# Images and flares of geodesic hotspots around a Kerr black hole

Jiewei Huang1 Zhenyu Zhang2 Minyong Guo1,3∗ Bin Chen2,4,5

###### Abstract

In this study, we develop a numerical method to generate images on an
observer’s screen, formed by radiation from hotspots on any timelike orbits
outside a black hole. This method uses the calculation of fractional numbers,
enabling us not only to produce the overall image but also to distinguish
between primary, secondary, and higher-order images. Building upon this, we
compute the images of hotspots from eight potential types of geodesic timelike
orbits outside a Kerr black hole, summarizing the properties of both the
overall and individual order images. Furthermore, we calculate the centroid
motion and lightcurve. Notably, we observe flare phenomena across all orbit
types and classify these flares into three categories based on the Doppler and
gravitational redshift effects.

1 Department of Physics, Beijing Normal University, Beijing 100875, P. R.
China

2Department of Physics, Peking University, No.5 Yiheyuan Rd, Beijing 100871,
P.R. China

3Key Laboratory of Multiscale Spin Physics, Ministry of Education, Beijing
100875, P. R. China

4Center for High Energy Physics, Peking University, No.5 Yiheyuan Rd, Beijing
100871, P. R. China

5 Collaborative Innovation Center of Quantum Matter, No.5 Yiheyuan Rd, Beijing
100871, P. R. China

$\ast$ Corresponding author<EMAIL_ADDRESS>
##  1 Introduction

In recent years, there has been a growing body of observational evidence for
the existence of supermassive black holes at the centers of galaxies. Apart
from the black hole image captured by the Event Horizon Telescope (EHT)
collaboration [1, 2], the GRAVITY collaboration has reported intriguing near-
infrared flare events originating from near the horizon of the black hole Sgr
A* at the center of the Milky Way [3, 4]. While extensive research has been
conducted on the black hole image, as referenced in [5, 6, 7, 8, 9, 10, 11],
our current work will focus on these flare events.

Bright flaring from accreting black holes can be observed across all
wavelengths, yet the mechanism driving near-infrared flares from the accretion
onto Sgr A* remains a significant point of contention [12]. The source of
these flares is commonly linked to the acceleration of electrons in a
localized flaring area that’s no larger than a few gravitational radii, rather
than a universal increase in the accretion rate or jet power [13]. There are
four main methods to study sources that can produce near-infrared flares: 1)
applying general relativistic magnetohydrodynamic simulations to a confined
area around the black hole [14, 15, 16, 12, 17, 18], 2) using semi-analytic
models to analyze the spectral energy distribution and multi-wavelength
variability of Sgr A* [19, 20, 21, 22], 3) running Particle-in-Cell (PIC)
simulations on a small portion of the accretion flow [23, 24, 25, 26, 27], and
4) performing calculations on hotspots orbiting the central black hole [28,
29, 30, 31, 32, 33, 34, 35, 36, 37].

Each of the methods mentioned above has its strengths and weaknesses. The
first method excels in simulating fluid dynamics around a black hole, but it
is computationally demanding and its results do not coincide with
observational data. The second method aligns with observational results, but
it doesn’t adequately explain the cause of flares and the distribution of
matter around the black hole. The third method is superior in understanding
the microphysical properties of the accretion flow, but its high computational
cost and limited calculation area pose challenges for studying sources of
realistic sizes. The fourth method, although deemed unable to explain the
flare-causing mechanism or the microphysical properties of the accretion flow,
has a smaller computational cost. This allows for the study of different black
hole parameters and hotspot orbits, theoretically enhancing our understanding
of flares [38, 39, 40]. In previous studies using the fourth method, semi-
analytical hotspot imaging models were commonly employed. These models
typically use numerical methods to calculate photon orbit trajectories, but
often confine the timelike orbits of hotspot motion to analytically solvable,
essentially circular, equatorial orbits. Recognizing that there’s no inherent
necessity for hotspots to remain on the equatorial plane, a significant
advancement was made in [41]. The introduction of the NERO code signified a
development in computing images for any particle orbits outside a Kerr black
hole, not just equatorial circular ones. Furthermore, they took into account
flares produced by hotspots maintaining a constant vertical component.

Inspired by this pioneering work and in view of a recent classification of
Kerr timelike geodesics [42], we aim to develop a numerical method capable of
calculating images for any timelike orbits in this work. We intend to compute
the images of all possible timelike hotspots outside a Kerr black hole.
Specifically, we differentiate primary images, secondary images, and higher-
order images by calculating the fractional number. We not only analyze the
centroid motion of the complete image but also study the characteristics of
different order images. We also calculate the lightcurve to investigate the
properties of flares. Particularly, we note that the intensity of the hotspot
image is mainly determined by the motion state of the hotspot and
gravitational redshift, i.e., the Doppler effect and gravitational redshift
jointly dominate the intensity. By analyzing, we ascertain how the Doppler
effect and gravitational redshift play roles when each type of orbit generates
a flare.

The structure of this paper is as follows. In Sec. 2, we revisit the
classification of timelike geodesics for a Kerr black hole. Sec. 3 introduces
the hotspot model and imaging methods we employ. In Sec. 4, we apply the model
from Sec. 3 to generate images from various viewing angles, demonstrating
results for all orbit types. We wrap up with a summary of our results in Sec.
5. Throughout this paper, we use the geometrized unit system where $G=c=1$.

##  2 Review on classification of radial Kerr geodesic motion

In this section, we aim to review the classification of radial Kerr geodesic
motion as developed in the paper [42]. The line element of the Kerr metric in
Boyer-Lindquist (BL) coordinates is presented in the following form

$\displaystyle\mathrm{d}s^{2}$ $\displaystyle=$
$\displaystyle-\mathopen{}\mathclose{{}\left(1-\frac{2Mr}{\Sigma}}\right)\mathrm{d}t^{2}+\frac{\Sigma}{\Delta}\mathrm{d}r^{2}+\Sigma\mathrm{d}\theta^{2}+\mathopen{}\mathclose{{}\left(r^{2}+a^{2}+\frac{2Mra^{2}}{\Sigma}\mathrm{sin}^{2}\theta}\right)\mathrm{d}\phi^{2}-\frac{4Mra}{\Sigma}\mathrm{sin}^{2}\theta\mathrm{d}t\mathrm{d}\phi\,,$

where

$\displaystyle\Delta=r^{2}-2Mr+a^{2}\,,\quad\quad\Sigma=r^{2}+a^{2}\cos^{2}\theta\,.$
(2.2)

From this point forward, for convenience and without loss of generality, we
set $M=1$. The outer event horizon of the black hole can then be expressed as:

$r_{+}=1+\sqrt{1-a^{2}}\,,$ (2.3)

which represents the larger root of the equation $\Delta=0$.

The timelike geodesic equations in Kerr spacetime can be fully integrated due
to the presence of four conserved quantities along the trajectory of a
particle. These quantities include: the mass111For the sake of simplifying
calculations, we can assume the mass of the timelike object to be unity,
without sacrificing generality. $\mu^{2}=-g_{\mu\nu}p^{\mu}p^{\nu}=1$, the
energy $E=-u\cdot\partial_{t}$, the angular momentum
$L=u\cdot\partial_{\phi}$, and the Carter constant $Q$. By utilizing these
conserved quantities, the equations of motion can be expressed as follows [43]

$\displaystyle u^{r}$ $\displaystyle=$
$\displaystyle\pm_{r}\frac{1}{\Sigma}\sqrt{\mathcal{R}(r)}\,,$ $\displaystyle
u^{\theta}$ $\displaystyle=$
$\displaystyle\pm_{\theta}\frac{1}{\Sigma}\sqrt{\Theta(\theta)}\,,$
$\displaystyle u^{\phi}$ $\displaystyle=$
$\displaystyle\frac{1}{\Sigma}\mathopen{}\mathclose{{}\left[\frac{a}{\Delta}[E(r^{2}+a^{2})-aL]+\frac{L}{\sin^{2}\theta}-aE}\right]\,,$
$\displaystyle u^{t}$ $\displaystyle=$
$\displaystyle\frac{1}{\Sigma}\mathopen{}\mathclose{{}\left[\frac{r^{2}+a^{2}}{\Delta}[E(r^{2}+a^{2})-aL]+a(L-aE\sin^{2}\theta)}\right]\,,$
(2.4)

where

$\displaystyle\mathcal{R}(r)$ $\displaystyle=$
$\displaystyle[E(r^{2}+a^{2})-aL]^{2}-\Delta[Q+(L-aE)^{2}+r^{2}]\,,$ (2.5)
$\displaystyle\Theta(\theta)$ $\displaystyle=$ $\displaystyle
Q+a^{2}(E^{2}-1)\cos^{2}\theta-L^{2}\cot^{2}\theta\,,$ (2.6)

are the radial and angular potentials respectively, with $\pm_{r}$ and
$\pm_{\theta}$ denote the signs of the radial and polar motions. The
classification of timelike geodesics is based on the root structure of the
radial potential $\mathcal{R}(r)$. We label the out-of-horizon roots of
$\mathcal{R}(r)$ as $r_{1}$, $r_{2}$ and $r_{3}$ where $r_{+}\leq
r_{1}<r_{2}<r_{3}$. urthermore, we adopt the notations from [42]: The symbols
$|$, $+$, $-$ and $\rangle$ denote the outer horizon $r_{+}$, a region where
motion is permitted ($\mathcal{R}(r)>0$), a region where motion is disallowed
($\mathcal{R}(r)<0$) and the radial infinity. The symbols $\bullet$,
$\bullet\bullet$, $\bullet\bullet\bullet$ and $|\bullet$ used to represent a
single root, a double root, a triple root, and a root at the outer horizon,
respectively.

Type | Radial range | Root structure | Energy range |  $\mathcal{N}$ (of free parameters)
---|---|---|---|---
$\mathcal{P}$ | $r_{+}\leq r<\infty$ | $|+\rangle$ | $E\geq 1$ | $3$
$\mathcal{D}$ | $r_{2}\leq r<\infty$ | $|+\bullet-\bullet+\rangle$ | $E\geq 1$ | $3$
$|\bullet-\bullet+\rangle$ | $E\geq 1$ | $2$
$\mathcal{T}$ | $r_{+}\leq r\leq r_{1}$ | $|+\bullet-\rangle$ | $0\leq E<1$ | $3$
$|+\bullet-\bullet+\bullet-\rangle$ | $E_{\text{ISCO}^{+}}<E<1$ | $3$
$|+\bullet-\bullet+\rangle$ | $E\geq 1$ | $3$
$|+\bullet-\bullet\bullet-\rangle$ | $E_{\text{ISCO}^{+}}<E<1$ | $2$
$\mathcal{B}$ | $r_{2}\leq r\leq r_{3}$ | $|+\bullet-\bullet+\bullet-\rangle$ | $E_{\text{ISCO}^{+}}<E<1$ | $3$
$|\bullet-\bullet+\bullet-\rangle$ | $E_{c}<E<1$ | $2$
$\mathcal{S}$ | $r=r_{1}=r_{2}$ | $|+\bullet\bullet+\bullet-\rangle$ | $E_{\text{ISCO}^{+}}<E<1$ | $2$
$|+\bullet\bullet+\rangle$ | $E\geq 1$ | $2$
$r=r_{2}=r_{3}$ | $|+\bullet-\bullet\bullet-\rangle$ | $E_{\text{ISCO}^{+}}<E<1$ | $2$
$|\bullet-\bullet\bullet-\rangle$ | $E_{c}\leq E<1$ | $1$
$r=r_{1}=r_{2}=r_{3}$ | $|+\bullet\bullet\bullet-\rangle$ | $E_{\text{ISCO}^{+}}<E<E_{\text{ISCO}^{-}}$ | $1$
$\mathcal{H}$ | $r_{1}<r\leq r_{2}$ | $|+\bullet\bullet+\bullet-\rangle$ | $E_{\text{ISCO}^{+}}<E<1$ | $2$
$\mathcal{WD}$ | $r_{1}<r<\infty$ | $|+\bullet\bullet+\rangle$ | $E\geq 1$ | $2$
$\mathcal{WT}$ | $r_{+}\leq r<r_{1}$ | $|+\bullet\bullet+\bullet-\rangle$ | $E_{\text{ISCO}^{+}}<E<1$ | $2$
$|+\bullet\bullet+\rangle$ | $E\geq 1$ | $2$
$|+\bullet\bullet\bullet-\rangle$ | $E_{\text{ISCO}^{+}}<E<E_{\text{ISCO}^{-}}$ | $1$
Table 1: Classification of timelike geodesics in the Kerr exterior.

As indicated in [42], timelike geodesics in the Kerr exterior are categorized
into eight classes with $E\geq 0$, as outlined in Table 1. This table can be
understood as follows: the first column represents the types of orbits, which
include plunging orbits $\mathcal{P}$, deflecting orbits $\mathcal{D}$,
trapped orbits $\mathcal{T}$, bounded orbits $\mathcal{B}$, spherical orbits
$\mathcal{S}$, homoclinic orbits $\mathcal{H}$, whirling deflecting orbits
$\mathcal{WD}$, and whirling trapped orbits $\mathcal{WT}$. Specifically,
plunging orbits refer to particles that either plunge into the black hole from
infinity or are emitted near the black hole and fly off to infinity.
Deflecting orbits describe particles that originate from infinity, reach a
turning point, and then rebound back to infinity. Trapped orbits represent
particles that are emitted near the black hole, reach a turning point at a
finite radius, and subsequently plunge into the black hole. Bounded orbits
refer to particles that oscillate between two radial turning points.
Furthermore, in spherical orbits, a particle maintains a constant orbital
radius $r$. In homoclinic orbits, a particle originates from an unstable
spherical orbit, bounces off a turning point, and then approaches the original
unstable spherical orbit again. Whirling deflecting particles originate from
an unstable spherical orbit and fly off to infinity, or alternatively, they
come from infinity and approach an unstable spherical orbit. Lastly, whirling
trapped particles either emit outward near the black hole, asymptotically
approaching a spherical orbit, or they originate from a spherical orbit before
plunging into the black hole.

The second column of Table 1 is relatively easy to comprehend, as it denotes
the range of the corresponding orbits in the radial coordinate. The third
column, on the other hand, may be slightly confusing. It represents the
complete root structure of the radial function $\mathcal{R}(r)=0$,
corresponding to the orbits outside the horizon. In other words, the range of
the corresponding orbits in the radial coordinate is a subinterval of the root
structure. Other intervals of the root structure may correspond to different
types of timelike orbits. The fourth column is also easy to understand; it
represents the range of energy that particles on the orbit can carry. The
fifth column $\mathcal{N}$ indicates the number of freely selectable values
among the three conserved quantities $E$, $L$, and $Q$ for particles on the
corresponding orbit. Given that the expression for $\mathcal{R}(r)$ includes
the three variable parameters $E$, $Q$, and $L$, aside from $r$, these
parameters can be freely chosen within permissible ranges when there are no
specific restrictions on the roots of $\mathcal{R}(r)$, leading to three
degrees of freedom222For a specific black hole, the spin parameter $a$ is
fixed. Furthermore, we have set the mass $\mu=1$. If we retain $\mu$ in the
calculations, the free parameters should be scaled as $E/\mu$, $L/\mu$ and
$Q/\mu^{2}$, which is equivalent to choosing $\mu=1$.. However, when the
requirement is for $\mathcal{R}(r)=0$ to have a double root $r_{\star}$ or a
root at $r=r_{+}$, satisfying $\mathcal{R}^{\prime}(r_{\star})=0$ or
$\mathcal{R}(r_{+})=0$ would impose a constraint, reducing the number of
freely selectable parameters to $2$. When $\mathcal{R}(r)=0$ has a triple
root, or a double root with another root precisely on the horizon, the number
of free parameters becomes $1$. Specifically, orbits with $\mathcal{N}=3$ are
referred to as generic orbits in [42], while those with $\mathcal{N}=2$ and
$\mathcal{N}=1$ are respectively termed codimension 1 and codimension 2
orbits. We would like to emphasize that the values of the free parameters are
not arbitrary. For a specific type of orbit, they must fall within a certain
range. While the expressions that describe these ranges are quite complex, we
will only provide the formulas for some key parameters immediately. For a more
detailed understanding, interested readers are encouraged to refer to [42] for
more information.

For the double root $r_{\star}$, the conditions
$\mathcal{R}(r_{\star})=\mathcal{R}^{\prime}(r_{\star})=0$ hold true. By
manipulating these equations, we can also express $Q$ and $L$ as functions of
$r_{\star}$ and $E$,

$\displaystyle Q_{\star}$
$\displaystyle=\dfrac{r_{\star}^{2}}{a^{2}\mathopen{}\mathclose{{}\left(r_{\star}-1}\right)^{2}}\bigg{(}-r_{\star}^{3}+3r_{\star}^{2}+\mathopen{}\mathclose{{}\left(a^{2}-4}\right)r_{\star}+a^{2}$
$\displaystyle+r_{\star}\mathopen{}\mathclose{{}\left(1-E^{2}}\right)\mathopen{}\mathclose{{}\left(r_{\star}^{3}-4r_{\star}^{2}+5r_{\star}-2a^{2}}\right)+2E\Delta\mathopen{}\mathclose{{}\left(r_{\star}}\right)\sqrt{r_{\star}\mathopen{}\mathclose{{}\left(1+\mathopen{}\mathclose{{}\left(E^{2}-1}\right)r_{\star}}\right)}\bigg{)}\,,$
(2.7) $\displaystyle L_{\star}$
$\displaystyle=\dfrac{E\mathopen{}\mathclose{{}\left(r_{\star}^{2}-a^{2}}\right)-\Delta\mathopen{}\mathclose{{}\left(r_{\star}}\right)\sqrt{r_{\star}\mathopen{}\mathclose{{}\left(1+\mathopen{}\mathclose{{}\left(E^{2}-1}\right)r_{\star}}\right)}}{a\mathopen{}\mathclose{{}\left(r_{\star}-1}\right)}\,.$
(2.8)

Take note that in this case $r_{\star}$ is not fixed and has a lower limit,
which is its minimum value. When $r_{\star}$ reaches this minimum, the
corresponding expression for energy $E_{c}$ can be represented as follows:

$E_{c}=\frac{r_{+}+2\sqrt{r_{+}-1}}{\sqrt{r_{+}(r_{+}+4\sqrt{r_{+}-1}+2)}}\,.$
(2.9)

On the other hand, if the orbit $r=r_{\star}$ is confined to the equatorial
plane, we have $Q_{\star}=0$. In this case, the particle follows a circular
trajectory. The corresponding energy can be calculated and expressed as

$E_{\pm}(r_{\star})=\frac{r_{\star}(r_{\star}-2)\pm
ar_{\star}^{1/2}}{\sqrt{r_{\star}^{3}(r_{\star}-3)\pm 2ar_{\star}^{5/2}}}\,,$
(2.10)

where the symbols $\pm$ represent whether the particle is moving in a prograde
or retrograde direction. If we further require the particle to move in the
innermost stable circular orbit (ISCO), i.e., where
$\mathcal{R}^{\prime\prime}(r)=0$ is satisfied, then the radius of the ISCO
can be determined as

$r_{\text{ISCO}^{\pm}}=3+Z_{2}\mp\sqrt{(3-Z_{1})(3+Z_{1}+2Z_{2})}\,,$ (2.11)

where

$Z_{1}=1+(1-a^{2})^{1/3}\mathopen{}\mathclose{{}\left[(1+a)^{1/3}+(1-a)^{1/3}}\right]\,,\qquad
Z_{2}=\sqrt{3a^{2}+Z_{1}^{2}}\,.$ (2.12)

Similarly, the symbols $\pm$ here represent prograde and retrograde motion,
respectively. Consequently, the energy for the corresponding prograde and
retrograde orbits is given by

$\displaystyle E_{\text{ISCO}^{+}}=E_{+}(r_{\text{ISCO}^{+}})\,,$ (2.13)
$\displaystyle E_{\text{ISCO}^{-}}=E_{-}(r_{\text{ISCO}^{-}})\,.$ (2.14)

By now, we have provided clear explanations for all the parameters mentioned
in Table 1. One should have a sufficient understanding of the orbit
classifications and the corresponding energy conditions outlined in the table.
We will also provide examples of each class later in Sec. 4.

##  3 Hotspot model and imaging method

In this section, we introduce our hotspot model and the associated imaging
method. Our work primarily focuses on the kinematic flare phenomena brought
about by gravitational redshift (blueshift) and Doppler shift. As such, we
dismiss relativistic magnetohydrodynamic effects. We represent the hotspot as
an opaque object traversing various types of geodesic paths outside a Kerr
black hole. For ease of computation, we model the shape of the hotspot as a
sphere with a small, fixed radius $b$ in the BL coordinates, with the radius
set to $b=0.25$ for this study. We do not aim to investigate the influence of
the radiation mechanism on imaging and flares. Therefore, our model does not
account for the detailed radiation spectrum of the hotspot’s emission. We thus
assume that the hotspot’s emission is isotropic and frequency-independent,
implying a broadband source with a flat spectrum.

To image the moving hotspot, we need to understand both the trajectory of the
light source and the radiation transfer from the source. We determine the
trajectory of the source by numerically solving the geodesic equation in the
Hamilton-Jacobi form. For the radiation transfer, we employ the numerical
backward ray-tracing method and the fisheye camera mode for imaging. The
specific technical details can be found in Appendix B of [44]. The camera is
set as a zero-angular-momentum observer (ZAMO), whose tetrads takes the
following form

$\displaystyle\hat{e}_{(0)}=\frac{g_{\phi\phi}\partial_{t}-g_{\phi
t}\partial_{\phi}}{\sqrt{g_{\phi\phi}\mathopen{}\mathclose{{}\left(g_{\phi
t}^{2}-g_{\phi\phi}g_{tt}}\right)}}\,,\quad\quad\hat{e}_{(1)}=-\frac{\partial_{r}}{\sqrt{g_{rr}}}\,,\quad\hat{e}_{(2)}=\frac{\partial_{\theta}}{\sqrt{g_{\theta\theta}}}\,,\quad\hat{e}_{(3)}=-\frac{\partial_{\phi}}{\sqrt{g_{\phi\phi}}}\,.$
(3.1)

We consider that the intensity of light emitted from the source and reaching
the observer is not absorbed by the medium during its journey. The intensity
divided by the frequency cubed, denoted as $I_{\nu}/\nu^{3}$, is conserved
along a light ray [45]. Thus, the observed and emitted specific intensities
$I_{\nu_{o}}$ and $I_{\nu_{s}}$ are related by radiative transport as

$\frac{I_{\nu_{o}}}{\nu_{o}^{3}}=\frac{I_{\nu_{s}}}{\nu_{s}^{3}}\,,$ (3.2)

where $\nu_{o}=-k_{\mu}\hat{e}_{(0)}^{\mu}$0 is the observed frequency on the
screen, with $k_{\mu}$ representing the 4-momentum of the photon. Meanwhile,
$\nu_{s}=-k_{\mu}u^{\mu}$ is the frequency measured by an observer comoving
with the source. After introducing the redshift factor $g=\nu_{o}/\nu_{s}$,
Eq. (3.2) can be rewritten as

$I_{\nu_{o}}=g^{3}I_{\nu_{s}}\,.$ (3.3)

Note that in our hotspot model, $I_{\nu_{s}}$ is a constant. Therefore, the
observed intensity is solely dependent on the redshift factor, $g$.

Figure 1: A diagram to illustrate the method to calculate the time

It is important to highlight a subtle issue that arises when handling the
imaging of a moving hotspot. The light that is emitted from the hotspot and
reaches the observer is not an instant reflection of the hotspot’s position.
Given that the hotspot is in motion, it will have moved to a new location by
the time the light reaches the observer, resulting in a delayed image. Given
the substantial distance between the observer and the hotspot, this time delay
is significant and cannot be overlooked. To tackle this issue, we have
implemented a specific strategy, illustrated in Fig. 1, which is consistent
with the one presented in the paper [41]. Let S represent the starting point
of the hotspot and O denote the observer. As the hotspot moves to point A and
intersects with the photon traced backward from the observer, we define the
coordinate time for the hotspot’s journey from S to A as $t_{\text{hs}}$, and
the coordinate time for the photon’s journey from O to A as $t_{\text{ph}}$,
all within the BL coordinate system. The total time of interest to us is
simply the sum of these two times, $t_{\text{hs}}+t_{\text{ph}}$. It is
crucial to note that we assume the observer’s proper time is equivalent to the
coordinate time, rendering this method effective only for observers
sufficiently distant from the black hole.

In addition, due to the strong gravitational lensing effects caused by black
holes, the imaging of the hotspot yields not just a primary image, but also
secondary and higher-order images. As a result, it becomes necessary for us to
distinguish between these different levels of images. To achieve this, we
employ a specific method that involves calculating two angular integrals over
the entire light path, stretching from the hotspot to the observer. The
definitions of these integrals are

$\displaystyle
G_{\theta}=\fint_{\theta_{s}}^{\theta_{o}}\dfrac{1}{\pm_{\theta}\sqrt{\Theta(\theta)}}\mathrm{d}\theta\,,\quad\quad\hat{G}_{\theta}=\int_{\theta_{-}}^{\theta_{+}}\dfrac{1}{\sqrt{\Theta}}\mathrm{d}\theta\,.$
(3.4)

The integral symbol $\displaystyle\fint$ denotes the integration along the
trajectory of a photon trajectory, while $\theta_{o}$ and $\theta_{s}$ are the
angular coordinates of the observer and the hotspot, respectively.
Furthermore, $\theta_{\pm}$ represents the larger and smaller angular turning
points, respectively. Next, we calculate the fractional number $n$ using the
formula

$\displaystyle n=\dfrac{G_{\theta}}{2\hat{G}_{\theta}}\,.$ (3.5)

When $n\leq 1/2$, it corresponds to the primary image. If $1/2<n\leq 1$, it
corresponds to the secondary image, and for $n>1$, it corresponds to higher-
order images.

Furthermore, despite the assumption that the emission rate of the hotspot is
constant within its local coordinate system, due to gravitational redshift
(blueshift) and the Doppler effect, the intensity of the image seen by the
observer varies at different times. To present the images more effectively, we
utilize the definition of $T$ as proposed in [46]:

$\displaystyle
T=\mathopen{}\mathclose{{}\left[{\log_{2}\mathopen{}\mathclose{{}\left(1+\dfrac{I_{o}^{\text{max}}}{I_{o}}}\right)}}\right]^{-1}\,,$
(3.6)

This definition is based on the expression for the brightness temperature
[47]. Here, $I_{o}^{\text{max}}$ is defined as the maximum value of $I_{o}$,
ensuring that $T$ ranges between $0$ and $1$. Additionally, at each moment, we
can compute the centroid position $\vec{x}_{c}$ of the flux across the camera
plane. Given the definition of the camera we have adopted [44], the flux
$F(i,\,j)$ of pixels $(i,\,j)$ observed at the screen can be calculated by

$\displaystyle
F(i,j)=I_{o}S_{0}\cos\mathopen{}\mathclose{{}\left[2\arctan\mathopen{}\mathclose{{}\left(\dfrac{1}{n}{\tan\mathopen{}\mathclose{{}\left(\frac{\alpha_{\text{fov}}}{2}}\right)}\sqrt{\mathopen{}\mathclose{{}\left(i-\frac{{n}+1}{2}}\right)^{2}+\mathopen{}\mathclose{{}\left(j-\frac{{n}+1}{2}}\right)^{2}}}\right)}\right]\,,$
(3.7)

where $S_{0}$ is the pixel size, $n$ is the number of horizontal or vertical
pixels, $i$ and $j$ are chosen in a range from $1$ to $n$, and
$\alpha_{\text{fov}}$ is the camera’s field of view. Once we have this
information, the centroid position for each image can be calculated using the
formula:

$\displaystyle\vec{x}_{c}(t)=\dfrac{\sum_{i,j}\vec{x}\mathopen{}\mathclose{{}\left(i,j}\right)F\mathopen{}\mathclose{{}\left(i,j}\right)}{\sum_{i,j}F\mathopen{}\mathclose{{}\left(i,j}\right)}\,,$
(3.8)

where $\vec{x}\mathopen{}\mathclose{{}\left(i,j}\right)$ represent the
coordinates of pixel $\mathopen{}\mathclose{{}\left(i,j}\right)$, and the
total flux of the snapshot, denoted as
$\sum_{i,j}F\mathopen{}\mathclose{{}\left(i,j}\right)$ can be regarged as the
flux for $\vec{x}_{c}(t)$. The images we obtained of the moving hotspot are a
series of snapshots. To conveniently identify the brighter snapshots, i.e.,
flares, we introduce the rescaled flux of $\vec{x}_{c}(t)$ as

$\displaystyle\bar{F}(\vec{x}_{c})=\frac{\sum_{i,j}F\mathopen{}\mathclose{{}\left(i,j}\right)}{\mathopen{}\mathclose{{}\left[\sum_{i,j}F\mathopen{}\mathclose{{}\left(i,j}\right)}\right]_{\text{max}}}\,.$
(3.9)

In the context of our model and the methodology we’ve implemented, we
ultimately determine the time evolution of both the centroid position and its
corresponding flux.

##  4 Results

In this section, we will display images of a hotspot orbiting around the Kerr
black hole with $a=0.94$, utilizing the orbit classifications discussed in the
previous section. Considering that the black hole (BH) is located a
significant distance from us in the universe, we set $r_{o}=300\gg r_{+}$ in
our numerical calculations. We are specifically examining two scenarios for
observational angles: one with $\theta_{o}=25^{\circ}$ and another with
$\theta_{o}=80^{\circ}$. Initially, we will investigate the characteristics of
hotspot imaging at the observational angle of $\theta_{o}=25^{\circ}$.
Afterwards, we will shift our focus to the observations made at
$\theta_{o}=80^{\circ}$. In the following calculations, we set the azimuthal
angle $\phi_{o}=0^{\circ}$.

Type | $E$ | $L$ | $Q$ | $r_{s}$ | Root structure | Root
---|---|---|---|---|---|---
$\mathcal{P}$ | $1.2$ | $1$ | $12$ | $40$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}|+\rangle}$ | $\diagdown$
$\mathcal{D}$ | $1.2$ | $3$ | $5$ | $40$ | $|+\bullet-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet+\rangle}$ | ${1.38,\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}2.38}}$
$\mathcal{T}$ | $0.95$ | $-1$ | $5$ | $15$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}|+\bullet}-\rangle$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}17.51}$
$\mathcal{B}$ | $0.95$ | $2$ | $5$ | $14$ | $|+\bullet-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet+\bullet}-\rangle$ | $1.53,\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}3.66},\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}14.78}$
$\mathcal{S}$ | $0.95$ | $-0.743$ | $12.82$ | $8$ | $|+\bullet-{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}-\rangle$ | $4.07,\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}8},\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}8}$
$\mathcal{H}$ | $0.95$ | $0.172$ | $11.86$ | $4.05$ | $|+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet+\bullet}-\rangle$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}4},\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}4},\,11.95$
$\mathcal{W}\mathcal{D}$ | $1.2$ | $-4.035$ | $22.47$ | $4.01$ | $|+{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet+\rangle}$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}4},\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}4}$
$\mathcal{W}\mathcal{T}$ | $0.95$ | $-1.256$ | $12.10$ | $5.99$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}|+\bullet}{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}\bullet}+\bullet-\rangle$ | ${\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}6},\,{\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}6},\,8.14$
Table 2: Parameters of different types of timelike geodesics.

For simplicity, we standardize the initial position to
$(t_{s},\,\theta_{s},\,\phi_{s})=(0,\,60^{\circ},\,0)$ when selecting
parameters for each orbit type. We opt for $\theta_{s}=60^{\circ}$ because our
established observation angles fall below $60^{\circ}$ and above $80^{\circ}$.
As a result, at the initial moment, the images of the hotspot will manifest in
the northern and southern hemispheres of the image plane respectively,
providing a highly representative view. Additionally, we set
$\phi_{s}=\phi_{o}=0^{\circ}$ to position the hotspot between the observer and
the black hole at the onset. We must underscore that different types of orbits
often correspond to more than a single root structure. Indeed, as illustrated
in Table 1, only the $\mathcal{P}$, $\mathcal{H}$ and $\mathcal{WD}$ orbits
are associated with a singular root structure, while all other orbit types are
not. However, as presented in Table 2, for each orbit type, we have
selectively chosen one parameter set for in-depth study. Our strategy for
selecting parameters involves opting for the scenario where $\mathcal{N}$ is
the largest for a given orbit type. This is because scenarios with smaller
$\mathcal{N}$ values correspond to orbits with repeated roots or roots that
fall on the horizon in a peculiar manner. Conversely, scenarios with the
largest $\mathcal{N}$ correspond to more common orbits. Additionally, for the
$\mathcal{T}$, $\mathcal{S}$, and $\mathcal{WT}$ orbits, the root structure is
still not unique when $\mathcal{N}$ is at its maximum. For the $\mathcal{T}$
orbit, given that its characteristic is being confined between $r_{+}$ and
$r_{1}$, we would like to choose orbits with $E<1$. Considering that the
$|+\bullet-\bullet+\bullet-\rangle$ orbit requires more stringent energy
conditions than the $|+\bullet-\rangle$ orbit, we opt for the latter. For the
S orbit, we naturally desire this orbit to be a stable one, so we select the
$|+\bullet-\bullet\bullet-\rangle$ orbit. For the $\mathcal{W}T$ orbit,
similar to the $\mathcal{T}$ orbit and considering that this orbit is confined
between $r_{+}$ and $r_{1}$, we choose the case with $E<1$, i.e., the
$|+\bullet\bullet+\bullet-\rangle$ orbit. In summary, the eight types of
orbits we select can be divided into two categories based on the value of
$\mathcal{N}$. The first category includes $\mathcal{P}$, $\mathcal{D}$,
$\mathcal{T}$, and $\mathcal{B}$ orbits, where $\mathcal{N}=3$. The second
category includes $\mathcal{S}$, $\mathcal{H}$, $\mathcal{WD}$, and
$\mathcal{WT}$ orbits, with $\mathcal{N}=2$. For orbits that can extend to
infinite distance, such as $\mathcal{P},\mathcal{D},\mathcal{WD}$, we set the
energy $E=1.2$ for these specific orbit types. For all other orbit types, we
fix the energy at $E=0.95$. Additional specific parameters can be referenced
in Table 2. Notably, within the ‘root structure’ column, regions of the
orbit’s existence are marked in blue. In the ‘root’ column, the black and blue
designations align with those in the ‘root structure’ column.

###  4.1 $\mathcal{N}=3$

In this subsection, we aim to investigate the imaging characteristics of
orbits where $\mathcal{N}=3$.

####  4.1.1 Plunging orbits

Figure 2: The upper row presents the trajectory and image of a plunging orbit,
while the lower row illustrates the centroid motion and the light curve of the
plunging particle. On the upper row, from left to right, we display: (Left)
The trajectory in 3D Cartesian coordinates, defined as
$x=r\sin\theta\cos\phi$, $y=r\sin\theta\sin\phi$, $z=r\cos\theta$. The red dot
represents the starting point of the trajectory. (Middle) The image of the
hotspot as observed by an observer. In this representation, the direct,
second, and higher order images are color-coded as red, blue, and orange,
respectively. (Right) The intensity map of the image, which offers a
visualization of the distribution of light intensity across the image. For the
lower row: (Left) The panel depicts the motion of the emission centroid. The
accompanying color bar indicates the time progression in units of minutes. The
time interval between two adjacent dots in the diagram is 10 minutes and the
first dot is 0 min. (Right) The panel presents the flux evolution
corresponding to the motion, providing a visual representation of how the
rescaled flux changes over time.

We initiate our study with Plunging orbits, denoted as $\mathcal{P}$. In Fig.
2, the upper panel displays the trajectory and image of a plunging orbit,
while the lower panel illustrates the centroid motion and the light curve for
a plunging particle. This scenario represents a particle plunging into the
black hole from infinity, observed at an inclination angle of
$\theta_{o}=25^{\circ}$.

In the left plot of the upper panel of Fig. 2, we observe a hotspot beginning
its journey from a position distant from the black hole and spiraling towards
it. Initially, when the hotspot is still far from the black hole, its orbital
speed is relatively slow. However, as it gets closer to the black hole, its
rotational speed dramatically increases. The corresponding image of this orbit
is illustrated in the middle plot of the upper panel. The primary image,
represented by the red line, shows the hotspot entering from the southern
hemisphere, spiraling, and eventually plunging into the black hole.
Additionally, the secondary image, marked by the blue line, is distinctly
visible in the northern hemisphere, while the higher order images, indicated
by the orange line, are barely noticeable around the shadow curve. The
corresponding intensities of these images are displayed in the right plot of
the upper panel. For the primary image, we note that as the hotspot moves away
from the observer and towards the black hole, its speed increases due to the
black hole’s gravitational attraction. Consequently, the brightness of the
hotspot’s image gradually decreases due to the Doppler redshift effect. As it
nears the black hole, the brightness of the spiraling image also decreases due
to the strong gravitational redshift effect. In contrast, for the secondary
image, the light is emitted in a direction roughly parallel to the hotspot’s
motion as it moves away from the observer and approaches very close to the
black hole. After circumnavigating the black hole due to the gravitational
lensing effect, the light finally reaches the observer. This scenario, due to
the Doppler blueshift effect, results in regions of significantly enhanced
brightness.

In the lower panel of Fig. 2, we observe that the trajectory of the hotspot
persists for more than 40 minutes on the observer’s screen. In the initial
phase, when the hotspot is positioned far from the black hole, the centroid is
predominantly determined by the primary image. However, as the hotspot
approaches the black hole, roughly 20 minutes into the observation, both the
primary and secondary images contribute to determining the centroid. At
approximately 28 minutes, there is a noticeable flare, characterized by a
sudden surge in the rescaled flux. Comparing this with the results from the
upper panel, we can ascertain that this flare is triggered by the Doppler
blueshift of the secondary image. Note that although the hotspot emits the
light forming the flare right at the onset of its motion, due to the lensing
effect, a substantial delay occurs before the flare manifests on the screen.
For ease of reference, we term this type of flare as a “Lensed Doppler
Blueshift Flare” (LDBF).

####  4.1.2 Deflecting orbits

Figure 3: The upper row in the illustration presents the trajectory and image
of a deflecting hotspot. Conversely, the lower row provides a depiction of the
centroid motion and its corresponding light curve for the deflecting orbit.
The interpretation of the points and lines in this figure aligns with those
outlined in Fig. 2.

Next, we turn our attention to the study of deflecting orbits, denoted as
$\mathcal{D}$. In Fig. 3, similar to Fig. 2, the upper row displays the
trajectory and image of a deflecting hotspot. The lower row, on the other
hand, delineates the motion of the emission centroid and its associated light
curve for the deflecting orbit. The observation is made at an inclination
angle of $\theta_{o}=25^{\circ}$.

From Fig. 3, it is observed that the hotspot initiates its journey from a
position far from the black hole, reaches a turning point, and then returns to
infinity. On the observer’s screen, the primary image of the hotspot departs
from the northern hemisphere towards the black shadow, circles within the
shadow, then flies into the southern hemisphere. The secondary images are
primarily distributed in the northern hemisphere near the shadow curve, while
the higher-order images clinging to the shadow curve are quite inconspicuous.
From the bottom-left graph, it can be deduced that the centroid motion is
determined by the primary image. The bottom-right graph reveals that the light
curve can be essentially divided into three segments. In the first segment,
the flux of the centroid decreases slightly. The second segment presents a
distinct flare, while the third segment is relatively stable, yet with a
higher flux than at the start. This phenomenon can be explained by the Doppler
effect. During the first segment, the hotspot is moving away from the
observer, causing a Doppler redshift. In contrast, during the third segment,
the hotspot is moving towards the observer, causing a Doppler blueshift.
However, as it is not directly facing the observer, the flux is only slightly
higher than in the first segment. The appearance of the flare is due to the
hotspot passing the turning point at this time. Its speed rapidly increases,
and due to its proximity to the black hole, the strong gravitational force can
cause a significant change in direction. Therefore, there is a moment when the
speed is directly or nearly directly facing the observer, causing a Doppler
blueshift that appears as a flare. In this paper, we refer to this type of
flare as a ‘turning Doppler blueshift flare’ (TDBF).

####  4.1.3 Trapped orbits

Figure 4: The upper row of the illustration showcases the trajectory and image
of a hotspot trapped in orbit. In contrast, the lower row visually represents
the centroid motion and the corresponding light curve for this trapped orbit.
The symbolism and interpretation of points and lines in this figure are
consistent with those defined in Fig. 2.

Then, we will turn our attention to the exploration of trapped orbits, denoted
as $\mathcal{T}$. In Fig. 4, akin to Fig. 2, the upper row illustrates the
trajectory and image of a hotspot trapped in orbit. Conversely, the lower row
outlines the movement of the emission centroid, along with its corresponding
light curve for this trapped orbit. All observations are made at an
inclination angle of $\theta_{o}=25^{\circ}$.

From Fig. 4, we observe that the hotspot embarks on its journey from $r_{s}$,
subsequently moving outward to reach $r_{1}$. After this, it changes
direction, circles the black hole several times, and ultimately falls into the
black hole. On the observer’s screen, the primary image of the hotspot starts
in the southern hemisphere, then moves downward. After reaching a turning
point, it changes direction and finally spirals into the black hole’s shadow.
The starting point of the secondary image, marked in blue, is near the
equatorial plane and close to the black hole’s shadow, and then it spirals and
falls into the black hole’s shadow. From the light intensity plot at the top
right and the light curve graph at the bottom right, we can see that the flux
of the hotspot’s image is highest when it starts moving, then decreases, rises
again after more than 20 minutes, and presents a noticeable flare at 60
minutes. The reason for this phenomenon is that initially, the hotspot is
moving away from both the black hole and the observer, and the angle between
the direction of the hotspot’s velocity and the direction pointing towards the
observer is increasing, leading to a gradual decrease in light intensity. As
the hotspot is captured and flies towards the black hole, the angle between
the direction of the hotspot’s velocity and the direction pointing towards the
observer decreases. During this process, the hotspot is far from the black
hole, so the effect of gravitational redshift is small, and the received light
intensity gradually increases. As the hotspot nears the black hole, despite
the increasing gravitational redshift, the impact of the Doppler effect on the
secondary image exceeds the effect of gravitational redshift, resulting in a
flare. This flare is similar to those observed in plunging orbits and is
characterized as a LDBF. Correspondingly, the centroid motion graph at the
bottom left can also be understood. Initially, the centroid is dominated by
the primary image. Subsequently, it is dominated by the secondary image, and
finally, it returns to being dominated by the primary image.

####  4.1.4 Bounded orbits

Figure 5: The upper row of the illustration showcases the trajectory and image
of a bounded hotspot. In contrast, the lower row visually represents the
centroid motion and the corresponding light curve for this bounded orbit. The
symbolism and interpretation of points and lines in this figure are consistent
with those defined in Fig. 2.

Next, we turn our attention to the images of bounded orbits, represented as
$\mathcal{B}$. As shown in Fig. 5 and similar to Fig. 2, the top row depicts
the trajectory and image of a bounded hotspot. The bottom row, on the other
hand, shows the path of the emission centroid and its corresponding light
curve for the same orbit. All these observations are conducted at an
inclination angle of $\theta_{o}=25^{\circ}$.

For the bounded orbit we’ve chosen, as outlined in Table 2, where
$r_{2}<r_{s}<r_{3}$, the hotspot oscillates between the two turning points at
$r_{2}$ and $r_{3}$. Fig. 5 depicts the hotspot’s motion path as $r_{s}\to
r_{3}\to r_{2}\to r_{3}\to r_{2}\to r_{e}<r_{3}$. Here, $r_{e}$ represents the
endpoint of the trajectory in the radial direction. Due to the extensive
trajectory of the hotspot’s motion, both the primary (red) and secondary
(blue) images on the observer’s screen are relatively complex. Instead of
detailing their intricate changes, we will focus on the motion of the centroid
and causes of flares. For the emission centroid, we observe it moving back and
forth within a finite radius, forming multiple rings. When the hotspot is far
from the black hole, the brightness of the primary image far exceeds that of
the secondary image, contributing more significantly to the emission centroid.
Therefore, the motion of the centroid mainly reflects the trajectory of the
primary image. As the hotspot approaches the vicinity of the black hole, the
positions of the primary and secondary images are roughly symmetrically
distributed about the black hole’s shadow (for instance, when the primary
image is above the shadow, the secondary image appears below it), and their
brightness is approximately equal. This results in the emission centroid
lingering within the black hole shadow. In terms of flux variation, we observe
two distinct flares around the 35 min and 130 min marks. Additionally, there
are two minor peaks noticeable around the 62 min and 100 min marks, connected
by a concave line in between. The flares observed at the 35 min and 130 min
points share a common origin, akin to the flare formation mechanism in
deflecting orbits. These flares emerge when the hotspot navigates past the
inner turning point, which is in close proximity to the black hole.
Consequently, the strong gravitational force induces a dramatic shift in
direction, and the hotspot’s speed escalates rapidly post this turning point.
Hence, we refer to these flares as TDBFs in our paper. The two minor peaks and
the intervening trough occur between the two flares, corresponding to the
hotspot’s motion from $r_{2}\to r_{3}\to r_{2}$. Within this range, and not
far from $r_{3}$, the radial motion direction of the hotspot forms an angle
less than $90^{\circ}$ with the direction towards the observer. Hence, it can
be essentially considered as moving towards the observer. Given that $r_{3}$
is relatively distant from the black hole, the gravitational influence is
minor, and there is minimal change in the direction of velocity. Consequently,
the primary factor affecting the flux is the magnitude of the velocity. In the
$r_{2}\to r_{3}\to r_{2}$ process, the hotspot’s velocity first increases,
then decreases, and then increases and decreases again.

###  4.2 $\mathcal{N}=2$

In this subsection, we will explore the features of images of orbits with
$\mathcal{N}=2$, as presented in Table 2.

####  4.2.1 Spherical orbits

Figure 6: The upper row of the illustration showcases the trajectory and image
of a spherical hotspot. In contrast, the lower row visually represents the
centroid motion and the corresponding light curve for this spherical orbit.
The symbolism and interpretation of points and lines in this figure are
consistent with those defined in Fig. 2.

First, we focus on the spherical orbits, denoted as $\mathcal{S}$. For these
orbits, the hotspot is fixed at $r=r_{s}=8$. Therefore, the aspects that
change in the trajectory are the polar and azimuthal coordinates. In Fig. 6,
we present an orbit with an azimuthal angle variation of $\Delta\varphi=4\pi$.
The top row of the illustration displays the trajectory and image of a
spherical hotspot, while the bottom row visually represents the centroid
motion and the corresponding light curve for this spherical orbit. The symbols
and interpretations of points and lines in this figure are consistent with
those defined in Fig. 2. The observation is conducted at an inclination angle
of $\theta_{o}=25^{\circ}$.

From the middle image in the first row of Fig. 6, it can be observed that the
primary image of the hotspot takes the form of a precessing cap, while the
secondary image appears on the opposite side of the primary image and has a
fuller shape. From the intensity map on the right side of the first row in
Fig. 6, it can be seen that although the overall brightness of the secondary
image is lower than that of the primary image, its brightness is still
significant and should not be overlooked. Hence, the centroid of the hotspot
image is influenced by both the primary and secondary images, resulting in an
irregular trajectory of the centroid motion. The right plot in the second row
of Fig. 6 displays a light curve with two prominent flares, each showing a
split. These flares arise from the angular motion of the hotspot, which has a
fixed radial position. When the hotspot moves to a certain angle, the Doppler
blue shift peaks, leading to a flare in the primary image. This is swiftly
followed by the blue-shift signal from the secondary image. The brief interval
between these two signals gives the impression of a broad flare split. Indeed,
when the time interval is reduced, the flare fully splits into two peaks,
though this result has not been further explored in the paper. Given the
spherical orbit’s periodic angular motion, a flare corresponding to the first
one around the 30 min mark is anticipated to appear around the 80 min mark,
also showing a split. However, due to the precession in the hotspot’s motion,
the second flare’s behavior doesn’t entirely mirror that of the first. This
type of flare is determined by both the angular Doppler blueshift and lensed
Doppler blueshift. Therefore, we can conveniently refer to it as ‘Mixed
Doppler Blueshift Flares’, abbreviated as ‘MDBF’.

####  4.2.2 Other orbits for $\mathcal{N}=2$

Figure 7: In the illustration, the top row highlights the trajectory and image
of a homoclinic hotspot. On the other hand, the bottom row graphically
illustrates the centroid motion and the associated light curve for this same
orbit. The symbols, points, and lines in this figure maintain consistency with
those defined in Fig. 2. Figure 8: The top row of the illustration
demonstrates the trajectory and image of a whirling deflecting hotspot,
whereas the bottom row provides a visual representation of the centroid motion
and the associated light curve for this whirling deflecting orbit. The
symbolism and interpretation of points and lines in this figure align with
those established in Fig. 2. Figure 9: In the illustration, the upper section
presents the trajectory and image of a whirling trapped hotspot, while the
lower section visually delineates the centroid motion and its corresponding
light curve for this whirling trapped orbit. The symbology and interpretation
of points and lines in this figure are in accordance with those specified in
Fig. 2.

Next, we shift our focus to other orbits when $\mathcal{N}=2$, namely, the
homoclinic orbits $\mathcal{H}$, the whirling deflecting orbits $\mathcal{WD}$
and the whirling trapped orbits $\mathcal{WT}$. We discuss the results of
these three types of orbits collectively because they can be viewed as
combinations of the previously mentioned orbits. The hotspot imaging and
flares can also be explained accordingly using the conclusions drawn earlier.

Homoclinic orbits

We know that in homoclinic orbits, a particle starts from an unstable
spherical orbit, deflects at a turning point, and then returns to the original
unstable spherical orbit. Clearly, the imaging characteristics during the
unstable spherical orbit phase should resemble those of $\mathcal{S}$.
Considering the turning point is relatively distant from the black hole, its
imaging characteristics should be similar to those near the outer turning
point of $\mathcal{B}$. In Fig. 7, the imaging outcomes for homoclinic orbits
are displayed, encompassing light intensity maps, graphs of centroid motion,
and light curves. In comparing Fig. 7 to Fig. 6 and Fig. 5, it’s evident that
aspects of both the light intensity maps and light curves from Fig. 6 and Fig.
5 are mirrored in Fig. 7. The flares in Fig. 7 align with those in Fig. 6, and
the dual minor peaks along with the intermediate trough in Fig. 7 bear
significant resemblance to their counterparts in Fig. 5. However, there are
some subtle discrepancies when Fig. 7 is juxtaposed with Fig. 6 and Fig. 5.
For example, the flare in Fig. 7 doesn’t display any splitting, and the second
minor peak in Fig. 7 is discernibly smaller than the first. Despite these
minor variations, they don’t alter the underlying cause of the flare,
confirming that for $\mathcal{H}$, these are also instances of MDBFs.

Whirling deflecting orbits

In Fig. 8, we present the results for a hotspot following a whirling
deflecting orbit. This particular trajectory is selected because the hotspot
originates from an unstable spherical orbit and then diverges toward infinity.
As such, it can be considered a combination of spherical and deflecting
orbits. Consequently, its imaging characteristics are a combination of the
features found in spherical and deflecting orbits. Upon examining the light
intensity plot in Fig. 8, it becomes evident that the hotspot mimics a
spherical motion before it separates from the black hole. The intensity
variation closely mirrors that of the spherical orbit represented in Fig. 6.
The phase of the hotspot’s departure from the black hole aligns with the
corresponding phase in Fig. 6. Looking at the changes in the light curve, we
notice two minor flares and one significantly larger flare. When these
observations are compared with the imaging results from spherical and
deflecting orbits, we find that the origins of the first two minor flares
coincide with the results from the spherical orbit, characterized as MDBFs.
The final and most substantial flare aligns with the results from the
deflecting orbit, identified as a TDBF.

Whirling trapped orbits

Finally, we present the imaging results for whirling trapped orbits in Fig. 9.
The whirling trapped orbit we chose for our work originates from a spherical
orbit and then plunges into the black hole. Based on the characteristics of
this orbit, we can consider it as a combination of spherical orbits and
trapped orbits. The first stage is an approximate spherical orbit, and the
second stage is a trapped orbit, which can be easily seen from the light
intensity map in Fig. 9. We can see a total of three cap rings, and the last
one has a noticeable deformation due to the hotspot deviating from the
spherical orbit significantly. Eventually, it falls into the black hole
shadow, consistent with the final stage of the light intensity map in Fig. 4.
From the light curve in Fig. 9, we can see three significant flares,
corresponding to the three cap rings generated by the approximate spherical
orbit. Therefore, the cause of these flares is consistent with the results
from the spherical orbit, which are MDBFs. The last small flare matches the
flare in Fig. 4, so it is a LDBF.

Figure 10: The results at $\theta_{o}=80^{\circ}$ are presented, with the
following order from top to bottom: plunging orbits $\mathcal{P}$, deflecting
orbits $\mathcal{D}$, trapped orbits $\mathcal{T}$, bounded orbits
$\mathcal{B}$, spherical orbits $\mathcal{S}$, homoclinic orbits
$\mathcal{H}$, whirling deflecting orbits $\mathcal{WD}$, and whirling trapped
orbits $\mathcal{WT}$. From left to right, they respectively represent the
image of the hotspot, light intensity map, centroid motion, and light curve.

###  4.3 Results at $\theta_{o}=80^{\circ}$

In this subsection, we also present the results for plunging orbits
$\mathcal{P}$, deflecting orbits $\mathcal{D}$, trapped orbits $\mathcal{T}$,
bounded orbits $\mathcal{B}$, spherical orbits $\mathcal{S}$, homoclinic
orbits $\mathcal{H}$, whirling deflecting orbits $\mathcal{WD}$, and whirling
trapped orbits $\mathcal{WT}$ at $\theta_{o}=80^{\circ}$ in Fig. 10.

We will not elaborate on the details of the image in Fig. 10. By comparing
Fig. 10 with the previous figures from Fig. 2 through to Fig. 9, we can see
that the main features of the light intensity and flares of the hotspot image
at an observation angle of $\theta_{o}=25^{\circ}$ still exist at an
observation angle of $\theta_{o}=80^{\circ}$. This suggests that the
characteristics of the hotspot image and flares we summarized earlier are not
sensitive to the observation angle and can be observed at different angles.

###  4.4 Einstein ring

Figure 11: The Einstein ring in the spherical case is shown. On the left is
the image of the hotspot, in the middle is the light intensity map, and on the
right is the plot of the centroid motion.

Interestingly, during our investigation of the imaging of a hotspot moving in
a spherical orbit, we discover that at specific instances, the hotspot aligns
directly opposite the observer on the line connecting with the black hole,
resulting in the appearance of an Einstein ring, as depicted in Fig. 11.
However, as this situation is not the primary focus of our study, we will not
delve into the details of this hotspot imaging, but merely present it here for
reference.

Type | $\mathcal{P}$ | $\mathcal{D}$ | $\mathcal{T}$ | $\mathcal{B}$ | $\mathcal{S}$ | $\mathcal{H}$ | $\mathcal{W}\mathcal{D}$ | $\mathcal{W}\mathcal{T}$
---|---|---|---|---|---|---|---|---
Flare number | $1$ | $1$ | $1$ | $2^{\ast}$ | $2^{\ast}$ | $2^{\ast}$ | $3^{\ast}$ | $4^{\ast}$
Causes of flare | LDBF | TDBF | LDBF | TDBF | MDBFs | MDBFs | TDBF | LDBF
MDBFs | MDBFs
Table 3: The number of flares and their causes for various types of orbits,
including plunging orbits $\mathcal{P}$, deflecting orbits $\mathcal{D}$,
trapped orbits $\mathcal{T}$, bounded orbits $\mathcal{B}$, spherical orbits
$\mathcal{S}$, homoclinic orbits $\mathcal{H}$, whirling deflecting orbits
$\mathcal{WD}$, and whirling trapped orbits $\mathcal{WT}$ are discussed here.
The ‘$\ast$’ denote that these numbers represent the number of flares
occurring within a particular period of interest in our study, rather than the
total number of flares throughout the entire history of the hotspot.

##  5 Summary

In our research, we explored the imaging of hotspots surrounding Kerr black
holes. The hotspot, characterized as a luminous sphere with a radius of
$b=0.25$, emits light uniformly and isotropically, independent of frequency.
We incorporated the hotspot’s movement along the geodesics outside a Kerr
black hole into our analysis. Building on the conclusions of [42], we
identified eight types of timelike geodesic orbits outside the Kerr black
hole: plunging orbits $\mathcal{P}$, deflecting orbits $\mathcal{D}$, trapped
orbits $\mathcal{T}$, bounded orbits $\mathcal{B}$, spherical orbits
$\mathcal{S}$, homoclinic orbits $\mathcal{H}$, whirling deflecting orbits
$\mathcal{WD}$, and whirling trapped orbits $\mathcal{WT}$. Importantly, our
study extended beyond the equatorial plane. Using numerical backward ray
tracing, we generated images of various orbital types. By calculating the
fractional number, we differentiated between primary, secondary, and higher-
order images and analyzed their distinct properties. We also tracked the
temporal flux variation received by each pixel on the observer’s image plane,
allowing us to determine and analyze the motion of the centroid position
across different orbits. Specifically, we computed the light curve of the
hotspot image, observed flaring phenomena, and performed a detailed
examination of these events.

Specifically, we observed flare phenomena across all orbit types. We
categorized flares into three types based on their origins: LDBFs, TDBFs, and
MDBFs. The potential for each orbit to produce these flare types is summarized
in Table 3. Note that the numbers marked with ‘$\ast$’ represent the count of
flares within a specific period of interest in our study, not the total flare
occurrences throughout the entire history of the hotspot. It’s worth noting
that for these orbit types, the majority of the motion characteristics remain
consistent throughout the entire history of the hotspot. For bounded orbits
$\mathcal{B}$, the hotspot oscillates between two turning points along the
radial direction; thus, analyzing one motion cycle suffices. For spherical
orbits $\mathcal{S}$, the hotspot continuously follows a spherical path, but
only a single complete cycle in the polar coordinate $\theta$ needs to be
analyzed. Similarly, for homoclinic orbits $\mathcal{H}$, whirling deflecting
orbits $\mathcal{WD}$, and whirling trapped orbits $\mathcal{WT}$, they all
involve numerous near-spherical movements. Therefore, a clear analysis of a
single cycle of near-spherical motion suffices.

We noted that LDBFs occur in orbits descending into the black hole, TDBFs in
orbits with turning points near the black hole, and MDBFs in orbits traversing
near the sphere. Furthermore, by comparing imaging results at $25^{\circ}$ and
$80^{\circ}$, we determined that the flare phenomena we observed and the
categorizations we established are not significantly influenced by the
observation angle. Our findings provide further insights into the causes of
flare formation and suggest that these common flare signals are likely
detectable in astronomical observations.

We conclude our paper by outlining several future perspectives. First, our
hotspot model has not yet accounted for the emission rates of the hotspot
across various frequency bands—an aspect that necessitates further
investigation. Second, considering the complex astronomical environment around
black holes, the hotspot’s trajectory might deviate from geodesics. Hence, it
could be beneficial to delve deeper into the imaging characteristics of non-
geodesic motion. Third, the existence of accretion disks and jets around the
black hole could potentially influence the flare signals produced by the
hotspot, a factor that deserves additional attention in future research.
Lastly, it’s important to note that the flares observed by the GRAVITY
collaboration also carry polarization information. As such, the theoretical
exploration of this polarization information is of substantial importance and
warrants further study.

## Acknowledgments

We thank Yehui Hou for helpful discussions. The work is partly supported by
NSFC Grant No. 12275004, 12205013 and 11873044. MG is also endorsed by ”the
Fundamental Research Funds for the Central Universities” with Grant No.
2021NTST13.

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