#  The feature of shadow images and observed luminosity of the Bardeen black
hole surrounded by different accretions

###### Abstract

In this paper, by exploring the photon motion in the region near the Bardeen
black hole, the shadow and observation properties of the black hole surrounded
by various accretion models are studied. We analyzed the changes in shadow
imaging and observation luminosity when the relevant physical parameters are
changed. For the different spherical accretions background, one can find that
the radius of shadow and the position of photon sphere do not change, but the
observation intensity of shadow in the infalling accretion model is
significantly lower than that of the static case. When the black hole is
surrounded by an optically and thin disk accretion, the contribution of the
photon rings, lensing rings and direct emission to the total observed flux has
also been studied. Under the different forms of the emission modes, the result
shows that the observed brightness is mainly determined by direct emission,
while the lensing rings will provide a small part of the observation flux and
the photon ring can provide a negligible observation flux. By comparing our
results with the Schwarzschild spacetime, it is found that the existence or
change of relevant status parameters will greatly affect the shape and
observation intensity of black hole shadow. These results support that the
change of state parameter will affect the spacetime structure, thus affecting
the observation feature of black hole shadows.

Ke-Jian He,1111<EMAIL_ADDRESS>Sen Guo,2222<EMAIL_ADDRESS>Shuang-Cheng
<EMAIL_ADDRESS>Guo-Ping Li∗444*corresponding author:
<EMAIL_ADDRESS>

1Department of Physics and Chongqing Key Laboratory for Strongly Coupled
Physics, Chongqing University, Chongqing 401331, China
2 Guangxi Key Laboratory for Relativistic Astrophysics, School of Physical
Science and Technology, Guangxi University, Nanning 530004, China
∗Physics and Space College, China West Normal University, Nanchong 637000,
China

##  1 Introduction

Since the concept of black hole was put forward, people have been trying to
find this mysterious physical object in the universe. Recently, the first
image of supermassive black holes at the center of M$87^{*}$ galaxy, which was
captured by the Event Horizon Telescope Collaboration (EHT), is a strong proof
of the existence of the black hole[1, 2, 3, 4, 5, 6]. One can observed that a
bright ring appears in the background of dark area in the image, where the
dark area in the center is called the shadow of black hole and the bright area
is photon sphere. It is commonly known that the strong gravity of black hole
deflects the light and forms the shadow of black hole, which was also known as
gravitational lensing effect[7]. With the deepening of the study of black
hole, people pay more attention to the observation of black hole shadow,
because it will bring new insights and development to the study of black
hole[8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,
27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40].

In our universe, the real astrophysical black holes in galaxies are surrounded
a large amount of high energy radiation material, which makes it possible to
indirectly observe the black hole[41]. The accretion flow around a black hole
is usually not spherically symmetric, but the simplified ideal spherical model
can provide strong support for the characteristics of black hole observation.
It is generally accepted that the shape of the black hole shadow is standard
circular geometry in the static spherically symmetric spacetime[42]. In [43],
Bardeen pointed out that the radius of shadow for Schwarzschild black hole is
$r_{s}=3M$, where $M$ is the mass of black hole. Meanwhile, Bardeen also
mentioned that the shadow shape of rotating black hole is closely related to
angular momentum. In 1979, the simulated image of a black hole is shown for
the first time, in which the black hole surrounded by optically thin disk was
considered [44]. Taking into account the model in which the black hole
surrounded by the optically thin spherical accretion, it is found that the
shadow is geometric characteristics of spacetime, and has nothing to do with
the details of the accretion process[45]. However, the observed luminosity is
affected by the accretion flow model [46, 47, 48, 49, 50]. When an optically
and geometrically thin disk accretion is located on the equatorial plane of
black hole, the observational peculiarity of black hole shadow observed by the
distant observer have been studied in[51]. Gralla et al. classified the
trajectories of light rays near the black hole, whereby divided the rings
outside the shadow area into the direct, photon ring, and lensing ring, and
proposed that the observation intensity is dominated by the direct case.
Furthermore, they also found that the size of the black center region depends
on the specific emission model of accretion flow. Using the different
accretion flow models, there is a series interesting observable features about
the shadow and rings of the black hole in other gravitational spacetime
background, which could refer [52, 53, 54, 55, 56, 57, 58, 61, 59, 60, 62].

Most often, the black hole has a singularity inside the horizon. While in
[63], the model of a black hole with regular non-singular geometry was
proposed by Bardeen. In this system, an energy-momentum tensor is introduced,
which is interpreted as the gravitational field of a nonlinear magnetic
monopole charge. Since then, the physical properties of Bardeen black hole has
aroused people’s interest, and involve a wide range of quasinormal modes,
energy distribution, radiation, thermodynamic behavior and so on [64, 65, 66,
67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77]. Indeed, the study of black hole
shadow can explore the basic physical properties of spacetime. Therefore, it
is necessary to study the shadow and observation characteristics in the
context of Bardeen spacetime. In this paper, we focus on the shadow and
observational luminosity of the Bardeen black hole surrounded by various
accretion models, i.e., the static, infalling spherical accretion and the
optically thin disk accretion, which were regarded as the only background
light source. By comparing the results in this work with Schwarzschild
spacetime, whereby we can study the influence of magnetic monopole charge on
the shadow and observation characteristics, due to the charge parameter is a
remarkable role of Bardeen spacetime. Therefore, it may provide a feasible
method to distinguish the Bardeen spacetime from Schwarzschild spacetime.

The paper is organized as follows. In section 2, we have studied the
associated trajectories of the light ray near the Bardeen black hole, as well
as the radius of photon sphere and the shadow of black hole when the state
parameters changed. In section 3, we analyzed the shadow image and luminosity
of the black hole surrounded by the different spherical accretion models. In
section 4, we considered thin disk emission near the black hole, and compared
the observational appearance with the different emission profiles. Finally, we
discussed our results and conclusions in section 5. In this paper, we use the
units $G=c=1$.

##  2 Null geodesic in the Bardeen space-time

We would like to consider the spherically symmetric spacetime, and the metric
of Bardeen regular black hole as follows[63],

$ds^{2}=-A(r)dt^{2}+B(r)dr^{2}+C(r)\left({d\theta}^{2}+\sin^{2}\theta{d\varphi}^{2}\right),$
(1)

with

$A(r)=1-\frac{2Mr^{2}}{\left(g^{2}+r^{2}\right)^{3/2}},\quad
B(r)=\frac{1}{A(r)},\quad C(r)=r^{2}.$ (2)

In which, $M$ is the mass of the black hole, and $g$ is the monopole charge of
a self-gravitating magnetic field described by a nonlinear electrodynamics
source. Noted that the spacetime structure could recover to the Schwarzschild
spacetime in the case of $g=0$. As mentioned in [64], the metric function can
be approximated as

$A(r)\sim
1-\frac{2M}{r}+\frac{3g^{2}M}{r^{3}}+{\mathcal{O}}(\frac{1}{r^{5}}).$ (3)

It is obvious that the form of the above equation is different from the
$Reissner-Nordstr\ddot{o}m$ spacetime. And, the horizon of the Bardeen black
hole can be obtain by

$A(r)=0,$ (4)

where the largest positive root is the event horizon of the black hole. It is
worth noting that the monopole charge $g$ should follow the condition
$g<\frac{4\surd 3}{9}M$, which is important threshold for the existence of
event horizon. For the case $g>\frac{4\surd 3}{9}M$, there is no horizon and
we do not consider in this paper. When the light ray passes from the vicinity
of a massive object it is deflected due to the interaction between light and
gravitational field of the massive object. In doing so, we can use the Euler-
Lagrange equation to investigate the motion behaviors of photons around the
Bardeen black hole, which reads as

$\frac{d}{d\lambda}\left(\frac{\partial\mathcal{L}}{\partial\dot{x}^{\mu}}\right)=\frac{\partial\mathcal{L}}{\partial
x^{\mu}},$ (5)

where $\lambda$ and $\dot{x}^{\mu}$ are the affine parameter and four-velocity
of the photon, respectively. For the metric function (1), the Lagrangian
$\mathcal{L}$ can be expressed as

$\mathcal{L}=\frac{1}{2}\mathit{g}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=\frac{1}{2}\left(-A(r)\dot{t}^{2}+B(r)\dot{r}^{2}+C(r)\left(\dot{\theta}^{2}+\sin\theta^{2}\varphi^{2}\right)\right).$
(6)

In the spherically symmetric spacetime, we discuss the motion of photons
confined to the equatorial plane of a black hole[42, 78, 10], and have the
restrictive conditions $\theta=\frac{\pi}{2}$ and $\dot{\theta}=0$. Due to the
$t$ and $\theta$ coordinates can not explicitly determine the coefficient of
the metric equation, there are two conserved quantities $E$ and $L$,
representing energy and angular momentum, respectively. That is

$\displaystyle
E=-\frac{\partial\mathcal{S}}{\partial\dot{t}}=A(r)\dot{t},\quad
L=\frac{\partial\mathcal{S}}{\partial\dot{\varphi}}=r^{2}\dot{\varphi}.$ (7)

For null geodesic $\mathit{g}_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=0$, we can
obtain the orbit equation

$\dot{r}^{2}+V_{eff}=\frac{1}{b^{2}}.$ (8)

In which, $V_{eff}$ is the effective potential, which is given by

$V_{eff}=\frac{1}{r^{2}}(1-\frac{2Mr^{2}}{\left(g^{2}+r^{2}\right)^{3/2}}),$
(9)

Moreover, $b$ is called the impact parameter, which is the ratio of angular
momentum to energy, i.e., $b=J/E=\frac{r^{2}\dot{\varphi}}{A(r)\dot{t}}$. In
the spacetime (1), there exist a critical impact curve with radius $b_{p}$,
and the light from this curve will asymptotically approach a bound photon
orbit. Indeed, the shadow is represented the interior of the critical curve,
and the surface of bound is the photon sphere. In the position of the photon
sphere, the effective potential should follow the condition $\dot{r}=0$ and
$\ddot{r}=0$, which means[47]

$\displaystyle V_{eff}=\frac{1}{b^{2}},\quad 2V^{\prime}_{eff}=0.$ (10)

From Eq.(10), one can find the relation between the photon sphere radius
$r_{p}$ and critical impact parameter $b_{p}$ is contented

$\displaystyle r_{p}{}^{2}-b_{p}^{2}A(r)=0,\quad
2b_{p}^{2}A(r)^{2}-r_{p}{}^{3}A^{\prime}(r)=0.$ (11)

For the distant observer ($r_{o}\rightarrow\infty$), $b_{p}$ is the value of
shadow radius that can be seen. For the different values of the monopole
charge $g$, we can get the different numerical results of the photon sphere
radius $r_{p}$, the shadow radius $b_{p}$ and the event horizon $r_{h}$ where
$M=1$, and listed in Table 1.

Table 1. The value of relevant physical quantities under the different values
of monopole charge $g$, in which $M=1$.
$g=0.1$ $g=0.2$ $g=0.3$ $g=0.4$ $g=0.475$ $g=0.6$ $g=0.7$ $g=0.75$ $r_{p}$
2.99164 2.96617 2.9224 2.85798 2.7937 2.64674 2.47495 2.35745 $b_{p}$ 5.18747
5.16108 5.11594 5.05508 4.98508 4.83975 4.67719 4.57212 $r_{h}$ 1.99247
1.96946 1.92962 1.87022 1.80981 1.66546 1.47436 1.29904

From Table 1, it can be find that all the related physical quantities $r_{p}$,
$b_{p}$ and $r_{h}$ show a decreasing trend with the increase of monopole
charge $g$. Therefore, the appearance of monopole charge $g$ will lead to the
reduction of the black hole area compared with the Schwarzschild black hole.
With the help of introducing parameter $u_{0}=1/r$, we can rewrite the motion
equation of photons, and use it to depict the trajectory of light ray, that is

$\displaystyle\mathcal{V}(u_{0})=\frac{du_{0}}{d\varphi}=\sqrt{\frac{1}{b^{2}}-u_{0}^{2}\left(1-\frac{2M}{u_{0}^{2}\left(g^{2}+\frac{1}{u_{0}^{2}}\right){}^{3/2}}\right)}.$
(12)

It is worth noting that the geometry of geodesics depends entirely on the
Eq.(12), which as a function of the impact parameter $b$. In the case of
$b<b_{p}$, the light ray is always trapped by the black hole and cannot reach
infinity distance. In the case of $b>b_{p}$, the light ray will be deflected
at the radial position $u_{0}$ where $\mathcal{V}(u_{0})=0$, and then move
away from the black hole reach infinity distance. When $b=b_{p}$, the photons
are in a state of rotation around the black hole, neither falling into the
black hole nor falling into the black hole. In the following sections, we
further study the shadow of black hole surrounded by various profiles of
accretion flow.

##  3 Image of the Bardeen black hole with spherical accretion

With the help of equation (12), on can find that the trajectory of a light ray
changed as the change of the value of monopole charge $g$ and impact parameter
$b$. Therefore, we simply describe the path of light rays under the relevant
physical parameter, as shown in Figure 1.

(a) $g=0.1$ (b) $g=0.475$ (c) $g=0.75$

Figure 1:  The trajectory of the light ray for the different values of
monopole charge $g$ in the polar coordinates $(r,\varphi)$ with $M=1$. In
which, the black hole is shown as a black disk, and the red dotted line
represents the position of the photon sphere.

In Figure 1, the red lines, green lines and black lines correspond to
$b=b_{p}$, $b>b_{p}$ and $b<b_{p}$, respectively. When the value of monopole
charge $g$ increases, it is obvious that the the radius of the photon sphere
$r_{p}$ decreases, as well as the event horizon $r_{h}$. On the other hand,
the deflection rate of light ray is different under the different values of
monopole charge $g$. Namely, the light density obtained by distance observers
is different, which naturally leads to the different observation intensity
casted by black hole shadow. For the distance observer, we want to study the
observation luminosity of the black hole shadow, in which the accretion flow
around the black hole is the only light source. Then, two simple relativistic
spherical accretion models is introduced, that is, the static spherical
accretion and the infalling spherical accretion.

###  3.1  The shadow and photon sphere with static spherical accretion flow

In the background of the black hole being wrapped by static spherical
accretion flow, we can integrate the specific emissivity along the photon path
$\gamma_{i}$, so as get the specific intensity $\mathcal{I}_{{obs}}$ for the
distance observer (usually measured in $\rm ergs^{-1}cm^{-2}str^{-1}Hz^{-1}$).
That is [8, 47]

$\mathcal{I}_{{obs}}(\nu_{0})=\int_{\gamma_{i}}{g_{s}}^{3}{j}(\nu_{e})dl_{{prop}},$
(13)

and

$g_{s}=\frac{\nu_{o}}{\nu_{e}}.$ (14)

It is worth mentioning that we focus on the case that the emitter is fixed in
the rest frame. In the above equation, $g_{s}$ is the redshift factor and
$\nu_{e}$ is the photon frequency at the emitter. In addition, ${j}(\nu_{e})$
is the emissivity per-unit volume at the emitter, $dl_{{prop}}$ is the
infinitesimal proper length, and $\nu_{0}$ is the observed photon frequency,
respectively. In the spacetime(1), the redshift factor $g_{s}=A(r)^{1/2}$. For
the specific emissivity ${j}(\nu_{e})$, we consider that the emission is
monochromatic with the emission frequency $\nu_{f}$ and the emission radial
profile is $1/r^{2}$, that is,

${j}(\nu_{e})\sim\frac{\delta\left(\nu_{e}-\nu_{f}\right)}{r^{2}},$ (15)

where $\delta$ is the delta function. In addition, we can obtain the proper
length $dl_{{prop}}$ is

$dl_{{prop}}=\sqrt{\frac{1}{A(r)}+r^{2}\left(\frac{d\varphi}{dr}\right)^{2}}dr.$
(16)

Then, the expression of the observed specific intensity can be rewritten as

$\mathcal{I}_{{obs}}(\nu_{o})=\int_{\gamma_{i}}\frac{(A(r))^{3/2}}{r^{2}}\sqrt{\frac{1}{A(r)}+r^{2}\left(\frac{d\varphi}{dr}\right)^{2}}dr.$
(17)

In Eq.(17), the total observed intensity $\mathcal{I}_{{obs}}(\nu_{o})$ as a
function of impact parameter $b$, which means that the change of relevant
physical quantity will directly affect the specific intensity observed by a
distant observer. We take $g=0$, $g=0.475$ and $g=0.75$ as examples, and the
change trend of observed intensity with the change $b$ is given in Figure 2.

Figure 2:  The observed specific intensity of the black hole is surrounded by
the static spherical accretion flow, in which the black line, green line and
red line correspond to $g=0$, $g=0.475$ and $g=0.75$. Here, we take M = 1.

It is clear that the observation intensity increases slowly with the increase
of $b$ ($b<b_{p}$), but increases rapidly near the photon sphere ($b=b_{p}$)
until reaches to the peak at the position of the photon sphere. After that,
the observation intensity decreases with the increase of $b$ ($b>b_{p}$) and
at the infinity $\mathcal{I}_{{obs}}(\nu_{o})\rightarrow 0$. In addition, the
larger value of $b$, the stronger specific intensity observed. For the case
$g=0$( the Schwarzschild spacetime), the peak of observation intensity is
obviously smaller than that when $g=0.45$ ( or $g=0.75$). In other words, the
observational intensity of the Bardeen black hole is stronger than that of the
Schwarzschild spacetime, and the appearance of magnetic monopole charge $g$
increases the light density, so that the observer can obtain more luminosity.
The two-dimensional intensity map in celestial coordinates casted by the black
hole is shown in Figure 3.

Figure 3:  Image of the black hole shadow with the static spherical accretion
for the different values of monopole charge $g$, in which $g=0$ (left
panel),$g=0.475$(mid panel) and $g=0.75$ (right panel).

From Figure 3, there exist a bright ring with the strongest luminosity, which
is the position of photon sphere. Due to the tiny fraction of the radiation
can escape from the black hole, the inner region of the photon sphere is not
completely black, and there is small observation luminosity near the photon
sphere. Obviously, the maximum luminosity of the shadow image enhanced with
the increase of monopole charge $g$, while the size of shadow and the radius
of photon sphere decreased.

###  3.2 The shadow and photon sphere with infalling spherical accretion flow

In the background of black hole surrounded by the infalling spherical
accretion, the radiating gas moves towards the black hole along the radial
direction. Indeed, the infalling spherical accretion is closer to the real
situation than the static spherical accretion model. We need to pay attention
to that the equation (17) is still suitable in the infalling accretion model,
but the redshift factor is different from the static case. In the infalling
accretion model, the redshift factor is related to the velocity of the
accretion flow, namely

$g_{i}=\frac{\mathcal{K}_{\rho}u_{0}^{\rho}}{\mathcal{K}_{\sigma}u_{e}^{\sigma}},$
(18)

and

$\mathcal{K}^{\mu}=\dot{x}_{\mu}.$ (19)

In which, $\mathcal{K}^{\mu}$ is the four-velocity of the photon, and
$u_{0}^{\mu}=(1,0,0,0)$ is the four-velocity of the static observer. Moreover,
the four-velocity of the infalling accretion correspond to $u_{e}^{\mu}$,
which is

$u_{e}^{t}={A(r)}^{-1},\quad u_{e}^{r}=-\sqrt{\frac{1-A(r)}{A(r)B(r)}},\quad
u_{e}^{\theta}=u_{e}^{\varphi}=0.$ (20)

Here, $\mathcal{K}_{t}$ is a constant of motion. For the photon,
$\mathcal{K}_{r}$ can be obtained by the equation
$\mathcal{K}_{\mu}\mathcal{K}^{\mu}=0$, and consequently

$\mathcal{K}_{t}=\frac{1}{b},\quad\frac{\mathcal{K}_{r}}{\mathcal{K}_{t}}=\pm\sqrt{B(r)\left(\frac{1}{A(r)}-\frac{b^{2}}{r^{2}}\right)}.$
(21)

In which, the sign $+$ or $-$ corresponds to the photon approaches to or goes
away from the black hole. Hence, the redshift factor of the infalling
accretion is

$g_{i}=\left(u_{e}^{t}+\left(\frac{\mathcal{K}_{r}}{\mathcal{K}_{e}}\right)u_{e}^{r}\right)^{-1}.$
(22)

On the other hand, the form of the proper distance should be rewritten as

$dl_{prop}=\mathcal{K}_{\mu}u_{e}^{\mu}d\lambda=\frac{\mathcal{K}_{t}}{g_{i}|\mathcal{K}_{r}|}dr.$
(23)

Considering the specific emissivity has the same form as in the equation (15),
the observed flux for the distance observer in the case of infalling spherical
accretion model is

$\mathcal{I}_{{obs}}({\nu^{i}_{o}})\propto\int_{\gamma_{i}}\frac{g_{i}^{3}\mathcal{K}_{t}dr}{r^{2}|\mathcal{K}_{r}|}.$
(24)

Based on the above equation, we also can investigate the shadow and
observation luminosity of the Bardeen black hole surrounded by the infalling
spherical accretion. Similarly, the observation luminosity distribution under
different monopole charge $g$ is given, which is shown in Figure 4.

Figure 4:  The observed specific intensity of the black hole is surrounded by
the infalling spherical accretion, in which the black line, green line and red
line correspond to $g=0$, $g=0.475$ and $g=0.75$, respectively. Here, we set M
= 1.

One can find that the maximum value of observed intensity still at the
position of photon sphere ($b\sim b_{p}$). In addition, the observed intensity
increases with the increase of $b$ in the region of $b<b_{p}$, and decreases
with the increase of $b$ in the region $b>b_{p}$. However, the peak value of
the observed intensity is much smaller than that of the static model, but the
difference of peak value is more obvious under the different values of
monopole charge $g$. In other words, the black hole shadow with the infalling
spherical accretion is darker than that of the static case. The two-
dimensional image of the observed intensity is also shown in Figure 5.

Figure 5:  Image of the black hole shadow with the infalling spherical
accretion for the different values of monopole charge $g$, in which $g=0$
(left panel), $g=0.475$ (mid panel) and $g=0.75$ (right panel).

Indeed, the central region inside the photon sphere in Figure 5 is evidently
darker than the corresponding region in Figure 3, and the intensity of the
bright ring is far less than that of static accretion. The obvious difference
between these two models is due to the Doppler effect, as near the event
horizon of the black hole, this effect is more obvious. In addition, it is can
be find that the radii of shadow and the position of photon sphere remain
unchanged in the different accretion progress. This suggests that the shadow
is an inherent property of spacetime, the behavior of accretion flow around
the black hole only affect the observation intensity.

##  4 The shadow and rings with thin disk accretion flow

###  4.1 The behavior of the trajectories of light rays

In this section, we are going to study the shadow when a black hole surrounded
by the thin disk accretion in the Bardeen black hole, where the optically and
geometrically thin disks is placed on the equatorial plane of black hole. Due
to the trajectories of light ray near the black hole are the important basis
for studying the shadow and observation feature of black hole. In [51], Gralla
et al. distinguish the trajectories of light rays near the black hole, and
they put forward the total number of orbits which is defined as

$n(b)=\frac{\varphi}{2\pi},$ (25)

which is a function of impact parameter $b$. Using equation (25), the
trajectories of the light ray is divided into three types, namely, the direct,
lensing and photon ring ones. That is, there are not only the photon rings but
also lensing rings outside of the black hole shadow. In the case of
$n(b)<3/4$, the light ray will intersect the equatorial plane only once,
corresponding to the direct emissions. In the case of $3/4<n(b)<5/4$, the
light ray crossing the equatorial plane at least twice, corresponding to the
lensing rings. In the case of $n(b)>5/4$, the light ray will intersect the
equatorial plane at least three times, and it corresponds to the photon rings.
For the different values of monopole charge $g$, the range of impact parameter
corresponding to direct, lensing rings and photon rings in the Baedeen
spacetime is shown in Table 2. And, we also show the relationship between the
total number of orbits $n(b)$ and impact parameter $b$ in Bardeen spacetime,
which is shown in Figure 6.

Table 2. The region of direct, lensing rings and photon rings corresponding to
impact parameters $b$ under the different values of monopole charge $g$, where
$M=1$.  g Direct Lensing rings Photon rings 0 $b<5.0267$ or $b>6.1927$
$5.0267<b<5.1893$ and $5.2305<b<6.1927$ $5.1893<b<5.2305$ 0.1 $b<5.0049$ or
$b>6.1623$ $5.0049<b<5.179$ and $5.2196<b<6.1623$ $5.179<b<5.2196$ 0.475
$b<4.7564$ or $b>6.0446$ $4.7564<b<4.9717$ and $5.0276<b<6.0446$
$4.9717<b<5.0276$ 0.75 $b<4.0769$ or $b>5.8461$ $4.0769<b<4.5142$ and
$4.6561<b<5.8461$ $4.5142<b<4.6561$

Figure 6:  Behavior of photons in Bardeen black hole as a function of impact
parameter $b$. The colors correspond to $g=0$ (black), $g=0.475$ (blue), and
$g=0.75$(red), respectively.

In Figure 6, it is found that the same value of $b$ may correspond to
different regions under the different values of monopole charge $g$, which is
corresponds to Table 2. In addition, the value of impact parameter $b$
corresponding to the photon rings and lensing rings are smaller than that of
the Schwarzschild spacetime ($g=0$), but the existence of monopole charge $g$
leads to the increase of the ratio of the region occupied by the photon rings
and lensing rings. When the impact parameter $b$ increases to a large enough
value, the trajectories light ray is direct case no matter what the value of
$g$ is. That is, both the monopole charge $g$ and impact parameter $b$ will
affect the observation characteristics of the Bardeen black hole. In order to
more clearly distinguish the distribution of light ray trajectories near the
black hole, we show the associated photon trajectories for Bardeen black hole
in the polar coordinates ($b,\varphi$), which is shown in Figure 7.

(a) $g=0.1$ (b) $g=0.475$ (c) $g=0.75$

Figure 7:  The selection of associated photon trajectories in the polar
coordinates $(b,\varphi)$, in which the red lines, blue lines and green lines
correspond to direct, lensing, and photon ring bands, in which the spacing in
impact parameter is 1/10, 1/100, and 1/1000 in the direct, lensing and photon
rings, respectively. And, the black hoke are shown as the black disks, the
dashed red line represent the photon ring.

###  4.2 Transfer functions

We consider that a distance static observer is located at the north pole, and
the thin disk accretion at the equatorial plane of the black hole. Moreover,
the lights emitted from the thin disk accretion is isotropically for the
static observer. In view of this, the specific intensity and frequency of the
emission are expressed as $I_{\nu}^{{em}}(r)$ and $\nu_{em}$, so as the
observed specific intensity and frequency are defined as
$I_{\nu^{\prime}}^{{obs}}(r)$ and $\nu_{obs}$. By considering
${I_{\nu}^{{em}}}/{\nu_{e}^{3}}$ is conserved along a ray from the Liouville
theorem, that is, one can obtain a correlation between the observed specific
intensity and the emissivity specific intensity [52], that is

$I_{\nu^{\prime}}^{{obs}}=(A(r))^{3/2}I_{\nu}^{{em}}(r).$ (26)

Thus, we can get the total observation and emitted intensity by integrating
for the full frequency, which reads as

$I_{{O}}=\int
I_{\nu^{\prime}}^{{obs}}(r)\,d\nu_{obs}^{\prime}=\int(A(r))^{2}I_{\nu}^{{em}}\,d\nu_{em}=(A(r))^{2}I_{\nu}^{{em}}(r),$
(27)

and

$I_{{E}}=\int I_{\nu}^{{em}}(r)d\nu_{em}.$ (28)

It is worth mentioning that the intensity of light emitted only from the
accretion disk, and the absorption or reflection of light and other factors
are not considered. When the trajectory of light ray passes through the
accretion disk on the equatorial disk, a certain luminosity will be obtained,
and transmitted to the observer at infinity. As discussed earlier, the
different values for the number of orbits $n(b)$ defined the number of times
that the ray path pass through the equatorial plane of the black hole. In the
case of $3/4<n(b)<5/4$, the light ray will bend around the black hole and
intersect with the back of the thin disk after the first intersection with the
thin disk. In this way, the light ray will pass through the thin disk twice.
When $n(b)>5/4$, the light passes through the back of the thin disk and then
through the front of the thin disk again, due to the light ray is more curved
around the black hole, i.e., the light will pass through the thin disk at
least three times. Hence, the sum of the intensity at each intersection is the
total intensity that the observer can observe, which is

$I_{{O}}(r)=\sum_{n}(A(r))^{2}I_{{E}}|_{r=r_{n}(b)}.$ (29)

Here, $r_{n}(b)$ is the transfer function, which represents the radial
position of the $n_{th}$ intersection with the thin emission disk, which means
that the relationship between radial coordinate $r$ and impact parameter $b$
can be expressed by the transfer function. In addition, $dr/db$ corresponds to
the slope of the transfer function, which is called the demagnification
factor. Actually, the slope of the transfer function reflects the demagnified
scale of the transfer function[51, 52]. In Figure 8, the graph of the transfer
functions is given under the different values of monopole charge $g$. For the
case of $g=0$(Schwarzschild spacetime), please refer[51].

(a) $g=0.1$ (b) $g=0.475$ (c) $g=0.75$

Figure 8:  The relationship between the first three transfer functions
$r_{n}(b)$ and $b$ in the Bardeen spacetime, in which the black, orange, and
red lines correspond to the direct, lensing rings and photon rings,
respectively.

The black line corresponds to the first transfer function($n=1$), and its
slope ($\frac{dr}{db}|_{1}$) is a small fixed value. And, the first transfer
function corresponds to the direct image of the thin emission disk, which is
the redshift of the source profile. The orange line corresponds to the second
transfer function ($n=2$), on can find that the slope ($\frac{dr}{db}|_{2}$)
is a very large value. Therefore, the observer will obtain a highly
demagnified image on the back of the thin disk, which corresponds to the
lensing rings. The red line represents the third transfer function ($n=3$),
whose slope ($\frac{dr}{db}|_{3}$) is close to
infinity($\frac{dr}{bd}|_{3}\sim\infty$). Namely, the observer will see an
extremely demagnified image of the front side of the disk, and the third
transfer function corresponds to the photon ring. As a result, the observation
luminosity provided by the photon rings and lensing rings only accounts for a
small proportion of the total observed flux, the direct emission is the main
measure of observed intensity. In addition, the image provided by the later
transfer function is more demagnetized and negligible, so we only consider the
first three transfer functions.

###  4.3 The observational appearance of Bardeen black hole surrounded by a
thin disk accretion

With the help of preparatory work, we can further study the specific intensity
of emission. For the emission form, we mainly consider three different models.
In the Model I, the peak of emission intensity at the position of the
innermost stable circular orbit ($r_{isco}$). There will be no emission inside
the innermost stable circular orbit, but exist a sharp downward trend outside
the innermost stable circular track, which is

$\displaystyle
I_{E}(r_{1})=\begin{cases}\left(\frac{1}{r-(r_{isco}-1)}\right)^{2},&r>r_{{isco}}\\\
0,&r<r_{{isco}}\end{cases}$ (30)

In the Model II, we assume that the position of the emission peak located at
the photon sphere ($r_{p}$), and the emission is a decay function of the power
of third, that is

$\displaystyle
I_{E}(r_{2})=\begin{cases}\left(\frac{1}{r-(r_{p}-1)}\right)^{3},&r>r_{p}\\\
0,&r<r_{p}\end{cases}$ (31)

In the Model III, there is the decaying gradually of the emission between the
event horizon $r_{h}$ and the innermost stable circular orbit $r_{isco}$, such
as

$\displaystyle
I_{E}(r_{3})=\begin{cases}\frac{1-\tan^{-1}(r-5)}{1-\tan^{-1}(r_{h}-5)},&r>r_{h}\\\
0,&r<r_{h}\end{cases}$ (32)

Taking the monopole charge $g=0.475$ as an example, the results about the
intensity of the emission and observation in Figure 9 are obtained. In Figure
9, the left column is the change trend of emission profiles $I_{E}(r)$ as the
increases of radius $r$, The middle column, which is shown that the
relationship between the observed intensity $I_{O}(r)$ and the impact
parameter $b$. And, the right column is the two-dimensional density plots of
observed intensities $I_{O}(r)$ in celestial coordinates. In addition, the
first, second and third row correspond to the emitted model I, II and III,
respectively. As can be readily seen from Figure 9, there is a disparity for
both the central black area and observation intensity in the different
emission models.

Figure 9:  Observational appearance of the thin disk with different emission
profiles near the black hole, viewed from a face-on orientation. In which, the
monopole charge $g=0.475$ and $M=1$.

In the Model I $I_{E}(r_{1})$ where the emission starts with the position of
innermost stable circular orbit, although the photon rings can get more
brightness from the thin disk, it is highly demagnetized. Which can be see
from the image of middle column, the observed intensity in the photon ring
region is very small and confined to a narrow region. Moreover, the two-
dimensional density map also shows that there is an almost invisible thin line
in the photon ring region. Therefore, the contribution of the photon ring to
the total observed flux is negligible. For the lensing rings, the observed
intensity is larger than that of the photon rings, but the area occupied by
the lensing rings is also very narrow, which leads to a bright line in the
area of the lensing rings in the two-dimensional density map. As the result,
the observation intensity obtained by the observer at infinity is mainly
provided by the direct emission.

In the Model II $I_{E}(r_{2})$ where the emission starts at the photon sphere,
the emission regions of the lensing rings and the photon rings overlap, which
resulting in a bright band outside the shadow has been shown in the two-
dimensional density map. The contribution of the lensing rings to the total
observed flux is higher than that of model I, while the contribution of the
photon ring is still negligible. That is to say, the observation intensity
obtained by the distant observer still directly determined by the direct
emission.

In the Model III $I_{E}(r_{3})$ where the emission starts from the event
horizon of black hole, the observation intensity increases with the increase
of $b$ from the outer edge of event horizon until it reaches to the peak near
the lensing ring. The photon rings and lensing ring completely coincide and
cannot be distinguished. In addition, the contribution of the lensing rings to
the total observed flux is partially helpful and the photon ring also makes a
negligible contribution, but the direct emission remains to dominate the total
observed intensity.

Then, we also plot the profiles of the emission and observed intensities when
the monopole charge $g=0.75$, which is shown in Figure 10. By comparing Figure
9 with Figure 10, it can be found that the observation intensity, range of the
center dark area and size of the light band outside shadow are all different
obviously. When the value of monopole charge $g$ increases, the range of the
center dark area and size of the light band outside shadow decrease. Moreover,
the observation intensity obtained by the distance observer also decreases.
Due to the proportion of photon rings and lensing rings area increases with
the larger monopole charge $g$, and the proportion of direct emission area
decreases relatively, which naturally leads to the weakening of the
observation intensity determined by direct emission. Therefore, the observed
intensity of the light band outside shadow is darker than that of the
Schwarzschild spacetime, which could be an effective characteristic to
distinguish the Bardeen black hole from Schwarzschild black hole.

Figure 10:  Observational appearance of the thin disk with different emission
profiles near the black hole, viewed from a face-on orientation. In which, the
monopole charge $g=0.75$ and $M=1$.

##  5 Conclusions and discussions

It is commonly known that the black hole is surrounded by the accretion
matter, and the distribution of accretion plays a key role in the imaging
results of black holes. In the context of Bardeen spacetime, we have
investigated the black hole shadow and observation luminosity under the
different accretion models. In particular, we pay close attention to the
different observation characteristics of the distant observer when the related
physical quantities is the different values.

When the black hole is surrounded by the spherically symmetric accretion flow,
we consider two behaviors of the accretion, i.e., the static and infalling
accretion model. For the case of static spherically symmetric accretions, one
can find that the observed luminosity increases with the increase of the
monopole charge $g$, while the photon sphere and radius of black hole
decrease. In addition, the maximum observation luminosity always appears in
the region of photon sphere under the different values of monopole charge $g$.
From the observed intensity density map, one can see that the central region
of the shadow is not completely dark, and the observation brightness of the
central region increases slightly with the increase of $g$. In the infalling
accretion model, the maximum observation luminosity is obviously lower than
that of the static model, which is the most prominent difference between these
two models. The Doppler effect of infalling matter leads to the difference
between the two kinds of spherical accretion. As a result, the observed image
in the infalling accretion model is darker than the static one under the same
state parameter. On the other hand, the change of observation intensity is
more obvious when the monopole charge $g$ takes more larger value. The maximum
observation luminosity with $g=0.75$ is about twice that of $g=0$
(Schwarzschild black hole). It must also be mentioned that in both spherical
accretion models, there is a great difference in the observed luminosity, but
the position of photon sphere and the shadow radii unchanged under the same
parameter, which means that the shadow is an inherent property of spacetime
and not affected by accretion flow behavior.

By considering the Bardeen black hole surrounded by a geometrically and
optically thin disk accretion, we have distinguished the trajectory of light
ray near black hole. Then, we take three different emission models for the
thin disk accretion flow. For the observer at infinity
($r_{0}\rightarrow\infty$), there is not only a dark central area, but also
the photon rings and lensing rings outside of black hole shadow. In the
different emission models, the maximum observation luminosity and the size of
central dark area are different. The thin disk serves as the only background
light source, the light ray in the region of photon rings can accretes through
the thin disk more than three times, and gains enough extra brightness.
However, the photon rings cannot be observed directly for the observer, due to
the image of the photon ring region is extremely demagnetized and the photon
ring region is very narrow. In addition, the image of lensing ring area with
the high demagnetization, leads to the lensing rings can only provide a very
small part of the total observed flux. Therefore, the observation intensity
are dominated by direct emission in these three different emission models.
Although these results are obtained with the ideal models, it is very
important to push forward the theoretical study of black hole shadow as far as
possible and lay the foundation for the observation results.

Acknowledgments

This work is supported by the National Natural Science Foundation of China
(Grant Nos. 11875095, 11903025), the Sichuan Youth Science and Technology
Innovation Research Team(21CXTD0038), and Basic Research Project of Science
and Technology Committee of Chongqing (Grant No. cstc2018jcyjA2480).

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