1. Introduction
Recently, the Event Horizon Telescope (EHT) Collaboration unveiled images of the supermassive black hole situated at the center of our galaxy [1–6]. These depictions consistently exhibit a luminous ring encircling a central dark region, similar to the M87* black hole images reported in 2019 [7–12]. In the presence of a light source behind or around the black hole, certain photons fall into the black hole, forming a central dark region known as the black hole shadow, while some undergo deflection, generating a radiant ring around the black hole shadow, referred to as the photon sphere [13]. The investigation of black hole shadows started from a Schwarzschild black hole [14]. Then Bardeen et al explored the shadow of a Kerr black hole [15]. Numerical calculation of the black hole shadow with a rotating accretion disk was started in [16]. The visible shapes of the brightest black holes accompanied by an accretion disk were comparable with the EHT observations for M87* and SgrA* [17, 18]. In recent years, research on black hole shadows has extended to black holes in Born–Infeld electromagnetic fields [19, 20], the Ayón–Beato–García black hole family [21] and black holes with multiple photon spheres [22–25]. Moreover, Euler–Heisenberg black holes [26], black holes immersed in quintessence dark energy [27], brane-world black holes [28] and others have also been studied [29–33]. Black hole shadows are important for verifying general relativity in strong gravitational fields and for studying cold dark matter perturbations in the early universe [34].
The ultimate phase of gravitational collapse is theorized to result in the formation of a Kerr–Newman black hole, regardless of the nature of the pre-existing objects involved in the collapse. This proposition implies that these outcomes can be uniquely defined based on their mass, angular momentum and charge properties [35]. The current view is that hairy black holes exist, described not only by mass, charge and angular momentum but also by some nonlinear fields or sources. One can consider a static hairy black hole, which is reasonable because the energy and angular momentum for a rotating one can be extracted by the Penrose superradiance process [36, 37] or the Blandford and Znajek mechanism [38]. Thus, the existence of non-rotating hairy black holes cannot be ignored. The hairy black hole solution is not easy to find because obtaining analytical solutions directly for the Einstein field equations is challenging. However, a notable exception that allows for direct solutions is the perfect fluid ${\tilde{T}}_{\mu \nu }$. Therefore, we consider employing the method of minimal geometric deformation (MGD) applied in brane-world scenarios [39, 40], which involves separately addressing and solving each term of the gravitational source in the equations. This provides a powerful tool for solving Einstein's field equations under nonlinear conditions. The MGD method is widely used in gravitational decoupling and has been verified by several problems in relativistic astrophysics [41–44]. Then a new static spherically symmetric hairy black hole solution is investigated by using gravitational decoupling [45–53].
To conduct our analysis, we will appropriately employ two types of accretion models, namely spherical accretion and disk accretion [54, 55]. Spherical accretion comprises stationary and infalling accretion. The former simulates the luminosity from a quiescent accretion flow while the latter simulates the luminosity from accretion flow radially infalling into the black hole. Disk accretion models include the exponential model, power-law exponential model, bell-shaped model and GLM model. In this context, the radiation from the exponential, power-law and bell-shaped models emanates respectively from the innermost stable circular orbit, the photon sphere and the event horizon. These models are considered to be in accordance with the observational evidence. Besides, the GLM model has been used in recent studies of general relativistic magnetohydrodynamic simulations of accretion flows [56]: under mild astrophysical assumptions, illuminated by a thin accretion disk, the photon rings are broken into an infinite sequence of concentric rings. These rings produce similar features on the observer's detector, potentially allowing differentiation between different black hole metrics [57]. However, due to the exponential decrease in their luminosities, inner rings are challenging to observe. In this paper, we theoretically predict the observed images of photon rings of hairy black holes under weak energy conditions (WEC). These might be observable in future instruments such as the next-generation Event Horizon Telescope [58].
This paper is organized as follows. In section 2 , we briefly review the method of gravitational decoupling and the work of Avalos and derive the black hole metric in WEC, then we derive the photon equation of motion and plot the trajectories for different values of the hairy parameter α. Then, in section 3 , we investigate the relationship between the light intensity I with respect to the radius r as well as the impact parameter b, and use the backward ray-tracing method to investigate the appearance of stationary and infalling accretion disks. Afterwards, in section 4 , we use the same method to simulate the shadows of the accretion disk under five different emission models. Finally, we outline our conclusions and present some perspectives for the future in section 5 .
2. Gravitational decoupling and geodesics
In this section, we present the concept of a hairy black hole in the framework of gravitational decoupling and proceed to examine the trajectories of massless and massive particles. In the case of spherically symmetric and stationary spacetime, an ansatz can be formulated
$\begin{eqnarray}{\rm{d}}{s}^{2}=-{{\rm{e}}}^{\nu (r)}{\rm{d}}{t}^{2}+{{\rm{e}}}^{\lambda (r)}{\rm{d}}{r}^{2}+{r}^{2}{\rm{d}}{{\rm{\Omega }}}^{2},\end{eqnarray}$
where ${\rm{d}}{{\rm{\Omega }}}^{2}={\rm{d}}{\theta }^{2}+{\sin }^{2}\theta {\rm{d}}{\phi }^{2}$. We can obtain the metric function [45] $\begin{eqnarray}F(r)={{\rm{e}}}^{\nu }={{\rm{e}}}^{-\lambda }=1-\displaystyle \frac{2M}{r}+\displaystyle \frac{5}{4}\displaystyle \frac{\alpha M}{{r}^{2}}+\displaystyle \frac{\alpha M}{{r}^{2}}\mathrm{ln}\displaystyle \frac{r}{{r}_{H}},\end{eqnarray}$
where M is the mass of the black hole, α is the hairy parameter, which represents the matter sector, rH is the horizon radius, which has the form ${r}_{H}=M+\sqrt{{M}^{2}-5/4\alpha {\rm{M}}}$, and 0 < α < 0.8M is necessary to avoid the appearance of a naked singularity.To conduct our investigation on the shadow of the black hole, it is natural to concentrate on the trajectory of the photons in the background spacetime. The equation of motion of the massless particles can be demonstrated as3 ) can be rewritten as
$\begin{eqnarray}-{g}_{\mu \nu }\displaystyle \frac{{\rm{d}}{x}^{\mu }}{{\rm{d}}\lambda }\displaystyle \frac{{\rm{d}}{x}^{\nu }}{{\rm{d}}\lambda }=0,\end{eqnarray}$
where λ is the affine parameter. We are more concerned with spherically symmetric spacetime, and one can find two Killing vectors $\begin{eqnarray}\begin{array}{l}{K}_{\mu }=(-F(r),0,0,0),\\ {R}_{\mu }=(0,0,0,{r}^{2}{\sin }^{2}\theta ).\end{array}\end{eqnarray}$
We could obtain the energy E and the angular momentum L for θ = π/2 along the geodesics $\begin{eqnarray}E=-{K}_{\mu }\displaystyle \frac{{\rm{d}}{x}^{\mu }}{{\rm{d}}\lambda }=F(r)\displaystyle \frac{{\rm{d}}t}{{\rm{d}}\lambda },\quad L={R}_{\mu }\displaystyle \frac{{\rm{d}}{x}^{\mu }}{{\rm{d}}\lambda }={r}^{2}\displaystyle \frac{{\rm{d}}\phi }{{\rm{d}}\lambda }.\end{eqnarray}$
Then the geodesics equation ( $\begin{eqnarray}-{E}^{2}+F(r)\displaystyle \frac{{L}^{2}}{{r}^{2}}+{\left(\displaystyle \frac{{\rm{d}}r}{{\rm{d}}\lambda }\right)}^{2}=0.\end{eqnarray}$
By setting the affine parameter λ into λ/L and defining the impact parameter b ≡ L/E, one can derive $\begin{eqnarray}\dot{t}=\displaystyle \frac{1}{{bF}(r)},\end{eqnarray}$
$\begin{eqnarray}\dot{\phi }=\pm \displaystyle \frac{1}{{r}^{2}},\end{eqnarray}$
$\begin{eqnarray}{\dot{r}}^{2}=\displaystyle \frac{1}{{b}^{2}}-\displaystyle \frac{F(r)}{{r}^{2}},\end{eqnarray}$
where the ± denotes the counterclockwise and clockwise geodesics of the photon. Subsequently, the equation for the trajectory of the photon can be deduced as $\begin{eqnarray}\displaystyle \frac{{\rm{d}}r}{{\rm{d}}\phi }=\pm {r}^{2}\sqrt{\displaystyle \frac{1}{{b}^{2}}-\displaystyle \frac{F(r)}{{r}^{2}}}.\end{eqnarray}$
One can set V(r) = F(r)/r2 as the effective potential of photons, which is shown in figure 1(a), and notice that when $\dot{r}{| }_{r={r}_{\mathrm{ph}}}=0,\ddot{r}{| }_{r={r}_{\mathrm{ph}}}=0$ photons can circle the black hole many times in unstable orbits, called the photon sphere, which satisfy10 ), as shown in figures 1(b)–(d) with different α. The green lines indicate photons that escape the black hole after being affected by its gravity, and the gray lines are photons that fall into the black hole. The red dashed line indicates the photon sphere, which is a very ideal situation. Note that the photon sphere is the division between photons that fall into the black hole and those that escape from the black hole.
$\begin{eqnarray}{b}_{\mathrm{ph}}=\displaystyle \frac{1}{\sqrt{V({r}_{\mathrm{ph}})}},\qquad \displaystyle \frac{{\rm{d}}V({r}_{\mathrm{ph}})}{{\rm{d}}r}=0,\qquad \displaystyle \frac{{{\rm{d}}}^{2}V({r}_{\mathrm{ph}})}{{\rm{d}}{r}^{2}}\leqslant 0.\end{eqnarray}$
We can obtain the figures of different parameters of the photon trajectory by numerically integrating equation (
Figure 1. The plot on the upper left is the effective potential V(r) of photon with respect to the radius r. The green, red and blue lines represent V(r) differing in the hairy parameter α. The other parts show the geodesic of photons for different α with fixed M = 1, where the green lines are the photons escaping from the black hole, the gray ones are photons falling in the black hole and the red dashed line indicates the photon sphere. |
Then we consider particles with unit mass. The geodesic equation implying the trajectories of particles with unit mass can be simply written as
$\begin{eqnarray}-{g}_{\mu \nu }\displaystyle \frac{{\rm{d}}{x}^{\mu }}{{\rm{d}}\lambda }\displaystyle \frac{{\rm{d}}{x}^{\nu }}{{\rm{d}}\lambda }=1,\end{eqnarray}$
where τ is the proper time of the particles. Similar to photons, one can derive the equation of motion in spacetime [59] $\begin{eqnarray}{\left(\displaystyle \frac{{\rm{d}}r}{{\rm{d}}\tau }\right)}^{2}={E}_{m}^{2}-F(r)\left(1+\displaystyle \frac{{L}_{m}^{2}}{{r}^{2}}\right),\end{eqnarray}$
where Em and Lm are the energy and angular momentum of the particle, respectively. Unlike for the definition of the effective potential of photons, we can simply set the effective potential of the massive particle $\begin{eqnarray}{V}_{\mathrm{eff}}(r)=\sqrt{F(r)\left(1+\displaystyle \frac{{L}_{m}^{2}}{{r}^{2}}\right)}.\end{eqnarray}$
There is a minimum radius at which stable circular motion remains feasible, defined as the innermost stable circular orbit (ISCO), and it satisfies $\begin{eqnarray}\displaystyle \frac{{\rm{d}}r}{{\rm{d}}\tau }=0,\qquad \displaystyle \frac{{{\rm{d}}}^{2}r}{{\rm{d}}{\tau }^{2}}=0.\end{eqnarray}$
Due to the complexity of the analytical solutions of rh, rph, bph and rISCO we list the numerical solutions in table 1.Table 1. Data for rh, rph, bph and rISCO for different α with fixed M = 1. For α → 0, the black hole reduces to the Schwarzschild type. |
| α = 0 | α = 0.3 | α = 0.4 | α = 0.5 | α = 0.6 | α = 0.7 | |
|---|---|---|---|---|---|---|
| rh | 2 | 1.790 57 | 1.707 11 | 1.612 37 | 1.5 | 1.353 55 |
| rph | 3 | 2.686 04 | 2.5692 | 2.418 91 | 2.250 46 | 2.030 96 |
| bph | 5.196 15 | 4.717 08 | 4.527 07 | 4.312 21 | 4.058 74 | 3.731 48 |
| rISCO | 6 | 5.271 45 | 4.984 17 | 4.660 41 | 4.279 80 | 3.789 92 |
Note that as the hairy parameter α increases, rh, rph and rISCO all decrease, and fewer photons fall into the black hole. As the impact parameter b is in a fixed interval, an increase in α causes the particles to undergo large-angle deflection. The effect of α is significant for the observational appearance.
3. Shadows and photon spheres with spherical accretions
A black hole accretes matter, which becomes the source of illumination. In this section, we study spherically symmetric accretion flows, categorized as stationary accretion and infalling accretion. The backward ray-tracing method is employed to study the specific intensity received by a distant observer [16].
3.1. Stationary spherical accretions
We start with the investigation of the hairy black hole shadows of stationary spherically symmetric accretions, and are generally concerned with the specific intensity observed by the static observer. The photon emissivity can be written as
$\begin{eqnarray}j({\nu }_{{\rm{e}}})\propto \rho (r)P({\nu }_{{\rm{e}}}),\end{eqnarray}$
where ρ(r) is the photon density. We assume that the density ρ(r) satisfies the logarithmic normal distribution due to spherical symmetry; this can be expressed as $\begin{eqnarray}\rho (r)=\displaystyle \frac{1}{r}\sqrt{\displaystyle \frac{\gamma }{\pi }}{{\rm{e}}}^{-\gamma {\mathrm{ln}}^{2}\tfrac{r}{{r}_{m}}},\end{eqnarray}$
The parameter γ determines the rate at which the photon density decays in relation to the distance from the sources and ${r}_{m}={r}_{\mathrm{ph}}{{\rm{e}}}^{\tfrac{1}{2\gamma }}$ is a parameter of the logarithmic normal distribution.One can employ a non-monochromatic light source satisfying a normal distribution with center frequency νc, and $P({\nu }_{c})={\int }_{{\nu }_{c}+\sqrt{2}\sigma }^{{\nu }_{c}-\sqrt{2}\sigma }\tfrac{1}{\sigma \sqrt{2\pi }}{{\rm{e}}}^{-\tfrac{{\left(\nu -{\nu }_{c}\right)}^{2}}{2{\sigma }^{2}}}{\rm{d}}\nu \approx 0.842\,70$. By integrating equation (16 ) along the null geodesic, the specific light intensity observed by the observer reads as [60]
$\begin{eqnarray}I(b)=\int {g}^{3}j({\nu }_{{\rm{e}}}){\rm{d}}{l}_{\mathrm{prop}}.\end{eqnarray}$
Here g is the redshift factor and dlprop is the infinitesimal proper line element. The infinitesimal proper length reads $\begin{eqnarray}{\rm{d}}{l}_{\mathrm{prop}}=\pm \sqrt{{g}_{{ij}}{\rm{d}}{x}^{i}{\rm{d}}{x}^{j}}=\pm \sqrt{\displaystyle \frac{1}{F(r)}+{r}^{2}{\left(\displaystyle \frac{{\rm{d}}\phi }{{\rm{d}}r}\right)}^{2}}{\rm{d}}r,\end{eqnarray}$
where the plus and minus signs of the fourth term indicate the photon's counterclockwise and clockwise motions, respectively. The redshift factor g can be determined by $g={p}_{\alpha }{u}_{{\rm{o}}}^{\alpha }/{p}_{\beta }{u}_{{\rm{e}}}^{\beta }$, where pα and pβ are the four-momentum of the photon at reception and emission and ${u}_{o}^{\alpha }\ \mathrm{and}\ {u}_{o}^{\beta }$ are the four-velocity of the observer and the accretion disk. Since the observer is located at an infinite distance, we have ${g}_{\mu \nu }{u}_{e}^{\mu }{u}_{e}^{\nu }=1$ and then ${u}_{{\rm{o}}}^{\alpha }=\left(1,0,0,0\right)$, ${u}_{{\rm{e}}}^{\beta }\,=(\sqrt{1/F(r)},0,0,0)$. By substituting pt = − E into ${g}_{\mu \nu }{p}_{e}^{\mu }{p}_{e}^{\nu }=0$, one can obtain $\begin{eqnarray}{p}_{\mu }=\left(-1,\pm \sqrt{\displaystyle \frac{1}{F(r)}\left(\displaystyle \frac{1}{F(r)}-\displaystyle \frac{{b}^{2}}{{r}^{2}}\right)},0,\pm b\right),\end{eqnarray}$
where the ± signs of the second term indicate the radial inward and outward motion of the photon, respectively.The specific intensity I with respect to the impact parameter b can be obtained by substituting equation (16 ) and equation (19 ) into equation (18 ). We set γ = 1 and M = 1 for simplicity. The blue line in the left column of figure 2 represents the specific intensity I(b) with respect to the impact parameter b. Therefore, we can obtain the shadow of a black hole with a static spherical accretion disk numerically by calculating equation (18 ) in figures 2(c), (f) and (i), corresponding to α = 0.3, 0.6 and 0.8, respectively. The color distribution in the diagrams represents the trend of I(b) with b.
Figure 2. Stationary and infalling spherical accretions with α = 0.3, α = 0.6 and α = 0.8 and fixed M = 1, γ = 1. The blue line in the plots in the left column represents the specific light intensity for static accretion and the red line represents the specific light intensity for infalling accretion. The two-dimensional diagrams in the middle column are for infalling accretion, while the two-dimensional diagrams on the right are for static accretion. |
One can see that the luminosity of stationary spherical accretion peaks near the photon sphere. As the hairy parameter α increases, the overall luminosity of stationary accretion decreases. The intensity inside the photon sphere does not completely disappear because some photons escape the black hole. Therefore, the shadow region of the black hole on the observer's screen is not completely dark [61].
3.2. Infalling spherical accretions
Accretion matter is always dynamic, and it is meaningful to study the shadows as well as the photon sphere of a black hole by the infalling spherical accretions. Similar to the stationary situation, we use equation (18 ) to describe the intensities of the shadows, and the redshift factor g has been investigated before. The four-velocity of the infalling accretion can be expressed as
$\begin{eqnarray}{u}_{{\rm{e}}}^{\beta }=\left(\displaystyle \frac{1}{F(r)},-\sqrt{1-F(r)},0,0\right),\end{eqnarray}$
then we can derive the expression for the redshift factor as $\begin{eqnarray}g=\displaystyle \frac{F(r)}{1\pm F(r)\sqrt{1-F(r)}\sqrt{1-\tfrac{F(r){b}^{2}}{{r}^{2}}}},\end{eqnarray}$
where the ± indicates a photon moving inward and outward, respectively. Red lines in figures 2(a), (d) and (g) show the specific intensity of infalling accretion, and figures 2(b), (e) and (h) demonstrate the luminance distribution of photons hitting the receiving screen for observers.Comparing the simulated illustrations of static and infalling accretion, we find some interesting facts. The specific light intensities of static and infalling accretion reach a maximum at the same impact parameter b. Subsequently, the specific light intensity of static accretion undergoes a significant decay, while that of infalling accretion is relatively mild; eventually they both converge to zero at infinity. Compared with stationary accretion, the luminosity of infalling accretion is slightly lower, attributed to the smaller redshift factor for matter falling into the accretion. As α increases, the overall luminosity of infalling accretion decreases.
4. Shadow and rings of a black hole with thin accretion
In this section we study thin disk accretion models of a hairy black hole in WEC. One can treat the accreted matter as the light source and study the shadow. Due to the rotation, accretion disks around massive celestial objects are typically located at the equatorial plane. If we consider an observer at infinity along the equatorial plane, details of the black hole's halo would be challenging to discern. For simplicity, we contemplate an observer at infinity perpendicular to the equatorial plane.
4.1. Observed specific intensities and transfer functions
Inspired by [60], we could use the orbital plane azimuthal angle φ to define the total number of orbits n = φ/2π, which depends on the impact parameter b. According to the different values of n, orbits can be categorized into distinct types: direct emission occurs when n < 3/4, resulting in a single intersection with the accretion disk; a lensing ring is formed when 3/4 < n < 5/4, leading to photons passing through the accretion disk twice; and a photon ring is established when n > 5/4, causing photons to traverse the accretion disk more than three times.
The range of the impact parameter b with respect to direct emission, a lensing ring and a photon ring are shown in table 2, and the visible appearance is shown in figure 3. In the upper row, the lines in red, blue and green represent direct emission, lensing ring and photon ring photons, respectively. Direct emission with a lower orbital number accounts for a large fraction of the photons, followed by the lensing ring, while the photon ring with the highest orbital number has rather few photons. At the same time, we find that the impact parameter of the photons in the lensing ring and photon ring decreases significantly as the hairy parameter α increases. In the lower row, we categorize the photons according to the number of orbits, redrawing the geodesic diagrams with the corresponding red, blue and green colors. We note with interest that an increase in α does not affect the proportional distribution of photons for the three orbital numbers, but it causes a decrease in the impact parameter b for the photon ring. This is a significant difference from several previously discovered hairy black holes [33, 62].
Figure 3. The visible ranges of direct emission, lensing ring and photon ring with respect to the impact parameter b. According to the number of loops, we replotted the geodesics. The red, blue and green lines are photons corresponding to direct emission, lensing ring and photon ring separately. |
Table 2. The ranges of impact parameter b with respect to direct emission, lensing ring and photon ring for different α and fixed M = 1. |
| Direct emission, n < 3/4 | Lensing ring, 3/4 < n < 5/4 | Photon ring, n > 5/4 | |
|---|---|---|---|
| α = 0.3 | 0 ≤ b < 4.54143, b > 5.61601 | 4.54143 < b < 4.70773, 4.74535 < b < 5.61601 | 4.70773 < b < 4.74535 |
| α = 0.6 | 0 ≤ b < 3.88749, b > 4.96069 | 3.88749 < b < 4.04903, 4.08953 < b < 4.96069 | 4.04903 < b < 4.08953 |
| α = 0.8 | 0 ≤ b < 2.78218, b > 4.02983 | 2.78218 < b < 2.95654, 3.01789 < b < 4.02983 | 2.95654 < b < 3.01789 |
It is assumed that the disk lies in the equatorial plane and the emissions from it are isotropic. The observed specific intensity with frequency νe can be written as [63]
$\begin{eqnarray}{I}_{{\rm{o}}}({\nu }_{{\rm{o}}},r)={g}^{3}{I}_{{\rm{e}}}({\nu }_{{\rm{e}}},r),\end{eqnarray}$
where νo is the observed photon frequency and Ie is the emitted specific intensity [32]. The total observed specific intensity can be expressed as $\begin{eqnarray}I(r)=\int {I}_{{\rm{o}}}(r){\rm{d}}\nu ={F}^{2}(r){I}_{{\rm{e}}}(r).\end{eqnarray}$
Since the thin disk is located in the equatorial plane, the photons emitted from the north pole will intersect the disk, and it will capture brightness each time it intersects. The total observed intensity should be considered as the sum of these intensities from each intersection, which can be reexpressed as $\begin{eqnarray}I(r)={F}^{2}(r)\displaystyle \sum _{n}{I}_{{\rm{e}}}(r){| }_{r={r}_{n}(b)},\end{eqnarray}$
where rn(b) is the transfer function with n = 1, 2, …. At each b, the slope of the transfer function dr/db represents the demagnification factor. We plot the first three transfer functions rn(b) in figure 4. The red, blue and green lines correspond to the transfer function of n = 1, 2 and 3, respectively. The slope of r1(b) is close to 1, which is considered to be associated with the redshift source. The slope of r2(b) is high, indicating that the lensing ring is the demagnified image of the backside of the accretion disk. The slope of r3(b) approaches infinity, suggesting that the photon ring is the extremely demagnified image of the accretion disk, and higher-order transfer functions theoretically exist but their low luminosity makes them almost impossible to observe.
Figure 4. The three transfer functions of a hairy black hole in WEC with a thin accretion disk. The red, blue and green lines represent the first, second and third transfer functions, respectively. |
4.2. Observational appearances
In this subsection, we will consider four emission models to work out the optical appearances of a hairy black hole in WEC. One can focus on four different emission positions of the accretion disk: ISCO, the photon sphere radius rph, the event horizon rh [19] and a certain position r = 17/3 outside the ISCO [55]. The first three positions are special and have applications in many articles studying black hole shadows. In contrast, 17/3 is a random position located beyond the innermost stabilizing circular orbit of the black hole, which is a very common situation, representing the emission of photons from matter on the accretion disk beyond the ISCO. We choose the GLM model to study this random position, considered by some of the literature to be the emission model that fits the observational reality [55]. These idealized models may not accurately represent reality but can still offer valuable perspectives on research into black hole shadow observations.
4.2.1. Exponential model
We employ the exponential model, whose emission of the accretion disk starts and peaks at the innermost stable circular orbit rISCO. The emission intensity is distributed exponentially with the radial coordinate r, which has a very sharp decay after the peak presented in figures 5(a)–(c). The formula is
$\begin{eqnarray}{I}_{{\rm{e}}1}(r)/{I}_{0}=\left\{\begin{array}{l}\exp \left[-(r-{r}_{\mathrm{ISCO}})\right],r\geqslant {r}_{\mathrm{ISCO}},\\ 0,r\lt {r}_{\mathrm{ISCO}},\end{array}\right.\end{eqnarray}$
where I0 is the maximum intensity. Ie1 has a very sharp decay after the peak, as shown in figures 5(a)–(c). For different α, there are three separate peaks observed in the specific light intensity, each corresponding to the intensities of direct emission, lensing ring and photon ring, respectively. These peaks are independent of each other and do not exhibit any superposition effects. For the cases with α = 0.3, 0.6 and 0.8, direct emission experiences sharp changes at b ≈ 6.2, 5.0 and 3.2, respectively. As a result of gravitational lensing effects, the intensity of the lensing ring is concentrated within a limited spectrum, which peaks at b ≈ 5.0, 4.2 and 3.3. Photon ring intensity is a very narrow peak with b ≈ 4.6, 4.0 and 3.0, contributing negligibly to the overall observed intensity.
Figure 5. Observational appearance of a thin disk in an exponential model with α = 0.3, 0.6 and 0.8 and fixed M = 1. |
The diagrams with respect to the special intensities are shown in figures 5(g)–(i), and the accretion structure are shown in figures 5(j)–(l). Within the brightest ring we can identify a sub-ring known as the lensing ring. Upon magnification, a very narrow photon ring is observed within the lensing ring. The overall brightness and width of the shadow are largely determined by direct emission. As the hairy parameter α increases, the overall luminosity of the accretion disk decreases and the lensing ring gets closer to direct emission. When α = 0.8, one can observe that coupling occurs between the lensing ring and direct emission, resulting in a significant emission peak due to the superposition of their intensities. As there is incomplete coupling between the direct emission and the lensing ring, we can still clearly see the photon ring by zooming in on figure 5(l).
4.2.2. Power-law exponential model
In this section we consider the exponential model for emission, but the difference from the normal exponential model is that we choose the start as well as the peak intensity at the photon sphere. The expression is
$\begin{eqnarray}{I}_{{\rm{e}}2}(r)/{I}_{0}=\left\{\begin{array}{l}\tfrac{1}{r-{r}_{\mathrm{ph}}+1}\exp \left[-(r-{r}_{\mathrm{ph}})\right],r\geqslant {r}_{\mathrm{ph}},\\ 0,r\lt {r}_{\mathrm{ph}},\end{array}\right.\end{eqnarray}$
where I0 represents the maximum normalized intensity. Note that the difference between this model and the previous one is that the observation intensity only peaks at two b values, which indicates that the three intensities are not independent of each other. The first peaks, located at b ≈ 3.5, 3.0 and 2.0 with respect to α = 0.3, 0.6 and 0.8, are contributed only by the direct intensity. Furthermore, the second peaks are located at b ≈ 4.9, 4.2 and 3.0 with respect to α = 0.3, 0.6 and 0.8, which are contributed by all three intensities.Two-dimensional representations of the total observed intensities can be found in figures 6 (g)–(i). The conjunction of the lensing ring and photon ring results in the creation of a notably bright yet exceptionally slender halo, with the two emissions being indistinguishable from each other. The direct emission exhibits a broader range despite having a lower observed intensity. In this theoretical framework, the overall brightness of the shadow is predominantly determined by the interaction between the photon ring and the lensing ring, with the direct emission mainly affecting the width of the shadow. As the hairy parameter α increases there is no significant decrease in the overall luminosity of the accretion disk. However, the size of rings undergoes a noticeable reduction.
Figure 6. Observational appearances of a thin disk in the power-law exponential model with α = 0.3, 0.6 and 0.8 and fixed M = 1. |
4.2.3. Bell-shaped model
Furthermore, with the assumption that the emission extends continuously to the outer event horizon rh, one can have the emission function moderately attenuated with a maximum at rh, and the explicit expression is given as
$\begin{eqnarray}{I}_{{\rm{e}}3}(r)/{I}_{0}=\left\{\begin{array}{l}\tfrac{1-\tanh \left[r-({r}_{\mathrm{ISCO}}-{r}_{{\rm{h}}}+0.5)\right]}{1-\tanh \left[{r}_{{\rm{h}}}-({r}_{\mathrm{ISCO}}-{r}_{{\rm{h}}}+0.5)\right]},r\geqslant {r}_{{\rm{h}}},\\ 0,r\lt {r}_{{\rm{h}}},\end{array}\right.\end{eqnarray}$
where I0 is also the maximum intensity. In figures 7(a)–(c), compared with the exponential and power-law exponential model, the direct emission intensities meet no abrupt end at any impact parameter b. They start from b ≈ 3.5, 2.9 and 2.0 with respect to α = 0.3, 0.6 and 0.8, and photon ring emission intensities hold at b ≈ 4.8, 4.0 and 2.9 while the lensing ring emission exists as b ∈ (4.7, 5.1), (4.1, 4.5) and (2.8, 3.4) [54].
Figure 7. Observational appearances of a thin disk in the bell-shaped model with α = 0.3, 0.6 and 0.8 and fixed M = 1. |
Two-dimensional representations of the total observed intensities are presented in figures 7(g)–(i). Similar to the power-law model, the lensing ring and the photon ring together create a thin and bright halo, while the direct emission forms a relatively dim but broader halo. The lensing ring is more pronounced, but direct emission still dominates while the photon ring remains insignificantly small. As the hairy parameter α increases, there is a significant decrease in the overall luminosity of the accretion disk, accompanied by a noticeable reduction in the size of the rings.
4.2.4. GLM model
The radiative intensity profile of the accretion disk is characterized using Johnson's standard unbounded model as a parameterization method, which has been previously employed in the literature within the context of reproducing general relativistic magnetohydrodynamic simulations of the accretion flow. We focus on the Gralla–Lupsasca–Marrone (GLM) model, and the radiation intensity profile of the GLM model reads as [63]
$\begin{eqnarray}{I}_{\mathrm{GLM}}=\displaystyle \frac{{{\rm{e}}}^{-\tfrac{1}{2}[\gamma +\mathrm{arcsinh}(\tfrac{r-\alpha }{\sigma })]}}{\sqrt{{\left(r-\alpha \right)}^{2}+{\sigma }^{2}}}.\end{eqnarray}$
This model assumes monochromatic emission (in the frame of the disk) characterized by three variable parameters: α, which is associated with the position of the emission peak and tends to shift the peak away from (or behind) the horizon for small (or large) values; γ, which controls the asymmetry of the profile, with negative (or positive) values favoring the positioning of the steeper part of the profile at the horizontal line (or outer side); σ (usually in units of M), which is related to the width of the outline, with small (or large) values often resulting in steeper (or wider) decay and narrower (or broader) rings.Thus, for this work, one can choose the GLM model [55] with γ = − 2, α = 17/3 and σ = 1/4. Specific intensities ${I}_{{e}_{4}}/{I}_{0}$ are shown in figures 8(a)–(c). They are the same because Ie does not concern α, and the specific observational intensities are demonstrated in figures 8(d)–(f). Note that in this model there are three peaks in each observational intensity corresponding to direct emission, the lensing ring and the photon ring respectively, which indicates that the three intensities are independent of each other and do not have superposition effects. The diagrams with respect to the special intensities are shown in figures 8(g)–(i), and the accretion structures are shown in figures 8(j)–(l). Similar to figure 5, the difference between the GLM model and the exponential model is that the direct emission rises and decays more moderately, causing the outermost ring to be brighter than in the exponential model; this is stable and does not change with α. One can also notice an extremely narrow ring of n = 3 inside the sub-ring (lensing ring) by magnification but, interestingly, this photon ring moves further away from the lensing ring with increasing α.
Figure 8. Observational appearances of a thin disk in the GLM3 model with α = 0.3, 0.6 and 0.8 and fixed M = 1, μ = 17/4, γ = − 2 and σ = 0.25. |
5. Conclusion and discussion
In this paper, we first derive the metric of the hairy black hole in WEC, and the hairy parameter α is interpreted as the matter sector independent of the global charge [45]. Angular momentum (L) and energy (E) are derived, and the impact parameter is defined as b ≡ L/E. Through numerical integration, we obtain the trajectories of photons and compare the images for different values of α. We observe that when the hairy parameter α increases, the ability of the black hole to deflect photons continuously decreases. Subsequently, we examine rISCO for hairy black holes and calculate its variation with α, which represents the minimum radius of stable orbits for massive particles.
Subsequently, the backward ray-tracing method is employed to investigate the hairy black hole shadow observed under spherical accretion. We study the emission intensity from the stationary spherical accretion. For simplicity, we assume that the emitted light frequency obeys a normal distribution and the photon density follows a logarithmic normal distribution. Then an examination of the redshift factor (g) and the infinitesimal proper line element is undertaken, ultimately deriving the integral formula for the observer's received light intensity. We simulated the projection of the shadow of stationary spherical accretion and the projection of spherical accretion under infalling conditions on the observer's screen. As a result, we observed that the overall light intensity for both types of spherical accretion decreases with an increase in the hairy parameter α. Under the same α, the light intensity for infalling is significantly lower than that for stationarity, attributed to the Doppler effect in the emission of infalling accretion material.
For a distant observer, the thin disk model is considered to be a theoretical model consistent with observed facts. In this sense, incident photons from the accretion disk can orbit the black hole a varying number of times before leaving the black hole, so that they may generate several light rings that confine the shadow. We investigate the classification of photon orbits, namely, direct emission, lensing ring and photon ring, and distinguish these three types of photon trajectories in the geodesic images. We summarize the relationship between emitted light intensity and the intensity observed. As the number of intersections between photons and the accretion disk increases, the light intensity received by the observer increases. Therefore, we sum the intensities and introduce the transfer function rn(b), depicting their profiles. The slopes of these images are referred to as demagnification factors. These ring images are presented in all four toy models. Next, considering the four models, we found that all four models consistently report a reduction in the size of the rings with increasing hairy parameter α. It is also observed that when the emission peaks occur at rph and rh, a coupling of three rings dominated by direct emission and the lensing ring becomes significant. Specifically, as the peak is at rph (power-law exponential model), the contribution of the lensing ring is minimal, while when the peak is at rh (bell-shaped model), the contribution of the lensing ring is significantly enhanced. The most prominent result we found is that when the emission peaks occur at rISCO and beyond (e.g. as mentioned at r = 17/3), one can clearly observe decoupled rings for n = 1, 2 and 3, especially the very fine n = 3 (photon ring) on the innermost side, which coincides with the expectations of the EHT discoveries.
With the improvement in astrophysical observation precision, rings with n = 2 and n = 3 may be detected by future detectors. This holds significant importance for validating the strong gravitational field predicted by general relativity.