# Orbits of particles with magnetic dipole moment around magnetized
Schwarzschild black holes: Applications to S2 star orbit

Uktamjon Uktamov<EMAIL_ADDRESS>Institute of Fundamental and
Applied Research, National Research University TIIAME, Kori Niyoziy 39,
Tashkent 100000, Uzbekistan  Mohsen Fathi*  —<EMAIL_ADDRESS>Vicerrectoría Académica, Universidad Central de Chile, Toesca 1783, Santiago
8320000, Chile  Facultad de Ingeniería y Arquitectura, Universidad Central de
Chile, Av. Santa Isabel 1186, 8330601, Santiago, Chile  Javlon Rayimbaev
<EMAIL_ADDRESS>School of Mathematics and Natural Sciences, New Uzbekistan
University, Movarounnahr St. 1, Tashkent 100007, Uzbekistan;  University of
Tashkent for Applied Sciences, Str. Gavhar 1, Tashkent 100149, Uzbekistan
National University of Uzbekistan, Tashkent 100174, Uzbekistan  Ahmadjon
Abdujabbarov<EMAIL_ADDRESS>Ulugh Beg Astronomical Institute, Astronomy
St 33, Tashkent 100052, Uzbekistan  Tashkent State Technical University,
Tashkent 100095, Uzbekistan

###### Abstract

This study provides a comprehensive analytical investigation of the bound and
unbound motion of magnetized particles orbiting a Schwarzschild black hole
immersed in an external asymptotically uniform magnetic field, which includes
all conceivable types of bounded and unbounded orbits. In particular, for
planetary orbits, we perform a comparative analysis of our findings with the
observed position of the S2 star carrying magnetic dipole moment around
Sagittarius A* (Sgr A*). We found maximum and minimum values for the parameter
of magnetic interaction between the magnetic dipole of the star and the
external magnetic field, as well as the energy and angular momentum of the S2
star. As a result, we obtain estimations of the magnetic dipole of the star in
order of $10^{6}\rm\ G\cdot cm^{3}$. Additionally, we explore deflecting
trajectories akin to gravitational Rutherford scattering. In obtaining the
solutions for the orbital equations, we articulate the elliptic integrals and
Jacobi elliptic functions, and our study is augmented by illustrative figures
and simulations.

###### pacs:

04.50.-h, 04.40.Dg, 97.60.Gb

##  I Introduction

The investigation of free-falling objects in gravitational fields links
geometry and gravity through the general theory of relativity, where the
curvature of spacetime plays the principal role. This fundamental
understanding gained popularity early once it was recognized that planets and
light follow geodesics. Outstanding discoveries, right after the proposition
of the Schwarzschild solution Schwarzschild (1916), was the verification of
light deflection during the 1919 solar eclipse expedition Dyson et al. (1920).
The precise evaluation of the anomalous precession in the perihelion of
Mercury Clemence (1947), which constitute the primary standard tests of
general relativity. However, the complex nature of the partial differential
equations governing particle dynamics in curved spacetimes necessitates
simplified or numerical solutions to compare theoretical results with
observational tests.

Analytical expressions have several advantages since they serve as benchmarks
for numerical methods, and hence, they facilitate comprehensive parameter
space exploration for predicting astrophysical observables. Efforts to find
exact analytical solutions for geodesic equations of particles have
intensified since Hagihara’s studies in 1931 Hagihara (1930), followed by
works by Darwin, Mielnik, and Plebański Darwin (1959, 1961); Mielnik and
Plebański (1962). These methods, which are based on the theories of elliptic
functions and modular forms, were studied by nominated nineteenth-century
mathematicians such as Jacobi Jacobi (2013), Abel Abel , Riemann Riemann
(1857, 1866), and Weierstrass Weierstrass (1854) (see also Ref. Baker (1995)
for a complete textbook review on these discoveries). Particularly noteworthy
is the application of modular forms in solving (hyper-)elliptic integrals
involved in particle trajectory studies, garnering significant attention in
recent decades. These methods, rooted in the theories of elliptic functions
and modular forms, have been the focus of numerous investigations analyzing
trajectory curves in black hole spacetimes inferred from general relativity
and its extensions (see, for example, Refs. Kraniotis and Whitehouse (2002,
2003); Kraniotis (2004, 2005); Cruz et al. (2005); Kraniotis (2007); Hackmann
and Lämmerzahl (2008a, b); Hackmann et al. (2009, 2010); Olivares et al.
(2011); Kraniotis (2011); Cruz et al. (2013); Villanueva et al. (2013);
Kraniotis (2014); Soroushfar et al. (2015); Villanueva and Olivares (2015);
Soroushfar et al. (2016); Hoseini et al. (2016, 2017); Fathi et al. (2020a, b,
2021a); González et al. (2021); Fathi et al. (2021b); Kraniotis (2021); Fathi
et al. (2022); Soroushfar and Afrooz (2022); Battista and Esposito (2022);
Fathi et al. (2023); Fathi (2023)).

Building upon this scientific foundation, contemporary research continues to
explore the properties and dynamics of particles falling in gravitational
fields in diverse astrophysical contexts. bound and unbound particle orbits,
in particular, represent fundamental aspects of motion that offer valuable
insights into the structure and evolution of the universe. Bound orbits,
characterized by closed trajectories around central masses, are found
extensively in celestial systems, ranging from planetary orbits in the solar
system to the motion of stars in galaxies Perek (1962); Barbieri and Gratton
(2002). These orbits contribute to astrophysical objects’ stability and long-
term dynamics and provide essential constraints on theoretical models of
gravitational dynamics and cosmological evolution.

On the other hand, unbound particle orbits extend indefinitely through space,
either escaping to infinity (deflecting trajectories) or falling into
gravitational singularities (plunge or captured orbits). These orbits play a
critical role in understanding cosmic phenomena such as the formation and
evolution of galaxies and the dynamics of stellar populations within galactic
clusters (see, for example, Ref. Zhong et al. (2023) and the book Binney and
Tremaine (2011)). They are also useful for modeling particle orbits within
dark matter halos Behroozi et al. (2013). Unbound orbits provide valuable
information about the escape velocity of celestial bodies, shedding light on
the mechanisms underlying stellar ejection and galactic interactions.
Moreover, the study of unbound orbits offers insights into the gravitational
interactions between celestial objects and their surrounding environments,
contributing to our understanding of the large-scale structure and evolution
of the universe Binney and Tremaine (2011).

The study of trajectories of magnetized particles around magnetized
astrophysical objects, particularly black holes, is of great importance in
understanding the nature of astrophysical phenomena. Magnetized black holes,
characterized by intense magnetic fields generated by rapidly rotating neutron
stars or stellar-mass black holes, exhibit intricate interactions between
gravitational and electromagnetic forces that give rise to a diverse range of
astrophysical phenomena Aliev and Gal’tsov (1989); Li (2002); Meier (2012);
Kruglov (2017); Ghosh et al. (2021); Kopáček and Karas (2023); Karas and
Stuchlík (2023); Khan et al. (2024). Charged particles, such as electrons and
protons, spiraling along magnetic field lines near these objects undergo
complex dynamics, including acceleration, deflection, and emission of
radiation. These processes play a crucial role in shaping the observational
properties of various astrophysical sources, including pulsars, gamma-ray
bursts, and active galactic nuclei Chang et al. (2022); Mészáros (2019);
Novikov and Stern (1986). By studying the trajectories of magnetized
particles, researchers can gain insights into the mechanisms driving particle
acceleration, the formation of relativistic jets, and the emission of high-
energy radiation observed across the electromagnetic spectrum. Furthermore,
understanding the interplay between gravitational and electromagnetic forces
near magnetized astrophysical objects offers valuable insights into
fundamental aspects of plasma physics, high-energy astrophysics, and general
relativity Nishikawa et al. (2005); Mościbrodzka et al. (2016); Mizuno (2022).

Given the importance of this subject, our study aims to investigate the
dynamics of magnetized particles affected by the magnetic field of an
astrophysical object, in this case, a magnetized Schwarzschild black hole. In
this study, we analyze the dynamics and motion of particles possessing
magnetic dipole moments analytically as they move in the exterior geometry of
a Schwarzschild black hole with an associated magnetic field. We utilize
elliptic integrals to express exact analytical solutions to the equations of
motion governing the behavior of these particles. Particularly, the deflecting
trajectories are considered akin to a type of gravitational Rutherford
scattering, which is also treated analytically.

To accomplish our objective, we structure this paper as follows: In Section
II, we introduce the spacetime under study and develop a Lagrangian formalism,
leading to the equations of motion for the test particles. In Section III, we
initiate our main study by providing a general classification of the possible
orbits based on the radial profile of the effective potential experienced by
the particles. In this section, we provide a precise derivation of the
planetary orbits with different eccentricities, also assessed by observational
tests regarding the S-star orbits at the center of the Milky Way. In Section
IV, we continue our study by investigating the unbound orbits, consisting of
deflecting trajectories. These trajectories are either deflected beyond the
black hole or inexorably fall into the event horizon. The former is termed
gravitational Rutherford scattering in Refs. Villanueva and Olivares (2015);
Fathi et al. (2021a). To complete this investigation, in Section V, we present
a fully analytical study of the purely radial trajectories. We conclude in
Section VI. Throughout the paper, we work in the geometrized system of units
where $G=c=1$, we adopt the sign convention $(-++\,+)$, and wherever primes
appear, they denote differentiations with respect to the radial coordinate.

##  II Motion of magnetized particles around a Schwarzschild black hole
immersed in magnetic fields

In this section, we derive the equations of motion governing magnetized test
particles orbiting a Schwarzschild black hole in the presence of an externally
applied asymptotically uniform magnetic field. The line element describing the
exterior spacetime of a Schwarzschild black hole with mass $M$ is given by

$ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\Omega^{2},$ (1)

in the usual Schwarzschild coordinates $x^{\alpha}=(t,r,\theta,\phi)$, with
$d\Omega^{2}=d\theta^{2}+\sin^{2}\theta d\phi^{2}$, where the lapse function
is given by

$f(r)=1-\frac{2M}{r}.$ (2)

###  II.1 Magnetization of Schwarzschild-like black holes

Here, we assume that the black hole is surrounded by an external,
asymptotically uniform magnetic field with a constant value of $B$ at
infinity. According to Wald’s approach, the four-potential, being solutions of
Maxwell’s equations, can be expressed as $A^{\mu}=\frac{B}{2}\xi^{\mu}_{\phi}$
Wald (1984), where $\xi_{\phi}^{\mu}=(0,0,0,1)$ represents a space-like
Killing vector. Consequently, we can immediately obtain the non-vanishing
components of the electromagnetic tensor as

$\displaystyle F_{r\phi}=Br\sin^{2}{\theta},$ (3a) $\displaystyle
F_{\theta\phi}=\frac{B}{2}r^{2}\sin{2\theta}.$ (3b)

The components of the magnetic field around the black hole, as measured by
zero angular momentum observers (ZAMOs), can be calculated as

$\displaystyle B^{i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\eta^{i\beta\sigma\mu}F_{\beta\sigma}w_{\mu},$ (4)

where $i=(r,\theta,\phi)$, $\bm{w}$ represents the four-velocity of the ZAMOs
with $w^{\mu}=1/\sqrt{f(r)}(1,0,0,0)$, and $\eta_{\alpha\beta\sigma\gamma}$
denotes the pseudo-tensorial form of the Levi-Civita symbol
$\epsilon_{\alpha\beta\sigma\gamma}$ with the following relations:

$\displaystyle\eta_{\alpha\beta\sigma\gamma}=\sqrt{-g}\,\epsilon_{\alpha\beta\sigma\gamma},\quad\eta^{\alpha\beta\sigma\gamma}=-\frac{1}{\sqrt{-g}}\epsilon^{\alpha\beta\sigma\gamma},$
(5)

where $g={{\rm{det}}|g_{\mu\nu}|}=-r^{4}\sin^{2}\theta$, corresponding to the
spacetime metric (1). Hence, one can obtain expressions for the non-zero
orthonormal components of the magnetic field using the tetrads carried by the
ZAMOs Rezzolla et al. (2001). Furthermore, the non-zero orthonormal components
of the magnetic field measured by the ZAMOs are calculated as

$\displaystyle B^{\hat{\theta}}=\sqrt{f(r)}\,B\sin{\theta},$ (6a)
$\displaystyle B^{\hat{r}}=B\cos{\theta}.$ (6b)

###  II.2 Magnetized particle dynamics

The dynamics of polarized particles within the spacetime described by the line
element (1) can be expressed using the Lagrangian Preti (2004)

$\displaystyle\mathscr{L}=\frac{1}{2}(m+U)g_{\mu\nu}u^{\mu}u^{\nu}-\frac{1}{2}\kappa
U,$ (7)

where $\kappa=1$ and the magnetic interaction term is represented by
$U=D^{\mu\nu}F_{\mu\nu}$. The calculation of the magnetic interaction term has
been undertaken by various authors (see, for example, Ref. de Felice and Sorge
(2003)). Specifically, it is given by $U=\mu_{\hat{\alpha}}B^{\hat{\alpha}}$.
Furthermore, suppose we limit the motion of the magnetized particles to the
equatorial plane and assume that the magnetic dipole of the particles is
perpendicular to this plane. In that case, we can express the components of
the magnetic moment four-vectors as $\mu^{\hat{\alpha}}=(0,0,\mu,0)$.
Consequently, we can derive the conjugate momenta as

$\displaystyle
p_{\mu}=\frac{\partial\mathscr{L}}{\partial\dot{x}^{\mu}}=(m+U)g_{\mu\nu}u^{\nu},$
(8)

based on which, the two constants of motion

$\displaystyle-\mathcal{E}$ $\displaystyle=$
$\displaystyle\bigr{[}1+\beta\mathcal{F}(r,\theta)\bigr{]}g_{tt}\dot{t},$ (9)
$\displaystyle l$ $\displaystyle=$
$\displaystyle\bigr{[}1+\beta\mathcal{F}(r,\theta)\bigr{]}g_{\phi\phi}\dot{\phi},$
(10)

can be defined in accordance with the symmetries of the spacetime, where
$\beta=\mu B/m$, and $\mathcal{F}(r,\theta)=\sqrt{f(r)}\sin{\theta}$. Here,
$\mathcal{E}=E/m$ and $l=L/m$ represent the magnetized particle’s specific
energy and specific angular momentum, respectively. Furthermore, one can
express the Hamilton-Jacobi equation while accounting for the magnetic field
as follows:

$\displaystyle g^{\mu\nu}\frac{\partial{\cal S}}{\partial
x^{\mu}}\frac{\partial{\cal S}}{\partial
x^{\nu}}=-m^{2}\left(1-\frac{U}{m}\right)^{2}.$ (11)

with $\mathcal{S}$ denoting the Jacobi action, and motion restricted to the
equatorial plane, the action can be expressed as ${\cal S}=-Et+L\phi+{\cal
S}_{r}(r)$, and in this context, we have $m=1$. Therefore, the radial motion
can be easily described in terms of the following equation:

$\displaystyle\dot{r}^{2}=\mathcal{E}^{2}-V_{\mathrm{eff}},$ (12)

in which

$\displaystyle
V_{\mathrm{eff}}(r;l)=f(r)\left(\left[1-{\beta\mathcal{F}(r,\pi/2)}\right]^{2}+\frac{l^{2}}{r^{2}}\right).$
(13)

is the effective potential felt by the test particles.

Figure 1: The radial profile of the effective potential is depicted for fixed
values of the magnetic coupling parameter $\beta$ (left panel), and the
specific angular momentum $l$ (middle panel). The minima ($r_{\text{min}}$) in
the effective potential corresponded to stable circular orbits, while the
maxima ($r_{\text{max}}$) correspond to unstable circular orbits.
Additionally, the turning points $r_{s}$, $r_{l}$, $r_{p}$, $r_{a}$, and the
radius of the unstable circular orbit $r_{u}$ are indicated in the right
panel.

In Fig. 1, we illustrate how the radial dependence of the effective potential
varies with the specific angular momentum (left panel) and the magnetic
interaction parameter $\beta$ (middle panel). The maximum of the effective
potential denotes instability. Furthermore, these plots indicate that
increasing the specific angular momentum $l$ of the magnetized particles and
the magnetic interaction parameter $\beta$ leads to an increase in the stable
circular orbits (at $r_{\text{min}}$) of the test particles with magnetic
dipole momentum. Conversely, an increase in the specific angular momentum of
the magnetized particle results in a decrease in the unstable circular orbits
(at $r_{\text{max}}$) of the particles with magnetic dipole moment. Moreover,
escalating the magnetic coupling parameter $\beta$ leads to an increase in the
unstable circular orbits of the magnetized particles. Additionally, Fig. 1
presents the radial profile of the effective potential with the specific
angular momentum $l_{\text{ISCO}}$ of the magnetized particles in the
Innermost Stable Circular Orbit (ISCO) for a given value of $\beta=0.1$.
Consequently, Fig. 1 indicates that for specific angular momentum values
$l>l_{\text{ISCO}}$, there are two extreme points in the effective potential,
namely $r_{\text{max}}$ corresponding to unstable circular orbits and
$r_{\text{min}}$ corresponding to stable circular orbits. For the specific
angular momentum value $l=l_{\text{ISCO}}$, there is only one extreme point
$r_{\text{min}}$, corresponding to the stable circular orbit. However, for
specific angular momentum values $l<l_{\text{ISCO}}$, there are no extreme
points in the effective potential, indicating the absence of any circular
orbits in this case. Additionally, insights into the orbits of magnetized test
particles can be obtained from the effective potential. In the right panel of
Fig. 1, we depict the turning points $r_{s}$ and $r_{l}$ corresponding to the
deflecting trajectories. Furthermore, concerning the planetary bound orbits,
we have also denoted the turning points $r_{p}$ (periapsis), the smallest
orbital separation, and $r_{a}$ (apoapsis), the largest orbital separation, of
particles possessing magnetic dipole moment orbiting a Schwarzschild Black
Hole within magnetized fields. We will later explore that test particles
establish bound orbits if their specific energy $\mathcal{E}$ is less than
$\sqrt{1-\beta}$ and unbound orbits if
$\mathcal{E}_{u}>\mathcal{E}\geq\sqrt{1-\beta}$, where
$\mathcal{E}_{u}^{2}=V_{\text{eff}}(r_{u})$ represents the energy
corresponding to the unstable circular orbits at $r_{u}=r_{\text{max}}$.

One can utilize Eqs. (9) and (12) to derive the general equation of motion in
the equatorial plane for angular trajectories. This yields the differential
equation

$\left(\frac{dr}{d\phi}\right)^{2}=\frac{r^{4}\left(1+\beta\sqrt{f(r)}\right)^{2}}{l^{2}}\\\
\times\left[\mathcal{E}^{2}-f(r)\left(1+\frac{l^{2}}{r^{2}}-\beta\sqrt{f(r)}\right)\right].$
(14)

In the forthcoming sections of the paper, we will employ this equation to
analytically analyse the bound and unbound orbits of magnetized test particles
in the spacetime under consideration.

##  III The bound orbits

Following the same procedure as in Ref. Lim and Yeo (2024) to determine the
($l,\mathcal{E}$) parameter space for circular orbits, in this section we will
investigate how this space varies with the magnetic coupling constant $\beta$.
Subsequently, by employing the conditions $\mathcal{E}^{2}=V_{\text{eff}}$ and
$V_{\text{eff}}^{\prime}=0$, we can ascertain the specific energy
$\mathcal{E}$ and specific angular momentum $l$ of the magnetized particles as

$\displaystyle l^{2}$ $\displaystyle=$
$\displaystyle\frac{Mr_{0}^{2}\left[\beta^{2}f(r_{0})+\left(2-3\beta\sqrt{f(r_{0})}\right)\right]}{2(r_{0}-3M)},$
(15) $\displaystyle\mathcal{E}^{2}$ $\displaystyle=$ $\displaystyle
f(r_{0})\left(1-\frac{1}{2}\beta\sqrt{f(r_{0})}\right)^{2}$ (16)
$\displaystyle+\frac{M\left[2-\beta\sqrt{f(r_{0})}\left(3-\beta\sqrt{f(r_{0})}\right)\right]}{2(r_{0}-3M)},$

corresponding to circular orbits at the radius $r=r_{0}$. It is well-known
that to identify stable circular orbits, one should examine the second
derivative of the effective potential, namely
$V_{\mathrm{eff}}^{\prime\prime}>0$ indicates stable circular orbits, while
$V_{\mathrm{eff}}^{\prime\prime}<0$ indicates unstable circular orbits.
However, our determination of stable and unstable circular orbits depends on
$\beta$. Therefore, in Table 1, we present the values of the critical radii of
the unstable circular orbits. It is evident that
$V_{\mathrm{eff}}^{\prime\prime}>0$ for $r_{0}>r_{\text{max}}$, corresponding
to stable orbits, and $V_{\mathrm{eff}}^{\prime\prime}<0$ for
$r_{\text{min}}<r_{0}<r_{\text{max}}$, corresponding to unstable orbits.
Surely, $r_{0}=r_{\text{min}}$ corresponds to the ISCO. Therefore, from Table
1, one can easily conclude that increasing the magnetic coupling constant
$\beta$ leads to an increase in the value of the ISCO radius. To provide more
precise insight, in Fig. 2, we

Figure 2:  Graphical representation of the ISCO radius $r_{\text{ISCO}}$, ISCO specific angular momentum $l_{\text{ISCO}}$, and the ISCO specific energy $\mathcal{E}_{\text{ISCO}}$ along the magnetic coupling parameter $\beta$.  $\beta$ | $r_{\min}$ | $r_{\max}$
---|---|---
0.0 | $3M$ | $6M$
0.1 | $3M$ | $6.09984M$
0.3 | $3M$ | $6.36414M$
0.5 | $3M$ | $6.78254M$
Table 1: The radii of stable and unstable circular orbits for different values
of the magnetic coupling constant $\beta$.

illustrate how the ISCO radius $r_{\text{ISCO}}$, the specific energy
$\mathcal{E}_{\text{ISCO}}$, and the specific angular momentum
$l_{\text{ISCO}}$ of the magnetized particle in the ISCO depend on the
magnetic coupling constant $\beta$. From Fig. 2, one can readily conclude that
increasing the magnetic coupling parameter $\beta$ results in a decrease in
the specific energy $\mathcal{E}_{\text{ISCO}}$ and the specific angular
momentum $l_{\text{ISCO}}$ of the magnetized particles in the ISCO. In
contrast, increasing the magnetic interaction term $\beta$ leads to an
increase in the value of the ISCO radius $r_{\text{ISCO}}$ of the particles
with the magnetic dipole moment. Additionally, in Fig. 3, we have plotted the
domain (for the quantities $\mathcal{E}$, $l/M$, and the line representing the
non-relativistic limit, i.e. $l/M\to\infty$), within which, magnetized
particles can possess circular orbits.

Figure 3: Values of the specific energy $\mathcal{E}$ and specific angular
momentum $l$ for circular orbits plotted for fixed values of the magnetic
coupling parameter $\beta$. Solid lines represent stable circular orbits,
while dashed lines represent the radius of unstable circular orbits.

###  III.1 Analytical solution for the angular trajectories

We now endeavor to utilize the classification scheme developed by Levin and
Perez-Giz (refer to Ref. Levin and Perez-Giz (2008)) to determine potential
values for specific energy $\mathcal{E}$ and specific angular momentum $l$ for
a given orbit. To begin this process, we introduce a new variable, $u=1/r$, in
Eq. (14), which yields

$\frac{du}{d\phi}=\pm\frac{\left(1+\beta\sqrt{1-2Mu}\,\right)}{l}\\\
\times\sqrt{\mathcal{E}^{2}-(1-2Mu)\left(1+l^{2}u^{2}-\beta\sqrt{1-2Mu}\right)}.$
(17)

Subsequently, employing the approximation $\beta\sqrt{1-2Mu}\approx\beta-\beta
Mu+\beta\mathcal{O}\left(u^{2}\right)$ and neglecting the $\beta^{2}u^{4}$
term, we arrive at the expression

$\left(\frac{du}{d\phi}\right)^{2}=(1+\beta)\left(au^{3}-bu^{2}+cu+d\right),$
(18)

where

$\displaystyle a=2M\left[(2\beta+1)-\frac{2\beta^{2}M^{2}}{l^{2}}\right],$
(19a) $\displaystyle b=(\beta+1)+\frac{2\beta(1-4\beta)M^{2}}{l^{2}},$ (19b)
$\displaystyle
c=\frac{M\left[2-\beta\left(5\beta+2\mathcal{E}^{2}-1\right)\right]}{l^{2}},$
(19c) $\displaystyle
d=\frac{(\beta+1)\left(\beta+\mathcal{E}^{2}-1\right)}{l^{2}}.$ (19d)

From Eq. (18), it is evident that for $\mathcal{E}^{2}<1-\beta$, bound orbits
occur, which is the primary focus of this section. We assume
$u_{1}<u_{2}<u_{3}$, where $\\{u_{1},u_{2},u_{3}\\}$ represent the real roots
of Eq. (18). In the scenario where all these roots are positive, in accordance
with the right panel of Fig. 1, the radii of the particle orbits vary between
$r_{a}={1}/{u_{1}}$ and $r_{p}={1}/{u_{2}}$ (that is, $u_{1}\leq u\leq
u_{2}$). This situation corresponds to planetary bound orbits (see, for
example, Ref. Chandrasekhar (1998)), wherein the radial position of the test
particles oscillates between the apoapsis $u_{1}$ and the periapsis $u_{2}$.
If one of these roots is negative (for instance, $u_{2}<0$ and
$0<u_{1}<u_{3}$), the test particles will either deflect to infinity at the
turning point $r_{l}=1/u_{1}$ or plunge into the black hole at the turning
point $r_{s}=1/u_{3}$, as denoted in the right panel of Fig. 1. The former
type of orbits is termed orbits of the first kind (OFK), while the latter is
designated as orbits of the second kind (OSK). These types of orbits are
discussed in the next section.

Now, to obtain a general analytical solution for the angular motion, we assume
that the magnetized particles commence their motion from $u=u_{1}$ and
$\phi=0$. Then, Eq. (18) can be expressed as

$\displaystyle\phi=\frac{1}{\sqrt{a(1+\beta)}}\int_{u_{1}}^{u}\frac{du}{\sqrt{(u-u_{1})(u-u_{2})(u-u_{3})}},$
(20)

which yields the solution

$\displaystyle\phi(u)=\frac{2F\Big{(}\arcsin{\sqrt{\frac{u-u_{1}}{u_{2}-u_{1}}}},k_{0}\Big{)}}{\sqrt{a(1+\beta)(u_{3}-u_{1})}},$
(21)

where

$k_{0}=\sqrt{\frac{u_{2}-u_{1}}{u_{3}-u_{1}}}.$ (22)

In Eq. (21), $F(x,k)$ is the incomplete elliptic integral of the first kind,
with argument $x$ and modulus $k$ Byrd and Friedman (1971). By performing the
inversion of the above solution and restoring the change of variable, we
arrive at the analytical expression

$\displaystyle
r(\phi)=\frac{1}{u_{1}+(u_{2}-u_{1})\operatorname{\mathrm{sn}}^{2}(\theta_{0},k_{0})},$
(23)

where
$\theta_{0}\equiv\theta_{0}(\phi)=\sqrt{a(1+\beta)(u_{3}-u_{1})}{\phi}/{2}$.
In the above relation, $\operatorname{\mathrm{sn}}(\theta,k)^{}$111The well-
known incomplete elliptic integral of the first kind is given by the general
form Byrd and Friedman (1971)
$\displaystyle\theta=F(\phi,k)=\int_{0}^{\phi}\frac{dt}{\sqrt{1-k^{2}\sin^{2}{t}}},$
where $\phi=F^{-1}(\theta,k)=\mathrm{am}(\theta,k)$ is the Jacobi amplitude.
This way, the Jacobi elliptic sine function is defined as
$\displaystyle\sin{\phi}=\sin{\big{(}\mathrm{am}(\theta,k)\big{)}=\operatorname{\mathrm{sn}}(\theta,k)}.$
denotes the Jacobi elliptic sine function, with argument $\theta$ and modulus
$k$.

###  III.2 Periodic orbits for fixed topological characteristics and the
impact of eccentricities

The roots of the characteristic polynomial (18) can be expressed in the
following form Chandrasekhar (1998):

$\displaystyle u_{1}=\frac{1-e}{\lambda},$ (24a) $\displaystyle
u_{2}=\frac{1+e}{\lambda},$ (24b) $\displaystyle
u_{3}=\frac{b}{(1+\beta)a}-\frac{2}{\lambda},$ (24c)

where $e$ and $\lambda$ denote eccentricity and the latus rectum,
respectively. For bound orbits, the condition $0\leq e<1$ applies. Then we
will have

$\displaystyle u_{3}-u_{1}=\frac{b}{(1+\beta)a}-\frac{3-e}{\lambda},$ (25a)
$\displaystyle u_{2}-u_{1}=\frac{2e}{\lambda}.$ (25b)

Accordingly, the exact solution for bound orbits can be written as

$\displaystyle
r(\phi)=\frac{\lambda}{1-e+2e\operatorname{\mathrm{sn}}^{2}(\theta_{1},k_{1})},$
(26)

in which

$\displaystyle\theta_{1}\equiv\theta_{1}(\phi)=\sqrt{b-\frac{a(1+\beta)(3-e)}{\lambda}}\,\frac{\phi}{2},$
(27a) $\displaystyle
k_{1}=\sqrt{\frac{2ea(1+\beta)}{\lambda-a(1+\beta)(3-e)}}.$ (27b)

Considering the oscillation of the radial coordinate between $u_{1}$ and
$u_{2}$ during one period of planetary orbits, one can determine the
associated change in the azimuth angle using Eq. (21). This calculation
results in,

$\displaystyle\Delta\phi_{u}=2\phi(u_{2})$ $\displaystyle=$
$\displaystyle\frac{4K(k_{0})}{\sqrt{a(1+\beta)(u_{3}-u_{1})}}$ (28)
$\displaystyle=$
$\displaystyle\frac{4K(k_{1})}{\sqrt{b-\frac{a(1+\beta)(3-e)}{\lambda}}},$

where $K(k)$ is the complete elliptic integral of the first kind Byrd and
Friedman (1971).

Levin and Perez-Giz Levin and Perez-Giz (2008) proposed a taxonomy scheme in
the form

$q+1=\omega+\frac{v}{z}+1=\frac{\Delta\phi_{u}}{2\pi},$ (29)

where $\omega$, $v$, and $z$ represent the whirl, vertex, and zoom numbers,
respectively. Subsequently, Equation (28) can be reformulated as

$\displaystyle
q+1=\frac{2K(k_{1})}{\pi\sqrt{b-\frac{a(1+\beta)(3-e)}{\lambda}}}.$ (30)

Furthermore, for each $e$ and $\lambda$, the corresponding specific angular
momentum and specific energy can be found using Eqs. (18) and (24) as

$\displaystyle\mathcal{E}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{\lambda\Psi}\Bigr{\\{}(1+\beta)\Big{[}2(1-\beta)\lambda^{3}$
(31)
$\displaystyle-4\beta^{2}\left(1-e^{2}\right)^{2}M^{3}\Big{]}+\Theta+\Phi\Bigr{\\}},$
$\displaystyle l^{2}$ $\displaystyle=$
$\displaystyle\frac{(\beta+1)M\Xi}{\Psi},$ (32)

where the coefficients involved are provided in appendix A.

Now, we can illustrate the periodic orbits of the magnetized particles for
selected values of the periodic orbit parameters $(z,\omega,v)$ and
eccentricity $e$. To achieve this, we numerically solve Eq. (30) to obtain
$\lambda$ and then determine the corresponding specific energy and specific
angular momentum using Eq. (32). In our numerical calculation of $\lambda$, we
have disregarded the term $\sim{\beta^{2}}/{l^{2}}$ in $a$ and, $b$ as it is
considerably smaller than $\sim\beta$. Subsequently, Tables (2), (3), and (4)
present numerical values of the latus rectum $\lambda$, along with the
corresponding specific energy $\mathcal{E}$ and specific angular momentum $l$,
for the periodic orbits $(4,0,1)$, $(3,0,1)$, and $(2,0,1)$, respectively.
These tables reveal that the magnetic interaction term $\beta$ leads to an
enlargement of the orbits of magnetized particles. This expansion can be
attributed to the repulsive interaction between magnetized particles and an
external uniform magnetic field $B$. Subsequently, plotting orbits for these
selected values is straightforward, as demonstrated in Cartesian coordinates
in Fig. 4. The diagrams in this figure illustrate how the magnetic interaction
term $\beta$ contributes to the expansion of periodic orbits for test
particles with dipole magnetic moment.

$\beta$ | $e$ | $\lambda$ | $l$ | $\mathcal{E}$
---|---|---|---|---
0.0 | 0.5 | $16.7637M$ | $4.56019M$ | 0.978644
0.0 | 0.8 | $16.9124M$ | $4.64227M$ | 0.989591
0.1 | 0.5 | $17.8396M$ | $4.39676M$ | 0.930824
0.1 | 0.8 | $18.2964M$ | $4.50733M$ | 0.940086
Table 2:  The specific energy $\mathcal{E}$ and specific angular momentum $l$ values corresponding to selected eccentricity $e$ and the periodic orbit $(4,0,1)$, with fixed values of the magnetic coupling parameter $\beta$. $\beta$ | $e$ | $\lambda$ | $l$ | $\mathcal{E}$
---|---|---|---|---
0.0 | 0.5 | $13.8243M$ | $4.25126M$ | 0.974472
0.0 | 0.8 | $13.9918M$ | $4.34876M$ | 0.987505
0.1 | 0.5 | $15.3592M$ | $4.17356M$ | 0.928205
0.1 | 0.8 | $15.8208M$ | $4.29611M$ | 0.938799
Table 3: The same table as in Table 2 for the periodic orbit $(3,0,1)$. $\beta$ | $e$ | $\lambda$ | $l$ | $\mathcal{E}$
---|---|---|---|---
0.0 | 0.5 | $10.9382M$ | $3.94488M$ | 0.968567
0.0 | 0.8 | $11.145M$ | $4.06822M$ | 0.984504
0.1 | 0.5 | $12.7528M$ | $3.94033M$ | 0.924563
0.1 | 0.8 | $13.2453M$ | $4.0825M$ | 0.936994
Table 4: The same table as in Table 2 for the periodic orbit $(2,0,1)$.

Figure 4: Orbits corresponding to various eccentricities: blue orbits
represent $\beta=0.1$, while black orbits represent $\beta=0.0$. The black
hole is assumed to be positioned at the origin.

###  III.3 Estimating the magnetic coupling parameter $\beta$ using observed
data from the S2 star

Studying the S2 star’s orbits provides a unique opportunity to test General
Relativity near Sgr A*. It offers insights into stellar dynamics and
gravitational forces in the galactic center. The proximity of S2 to the black
hole Sgr A* allows exploration of supermassive black hole properties, such as
mass and spin. Tracking S2’s orbit over time informs our understanding of
black hole characteristics and the surrounding environment. Hence, observation
data of the motion of the S2 star near Sgr A* offer a pathway to evaluate
various physical theories (see, for example, Lacroix (2018)). In this
subsection, we aim to determine the potential values of the magnetic coupling
parameter $\beta$ based on the observation data provided in Ref. Do et al.
(2019). To achieve this, we utilize Eq. (30) to compute numerical values of
$\beta$. Given numerical values of eccentricity $e$ and latus rectum
$\lambda$, we can derive numerical values of $\beta$. For a simple elliptic
trajectory, the orbit of the S2 star should be $(1,0,0)$. Thus, Eq. (30) can
be represented as

$\displaystyle\frac{\pi}{2}=\frac{K(k_{1})}{\sqrt{b-\frac{a(1+\beta)(3-e)}{\lambda}}}.$
(33)

Figure 5: Observed trajectory of the star S2 orbiting SgrA*.

We have depicted the periodic orbit corresponding to the observed trajectory
of the S2 star orbiting Sgr A* in Fig. 5. In this coordinate system, the $x$
and $y$ axes correspond to offsets in the right ascension
($\Delta\mathrm{RA}/M$) and declination ($\Delta\mathrm{{Dec}}/{M}$) of Sgr
A*. Additionally, the origin of the coordinate system coincides with the
fitted center of mass of Sgr A*.

$\beta_{\max}$ | $\beta_{\min}$ | $l_{\max}$ | $l_{\min}$ | $\mathcal{E}_{\max}$ | $\mathcal{E}_{\min}$
---|---|---|---|---|---
0.419751 | 0.14249 | $5.02989M$ | $7.4095M$ | 0.871573 | 0.923948
Table 5: The magnetic coupling parameter $\beta$ values yield periodic orbits
for the observed trajectory of the S2 star orbiting Sgr A*, alongside the
respective specific energy and specific angular momentum of the S2 star.

Using Eq. (33) together with Eqs. (31) and (32), we computed the possible
values of the magnetic coupling parameter $\beta_{\text{min,max}}$ and,
correspondingly, the specific energy $\mathcal{E}_{\text{min,max}}$ and the
specific angular momentum $l_{\text{min,max}}$ of the S2 star (see Table 5).
Our calculations indicate that the specific energy and specific angular
momentum of the S2 star lie within the ranges $0.871573\leq\mathcal{E}\leq
0.923948$ and $5.02989M\leq l\leq 7.4095M$, respectively.

Our findings may help us to determine the limits for the magnetic dipole
moment of the S2 star with the mass $\approx 14M_{\odot}$ Habibi et al.
(2017). The magnetic field in the vicinity of Sgr A* (at the ISCO
$6M_{\mathrm{SgrA*}}\simeq 0.24$ AU) is approximately 5-100 G Kunneriath et
al. (2010); Eckart et al. (2012), and near the orbit of the magnetar PSR
1745-2900 is about mG Eatough et al. (2013). To estimate the magnetic dipole
moment of the magnetized S2 star, we assume the magnetic field near its orbit
is about 1 G. The estimated value of the magnetic interaction parameter
$\beta=\mu B/m$ is

$\mu\simeq 2\beta\times
10^{6}\left(\frac{m}{14M_{\odot}}\right)\left(\frac{B}{1G}\right)^{-1}{\rm
G\cdot cm^{3}}.$ (34)

One may have the possible range of the maximum and minimum of the magnetic
dipole using the obtained $\beta_{\max,\min}$ as

$\frac{1}{3}\leq\left(\frac{\mu}{10^{6}\rm G\cdot
cm^{3}}\right)\leq\frac{1}{2}.$ (35)

##  IV unbound orbits

In this section, we will shift our focus from analyzing bound orbits to
exploring unbound trajectories, particularly those involving deflection, which
hold significant implications for understanding the dynamics of magnetized
particles in black hole spacetimes.

###  IV.1 Unstable circular orbits

According to the right panel of Fig. 1, unstable circular orbits occur at the
maximum of the effective potential, where the condition
$V^{\prime}_{\mathrm{eff}}=0$ is satisfied. Figure 6 illustrates the
relationship between the radius of unstable circular orbits, $r_{u}$, and the
specific angular momentum, considering fixed values of the coupling parameter
$\beta$. As the diagram shows, as the $\beta$ parameter increases, the mutual
dependence of the above quantities becomes less noticeable. In other words,
for larger values of $\beta$, the changes in $r_{u}$ depend less strongly on
the changes in $l$.

Figure 6: The relationship between the radius of unstable circular orbits and
specific angular momentum for fixed values of the magnetic coupling constant
$\beta$.

Furthermore, utilizing Eq. (9), we can determine the period of unstable
circular orbits as measured by both the comoving (proper) and distant
observers for a long-term circular orbit. When considering a complete orbit
(i.e. $\phi_{u}=2\pi$), one can obtain the following:

$\displaystyle T_{\tau}=\frac{2\pi
r_{u}^{2}}{l_{u}}\Big{[}1+\beta\mathcal{F}(r_{u})\Big{]},\quad
T_{t}=\frac{2\pi r_{u}^{2}\mathcal{E}_{u}}{l_{u}f(r_{u})}.$ (36)

Furthermore, $l_{u}$ can be derived using the condition
$V^{\prime}_{\mathrm{eff}}=0$. Furthermore, the expression $\dot{r}=0$ or
$\mathcal{E}^{2}_{u}=V_{\mathrm{eff}}(r_{u})$ leads to the determination of
$\mathcal{E}_{u}$. Therefore, we obtain,

$\displaystyle T_{\tau}=4\pi
r_{u}\left[1+\sqrt{f(r_{u})}\,\beta\right]\Omega(r_{u}),$ (38) $\displaystyle
T_{t}=4\pi r_{u}\Omega(r_{u})\Lambda(r_{u}),$

in which

$\displaystyle\Omega^{2}(r)=\frac{r-3M}{M\left(4-3\beta\sqrt{f(r)}\right)},$
(39a)
$\displaystyle\Lambda^{2}(r)=\frac{1}{f(r)}\left(\frac{1+l^{2}}{r^{2}}-\frac{1}{2}\beta\sqrt{f(r)}\right).$
(39b)

We continue this section by exploring the deflecting trajectories.

###  IV.2 OFK

From Eq. (18), it is evident that for $\mathcal{E}^{2}\geq 1-\beta$,
magnetized particles follow unbound orbits. Utilizing the condition
${dr}/{d\phi}\mid_{r_{t}}=0$ of a general turning point $r_{t}$, and Eq. (14),
we can find the turning points of the particles, which helps to distinguish
between OFK and OSK. Subsequently, in Fig. 7, we illustrate how the radii of
the turning points depend on the angular momentum for fixed values of the
specific energy $\mathcal{E}$ and magnetic coupling parameter $\beta$.

Figure 7: The variation of turning point radii with specific angular momentum
for fixed values of specific energy $\mathcal{E}$ (left panel), and coupling
parameter $\beta$ (right panel). Red points represent the smallest orbital
separation $r_{s}$, while gray points denote the largest orbital separation
$r_{l}$.

From these diagrams, it is evident that particles exhibit two distinct turning
point radii: the smallest orbital separation $r_{s}$ and the largest orbital
separation $r_{l}$. The latter, $r_{l}$, corresponds to the OFK. Particles
reaching this orbit either proceed to infinity following hyperbolic motion in
the $\mathcal{E}^{2}_{u}>\mathcal{E}^{2}\geq 1-\beta$ case or engage in
periodic orbits when $r_{l}=r_{a}$, as observed previously, in the
$\mathcal{E}^{2}<1-\beta$ scenario. Notably, the equation of motion governing
OFK is almost identical to that of bound orbits. However, it is crucial to
consider that this is one of the roots of Eq. (18) is negative for unbound
orbits. Consequently, assuming that a magnetized particle initiates its motion
from $u=u_{2}$ and $\phi=0$, Eq. (21) can be expressed as

$\displaystyle\phi(u)=\frac{2F\left(\arcsin{\sqrt{\frac{u-u_{1}}{u_{2}-u_{1}}}},k_{0}\right)-2K\left(k_{0}\right)}{\sqrt{a(1+\beta)(u_{3}-u_{1})}}.$
(40)

Following some straightforward calculations, we obtain the trajectory of the
magnetized particles as

$\displaystyle
r(\phi)=\frac{1}{u_{1}+(u_{2}-u_{1})\operatorname{\mathrm{sn}}^{2}\Big{(}\theta_{0}+K(k_{0}),k_{0}\Big{)}}.$
(41)

Similarly to Eqs. (24), we can express the roots of Eq. (18) in the same
manner, however, here $e\geq 1$ as we are dealing with hyperbolic orbits. Then
Eq. (41) can be expressed as

$r(\phi)=\frac{\lambda}{1-e+2e\operatorname{\mathrm{sn}}^{2}\Big{(}\theta_{1}+K(k_{1}),k_{1}\Big{)}}.$
(42)

We have determined the corresponding specific energy and angular momentum for
each $e$ and $\lambda$ in Eq. (32). Furthermore, in Table 6, we have provided
the values of the latus rectum and the specific angular momentum for the
selected values of specific energy $\mathcal{E}$, eccentricity $e$, and
magnetic coupling constant $\beta$.

$\beta$ | $e$ | $\lambda$ | $l$ | $\mathcal{E}$
---|---|---|---|---
0.0 | 1.5 | $6.28198M$ | $6.18387M$ | 1.2
0.0 | 1.5 | $5.52617M$ | $10.5156M$ | 1.5
0.0 | 2.0 | $11.4327M$ | $5.43019M$ | 1.2
0.0 | 2.0 | $8.23412M$ | $7.41205M$ | 1.5
0.3 | 1.5 | $8.56076M$ | $13.1119M$ | 1.2
0.3 | 1.5 | $8.43699M$ | $19.1212M$ | 1.5
0.3 | 2 | $11.9084M$ | $8.70001M$ | 1.2
0.3 | 2 | $11.3611M$ | $12.3606M$ | 1.5
Table 6: The specific angular momentum and latus rectum values corresponding
to selected specific energy and eccentricity, with fixed magnetic coupling
parameter.

Hence, plotting the hyperbolic trajectories of the magnetized particles is now
straightforward. In Fig. 8, we depict the OFK for the particles around the
black hole. Given the behavior of particles and their deflection from the
black hole, the concept of OFK in this context aligns with the magnetic
counterpart of the gravitational Rutherford scattering (see Refs. Villanueva
and Olivares (2015); Fathi et al. (2021a)). The scattering process can be
either repulsive or attractive, depending on the eccentricity $e$ and,
correspondingly, the specific energy $\mathcal{E}$ and specific angular
momentum $l$ associated with this eccentricity.

Figure 8: The gravitational Rutherford scattering plotted for $e=1.5$ (left
panel), and for $e=2$ (right panel).

The diagrams show that neutral particles approach the black hole more closely
than magnetized particles just before heading toward infinity. Similarly to
bound orbits, this phenomenon can be attributed to the repulsive nature of the
magnetic interaction between magnetized test particles and the external
magnetic field $B$. Furthermore, for identical specific energy values, the
influence of eccentricity becomes apparent in the transition between
attractive and repulsive OFK. An increase in this parameter enhances the
tendency of particles to move away from the black hole.

###  IV.3 OSK

As depicted in Fig. 7, magnetized particles exhibit two distinct turning
points during their unbound orbits. In particular, the smallest orbital
separation, denoted as $r_{s}$, corresponds to the OSK. In this scenario,
particles reaching $r_{s}$ are drawn into the event horizon by spiraling
motion. Assuming particles commence their motion from the point
$u=u_{3}={1}/{r_{s}}$ and $\phi=0$, integration of Eq. (20) yields

$\phi(u)=\frac{2}{\sqrt{a(1+\beta)(u_{3}-u_{1})}}\left[F\left(\arcsin{\sqrt{\frac{u-u_{1}}{u_{2}-u_{1}}}},k_{0}\right)\right.\\\
\left.-F\left(\arcsin{\frac{1}{k_{0}}},k_{0}\right)\right],$ (43)

which enables us to find the analytical solution for the OSK as

$r(\phi)\\\
=\frac{1}{u_{1}+(u_{2}-u_{1})\operatorname{\mathrm{sn}}^{2}\Big{(}\theta_{0}+F\left(\arcsin{\frac{1}{k_{0}}},k_{0}\right),k_{0}\Big{)}}.$
(44)

In Fig. 9, we illustrate the trajectories of magnetized particles approaching
the black hole from a distance $r_{s}$, with fixed values of $\beta$ and the
impact parameter $b={l}/{\mathcal{E}}$. From the figure, we infer that a
larger impact parameter results in a more noticeable bending of the
trajectories as particles approach the black hole, with the turning point
positioned at a smaller distance. The magnetic interaction further enhances
this bending, especially evident in the final segment before the particles
fall into the black hole. Conversely, smaller impact parameters cause the
trajectories to start their approach from a more distant location, indicating
a turning point at a greater distance from the black hole.

Figure 9: The OSK for different values of the $\beta$-parameter and the impact
parameter $b$.

So far, we have comprehensively analyzed all possible angular trajectories for
magnetized test particles and presented fully analytical solutions to the
equations of motion. For the completeness of our study, in the next section,
we proceed with the analysis of purely radial trajectories.

##  V Radial trajectories

Studying particles in free fall with zero angular momentum offers several
advantages for standard general relativistic tests. One key area of
investigation is the gravitational clock effect experienced by falling
observers in gravitational fields, which forms the theoretical basis for
phenomena such as the gravitational redshift-blueshift of light rays passing
near a black hole. Another significant aspect is the phenomenon of frozen
infalling objects, observed by distant observers as they approach the event
horizon of a black hole. This stems from the disparities in time measurements
between distant observers ($t$) and falling observers ($\tau$), and provides
valuable insight into gravitational effects and their implications for our
understanding of spacetime dynamics Zel’dovich et al. (1972); Ryder (2009);
Zel’dovich and Novikov (2014). Hence, the radial motion of magnetized
particles without angular momentum also presents an interesting subject to
explore.

For purely radial orbits, the effective potential (13) adopts the following
form:

$\displaystyle V_{\mathrm{eff}}=f(r)\bigr{[}1-\beta\mathcal{F}(r)\bigr{]},$
(45)

which, together with Eqs. (11) and (13) enables us to establish relationships
for the radial velocities as

$\displaystyle\left(\frac{dr}{d\tau}\right)^{2}$ $\displaystyle=$
$\displaystyle\mathcal{E}^{2}-f(r)\bigr{[}1-\beta\mathcal{F}(r)\bigr{]},$ (46)
$\displaystyle\left(\frac{dr}{dt}\right)^{2}$ $\displaystyle=$
$\displaystyle\Bigr{\\{}f(r)\bigr{[}1+\beta\mathcal{F}(r)\bigr{]}\Bigr{\\}}^{2}$
(47)
$\displaystyle\times\Bigr{\\{}1-\frac{f(r)}{\mathcal{E}^{2}}\bigr{[}1+\beta\mathcal{F}(r)\bigr{]}\Bigr{\\}}.$

In obtaining analytical representations for the time parameters, we derived
power series expansions for Eqs. (46) and (47) centered at $\beta=0$ up to the
second order. Upon integrating these radial velocities, we can determine the
radial dependence of the time parameters for magnetized particles that
initiate their motion from the point $r_{0}$. This procedure yields

$\displaystyle\tau(r)=\mathcal{T}_{\tau}(r_{0})-\mathcal{T}_{\tau}(r),$ (48)
$\displaystyle t(r)=\mathcal{T}_{t}(r_{0})-\mathcal{T}_{t}(r),$ (49)

where

$\displaystyle\mathcal{T}_{\tau}(r)=\frac{1}{8}\Big{[}\mathcal{A}_{\tau}+\mathcal{B}_{\tau}-\mathcal{C}_{\tau}+\mathcal{D}_{\tau}\Big{]},$
(50a)
$\displaystyle\mathcal{T}_{t}(r)=\frac{1}{8}\mathcal{E}\Big{[}\mathcal{A}_{t}+\mathcal{B}_{t}-\mathcal{C}_{t}-\mathcal{D}_{t}\Big{]}.$
(50b)

The expansion coefficients are shown in the Appendix B. In Fig. 10, we depict
the dependency of the proper time and coordinate time of magnetized particles
on their distance from the black hole. This illustration highlights that while
a falling observer may perceive particles crossing the event horizon, the
process takes an infinite amount of time for a distant observer.

Figure 10: Radial profiles of proper time (dot-dashed curves) and coordinate
time (solid curves) for test particles on radial orbits, plotted for various
values of the $\beta$-parameter and specific energy.

Having thoroughly investigated the angular and radial motion of magnetized
particles through comprehensive analytical methods, we have explored various
aspects of their trajectory dynamics. With this in mind, we close our
discussion at this point and proceed to summarize our findings in the next
section.

##  VI Summary and conclusions

This study investigated particles’ bound and unbound orbits with a dipole
moment as they approached a magnetized Schwarzschild black hole. Magnetization
of the black hole was achieved by assuming proportionality between the four-
potential and a space-like Killing vector. We constructed the Lagrangian
dynamics for test particles, where the magnetic moment directly contributed to
their conjugate momenta, with a magnetic interaction parameter $\beta$ playing
a crucial role in determining the magnetic field’s effects on particle motion.
We derived and analyzed the effective potential experienced by approaching
particles, classifying possible bound and unbound orbits based on their
specific energy and angular momentum. Bound orbits resembled planetary orbits
oscillating between apoapsis and periapsis, with the possibility of complete
circularity at the minimum of the effective potential, determining the ISCO.
We noted that while the ISCO radius increased with $\beta$, it caused
decreases in the specific energy and angular momentum of particles on the
ISCO. Analytical solutions to the angular equations of motion for bound orbits
were expressed in terms of elliptic integrals and Jacobi elliptic functions,
with a precise classification of orbits based on topological characterization.
Simulating various bound orbits with different eccentricities revealed
topological variations and geometric extensions in the exterior spacetime of
the black hole. Comparisons between observed positions of the S2 star around
Sgr A* and theoretical predictions were made for two different values of
$\beta$, calculating minimum and maximum values for this parameter, as well as
specific energy and specific angular momentum of the S2 star. We found that
the magnetic dipole moment associated with the S2 star was on the order of
$10^{6}\,\mathrm{G\cdot cm^{3}}$. Unbound orbits, classified as unstable
circular orbits, OFK, and OSK, either deflected from the black hole to
infinity or fell inexorably onto the event horizon. The period of unstable
circular orbits was calculated for comoving and distant observers, and the OFK
was likened to the magnetic counterpart of gravitational Rutherford
scattering. For such deflecting trajectories, the characteristic polynomial of
the equation of motion had only two positive roots, leading to analytical
solutions differing in elliptic integrals’ arguments. The scattering of test
particles could be attractive or repulsive for the OFK, depending on
eccentricity. The OSK trajectory curvature increased with the magnetic
interaction parameter, resulting in a smaller turning point. Lastly, we
explored purely radial orbits for magnetized particles, obtaining the radial
profile of proper time for infalling observers and coordinate time for distant
observers. The study of magnetized particles in black hole spacetimes offers
new avenues for investigating relativistic stars at galaxy centers potentially
associated with magnetic fields. Expanding models to include black holes with
more degrees of freedom could advance research in relativistic astrophysics, a
direction for future studies.

## Acknowledgements

M.F. is supported by Universidad Central de Chile through the project No.
PDUCEN20240008. J.R. & A.A. thank to the Grant F-FA-2021-510 of the Uzbekistan
Ministry for Innovative Development.

##  Appendix A The coefficients of Eqs. (31) and (32)

These coefficients are as follows:

$\displaystyle\Phi=4\left(\beta+1\right)\left(2\beta+1\right)\left[\beta(5\beta-1)-2\right]\left(e^{2}+1\right)\lambda
M^{2},$ (51a)
$\displaystyle\Psi=8\beta\left(\beta+1\right)\left(2\beta+1\right)\left(1-e^{2}\right)M^{2}-2\left(\beta+1\right)\lambda^{2}$
$\displaystyle\qquad\qquad\qquad-2\lambda
M\Bigr{\\{}\beta\Big{[}3\beta(2\beta+5)+13\Big{]}+\Big{[}\beta(\beta+1)(2\beta+3)+1\Big{]}e^{2}+3\Bigr{\\}},$
(51b)
$\displaystyle\Xi=\left(-3\beta^{2}-\beta+2\right)\lambda^{3}+16\beta^{3}\left(1-e^{2}\right)M^{3}-4\left(\beta+1\right)\beta^{2}\left(e^{2}+3\right)\lambda
M^{2},$ (51c)
$\displaystyle\Theta=\lambda^{2}M\Bigr{\\{}\beta\Big{[}\beta\Big{(}6\beta(2\beta+3)-1\Big{)}+\Big{(}\beta\left(4\beta^{2}+6\beta-7\right)-5\Big{)}e^{2}-19\Big{]}-8\Bigr{\\}}.$
(51d)

##  Appendix B The coefficients of Eqs. (50)

These coefficients are as follows:

$\displaystyle\mathcal{A}_{\tau}=\frac{r\sqrt{\mathcal{E}^{2}-f(r)}\left[-4M^{2}\left(3\beta^{2}\mathcal{E}_{1}-8\mathcal{E}^{2}_{0}\right)-4Mr\mathcal{E}_{0}\left(\beta^{2}\mathcal{E}_{2}-8\mathcal{E}^{2}_{0}\right)+r^{2}\mathcal{E}_{0}\left(3\beta^{2}+8\mathcal{E}_{0}^{2}\right)\right]}{\mathcal{E}_{0}^{3}\left(2M+r\mathcal{E}_{0}\right)^{2}},$
(52a)
$\displaystyle\mathcal{B}_{\tau}=\frac{M\left[3\beta^{2}\left(6\mathcal{E}^{2}-1\right)+8\mathcal{E}_{0}^{2}\right]\ln\left(\frac{2\left(M+r\mathcal{E}_{0}\right)}{\sqrt{\mathcal{E}_{0}}}+2r\sqrt{\mathcal{E}^{2}-f(r)}\right)}{\mathcal{E}_{0}^{7/2}},$
(52b) $\displaystyle\mathcal{C}_{\tau}=\frac{4\beta
r\sqrt{f(r)}\sqrt{\mathcal{E}^{2}-f(r)}\left(M\mathcal{E}_{3}+r\mathcal{E}_{0}\right)}{\mathcal{E}_{0}^{2}\left(2M+r\mathcal{E}_{0}\right)},$
(52c) $\displaystyle\mathcal{D}_{\tau}=\frac{12\beta
M\mathcal{E}^{2}\ln\left(2r\sqrt{f(r)}\sqrt{\mathcal{E}^{2}-f(r)}-\frac{2\left[M\left(\mathcal{E}^{2}-2\right)-r\mathcal{E}_{0}\right]}{\sqrt{\mathcal{E}_{0}}}\right)}{\mathcal{E}_{0}^{5/2}},$
(52d)
$\displaystyle\mathcal{A}_{t}=\frac{r\sqrt{\mathcal{E}^{2}-f(r)}\left[4M^{2}\left(\beta^{2}\mathcal{E}_{4}+8\mathcal{E}_{0}^{2}\right)+4Mr\mathcal{E}_{0}\left(\beta^{2}\mathcal{E}_{5}+8\mathcal{E}_{0}^{2}\right)+r^{2}\mathcal{E}_{0}^{2}\left(\beta^{2}\mathcal{E}_{6}+8\mathcal{E}_{0}^{4}\right)\right]}{\mathcal{E}_{0}^{3}\left(2M+r\mathcal{E}_{0}\right)^{2}},$
(52e)
$\displaystyle\mathcal{B}_{t}=\frac{M\left[\beta^{2}\mathcal{E}_{4}-8\mathcal{E}_{0}^{2}\mathcal{E}_{7}\right]\ln\left(\frac{2\left(M+r\mathcal{E}_{0}\right)}{\sqrt{\mathcal{E}_{0}}}+2r\sqrt{\mathcal{E}^{2}-f(r)}\right)}{\mathcal{E}_{0}^{7/2}}-\frac{4\beta
r\sqrt{f(r)(\mathcal{E}^{2}-f(r))}\left(M\mathcal{E}_{3}+r\mathcal{E}_{8}\right)}{\mathcal{E}_{0}^{2}\left(2M+r\mathcal{E}_{0}\right)},$
(52f) $\displaystyle\mathcal{C}_{t}=\frac{4\beta
M\mathcal{E}_{9}\ln\left(2r\sqrt{f(r)(\mathcal{E}^{2}-f(r))}-\frac{2\left[M\left(\mathcal{E}^{2}-2\right)-r\mathcal{E}_{0}\right]}{\sqrt{\mathcal{E}_{0}}}\right)}{\mathcal{E}_{0}^{5/2}},$
(52g)
$\displaystyle\mathcal{D}_{t}=\frac{16M\ln\left(r\left[2\mathcal{E}\sqrt{\mathcal{E}^{2}-f(r)}+2\mathcal{E}_{0}\right]+2M\right)}{\mathcal{E}}+\frac{16M\ln
rf(r)}{\mathcal{E}},$ (52h)

in which we have defined

$\displaystyle\mathcal{E}_{0}=\mathcal{E}^{2}-1,$ (53a)
$\displaystyle\mathcal{E}_{1}=2\mathcal{E}^{6}-6\mathcal{E}^{4}-1,$ (53b)
$\displaystyle\mathcal{E}_{2}=2\mathcal{E}^{6}-9\mathcal{E}^{4}-3,$ (53c)
$\displaystyle\mathcal{E}_{3}=4\mathcal{E}^{2}+2,$ (53d)
$\displaystyle\mathcal{E}_{4}=16\mathcal{E}^{4}-8\mathcal{E}^{2}+7,$ (53e)
$\displaystyle\mathcal{E}_{5}=13\mathcal{E}^{4}-10\mathcal{E}^{2}+7,$ (53f)
$\displaystyle\mathcal{E}_{6}=8\mathcal{E}^{4}-12\mathcal{E}^{2}+7,$ (53g)
$\displaystyle\mathcal{E}_{7}=2\mathcal{E}^{2}-3,$ (53h)
$\displaystyle\mathcal{E}_{8}=2\mathcal{E}^{4}-3r\mathcal{E}^{2}+r,$ (53i)
$\displaystyle\mathcal{E}_{9}=2\mathcal{E}^{4}-7\mathcal{E}^{2}+2.$ (53j)

## References

  * Schwarzschild (1916) K. Schwarzschild, Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften pp. 189–196 (1916).
  * Dyson et al. (1920) F. W. Dyson, A. S. Eddington, and C. Davidson, Philosophical Transactions of the Royal Society of London Series A 220, 291 (1920).
  * Clemence (1947) G. M. Clemence, Rev. Mod. Phys. 19, 361 (1947), URL https://link.aps.org/doi/10.1103/RevModPhys.19.361.
  * Hagihara (1930) Y. Hagihara, Japanese Journal of Astronomy and Geophysics 8, 67 (1930).
  * Darwin (1959) C. G. Darwin, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 249, 180 (1959), ISSN 0080-4630, 2053-9169, URL https://royalsocietypublishing.org/doi/10.1098/rspa.1959.0015.
  * Darwin (1961) C. G. Darwin, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 263, 39 (1961), ISSN 0080-4630, 2053-9169, URL https://royalsocietypublishing.org/doi/10.1098/rspa.1961.0142.
  * Mielnik and Plebański (1962) B. Mielnik and J. Plebański, Acta Phys. Pol. 21, 239 (1962).
  * Jacobi (2013) C. G. J. Jacobi, _C. G. J. Jacobi’s Gesammelte Werke: Herausgegeben auf Veranlassung der königlich preussischen Akademie der Wissenschaften_ , Cambridge Library Collection - Mathematics (Cambridge University Press, 2013).
  * (9) N. H. Abel, _Oeuvres complètes de Niels Henrik Abel: Nouvelle édition_ , vol. 2 of _Cambridge Library Collection - Mathematics_ (Cambridge University Press, ????).
  * Riemann (1857) B. Riemann, Journal für die reine und angewandte Mathematik (Crelles Journal) 1857, 115 (1857), ISSN 0075-4102, 1435-5345, URL https://www.degruyter.com/document/doi/10.1515/crll.1857.54.115/html.
  * Riemann (1866) B. Riemann, Journal für die reine und angewandte Mathematik (Crelles Journal) 1866, 161 (1866), URL https://doi.org/10.1515/crll.1866.65.161.
  * Weierstrass (1854) C. Weierstrass, Journal für die reine und angewandte Mathematik (Crelles Journal) 1854, 289 (1854), URL https://doi.org/10.1515/crll.1854.47.289.
  * Baker (1995) H. F. Baker, _Abelian functions: Abel’s theorem and the allied theory of theta functions_ , Cambridge mathematical library (Cambridge University Press, Cambridge, [Eng.] ; New York, 1995), ISBN 9780521498777.
  * Kraniotis and Whitehouse (2002) G. V. Kraniotis and S. B. Whitehouse, Classical and Quantum Gravity 19, 5073 (2002), ISSN 0264-9381, URL https://iopscience.iop.org/article/10.1088/0264-9381/19/20/304.
  * Kraniotis and Whitehouse (2003) G. V. Kraniotis and S. B. Whitehouse, Classical and Quantum Gravity 20, 4817 (2003), ISSN 0264-9381, 1361-6382, URL https://iopscience.iop.org/article/10.1088/0264-9381/20/22/007.
  * Kraniotis (2004) G. V. Kraniotis, Classical and Quantum Gravity 21, 4743 (2004), ISSN 0264-9381, 1361-6382, URL https://iopscience.iop.org/article/10.1088/0264-9381/21/19/016.
  * Kraniotis (2005) G. V. Kraniotis, Classical and Quantum Gravity 22, 4391 (2005), ISSN 0264-9381, 1361-6382, URL https://iopscience.iop.org/article/10.1088/0264-9381/22/21/001.
  * Cruz et al. (2005) N. Cruz, M. Olivares, and J. R. Villanueva, Classical and Quantum Gravity 22, 1167 (2005), ISSN 0264-9381, 1361-6382, URL https://iopscience.iop.org/article/10.1088/0264-9381/22/6/016.
  * Kraniotis (2007) G. V. Kraniotis, Classical and Quantum Gravity 24, 1775 (2007), ISSN 0264-9381, 1361-6382, URL https://iopscience.iop.org/article/10.1088/0264-9381/24/7/007.
  * Hackmann and Lämmerzahl (2008a) E. Hackmann and C. Lämmerzahl, Physical Review Letters 100, 171101 (2008a), ISSN 0031-9007, 1079-7114, URL https://link.aps.org/doi/10.1103/PhysRevLett.100.171101.
  * Hackmann and Lämmerzahl (2008b) E. Hackmann and C. Lämmerzahl, Physical Review D 78, 024035 (2008b), ISSN 1550-7998, 1550-2368, URL https://link.aps.org/doi/10.1103/PhysRevD.78.024035.
  * Hackmann et al. (2009) E. Hackmann, V. Kagramanova, J. Kunz, and C. Lämmerzahl, EPL (Europhysics Letters) 88, 30008 (2009), ISSN 0295-5075, 1286-4854, URL https://iopscience.iop.org/article/10.1209/0295-5075/88/30008.
  * Hackmann et al. (2010) E. Hackmann, B. Hartmann, C. Lämmerzahl, and P. Sirimachan, Physical Review D 81, 064016 (2010), ISSN 1550-7998, 1550-2368, URL https://link.aps.org/doi/10.1103/PhysRevD.81.064016.
  * Olivares et al. (2011) M. Olivares, J. Saavedra, C. Leiva, and J. R. Villanueva, Modern Physics Letters A 26, 2923 (2011), ISSN 0217-7323, 1793-6632, URL https://www.worldscientific.com/doi/abs/10.1142/S0217732311037261.
  * Kraniotis (2011) G. V. Kraniotis, Classical and Quantum Gravity 28, 085021 (2011), ISSN 0264-9381, 1361-6382, URL https://iopscience.iop.org/article/10.1088/0264-9381/28/8/085021.
  * Cruz et al. (2013) N. Cruz, M. Olivares, and J. R. Villanueva, The European Physical Journal C 73, 2485 (2013), ISSN 1434-6044, 1434-6052, URL http://link.springer.com/10.1140/epjc/s10052-013-2485-8.
  * Villanueva et al. (2013) J. R. Villanueva, J. Saavedra, M. Olivares, and N. Cruz, Astrophysics and Space Science 344, 437 (2013), ISSN 0004-640X, 1572-946X, URL http://link.springer.com/10.1007/s10509-012-1333-x.
  * Kraniotis (2014) G. V. Kraniotis, General Relativity and Gravitation 46, 1818 (2014), ISSN 0001-7701, 1572-9532, URL http://link.springer.com/10.1007/s10714-014-1818-8.
  * Soroushfar et al. (2015) S. Soroushfar, R. Saffari, J. Kunz, and C. Lämmerzahl, Physical Review D 92, 044010 (2015), ISSN 1550-7998, 1550-2368, URL https://link.aps.org/doi/10.1103/PhysRevD.92.044010.
  * Villanueva and Olivares (2015) J. R. Villanueva and M. Olivares, The European Physical Journal C 75, 562 (2015), ISSN 1434-6044, 1434-6052, URL http://link.springer.com/10.1140/epjc/s10052-015-3794-x.
  * Soroushfar et al. (2016) S. Soroushfar, R. Saffari, S. Kazempour, S. Grunau, and J. Kunz, Physical Review D 94, 024052 (2016), ISSN 2470-0010, 2470-0029, URL https://link.aps.org/doi/10.1103/PhysRevD.94.024052.
  * Hoseini et al. (2016) B. Hoseini, R. Saffari, S. Soroushfar, S. Grunau, and J. Kunz, Physical Review D 94, 044021 (2016), ISSN 2470-0010, 2470-0029, URL https://link.aps.org/doi/10.1103/PhysRevD.94.044021.
  * Hoseini et al. (2017) B. Hoseini, R. Saffari, and S. Soroushfar, Classical and Quantum Gravity 34, 055004 (2017), ISSN 0264-9381, 1361-6382, URL https://iopscience.iop.org/article/10.1088/1361-6382/aa5a63.
  * Fathi et al. (2020a) M. Fathi, M. Kariminezhaddahka, M. Olivares, and J. R. Villanueva, The European Physical Journal C 80, 377 (2020a), ISSN 1434-6044, 1434-6052, URL https://link.springer.com/10.1140/epjc/s10052-020-7945-3.
  * Fathi et al. (2020b) M. Fathi, M. Olivares, and J. R. Villanueva, The European Physical Journal C 80, 51 (2020b), ISSN 1434-6044, 1434-6052, URL http://link.springer.com/10.1140/epjc/s10052-020-7623-5.
  * Fathi et al. (2021a) M. Fathi, M. Olivares, and J. R. Villanueva, The European Physical Journal Plus 136, 420 (2021a), ISSN 2190-5444, URL https://link.springer.com/10.1140/epjp/s13360-021-01441-9.
  * González et al. (2021) P. A. González, M. Olivares, Y. Vásquez, and J. R. Villanueva, The European Physical Journal C 81, 236 (2021), ISSN 1434-6044, 1434-6052, URL https://link.springer.com/10.1140/epjc/s10052-021-09024-9.
  * Fathi et al. (2021b) M. Fathi, M. Olivares, and J. R. Villanueva, The European Physical Journal C 81, 987 (2021b), ISSN 1434-6044, 1434-6052, URL https://link.springer.com/10.1140/epjc/s10052-021-09787-1.
  * Kraniotis (2021) G. V. Kraniotis, The European Physical Journal C 81, 147 (2021), ISSN 1434-6044, 1434-6052, URL https://link.springer.com/10.1140/epjc/s10052-021-08911-5.
  * Fathi et al. (2022) M. Fathi, M. Olivares, and J. R. Villanueva, The European Physical Journal C 82, 629 (2022), ISSN 1434-6052, URL https://link.springer.com/10.1140/epjc/s10052-022-10600-w.
  * Soroushfar and Afrooz (2022) S. Soroushfar and M. Afrooz, Indian Journal of Physics 96, 593 (2022), ISSN 0973-1458, 0974-9845, URL https://link.springer.com/10.1007/s12648-020-01971-5.
  * Battista and Esposito (2022) E. Battista and G. Esposito, The European Physical Journal C 82, 1088 (2022), ISSN 1434-6052, URL https://link.springer.com/10.1140/epjc/s10052-022-11070-w.
  * Fathi et al. (2023) M. Fathi, M. Olivares, and J. R. Villanueva, The European Physical Journal Plus 138, 7 (2023), ISSN 2190-5444, URL https://link.springer.com/10.1140/epjp/s13360-022-03538-1.
  * Fathi (2023) M. Fathi, Annals of Physics 457, 169401 (2023), ISSN 0003-4916, URL https://www.sciencedirect.com/science/article/pii/S0003491623001872.
  * Perek (1962) L. Perek, Vistas in Astronomy 5, 28 (1962), ISSN 00836656, URL https://linkinghub.elsevier.com/retrieve/pii/008366566290003X.
  * Barbieri and Gratton (2002) M. Barbieri and R. G. Gratton, Astronomy & Astrophysics 384, 879 (2002), ISSN 0004-6361, 1432-0746, URL http://www.aanda.org/10.1051/0004-6361:20020011.
  * Zhong et al. (2023) S. Zhong, K. Hayasaki, S. Li, P. Berczik, and R. Spurzem, The Astrophysical Journal 959, 19 (2023), ISSN 0004-637X, 1538-4357, URL https://iopscience.iop.org/article/10.3847/1538-4357/ad0122.
  * Binney and Tremaine (2011) J. Binney and S. Tremaine, _Galactic Dynamics: Second Edition_ (Princeton University Press, 2011), ISBN 9781400828722 9780691130262, URL http://www.jstor.org/stable/10.2307/j.ctvc778ff.
  * Behroozi et al. (2013) P. S. Behroozi, A. Loeb, and R. H. Wechsler, Journal of Cosmology and Astroparticle Physics 2013, 019 (2013), ISSN 1475-7516, URL https://iopscience.iop.org/article/10.1088/1475-7516/2013/06/019.
  * Aliev and Gal’tsov (1989) A. N. Aliev and D. V. Gal’tsov, Soviet Physics Uspekhi 32, 75 (1989), ISSN 0038-5670, URL https://iopscience.iop.org/article/10.1070/PU1989v032n01ABEH002677.
  * Li (2002) L.-X. Li, Astronomy & Astrophysics 392, 469 (2002), ISSN 0004-6361, 1432-0746, URL http://www.aanda.org/10.1051/0004-6361:20020958.
  * Meier (2012) D. L. Meier, _Black Hole Astrophysics: The Engine Paradigm_ (Springer Berlin Heidelberg, Berlin, Heidelberg, 2012), ISBN 9783642019357 9783642019364, URL https://link.springer.com/10.1007/978-3-642-01936-4.
  * Kruglov (2017) S. I. Kruglov, International Journal of Modern Physics A 32, 1750147 (2017), ISSN 0217-751X, 1793-656X, URL https://www.worldscientific.com/doi/abs/10.1142/S0217751X17501470.
  * Ghosh et al. (2021) D. Ghosh, A. Thalapillil, and F. Ullah, Physical Review D 103, 023006 (2021), ISSN 2470-0010, 2470-0029, URL https://link.aps.org/doi/10.1103/PhysRevD.103.023006.
  * Kopáček and Karas (2023) O. Kopáček and V. Karas, in _The Sixteenth Marcel Grossmann Meeting_ (WORLD SCIENTIFIC, Online, 2023), pp. 3999–4009, ISBN 9789811269769 9789811269776, URL https://www.worldscientific.com/doi/10.1142/9789811269776_0333.
  * Karas and Stuchlík (2023) V. Karas and Z. Stuchlík, Universe 9, 267 (2023), ISSN 2218-1997, URL https://www.mdpi.com/2218-1997/9/6/267.
  * Khan et al. (2024) S. U. Khan, U. Uktamov, J. Rayimbaev, A. Abdujabbarov, I. Ibragimov, and Z.-M. Chen, Eur. Phys. J. C 84, 203 (2024).
  * Chang et al. (2022) S. Chang, L. Zhang, Z. Jiang, and X. Li, Monthly Notices of the Royal Astronomical Society 513, 925 (2022), ISSN 0035-8711, eprint https://academic.oup.com/mnras/article-pdf/513/1/925/43465112/stac931.pdf, URL https://doi.org/10.1093/mnras/stac931.
  * Mészáros (2019) P. Mészáros, in _INTEGRAL CONFERENCE AND AHEAD GAMMA-RAY WORKSHOP INTEGRAL LOOKS AHEAD TO MULTI-MESSENGER ASTRONOMY_ (Mem. Soc. Ast. It. 90, 1-2, 2019), pp. 57–66, URL https://www.memsait.it/volumi/Vol.%2090%20n.%201-2,%202019/2019MmSAI..90...57M.pdf.
  * Novikov and Stern (1986) I. D. Novikov and B. E. Stern, _The Possible Mechanism of the Formation of the Hard Spectrum of Active Galactic Nuclei_ (Springer Netherlands, Dordrecht, 1986), pp. 149–171, ISBN 978-94-009-4562-3, URL https://doi.org/10.1007/978-94-009-4562-3_10.
  * Nishikawa et al. (2005) K. Nishikawa, G. Richardson, S. Koide, K. Shibata, T. Kudoh, P. Hardee, and G. J. Fishman, The Astrophysical Journal 625, 60 (2005), ISSN 0004-637X, 1538-4357, URL https://iopscience.iop.org/article/10.1086/429360.
  * Mościbrodzka et al. (2016) M. Mościbrodzka, H. Falcke, and H. Shiokawa, Astronomy & Astrophysics 586, A38 (2016), ISSN 0004-6361, 1432-0746, URL http://www.aanda.org/10.1051/0004-6361/201526630.
  * Mizuno (2022) Y. Mizuno, Universe 8, 85 (2022), ISSN 2218-1997, URL https://www.mdpi.com/2218-1997/8/2/85.
  * Wald (1984) R. M. Wald, _General relativity_ (University of Chicago Press, Chicago, 1984), ISBN 9780226870328 9780226870335.
  * Rezzolla et al. (2001) L. Rezzolla, B. J. Ahmedov, and J. C. Miller, Monthly Notices of the Royal Astronomical Society 322, 723 (2001), ISSN 00358711, 13652966, URL https://academic.oup.com/mnras/article/322/4/723/1016988.
  * Preti (2004) G. Preti, Phys. Rev. D 70, 024012 (2004).
  * de Felice and Sorge (2003) F. de Felice and F. Sorge, Classical and Quantum Gravity 20, 469 (2003).
  * Lim and Yeo (2024) Y.-K. Lim and Z. C. Yeo, Phys. Rev. D 109, 024037 (2024), eprint 2401.13894.
  * Levin and Perez-Giz (2008) J. Levin and G. Perez-Giz, Phys. Rev. D 77, 103005 (2008), eprint 0802.0459.
  * Chandrasekhar (1998) S. Chandrasekhar, _The Mathematical Theory of Black Holes_ (1998).
  * Byrd and Friedman (1971) P. F. Byrd and M. D. Friedman, _Handbook of Elliptic Integrals for Engineers and Scientists_ (Springer Berlin Heidelberg, Berlin, Heidelberg, 1971), ISBN 9783642651403 9783642651380, URL http://link.springer.com/10.1007/978-3-642-65138-0.
  * Lacroix (2018) T. Lacroix, A& A 619, A46 (2018), eprint 1801.01308.
  * Do et al. (2019) T. Do, A. Hees, A. Ghez, G. D. Martinez, D. S. Chu, S. Jia, S. Sakai, J. R. Lu, A. K. Gautam, K. K. O’Neil, et al., Science 365, 664 (2019), eprint 1907.10731.
  * Habibi et al. (2017) M. Habibi, S. Gillessen, F. Martins, F. Eisenhauer, P. M. Plewa, O. Pfuhl, E. George, J. Dexter, I. Waisberg, T. Ott, et al., Astrophys. J. 847, 120 (2017), eprint 1708.06353.
  * Kunneriath et al. (2010) D. Kunneriath, G. Witzel, A. Eckart, M. Zamaninasab, R. Gießübel, R. Schödel, F. K. Baganoff, M. R. Morris, M. Dovčiak, W. J. Duschl, et al., Atsron. Astrophys 517, A46 (2010), eprint 1008.1560.
  * Eckart et al. (2012) A. Eckart, M. García-Marín, S. N. Vogel, P. Teuben, M. R. Morris, F. Baganoff, J. Dexter, R. Schödel, G. Witzel, M. Valencia-S., et al., Astron.Astrphys 537, A52 (2012).
  * Eatough et al. (2013) R. P. Eatough, H. Falcke, R. Karuppusamy, K. J. Lee, D. J. Champion, E. F. Keane, G. Desvignes, D. H. F. M. Schnitzeler, L. G. Spitler, M. Kramer, et al., Nature 501, 391 (2013), eprint 1308.3147.
  * Zel’dovich et al. (1972) Y. B. Zel’dovich, I. D. Novikov, and J. Silk, Physics Today 25, 63 (1972).
  * Ryder (2009) L. Ryder, _Introduction to General Relativity_ (Cambridge University Press, 2009).
  * Zel’dovich and Novikov (2014) Y. B. Zel’dovich and I. D. Novikov, _Stars and relativity_ (Dover Publications, Mineola, N.Y, 2014), ISBN 9780486694245.