# Evaluation of plasma on deflection angle and shadow by a black hole solution
immersed in perfect fluid in Rastall theory

Riasat Ali<EMAIL_ADDRESS>Department of Mathematics, Shanghai
University and Newtouch Center for Mathematics of Shanghai University,
Shanghai-200444, People’s Republic of China  Xia Tiecheng<EMAIL_ADDRESS>Department of Mathematics, Shanghai University and Newtouch Center for
Mathematics of Shanghai University, Shanghai-200444, People’s Republic of
China  Rimsha Babar<EMAIL_ADDRESS>Department of Mathematics, GC
University Faisalabad Layyah Campus, Layyah-31200, Pakistan  Ali Övgün
<EMAIL_ADDRESS>Physics Department, Eastern Mediterranean University,
Famagusta, 99628 North Cyprus, via Mersin 10, Turkiye  Center for Theoretical
Physics, Khazar University, 41 Mehseti Street, Baku, AZ1096, Azerbaijan

###### Abstract

The present study examines the gravitational deflection of particles in curved
space-times immersed in perfect fluid in the context of Rastall theory,
applying the Gibbons-Werner useful technique. In its application to integral
areas inside a four-dimensional space-time, the Gauss-Bonnet theorem studies
the computation expression of the deflection angle. The Gibbons-Werner
technique has two limitations in Rastall theory for space-times immersed in
perfect fluid: the integral region is generally infinite and the integral
complicates calculation, particularly for complex space-times and extremely
accurate solutions. An infinite region approach to Gibbons-Werner is proposed
to avoid singularity. For demonstrating the Gibbons-Werner method, we use a
complete Rastall theory framework. It studies black hole solutions like the
dust field, radiation field, quintessence field, cosmological constant field
and phantom field using the Gibbons-Werner approach. Additionally, we check
the deflection angle from these space-times under the influence of plasma.
Furthermore, we compute analytically the influence of a plasma on a black hole
shadow by using a ray-tracing approach. In the Hamiltonian equation, our model
describes the plasma, so the light ray motion equations are independent of the
plasma’s velocity. It is assumed that plasma is a dispersive medium, pressure-
less and non-magnetized. We investigate the perfect fluid in Rastall theory in
further depth, where the plasma particle density corresponds to particle
accumulation. The supermassive black hole shadows are explored in the case
when plasma falls radially from infinity onto a black hole.

Black hole solutions in Rastall theory; Gibbons-Werner technique; deflection
angle; ray-tracing approach; shadow

##  I Introduction

In general relativity (GR), one of the most significant fields of study is the
movement of objects in intense gravitational fields. An object’s path can be
examined using the geodesic expression of the intense gravitational field. By
considering it as a test particle, its gravitational field is ignorable. An
important consideration in observing the test particle’s velocity is the
trajectory’s deflection. In general, there are two types of particles: massive
particles and massless particles, especially photons. Gravitational theories
R1  such as GR are verified in significant part for massless particles through
gravitational deflection. Many techniques have been suggested R2 -R22  to
study the deflection angle (DA) of photons. For massive particles, they may
function as representatives of all that exists. Later on, Eddington’s
detection justified R23  the deflection of light moving by the sun.
Gravitational lensing has been extensively studied as a powerful tool in
various fields of cosmology and astronomy. For neutrinos, gravitons, neutrons
and celestial events of high energy ($\pi$-mesons, K-mesons, $\mu$ons, etc.)
are examples. There are also theoretically weakly associating massive
particles and their axions R24 . Important details as to the source, lens,
trajectory context, and particles can actually be obtained from the
gravitational deflection of these significant particles R25 ; R26 ; R27 ; R28
; R29 ; R30 ; R31 ; R32 ; R33 ; R34 ; R35 ; R36 . The Gibbons-Werner (GW)
approach, originally proposed by Gibbons-Werner in 2008 R36  and then modified
by scientists in more recent times R37 ; R38 ; R39 ; R40 ; R41 ; R43 ; R44 ,
helps in the geometric explanation and determination of the DA for both
massless and massive particles. The development of a four-dimensional
spacetime that can be utilised to explain the motion and integral region of
the particle on a two-dimensional Riemann manifold, the trajectory of the
particle, an auxiliary circular curve, a radial upward curve that passes
through the source, and a radial outward curve that passes through the
observer are all part of the fundamental technique of the GW approach. The
Gauss-Bonnet theorem (GBT) can be applied to the integral region for the DA in
the form of geometric parameters.

Consider a black hole (BH), which has a consistent spacetime structure but can
be lighted by external electromagnetic radiation with varying shapes, colours,
and behaviours throughout time. The BH can be detected by many mathematical
approaches, but its impact on light distribution can be observed. The Event
Horizon Telescope Collaboration R45  appears as the first image of a BH. The
image shows a continuous spacetime structure illuminated by a time-varying
emission zone, encouraging more investigation and explanation. Several of the
concepts employed by the Event Horizon Telescope (EHT), such as BH shadows,
have been given multiple explanations in the literature. In R46 , Synge
calculated the angular radius of the Schwarzschild BH’s shadow using a static
observer model. The investigation has been done on the various geometries of
BH shadows in Refs R47  -R76 . As the magnetic field parameter grows, the
shadows of the Schwarzschild and Kerr BHs immersed in the Melvin magnetic
field grow and elongate in a horizontal direction. Black hole shadows can only
be predicted by employing a ray-tracing approach R57 ; R58 ; R62  for light
motion systems with non-integrability.

However,the work has focused on the direct local effects of cosmic models on
the known BH solutions. Babichev et al. R77  demonstrate that in a scenario
with a phantom field, the accreting particles of the phantom scalar field into
the central BH cause the BH mass to decrease. But this does have a broad
influence. A modified metric that includes the BHs surrounding spacetime can
be used to determine the local changes in the spacetime geometry surrounding
the fundamental BH. In this context, Kiselev R78  has achieved an analytically
static, spherically symmetric solution to Einstein’s equations. The basis of
the Rastall theory is related to high-curvature situations; hence, the physics
of BHs may provide a suitable framework for further exploration of this
theory. Thus, as a novel class of non-vacuum BH solutions to this theory, our
goal in this study is to find the surrounding Kiselev-like BH solutions. The
properties of the BHs surrounding fields, which are typically such as
radiation, dust, or a dark energy component, define this solution R78 ; R79 .
The Kiselev every field solution’s special cases (a) Schwarzschild BH
encompassed by quintessence, BH encompassed by magnetised Ernst field and a
Reissner-Nordstrom BH encircled by radiation and dust (b), as well as its
phase changes and thermodynamical quantities are examined in R79 ; R80  and
R81  describes the dynamics of a neutral and a charged particle surrounding
the Schwarzschild black encircled by quintessence matter.

In order to avoid spacetime singularity, one of the main reasons to research
the BH surrounding field in Rastall theory. Although there is a well-developed
Rastall theory at this point, there are numerous attempts, such as black rings
field theory and BH geometry. The expectation that there should be an
intrinsic extended structure in spacetime is acknowledged by almost all of
this approach (Gibbons-Werner approach). Stable-orbit photons are in constant
motion around a BH; they are unable to leave the BH or travel to infinity. In
this instance, the prograde stable photon orbits correspond to the dark region
that only partially emerges from the main shadow. The BH shadow is half-
panoramic (equatorial), due to the lack of backward unstable (stable) light
rings. In this instance, the BH shadow turns into an equatorial, panoramic
shadow without a grey area for stable photon orbits. The expectation that
there should be an intrinsic extended structure in spacetime is acknowledged
by almost all of this approach (ray-tracing approach). It is studied that the
BH shadows are immersed in charge fields and that the BH shadow is affected by
the perfect fluid in the context of Rastall theory.

This is way the paper is formatted. Section $2$ discusses an overview of the
analytical technique for spherically symmetric BHs in Rastall theory. Section
$3$ study the DA for BHs occupied by dust, quintessence, radiation, phantom
fields and cosmological constant and also their graphical analysis. In section
$4$, we investigate the results of DA and graphical behavior in the background
of plasma frame for associated BHs. Section $5$ comprises the shadow cast and
contour plots for BHs surrounded by dust, quintessence, radiation, phantom
fields and cosmological constant. In section $6$, we summarize the results of
our work.

##  II Introductory Review of black hole in the background of Rastall Theory

Within the framework of the Rastall theory of gravity, we seek the general
non-vacuum spherically symmetric static uncharged BH solutions in this
section. Utilizing Rastall’s idea R82 ; R83 ; A1 , we obtain the following for
a spacetime where an energy-momentum source of $T_{ab}$ occupies the Ricci
scalar $R$ as

$\lambda R^{,b}=T^{ab};a,$ (1)

with $\lambda$ represents the Rastall parameter, which is the conventional GR
conservation law that has deviated. The Rastall field equations can be
expressed as

$kT_{ab}=G_{ab}+k\lambda g_{ab}R,$ (2)

where the gravitational constant of Rastall coupling is denoted by $k$. In the
limit of $\lambda\rightarrow 0$ and $k=8\pi G$, these field equations reduce
to GR field equations, with $G$ is the gravitational constant of Newton
coupling. To derive BH solutions, we take the usual Schwarzschild coordinates
general spherical symmetric metric as A1

$ds^{2}=-A(r)dt^{2}+\frac{dr^{2}}{A(r)}+r^{2}d\Omega^{2},$ (3)

with the two-dimensional unit sphere is represented by the equation
$d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$, and $f(r)$ is a
fundamental metric function that depends on the radial coordinate. We can
derive non-vanishing components of the Rastall tensor, defined as
$H_{ab}=G_{ab}+k\lambda g_{ab}R$, can be followed by applying this metric, we
get A1

$\displaystyle H^{0}_{0}$ $\displaystyle=$ $\displaystyle G^{0}_{0}+k\lambda
R=-\frac{1}{A}G_{00}+k\lambda R=\frac{1}{r^{2}}(\acute{A}r-1+A)+k\lambda R,$
$\displaystyle H^{1}_{1}$ $\displaystyle=$ $\displaystyle G^{1}_{1}+k\lambda
R=AG_{11}+k\lambda R=\frac{1}{r^{2}}(\acute{A}r-1+A)+k\lambda R,$
$\displaystyle H^{2}_{2}$ $\displaystyle=$ $\displaystyle G^{2}_{2}+k\lambda
R=-\frac{1}{r^{2}}G_{22}+k\lambda
R=\frac{1}{r^{2}}(\acute{A}r+\frac{1}{2}r^{2}\acute{\acute{A}})+k\lambda R,$
$\displaystyle H^{3}_{3}$ $\displaystyle=$ $\displaystyle G^{3}_{3}+k\lambda
R=-\frac{1}{r^{2}\sin^{2}\theta}G_{33}+k\lambda
R=\frac{1}{r^{2}}(\acute{A}r+\frac{1}{2}r^{2}\acute{\acute{A}})+k\lambda R.$
(4)

In this case, the Ricci scalar is defined as

$R=-\frac{1}{r^{2}}(r^{2}\acute{\acute{A}}+4r\acute{A}+2+2A),$ (5)

with the derivative with regard to the radial coordinate r is represented by
the prime sign. In relation to the Rastall tensor ($H^{a}_{b}$) non-vanishing
components, then the following diagonal form should be present in the entire
energy-momentum tensor that governs this spacetime.

$T^{a}_{b}=\begin{pmatrix}T^{0}_{0}&0&0&0\\\ 0&T^{1}_{1}&0&0\\\
0&0&T^{2}_{2}&0\\\ 0&0&0&T^{3}_{3}\end{pmatrix},$ (6)

with it is satisfied with the Rastall tensor $H^{a}_{b}$ symmetry features.
With respect to solutions of Rastall tensor, the equality conditions
$H^{0}_{0}=H^{1}_{1}$ and $H^{2}_{2}=H^{3}_{3}$ must also have
$T^{0}_{0}=T^{1}_{1}$ and $T^{2}_{2}=T^{3}_{3}$, respectively. After that, a
general energy and momentum tensor T with these symmetry features can be
generated in the manner shown as

$T^{a}_{b}=\tau^{a}_{b}+E^{a}_{b},$ (7)

with $E^{a}_{b}$ is the trace free of Maxwell tensor defined by

$E^{a}_{b}=\frac{2}{k}(F_{a\mu}F^{\mu}_{b}-\frac{1}{4}g_{ab}F^{\mu\nu}F_{{\mu\nu}}).$
(8)

So that $F_{ab}$ represents the anti-symmetric Faraday tensor obeying the
corresponding vacuum Maxwell expression as

$\displaystyle F^{ab}_{;a}$ $\displaystyle=$ $\displaystyle 0,$
$\displaystyle\partial[\alpha F_{ab}]$ $\displaystyle=$ $\displaystyle 0.$ (9)

Given that the spacetime metric (3) has spherical symmetry, the only non-
vanishing Faraday tensor $F^{ab}$ of components is imposed to be
$F^{01}=-F^{10}$. Next, using the equations in (9), one can get

$F^{01}=\frac{Q}{r^{2}},$ (10)

where $Q$ represents an integration constant behaving as an electrostatic
charge, the only Maxwell tensor $E^{a}_{b}$ non-vanishing components are given
by equations (3), (8), and (10) as

$E^{a}_{b}=\frac{Q^{2}}{kr^{4}}\textit{diagonal}(-1,-1,1,1),$ (11)

possessing the symmetries in the $H^{a}_{b}$ tensor and evidently expressing
an electrostatic field. Conversely, $\tau^{a}_{b}$ denotes the surrounding
field’s energy–momentum tensor, which is defined R84  as

$\displaystyle\tau^{0}_{0}$ $\displaystyle=$ $\displaystyle-\rho_{s}(r)$
$\displaystyle\tau^{j}_{i}$ $\displaystyle=$
$\displaystyle-\rho_{s}(r)\mu\big{(}-\frac{r_{j}r^{i}}{r_{n}r^{n}}-3\nu\frac{r_{j}r^{i}}{r_{n}r^{n}}+\nu\delta^{j}_{i}\big{)}.$
(12)

With respect to the arbitrary parameters $\mu$ and $\nu$, which correspond to
the internal structure of the BH surrounding the field, this model of
$\tau^{a}_{b}$ implies that the space component is proportionate to the time
category, which represents the energy density. The surrounding field in this
case is indicated by the subscript (s), which is usually any combination of
dust, radiation, quintessence, cosmological constant, and phantom field. We
can determine the isotropic average over the angles by using R84  the

$<\tau^{j}_{i}>=\frac{1}{3}\rho_{s}\delta^{j}_{i}=\rho_{s}\delta^{j}_{i},$
(13)

as it is assumed that
$<r^{j}r_{i}>=\frac{\mu}{3}\rho_{s}\delta^{j}_{i}r_{n}r^{n}.$ The barotropic
expression of equilibrium for the surrounding field thus becomes

$p_{s}=\rho_{s}\omega_{s},~{}~{}~{}~{}\omega_{s}=\frac{\mu}{3},$ (14)

with $\omega_{s}$ and $\rho_{s}$ indicate the equation of the state parameter
and the pressure, respectively. The principle of the addition and linearity
scenario assumed in reference R84  to find the free parameter $\nu$ of the
energy momentum tensor of the surrounding field as suggested is thus precisely
provided by the field expressions (4) with respect to the entire energy-
momentum tensor in (7), (11) and (12) as

$\nu=-\frac{3\omega_{s}+1}{6\omega_{s}}.$ (15)

Then, the following form can be used to derive the non-vanishing components of
the $\tau_{ab}$ tensor

$\displaystyle\tau^{0}_{0}$ $\displaystyle=$
$\displaystyle\tau^{1}_{1}=-\rho_{s}$ $\displaystyle\tau^{2}_{2}$
$\displaystyle=$
$\displaystyle\tau^{3}_{3}=\frac{1}{2}(3\omega_{s}+1)\rho_{s},$ (16)

which in the Rastall tensor $H^{a}_{b}$ also have identical symmetries.
Consequently, all of $H^{a}_{b}$ symmetry is admitted by our whole constructed
energy-momentum tensor in (7). The Rastall field equations $\tau^{a}_{b}$ can
be regarded as the only energy-momentum tensor of assistance. The solutions
produced will thus explain the encircled uncharged BH solutions in the Rastall
theory context, which are not the same as the ones in GR. Within the context
of this theory, the most generic class of static surrounding charged BH
solutions can be obtained by including the Maxwell tensor $E^{a}_{b}$ in
$T^{a}_{b}$. The field equations are solved and their general solution is
obtained in the following. Next, we tackle the two uncharged/charged
solutions. From the $H^{0}_{0}=T^{0}_{0}$ and $H^{1}_{1}=T^{1}_{1}$ components
of the Rastall field expression, the differential equation that follows is
obtained as

$\frac{1}{r^{2}}(r\acute{A}+A-1)-\frac{k\lambda}{r^{2}}(r^{2}\acute{\acute{A}}+4r\acute{A}+2\acute{A}-2)=-\frac{Q^{2}}{r^{4}}-k\rho_{s},$
(17)

and $H^{2}_{2}=T^{2}_{2}$ and $H^{3}_{3}=T^{3}_{3}$ components are interpreted
as

$\frac{1}{r^{2}}(r\acute{A}+\frac{1}{2}r^{2}\acute{\acute{A}})-\frac{k\lambda}{r^{2}}(r^{2}\acute{\acute{A}}+4r\acute{A}+2A-2)=-\frac{Q^{2}}{r^{4}}+\frac{k\rho_{s}}{2}(1+3\omega_{s}),$
(18)

As a result A1 , the two differential equations (17) and (18) allow us to
analytically compute the one unknown functions $A(r)$ as

$A(r)=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{N_{s}}{r^{\frac{1-3\omega_{s}-6k\lambda(1+\omega_{s})}{1-3k\lambda(1+\omega_{s})}}},$
(19)

where the BH mass and the surrounding field structure parameter are
represented by two integration constants, $M$ and $N_{s}$, respectively. The
parameters $k$ and $\lambda$ represents the Rastall geometric parameters and
$\omega_{s}$ denotes the equation of state parameter of BH surrounding field.
Keep in mind that the characteristics of the surrounding field are represented
by the integration constant $N_{s}$. Any combination of $k$, $\lambda$ and
$\omega_{s}$ parameters can accept various positive or negative $N_{s}$
values. We recover the Reissner-Nordström BH occupied by a surrounding field
in GR, which was initially discovered by Kiselev R84 , in the limit of
$\lambda\rightarrow 0$ and $k=8\pi GN$. The metric in Eq. (19) is a new static
solution with some intriguing features. Now, we will study the surrounding BH
by the dust radiation, quintessence, cosmological constant and phantom fields,
as the sub-classes of the general solution of Eq. (19) and their interesting
aspects in detail. The two differential equations (17) and (18) allow us to
analytically compute other one unknown function $\rho_{s}(r)$ as

$\rho_{s}(r)=-\frac{3W_{s}N_{s}}{kr^{\frac{3(1+\omega_{s})-12k\lambda(1+\omega_{s})}{1-3k\lambda(1+\omega_{s})}}},$
(20)

where the field structure parameter can be represented by

$W_{s}(r)=-\frac{(1-4k\lambda)(k\lambda(1+\omega_{s})-\omega_{s})}{(1-3k\lambda(1+\omega_{s}))^{2}}.$
(21)

Keep in mind that the characteristics of the surrounding field are represented
by the integration constant $N_{s}$. We have
$\rho_{s}(r)=-\frac{3}{k}W_{s}N_{s}r^{-3(1+\omega_{s})}$ where
$W_{s}=\omega_{s}$ for $\lambda=0$, in the GR limit. The BH metric (19)
surrounded by the dust field in Rastall theory can be presented as

$ds^{2}=A(r)dt^{2}+B(r)dr^{2}+C(r)d\theta^{2}+D(r)d\phi^{2},$ (22)

where

$A(r)=\frac{1}{B(r)}=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{N_{d}}{r^{\frac{1-6k\lambda}{1-3k\lambda}}},~{}~{}~{}C(r)=r^{2},~{}~{}~{}D(r)=r^{2}\sin^{2}\theta,$
(23)

and $M$ is a mass of BH, $N_{d}$ is the dust field structure parameter and $Q$
is a charge of BH. This metric is not the same as the metric of the charged BH
in GR R84  that is encircled by a dust field. As one can see, the BH in the
dust background in GR appears as a charged BH with an effective mass
$M_{e}=2M+N_{d}$ in the limit of $\lambda\rightarrow 0$ and $k=8\pi GN$. We
may observe that, for $k\lambda\neq 0$, the Rastall theory’s geometric
parameters $k$ and $\lambda$ can be crucial in determining different solutions
with respect to GR. For $k\lambda\neq 0$, the Rastall correction term gives
the BH a distinct character that is incomparable to the mass and charge terms,
and it never behaves like the mass or charge terms. Due to the Rastall
geometric parameters, the presence of such nontrivial character might
significantly alter the thermodynamics, causal structure, and Penrose diagrams
in comparison to those of GR. For this case, the geometric parameter $W_{d}$
can be defined as

$W_{d}=\frac{k\lambda(1-4k\lambda)}{(1-3k\lambda)^{2}}.$ (24)

Next, with respect to the weak energy condition denoted by the relation
$W_{s}N_{s}\leq 0$, we require $N_{d}>0$ for $0\leq k\lambda<\frac{1}{4}$ and
$N_{d}<0$ for the field structure constant for $k\lambda>\frac{1}{4}$. In this
instance, $W_{d}$ and $\rho_{d}$ are essentially distinct from their GR
versions and the density $\rho_{d}$ can be given as

$\rho_{d}=\frac{3\lambda(1-4k\lambda)N_{d}}{(1-3k\lambda)^{2}}r^{-\frac{3-12k\lambda}{1-3k\lambda}}.$
(25)

Moreover, the geometry of the Rastall theory allows us to derive an effective
equation of state parameter $\omega_{eff}$ for the modification term can be
written as

$\omega_{eff}=\frac{1}{3}\left(-1+\frac{1-6k\lambda}{1-3k\lambda}\right).$
(26)

It may be observed that, with the exception of the $k\lambda=0$ corresponding
to the GR limit, $\omega_{eff}$ can never be zero. Therefore, this theory’s
solutions essentially diverge from GR’s. The Rastall solution of BH
thermodynamical quantities are examined in R85a ; R85b . Regarding Eq. (26),
there are two distinct and intriguing groups. For the case,
$\frac{1}{6}<k\lambda<\frac{1}{3}$ that implies to
$\omega_{eff}\leq\frac{1}{3}$. Here, dark energy is represented by an
effective surrounding fluid with an effective equation of state parameter
$\omega_{e}$, which results in an effective repulsive gravitational pull. It
is noteworthy that the effective equation of state $\omega_{eff}$ for
$k\lambda=\frac{2}{10}$ and $\frac{2}{9}$ falls inside the quintessence range,
whereas for $k\lambda=\frac{3}{10}$, it falls into the strong phantom range.
This illustrates the idea that, for a given $k$, the stronger the acceleration
phase, or the stronger coupling $g_{\mu\nu}R$ in Rastall theory, the larger
values of $\lambda$. When $k\lambda<\frac{1}{6}\bigcup k\lambda>\frac{1}{3}$
implies to $\omega_{eff}\geq-\frac{1}{3}$ is the result. Here, we have an
effective surrounding fluid that possesses the usual attractive gravitational
effect and whose equation of state parameter respects the strong energy
condition. Depending on the value of the effective equation of state
parameter, this could lead to the universe expanding more slowly or even
contracting.

##  III  DEFLECTION ANGLE IN NON-PLASMA MEDIUM

This section is based on the study of DA in a non-plasma frame for a BH
surrounded by dust field in Rastall theory. The generalized GW technique R37 ;
R41  significantly improves the associated computation and provides a thorough
framework for describing the GW approach for a wide range of scenarios. We
think that the GW method’s implementation in BH physics will be made much
easier by our study. When the GW approach is applied and developed for
spacetimes, there is an ill-defined infinite integral region for some
spacetimes expressing singular behavior, and the computation required is
complex. In particular, we study that the radial coordinate of the additional
circular arc can be chosen freely by carefully examining the integrals of
geodesic curvature along the support circular arc and of Gaussian curvature
throughout the integral region. As a result, the ill-defined condition is
resolved, and the integral region without singular performance can be formed.
With complex computation, we construct a simplified formula that allows us to
determine the DA in a few steps. This formula is based on the free choice of
the auxiliary circular arc and the reduction of the integral of geodesic
curvature along the path. The efficiency and effectiveness of our approach are
persuasively validated when we finally compute the DA of particles in Kiselev
spacetime in Rastall gravity. Furthermore, we provide the DA of particles for
the charged solution in Rastall gravity for the first time. The expression
that follows can be used to find the DA for the Kiselev solution using GBT in
the non-singular domain region. When we consider the null condition $ds^{2}=0$
at equatorial plane $(\theta=\frac{\pi}{2})$ for the metric (22), when the
tropical area is where the source, observer, and null photon are all located
then the optical metric can be obtained as

$dt^{2}=\frac{B(r)}{A(r)}dr^{2}+\frac{r^{2}}{A(r)}d{\phi}.$ (27)

We rewrite the above metric in the form

$dt^{2}=Y(r)dr^{2}+Z(r)d\phi,$ (28)

where

$Y(r)=\frac{B(r)}{A(r)},~{}~{}~{}~{}~{}Z=\frac{r^{2}}{A(r)}.$ (29)

In order to calculate the Ricci scalar for the given metric, at first, we
compute the non-zero Christoffel symbols in the following manner

$\Gamma^{1}_{11}=\frac{Z^{\prime}(r)}{2Z(r)}-\frac{Y^{\prime}(r)}{2Y(r)},~{}~{}~{}\Gamma^{2}_{12}=\frac{Y(r)}{r}-\frac{Y^{\prime}(r)}{2},~{}~{}~{}\Gamma^{1}_{22}=\frac{r^{2}Y^{\prime}(r)}{2Y(r)Z(r)}-\frac{r}{Z(r)},$
(30)

where $1$ and $2$, represent the $r$ and $\phi$ coordinates, respectively. The
Ricci scalar is obtained as follows

$R=\frac{2Z^{{}^{\prime\prime}}(r)Y(r)Z(r)-Z^{\prime}(r)Y^{\prime}(r)Z(r)-Z^{\prime
2}(r)Y(r)}{2Y^{2}(r)Z^{2}(r)}.$ (31)

The Gaussian curvature can be derived by using the following formula

$K=\frac{R}{2}.$ (32)

Using Eq. (33), the Gaussian curvature for a BH surrounded by dust field is
calculated as

$K\approx\frac{2M}{r^{3}}+\frac{(1+6k\lambda)N_{d}}{r^{3}}-\frac{3Q^{2}}{r^{4}}+\frac{3(2k\lambda-1)MN_{d}}{r^{4}}+\frac{6MQ^{2}}{r^{5}}+\frac{3(2-\lambda)N_{d}Q^{2}}{2r^{5}}+O(M^{2},Q^{3},\lambda^{2}).$
(33)

The DA is determined by applying the optical Gaussian curvature to a non-
singular region $W_{e}$, bounded by $\partial W_{e}=\omega_{\tilde{g}}\bigcup
U_{e}$. As an alternative, a non-singular domain with the Euler characteristic
$\alpha(W_{e})=1$ outside of the light trajectory can be used. The GBT for
this area can be expressed as R37

$\int\int_{W_{e}}{KdS}+\oint_{\partial W_{e}}{\kappa
dt}+\sum_{i}\epsilon_{i}=2\pi\alpha(W_{e}).$ (34)

As $k=\bar{g}(\nabla_{\dot{\varpi}}\dot{\varpi},\ddot{\varpi})$ and
$\bar{g}(\dot{\varpi},\dot{\varpi})=1,~{}\ddot{\varpi}$ imply unite
acceleration vector, and $\epsilon_{i}i$ expresses the exterior angle at the
$ith$ vertex, $\kappa$ represents geodesic curvature in the above expression.
The associated jump angles drop to $\frac{\pi}{2}$ as $e\rightarrow\infty$,
yielding $\theta_{O}+\theta_{s}\rightarrow\pi$. Thus,

$\int\int_{W_{e}}{KdS}+\oint_{\partial W_{e}}{\kappa
dt}+\epsilon_{i}=2\pi\alpha(W_{e}),$ (35)

here, jump angle is represented by $\epsilon_{i}=\pi$. Obtaining the geodesic
curvature when $e\rightarrow\infty$ in the following way

$\kappa(U_{e})=\mid\nabla\dot{{}_{U_{e}}}\dot{U_{e}}\mid.$ (36)

Since the radial component of geodesic curvature is computed as

$(\nabla_{\dot{U_{e}}}\dot{U_{e}})^{r}=\dot{U^{\phi}_{e}}\partial_{\phi}\dot{U_{e}^{r}}+\Gamma^{1}_{22}(\dot{U}^{\phi}_{e})^{2}.$
(37)

For large value of $e,U_{e}:=r(\phi)=e=\text{constant}$, then the result is

$\lim_{e\rightarrow\infty}\kappa(U_{e})=\lim_{e\rightarrow\infty}(\nabla_{\dot{U_{e}}}\dot{U_{e}})^{r}\rightarrow\frac{1}{e}.$
(38)

Since there is no topological defect in the geodesic curvature,
$K(U_{e})\rightarrow e^{-1}$. However, by using the optical metric Eq. (27),
it can be written as follows

$\lim_{e\rightarrow\infty}dt=ed\phi.$ (39)

Using Eq. (38) and (39), we obtain

$K(U_{e})dt=d\phi.$ (40)

Now, using Eq. (40), we can get the following equation

$\int\int_{W_{e}}{KdS}+\oint_{\partial W_{e}}{\kappa
dt}{=}^{e\rightarrow\infty}\int\int_{W\infty}{KdS}+\int^{\pi+\psi}_{0}d\phi.$
(41)

We employ the straight-line approximation, $r=\frac{b}{\sin\phi}$, where $b$
represents the impact parameter. The Gibbons and Werner approach uses the
Gauss-Bonnet theorem to determine the DA as R37

$\psi\approx-\int^{\pi}_{0}\int^{\infty}_{\frac{b}{sin\phi}}K\sqrt{\det\vec{g}}drd\phi,$
(42)

where $\sqrt{\det\vec{g}}$ is computed as

$\sqrt{\det\vec{g}}=\frac{3M}{r^{2}}+\frac{1}{r^{3}}-\frac{3Q^{2}}{2r^{3}}+\frac{3MN_{d}}{2r^{1+\frac{1-6k\lambda}{1-3k\lambda}}}.$
(43)

To compute the angle of deflection, use the Gaussian curvature up to the
leading order terms and obtain

$\psi_{1}\approx\frac{4M}{b}-\frac{2N_{d}}{b}-\frac{12N_{d}k\lambda}{b}-\frac{3Q^{2}\pi}{4b^{2}}-\frac{3MN_{d}\pi}{2b^{2}}-\frac{3MN_{d}k\lambda\pi}{2b^{2}}-\frac{4NQ^{2}}{3b^{3}}+\frac{8MQ^{2}}{3b^{3}}+\frac{2N_{d}Q^{2}k\lambda}{3b^{3}}+O(M^{2},Q^{4},\lambda^{2}).$
(44)

The DA for a BH surrounded by a dust field is dependent on impact parameter
$b$, BH mass $M$, charge $Q$, Rastall geometric parameters $k$, $\lambda$ and
dust field structure parameter $N_{d}$. We can also observe that the impact
parameter is in an inverse relation with the angle $\psi$. Moreover, it has
worth to mention here that, in the absence of structure parameter $N_{d}=0$
and Rastall parameters $k\lambda=0$, the above result reduce into DA of
Reissner–Nordström BH W1 ; W2  as well as in the absence of charge $Q=0$, we
recover the DA of Schwarzschild BH $\psi_{Sch}=4M/b$ A1 .

Figure 1: Deflection angle $\psi_{1}$ versus impact parameter $b$ with
variations of charge $Q$ and Rastall parameters $k\lambda$ with fixed mass
$M=1$ and dust field structure parameter $N_{d}=0.1$

.

In Fig. 1: the left panel shows the variations of charge $Q$ with fixed mass
$M=1$, Rastall parameters $k\lambda=0.5$ and dust field structure parameter
$N_{d}=0.1$ and right panel represents the variations of Rastall parameters
$k\lambda$ with fixed mass $M=1$, charge $Q=0.5$ and dust field structure
parameter $N_{d}=0.1$. In both panels the black curve gives the Schwarzschild
solution. We can observe that the DA for rising values of charge and Rastall
parameters from a BH surrounded by a dust field is decreasing via rising
impact parameter and it attains an asymptotically flat form till
$b\rightarrow\infty$. A strong deflection can be observed for small value of
impact parameter $b=1.2$. Graphically, we can also see the inverse
relationship between impact parameter $b$ and DA $\psi_{1}$.

###  III.1 BLACK HOLE SURROUNDED BY THE QUINTESSENCE FIELD

The metric functions for a BH surrounded by a quintessence field are given as

$A(r)=\frac{1}{B(r)}=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{N_{q}}{r^{\frac{-1-2k\lambda}{1-k\lambda}}}~{}~{}C(r)=r^{2},~{}~{}D(r)=r^{2}\sin^{2}\theta.$
(45)

Using Eqs. (31) and (32), the Gaussian curvature for a BH surrounded by a
quintessence field can be defined as

$K\approx\frac{2M}{r^{3}}+\frac{4Q^{2}N_{q}}{r^{3}}+\frac{18Q^{2}k\lambda
N_{q}}{4r^{3}}-\frac{3Q^{2}}{r^{4}}+\frac{6MQ^{2}}{r^{5}}+O(M^{2},Q^{4},\lambda^{2}).$
(46)

The $\sqrt{\det\vec{g}}$ for the corresponding BH can be calculated as

$\sqrt{\det\vec{g}}=\frac{3M}{r^{2}}+\frac{1}{r^{3}}-\frac{3Q^{2}}{2r^{3}}+\frac{3N_{q}}{2r^{1+\frac{-1-2k\lambda}{1-k\lambda}}}.$
(47)

Using Eqs. (46) and (47) into Eq. (42), we obtain the DA for BH surrounded by
quintessence field as follows

$\psi_{2}\approx\frac{4M}{b}+\frac{8N_{q}Q^{2}}{b}+\frac{9N_{q}Q^{2}k\lambda}{b}-\frac{3Q^{2}\pi}{4b^{2}}+\frac{8MQ^{2}}{3b^{3}}+O(M^{2},Q^{4},\lambda^{2}).$
(48)

The DA in non-plasma medium for a BH surrounded by a quintessence field
depends on impact parameter $b$, BH mass $M$, charge $Q$, Rastall geometric
parameters $k$, $\lambda$ and quintessential field structure parameter
$N_{q}$. Furthermore, it is important to note that the above result reduces to
the DA of Reissner–Nordström BH in the absence of the structure parameter
$N_{q}=0$ and the Rastall parameters $k\lambda=0$. In addition, we recover the
DA of Schwarzschild BH $\psi_{Sch}=4M/b$ in the absence of charge $Q=0$.

Figure 2: Deflection angle $\psi_{2}$ via impact parameter $b$ with variations
of charge $Q$ and Rastall parameters $k\lambda$ with fixed mass $M=1$ and
quintessential field structure parameter $N_{q}=0.5$.

In Fig. 2: the left panel displays variations in charge $Q$ with fixed mass
$M=1$, Rastall parameters $k\lambda=0.1$, and quintessential field structure
parameter $N_{q}=0.5$, whereas Rastall parameters $k\lambda$ with fixed mass
$M=1$, charge $Q=0.3$, and quintessential field structure parameter
$N_{q}=0.5$ are represented in the right panel. In both panels, the
Schwarzschild solution is indicated by the black curve. From a BH surrounded
by a quintessential field, we can see that the DA for rising values of charge
and Rastall parameters is decreasing via rising impact parameter, reaching an
asymptotically flat form till $b\rightarrow\infty$. A significant deflection
is seen for small value of impact parameter. Additionally, the inverse
relationship between impact parameter $b$ and DA $\psi_{2}$ is also observed.

###  III.2 BLACK HOLE SURROUNDED BY THE RADIATION FIELD

The metric function for a BH surrounded by a radiation field is defined as

$A(r)=\frac{1}{B(r)}=1-\frac{2M}{r}+\frac{Q^{2}-N_{r}}{r^{2}},~{}~{}~{}~{}~{}~{}C(r)=r^{2},~{}~{}~{}~{}~{}~{}D(r)=r^{2}\sin^{2}\theta.$
(49)

Utilizing Eqs. (31) and (32), the Gaussian curvature for a BH surrounded by
the radiation field can be computed as follows

$K\approx\frac{2M}{r^{3}}-\frac{3Q^{2}}{r^{4}}+\frac{3N_{r}}{r^{4}}+\frac{6MQ^{2}}{r^{5}}-\frac{6MN_{r}}{r^{5}}+\frac{4N_{r}Q^{2}}{r^{6}}+O(M^{2},Q^{4}).$
(50)

The value of $\sqrt{\det\vec{g}}$ for associated BH is given by

$\sqrt{\det\vec{g}}=\frac{3M}{r^{2}}+\frac{1}{r^{3}}-\frac{3Q^{2}}{2r^{3}}+\frac{3N_{r}}{2r^{3}}.$
(51)

Using Eqs. (50) and (51) into Eq. (42), we derive the DA for BH surrounded by
radiation field as

$\psi_{3}\approx\frac{4M}{b}-\frac{3Q^{2}\pi}{4b^{2}}+\frac{3N_{r}\pi}{4b^{2}}+\frac{8MQ^{2}}{3b^{3}}-\frac{8MN_{r}}{3b^{3}}+\frac{3N_{r}Q^{2}\pi}{8b^{4}}+O(M^{2},Q^{4}).$
(52)

The DA in non-plasma frame for a BH surrounded by a radiation field is
dependent on impact parameter $b$, BH mass $M$, charge $Q$ and radiation field
structure parameter $N_{r}$. It is important to note that the above result
reduces to the DA of Reissner–Nordström BH in the absence of the structure
parameter $N_{r}=0$. In addition, we recover the DA of Schwarzschild BH
$\psi_{Sch}=4M/b$ in the absence of charge $Q=0$.

Figure 3: Deflection angle $\psi_{3}$ with respect to impact parameter $b$
with variations of charge $Q$ and radiation field structure parameter $N_{r}$
with fixed mass $M=1$.

Fig. 3: in the left panel charge $Q$ varied with fixed mass and the radiation
field structure parameter $M=1=N_{r}$, while the right panel shows variations
of radiation field structure parameter $N_{r}$ with fixed mass $M=1$ and
charge $Q=0.5$. The black curve in both panels indicates the Schwarzschild
solution. The DA with increasing charge and structure parameter values is
diminishing by growing impact parameter from a BH surrounded by a radiation
field, achieving an asymptotically flat form till $b\rightarrow\infty$. A
notable deviation is observed when the impact parameter is minimal.
Furthermore, the impact parameter $b$ and DA $\psi_{3}$ are found to be
inversely related. It is important to mention here that, at some points in
left plot the light deflects with the same angle for different variations of
charge. Moreover, a strong deflection can be observed in the presence of
radiation field as compared to Schwarzschild case.

###  III.3 BLACK HOLE SURROUNDED BY THE PHANTOM FILED

The metric functions for a BH surrounded by a phantom field are given by

$A(r)=\frac{1}{B(r)}=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-\frac{N_{p}}{r^{\frac{-3+2k\lambda}{1+k\lambda}}},~{}~{}~{}~{}~{}C(r)=r^{2},~{}~{}~{}~{}~{}D(r)=r^{2}\sin^{2}\theta.$
(53)

Considering Eqs. (31) and (32), the Gaussian curvature for a BH surrounded by
the phantom field can be given as follows

$K\approx\frac{2M}{r^{3}}+\frac{4k\lambda
M}{r^{3}}-\frac{3Q^{2}}{r^{4}}-\frac{35N_{p}k\lambda
Q^{2}}{2r^{4}}+\frac{6MQ^{2}}{r^{5}}-\frac{11N_{p}M}{r^{5}}+O(M^{2},Q^{4},\lambda^{2}).$
(54)

The $\sqrt{\det\vec{g}}$ for given BH is computed as

$\sqrt{\det\vec{g}}=\frac{3M}{r^{2}}+\frac{1}{r^{3}}-\frac{3Q^{2}}{2r^{3}}+\frac{3N_{p}}{2r^{1+\frac{-3+2k\lambda}{1+k\lambda}}}.$
(55)

Utilizing Eqs. (54) and (55) into Eq. (42), we derive the DA for BH surrounded
by phantom field as follows

$\psi_{4}\approx\frac{4M}{b}+\frac{8Mk\lambda}{b}-\frac{35N_{p}k\lambda
Q^{2}\pi}{8b^{2}}-\frac{3Q^{2}\pi}{4b^{2}}+\frac{8MQ^{2}}{3b^{3}}-\frac{44N_{p}M}{9b^{3}}+O(M^{2},Q^{4},\lambda^{2}).$
(56)

The DA in non-plasma medium for a BH surrounded by a phantom field depends on
impact parameter $b$, BH mass $M$, charge $Q$, Rastall geometric parameters
$k$, $\lambda$ and phantom field structure parameter $N_{p}$. It is
significant to note that in the absence of the structure parameter $N_{p}=0$
and the Rastall parameters $k\lambda=0$, the above result simplifies to the DA
of Reissner–Nordström BH. Furthermore, in the absence of charge $Q=0$, we
recover the DA of Schwarzschild BH $\psi_{Sch}=4M/b$.

Figure 4: Deflection angle $\psi_{4}$ versus impact parameter $b$ with
variations of charge $Q$ and Rastall parameters $k\lambda$ with fixed mass
$M=1$ and phantom field structure parameter $N_{p}=0.5$.

Fig. 4: in the left panel charge $Q$ shows variations with fixed mass, Rastall
parameters and the phantom field structure parameter $k\lambda=0.5=N_{p}$,
while the right panel represents the variations of Rastall parameters
$k\lambda$ with fixed mass $M=1$, charge $Q=0.1$ and phantom field structure
parameter $N_{p}=0.5$. The Schwarzschild solution is shown by the black curve
in both images. As the charge and Rastall parameters values increase, the DA
decreases due to an increasing impact parameter from a BH encircled by a
phantom field. A strong deflection is shown when the impact parameter is
small. Moreover, an inverse relationship between the impact parameter $b$ and
the DA $\psi_{4}$ is discovered. It is significant to note that, for various
variations in charge, the light deflects at certain spots in the left plot
with the same angle for Schwarzschild. Furthermore, a significant deviation
from the Schwarzschild scenario can be seen when a phantom field is present.

###  III.4 BLACK HOLE SURROUNDED BY THE COSMOLOGICAL CONSTANT

The metric functions for a BH surrounded by a cosmological constant is defined
as

$A(r)=\frac{1}{B(r)}=1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}-N_{c}r^{2},~{}~{}~{}~{}~{}C(r)=r^{2},~{}~{}~{}~{}~{}D=r^{2}\sin^{2}\theta.$
(57)

Considering Eqs. (31) and (32), the Gaussian curvature for a BH surrounded by
the cosmological constant is computed as

$K\approx\frac{2M}{r^{3}}+\frac{N_{c}}{r^{3}}-\frac{3Q^{2}}{r^{4}}-\frac{6MN_{c}}{r^{4}}+\frac{6MQ^{2}}{r^{5}}+\frac{6N_{c}Q^{2}}{r^{5}}+O(M^{2},Q^{4}).$
(58)

The value of $\sqrt{\det\vec{g}}$ for BH surrounded by cosmological constant
is calculated in the following way

$\sqrt{\det\vec{g}}=\frac{3M}{r^{2}}+\frac{1}{r^{3}}-\frac{3Q^{2}}{2r^{3}}+\frac{3rN_{c}}{2}.$
(59)

Utilizing Eqs. (58) and (59) into Eq. (42), we investigate the DA for BH
surrounded by cosmological constant as follows

$\psi_{5}\approx\frac{4M}{b}+\frac{2N_{c}}{b}-\frac{3MN_{c}\pi}{2b^{2}}-\frac{3Q^{2}\pi}{4b^{2}}+\frac{8N_{c}Q^{2}}{3b^{3}}+\frac{8MQ^{2}}{3b^{3}}+O(M^{2},Q^{4}).$
(60)

The DA in non-plasma medium for a BH surrounded by a cosmological constant
depends upon impact parameter $b$, BH mass $M$, charge $Q$ and cosmological
field structure parameter $N_{c}$. The above conclusion simplifies to the DA
of Reissner-Nordström BH in the absence of the structure parameter $N_{c}=0$.
Furthermore, we obtain the DA of Schwarzschild BH $\psi_{Sch}=4M/b$, in the
absence of charge $Q=0$.

Figure 5: Deflection angle $\psi_{5}$ via impact parameter $b$ with variations
of charge $Q$ and cosmological field structure parameter $N_{c}$ with fixed
mass $M=1$.

Fig. 5: in the left panel charge $Q$ varied with fixed mass and the radiation
field structure parameter $N_{c}=0.1$, while the right panel gives variations
of cosmological field structure parameter $N_{c}$ with fixed mass $M=1$ and
charge $Q=0.5$.

The Schwarzschild solution is shown in both panels by the black curve. As the
impact parameter grows from a BH surrounded by cosmological field, the DA
decreases with increasing charge and structure parameter values, reaching an
asymptotically flat form until $b\rightarrow\infty$. When the impact parameter
is small, a strong deflection is seen. In addition, an inverse relationship is
established between the impact parameter $b$ and the DA $\psi_{5}$. It should
be noted that, for varying charges, the light deflects at certain spots in the
right plot at the same angle as compared to the Schwarzschild case, a
significant deflection is shown in the presence of cosmological field.

##  IV  DEFLECTION ANGLE IN THE CONTEXT OF PLASMA MEDIUM

This section studies the computation of the DA in the context of plasma
medium. Plasma is another important component that influences a gravitational
lens R58 ; Tsupko:2021yca . Refraction in plasma causes increased deflection.
The refractive index attempts to obtain auxiliary components and is especially
important in the radio regime. Consider $v$ be the velocity of light as it
travels through hot, ionised gas to understand the impacts of plasma. The
refractive index, $n(r)=c/v$ for a BH surrounded by dust field is given by R41
.

$n(r)=\sqrt{1-\frac{\omega^{2}_{e}}{\omega^{2}_{\infty}}A(r)},$ (61)

where $c=1$ as well as $\omega^{2}_{\infty}$ is the photon frequency measured
by an observer at infinity and $\omega^{2}_{e}$ is the electron plasma
frequency. The optical metric in terms of plasma medium can be defined as

$d\sigma^{2}=g^{opt}_{ij}dx^{i}dx^{j}=n^{2}\left(\frac{B(r)}{A(r)}dr^{2}+\frac{r^{2}}{A(r)}d\phi^{2}\right).$
(62)

The optical Gaussian curvature for a BH surrounded by dust field in terms of
plasma frame can be calculated as follows

$\displaystyle K$ $\displaystyle\approx$
$\displaystyle\frac{2M}{r^{3}}+\frac{(1+6k\lambda)N_{d}}{r^{3}}+\frac{12kM\lambda}{r^{3}}-\frac{3Q^{2}}{r^{4}}+\frac{3(2k\lambda-1)MN_{d}}{r^{4}}+\frac{6MQ^{2}}{r^{5}}+\frac{3(2-\lambda)N_{d}Q^{2}}{2r^{5}}+\frac{6M\omega^{2}_{e}}{r^{3}\omega^{2}_{\infty}}+\frac{3N_{d}\omega^{2}_{e}}{r^{3}\omega^{2}_{\infty}}$
(63) $\displaystyle-$ $\displaystyle\frac{15\lambda
kN_{d}\omega^{2}_{e}}{2r^{3}\omega^{2}_{\infty}}-\frac{12MN_{d}\omega^{2}_{e}}{r^{4}\omega^{2}_{\infty}}-\frac{5Q^{2}\omega^{2}_{e}}{r^{4}\omega^{2}_{\infty}}+\frac{30kM\lambda
N_{d}\omega^{2}_{e}}{r^{4}\omega^{2}_{\infty}}+\frac{13N_{d}Q^{2}\omega^{2}_{e}}{r^{5}\omega^{2}_{\infty}}+\frac{26Q^{2}M\omega^{2}_{e}}{r^{5}\omega^{2}_{\infty}}-\frac{15\lambda
kN_{d}Q^{2}\omega^{2}_{e}}{r^{5}\omega^{2}_{\infty}}$ $\displaystyle+$
$\displaystyle O(M^{2},Q^{3},\lambda^{2},\omega^{4}_{e}).$

Using the GBT, we calculate the bending angle. To do this, we use the straight
line approximation $e=\frac{b}{\sin\phi}$ at zeroth order, and the DA is
obtained as

$\psi_{1}\approx-\int^{\pi}_{0}\int^{\infty}_{b/\sin\phi}K\sqrt{det\vec{g}}drd\phi.$
(64)

The DA of the BH surrounded by dust field in terms of plasma frame for the
leading order terms is computed as

$\displaystyle\psi_{1}$ $\displaystyle\approx$
$\displaystyle\frac{4M}{b}-\frac{2N_{d}}{b}-\frac{12N_{d}k\lambda}{b}+\frac{24Mk\lambda}{b}-\frac{3Q^{2}\pi}{4b^{2}}-\frac{3MN_{d}\pi}{2b^{2}}-\frac{3MN_{d}k\lambda\pi}{2b^{2}}-\frac{4NQ^{2}}{3b^{3}}+\frac{8MQ^{2}}{3b^{3}}+\frac{2N_{d}Q^{2}k\lambda}{3b^{3}}+\frac{6M\omega^{2}_{e}}{b\omega^{2}_{\infty}}$
(65) $\displaystyle+$
$\displaystyle\frac{6N_{d}\omega^{2}_{e}}{b\omega^{2}_{\infty}}-\frac{15N_{d}k\lambda\omega^{2}_{e}}{b\omega^{2}_{\infty}}-\frac{5Q^{2}\pi\lambda\omega^{2}_{e}}{4b^{2}\omega^{2}_{\infty}}+\frac{15MN_{d}k\lambda\omega^{2}_{e}}{2b^{2}\omega^{2}_{\infty}}-\frac{21MN_{d}\pi\omega^{2}_{e}}{4b^{2}\omega^{2}_{\infty}}+\frac{52N_{d}Q^{2}\omega^{2}_{e}}{9b^{3}\omega^{2}_{\infty}}+\frac{104MQ^{2}\omega^{2}_{e}}{9b^{3}\omega^{2}_{\infty}}-\frac{20N_{d}Q^{2}k\lambda\omega^{2}_{e}}{3b^{3}\omega^{2}_{\infty}}$
$\displaystyle+$ $\displaystyle O(M^{2},Q^{4},\lambda^{2},\omega^{4}_{e}).$

The DA for a BH surrounded by a dust field depends on impact parameter $b$, BH
mass $M$, charge $Q$, Rastall geometric parameters $k$ and $\lambda$, field
structure parameters $N_{d}$ as well as plasma frequencies $\omega_{\infty}$ &
$\omega_{e}$. Moreover, it is important to mention here that, when, we ignore
the plasma effects in Eq. (65) such that
$\omega^{2}_{e}/\omega^{2}_{\infty}\rightarrow 0$, then, we recover the
results of Eq. (44) in a non-plasma medium.

Figure 6: Deflection angle $\psi_{1}$ w.r.t impact parameter $b$ with
variations of ratio of plasma frequencies $\omega^{2}_{e}/\omega^{2}_{\infty}$
and Rastall parameters $k\lambda$ with fixed mass $M=1$.

In Fig. 6: the left panel shows the variations of ratio of plasma frequencies
$\omega^{2}_{e}/\omega^{2}_{\infty}$ with fixed mass $M=1$, charge $Q=0.2$,
Rastall parameters $k\lambda=0.1$ and dust field structure parameter
$N_{d}=0.2$ and right panel represents the variations of Rastall parameters
$k\lambda$ with fixed mass $M=1$, charge $Q=0.2$, ratio of plasma frequencies
$\omega^{2}_{e}/\omega^{2}_{\infty}=0.8$ and dust field structure parameter
$N_{d}=0.5$. It is evident that when the impact parameter rises, the DA for
increasing plasma frequencies and Rastall parameters from a BH encircled by a
dust field decreases. A significant inclination is observable at low impact
parameter values where we observe the strongest deflection. The inverse link
between impact parameter $b$ and DA $\psi_{1}$ is also visually apparent.

###  IV.1 BLACK HOLE SURROUNDED BY THE QUINTESSENCE FIELD

The Gaussian curvature for a BH surrounded by a quintessence field under the
influence of plasma medium can be given as

$\displaystyle K$ $\displaystyle\approx$
$\displaystyle\frac{2M}{r^{3}}+\frac{4Q^{2}N_{q}}{r^{3}}+\frac{9Q^{2}k\lambda
N_{q}}{2r^{3}}-\frac{3Q^{2}}{r^{4}}+\frac{6MQ^{2}}{r^{5}}+\frac{3M\omega^{2}_{e}}{r^{3}\omega^{2}_{\infty}}+\frac{9N_{q}k\lambda
Q^{2}\omega^{2}_{e}}{r^{3}\omega^{2}_{\infty}}+\frac{11N_{q}Q^{2}\omega^{2}_{e}}{r^{3}\omega^{2}_{\infty}}-\frac{5Q^{2}\omega^{2}_{e}}{r^{4}\omega^{2}_{\infty}}+\frac{26MQ^{2}\omega^{2}_{e}}{r^{5}\omega^{2}_{\infty}}$
(66) $\displaystyle+$ $\displaystyle
O(M^{2},Q^{4},\lambda^{2},\omega^{4}_{e}).$

The DA of a BH surrounded by the quintessence field in a plasma frame for the
leading order terms is calculated as

$\displaystyle\psi_{2}$ $\displaystyle\approx$
$\displaystyle\frac{4M}{b}+\frac{8N_{q}Q^{2}}{b}+\frac{9N_{q}Q^{2}k\lambda}{b}-\frac{3Q^{2}\pi}{4b^{2}}+\frac{8MQ^{2}}{3b^{3}}+\frac{6M\omega^{2}_{e}}{b\omega^{2}_{\infty}}+\frac{22N_{q}Q^{2}\omega^{2}_{e}}{b\omega^{2}_{\infty}}+\frac{18N_{q}k\lambda
Q^{2}\omega^{2}_{e}}{b\omega^{2}_{\infty}}-\frac{5Q^{2}\pi\omega^{2}_{e}}{4b^{2}\omega^{2}_{\infty}}$
(67) $\displaystyle+$
$\displaystyle\frac{104MQ^{2}\omega^{2}_{e}}{9b^{3}\omega^{2}_{\infty}}+O(M^{2},Q^{4},\lambda^{2},\omega^{4}_{e}).$

The DA in plasma medium for a BH surrounded by a quintessence field depends on
impact parameter $b$, BH mass $M$, charge $Q$, Rastall geometric parameters
$k$ and $\lambda$, field structure parameters $N_{q}$ as well as plasma
frequencies $\omega_{\infty}$ & $\omega_{e}$. It has also worth to note that,
when, we ignore the plasma effects in Eq. (67) such that
$\omega^{2}_{e}/\omega^{2}_{\infty}\rightarrow 0$, then, we recover the
results of Eq. (48) in a non-plasma medium.

Figure 7: Deflection angle $\psi_{2}$ versus impact parameter $b$ with
variations in ratio of plasma frequencies $\omega^{2}_{e}/\omega^{2}_{\infty}$
and Rastall parameters $k\lambda$ with fixed mass, quintessential field
structure parameter $M=1=N_{d}$ and charge $Q=0.5$.

In Fig. 7: the left panel displays variations in ratio of plasma frequencies
$\omega^{2}_{e}/\omega^{2}_{\infty}$ with fixed mass, quintessential field
structure parameter $M=1=N_{d}$, charge $Q=0.5$ and Rastall parameters
$k\lambda=0.1$, whereas variations in Rastall parameters $k\lambda$ with with
fixed mass, quintessential field structure parameter $M=1=N_{d}$, charge
$Q=0.5$ and ratio of plasma frequencies
$\omega^{2}_{e}/\omega^{2}_{\infty}=0.8$ are represented in the right panel.
The DA for increasing plasma frequencies and Rastall parameter values is shown
to decrease via growing impact parameter from a BH surrounded by a
quintessential field, eventually achieving an asymptotically flat form until
$b\rightarrow\infty$. A notable deviation with strong deflection is observed
when the impact parameter is minimal. Furthermore, the inverse link between DA
$\psi_{2}$ and impact parameter $b$ is also noted.

###  IV.2 BLACK HOLE SURROUNDED BY THE RADIATION FIELD

The Gaussian curvature for the BH metric surrounded by a radiation field in
the presence of plasma can be calculated as follows

$\displaystyle K$ $\displaystyle\approx$
$\displaystyle\frac{2M}{r^{3}}-\frac{3Q^{2}}{r^{4}}+\frac{3N_{r}}{r^{4}}+\frac{6MQ^{2}}{r^{5}}-\frac{6MN_{r}}{r^{5}}+\frac{4N_{r}Q^{2}}{r^{6}}+\frac{3M\omega^{2}_{e}}{r^{3}\omega^{2}_{\infty}}-\frac{5Q^{2}\omega^{2}_{e}}{r^{4}\omega^{2}_{\infty}}+\frac{5N_{r}\omega^{2}_{e}}{r^{4}\omega^{2}_{\infty}}+\frac{26MQ^{2}\omega^{2}_{e}}{r^{5}\omega^{2}_{\infty}}-\frac{26MN_{r}\omega^{2}_{e}}{r^{5}\omega^{2}_{\infty}}$
(68) $\displaystyle+$
$\displaystyle\frac{20Q^{2}N_{r}\omega^{2}_{e}}{r^{6}\omega^{2}_{\infty}}+O(M^{2},Q^{4},\omega^{4}_{e}).$

The DA of a BH surrounded by the radiation field in a plasma medium can be
computed as

$\displaystyle\psi_{3}$ $\displaystyle\approx$
$\displaystyle\frac{4M}{b}-\frac{3Q^{2}\pi}{4b^{2}}+\frac{3N_{r}\pi}{4b^{2}}+\frac{8MQ^{2}}{3b^{3}}-\frac{8MN_{r}}{3b^{3}}+\frac{3N_{r}Q^{2}\pi}{8b^{4}}+\frac{6M\omega^{2}_{e}}{b\omega^{2}_{\infty}}+\frac{5N_{r}\pi\omega^{2}_{e}}{4b^{2}\omega^{2}_{\infty}}-\frac{5Q^{2}\pi\omega^{2}_{e}}{4b^{2}\omega^{2}_{\infty}}-\frac{104MN_{r}\omega^{2}_{e}}{9b^{3}\omega^{2}_{\infty}}$
(69) $\displaystyle+$
$\displaystyle\frac{104MQ^{2}\omega^{2}_{e}}{9b^{3}\omega^{2}_{\infty}}+\frac{15NQ^{2}\pi\omega^{2}_{e}}{8b^{4}\omega^{2}_{\infty}}+O(M^{2},Q^{4},\omega^{4}_{e}).$

The DA in plasma frame for a BH surrounded by a radiation field is dependent
on impact parameter $b$, BH mass $M$, charge $Q$, radiation field structure
parameter $N_{r}$ as well as plasma frequencies $\omega_{\infty}$ &
$\omega_{e}$. It has also worth to mention here that, when, we neglect the
plasma effects in Eq. (69) i. e.,
$\omega^{2}_{e}/\omega^{2}_{\infty}\rightarrow 0$, then, we obtain the results
of Eq. (52) in a non-plasma frame.

Figure 8: Deflection angle $\psi_{3}$ via impact parameter $b$ with variations
of plasma frequencies ratio $\omega^{2}_{e}/\omega^{2}_{\infty}$ and charge
$Q$ with fixed mass $M=1$ and radiation field structure parameter $N_{r}$.

Fig. 8: in the left panel charge plasma frequencies ratio
$\omega^{2}_{e}/\omega^{2}_{\infty}$ varies with fixed mass and the radiation
field structure parameter $M=1$, charge $Q=0.5$ and radiation field structure
parameter $N_{r}=0.1$, whereas the right panel shows variations of charge $Q$
with fixed mass $M=1$, plasma frequencies ratio
$\omega^{2}_{e}/\omega^{2}_{\infty}=0.8$ and radiation field structure
parameter $N_{r}=0.1$. As the impact parameter grows from a BH surrounded by a
radiation field, the DA with increasing charge and plasma frequencies ratio
diminishes, reaching an asymptotically flat form until $b\rightarrow\infty$. A
significant deflection is shown when the impact parameter is small. Moreover,
an inverse relationship between the impact parameter $b$ and DA $\psi_{3}$ is
discovered. A strong deflection occurs with the increasing ratio of plasma
frequencies.

###  IV.3 BLACK HOLE SURROUNDED BY THE PHANTOM FILED

In the presence of plasma frame, the curvature for a BH surrounded by a
phantom field can be computed in the following way

$\displaystyle K$ $\displaystyle\approx$
$\displaystyle\frac{2M}{r^{3}}+\frac{4k\lambda
M}{r^{3}}-\frac{3Q^{2}}{r^{4}}-\frac{35N_{p}k\lambda
Q^{2}}{2r^{4}}+\frac{6MQ^{2}}{r^{5}}-\frac{11N_{p}M}{r^{5}}+\frac{3M\omega^{2}_{e}}{r^{3}\omega^{2}_{\infty}}-\frac{4Mk\lambda\omega^{2}_{e}}{r^{3}\omega^{2}_{\infty}}-\frac{5Q^{2}\omega^{2}_{e}}{r^{4}\omega^{2}_{\infty}}-\frac{55N_{p}k\lambda
Q^{2}\omega^{2}_{e}}{r^{4}\omega^{2}_{\infty}}$ (70) $\displaystyle-$
$\displaystyle\frac{36MN_{p}\omega^{2}_{e}}{r^{5}\omega^{2}_{\infty}}+\frac{26MQ^{2}\omega^{2}_{e}}{r^{5}\omega^{2}_{\infty}}+O(M^{2},Q^{4},\lambda^{2},\omega^{4}_{e}).$

The DA in a plasma medium of the BH surrounded by a phantom field is derived
as

$\displaystyle\psi_{4}$ $\displaystyle\approx$
$\displaystyle\frac{4M}{b}+\frac{8Mk\lambda}{b}-\frac{35N_{p}k\lambda
Q^{2}\pi}{8b^{2}}-\frac{3Q^{2}\pi}{4b^{2}}+\frac{8MQ^{2}}{3b^{3}}-\frac{44N_{p}M}{9b^{3}}+\frac{6M\omega^{2}_{e}}{b\omega^{2}_{\infty}}-\frac{8Mk\lambda\omega^{2}_{e}}{b\omega^{2}_{\infty}}-\frac{5Q^{2}\pi\omega^{2}_{e}}{4b^{2}\omega^{2}_{\infty}}$
(71) $\displaystyle-$ $\displaystyle\frac{55N_{p}k\lambda
Q^{2}\pi\omega^{2}_{e}}{4b^{2}\omega^{2}_{\infty}}-\frac{16N_{p}M\omega^{2}_{e}}{b^{3}\omega^{2}_{\infty}}+\frac{104MQ^{2}\omega^{2}_{e}}{9b^{3}\omega^{2}_{\infty}}+O(M^{2},Q^{4},\lambda^{2},\omega^{4}_{e}).$

The DA in plasma medium for a BH surrounded by a phantom field depends upon
impact parameter $b$, BH mass $M$, charge $Q$, Rastall geometric parameters
$k$ and $\lambda$, field structure parameters $N_{p}$ as well as plasma
frequencies $\omega_{\infty}$ & $\omega_{e}$. It has important to mention here
that, when, we neglect the plasma effects in Eq. (71) i. e.,
$\omega^{2}_{e}/\omega^{2}_{\infty}\rightarrow 0$, then, we obtain the results
of Eq. (56) in a non-plasma medium.

Figure 9: Deflection angle $\psi_{4}$ w.r.t impact parameter $b$ with
variations of plasma frequencies ratio $\omega^{2}_{e}/\omega^{2}_{\infty}$
and Rastall parameters $k\lambda$ with fixed mass $M=1$, charge $Q=0.5$ and
phantom field structure parameter $N_{p}=0.5$.

Fig. 9: in the left panel charge plasma frequencies ratio
$\omega^{2}_{e}/\omega^{2}_{\infty}$ show variations with fixed mass $M=1$,
charge $Q=0.3$, Rastall parameters $k\lambda=0.1$ and the phantom field
structure parameter $N_{p}=0.5$, while the right panel represents the
variations of Rastall parameters $k\lambda$ with fixed mass $M=1$, charge
$Q=0.3$, plasma frequencies ratio $\omega^{2}_{e}/\omega^{2}_{\infty}=0.1$ and
phantom field structure parameter $N_{p}=0.5$. Due to a rising impact
parameter from a BH surrounded by a phantom field, the DA falls as the values
of the plasma frequencies ratio and Rastall parameters rise. When the impact
parameter is modest, a significant deflection is observed. Furthermore, a
negative correlation is seen between the DA $\psi_{4}$ and the impact
parameter $b$.

###  IV.4 BLACK HOLE SURROUNDED BY THE COSMOLOGICAL CONSTANT

In order to calculate the DA for BH surrounded by cosmological constant, we
first compute the Gaussian curvature in the following way

$\displaystyle K$ $\displaystyle\approx$
$\displaystyle\frac{2M}{r^{3}}+\frac{N_{c}}{r^{3}}-\frac{3Q^{2}}{r^{4}}-\frac{6MN_{c}}{r^{4}}+\frac{6N_{c}Q^{2}}{r^{5}}+\frac{6MQ^{2}}{r^{5}}+\frac{3M\omega^{2}_{e}}{r^{3}\omega^{2}_{\infty}}+\frac{3N_{c}\omega^{2}_{e}}{r^{3}\omega^{2}_{\infty}}-\frac{5Q^{2}\omega^{2}_{e}}{r^{4}\omega^{2}_{\infty}}+\frac{16N_{c}Q^{2}\omega^{2}_{e}}{r^{5}\omega^{2}_{\infty}}+\frac{18MN_{c}\omega^{2}_{e}}{r^{4}\omega^{2}_{\infty}}$
(72) $\displaystyle+$
$\displaystyle\frac{26MQ^{2}\omega^{2}_{e}}{r^{5}\omega^{2}_{\infty}}+O(M^{2},Q^{4},\omega^{4}_{e}).$

In a plasma frame, the DA of the BH surrounded by the cosmological constant
for the leading order terms is calculated as

$\displaystyle\psi_{5}$ $\displaystyle\approx$
$\displaystyle\frac{4M}{b}+\frac{2N_{c}}{b}-\frac{3MN_{c}\pi}{2b^{2}}-\frac{3Q^{2}\pi}{4b^{2}}+\frac{8N_{c}Q^{2}}{3b^{3}}+\frac{8MQ^{2}}{3b^{3}}+\frac{6M\omega^{2}_{e}}{b\omega^{2}_{\infty}}+\frac{6N_{c}\omega^{2}_{e}}{b\omega^{2}_{\infty}}-\frac{5Q^{2}\pi\omega^{2}_{e}}{4b^{2}\omega^{2}_{\infty}}+\frac{9MN_{c}\pi\omega^{2}_{e}}{2b^{2}\omega^{2}_{\infty}}+\frac{4N_{c}Q^{2}\omega^{2}_{e}}{b^{3}\omega^{2}_{\infty}}$
(73) $\displaystyle+$
$\displaystyle\frac{104MQ^{2}\omega^{2}_{e}}{9b^{3}\omega^{2}_{\infty}}+O(M^{2},Q^{4},\omega^{4}_{e}).$

The DA in plasma medium for a BH surrounded by a cosmological constant depends
upon impact parameter $b$, BH mass $M$, charge $Q$, cosmological field
structure parameter $N_{c}$ as well as plasma frequencies $\omega_{e}$ &
$\omega_{\infty}$. It has also worth to mention here that, when, we neglect
the plasma effects in Eq. (73) i. e.,
$\omega^{2}_{e}/\omega^{2}_{\infty}\rightarrow 0$, then, we get the results of
Eq. (60) in a non-plasma frame.

Figure 10: Deflection angle $\psi_{5}$ via impact parameter $b$ with
variations of plasma frequencies ratio $\omega^{2}_{e}/\omega^{2}_{\infty}$
and charge $Q$ with fixed mass $M=1$ and cosmological field structure
parameter $N_{c}=0.5$.

Fig. 10: in the left panel plasma frequencies ratio varies with fixed mass
$M=1$, charge $Q=0.3$ and the radiation field structure parameter $N_{c}=0.5$,
while the right panel gives variations of charge $Q$ with fixed mass $M=1$,
cosmological field structure parameter $N_{c}=0.5$ and plasma frequencies
ratio $\omega^{2}_{e}/\omega^{2}_{\infty}=0.8$.

The DA reduces with increasing plasma frequencies ratio and charge, achieving
an asymptotically flat form until $b\rightarrow\infty$. This drop occurs as
the impact parameter grows from a BH surrounded by cosmological field. There
is a significant strong deflection observed when the impact parameter is
modest. Furthermore, a negative correlation is seen between the DA $\psi_{5}$
and the impact parameter $b$.

##  V EQUATIONS OF MOTION FOR LIGHT RAYS IN A NON-MAGNETIZED PLASMA

This study examines the shadow cast by a BH surrounded by a dust field given
in Eq. (22) and the impact of Rastall parameters on it. The core of the
critical curve, also known as the apparent boundary, is represented by a BH
shadow. This curve has been optimised so that light rays that are a part of it
approach a bound orbit of photons asymptotically when a distant observer
follows them back to the BH Perlick:2021aok ; R58 .

For a photon, the Hamilton-Jacobi method in the equatorial plane
$\theta=\frac{\pi}{2}$ is given as

$H=\frac{1}{2}\left(g^{ik}j_{i}j_{k}+\omega_{p}(r^{2})\right)=\frac{1}{2}\left(-\frac{y^{2}_{t}}{A(r)}+\frac{y^{2}_{r}}{B(r)}+\frac{y^{2}_{\phi}}{D(r)}+\omega_{p}(r^{2})\right),$
(74)

where $j_{i}$ represents the momentum of the photon, $y_{\phi}=j_{\phi}$ is
its angular momentum and $y_{t}=-j_{t}$ is the photon’s energy. The light rays
are the solutions to Hamilton’s equations for a derivation of the Hamiltonian
Eq. (74) from Maxwell’s equations with a two-fluid source as

$\dot{y_{i}}=-\frac{\partial H}{\partial
x^{i}},~{}~{}~{}~{}~{}~{}~{}~{}~{}\dot{x^{i}}=-\frac{\partial H}{\partial
y_{i}}.$ (75)

The above equations implies

$\displaystyle\dot{y_{t}}$ $\displaystyle=$ $\displaystyle-\frac{\partial
H}{\partial t}=0,~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\dot{y_{\phi}}=-\frac{\partial
H}{\partial\psi}=0,$ (76) $\displaystyle\dot{y_{r}}$ $\displaystyle=$
$\displaystyle-\frac{\partial H}{\partial
r}=\frac{1}{2}\left(-\frac{y^{2}_{t}A^{\prime}(r)}{A(r)^{2}}+\frac{y^{2}_{r}B^{\prime}(r)}{B(r)^{2}}+\frac{y^{2}_{\phi}D^{\prime}(r)}{D(r)^{2}}-\frac{d}{dr}\omega_{p}(r^{2})\right),$
(77) $\displaystyle\dot{t}$ $\displaystyle=$ $\displaystyle\frac{\partial
H}{\partial y_{t}}=-\frac{y_{t}}{A(r)},$ (78) $\displaystyle\dot{\phi}$
$\displaystyle=$ $\displaystyle\frac{\partial H}{\partial
y_{\phi}}=-y_{\phi},$ (79) $\displaystyle\dot{r}$ $\displaystyle=$
$\displaystyle\frac{\partial H}{\partial y_{r}}=-\frac{y_{r}}{B(r)}.$ (80)

In this case, a prime denotes differentiation with regard to $r$, while a dot
denotes differentiation with respect to an affine parameter $\lambda$.

By setting $H=0$ in Eq. (74), we get the result in this form

$0=-\frac{y^{2}_{t}}{A(r)}+\frac{y^{2}_{r}}{B(r)}+\frac{y^{2}_{\phi}}{D(r)}+\omega_{p}(r^{2}).$
(81)

From Eq. (76), it is followed that the $y_{t}$ and $y_{\phi}$ are constants of
motion. We consider $\omega_{0}=-j_{t}$. Given a fixed $\omega_{0}$ and an
asymptotically flat spacetime (e.g., $A(r)\rightarrow 1$ as
$r\rightarrow\infty$), the gravitational redshift formula transforms the
frequency $\omega$ as measured by a static observer into a function of $r$ in
the given form

$\omega(r)=\frac{\omega_{0}}{\sqrt{A(r)}}.$ (82)

As a result of Eq. (81), a light beam with constant speed $\omega_{0}$ is
limited to the area where

$\frac{\omega^{2}_{0}}{A(r)}>\omega_{p}(r)^{2}.$ (83)

The constraint (82) states that the photon frequency $\omega(r)$ must exceed
the plasma frequency $\omega_{p}(r)$ at a given place. For light to propagate
through a plasma, this is always true. In order to obtain the orbit equation,
we use Eq. (78) and Eq. (79) and get

$\frac{dr}{d\phi}=\frac{\dot{r}}{\dot{\phi}}=\frac{D(r)y_{r}}{B(r)y_{\phi}}.$
(84)

Using $y_{r}$ from Eq. (81), we get the result in this form

$\frac{dr}{d\phi}=\pm\frac{\sqrt{D(r)}}{\sqrt{B(r)}}\sqrt{\frac{\omega^{2}_{0}v(r)^{2}}{y^{2}_{\phi}}-1},$
(85)

where

$v(r)^{2}=\frac{D(r)}{A(r)}\left(1-A(r)\frac{\omega_{p}^{2}}{\omega_{0}^{2}}\right).$
(86)

Since $X$ is the turning point of the trajectory, the condition
$\frac{dr}{d\phi}\Big{|}_{X}=0$ must hold. This equation connects $X$ to the
constant of motion, $\frac{y_{\phi}}{\omega_{0}}$ as follows

$v(X)^{2}=\frac{y_{\psi}^{2}}{\omega_{0}^{2}}.$ (87)

The shadow’s radius is defined by the original direction of the photon’s
light, which is asymptotically radial towards the outer photon sphere. In
order to calculate the shadow of the BH surrounded by a dust field, we
consider a light beam projected with an angle $\gamma$ in the radial direction
from the observer point $r_{0}$. We define $\gamma$ in the following way

$\cot\gamma=\pm\frac{\sqrt{g_{rr}}}{\sqrt{g_{\phi\phi}}}\frac{dr}{d\phi}\Big{|}_{r=r_{0}}=\pm\frac{\sqrt{B(r)}}{\sqrt{D(r)}}\frac{dr}{d\phi}\Big{|}_{r=r_{0}}.$
(88)

We rewrite the orbit equation Eq. (85) after attaining a minimum radius $R$ by
using Eq. (87) in the form

$\frac{dr}{d\phi}=\pm\frac{\sqrt{D(r)}}{\sqrt{B(r)}}\sqrt{\frac{v(r_{0})^{2}}{v(X)^{2}}-1}.$
(89)

Using Eq. (89) into Eq. (88) for the angle $\gamma$, we get

$\cot^{2}\gamma={\frac{v(r_{0})^{2}}{v(X)^{2}}-1}.$ (90)

By using the identity, we obtain

$1+\cot^{2}\gamma=\frac{1}{\sin^{2}\gamma}.$ (91)

Using Eq. (91) into (90), we get the result

$\sin^{2}\gamma=\frac{v(X)^{2}}{v(r_{0})^{2}}.$ (92)

Light rays spiralling asymptotically towards a circular light orbit at radius
$r_{ph}$ define the boundary of the shadow $\gamma$. Consequently,
$X\rightarrow r_{ph}$ in Eq. (92) provides the angular radius of the shadow as

$\sin^{2}\gamma=\frac{v(r_{ph})^{2}}{v(r_{0})^{2}},$ (93)

where $v(r_{0})$ can be represented by the formula in Eq. (85) at $r=r_{0}$.
In numerous cases, it is possible to presume that the observer is situated in
an area with a negligible plasma density. Then Eq. (85) implies

$v(r_{0})^{2}=\frac{D(r_{0})}{A(r_{0})},$ (94)

and Eq. (93) can be written as follows

$\sin^{2}\gamma=\frac{A(r_{0})D(r_{ph})}{A(r_{ph})D(r_{0})}\left(1-\frac{A(r_{ph})\omega_{p}^{2}(r_{ph})}{\omega^{2}_{o}}\right).$
(95)

After substituting the values of metric function into the above equation, we
obtain the result into following manner

$\sin^{2}\gamma=\frac{r^{2}_{ph}\left(2Mr_{0}-Q^{2}-r^{2}_{0}+N_{d}r^{\frac{1}{1-3k\lambda}}_{0}\right)\left(Q^{2}\omega^{2}_{p}-2Mr_{ph}\omega^{2}_{p}+r^{2}_{ph}(\omega^{2}_{P}-\omega^{2}_{0})-N_{d}r^{\frac{1}{1-3k\lambda}}_{ph}\omega^{2}_{p}\right)}{r^{4}_{0}\left(Q^{2}-2Mr_{ph}+r^{2}_{ph}-N_{d}r^{\frac{1}{1-3k\lambda}}_{ph}\right)\omega^{2}_{0}}.$
(96)

The shadow of the BH surrounded by a dust field is dependent on BH mass $M$,
charge $Q$, Rastall parameters $k,~{}\lambda$, dust field structure parameter
$N_{d}$, plasma frequencies $\omega_{p},~{}\omega_{0}$, photon radius $r_{ph}$
and observer radius $r_{0}$.

Figure 11: Shadow of BH surrounded by dust field in celestial plane
$(\alpha,\beta)$ for variations of photon radius $r_{ph}$ and observer radius
$r_{0}$.

Fig. 11: the left panel shows the contour plots of shadow for variations of
photon radius $r_{ph}=0.5$ (purple), $0.6$ (yellow), $0.7$ (green), $0.8$
(orange), $0.9$ (blue) and $1$ (red) with fixed values of mass $M=1$, observer
radius $r_{0}=1$, charge $Q=0.3$, plasma frequencies
$\omega_{p}=0.1,~{}\omega_{0}=0.5$, Rastall parameters $k\lambda=0.1$ and dust
field structure parameter $N_{d}=0.5$. One can see that when the photon radius
rises, the shadow radius size also increases.

The right panel gives the contour plots of shadow for different choices of
radius $r_{0}=0.5$ (purple), $0.6$ (yellow), $0.7$ (green), $0.8$ (orange),
$0.9$ (blue), $1$ (red) and fixed mass $M=1$, charge $Q=0.3$, photon radius
$r_{ph}=1$, plasma frequencies $\omega_{p}=0.1,~{}\omega_{0}=0.5$, Rastall
parameters $k\lambda=0.1$ and dust field structure parameter $N_{d}=0.5$. It
can be observed that the size of shadow radius decreases with increasing
observers radius $r_{0}$ and it is bigger than photon radius.

###  V.1 BLACK HOLE SURROUNDED BY THE QUINTESSENCE FIELD

After putting the values of metric function from Eq. (45) into Eq. (95), we
get the result in the given form

$\sin^{2}\gamma_{1}=\frac{r^{2}_{ph}\left(2Mr_{0}-Q^{2}-r^{2}_{0}+N_{q}r^{1+\frac{3k\lambda}{-1+k\lambda}}_{0}\right)\left(Q^{2}\omega^{2}_{p}-2Mr_{ph}\omega^{2}_{p}+r^{2}_{ph}(\omega^{2}_{P}-\omega^{2}_{0})-N_{q}r^{1+\frac{3k\lambda}{-1+k\lambda}}_{ph}\omega^{2}_{p}\right)}{r^{4}_{0}\left(Q^{2}-2Mr_{ph}+r^{2}_{ph}-N_{q}r^{1+\frac{3k\lambda}{-1+k\lambda}}_{ph}\right)\omega^{2}_{0}}.$
(97)

The shadow of the BH surrounded by a quintessence field is dependent on BH
mass $M$, charge $Q$, Rastall parameters $k,~{}\lambda$, quintessence field
structure parameter $N_{q}$, plasma frequencies $\omega_{p},~{}\omega_{0}$,
photon radius $r_{ph}$ and observer radius $r_{0}$.

Figure 12: Shadow of BH surrounded by quintessential field in celestial plane
$(\alpha,\beta)$ for variations of photon radius $r_{ph}$ and observer radius
$r_{0}$.

Fig. 12: the left panel represents the contour plots of shadow for variations
of photon radius $r_{ph}=0.5$ (purple), $0.6$ (yellow), $0.7$ (green), $0.8$
(orange), $0.9$ (blue) and $1$ (red) with fixed values of mass $M=1$, observer
radius $r_{0}=1$, charge $Q=0.3$, plasma frequencies
$\omega_{p}=0.1,~{}\omega_{0}=0.5$, Rastall parameters $k\lambda=0.1$ and dust
field structure parameter $N_{q}=0.5$. We can observe that when the photon
radius rises, the shadow radius size also increases.

The right panel gives the contour plots of shadow for different choices of
radius $r_{0}=0.5$ (purple), $0.6$ (yellow), $0.7$ (green), $0.8$ (orange),
$0.9$ (blue), $1$ (red) and fixed mass $M=1$, charge $Q=0.3$, photon radius
$r_{ph}=1$, plasma frequencies $\omega_{p}=0.1,~{}\omega_{0}=0.5$, Rastall
parameters $k\lambda=0.1$ and dust field structure parameter $N_{q}=0.5$. It
can be observed that the size of shadow radius decreases with increasing
observers radius $r_{0}$.

###  V.2 BLACK HOLE SURROUNDED BY THE RADIATION FIELD

After substituting the values of metric function from Eq. (49) into Eq. (95),
we obtain the result in the given form

$\sin^{2}\gamma_{2}=\frac{r^{2}_{ph}\left(2Mr_{0}-Q^{2}-r^{2}_{0}+N_{r}\right)\left(Q^{2}\omega^{2}_{p}-2Mr_{ph}\omega^{2}_{p}+r^{2}_{ph}(\omega^{2}_{P}-\omega^{2}_{0})-N_{r}\omega^{2}_{p}\right)}{r^{4}_{0}\left(Q^{2}-2Mr_{ph}+r^{2}_{ph}-N_{r}\right)\omega^{2}_{0}}.$
(98)

The shadow of the BH surrounded by a phantom field depends upon BH mass $M$,
charge $Q$, radiation field structure parameter $N_{r}$, plasma frequencies
$\omega_{p},~{}\omega_{0}$, photon radius $r_{ph}$ and observer radius
$r_{0}$.

Figure 13: Shadow of BH surrounded by radiation field in celestial plane
$(\alpha,\beta)$ for variations of photon radius $r_{ph}$ and observer radius
$r_{0}$.

Fig. 13: the left panel illustrate the contour plots of shadow for variations
of photon radius $r_{ph}=0.5$ (purple), $0.6$ (yellow), $0.7$ (green), $0.8$
(orange), $0.9$ (blue) and $1$ (red) with fixed values of mass $M=1$, observer
radius $r_{0}=1$, charge $Q=0.3$, plasma frequencies
$\omega_{p}=0.1,~{}\omega_{0}=0.5$, Rastall parameters $k\lambda=0.1$ and dust
field structure parameter $N_{r}=0.5$. We can see that the shadow radius size
increases along with the photon radius.

The right panel gives the contour plots of shadow for different choices of
radius $r_{0}=0.5$ (purple), $0.6$ (yellow), $0.7$ (green), $0.8$ (orange),
$0.9$ (blue), $1$ (red) and fixed mass $M=1$, charge $Q=0.3$, photon radius
$r_{ph}=1$, plasma frequencies $\omega_{p}=0.1,~{}\omega_{0}=0.5$, Rastall
parameters $k\lambda=0.1$ and dust field structure parameter $N_{r}=0.5$. It
is shown that when the observers radius $r_{0}$ increases, the shadow radius’s
size reduces.

###  V.3 BLACK HOLE SURROUNDED BY THE PHANTOM FILED

After putting the values of metric function from Eq. (53) into Eq. (95), we
get the result as follows

$\sin^{2}\gamma_{3}=\frac{r^{2}_{ph}\left(2Mr_{0}-Q^{2}-r^{2}_{0}+N_{p}r^{\frac{5}{1+k\lambda}}_{0}\right)\left(Q^{2}\omega^{2}_{p}-2Mr_{ph}\omega^{2}_{p}+r^{2}_{ph}(\omega^{2}_{P}-\omega^{2}_{0})-N_{p}r^{\frac{5}{1+k\lambda}}_{ph}\omega^{2}_{p}\right)}{r^{4}_{0}\left(Q^{2}-2Mr_{ph}+r^{2}_{ph}-N_{p}r^{\frac{5}{1+k\lambda}}_{ph}\right)\omega^{2}_{0}}.$
(99)

The shadow of the BH surrounded by a phantom field depends on BH mass $M$,
charge $Q$, Rastall parameters $k,~{}\lambda$, phantom field structure
parameter $N_{p}$, plasma frequencies $\omega_{p},~{}\omega_{0}$, photon
radius $r_{ph}$ and observer radius $r_{0}$.

Figure 14: Shadow of BH surrounded by phantom field in celestial plane
$(\alpha,\beta)$ for variations of photon radius $r_{ph}$ and observer radius
$r_{0}$.

Fig. 14: the left panel depicts the contour plots of shadow for different
choices of photon radius $r_{ph}=0.5$ (purple), $0.6$ (yellow), $0.7$ (green),
$0.8$ (orange), $0.9$ (blue) and $1$ (red) with fixed values of mass $M=1$,
observer radius $r_{0}=1$, charge $Q=0.3$, plasma frequencies
$\omega_{p}=0.1,~{}\omega_{0}=0.5$, Rastall parameters $k\lambda=0.1$ and dust
field structure parameter $N_{p}=0.5$. It is evident that as the photon radius
rises, so does the shadow radius size.

The right panel shows the contour plots of shadow for variations of radius
$r_{0}=0.5$ (purple), $0.6$ (yellow), $0.7$ (green), $0.8$ (orange), $0.9$
(blue), $1$ (red) and fixed mass $M=1$, charge $Q=0.3$, photon radius
$r_{ph}=1$, plasma frequencies $\omega_{p}=0.1,~{}\omega_{0}=0.5$, Rastall
parameters $k\lambda=0.1$ and dust field structure parameter $N_{p}=0.5$. It
is demonstrated that the size of the shadow radius decreases with increasing
observers radius $r_{0}$.

###  V.4 BLACK HOLE SURROUNDED BY THE COSMOLOGICAL CONSTANT

After using the values of metric function from Eq. (57) into Eq. (95), we
acquire the result in the given form

$\sin^{2}\gamma_{4}=\frac{r^{2}_{ph}\left(2Mr_{0}-Q^{2}-r^{2}_{0}+N_{c}r^{4}_{0}\right)\left(Q^{2}\omega^{2}_{p}-2Mr_{ph}\omega^{2}_{p}+r^{2}_{ph}(\omega^{2}_{P}-\omega^{2}_{0})-N_{c}r^{4}_{ph}\omega^{2}_{p}\right)}{r^{4}_{0}\left(Q^{2}-2Mr_{ph}+r^{2}_{ph}-N_{c}r^{4}_{ph}\right)\omega^{2}_{0}}.$
(100)

The shadow of the BH surrounded by a cosmological constant depends on BH mass
$M$, charge $Q$, cosmological field structure parameter $N_{c}$, plasma
frequencies $\omega_{p},~{}\omega_{0}$, photon radius $r_{ph}$ and observer
radius $r_{0}$.

Figure 15: Shadow of BH surrounded by cosmological field in celestial plane
$(\alpha,\beta)$ for variations of photon radius $r_{ph}$ and observer radius
$r_{0}$.

Fig. 15: the left panel represents the contour plots of shadow for different
choices of photon radius $r_{ph}=0.5$ (purple), $0.6$ (yellow), $0.7$ (green),
$0.8$ (orange), $0.9$ (blue) and $1$ (red) with fixed values of mass $M=1$,
observer radius $r_{0}=1$, charge $Q=0.3$, plasma frequencies
$\omega_{p}=0.1,~{}\omega_{0}=0.5$, Rastall parameters $k\lambda=0.1$ and dust
field structure parameter $N_{c}=0.5$. It is clear that the shadow radius size
increases with the photon radius.

The right panel shows the contour plots of shadow for variations of radius
$r_{0}=0.5$ (purple), $0.6$ (yellow), $0.7$ (green), $0.8$ (orange), $0.9$
(blue), $1$ (red) and fixed mass $M=1$, charge $Q=0.3$, photon radius
$r_{ph}=1$, plasma frequencies $\omega_{p}=0.1,~{}\omega_{0}=0.5$, Rastall
parameters $k\lambda=0.1$ and dust field structure parameter $N_{c}=0.5$. It
is shown that as the observers radius $r_{0}$ increases, the shadow radius’s
size decreases.

##  VI Conclusions

In the framework of Rastall theory, we have studied generic charged BH
solutions surrounded by an ideal fluid. Next, we have looked more closely into
the particular situations of BHs encircled by dust, quintessence, radiation,
phantom fields and cosmological constant. In the generalized theory, physical
parameters are studied by applying the weak energy condition, which signifies
a positive energy density. An effective behaviour of the BH surrounding the
field is realized by comparing the solution in GR with the new term in the
metric that arose from the Rastall theory. The ray tracing and generalized GW
methods offer a comprehensive Rastall framework for characterizing strategies
for the Kiselev BH solution. In the non-plasma, plasma and perfect fluid
mediums, we have talked about the Rastall solution and derived DA by using the
GW technique. The Gaussian optical curvature is determined for this purpose
using an optical metric and we have then examined the DA using GBT.

Firstly, we investigated the DA for BHs surrounded by dust, quintessence,
radiation, phantom and cosmological fields in a non-plasma medium. We have
concluded from our results that BHs surrounded by dust, quintessence and
phantom fields are dependent on impact parameter $b$, BH mass $M$, charge $Q$,
Rastall geometric parameters $k$, $\lambda$ and field structure parameters
$N_{d},~{}N_{q}$ and $N_{p}$, respectively. Moreover, the DA for BHs
surrounded by radiation field and cosmological constant depends upon on impact
parameter $b$, BH mass $M$, charge $Q$ and field structure parameters $N_{r}$
and $N_{c}$, respectively. We also observed analytically and graphically that
the impact parameter $b$ is in an inverse relation with the DA of
corresponding BHs. Moreover, it has worth to mention here that, in the absence
of structure and Rastall parameters, the results of DA of our corresponding
BHs reduce into DA of Reissner–Nordström BH as well as in the absence of
charge $Q=0$, we recovered the DA of Schwarzschild BH $\psi_{Sch}=4M/b$.

With the help of graphical interpretation, we showed that the impact parameter
$b$ has an inverse relation with DA. Moreover, by comparing the graphical
conduct of DA of our corresponding BHs with Schwarzschild case, we concluded
that BHs surrounded by dust, quintessence fields and cosmological constants
have less deflection as compared to Schwarzschild BH. In addition, we observed
strong deflection is seen for small value of impact parameter in all plots of
non-plasma medium.

Secondly, we examined the DA for BHs surrounded by dust, quintessence,
radiation, phantom and cosmological fields in a plasma medium and concluded
that the DA for BHs surrounded by dust, quintessence and phantom fields
depends upon impact parameter $b$, BH mass $M$, charge $Q$, Rastall geometric
parameters $k$ and $\lambda$, as well as plasma frequencies $\omega_{\infty}$
& $\omega_{e}$ and field structure parameters $N_{d},~{}N_{q}$ and $N_{p}$,
respectively. Moreover, the DA for BHs surrounded by radiation field and
cosmological constant depends upon on impact parameter $b$, BH mass $M$,
charge $Q$, plasma frequencies $\omega_{\infty}$ & $\omega_{e}$ and field
structure parameters $N_{r}$ and $N_{c}$, respectively. It has important to
mention here that, when, we neglect the plasma effects in our results i. e.,
$\omega^{2}_{e}/\omega^{2}_{\infty}\rightarrow 0$, then, we obtained the
results of DA in a non-plasma medium for corresponding BHs.

After that, we analyzed the graphical results for DA in the presence of plasma
for variations of plasma frequencies ratio
$\omega^{2}_{e}/\omega^{2}_{\infty}$, Rasatll parameters $k\lambda$ and charge
$Q$. Furthermore, from our graphical analysis, we concluded that the DA for
increasing plasma frequencies and Rastall parameter values is shown to
decrease via growing impact parameter from a BH surrounded by a dust,
quintessential and phantom field that eventually achieving an asymptotically
flat form until $b\rightarrow\infty$. A notable deviation with strong
deflection is observed when the impact parameter is minimal. Furthermore, the
inverse link between DA and impact parameter $b$ is also noted in the presence
of plasma medium.

At last, we investigated the shadows of BHs defined by the Kiselev solution in
the Rastall theory. A Hamiltonian equation that is variable-separable and a
light-ray motion equation are independent of the plasma’s velocity in the
presence of a pressure-less and non-magnetized plasma. We explored the perfect
fluid in Rastall theory, where particle accumulation is correlated with the
plasma frequency. The scenario in which plasma falls radially upon a BH from
infinity is investigated in terms of dust, quintessence, radiation, phantom
fields and cosmological constants encircled by a Rastall solution. Moreover,
the shadow of our corresponding BHs depends on BH mass $M$, charge $Q$,
Rastall parameters $k,~{}\lambda$, plasma frequencies
$\omega_{p},~{}\omega_{0}$, photon radius $r_{ph}$, observer radius $r_{0}$
and their field structure parameters $N_{d},~{}N_{q},~{}N_{r},~{}N_{p}$ and
$N_{c}$, respectively. We plotted the contour plots for corresponding BHs for
their shadows with variations of photon radius $r_{ph}$ as well as observer
radius $r_{0}$. We concluded that the size of shadow radius increases with the
rise in photon radius whereas the size of shadow decreases with rise in values
of observer’s radius.

The presence of the conical singularity on the symmetric axis in the
background of Rastall spacetime is responsible for the appearance of the BHs
encircled by phantom fields, quintessence, cosmological constants, radiation,
and dust. Examining how a conical singularity affects the DA and shadow
variables is interesting. Future reports on this work will be provided.

## Acknowledgement

The paper was funded by the National Natural Science Foundation of China
11975145. A. Ö. would like to acknowledge the contribution of the COST Action
CA21106 - COSMIC WISPers in the Dark Universe: Theory, astrophysics and
experiments (CosmicWISPers) and the CA22113 - Fundamental challenges in
theoretical physics (THEORY-CHALLENGES).

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