1. Introduction
As one of the predictions of general relativity, black holes exist in our universe as extremely dense objects. Moreover, with the improvement of observation technology, people have been able to observe this mysterious object in the Universe. In 2015, gravitational waves from the merger of two black holes were monitored by the Laser Interferometer Gravitational Wave Observatory (LIGO) [1]. In 2019, the Event Horizon Telescope (EHT) collaboration released an image of a supermassive black hole located at the center of galaxy M87 [2–7]. And recently, the EHT Collaboration released the first image of a supermassive black hole at the center of the Milky Way [8]. These works strongly proved the existence of black holes and successfully confirmed the prediction of general relativity. In the two images released by EHT, similar bright rings and dark areas appeared. This bright ring is called a photon ring, and the dark central region is the shadow of black holes. Due to the gravitational lens effect, the shadows of black holes are formed, and some of the light rays near the black hole deflect and rotate around the black hole, which leads to the formation of a photon ring [9–12]. The shadows of black holes are widely studied, which will further reveal the characteristics of black holes [13–41].
In the process of observing black holes, the accretion matter around black holes plays an indispensable role. In the early 1970s, it was thought that the outward transfer of angular momentum of the accretion matter would cause an accretion disk to form around the black hole [42–44]. This type of accretion disk mentioned in the [42–44] is geometrically thin and is called the thin disk. The gas of the thin disk is optically thick and has a very low temperature compared to the Verri temperature [45]. In 1976, a geometrically thin and optically thin accretion model was proposed by Shapiro [46]. Composed of high-temperature gas in the inner region, this model is able to explain the x-ray emission seen in some black hole sources, but it is unstable [47]. Immediately after, the thick disk was also studied [48–51]. In 1988, the slim disk was proposed considering the case where the accretion gas is optically too thick so that the thin disk is no longer applicable [45, 52]. The Lorentz violation parameter l was shown to have a large effect on the energy flux, temperature distribution and emission spectrum of the thin disk [53]. Faraji et al in their recent paper confirmed that the quadrupole parameter plays a non-negligible role in the physical properties of the thin disk solution around twisted Schwarzschild spacetime [54]. Recently, efforts have been devoted to studying the shadows and the observational appearance of black holes which are surrounded by spherically symmetric accretion and thin disk accretion [21–36]. Considering both the static and infalling spherically symmetric accretion models, the inner region of the infalling one is darker than that of the static one due to the Doppler effect [23, 24]. Also, under the background that black holes are surrounded by the thin disk accretion, the light rays around black holes can be divided into three types, i.e., direct emission, lens ring and photon ring [23, 24, 27, 30, 32–36]. Moreover, there are differences in the contributions of direct emission, lens rings and photon rings to the observed intensity of black holes. It can be seen that studying the observational appearance of a black hole surrounded by accretion can reflect some characteristics of the black hole, and these studies can provide theoretical support for further observations of black holes in the Universe.
It is well known that the singularity problem is an important issue in general relativity. Among them, Ayón Beato and García coupled nonlinear electrodynamics to the Einstein field equation to obtain some regular black hole solutions which have no singularities and their gravitational field is regularized [55–58]. When the charge q = 0, this class of black hole solutions transfers to singular Schwarzschild solutions. When the charge q ≠ 0, the charge q appears as a nonlinear electromagnetic field Fμν, while this class of black holes is obtained without singularities, and the associated gravitational field is regularized. In [59], Cai et al studied a class of ABG-related black hole solutions containing mass m, charge q, and three terms α, β, and γ associated with nonlinear electrodynamics. The charge q and the term γ are shown to have a strong influence on the Hawking temperature and the radius of an event horizon. Furthermore, Cai et al obtained an upper limit on the charge q based on the analysis of data from the shadow of M87* [59]. That is, in the 1 σ confidence region, the upper limit of q is 0.7. While in the 2 σ confidence region, the upper limit of q is always below 1. Recently, the thermodynamic phase transition of this generalized ABG black hole was studied by Ghasemi et al. The results suggest that the thermodynamic system of this ABG black hole is involved in the Hawking-Page phase transition, in which the ABG black hole of unbalanced evaporation eventually reaches the AdS vacuum space [60]. Obviously, this ABG black hole has many properties that deserve to be studied.
Based on the previous work, it is an interesting practice to explore the shadow and observational appearance of the ABG black hole surrounded by the thin disk accretion. In this paper, when a thin disk exists around the ABG black hole, we have studied its shadow by using a ray-tracing method. First, by analyzing the behavior of photons around the ABG black hole, we obtained the trajectories of light rays in the vicinity of the ABG black hole surrounded by the thin disk accretion. Then, the trajectories of light rays emitted from the north pole direction of the ABG black hole are analyzed. The results show that the light rays are distinguished into three types, i.e., direct emission, lens rings and photon rings. Further, the characteristics of the three rings and differences in their contribution to the observed intensity are studied. Especially, with the emission function at different positions, we also study whether changes in the charge q and the term γ have an impact on the shadow and observational appearance of the ABG black hole. This could help us to understand the importance of q and γ for the ABG black hole.
The paper is structured as follows. In section 2 , the effective potential and photon orbits of the ABG black hole are discussed. In section 3 , when the ABG black hole is surrounded by a thin disk, we studied its shadows and rings. Section 4 concerns summary and discussion.
2. The effective potential and photon orbits of the ABG black hole
In this section, we will discuss the effective potential and photon orbits of the ABG black hole. The premise of studying the deflection of a light beam near the ABG black hole is to understand the motion of photons around the ABG black hole. The nonlinear Einstein Maxwell action function is obtained by coupling the nonlinear electrodynamics to the Einstein field equations by Ayón Beato and García [55], which is2 ) do not depend on time t and azimuth φ. Therefore, there are two conserved quantities, energy E and angular momentum L, which are denoted as4 )–(6 ), (7 ) and ${g}_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=0$ for the null geodesic, the expression can be obtained as8 ) can be obtained, which is
$\begin{eqnarray}S=\int {{\rm{d}}}^{4}x\sqrt{-g}\left[\displaystyle \frac{R}{16\pi }-\displaystyle \frac{L(P)}{4\pi }\right].\end{eqnarray}$
Here, R is the scalar curvature, L(P) is a function of the invariant scalar $P\equiv \tfrac{1}{4}{P}_{\mu \nu }{P}^{\mu \nu }$ and the nonlinear antisymmetric tensor ${P}_{\mu \nu }\equiv \tfrac{{F}_{\mu \nu }}{{H}_{P}}$, L(P) = 2PHP − H(P) is the nonlinear electromagnetic field Lagrangian density. Also, ${H}_{P}\equiv \tfrac{{\rm{d}}H(P)}{{\rm{d}}P}$ is the structure function of the nonlinear electrodynamic theory, Fμν ≡ ∂μAν − ∂νAμ is the nonlinear electromagnetic tensor, and Aμ is the electromagnetic potential [55]. Considering the role of static spherically symmetric black holes in four-dimensional spacetime, the form is $\begin{eqnarray}{\rm{d}}{s}^{2}=-f(r){\rm{d}}{t}^{2}+\displaystyle \frac{1}{f(r)}{\rm{d}}{r}^{2}+{r}^{2}{\rm{d}}{\theta }^{2}+{r}^{2}{\sin }^{2}\theta {\rm{d}}{\phi }^{2}\end{eqnarray}$
with $\begin{eqnarray}f(r)=1-\displaystyle \frac{2{{mr}}^{\tfrac{\alpha \gamma }{2}-1}}{{\left({r}^{\gamma }+{q}^{\gamma }\right)}^{\tfrac{\alpha }{2}}}+\displaystyle \frac{{q}^{2}{r}^{\tfrac{\beta \gamma }{2}-2}}{{\left({r}^{\gamma }+{q}^{\gamma }\right)}^{\tfrac{\beta }{2}}}.\end{eqnarray}$
It is a class of Ayón-Beato-García (ABG) related black hole solutions with mass m, charge q and three terms α, β and γ associated with nonlinear electrodynamics [59]. This class of black hole solutions is regular under the condition, that is, αγ ≥ 6, βγ ≥ 8 and γ > 0. According to the saturation cases $\alpha =\tfrac{6}{\gamma }$ and $\beta =\tfrac{8}{\gamma }$, the metric function appears in the following form $\begin{eqnarray}f(r)=1-\displaystyle \frac{2{{mr}}^{2}}{{\left({r}^{\gamma }+{q}^{\gamma }\right)}^{\tfrac{3}{\gamma }}}+\displaystyle \frac{{q}^{2}{r}^{2}}{{\left({r}^{\gamma }+{q}^{\gamma }\right)}^{\tfrac{4}{\gamma }}}.\end{eqnarray}$
With the help of the Euler–Lagrange equation, we can study the null geodesic of the ABG black hole, which is $\begin{eqnarray}\displaystyle \frac{{\rm{d}}}{\lambda }\left(\displaystyle \frac{\partial { \mathcal L }}{\partial {\dot{x}}^{\mu }}\right)=\displaystyle \frac{\partial { \mathcal L }}{\partial {x}^{\mu }},\end{eqnarray}$
where λ is the affine parameter and ${\dot{x}}^{\mu }$ is the four-dimensional velocity of the photon. In this spacetime, the Lagrangian quantity of the particle is $\begin{eqnarray}\begin{array}{rcl}{ \mathcal L } & = & \displaystyle \frac{1}{2}{g}_{\mu \nu }{\dot{x}}^{\mu }{\dot{x}}^{\nu }=\displaystyle \frac{1}{2}\left[-f(r){\dot{t}}^{2}+\displaystyle \frac{{\dot{r}}^{2}}{f(r)}\right.\\ & & \left.+{r}^{2}\left({\dot{\theta }}^{2}+{\sin }^{2}\theta {\dot{\phi }}^{2}\right)\Space{0ex}{3.19ex}{0ex}\right].\end{array}\end{eqnarray}$
Under the condition of setting $\theta =\tfrac{\pi }{2}$, the motion of the photon is fixed on the equatorial plane. In addition, the metric coefficients in equation ( $\begin{eqnarray}E=-\displaystyle \frac{\partial { \mathcal L }}{\partial \dot{t}}=f(r)\dot{t},L=\displaystyle \frac{\partial { \mathcal L }}{\partial \dot{\phi }}={r}^{2}\dot{\phi }.\end{eqnarray}$
According to equations ( $\begin{eqnarray}\dot{t}=\displaystyle \frac{1}{{b}_{c}f(r)},\dot{\phi }=\pm \displaystyle \frac{1}{{r}^{2}},{\dot{r}}^{2}=\displaystyle \frac{1}{{b}_{c}^{2}}-\displaystyle \frac{1}{{r}^{2}}f(r).\end{eqnarray}$
It is worth noting that the affine parameter is taken as $\lambda \to \tfrac{\lambda }{| L| }$. In the above equation, bc is called the impact parameter, which is the ratio of the absolute value of the angular momentum to the energy, i.e., ${b}_{c}=\tfrac{| L| }{E}$. Therefore, another form of equation ( $\begin{eqnarray}{\dot{r}}^{2}+{V}_{\mathrm{eff}}=\displaystyle \frac{1}{{b}_{c}^{2}},\end{eqnarray}$
where ${V}_{\mathrm{eff}}=\tfrac{1}{{r}^{2}}f(r)$ is the effective potential of the ABG black hole. At the position of the photon sphere, the effective potential should satisfy the conditions of ${V}_{\mathrm{eff}}=\tfrac{1}{{b}_{c}^{2}}$ and ${V}_{\mathrm{eff}}^{{\prime} }=0$. In the background of four-dimensional symmetric spacetime, the radius rp and the impact parameter bp of the photon sphere satisfy the conditions of ${r}_{p}^{2}={b}_{p}^{2}f(r)$ and $2{b}_{p}^{2}f{\left(r\right)}^{2}={r}^{3}f^{\prime} (r)$. When the charge q and the term γ associated with nonlinear electrodynamics take different values, based on the conditions satisfied by rp and bp, we can obtain the relevant physical quantities r+, rp and bp, which are given in tables 1 and 2.Table 1. The event horizon r+, the radius rp and the impact parameter bp of the photon sphere when q takes different values, where m = 1 and γ = 1. |
| q = 0.01 | q = 0.02 | q = 0.04 | q = 0.08 | q = 0.1 | q = 0.15 | q = 0.2 | q = 0.25 | |
|---|---|---|---|---|---|---|---|---|
| r+ | 1.96980 | 1.93917 | 1.87659 | 1.74532 | 1.67612 | 1.48944 | 1.27371 | 0.995778 |
| rp | 2.95973 | 2.91890 | 2.83547 | 2.66061 | 2.56853 | 2.32108 | 2.03865 | 1.69121 |
| bp | 5.14384 | 5.09081 | 4.98244 | 4.75548 | 4.63613 | 4.31635 | 3.95483 | 3.52192 |
As can be seen from table 1, the event horizon r+, the radius rp and the impact parameter bp of the photon sphere decrease as the charge q increases. In table 2, the event horizon r+, the radius rp and the impact parameter bp of the photon sphere increase as the term γ associated with nonlinear electrodynamics increases. However, after γ increases to 4, the values of r+, rp and bp no longer increase. In addition, the effective potential Veff is plotted when q and γ take different values, as shown in figure 1.
Figure 1. The profiles of the effective potential Veff and r, in which m = 1. The left figure corresponds to the case where γ is fixed and q varies, and the right figure corresponds to the case where q is fixed and γ varies. |
Table 2. The event horizon r+, the radius rp and the impact parameter bp of the photon sphere when γ takes different values, where m = 1 and q = 0.04. |
| γ = 1 | γ = 2 | γ = 3 | γ = 4 | γ = 5 | γ = 8 | γ = 10 | γ = 20 | |
|---|---|---|---|---|---|---|---|---|
| r+ | 1.87659 | 1.99800 | 1.99918 | 1.99920 | 1.99920 | 1.99920 | 1.99920 | 1.99920 |
| rp | 2.83547 | 2.99760 | 2.99892 | 2.99893 | 2.99893 | 2.99893 | 2.99893 | 2.99893 |
| bp | 4.98244 | 5.19338 | 5.19475 | 5.19477 | 5.19477 | 5.19477 | 5.19477 | 5.19477 |
The relationship between effective potential Veff and r is shown in figure 1. It should be noted that Veff = 0 corresponds to the position of the event horizon. As r increases, the value of Veff begins to increase, and reaches the maximum at the photon sphere where r ∼ rp. In subfigure (a), when the term γ associated with nonlinear electrodynamics is fixed, the maximum value of Veff increases with the increase of q, while the corresponding event horizon decreases with the increase of q. In subfigure (b), when q is fixed, the maximum value of Veff decreases with the increase of γ. Here, an interesting phenomenon emerges, when γ = 2, γ = 3, γ = 8 and γ = 10, the lines of Veff have almost the same trend and are difficult to distinguish. In other words, when γ increases to a certain value, the change of Veff is very small, and the corresponding event horizon almost does not change. Comparing the two subfigures in figure 1, it can be found that the charge q has a greater effect on Veff, while the term γ associated with nonlinear electrodynamics has a smaller effect on Veff.
Now, let us focus on the trajectory of the light ray. Based on equation (8 ), the resulting equation is10 ), it can be rewritten as
$\begin{eqnarray}\displaystyle \frac{{\rm{d}}r}{{\rm{d}}\phi }=\pm {r}^{2}\sqrt{\displaystyle \frac{1}{{b}_{c}^{2}}-\displaystyle \frac{1}{{r}^{2}}\left[1-\displaystyle \frac{2{{mr}}^{2}}{{\left({r}^{\gamma }+{q}^{\gamma }\right)}^{\tfrac{3}{\gamma }}}+\displaystyle \frac{{q}^{2}{r}^{2}}{{\left({r}^{\gamma }+{q}^{\gamma }\right)}^{\tfrac{4}{\gamma }}}\right]}.\end{eqnarray}$
By redefining a parameter u = 1/r in equation ( $\begin{eqnarray}\begin{array}{l}{ \mathcal R }(u,b)\equiv \displaystyle \frac{{\rm{d}}u}{{\rm{d}}\phi }\\ \quad =\sqrt{\displaystyle \frac{1}{{b}_{c}^{2}}-{u}^{2}\left[1-\displaystyle \frac{2m}{{u}^{2}{\left({\left(\tfrac{1}{u}\right)}^{\gamma }+{q}^{\gamma }\right)}^{\tfrac{3}{\gamma }}}+\displaystyle \frac{{q}^{2}}{{u}^{2}{\left({\left(\tfrac{1}{u}\right)}^{\gamma }+{q}^{\gamma }\right)}^{\tfrac{4}{\gamma }}}\right]}.\end{array}\end{eqnarray}$
With the help of the ray-tracing method, the trajectories of light rays are obtained for different values of the charge q and the term γ, as shown in figure 2.
Figure 2. The trajectories of light rays in the polar coordinates (r, φ). Here, the black disk represents the ABG black hole and the gray dotted circle represents the photon sphere. |
In figure 2, these different colored lines have different physical meanings. It is true that when bc = bp, the light ray will always surround the black hole infinite times, and neither be deflected nor fall into the black hole. Due to the numerical accuracy in the mathematical program, it is difficult for us to accurately show the light rays for the case bc = bp. So, we selected the case that bc is very close to but smaller than bp to show the light ray that is very close to the photon sphere, which corresponds to the red line in figure 2. When bc > bp, the light rays correspond to the green line, and the light rays are deflected to form the shadow of the ABG black hole. When bc < bp, the light ray corresponds to the black line, and the light rays fall into the ABG black hole. From figure 2, we can see that the trajectories of the photons are different when the charge q and the term γ associated with nonlinear electrodynamics are taken to different values. It is suggested that changes in q and γ cause changes in the geometric structure of spacetime, which also causes the appearance of black holes observed by distant observers to be different.
3. Shadows and rings of the ABG black hole
3.1. Direct emission, lens ring and photon ring
In this section, we will study the observational appearance of the ABG black hole surrounded by the thin disk accretion. In this model, we consider the thin disks are horizontally placed on the equatorial plane of the ABG black hole3(3Obviously, it is also very interesting to discuss a tipped disk, which we leave for future work.). The trajectory of the light ray near a black hole is an important basis for studying the shadow and rings of a black hole. Gralla et al suggest that light rays will have different types when light rays reach the vicinity of a black hole [32]. Therefore, following the method in [32], we use the total number of orbits $n=\tfrac{\varphi }{2\pi }$ to distinguish the trajectories of the light rays as direct emission, lens ring and photon ring. The total number of orbits $n\lt \tfrac{3}{4}$ corresponds to the direct emission, where the trajectories of the light rays will intersect the equatorial plane once. When the total number of orbits $\tfrac{3}{4}\lt n\lt \tfrac{5}{4}$ corresponds to the lens ring, the trajectories of the light rays will intersect the equatorial plane at least twice. When $n\gt \tfrac{5}{4}$ corresponds to the photon ring, the trajectories of the light rays will intersect the equatorial plane at least three times. For the ABG black hole, the regions of direct emission, lens ring and photon ring with respect to the impact parameter bc, are given in table 3 and figure 3 when q and γ take different values.
Figure 3. The relationship between the total number of orbits n and the impact parameter bc. |
Table 3. The regions of direct emission, lens rings and photon rings relative to the impact parameter bc. |
| γ = 1 and q = 0.04 | γ = 1 and q = 0.08 | γ = 3 and q = 0.04 | |
|---|---|---|---|
| Direct emission | bc < 4.78943 | bc < 4.54746 | bc < 5.01366 |
| $n\lt \tfrac{3}{4}$ | bc > 5.98492 | bc > 5.79496 | bc > 6.16637 |
| | |||
| Lens ring | 4.78943 < bc < 4.97268 | 4.54746 < bc < 4.74376 | 5.01366 < bc < 5.18640 |
| $\tfrac{3}{4}\lt n\lt \tfrac{5}{4}$ | 5.01793 < bc < 5.98492 | 4.79575 < bc < 5.79496 | 5.22656 < bc < 6.16637 |
| | |||
| Photon ring | 4.97268 < bc < 5.01793 | 4.74376 < bc < 4.79575 | 5.18640 < bc < 5.22656 |
| $n\gt \tfrac{5}{4}$ | |||
As can be seen from table 3, the regions of direct emission, lens ring and photon ring change as the charge q and the term γ associated with nonlinear electrodynamics increase. In figure 3, the relationship between the total number of orbits n and the impact parameter bc is shown in the coordinate system and the regions of direct emission, lens ring and photon ring are also shown. In figure 3, the value of bc decreases when γ is fixed and q increases, and increases when q is fixed and γ increases. Thus, changes in q and γ have opposite effects on the position of these three regions. To visualize the effect of a change in q and γ on these three regions, we give the trajectories of photons near the ABG black hole in figure 4.
Figure 4. Trajectories of photons near the ABG black hole. Here, the gray dashed line represents the photon sphere, the black disk represents the ABG black hole, the red line represents the direct emission, the blue line represents the lens ring and the green line represents the photon ring. |
In figure 4, the trajectories of photons near the ABG black hole are shown in the polar coordinate system (bc, φ). From figure 4, we find that γ is fixed while q increases and the area of the lens ring and photon ring becomes wider. The opposite is the case, where q is fixed and γ increases, and the area of the lens ring and photon ring becomes narrower. Obviously, the charge q and the term γ associated with nonlinear electrodynamics have a great influence on the trajectories of light rays.
3.2. Observed specific intensity and transfer function
We can study the observed specific intensity under the model of a thin disk surrounding the ABG black hole. It is assumed that the thin disk accretion locates at the stationary frame of the static world line and that the photons emitted from the disk are isotropic. When the static observer locates at the north pole of the ABG black hole, the observed specific intensity can be expressed as [23]
$\begin{eqnarray}{I}_{\mathrm{obs}}(r)=f{\left(r\right)}^{\tfrac{3}{2}}{I}_{\mathrm{emi}}(r),\end{eqnarray}$
where Iobs(r) is the observed specific intensity, and frequency ν and Iemi(r) is the emitted specific intensity with frequency νe. The total specific intensity I(r) is obtained by integrating over all frequencies of Iobs(r), which is denoted as $\begin{eqnarray}\begin{array}{rcl}I(r) & = & \displaystyle \int {I}_{\mathrm{obs}}(r){\rm{d}}\nu =\displaystyle \int f{\left(r\right)}^{2}{I}_{\mathrm{emi}}(r){\rm{d}}{\nu }_{e}\\ & = & f{\left(r\right)}^{2}{I}_{\mathrm{em}}(r).\end{array}\end{eqnarray}$
In the above equation, Iem(r) = ∫Iemi(r)dνe is the total emission specific intensity and $\nu =f{\left(r\right)}^{\tfrac{1}{2}}{\nu }_{e}$.As the light ray reaches the vicinity of the ABG black hole, it will intersect with the thin disk located in the equatorial plane of the ABG black hole. The light ray intersects the thin disk once, which corresponds to the case of direct emission. The light ray intersects the thin disk twice, which forms the lens ring. The light ray intersects the thin disk three times, which forms the photon ring. The additional luminosity gained by the photon ring increases as the number of times the light ray intersects the thin disk increases. Therefore, the observed intensity is the sum of the intensities at these intersections, which is
$\begin{eqnarray}I(r)=\sum f{\left(r\right)}^{2}{I}_{\mathrm{em}}(r){| }_{r={r}_{n}({b}_{c})}.\end{eqnarray}$
Here, rn(bc) is defined as the transfer function, which is used to represent the radial position of the thin disk plane with the nth intersection point. As discussed above, different light rays will intersect with the thin disk different times. Each intersection will make the light rays obtain the additional luminosity and this luminosity is dependent on the radial positions, which will give rise to the different appearances of ABG black holes. Therefore, it is only when the transfer function is considered that the lens ring and the photon ring can be formed, which will be shown in a later section. It is worth noting that we do not consider the absorption of a light ray by the thin disk, as this would lead to a reduction in the observed intensity. In addition, the slope of the transfer function determines the amplification ratio. For different q and γ, the relationship between the transfer function rn(bc) and the impact parameter bc is plotted in figure 5.
Figure 5. Relationship between the transfer function rn(bc) and the impact parameter bc. Here, the red, blue and green lines represent the radial coordinates of the first, second and third transfer functions intersecting the thin disk. |
As can be seen from figure 5, the slope of the red line is small, the slope of the blue line is large and the slope of the green line tends to infinity. Here, the red line represents the radial coordinate of the first transfer function r1(bc) intersecting the thin disk, which corresponds to the direct emission and is related to the redshift source. The blue line represents the radial coordinates of the second transfer function r2(bc) intersecting the thin disk, which represents the lens ring. The green line represents the radial coordinates of the third transfer function r3(bc) intersecting the thin disk, which represents the photon ring.
3.3. Observational characteristics of direct emission and ring
Once the form and position of a typical emission function have been determined, we can study the observed specific intensity. Considering the fact that universality of exponential decay, we assume that the shadow luminosity intensity decreases exponentially, which is14 ) and (15 ).
$\begin{eqnarray}{I}_{\mathrm{em}}(r)=\left\{\begin{array}{ll}\exp \left[\displaystyle \frac{-{\left(r-{r}_{\mathrm{in}}\right)}^{2}}{10}\right], & r\gt {r}_{\mathrm{in}}\\ 0, & r\leqslant {r}_{\mathrm{in}}\end{array}\right..\end{eqnarray}$
Here, rin is the innermost position of the thin disk accretion, which is assumed to have three cases. For the first case, the thin disk accretion is located at the innermost stable circular orbit with rin = risco, where risco is the innermost stable circular orbit of the ABG black hole. For the second case, the thin disk accretion is located at the photon sphere with rin = rp. For the third case, the thin disk accretion is located at the event horizon with rin = r+. When q and γ take different values, the observed specific intensity and its optical appearance are plotted in figures 6, 7 and 8 with the help of equations (
Figure 6. The observed specific intensity and its optical appearance, γ = 1 and q = 0.04. The top row is the relation about I(r) versus bc. The bottom row is the two-dimensional of the observed specific intensity. |
The first, second and third columns correspond to the observed intensity and the optical appearance of the observed intensity when the thin disk accretion located at risco, rp and r+, respectively. In the first column of figures 6, 7 and 8, the thin disk accretion is located at the innermost stable circular orbit. The top row shows that for γ = 1 and q = 0.04, the first peak of the observed intensity is 0.38 at bc ≃ 5m (0.1 at bc ≃ 4.8m for γ = 1 and q = 0.08) (0.41 at bc ≃ 5.2m for γ = 3 and q = 0.04), which corresponds to the photon ring. Then, the second peak of the observed intensity is 0.46 at bc ≃ 5.4m (0.44 at bc ≃ 5.2m for γ = 1 and q = 0.08) (0.47 at bc ≃ 5.6m for γ = 3 and q = 0.04), which corresponds to the lens ring. Finally, the third peak is 0.45 at bc ≃ 7.3m (0.44 at bc ≃ 7m for γ = 1 and q = 0.08) (0.46 at bc ≃ 7.5m for γ = 3 and q = 0.04), which corresponds to direct emission. It can be seen from the bottom row of the first column of figures 6, 7 and 8, that the lens ring is a thin ring and the photon ring is a very narrow ring. In the second columns of figures 6, 7 and 8, the thin disk accretion is located at the photon sphere. For the cases of γ = 1 and q = 0.04 and γ = 3 and q = 0.04, the observed intensity has two peaks. As can be seen from the bottom rows of figures 6 and 8, the lens ring and the photon ring can be distinguished. However, for the case of γ = 1 and q = 0.08, there is only one peak of the observed intensity. The lens ring and photon ring are superimposed in the bottom row of figure 7. In the third columns of figures 6, 7 and 8, the thin disk accretion is located at the event horizon. For the cases of γ = 1 and q = 0.04 and γ = 3 and q = 0.04, the lens ring and the photon ring can be distinguished. However, for the case of γ = 1 and q = 0.08, the lens ring and photon ring are superimposed. In summary, the direct emission determines the total observed intensity, the lens ring makes a small contribution to the observation, and the photon ring makes almost no contribution.
Figure 7. The observed specific intensity and its optical appearance, γ = 1 and q = 0.08. The top row is the relation regarding I(r) versus bc. The bottom row is the two-dimensional of the observed specific intensity. |
Figure 8. The observed specific intensity and its optical appearance, γ = 3 and q = 0.04. The top row is the relation regarding I(r) versus bc. The bottom row is the two-dimensional of the observed specific intensity. |
Comparing these three sets of figures, some meaningful conclusions can be drawn. Firstly, by comparing figure 6 and figure 7, it can be seen that all the peaks and positions of the peaks of the observed intensity decrease as the charge q increases. In particular, for the case of q = 0.08, the corresponding lens ring and photon ring are overlapped when the thin disk accretion is located at rp and r+, which is a very different result from q = 0.04. This suggests that the charge q has a significant effect on the observed appearance of the ABG black hole. Secondly, all the peaks of the observed intensity and the positions of the peaks increase with the increase of the term γ associated with nonlinear electrodynamics. Meanwhile, the brightness of the ring with γ = 3 is higher than that of γ = 1. This indicates that the brightness of the ring is closely related to γ. Thirdly, the observed appearances of the ABG black hole are different when the thin disk accretion is located at different positions. In particular, shadows and photon rings are more easily distinguished when the thin disk accretion is located at risco. Obviously, the charge q and the term γ associated with nonlinear electrodynamics have a great influence on the observation of the ABG black hole, which may help us to distinguish the ABG black hole from other black holes.
4. Conclusion and discussion
In this paper, we have studied the shadows and observational appearance of the ABG black hole surrounded by the thin disk accretion using a ray-tracing method. For the ABG black hole, the effective potential and the motion of photons in the vicinity of the ABG black hole are discussed. When the ABG black hole is surrounded by the thin disk accretion, we find that the trajectories of light rays are redefined as direct emission, lens ring and photon ring. After introducing the transfer and emission functions, we further study the observational appearance of the ABG black hole surrounded by the thin disk accretion.
After completing the above works, we have obtained some interesting conclusions. Firstly, the event horizon r+, the radius rp and the impact parameter bp of the photon sphere decrease as the charge q increases, while the maximum value of the effective potential Veff increases. However, the event horizon r+, the radius rp, and the impact parameter bp of the photon sphere increase with the increases of the term γ associated with nonlinear electrodynamics, and the maximum value of the effective potential Veff decreases. Secondly, the effects of the charge q and the term γ associated with nonlinear electrodynamics on the trajectories of light rays are significant. The regions of direct emission, lens ring and photon ring all change with q and γ. This means that q and γ also have an effect on the observational appearance. In particular, as the charge q increases, the peak and the position of the peak decreases for all observed intensities. Also, we note that the lens ring and the photon ring for γ = 1 and q = 0.08 are overlapped when the thin disk accretion located at the photon sphere and the event horizon, while the lens ring and the photon ring for γ = 1 and q = 0.04 are not overlapped. When the charge q is fixed, the peak and the position of the peak of all observed intensities corresponding to γ = 3 are always greater than that of γ = 1. In particular, the brightness of the photon ring corresponding to γ = 3 and q = 0.04 is higher than that of γ = 1 and q = 0.04. Finally, the position of the thin disk accretion also has a great influence on the observation of the ABG black hole. The size of the shadows and the observed appearance of the ABG black hole are different when the thin disk accretion is located at different positions.
In addition, the direct emission plays a decisive role in the total intensity of the observation, while the lens ring makes a small contribution to the observation and the photon ring makes almost no contribution. It is worth noting that the charge q and the term γ associated with nonlinear electrodynamics have a strong influence on the shadows of the ABG black hole, which will probably help us to distinguish the ABG black hole from other black holes in different gravitational contexts. As a brief outlook, it will be interesting to study the shadow of the ABG black hole when it is surrounded by the spherically symmetric accretion, which may further reveal more characteristics of the ABG black hole.