1. Introduction
Black holes are regions in space where gravity is so strong that nothing, not even light, can escape from them. They are crucial to our understanding of fundamental physics, as they challenge our understanding of gravity, space, and time. Studying black holes provides insights into extreme conditions in the Universe, helps us probe the nature of spacetime, and contributes to theories like general relativity and quantum mechanics. Additionally, they play a vital role in shaping galaxies and the Universe's evolution. Black holes are typically categorized into several types: uncharged and stationary (Schwarzschild), rotating uncharged (Kerr), rotating charged (Kerr–Newman), and stationary charged (Reissner–Nordström), among others. Recent research, as evidenced by studies [1–3], strongly reinforces the existence of spinning black holes within the M87 and Sgr A regions.
A key issue in Einstein's general theory of relativity involves the singularities found at the start of the Universe and within black hole solutions. Similar singularities also exist in Maxwell's theory of Electrodynamics. To solve these problems, nonlinear electrodynamics (NED) offers several advantages over standard electrodynamics. The first NED model, proposed by Born and Infeld (BI) [4], avoids infinities at the core of point-like particles and has finite self-energy for charges [5]. However, standard quantum electrodynamics introduces nonlinearities that break some desirable properties. A specific NED model, known as ModMax [6], which is a low-energy limit of a one-parameter generalization of BI, and preserves these positive features which consider gravitating configurations in electrovacuum with a nonlinear constitutive relation and matter equations invariant under Hodge duality rotations and conformal transformations. This conformally extended duality-invariant theory [7]. We investigated this spacetime metric and obtained its rotating solution using the Newman–Janis algorithm, and particle dynamics, thermodynamic analysis, and shadow of the ModMax rotating black hole are also studied in [8, 9].
Gravitational lensing theory unveils the remarkable phenomenon of light bending under the influence of gravity. It encompasses a variety of effects, including the deflection of light from distant objects, a change of apparent angular position of the source. In some instances, this bending can even produce multiple, distorted images of the same source. Furthermore, the lensing effect can magnify faint objects, allowing us to peer deeper into the cosmos. Additionally, it can distort the shapes of objects behind the lensing mass, revealing valuable information about the intervening matter. Even time itself is affected, with light experiencing slight delays depending on the path taken due to lensing. This powerful tool has become a cornerstone of astrophysics, aiding in investigations of distant objects, mapping the distribution of elusive dark matter, probing the large-scale structure of the Universe, analyzing the cosmic microwave background, and even detecting exoplanets. Moreover, gravitational lensing serves as a crucial test of Einstein's theory of general relativity. By delving into both classical and recent research, we can gain a deeper understanding of this fascinating phenomenon and its ongoing advancements [10–33].
In the context of gravitational lensing, where geometric optics often dominate, the primary effect of interest is the deflection angle experienced by light rays. However, the presence of plasma, with its inherent dispersion properties, introduces the possibility of chromatic effects [34]. It is widely known that light rays in a transparent, inhomogeneous medium propagate along curve trajectories [35]. There are also extended works carried out by a number of authors in plasma for the weak-field regime in various spacetime metrics [36–47].
The phenomena of gravitational lensing and black hole shadows provide profound insights into spacetime and general relativity. Gravitational lensing reveals dark matter distribution and probes the Universe's structure, serving as a critical test for Einstein's theories. Black hole shadows offer direct evidence of black holes and allow us to test general relativity in the strong-field regime. This paper studies weak gravitational lensing and shadow radius in Dyonic ModMax (DM) spacetime, considering uniform and non-uniform plasma distributions. By comparing corrections to vacuum lensing due to gravitational effects and plasma inhomogeneity, we reveal how plasma influences photon orbits, shadow properties, and image magnification. Utilizing Event Horizon Telescope (EHT) data for M87* and Sgr A*, we constrain modified gravity parameters, bridging theoretical models with empirical observations and enhancing our understanding of black holes and their environments. Moreover, this paper investigates how the presence of plasma, a hot ionized gas found throughout the Universe, can affect various aspects of gravitational lensing. Light rays traveling through space can encounter plasma, which can influence their path due to phenomena like absorption, scattering, and refraction.
This paper is organized as follows: in section 2 , we provide a brief review of the DM black hole. In section 3 , we give null geodesic equations in DM spacetime. In section 4 , we probe the DM metric with BH's shadow, In section 5 , we calculate the deflection angle and magnification in weak-field limit surrounded plasma. 5, Finally, in section 6 , we summarize the obtained results.
2. Brief review of DM black holes
Now we briefly discuss ModMax field equations and it is given as [7]:
$\begin{eqnarray}{{\rm{\nabla }}}_{\mu }{P}^{\mu \nu }=0,\end{eqnarray}$
where Pμν is dual variable given by [48] and the Einstein equations are $\begin{eqnarray}{R}_{\nu }^{\mu }-\displaystyle \frac{1}{2}{\delta }_{\nu }^{\mu }R+{\rm{\Lambda }}{\delta }_{\nu }^{\mu }=8\pi {T}_{\nu }^{\mu }.\end{eqnarray}$
Just prior to this, the energy momentum tensor T is provided as [7, 48] $\begin{eqnarray}\begin{array}{l}8\pi {T}_{\nu }^{\mu }=-{F}^{\mu \beta }{P}_{\beta \nu }+{\delta }_{\nu }^{\mu }L=\,{F}^{\mu \beta }({L}_{x}{F}_{\beta \nu }+{L}_{y}{\tilde{F}}_{\beta \nu })+{\delta }_{\nu }^{\mu }L\\ \,=\,{\hat{L}}_{s}{P}_{\nu \beta }^{\mu \beta }+{\delta }_{\nu }^{\mu }(2s{\hat{L}}_{s}+t{\hat{L}}_{t}-\hat{L})\end{array}\end{eqnarray}$
with ${\hat{L}}_{s}\,:=\,\partial L/\partial s$ and ${\hat{L}}_{t}\,:=\,\partial L/\partial t$.When using the duality variables, solving the conservation laws is straightforward. However, for the electrically charged case, the natural variables to use are Pμν, while for the magnetically charged case, the more appropriate components are Fμν [49].
Let us examine a metric with spherical symmetrys
$\begin{eqnarray}{\rm{d}}{s}^{2}=-f(r){\rm{d}}{t}^{2}+f{\left(r\right)}^{-1}{\rm{d}}{r}^{2}+{r}^{2}({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2}).\end{eqnarray}$
Throughout this section we focus on asymptotically flat configurations and thus choose Λ = 0.To obtain dyon solutions, we get first ${A}_{\mu }=({\rm{\Phi }}(r),0,0,{Q}_{m}\cos \theta )$. It follows that12 ) square root should be positive, we will find the maximum value of black holes charge:
$\begin{eqnarray}{F}_{\mu \nu }=-{{\rm{\Phi }}}_{I,r}{\delta }_{[\mu }^{0}{\delta }_{\nu ]}^{1}-{Q}_{m}\sin \theta {\delta }_{[\mu }^{2}{\delta }_{\nu ]}^{3}.\end{eqnarray}$
In consequence $\begin{eqnarray}\begin{array}{l}x=\displaystyle \frac{1}{2}(-{{\rm{\Phi }}}_{,r}^{2}+\displaystyle \frac{{Q}_{m}^{2}}{{r}^{4}}),y=-\displaystyle \frac{{{\rm{\Phi }}}_{,r}{Q}_{m}}{{r}^{2}},\\ \,\,{\left({x}^{2}+{y}^{2}\right)}^{1/2}=\displaystyle \frac{1}{2}(-{{\rm{\Phi }}}_{,r}^{2}+\displaystyle \frac{{Q}_{m}^{2}}{{r}^{4}}),\end{array}\end{eqnarray}$
and after simplifying above expressions, the only non-vanishing duality variables are $\begin{eqnarray}\begin{array}{ccl}{P}_{01} & = & -(\cosh \gamma +\sinh \gamma ){{\rm{\Phi }}}_{,r}=-{{\rm{e}}}^{\gamma }{\rm{\Phi }},r\\ {P}_{23} & = & -(\cosh \gamma -\sinh \gamma ){Q}_{m}\sin \theta =-{{\rm{e}}}^{-\gamma }\sin \theta .\end{array}\end{eqnarray}$
The expression for P23 automatically satisfies one of the conservation equations. The electric potential is written as $\begin{eqnarray}{\rm{\Phi }}(r)=\frac{{Q}_{e}{{\rm{e}}}^{-\gamma }}{r},\end{eqnarray}$
implying that the conservation law associated to P01 also holds; furthermore, we have P01 = Qe/r2. With these choices, the gravitational field equations read $\begin{eqnarray}\begin{array}{rcl}-\displaystyle \frac{{m}_{,r}}{{r}^{2}} & = & -\displaystyle \frac{({Q}_{{\rm{e}}}^{2}+{Q}_{{\rm{m}}}^{2}){{\rm{e}}}^{-\gamma }}{2{r}^{4}},\\ -\displaystyle \frac{{m}_{,{rr}}}{2r} & = & \displaystyle \frac{({Q}_{{\rm{e}}}^{2}+{Q}_{{\rm{m}}}^{2}){{\rm{e}}}^{-\gamma }}{2{r}^{4}},\end{array}\end{eqnarray}$
it follows that $\begin{eqnarray}m(r)=M-\displaystyle \frac{({Q}_{{\rm{e}}}^{2}+{Q}_{{\rm{m}}}^{2}){{\rm{e}}}^{-\gamma }}{2r},\end{eqnarray}$
thus the metric coefficient is written as $\begin{eqnarray}f(r)=1-\displaystyle \frac{2M}{r}+\displaystyle \frac{({Q}_{{\rm{e}}}^{2}+{Q}_{{\rm{m}}}^{2})\cdot \exp (-\gamma )}{{r}^{2}},\end{eqnarray}$
where Qe and Qm are the electric and magnetic charge of the black hole respectively, γ is connected to a fixed screening factor for the black hole's charge, corresponds to the nonlinear parameter [7, 50]. Maxwell's theory is reduced when the nonlinear parameter γ vanishes. The Schwarzschild metric is recovered when Qe = 0 and Qm = 0. From f(r) = 0 we can find the event horizon radius: $\begin{eqnarray}{r}_{h}=M\pm \sqrt{\left({M}^{2}\exp (-\gamma )-({Q}_{{\rm{e}}}^{2}+{Q}_{{\rm{m}}}^{2}\right)\exp (-\gamma )}.\end{eqnarray}$
If we mark ${Q}_{{\rm{e}}}^{2}+{Q}_{{\rm{m}}}^{2}={Q}^{2}$ and in equation ( $\begin{eqnarray}0\lt Q\lt \sqrt{{M}^{2}\cdot {{\rm{e}}}^{\gamma }}.\end{eqnarray}$
In figure 1 (left panel) given the radial dependence of f(r) and event horizon radius dependence to both the total charge of the black hole and γ parameter. Since Q and γ have such values that we may observe one, two, or no event horizon at all, further, the DM's permitted range for the parameters Q/M and γ may be obtained, as seen in figure 1 (right panel). Using a DM spacetime, we determined the existence of black holes and obtained the dependency of the border between them on the charge of black hole Q/M and γ parameters. From figure 1 (right panel) it is also easy to obtain the value of the γ parameter. As we get the charge of black hole Q/M > 0.7, parameter γ can take this value γ > −0.5 From figure 2, one can see that increasing the γ (left panel) parameter leads to the radius of the event horizon also increasing. While increasing the value of the Q/M causes the radius of the event horizon to decrease.
Figure 1. This graph shows a variation of f(r) (left panel) and event horizon radius due to the Dyonic ModMax black hole and region (right panel), where a black hole can exist or not. |
Figure 2. Plots a variation of event horizon radius rh/M with respect to Q/M (left panel) and γ (right panel). |
3. Null geodesic in DM spacetime
When a photon passes through a plasma, the Hamiltonian is identified as [51]16 ) is valid only when ωp/ω < 1, otherwise the photon does not propagate in a plasma medium. If ω0 ≃ ω, the deflection angle is much larger than the vacuum case (ωp = 0), α ≫ 2R/b, where b is the impact parameter. By using equations (14 ) and (16 ) and xα = δH/δpα relationship the components of the four-velocity vector for photons in the equatorial plane $(\theta =\tfrac{\pi }{2},{p}_{\theta }=0)$ can be written as follows19 ) and (20 ), we derive an equation for the phase trajectory of light23 ) into (24 ) We can obtain the mathematical expression for radius of photon rph within a plasma medium as
$\begin{eqnarray}H({x}^{\alpha },{p}_{\alpha })=\displaystyle \frac{1}{2}{\tilde{g}}^{\alpha \beta }{p}_{\alpha }{p}_{\beta },\end{eqnarray}$
where xα describes the coordinates of spacetime and ${\tilde{g}}_{\alpha \beta }$ serves as the effective metric and can be expressed in the form of $\begin{eqnarray}{\tilde{g}}^{\alpha \beta }={g}^{\alpha \beta }-({n}^{2}-1){u}^{\alpha }{u}^{\beta },\end{eqnarray}$
where n is the refractive index of the plasma, pα and uβ inform four-momentum and four-velocity of the photon, respectively. The refraction index n of plasma is [52]: $\begin{eqnarray}{n}^{2}=1-\displaystyle \frac{{\omega }_{p}{\left(r\right)}^{2}}{\omega {\left(r\right)}^{2}},\end{eqnarray}$
where ${\omega }_{p}^{2}(r)=4\pi {e}^{2}N(r)/{m}_{{\rm{e}}}$ (e and me are the electron charge and mass, respectively, and N is the number density of electrons) is the frequency of electrons in the plasma and the photon frequency ω(r) measured by a static observer is measured by using the gravitational redshift formula $\begin{eqnarray}\omega (r)=\displaystyle \frac{{\omega }_{0}}{\sqrt{f(r)}}.\end{eqnarray}$
At infinity, ${\omega }_{0}=\mathrm{const}$ is the frequency (f( ∞ ) = 1) and ω0 = ω( ∞ ) = −pt, which denotes the energy of the photon at spatial infinity [53]. Light can propagate in the plasma only if its frequency is larger than the plasma frequency, hence equation ( $\begin{eqnarray}\dot{t}\equiv \displaystyle \frac{{\rm{d}}t}{{\rm{d}}\lambda }=\displaystyle \frac{-{p}_{t}}{f(r)},\end{eqnarray}$
$\begin{eqnarray}\dot{r}\equiv \displaystyle \frac{{\rm{d}}r}{{\rm{d}}\lambda }={p}_{r}f(r),\end{eqnarray}$
$\begin{eqnarray}\dot{\phi }\equiv \displaystyle \frac{{\rm{d}}\phi }{{\rm{d}}\lambda }=\displaystyle \frac{{p}_{\phi }}{{r}^{2}}.\end{eqnarray}$
From equations ( $\begin{eqnarray}\displaystyle \frac{{\rm{d}}r}{{\rm{d}}\phi }=\displaystyle \frac{\dot{r}}{\dot{\phi }}=\displaystyle \frac{f(r){r}^{2}{p}_{r}}{{p}_{\phi }}.\end{eqnarray}$
Using the constraint H = 0 for the photon motion, we can get following equation $\begin{eqnarray}\displaystyle \frac{{\rm{d}}r}{{\rm{d}}\phi }=\pm r\sqrt{f(r)}\sqrt{{h}^{2}(r)\displaystyle \frac{{\omega }_{0}^{2}}{{p}_{\phi }^{2}}-1},\end{eqnarray}$
where we define [53] $\begin{eqnarray}{h}^{2}(r)\equiv {r}^{2}\left[\displaystyle \frac{1}{f(r)}-\displaystyle \frac{{\omega }_{p}^{2}(r)}{{\omega }_{0}^{2}}\right].\end{eqnarray}$
The radius of a circular light orbit around a black hole, specifically forming a photon sphere with radius rph, is found by solving the following equation [53] $\begin{eqnarray}{\left.\displaystyle \frac{{\rm{d}}({h}^{2}(r))}{{\rm{d}}r}\right|}_{r={r}_{\mathrm{ph}}}=0\end{eqnarray}$
After putting equations ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{r}_{{\rm{ph}}}^{2}({r}_{{\rm{ph}}}^{2}-3{{Mr}}_{{\rm{ph}}}+3{Q}^{2}\exp (-\gamma ))}{({r}_{{\rm{ph}}}^{2}-2{{Mr}}_{{\rm{ph}}}+{Q}^{2}\exp (-\gamma ))}\\ \quad \,=\,\displaystyle \frac{{\omega }_{p}^{2}({r}_{{\rm{ph}}})}{{\omega }_{0}^{2}}+\displaystyle \frac{{r}_{{\rm{ph}}}{\omega }_{p}({r}_{{\rm{ph}}})\omega {{\prime} }_{p}({r}_{{\rm{ph}}})}{{\omega }_{0}^{2}},\end{array}\end{eqnarray}$
where the prime symbol indicates differentiation with respect to the radial coordinate r. The roots of the equation cannot be solved analytically for most variants of ωp(r); however, we will get a few simplified cases below.3.1. Homegeneous plasma
In the special case of a homogeneous plasma with constant plasma ${\omega }_{p}^{2}(r)=\mathrm{const}$ frequency throughout the medium. Equation (25 ) has been numerically solved, the result is plotted in figure 3. One can easily notice that the radius of the photon around the black hole decreases with increasing the charge of the black hole While increasing the radius of the photon corresponds to increasing the value of the γ parameter. Additionally, the plasma environment enlarges the photon sphere radius.
Figure 3. This graph illustrates the relationship between the radius of the photon sphere rph/M and the ${\omega }_{p}^{2}=\mathrm{const}$ frequency in a homogeneous plasma concerning Q (left panel) and γ (right panel). |
3.2. Inhomogeneous plasma
We are currently examining photon spheres in an inhomogeneous plasma, where the plasma frequency must adhere to a straightforward power-law relationship form [36, 54] 26 ) into (25 ). We get a numerical expression for the radius of the photon sphere in the inhomogeneous plasma medium, as plotted in figure 4. One can notice that the radius of the photon sphere decreases when the Q/M parameter increases. However, increasing the value of γ leads to the photon sphere's also increasing. Nevertheless, the plasma medium leads to the photon sphere's radius increasing if its distribution is given as the ${\omega }_{p}^{2}(r)={z}_{0}/r$ law, the presence of plasma around the black hole slightly decreases the photon radius. Furthermore, it is difficult to notice a difference in effects on photon radius with ${\omega }_{p}^{2}(r)={z}_{0}/r$ and ${\omega }_{p}^{2}(r)=\mathrm{const}$ plasma. It implies that testing and differentiating between homogeneous and non-homogeneous plasma around black holes using their shadows will be quite difficult.
$\begin{eqnarray}{\omega }_{p}^{2}(r)=\displaystyle \frac{{z}_{0}}{r},\end{eqnarray}$
where z0 is the free parameter. To examine the primary features of the power-law model, we restrict ourselves to this case: ${\omega }_{p}^{2}(r)\sim \tfrac{1}{r}$ [54]. After putting equations (
Figure 4. Illustrates the radius of the photon sphere rph/M the variation of Q (left panel) and γ (right panel) parameters for the case ${\omega }_{p}^{2}={z}_{0}/r$. |
4. Shadow of a black hole surrounded by plasma medium
In this section, we investigate the radius of the shadow of DM spacetime metric in the presence of uniform and non-uniform plasma medium. The angular radius αsh of the black hole shadow is defined by a geometric approach which results in [43, 53]
$\begin{eqnarray}{\sin }^{2}\,{\alpha }_{{\rm{sh}}}=\displaystyle \frac{{h}^{2}({r}_{{\rm{p}}})}{{h}^{2}({r}_{0})}=\displaystyle \frac{{r}^{2}\left[\tfrac{1}{f({r}_{{\rm{p}}})}-\tfrac{{\omega }_{{\rm{p}}}^{2}({r}_{{\rm{p}}})}{{\omega }_{0}^{2}}\right]}{{r}_{0}^{2}\left[\tfrac{1}{f({r}_{0})}-\tfrac{{\omega }_{{\rm{p}}}^{2}({r}_{0})}{{\omega }_{0}^{2}}\right]},\end{eqnarray}$
where r0 and rp indicate the positions of the observer and the radius of photon sphere, respectively. If the observer is positioned far enough away from the black hole, we can obtain an approximate expression for the radius of the black hole shadow [53] $\begin{eqnarray}{R}_{{\rm{sh}}}\approx {r}_{0}\,\sin \,{\alpha }_{{\rm{sh}}}=\sqrt{{r}_{{\rm{p}}}^{2}\left[\displaystyle \frac{1}{f({r}_{{\rm{p}}})}-\displaystyle \frac{{\omega }_{{\rm{p}}}^{2}({r}_{{\rm{p}}})}{{\omega }_{0}^{2}}\right]}.\end{eqnarray}$
This is based on the fact that h(r) → r, which follows from equation (23 ) at spatial infinity for both the models of plasma. For vacuum ω(r) ≡ 0, we recover the radius of the Schwarzshild black hole shadow, ${R}_{{\rm{sh}}}=3\sqrt{3}M$, when rp = 3M. The radius of the black hole shadow is illustrated for different parameters Q/M and γ in figure 5 surrendered homogeneous plasma. This figure illustrates that the shadow radius decreases much more steeply with Q/M as well as plasma frequency While the increasing value of γ corresponds to also increasing the shadow radius. Furthermore, we have investigated inhomogeneous plasma with ${\omega }_{p}^{2}(r)={z}_{0}/r$ in equation (26 ). From figure 6, one can see that due to the presence of plasma radius, the black hole's shadow is decreasing.
Figure 5. This graph shows the relationship between the radius of the photon sphere rph/M and the frequency in a homogeneous plasma medium with a ${\omega }_{{\rm{p}}}^{2}=\mathrm{const}$ concerning Q (left panel) and γ (right panel). |
Figure 6. This graph illustrates variation of the radius of shadow Rsh/M concerning Q/M and γ (right panel)for the case ${\omega }_{{\rm{p}}}^{2}={z}_{0}/r$. |
We consider supermassive black holes M87* and Sgr A* as spherically symmetric and static. However, observations from the EHT do not fully support this assumption. Despite that, the study investigates constraints on the parameters Q and γ using data from the EHT project. To determine the parameters Q and γ, the EHT project utilizes observational data from the black hole shadows of supermassive black holes M87* and Sgr A*. The angular diameter of the shadow, the distance from the Sun system and the mass of the black hole at the center of the Galaxy M87, are ΩM87* = 42 ± 3μas, D = 16.8 ± 0.8 Mpc and MM87* = (6.5 ± 0.7) × 109 M⊙, respectively [3]. For the Sgr A* the data recently obtained by the EHT project is ΩSgr A* = 48.7 ± 7 μas, D = 8277 ± 9 ± 33 pc and MSgr A* = (4.297 ± 0.013) × 106 M⊙ [2]. In figure 7 we can observe that the spacetime parameters of the DM black hole exhibit certain values that could potentially coincide with observations. The accompanying image represents these values in parametric space. Black lines are responsible for the angular diameter of the shadow of black hole θsh = 39μas for M87* (left panel) and θsh = 41.7 μas for Sgr A* (right panel), respectively. This means that As we get Q/M = 1, parameter γ ≥ 0 for Sgr A* (right panel).
Figure 7. We are restricting parameters based on the observational angular diameter of the shadows of the black holes M87* (left panel) and Sgr A* (right panel). The black lines represent the angular diameters of the black hole shadows, with θsh = 39 μas for M87* (left panel) and θsh = 41.7 μas for Sgr A* (right panel). The areas within these lines correspond to the shadow constraints for the M87* (left panel) and Sgr A* (right panel) black holes. |
5. Gravitational weak lensing in the presence of the plasma medium
In this part, we focus on examining the impacts of gravitational lensing in the DM black hole surrounded by a plasma considering a weak-field approximation defined as follows [34]
$\begin{eqnarray}{g}_{\alpha \beta }={\eta }_{\alpha \beta }+{h}_{\alpha \beta },\end{eqnarray}$
where ηαβ and hαβ represent the Minkowski metric and the perturbation metric, respectively, along with their characteristics [34] $\begin{eqnarray}{\eta }_{\alpha \beta }=\mathrm{diag}(-1,1,1,1),\end{eqnarray}$
$\begin{eqnarray}{h}_{\alpha \beta }\lt \,\lt 1,\hspace{0.5cm}{h}_{\alpha \beta }\to 0\hspace{0.5cm}\mathrm{under}\hspace{0.5cm}{x}^{\alpha }\to \infty \end{eqnarray}$
$\begin{eqnarray}{g}^{\alpha \beta }={\eta }^{\alpha \beta }-{h}^{\alpha \beta },\hspace{0.27cm}{h}^{\alpha \beta }={h}_{\alpha \beta }.\end{eqnarray}$
Now we want to examine the plasma effects on the bending angle of the light rays. In the case of plasma medium, the deflection angle can be written as [34, 38] $\begin{eqnarray}\begin{array}{rcl}{\hat{\alpha }}_{i} & = & \pm \displaystyle \frac{1}{2}{\displaystyle \int }_{-\infty }^{\infty }\left[\displaystyle \frac{1}{2}\left({h}_{33,i}+\displaystyle \frac{{\omega }^{2}}{{\omega }^{2}-{\omega }_{e}^{2}}{h}_{00,i}\right.\right.\\ & & \left.\left.-\,\displaystyle \frac{{K}_{e}}{{\omega }^{2}-{\omega }_{e}^{2}}{\omega }^{2}{N}_{,i}\right)\right]{\rm{d}}z.\end{array}\end{eqnarray}$
N(xi) indicates the number density of the particles in the plasma around the black hole and Ke = 4πe2/me is a constant parameter. The ± signs of $\hat{{\alpha }_{i}}$ indicate deflection towards or away from the central object, respectively. At large distances, the black hole metric can be approximated as $\begin{eqnarray}{\rm{d}}{s}^{2}={\rm{d}}{s}_{0}^{2}+...{\rm{d}}{t}^{2}+...{\rm{d}}{r}^{2},\end{eqnarray}$
where ${\rm{d}}{s}^{2}=-{\rm{d}}{t}^{2}+{\rm{d}}{r}^{2}+{r}^{2}({\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2})$ and Rs is the radius of Schwarzschild. In the Cartesian coordinates the components hαβ can be given as $\begin{eqnarray}{h}_{00}=\left(\displaystyle \frac{{R}_{{\rm{s}}}}{r}-\exp (-\gamma )\displaystyle \frac{{Q}^{2}}{{r}^{2}}\right),\end{eqnarray}$
$\begin{eqnarray}{h}_{{jk}}=\left(\displaystyle \frac{{R}_{{\rm{s}}}}{r}-\exp (-\gamma )\displaystyle \frac{{Q}^{2}}{{r}^{2}}\right){n}_{j}{n}_{k},\end{eqnarray}$
$\begin{eqnarray}{\,h}_{33}=\left(\displaystyle \frac{{R}_{{\rm{s}}}}{r}-\exp (-\gamma )\displaystyle \frac{{Q}^{2}}{{r}^{2}}\right){\cos }^{2}x,\end{eqnarray}$
where $\cos x=\tfrac{z}{\sqrt{{b}^{2}+{z}^{2}}}$ and $r=\sqrt{{b}^{2}+{z}^{2}}$, b is the impact parameter signifying the closest approach of the photons to the black hole. Using the above expressions, we can calculate the light deflection angle concerning b for a black hole in a plasma $\begin{eqnarray}\begin{array}{l}{\hat{\alpha }}_{b}={\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{1}{2}\left[{\partial }_{b}\left(\left(\displaystyle \frac{{R}_{{\rm{s}}}}{r}-\exp (-\gamma )\displaystyle \frac{{Q}^{2}}{{r}^{2}}\right){\cos }^{2}x\right)\right.\\ \qquad +\,{\partial }_{b}\left(\displaystyle \frac{{R}_{{\rm{s}}}}{r}-\exp (-\gamma )\displaystyle \frac{{Q}^{2}}{{r}^{2}}\right)\\ \qquad \left.\times \,\displaystyle \frac{{\omega }^{2}}{{\omega }^{2}-{\omega }_{e}^{2}}-\displaystyle \frac{{K}_{e}}{{\omega }^{2}-{\omega }_{e}^{2}}{\partial }_{b}N\right]{\rm{d}}z.\end{array}\end{eqnarray}$
after putting this expression ${\partial }_{b}=\tfrac{b}{r}{\partial }_{r}$ into above equation, we can get following equation $\begin{eqnarray}\begin{array}{l}{\hat{\alpha }}_{b}={\displaystyle \int }_{-\infty }^{\infty }\displaystyle \frac{b}{2r}\left[{\partial }_{r}\left(\left(\displaystyle \frac{{R}_{{\rm{s}}}}{r}-\exp (-\gamma )\displaystyle \frac{{Q}^{2}}{{r}^{2}}\right){\cos }^{2}x\right)\right.\\ \qquad +\,{\partial }_{r}\left(\displaystyle \frac{{R}_{{\rm{s}}}}{r}-\exp (-\gamma )\displaystyle \frac{{Q}^{2}}{{r}^{2}}\right)\\ \,\left.\times \,\displaystyle \frac{{\omega }^{2}}{{\omega }^{2}-{\omega }_{e}^{2}}-\displaystyle \frac{{K}_{e}}{{\omega }^{2}-{\omega }_{e}^{2}}{\partial }_{r}N\right]{\rm{d}}z.\end{array}\end{eqnarray}$
5.1. Uniform plasma
First of all, we study uniform plasma with ${\omega }_{0}^{2}=\mathrm{const}$. In this case, the refractive index is not directly tied to spatial coordinates, allowing us to disregard its refractive effect. In simpler terms, we omit the last term of the equation (39 ). Integrating equation (39 ) gives us the following result for the deflection angle
$\begin{eqnarray}\begin{array}{c}{\hat{\alpha }}_{\mathrm{uni}}=\left(\frac{{R}_{s}}{b}-\exp (-\gamma )\frac{\pi {Q}^{2}}{4{b}^{2}}\right)\\ \,+\,\left(\frac{{R}_{s}}{b}-\exp (-\gamma )\frac{\pi {Q}^{2}}{2{b}^{2}}\right)\frac{1}{\left(1-\frac{{\omega }_{0}^{2}}{{\omega }^{2}}\right)}.\end{array}\end{eqnarray}$
Figure 8 shows the impact parameter b for the charge of the black hole Q (left panel) and γ parameter (right panel) depending on plasma parameters $\tfrac{{\omega }_{0}^{2}}{{\omega }^{2}}$. Decreasing the impact parameter b leads to a deflection angle increase. Figure 9 is a visualization of the deflection angle distinctively concerning the charge of the black hole Q (left panel) and γ parameter(right panel) depending on $\tfrac{{\omega }_{0}^{2}}{{\omega }^{2}}$. The bending angle reaches maximum value thanks to high plasma distribution (right panel) and is seen to be strictly decreasing against an increasing γ parameter (left panel), for instance, taking Q and γ = 0 the Schwarzschild gravity ensures the highest degree of deviation ${\hat{\alpha }}_{\mathrm{uni}}$. We can easily notice that the presence of plasma around the black hole in comparison with the vacuum case $\tfrac{{\omega }_{0}^{2}}{{\omega }^{2}}=0$ contributes to the photon motion.
Figure 8. Plot of the deflection angle αuni as a function of the impact parameter b for different Q charge of the black hole (left panel) and γ parameter (right panel) in the case of constant plasma parameters $\tfrac{{\omega }_{0}^{2}}{{\omega }^{2}}$ . |
Figure 9. This graph describes the deflection angle αuni as a function of the charge of the black hole Q (left panel) and γ parameter(right panel) depending on $\tfrac{{\omega }_{0}^{2}}{{\omega }^{2}}$ with a fixed parameter b/M = 7. |
5.2. Singular isothermal sphere
The singular isothermal sphere (SIS) is the most appropriate model for comprehending the characteristics of a gravitational lens photon. It was primarily used to explore the properties of lenses and clusters. In general, the SIS is a spherical cloud of gas with infinity at its center. The distribution of density of an SIS is written as the following [34, 46]
$\begin{eqnarray}\rho (r)=\displaystyle \frac{{\sigma }_{\nu }^{2}}{2\pi {r}^{2}},\end{eqnarray}$
where ${\sigma }_{\nu }^{2}$ implies a one-dimensional velocity. The plasma concentration can be described by the following analytic dispersion [34, 46] $\begin{eqnarray}N(r)=\displaystyle \frac{\rho (r)}{\kappa {m}_{{\rm{p}}}},\end{eqnarray}$
where mp is the mass of the proton and k is a dimensionless constant coefficient generally connected to the dark matter universe. Using the frequency of plasma takes this form $\begin{eqnarray}{\omega }_{e}^{2}={K}_{e}N(r)=\displaystyle \frac{{K}_{e}{\sigma }_{\nu }^{2}}{2\pi \kappa {m}_{{\rm{p}}}{r}^{2}}.\end{eqnarray}$
We consider the above-mentioned properties of the SIS and calculate the bending of the light ${\hat{\alpha }}_{\mathrm{SIS}}$ as below $\begin{eqnarray}\begin{array}{c}{\hat{\alpha }}_{\mathrm{SIS}}=\left(\frac{2{R}_{s}}{b}-\exp (-\gamma )\frac{3\pi {Q}^{2}}{4{b}^{2}}\right)\\ \quad +\,\frac{{R}_{s}^{2}{\omega }_{c}^{2}}{{b}^{2}{\omega }^{2}}\left(\frac{2}{3\pi b}-\frac{1}{2}\right),\end{array}\end{eqnarray}$
where ${\omega }_{c}^{2}$ is a supplementary plasma constant which is given the following analytic expression [46] $\begin{eqnarray}{\omega }_{c}^{2}=\displaystyle \frac{{K}_{e}{\sigma }_{\nu }^{2}}{2\pi \kappa {m}_{{\rm{p}}}{R}_{s}^{2}}.\end{eqnarray}$
To examine the effect of SIS on the trajectory of a photon, we showed the angle of deflection ${\hat{\alpha }}_{\mathrm{SIS}}$ as a function of the impact parameter b, see figure 10, interestingly, we can notice that the uniform plasma and SIS medium share similar features concerning the critical parameter b. Note that the quantity $\tfrac{{\omega }_{c}^{2}}{{\omega }^{2}}$ indicates the SIS distribution around the black hole. As a result, we detected the photon sensitivity to specified parameters along with black hole parameters Q and γ, using graphical analysis in figure 11. We saw that ${\hat{\alpha }}_{\mathrm{SIS}}$ decreases when the charge of the black hole Q increases (left panel) and, conversely, ${\hat{\alpha }}_{\mathrm{SIS}}$ increases when the γ parameter increases (right panel). Hence, the presence of SIS around a black hole partially affects massless particles.
Figure 10. This graph describes a variation of ${\hat{\alpha }}_{\mathrm{sis}}$ with respect to b in different charges of black hole Q/M (left panel) and γ (right panel) parameters. |
Figure 11. This graph illustrates a variation of ${\hat{\alpha }}_{\mathrm{sis}}$ concerning the charge of black hole Q/M (left panel) and γ (right panel) parameters in different $\tfrac{{\omega }_{0}^{2}}{{\omega }^{2}}$. |
5.3. Non-singular isothermal sphere
Next, we will examine the photon motions in the presence of a non-singular isothermal sphere (NSIS), as this provides a more realistic and practical framework for our analysis. Compared to the SIS, in this plasma model the singularity is restricted by a finite core at the center of the gas cloud where the distribution of density is given as a following42 ) takes the form43 ), (45 ), and (47 ) as follows
$\begin{eqnarray}\rho (r)=\displaystyle \frac{{\sigma }_{\nu }^{2}}{2\pi ({r}^{2}+{r}_{{\rm{c}}}^{2})}=\displaystyle \frac{{\rho }_{o}}{1+\tfrac{{r}^{2}}{{r}_{{\rm{c}}}^{2}}};\qquad {\rho }_{0}=\displaystyle \frac{{\sigma }_{\nu }^{2}}{2\pi {r}_{{\rm{c}}}^{2}},\end{eqnarray}$
where the radius of the core is written by rc. The plasma concentration in NSIS model using equation ( $\begin{eqnarray}N(r)=\displaystyle \frac{{\sigma }_{\nu }^{2}}{2\pi {{km}}_{{\rm{p}}}({r}^{2}+{r}_{{\rm{c}}}^{2})}.\end{eqnarray}$
We derive the frequency of plasma from equations ( $\begin{eqnarray}{\omega }_{{\rm{c}}}^{2}=\displaystyle \frac{{K}_{e}{\sigma }_{\nu }^{2}}{2\pi \kappa {m}_{{\rm{p}}}({r}^{2}+{r}_{{\rm{c}}}^{2})}.\end{eqnarray}$
After some analytic calculations, we can take following expressions for the deflection angle in the NSIS model $\begin{eqnarray}\begin{array}{c}{\hat{\alpha }}_{\mathrm{NSIS}}=\frac{2{R}_{{\rm{s}}}}{b}-\exp (-\gamma ){Q}^{2}\times \,\frac{5\pi }{4{b}^{2}}+\frac{{\omega }_{{\rm{c}}}^{2}{R}_{{\rm{s}}}^{2}b}{{\omega }^{2}{r}_{{\rm{c}}}^{4}\sqrt{{b}^{2}+{r}_{{\rm{c}}}^{2}}}\\ \,+\,\frac{{R}_{{\rm{s}}}^{2}{\omega }_{{\rm{c}}}^{2}({r}_{{\rm{c}}}^{2}-2{b}^{2})}{2{b}^{2}{\omega }^{2}{r}_{{\rm{c}}}^{4}}+{R}_{{\rm{s}}}^{2}\frac{{\omega }_{{\rm{c}}}^{2}}{{\omega }^{2}}\frac{{R}_{{\rm{s}}}}{\pi {{br}}_{{\rm{c}}}^{2}}-\frac{b}{2{\left({b}^{2}+{r}_{{\rm{c}}}^{2}\right)}^{\frac{3}{2}}}\\ \quad -\,\frac{{{bR}}_{{\rm{s}}}{\tanh }^{-1}\frac{{r}_{{\rm{c}}}}{\sqrt{{b}^{2}+{r}_{{\rm{c}}}^{2}}}}{\pi {r}_{{\rm{c}}}^{2}\sqrt{{b}^{2}+{r}_{{\rm{c}}}^{2}}}.\end{array}\end{eqnarray}$
Here, NSIS distribution is related to the parameter $\tfrac{{\omega }_{c}^{2}}{{\omega }^{2}}$. One can see from figures 12 and 13 that the behavior of the impact parameter b, the charge of the black hole Q, γ parameter, and it is difficult to notice the influence of $\tfrac{{\omega }_{{\rm{c}}}^{2}}{{\omega }^{2}}$ from a specific point of view when comparing with the homogeneous plasma and SIS case [55]. Nevertheless, figure 14 illustrates a visual juxtaposition of the ${\hat{\alpha }}_{\mathrm{uni}},{\hat{\alpha }}_{\mathrm{SIS}}$, and ${\hat{\alpha }}_{\mathrm{Nsis}}$ depending on the impact of parameter b with fixed charge of the black hole and γ parameters. It is noticeable that the deflection angle is maximum in a homogenous plasma medium in comparison with other models. The final result is written in a mathematical expression as, ${\hat{\alpha }}_{\mathrm{uni}}\gt {\hat{\alpha }}_{\mathrm{Nsis}}\gt {\hat{\alpha }}_{\mathrm{SIS}}$.
Figure 12. Plots of a variation of ${\hat{\alpha }}_{\mathrm{Nsis}}$ depending on impact parameter b in various Q/M (left panel) and γ (right panel) parameters depending on $\tfrac{{\omega }_{0}^{2}}{{\omega }^{2}}$. |
Figure 13. This figure illustrates a variation of ${\hat{\alpha }}_{\mathrm{Nsis}}$ with respect to the charge of black hole Q/M (left panel) and γ (right panel) parameter in different plasma frequency $\tfrac{{\omega }_{0}^{2}}{{\omega }^{2}}$. |
Figure 14. Comparison of the bending of the light ${\hat{\alpha }}_{b}$ as a function of the impact parameter b. The fixed parameters are used as follows$\tfrac{{\omega }_{0}^{2}}{{\omega }^{2}}=0.5,\tfrac{{\omega }_{{\rm{c}}}^{2}}{{\omega }^{2}}=0.5,{r}_{{\rm{c}}}=3$. |
5.4. Lens equation and magnification
Now we turn our attention to the observable effects of gravitational lensing, specifically the magnification of the image source brightness in the existence of plasma. We consider the angle of deflection $\hat{\alpha }$, as discussed in our above sections, with particular attention paid to uniform plasma. Figure 15 shows a schematic diagram of the gravitational lensing system illustrating the source, the black hole as the lens, and the observer. To obtain the image magnification of sources, it is better to use the lens equation, which is associated with the source angular position β and image θ and distances from the source to the observer Ds and the lens Dls and the angle of deflection [15, 17, 56–58]:40 ) into equation (51 ) we can change equation (51 ) to the following form52 ) to this equation53 ) which is written as [59, 60]
$\begin{eqnarray}\theta {D}_{{\rm{s}}}=\beta {D}_{{\rm{s}}}+\hat{\alpha }{D}_{{\rm{ls}}}.\end{eqnarray}$
In the weak gravity, we are able to approximate b ≈ Ddθ relation. Then we can get the following expression $\begin{eqnarray}\beta =\theta -\displaystyle \frac{{D}_{{ds}}}{{D}_{s}}\hat{\alpha }.\end{eqnarray}$
In the state of homogeneous plasma, after substituting equation ( $\begin{eqnarray}\begin{array}{l}\beta =\theta -\displaystyle \frac{{D}_{{ds}}}{{D}_{s}{D}_{d}}\left(1+\displaystyle \frac{1}{1-\tfrac{{\omega }_{0}^{2}}{{\omega }^{2}}}\right)\displaystyle \frac{{R}_{s}}{\theta }\\ \quad +\,\displaystyle \frac{{D}_{{ds}}}{{D}_{s}{D}_{d}^{2}}\left(1+\displaystyle \frac{2}{1-\tfrac{{\omega }_{0}^{2}}{{\omega }^{2}}}\right)\displaystyle \frac{\pi {Q}^{2}\exp (-\gamma ){R}_{s}^{2}}{4{D}_{d}{\theta }^{2}}.\end{array}\end{eqnarray}$
After creating new variable $x=\theta -\tfrac{\beta }{3}$, we can bring equation ( $\begin{eqnarray}{x}^{3}+p\ x+q=0,\end{eqnarray}$
where $\begin{eqnarray}p=-\displaystyle \frac{{\beta }^{2}}{3}-\displaystyle \frac{1}{2}\left(1+\displaystyle \frac{1}{1-\tfrac{{\omega }_{0}^{2}}{{\omega }^{2}}}\right){\theta }_{E}^{2},\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}q=-2\displaystyle \frac{{\beta }^{3}}{27}-\displaystyle \frac{\beta }{6}\left(1+\displaystyle \frac{1}{1-\tfrac{{\omega }_{0}^{2}}{{\omega }^{2}}}\right){\theta }_{E}^{2}\\ \qquad +\,\displaystyle \frac{\pi {Q}^{2}\exp (-\gamma ){R}_{{\rm{s}}}}{32{D}_{d}}\left(1+\displaystyle \frac{2}{1-\tfrac{{\omega }_{0}^{2}}{{\omega }^{2}}}\right){\theta }_{E}^{2},\end{array}\end{eqnarray}$
$\begin{eqnarray}{\theta }_{E}=\sqrt{\displaystyle \frac{2{R}_{{\rm{s}}}{D}_{{ds}}}{{D}_{{\rm{s}}}{D}_{d}}}.\end{eqnarray}$
Then we obtain the solution of equation ( $\begin{eqnarray}x=2{s}^{1/3}\cos \displaystyle \frac{\phi +2k\pi }{3},\end{eqnarray}$
where $\begin{eqnarray}s=\sqrt{-\displaystyle \frac{{p}^{3}}{27}}\,\,\phi =\arccos \left(-\displaystyle \frac{q}{2s}\right).\end{eqnarray}$
Figure 15. Schematic view of the gravitational lensing system. |
The total magnification of the images is determined by the ratio of the observer's solid angles to the source, summed for each image. In the weak-field approximation, the total magnification, denoted as μΣ, is approximately given as [37]
$\begin{eqnarray}\begin{array}{rcl}{\mu }_{{\rm{\Sigma }}} & = & \displaystyle \frac{{I}_{\mathrm{tot}}}{{I}_{* }}=\displaystyle \sum _{k}\left|\left(\displaystyle \frac{{\theta }_{k}}{\beta }\right)\left(\displaystyle \frac{{\rm{d}}{\theta }_{k}}{{\rm{d}}\beta }\right)\right|,\\ k & = & 1,2,\ldots ,n\end{array}\end{eqnarray}$
where θk = x + β/3. There are three components of the magnification of the images which are ordinary two images and relativistic images that appeared due to the presence of the ε parameter.By using some mathematical calculations we can obtain a numerical expression for the total magnification of the image. Figure 16 (left panel) shows the total magnification of the image as a function of source angular position β for different values of the black hole charge Q and γ quantities. The increasing value of the β position leads to a total magnification decrease. Additionally, one can notice that the existence of the plasma medium affects the growth of the magnification (for the right panel). From these pictures, it is possible to see that contrasting magnification rates in uniform and vacuum cases, the rate of primary image is a bigger comparison with secondary and relativistic images. In figure 17 the plot of total magnification as a function of source position β for different values γ (left panel) and Q (right panel) parameters. It is possible to see that an increase of both parameters leads to a total magnification to decrease
Figure 16. (Left panel) Total magnification as a function of source position β for Q/M = 0.6, γ = –0.55, and M/Dd = 22.6 · 10−12. (Right panel) The rate of magnification of the image source as a function of source position β for ${\omega }_{0}^{2}/{\omega }^{2}=0.5$ and M/Dd = 2.26 × 10−11. Black thick and red dashed curves identify the rate of magnification of primary and secondary images, respectively, while blue curves are relativistic images thanks to the existence of the charge of the black hole Q and γ parameters. |
Figure 17. Plot of total magnification as a function of source position β for Q/M = 0.6 (left panel), γ = –0.55(right panel) in fixed ${\omega }_{o}^{2}$ /ω2 = 0.75 and M/Dd = 22.6 × 10−12. |
In table 1, we compare image magnification around uniform plasma and vacuum cases for Sgr A*. It is noticeable that the magnification of the third image decreases with the existence of plasma. In contrast, decreasing plasma frequencies cause growth in the magnification of primary and secondary images. Moreover, we can determine values of the position of the image, angle of deflection, and magnification of the images for various galaxies. Additionally, we found values of image positions and magnification of these images in fixed DM parameters. Table 2 shows the value of these quantities for various galaxies.
Table 1. This table compare the rate of magnification of images in uniform plasma and vacuum cases. In this table M/Dd = 2.26 × 10−11 is used which corresponds to the supermassive black hole in the center of the Milky Way, β = 1 μas and Ds/Dds = 2 (these parameters have been taken from [17]). |
| μn/μnvac | ${\omega }_{0}^{2}/{\omega }^{2}=0.1$ | ${\omega }_{0}^{2}/{\omega }^{2}=0.5$ | ${\omega }_{0}^{2}/{\omega }^{2}=0.9$ | ${\mu }_{n}^{\mathrm{vac}}$ |
|---|---|---|---|---|
| μ1/μ1vac | 1.002 16 | 1.068 72 | 1.662 08 | 3.889 37 |
| μ2/μ2vac | 1.002 91 | 1.092 51 | 1.891 22 | −2.88937 |
| μ3/μ3vac | 0.999 983 | 0.985 518 | 0.589 747 | −1.001359 × 10−17 |
Table 2. Approximated value of angular positions, magnifications, of primary, secondary, and relativistic images at centers of many galaxies. θ and μ stand, respectively, for angular position and magnifications. The first column gives the names of galaxies. Subscripts sec, pri, and rel attached to them stand, respectively, for secondary, primary, and relativistic images. whereas all of the angular quantities are given by μas. The values of distance M/Dd are taken from [17]. The angular position of the source. Relativistic images are on the same side as the primary image. All calculation is done with β = 1(μas), Q/M = 0.6, γ = 0.5 and ${\omega }_{o}^{2}$ /ω2 = 0.5. |
| Secondary* | Primary* | Relativistic* | ||||
|---|---|---|---|---|---|---|
| MDO in galaxy | ${\theta }_{\sec }$ | ${\mu }_{\sec }$ | θpri | μpri | θrel | μrel |
| Milky Way | −6.77897 | 3.156 66 | 7.778 96 | 4.156 66 | 8.70217 × 10−6 | 1.24965 × 10−17 |
| NGC4649 | −3.17786 | 1.372 92 | 4.177 86 | 2.37292 | 2.19094 × 10−6 | 7.92244 × 10−19 |
| NGC4594 | −3.07328 | 1.321 62 | 4.07328 | 2.321 62 | 2.0658 × 10−6 | 7.04219 × 10−19 |
| NGC3115 | −2.92767 | 1.2503 | 3.927 67 | 2.2503 | 1.89758 × 10−6 | 5.94204 × 10−19 |
| NGC224(M2) | −2.74398 | 1.160525 | 3.743 98 | 2.160 52 | 1.69534 × 10−6 | 4.74221 × 10−19 |
| IC1459 | −2.63044 | 1.105 15 | 3.63044 | 2.105 15 | 1.5759 × 10−6 | 4.09932 × 10−19 |
| NGC4374 | −2.07365 | 0.835 395 | 3.07365 | 1.8354 | 1.0518 × 10−6 | 1.82633 × 10−19 |
| NGC4374(M84) | −2.01236 | 0.805 932 | 3.01236 | 1.805 93 | 1.00035 × 10−6 | 1.65148 × 10−19 |
6. Conclusion
In this paper, we have studied the DM black hole by examining null geodesics, shadows, and weak gravitational lensing in both homogeneous and inhomogeneous plasma environments. From the above calculations, we can conclude as follows:
| • | Testing the null geodesic we can notice that photon orbits decrease owing to the existence of the parameters Q and γ. Also, the radius of the black hole's shadow decreases with increasing of the parameters Q and γ. |
| • | While examining weak gravitational field limit, it is possible to see that the existence of DM parameters Q and γ leads to the bending of light to decrease. |
| • | Also, we have studied the influence of plasma on the radius of the photon sphere, shadow, angle of deflection and magnification rate. Increasing the plasma parameter causes the radius of the photon and shadow to decrease in the case of inhomogenous plasma medium when the parameter of plasma is growing. Additionally, the angle of deflection and magnification of the images increase in the existence of plasma. |