To determine the critical points, we assume $D_{1}=D_{2}=0$ and we obtained

$\displaystyle V_{c}^{2}=\frac{rf^{\prime}(r)}{4f(r)+rf^{\prime}(r)},$ (71)

and

$\displaystyle u_{c}^{2}=\frac{1}{4}rf^{\prime}(r).$ (72)

The index $c$ is referred to as the critical point. Note that the right-hand
side of Eq. (70) is always positive. We compute the critical radius range by
using the subsequent expression

$\displaystyle 4f(r)+rf^{\prime}(r)>0.$ (73)

By utilizing Eq. (58), we acquire

$\displaystyle c_{s}^{2}=C_{4}\sqrt{[u^{2}+f(r)]^{-1}}-1.$ (74)

The relation that represents the speed of sound is given by
$c_{s}^{2}=\frac{dp}{d\rho}$.

##  V Circular equatorial geodesics

a)  b)
c)

Figure 4:  The illustration of $V_{eff}$ is a function of $r$. (a) for
$\zeta=-0.2$, $q=0.1$, and altered values of $L$ (b) for $q=1$, $L=10$ and
various values of $\zeta$ c) for $\zeta=-1.5$, $L=10$ and distinct values of
$q$.

The effective potential is necessary to examine circular geodesics in the
plane $\theta=\frac{\pi}{2}$, which is obtained from Eq. (18) as

$\displaystyle V_{eff}=\left(1-\frac{2M}{r}+\frac{q^{2}}{r^{2}}-\frac{4\zeta
q\sqrt{q}\ln r}{3r}\right)\left(1+\frac{L^{2}}{r^{2}}\right).$

The graphs presented in Fig. 4 illustrate the behaviors of the effective
potential as a function of the radial coordinate $r$ for different values of
the specific angular momentum $L$, electric charge $q$, and the NED parameter
$\zeta$. In Fig. 4 a, one can observe the first extrema exist at $L=5$ and no
other extrema arise for $L<5$. Additionally, the effective potential increases
when angular momentum $L$ rises. The black dot within Fig. 4 a gives the
precise location of the ISCO, situated at $r=5.97347$. Furthermore, the
effective potential $V_{eff}$ reveals two extrema for higher values of $L$.
The stable and unstable circular orbits lie at the minimum and maximum of
$V_{eff}$, respectively. From Fig. 4 b, it is clear that as the value of BH
parameter $\zeta$ rises, the effective potential decreases. In addition, from
Fig. 4 c, we observe that how the BH charge $q$ affects the effective
potential $V_{eff}$ along $r$. In Fig. 4 c, we can see $V_{eff}$ directly
proportional to the charge $q$. We are currently interested in calculating
ISCO for RN-BH coupled with NED because it forms the inner edges of the
accretion disc, so the role of ISCO is significant. Unfortunately, we are
unable to determine the ISCO analytically. Therefore, we will have to turn to
numerical approaches by utilizing the general formula for ISCO provided in
reference [1]. Whenever $\zeta=-0.2$, $M=1$, and the charge $q=0.1$, the ISCO
of BH is $r_{isco}=5.97347$. More detail regarding this can be found in Table
1.

In order to thoroughly investigate the accretion process, the ISCO is
significant. It is also mandatory to conduct an analysis of other radii to
obtain a comprehensive understanding. As previously mentioned, a circular
orbit is present when the value of $r>r_{ph}$. The particle’s motion will
demonstrate instability for small perturbations when the radial distance lies
within the range of $r_{ph}<r<r_{isco}$. This relationship suggests that
particles either escape to infinity or are dragged into the BH. If
$r>r_{isco}$., the particle proceeds to move in stable circular orbits. Also,
the photon sphere $r_{ph}$, circular orbit $r_{isco}$ and marginally bound
orbit $r_{mb}$ are given in Table 1.

(a) (b) (c) (d) .

Figure 5:  The profile of energy illustrated in left plots while in right
plots angular momentum is depicted along $r$ for numerous values of BH
parameters $\zeta$ and $q$.

In the plane $\theta=\frac{\pi}{2}$, one can derive the formulas for the
following quantities

$\displaystyle
E^{2}=\frac{1}{3r^{2}}\frac{\left[3\left(r(r-2M)+q^{2}\right)-4\zeta
q^{3/2}r\ln r\right]^{2}}{3r(r-3M)+2\zeta q^{3/2}r-6\zeta q^{3/2}r\ln
r+6q^{2}},$ $\displaystyle L^{2}=\frac{3r^{2}\left(3Mr-2\zeta q^{3/2}r+2\zeta
q^{3/2}r\ln r-3q^{2}\right)}{3r(r-3M)+2\zeta q^{3/2}r-6\zeta q^{3/2}r\ln
r+6q^{2}},$ (77)

and

$\displaystyle\Omega_{\phi}^{2}=\frac{3Mr-2\zeta q^{3/2}r+2\zeta q^{3/2}r\ln
r-3q^{2}}{3r^{4}}.$ (78)

Figure. 5 depicts the specific energy and specific angular momentum profile
along $r$. Also, the effects of BH parameters $\zeta$ and charge $q$ have been
investigated in the given plots. Presently, our focus to analyze the following
quantities $E_{isco}$, $L_{isco}$, $\Omega_{isco}$, and $l_{isco}$ in ISCO.
However, it is worth noting that an analytical identification of all these
quantities is not possible. Therefore, the numerical calculation is outlined
in the provided table.

###  V.1 Radiant energy flux

We begin to investigate the flux radiation emanating from the outermost layer
of the disc in the plane $\theta=\frac{\pi}{2}$ by the utilization of the
corresponding quantities $E$, $L$, and $\Omega_{\phi}$. The flux radiant
energy associated with the accretion disc can be investigated through Eqs.
(LABEL:c7) and (31), given as

$\displaystyle K(r)$ $\displaystyle=$
$\displaystyle-\Bigg{\\{}\sqrt{3}\dot{M}\sqrt{\frac{r^{4}}{3Mr-2\zeta
q^{3/2}r+2\zeta q^{3/2}r\ln r-3q^{2}}}\Big{(}9Mr-8\zeta q^{3/2}r+6\zeta
q^{3/2}r\ln r-12q^{2}\Big{)}$ (79)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\Big{[}3r(-2Mr-M+r^{2})+2\zeta
q^{3/2}r-2\zeta q^{3/2}r(2r+1)\ln r+3q^{2}(r+1)\Big{]}\Bigg{\\}}$
$\displaystyle~{}~{}\times\Big{[}8\pi r^{5}(3r(-2Mr+M+r^{2})-2\zeta
q^{3/2}r+2\zeta q^{3/2}r(1-2r)\ln
r+3q^{2}(r-1))^{2}\Big{]}^{-1}\int^{r}_{r_{isco}}F(r)dr,$

where

$\displaystyle F(r)$ $\displaystyle=$
$\displaystyle\Bigg{\\{}\sqrt{\frac{r^{4}}{9Mr-6\zeta q^{3/2}r+6\zeta
q^{3/2}r\ln r-9q^{2}}}$ (81)
$\displaystyle~{}~{}~{}\times\Big{[}3r(-2Mr+M+r^{2})-2\zeta q^{3/2}r+2\zeta
q^{3/2}r(1-2r)\ln r+3q^{2}(r-1)\Big{]}$
$\displaystyle~{}~{}~{}\times\Big{[}2\zeta
q^{3/2}r\ln(r)(3r(r^{2}-4M(r+2))+16\zeta q^{3/2}r-4\zeta q^{3/2}r(r+2)\ln
r+3q^{2}(r+8))$ $\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}-48\zeta
q^{3/2}r(q^{2}-Mr)+9(Mq^{2}r(r+8)+Mr^{2}(r^{2}-2M(r+2))-4q^{4})-16\zeta^{2}q^{3}r^{2}\Big{]}\Bigg{\\}}$
$\displaystyle~{}~{}~{}~{}\times\Big{(}2r^{3}(3r(-2Mr-M+r^{2})+2\zeta
q^{3/2}r-2\zeta q^{3/2}r(2r+1)\ln r+3q^{2}(r+1))^{2}\Big{)}^{-1}.$

In order to analyze the radiation flux behavior of the accretion disc
surrounding BH for numerous values of the coupling parameter $\zeta$, as
depicted in Fig. 6, it is observed that as the parameter $\zeta$ grows, the
flux energy of the accretion disc decreases.

Figure 6: The profile of energy flux $K$ along $r$, for different value of
$\zeta$.

###  V.2 Radiant temperature

It is speculated that the accretion disc is in thermal equilibrium, so the
emission of radiation to the disc follows black body radiation principles. The
Stefan-Boltzmann law, $K(r)=\sigma T^{4}$, established a relationship between
energy flux and temperature. Here, $\sigma$ represents the Stefan-Boltzmann
constant. The disc temperature is evaluated through the utilization of the BH
parameter $\zeta$. In Fig. 7, we observe the temperature distribution on a
disc for different values of $\zeta$ while keeping the parameter $q=0.1$
fixed. It is observed that the disc temperature drops as the parameter $\zeta$
assumes progressively larger values.

Figure 7: The radiation temperature profile for various values of $\zeta$.

###  V.3 Radiative efficiency

The emission of radiation produced by the transformation of gravitational
energy arises together with the gradual inward spiraling of the material
comprising the disc towards its central region. The specific energy within the
ISCO radius can be used to figure out the radiative efficiency, which is the
ability of the central entity to convert mass at rest into radiation

$\displaystyle\eta=1-E_{isco}.$ (82)

The numerical outcomes of ISCO, the photon sphere radius $r_{ph}$, the
marginally bound orbit $r_{mb}$, the specific energy at ISCO denoted as
$E^{2}_{isco}$, the specific angular momentum at ISCO indicated as
$L^{2}_{isco}$, the specific angular velocity at ISCO referred to as
$\Omega^{2}_{isco}$, the specific angular momentum at ISCO symbolized as
$l^{2}_{isco}$, the maximum energy flux and the maximum temperature
distribution are presented in Table 1.

Table 1: The numerical results are presented for $M=1$, $q=0.1$ and various values of BH parameter $\zeta$. $\zeta$ | ISCO | $r_{mb}$ | $r_{ph}$ | $E^{2}_{isco}$ | $L^{2}_{isco}$ | $\Omega^{2}_{isco}$ | $l^{2}_{isco}$ | $K_{max}(r)$ | $T_{max}(r)$ | $\eta$
---|---|---|---|---|---|---|---|---|---|---
-0.3 | 5.967814 | 3.98041 | 2.9789 | 0.890526 | 11.7976 | 0.00467366 | 13.248 | $2.559\times 10^{13}$ | $1.439\times 10^{5}$ | 0.0563234
-0.2 | 5.9734741 | 3.98355 | 2.98369 | 0.889895 | 11.8536 | 0.00466815 | 13.3203 | $2.551\times 10^{13}$ | $1.43\times 10^{5}$ | 0.0566574
-0.1 | 5.979193 | 3.98674 | 2.98849 | 0.889267 | 11.91 | 0.00466253 | 13.393 | $2.54\times 10^{13}$ | $1.4215\times 10^{5}$ | 0.0569904

###  V.4 Epicyclic frequencies

In the presence of perturbations within the equatorial plane, when the
particles move along a circular orbit, that will produce small oscillations in
the vertical and radial directions. The radial and vertical epicyclic
frequencies are determined from Eqs. (39) and (40). Since the expressions of
these quantities are a little bit lengthy, we are not going to present them
explicitly here. Instead, we calculate them numerically and presented the
results in In Fig. 8. In Fig. 8, the profile of epicyclic frequencies
$(\Omega_{\theta},\Omega_{r})$ can be examined along the dimensionless radial
coordinate $r$ for numerous values of coupling parameter $\zeta$ and charge
$q$ of BH. In Fig. 8, the cyan curve indicates the behavior of the vertical
epicyclic frequency along radial coordinates $r$. It is worthwhile to note
that the vertical frequency goes down as the radius $r$ goes up, while the
remaining curves express patterns of radial frequency along with radial
coordinate $r$. From the given figure, it is clear that the radial frequency
increases to its maximum for small radii and then starts decreasing towards
equilibrium for large radii. The radial frequency is maximum at $\zeta=-0.4$
in the given domain. Moreover, in Fig. 8, it can be seen that vertical
frequency has a similar behavior as discussed in Fig. 8. While the radial
frequency gives reversal behavior due to charge parameter $q$ as mentioned in
Fig. 8.

a) b)

Figure 8: The epicyclic frequencies depicted as a function of $r$ for altered
values of BH parameter $\zeta$ in left plot and $q$ in right plot.

##  VI Conclusions

We investigate the process of accretion and particle geodesic motion
surrounding the Reissner-Nordström BH coupled with NED parameter $\zeta$, in
the equatorial plane. The stability and circular geodesics of their orbits
have been investigated, examining the oscillations that arise from
perturbations, the existence of unstable orbits, and ultimately enabling the
construction of a fundamental formulation for understanding accretion flow
near the BH. Furthermore, the dynamical parameters, effective potential,
typical radius, specific energy, epicyclic frequencies, specific angular
momentum, emission rate, and mass accretion rate of the BH are determined. By
establishing the state function $p=k\rho$ for isothermal fluid, one can deduce
the general solutions within the framework of a Reissner-Nordström BH coupled
with (NED) parameter $\zeta$.

The transformation of the loci of unstable and stable circular orbits is the
obvious outcome of the impact of the BH parameter $\zeta$ on the effective
potential, as observed in our investigations. As the parameter $\zeta$ is
enhanced, the $V_{eff}$ has associated decline, thereby allowing us to
identify the precise location of the ISCO as depicted in Fig. 4 a. The
location of the radii, namely $r_{isco}$, $r_{ph}$, $r_{sin}$, and $r_{mb}$,
within this particular space-time exhibits significant deviations from the
Schwarzschild solutions. In Table 1, we presented an examination of the
numerical outcomes associated with the ISCO, the radius of the photon sphere,
the marginally bound, $E_{isco}$, $l_{isco}$, $\Omega_{isco}$ and the
radioactive efficiency. In Fig. 5, we observe how the parameter $\zeta$ and
charge $q$ affected the angular momentum and energy of the BH. Furthermore, we
observe that as $\zeta$ increases, the efficacy of the accretion process
increases. It is noticeable that as the BH parameter $\zeta$ grows the flux of
radiation and the radiant temperature has associated decline, while the
radiative efficiency increases.

We examine the characteristics of the fluid particle density, radial velocity,
and accretion processes by employing the state parameter $k=0.5$ with
considering isothermal fluid. It has been observed that the radial velocity
attains maximum value at small radii for the parameters $\zeta$ and $q$ near
the BH but farthest from the BH fluid has no radial velocity. The accretion
process happens when fluid traverses from the critical points its speed
matches with sound speed. So prior to the critical point, the flow of the
fluid has a subsonic regime. So prior to the critical point, the flow of the
fluid has a subsonic regime, but the flow becomes supersonic as it crosses
that point near the BH due to a strong gravitational field. Upon examining the
rate of accretion, we determined that its behavior extensively relies on fluid
nature and BH parameters $\zeta$ and charge $q$. In the scenario of a normal
fluid, the growth in mass accretion happens due to the immense gravitational
field, and it has a maximum value near the BH. The rise in mass accretion
occurs due to a positive deviation in the Schwarzschild BH scenario. Finally,
circular orbits with their properties and epicyclic frequencies are examined
in this paper. The vertical epicyclic frequency is a decreasing function of
the radial distance $r$, without any extrema. The impact of BH parameters
$\zeta$ and $q$ on radial frequency is considerable. We can see that the
radial frequency attains its maximum value at a small BH radius $r$ as
illustrated in Fig. 8. Also, we observe that radial frequency decreases due to
enhancement in BH parameter $\zeta$ but as the value of charge $q$ grows, the
radial frequency will also increase.

## Acknowledgements

The work of G. Abbas has been partially supported by the National Natural
Science Foundation of China under project No. 11988101. He is grateful to the
compact objects and diffused medium Research Group at NAOC led by Prof. JinLin
Han for the excellent hospitality and friendly environment. He is also
thankful to The Islamia University of Bahawalpur, Pakistan for the grant of
study leave. Tao Zhu is supported by the Zhejiang Provincial Natural Science
Foundation of China under Grant No. LR21A050001 and LY20A050002, the National
Key Research and Development Program of China Grant No.2020YFC2201503, the
National Natural Science Foundation of China under Grant No. 12275238 and No.
11675143, and the Fundamental Research Funds for the Provincial Universities
of Zhejiang, China under Grants No. RF-A2019015.

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