# The quasi-Keplerian motion in the regular Bardeen spacetime

Jie Li  School of Nuclear Science and Technology, University of South China,
Hengyang, 421001, China  Bo Yang<EMAIL_ADDRESS>School of Mathematics and
Physics, University of South China, Hengyang, 421001, China  Wenbin Lin
<EMAIL_ADDRESS>School of Mathematics and Physics, University of South China,
Hengyang, 421001, China  School of Physical Science and Technology, Southwest
Jiaotong University, Chengdu, 610031, China

###### Abstract

The second post-Newtonian solution for the quasi-Keplerian motion of a test
particle in the gravitational field of regular Bardeen black hole is derived.
The solution is formulated in terms of the test particle’s orbital energy and
angular momentum, as well as the mass and magnetic charge of the Bardeen black
hole. The leading effects of the magnetic charge on the test particle’s orbit
and motion including perihelion precession are displayed explicitly. In
particular, it is shown that to the second post-Newtonian order the magnetic
charge does not affect the test particle’s orbital period.

##  I introduction

Black hole is one of the most prominent predictions of general relativity.
With the development of science and technology, the gravitational wave
generated by the collision of two black holes was detected Abbott  _et al._
(2016); Cervantes-Cota  _et al._ (2016); Królak and Verma (2021) and the
shadow of black holes was revealed by Event Horizon Telescope collaboration
Akiyama  _et al._ (2019), which directly confirmed the existence of black
holes, greatly stimulated people’s interest in the physics of black holes.
However, spacetime singularity is one of the most intriguing problems.
According to Penrose and Hawking’s singularity theorems, every black hole
inevitably contains a singularity within the framework of Einstein’s general
relativity as long as some certain conditions are satisfied Hawking and
Penrose (1970). At the singularity of spacetime, physical quantities and
geometry diverge, and the laws of physics lose their power completely. To
avoid the singularity problem, Bardeen Bardeen (1968) first proposed a regular
black hole model named the Badeen black hole, which was later proved to be the
exact solution of the Einstein’s field equations coupling with nonlinear
electrodynamics Ayón-Beato and García (2000). Subsequently, a number of other
solutions to regular black holes such as Hayward black hole Hayward (2006),
Ayón-Beato and García black hole Ayón-Beato and García (1998) and Simpson-
Visser black holes Simpson and Visser (2019) have been proposed, and the
rotating solutions of these black holes has been obtained in Toshmatov  _et
al._ (2014); Abdujabbarov  _et al._ (2016); Mazza  _et al._ (2021).

Up to now, Bardeen black hole has attracted extensive attention. Bretón et al
studied the stability of black holes with nonlinear electromagnetic fields
Bretón and Bergliaffa (2015) and Quasinormal modes was investigated by
Fernando and Correa Fernando and Correa (2012). Gravitational lensing by black
holes with nonlinear electromagnetic fields was investigated in Eiroa and
Sendra (2011); Ghaffarnejad and Niad (2016). By analyzing the behavior of the
effective potential, Zhou et al investigated the time-like and null geodesic
structures Zhou  _et al._ (2012). Abdujabbarov et al studied the Shadow of
rotating regular black holes, and found that magnetic charge decreases the
size of the shadow Abdujabbarov  _et al._ (2016). Gao et al studied the bound
orbits around the Badeen black hole Gao and Deng (2020) and greybody factor
are studied in Sharif and Ama-Tul-Mughani (2019); Ama-Tul-Mughani  _et al._
(2021); Sharif and Shaukat (2022). The thermodynamic properties of the Bardeen
black holes are discussed in Man and Cheng (2014). Some properties of Bardeen
black holes in other modified theories have also been investigated Mehdipour
and Ahmadi (2016); Ghaderi and Malakolkalami (2018); Saleh  _et al._ (2018a,
b); Pourhassan and Debnath (2019); Singh and Siwach (2020); Singh  _et al._
(2020); Kumar  _et al._ (2022).

The motion of bodies in the vicinity of black holes is an important problem of
relativistic astrophysics. To solve the motion of the bodies in the strong
gravitational fields, the post-Newtonian (PN) approximations have been
extensively applied, and various analytical solutions have been obtained,
including the first and higher PN effects of the mass Brumberg (1972); Soffel
_et al._ (1987); Soffel (1989); Klioner and Kopeikin (1994); Kopeikin  _et
al._ (2012); Damour and Schäfer (1988); Schäfer and Wex (1993); Memmesheimer
_et al._ (2004); Boetzel  _et al._ (2017); Cho  _et al._ (2018); Soffel and
Han (2019); Yang and Lin (2020a, b), the leading order (1.5PN) spin effects
Wex (1995); Gergely  _et al._ (1998a, b, c); K$\ddot{o}$nigsd$\ddot{o}$rffer
and Gopakumar (2005); Keresztes  _et al._ (2005);
K$\ddot{o}$nigsd$\ddot{o}$rffer and Gopakumar (2006); Gopakumar and Schäfer
(2011); Bohé  _et al._ (2013); Gergely and Keresztes (2015); Mikóczi (2017) on
the general motion, as well as the leading order (2PN) mass quadrupole effects
on the circular motion Poisson (1998).

In our previous work Yang and Lin (2019); Yang  _et al._ (2022), we have
derived the quasi-Keplerian motion for the neutral test particle in the
Reissner-Nordström spacetime, and that for the charged test particle. We
obtained the effects of the spin-induced quadrupole on the equatorial motion
of a test particle in the Kerr spacetime Yang and Lin (2020c) and the next-to-
leading order spin-orbit effects on the test particle’s generally inclined
motion Yang and Lin (2021). In this work, we will derive the quasi-Keplerian
motion of a test particle in the gravitational field of regular Bardeen black
hole under the standard spherical coordinates. The magnetic charge effects of
the Bardeen black hole on the quasi-Keplerian motion including the orbital
perihelion precession are displayed explicitly.

The rest of this paper is organized as follows. Section II introduces the
standard metric of Bardeen black hole in the 2PN approximation, the
corresponding Lagrangian, as well as the orbital energy and angular momentum.
In Section III we present the detail derivation on the 2PN solution for the
quasi-Keplerian motion. Summary is given in section IV.

##  II The 2PN Lagrangian, energy, angular momentum in Bardeen spacetime

Assume the regular Bardeen black hole is located at the coordinate origin. In
the standard spherical coordinates, the exact metric of Bardeen black hole can
be expressed as

$\displaystyle
ds^{2}=-f(r)dt^{2}+f(r)^{-1}dr^{2}+r^{2}d\theta^{2}+r^{2}\sin\theta^{2}d\phi^{2},$
(1)

where the lapse function $f(r)$ is given by

$\displaystyle f(r)=1-\frac{2mr^{2}}{(r^{2}+g^{2})^{3/2}}~{},$ (2)

here $m$ denotes the standard gravitational mass parameter, while $g$
corresponds to the magnetic charge parameter measured in units of $m$. When
$g=0$, this metric behaves as a Schwarzschild metric. When
$g^{2}\leq\frac{16}{27}m^{2}$, a no-singularity black hole is described. In
particular, when $g^{2}=\frac{16}{27}m^{2}$, single horizon solution is
allowed, i.e., regular extremal black hole as shown in Fig 1. The
gravitational constant and the speed of light in vacuum are set as 1.

Figure 1:  Horizons of the Bardeen black hole with different values of $g$.

According to the theory of the PN approximation, this metric can be expanded
into the power of $\frac{m}{r}$ with $r\\!=\\!|\bm{x}|$. Keeping all terms to
the 2PN order for the dynamics of the massive particle, we have

$\displaystyle g_{00}=-1+\frac{2m}{r}-\frac{3g^{2}m}{r^{3}}~{},$ (3)
$\displaystyle g_{0i}=0~{},$ (4) $\displaystyle
g_{ij}=\delta_{ij}+\Big{(}\frac{2m}{r}+\frac{4m^{2}}{r^{2}}\Big{)}\frac{x^{i}x^{j}}{r^{2}},$
(5)

where $i$ and $j$ running from 1 to 3, $\epsilon_{ijk}$ is Levi-Civita symbol
Weinberg (1972).

With the 2PN metric Eqs. (3)-(5), the Lagrangian of the test particle can be
written as

$\displaystyle{\rm
L}=\frac{1}{2}\bm{v}^{2}+\frac{m}{r}+\frac{1}{8}\bm{v}^{4}+\frac{1}{2}\frac{m}{r}\bm{v}^{2}+\frac{1}{2}\frac{m^{2}}{r^{2}}+\frac{m(\bm{v}\\!\cdot\\!\bm{x})^{2}}{r^{3}}+\frac{1}{2}\frac{m^{3}}{r^{3}}\Big{(}1-\frac{3g^{2}}{m^{2}}\Big{)}$
$\displaystyle~{}~{}~{}~{}~{}~{}+\frac{1}{16}\bm{v}^{6}+\frac{3}{4}\frac{m^{2}}{r^{2}}\bm{v}^{2}+\frac{3}{8}\frac{m}{r}\bm{v}^{4}+\frac{3m^{2}(\bm{v}\\!\cdot\\!\bm{x})^{2}}{r^{4}}+\frac{m(\bm{v}\\!\cdot\\!\bm{x})^{2}\bm{v}^{2}}{2r^{3}}~{},$
(6)

where $\bm{v}$ denotes the the test particle’s velocity. According to this
Lagrangian, the energy $\mathcal{E}$ and the angular momentum $\mathcal{J}$ of
the equatorial motion can be obtained as

$\displaystyle\mathcal{E}=\frac{1}{2}\bm{v}^{2}-\frac{m}{r}+\frac{3}{8}\bm{v}^{4}+\frac{1}{2}\frac{m}{r}\bm{v}^{2}-\frac{1}{2}\frac{m^{2}}{r^{2}}+\frac{m(\bm{v}\\!\cdot\\!\bm{x})^{2}}{r^{3}}-\frac{1}{2}\frac{m^{3}}{r^{3}}\Big{(}1-\frac{3g^{2}}{m^{2}}\Big{)}\\!$
$\displaystyle~{}~{}~{}~{}~{}~{}+\frac{5}{16}\bm{v}^{6}+\frac{3}{4}\frac{m^{2}}{r^{2}}\bm{v}^{2}+\frac{9}{8}\frac{m}{r}\bm{v}^{4}+\frac{3m^{2}(\bm{v}\\!\cdot\\!\bm{x})^{2}}{r^{4}}+\frac{3}{2}\frac{m(\bm{v}\\!\cdot\\!\bm{x})^{2}\bm{v}^{2}}{r^{3}}~{},$
(7)
$\displaystyle\mathcal{J}=|\bm{x}\\!\times\\!\bm{v}|\Big{[}1+\frac{1}{2}\bm{v}^{2}+\frac{m}{r}+\frac{3}{8}\bm{v}^{4}+\frac{3}{2}\frac{m^{2}}{r^{2}}+\frac{3}{2}\frac{m}{r}\bm{v}^{2}+\frac{m(\bm{v}\\!\cdot\\!\bm{x})^{2}}{r^{3}}\Big{]}~{}.$
(8)

##  III The quasi-Keplerian motion in the 2PN approximation

We deduce the analytical solution for the quasi-Keplerian motion according to
the same procedure given by Brumberg Brumberg (1972).

The trajectory of the test particle in the equatorial plane of the Bardeen
black hole can be expressed as

$\bm{x}=r(\cos\phi\,\bm{e}_{x}+\sin\phi\,\bm{e}_{y})~{},$ (9)

where $\phi$ is the azimuthal angle. $\bm{e}_{x}$ and $\bm{e}_{y}$ are the
unit vectors of the $x$-axis and $y$-axis.

We can re-write the orbital energy and angular momentum in Eqs. (7)-(8) as

$\displaystyle\hskip
0.0pt\mathcal{E}=\frac{1}{2}(\dot{r}^{2}\\!+\\!r^{2}\dot{\phi}^{2})\\!-\\!\frac{m}{r}\\!+\\!\frac{3}{8}(\dot{r}^{2}\\!+\\!r^{2}\dot{\phi}^{2})^{2}\\!+\\!\frac{1}{2}\frac{m}{r}(\dot{r}^{2}\\!+\\!r^{2}\dot{\phi}^{2})\\!-\\!\frac{m^{2}}{2r^{2}}\\!+\\!\frac{m}{r}\dot{r}^{2}\\!-\\!\frac{m^{3}}{2r^{3}}\Big{(}1\\!-\\!\frac{3g^{2}}{m^{2}}\Big{)}$
$\displaystyle\hskip
28.45274pt+\frac{5}{16}(\dot{r}^{2}\\!+\\!r^{2}\dot{\phi}^{2})^{3}\\!+\\!\frac{3}{4}\frac{m^{2}}{r^{2}}(\dot{r}^{2}\\!+\\!r^{2}\dot{\phi}^{2})\\!+\\!\frac{9}{8}\frac{m}{r}(\dot{r}^{2}\\!+\\!r^{2}\dot{\phi}^{2})^{2}\\!+\\!\frac{3m^{2}}{r^{2}}\dot{r}^{2}\\!+\\!\frac{3}{2}\frac{m}{r}\dot{r}^{2}(\dot{r}^{2}\\!+\\!r^{2}\dot{\phi}^{2}),$
(10) $\displaystyle\hskip
0.0pt\mathcal{J}=r^{2}\dot{\phi}\Big{[}1\\!+\\!\frac{1}{2}(\dot{r}^{2}\\!+\\!r^{2}\dot{\phi}^{2})\\!+\\!\frac{m}{r}\\!+\\!\frac{3}{8}(\dot{r}^{2}\\!+\\!r^{2}\dot{\phi}^{2})^{2}\\!+\\!\frac{3}{2}\frac{m}{r}(\dot{r}^{2}\\!+\\!r^{2}\dot{\phi}^{2})\\!+\\!\frac{3}{2}\frac{m^{2}}{r^{2}}\\!+\\!\frac{m}{r}\dot{r}^{2}\Big{]}~{},$
(11)

where the dot means the derivative with respect to the time. From these two
expressions we can obtain

$r^{4}\dot{\phi}^{2}=\mathcal{J}^{2}\Big{(}1\\!-\\!2\,\mathcal{E}\\!-\\!\frac{4m}{r}\\!+\\!3\,\mathcal{E}^{2}\\!+\\!\frac{4m^{2}}{r^{2}}\\!+\\!8\,\mathcal{E}\frac{m}{r}\Big{)}~{},$
(12)

and

$\dot{r}^{2}=A+\frac{B}{r}+\frac{C}{r^{2}}+\frac{D}{r^{3}}+\frac{E}{r^{4}}~{},$
(13)

with

$\displaystyle
A=2\mathcal{E}\Big{(}1\\!-\\!\frac{3}{2}\,\mathcal{E}\\!+\\!2\,\mathcal{E}^{2}\Big{)}~{},$
(14) $\displaystyle
B=2m\Big{(}1\\!-\\!6\,\mathcal{E}\\!+\\!9\,\mathcal{E}^{2}\Big{)}~{},$ (15)
$\displaystyle
C=-\mathcal{J}^{2}\Big{(}1\\!-\\!2\,\mathcal{E}\\!+\\!\frac{8m^{2}}{\mathcal{J}^{2}}\\!-\\!24\frac{m^{2}\mathcal{E}}{\mathcal{J}^{2}}\\!+\\!3\,\mathcal{E}^{2}\Big{)}~{},$
(16) $\displaystyle
D=6m\mathcal{J}^{2}\Big{[}1\\!-\\!2\,\mathcal{E}\\!+\\!\frac{4}{3}\frac{m^{2}}{\mathcal{J}^{2}}\Big{(}1\\!-\\!\frac{3g^{2}}{8m^{2}}\Big{)}\Big{]}~{},$
(17) $\displaystyle E=-12m^{2}\mathcal{J}^{2}~{}.$ (18)

Throughout this paper, the signs “$\pm$” denote the anti-clockwise motion and
the clockwise motion, respectively.

utilizing the relation

$\dot{r}^{2}=\Big{[}\frac{d(1/r)}{d\phi}\Big{]}^{2}(r^{4}\dot{\phi}^{2})~{},$
(19)

and plugging Eqs. (12)-(13) into (19), we can express the radial equation in
the form

$\displaystyle\Big{[}\frac{d(1/r)}{d\phi}\Big{]}^{2}=A^{\prime}+\frac{B^{\prime}}{r}+\frac{C^{\prime}}{r^{2}}+\frac{D^{\prime}}{r^{3}}+\frac{E^{\prime}}{r^{4}}~{},$
(20)

with

$\displaystyle
A^{\prime}=\frac{2\mathcal{E}}{\mathcal{J}^{2}}\Big{(}1\\!+\\!\frac{1}{2}\,\mathcal{E}\Big{)}~{},$
(21) $\displaystyle B^{\prime}=\frac{2m}{\mathcal{J}^{2}}~{},$ (22)
$\displaystyle C^{\prime}=-1~{},$ (23) $\displaystyle
D^{\prime}=2m\Big{(}1\\!-\\!\frac{3g^{2}}{2\mathcal{J}^{2}}\Big{)}~{},$ (24)
$\displaystyle E^{\prime}=0~{}.$ (25)

Since the right hand side of Eq. (20) is a fourth-order polynomial in
$r^{-1}$, we can further re-write it as

$\displaystyle\Big{[}\frac{d(1/r)}{d\phi}\Big{]}^{2}=\Big{[}\frac{1}{r}\\!-\\!\frac{1}{a_{r}(1+e_{r})}\Big{]}\Big{[}\frac{1}{a_{r}(1-e_{r})}\\!-\\!\frac{1}{r}\Big{]}\Big{(}C_{1}+\frac{C_{2}}{r}+\frac{C_{3}}{r^{2}}\Big{)}~{}.$
(26)

Comparing the coefficients between Eq. (20) and Eq. (26), we have

$\displaystyle
a_{r}=\frac{m}{-2\mathcal{E}}\Big{[}1\\!+\\!\frac{3}{2}\mathcal{E}\\!+\\!\frac{1}{4}\mathcal{E}^{2}\\!+\\!8\frac{m^{2}\mathcal{E}}{\mathcal{J}^{2}}\Big{(}1\\!-\\!\frac{3g^{2}}{8m^{2}}\Big{)}\Big{]}~{},$
(27) $\displaystyle
e_{r}^{2}=1\\!+\\!\frac{2\mathcal{E}\mathcal{J}^{2}}{m^{2}}\\!-\\!\mathcal{E}\Big{(}8\\!+\\!7\frac{\mathcal{E}\mathcal{J}^{2}}{m^{2}}\Big{)}\\!+\\!\,\mathcal{E}^{2}\Big{[}16\frac{\mathcal{E}\mathcal{J}^{2}}{m^{2}}\\!-\\!20\Big{(}1\\!-\\!\frac{3g^{2}}{5m^{2}}\Big{)}\\!-\\!32\frac{m^{2}}{\mathcal{E}\mathcal{J}^{2}}\Big{(}1\\!-\\!\frac{3g^{2}}{8m^{2}}\Big{)}\Big{]}~{},$
(28) $\displaystyle
C_{1}=1\\!-\\!\frac{4m^{2}}{\mathcal{J}^{2}}\\!-\\!8\frac{m^{2}\mathcal{E}}{\mathcal{J}^{2}}\\!-\\!\frac{16m^{4}}{\mathcal{J}^{4}}\Big{(}1\\!-\\!\frac{3g^{2}}{8m^{2}}\Big{)}~{},$
(29) $\displaystyle
C_{2}=-2m\Big{(}1\\!-\\!\frac{3g^{2}}{2\mathcal{J}^{2}}\Big{)}~{},$ (30)
$\displaystyle C_{3}=0~{}.$ (31)

It can be seen from Eq. (26) that $r_{\pm}=a_{r}(1\pm e_{r})$ represent the
maximal and minimal values for $r$. Hence, $a_{r}$ and $e_{r}$ can be regarded
as the semi-major axis and the eccentricity of the quasi-Keplerian orbit.

The solution of Eq. (26) can be written as:

$\displaystyle r=\frac{a_{r}(1-e_{r}^{2})}{1+e_{r}\cos{f}}~{},$ (32)

with $f$ being the true anomaly for the quasi-Keplerian orbit and satisfying

$\displaystyle\Big{(}\frac{df}{d\phi}\Big{)}^{2}=C_{1}+\frac{C_{2}}{r}+\frac{C_{3}}{r^{2}}~{}.$
(33)

Substituting Eqs. (29)-(32) into Eq. (33), we have

$\displaystyle\frac{df}{d\phi}=F\Big{\\{}1\\!-\\!\frac{m^{2}}{\mathcal{J}^{2}}\Big{[}1\\!+\\!2\mathcal{E}\\!+\\!10\frac{m^{2}}{\mathcal{J}^{2}}\Big{(}1\\!-\\!\frac{3g^{2}}{20m^{2}}\Big{)}\Big{]}e_{r}\cos
f\\!-\\!\frac{m^{4}}{4\mathcal{J}^{4}}e_{r}^{2}\cos 2f\Big{\\}}~{},$ (34)

with

$\displaystyle
F=\Big{[}1\\!-\\!\frac{3m^{2}}{\mathcal{J}^{2}}\\!-\\!\frac{13}{2}\frac{m^{2}\mathcal{E}}{\mathcal{J}^{2}}\\!-\\!\frac{67}{4}\frac{m^{4}}{\mathcal{J}^{4}}\Big{(}1\\!-\\!\frac{18g^{2}}{67m^{2}}\Big{)}\Big{]}~{}.$
(35)

Making integration of Eq. (34), we can obtain

$\phi\Big{(}\frac{2\pi}{\Phi}\Big{)}=f+\frac{m^{2}}{\mathcal{J}^{2}}\Big{[}1\\!+\\!2\mathcal{E}\\!+\\!10\frac{m^{2}}{\mathcal{J}^{2}}\Big{(}1\\!-\\!\frac{3g^{2}}{20m^{2}}\Big{)}\Big{]}e_{r}\sin
f\\!+\\!\frac{3}{8}\frac{m^{4}}{\mathcal{J}^{4}}e_{r}^{2}\sin 2f~{},$ (36)

with

$\Phi=2\pi\Big{[}1\\!+\\!3\frac{m^{2}}{\mathcal{J}^{2}}\\!+\\!\frac{15}{2}\frac{m^{2}\mathcal{E}}{\mathcal{J}^{2}}\\!+\\!\frac{105}{4}\frac{m^{4}}{\mathcal{J}^{4}}\Big{(}1\\!-\\!\frac{6g^{2}}{35m^{2}}\Big{)}\Big{]}~{}.$
(37)

Finally, we derive the time dependence of the quasi-Keplerian motion.
Combining Eqs. (12) and (34)-(35), we have

$\displaystyle\hskip
0.0ptr^{2}\dot{f}=\mathcal{J}\Big{\\{}1\\!-\\!\mathcal{E}\\!-\\!2\frac{m}{r}\\!-\\!3\frac{m^{2}}{\mathcal{J}^{2}}\\!+\\!\mathcal{E}^{2}\\!+\\!2\mathcal{E}\frac{m}{r}\\!-\\!\frac{7}{2}\frac{m^{2}\mathcal{E}}{\mathcal{J}^{2}}\\!+\\!6\frac{m}{r}\frac{m^{2}}{\mathcal{J}^{2}}\\!-\\!\frac{m^{4}}{\mathcal{J}^{4}}\Big{(}\frac{67}{4}\\!-\\!\frac{9g^{2}}{2m^{2}}\Big{)}$
$\displaystyle\hskip
42.67912pt-\frac{m^{2}}{\mathcal{J}^{2}}\Big{[}1\\!+\\!\mathcal{E}\\!-\\!2\frac{m}{r}\\!+\\!\frac{m^{2}}{\mathcal{J}^{2}}\Big{(}7\\!-\\!\frac{3g^{2}}{2m^{2}}\Big{)}\Big{]}e_{r}\cos
f\\!-\\!\frac{1}{4}\frac{M^{4}}{\mathcal{J}^{4}}e_{r}^{2}\cos 2f\Big{\\}}.$
(38)

Introducing the post-Newtonian eccentric anomaly $u$ by the relations

$\hskip 0.0pt\sin f\\!=\\!\frac{(1\\!-\\!e_{r}^{2})^{\frac{1}{2}}\sin
u}{1\\!-\\!e_{r}\cos u};\,\,\cos f\\!=\\!\frac{\cos
u\\!-\\!e_{r}}{1\\!-\\!e_{r}\cos
u};\,\,f\\!=\\!2\\!\arctan\\!\Big{(}\sqrt{\frac{1\\!+\\!e_{r}}{1\\!-\\!e_{r}}}\tan\frac{u}{2}\Big{)},$
(39)

we have

$\displaystyle\frac{df}{dt}=\frac{(1-e_{r}^{2})^{1/2}}{1-e_{r}\cos
u}\frac{du}{dt}~{},$ (40)

and we can formulate the orbit given in Eq. (32) in terms of $u$ as

$r=a_{r}(1-e_{r}\cos u)~{}.$ (41)

Integrating Eq. (38) and making use of Eqs. (39)-(41), we can achieve the
final piece of the 2PN closed-form solution for the equatorial motion in
Bardeen spacetime.

$t\Big{(}\frac{2\pi}{{\rm
T}_{u}}\Big{)}=u-e_{t}\sin{u}+\frac{30m\,\mathcal{E}^{2}}{\sqrt{-2\,\mathcal{E}\mathcal{J}^{2}}}(f-u)~{},$
(42)

with ${\rm T}_{\\!u}$ being the period for the eccentric anomaly $u$ of the
quasi-Keplerian motion

${\rm T}_{u}=\frac{2\pi
m}{(-2\mathcal{E})^{\frac{3}{2}}}\Big{[}1\\!-\\!\frac{15}{4}\mathcal{E}\\!-\\!\frac{105}{32}\mathcal{E}^{2}\\!+\\!\frac{30m\mathcal{E}^{2}}{\sqrt{-2\mathcal{E}\mathcal{J}^{2}}}\Big{]}~{},$
(43)

and $e_{t}$ being the time eccentricity

$e_{t}=e_{r}\Big{[}1\\!+\\!6\,\mathcal{E}\\!+\\!27\,\mathcal{E}^{2}\\!-\\!\frac{30m\mathcal{E}^{2}}{\sqrt{-2\mathcal{E}\mathcal{J}^{2}}}\\!+\\!8\frac{m^{2}\mathcal{E}}{\mathcal{J}^{2}}\Big{(}1\\!-\\!\frac{3g^{2}}{8m^{2}}\Big{)}\Big{]}~{}.$
(44)

In the literature, one usually replace the true anomaly $f$ in the formula of
quasi-Keplerian equation with another true anomaly $\upsilon$, which requires
that $\sin\upsilon$ contribution vanishes at each PN order in
$\phi(\frac{2\pi}{\Phi})$ Memmesheimer  _et al._ (2004);
K$\ddot{o}$nigsd$\ddot{o}$rffer and Gopakumar (2005); Tessmer  _et al._
(2010).

Following the same method given in Ref Tessmer  _et al._ (2010), we set

$\upsilon=2\arctan\Big{(}\sqrt{\frac{1+e_{\phi}}{1-e_{\phi}}}\tan\frac{u}{2}\Big{)}~{},$
(45)

with

$e_{\phi}=e_{r}(1+\epsilon\,c_{1}+\epsilon^{2}\,c_{2})~{},$ (46)

differing from the radial eccentricity $e_{r}$ by some 1PN and 2PN level
corrections $c_{1}$ and $c_{2}$. Here $\epsilon$ only denotes the PN order and
does not have any value. Eliminating $u$ in Eq. (39) with the help of Eq.
(45), we have Tessmer  _et al._ (2010)

$\hskip-1.42271ptf\\!=\\!\upsilon+\epsilon\,c_{1}\frac{e_{r}}{e_{r}^{2}\\!-\\!1}\sin\upsilon+\epsilon^{2}\,\Big{[}\Big{(}c_{2}\\!-\\!c_{1}^{2}\frac{e_{r}^{2}}{e_{r}^{2}\\!-\\!1}\Big{)}\frac{e_{r}}{e_{r}^{2}\\!-\\!1}\sin\upsilon+\frac{c_{1}^{2}}{4}\frac{e_{r}^{2}}{(e_{r}^{2}\\!-\\!1)^{2}}\sin
2\upsilon\Big{]}.$ (47)

Substituting this result into Eq. (36) and requiring the $\sin\upsilon$ term
to vanish in $\phi(\frac{2\pi}{\Phi})$, we can obtain

$\displaystyle c_{1}=-2\mathcal{E}~{},$ (48) $\displaystyle
c_{2}=-\\!18\mathcal{E}\frac{m^{2}}{\mathcal{J}^{2}}\Big{(}1\\!-\\!\frac{g^{2}}{6m^{2}}\Big{)}~{},$
(49)

which lead to

$e_{\phi}=e_{r}\Big{[}1\\!-\\!2\mathcal{E}\\!-\\!18\frac{m^{2}\mathcal{E}}{\mathcal{J}^{2}}\Big{(}1\\!-\\!\frac{g^{2}}{6m^{2}}\Big{)}\Big{]}~{},$
(50)
$\phi\Big{(}\frac{2\pi}{\Phi}\Big{)}=\upsilon\\!+\\!\frac{1}{8}\frac{m^{4}}{\mathcal{J}^{4}}e_{r}^{2}\sin
2\upsilon~{}.$ (51)

With the true anomaly $\upsilon$, we can re-express the time dependance of the
quasi-Keplerian motion Eq. (42) in the form of

$t\Big{(}\frac{2\pi}{{\rm
T}_{u}}\Big{)}=u-e_{t}\sin{u}+\frac{30m\,\mathcal{E}^{2}}{\sqrt{-2\,\mathcal{E}\mathcal{J}^{2}}}(\upsilon-u)~{}.$
(52)

Notice that all the formulas in this work are valid up to the 2PN accuracy.

##  IV Summary

We start with the 2PN metric of the Bardeen black hole in the standard
coordinates, calculate the corresponding orbital energy and angular momentum
of the test particle in the equatorial plane, and then through an iterative
method and function fitting method, derive the 2PN solution for the quasi-
Keplerian motion in the Bardeen spacetime. We obtain two slightly different
but equivalent formulations in the 2PN approximation.

The results are summarized as follows.

The first formulation can be expressed as

$\displaystyle\bm{x}=r(\cos\phi\,\bm{e}_{x}+\sin\phi\,\bm{e}_{y})~{},$
$\displaystyle r=a_{r}(1-e_{r}\cos u)~{},$
$\displaystyle\phi\Big{(}\frac{2\pi}{\Phi}\Big{)}=f+N_{0}\sin f+N_{1}\sin
2f~{},$ $\displaystyle
f=2\arctan\Big{(}\sqrt{\frac{1+e_{r}}{1-e_{r}}}\tan\frac{u}{2}\Big{)}~{},$
$\displaystyle t\Big{(}\frac{2\pi}{{\rm
T}_{u}}\Big{)}=u-e_{t}\sin{u}+N_{2}(f-u)~{},$

and the second formulation can be expressed as

$\displaystyle\bm{x}=r(\cos\phi\,\bm{e}_{x}+\sin\phi\,\bm{e}_{y})~{},$
$\displaystyle r=a_{r}(1-e_{r}\cos u)~{},$
$\displaystyle\phi\Big{(}\frac{2\pi}{\Phi}\Big{)}=\upsilon+N_{3}\sin
2\upsilon~{},$
$\displaystyle\upsilon=2\arctan\Big{(}\sqrt{\frac{1+e_{\phi}}{1-e_{\phi}}}\tan\frac{u}{2}\Big{)}~{},$
$\displaystyle t\Big{(}\frac{2\pi}{{\rm
T}_{u}}\Big{)}=u-e_{t}\sin{u}+N_{2}(\upsilon-u)~{},$

where

$\displaystyle
a_{r}=\frac{m}{-2\mathcal{E}}\Big{[}1\\!+\\!\frac{3}{2}\mathcal{E}\\!+\\!\frac{1}{4}\mathcal{E}^{2}\\!+\\!8\frac{m^{2}\mathcal{E}}{\mathcal{J}^{2}}\Big{(}1\\!-\\!\frac{3g^{2}}{8m^{2}}\Big{)}\Big{]}~{},$
$\displaystyle
e_{r}^{2}=1\\!+\\!\frac{2\mathcal{E}\mathcal{J}^{2}}{m^{2}}\\!-\\!\mathcal{E}\Big{(}8\\!+\\!7\frac{\mathcal{E}\mathcal{J}^{2}}{m^{2}}\Big{)}\\!+\\!\,\mathcal{E}^{2}\Big{[}16\frac{\mathcal{E}\mathcal{J}^{2}}{m^{2}}\\!-\\!20\Big{(}1\\!-\\!\frac{3g^{2}}{5m^{2}}\Big{)}\\!-\\!32\frac{m^{2}}{\mathcal{E}\mathcal{J}^{2}}\Big{(}1\\!-\\!\frac{3g^{2}}{8m^{2}}\Big{)}\Big{]}~{},$
$\displaystyle
e_{t}=e_{r}\Big{[}1\\!+\\!6\,\mathcal{E}\\!+\\!27\,\mathcal{E}^{2}\\!-\\!\frac{30m\mathcal{E}^{2}}{\sqrt{-2\mathcal{E}\mathcal{J}^{2}}}\\!+\\!8\frac{m^{2}\mathcal{E}}{\mathcal{J}^{2}}\Big{(}1\\!-\\!\frac{3g^{2}}{8m^{2}}\Big{)}\Big{]}~{},$
$\displaystyle
e_{\phi}=e_{r}\Big{[}1\\!-\\!2\mathcal{E}\\!-\\!18\frac{m^{2}\mathcal{E}}{\mathcal{J}^{2}}\Big{(}1\\!-\\!\frac{g^{2}}{6m^{2}}\Big{)}\Big{]}~{},$
$\displaystyle\Phi=2\pi\Big{[}1\\!+\\!3\frac{m^{2}}{\mathcal{J}^{2}}\\!+\\!\frac{15}{2}\frac{m^{2}\mathcal{E}}{\mathcal{J}^{2}}\\!+\\!\frac{105}{4}\frac{m^{4}}{\mathcal{J}^{4}}\Big{(}1\\!-\\!\frac{6g^{2}}{35m^{2}}\Big{)}\Big{]}~{},$
$\displaystyle
N_{0}=\frac{m^{2}}{\mathcal{J}^{2}}\Big{[}1\\!+\\!2\mathcal{E}\\!+\\!10\frac{m^{2}}{\mathcal{J}^{2}}\Big{(}1\\!-\\!\frac{3g^{2}}{20m^{2}}\Big{)}\Big{]}\Big{(}1\\!+\\!\frac{2\mathcal{E}\mathcal{J}^{2}}{m^{2}}\Big{)}^{\\!\frac{1}{2}}~{},$
$\displaystyle
N_{1}=\frac{3}{8}\frac{m^{4}}{\mathcal{J}^{4}}\Big{(}1+\frac{2\mathcal{E}\mathcal{J}^{2}}{m^{2}}\Big{)}~{},$
$\displaystyle
N_{2}=\frac{30m\,\mathcal{E}^{2}}{\sqrt{-2\,\mathcal{E}\mathcal{J}^{2}}}~{},$
$\displaystyle
N_{3}=\frac{1}{8}\frac{m^{4}}{\mathcal{J}^{4}}\Big{(}1+\frac{2\mathcal{E}\mathcal{J}^{2}}{m^{2}}\Big{)}~{},$
$\displaystyle{\rm T}_{u}=\frac{2\pi
m}{(-2\mathcal{E})^{\frac{3}{2}}}\Big{(}1\\!-\\!\frac{15}{4}\mathcal{E}\\!-\\!\frac{105}{32}\mathcal{E}^{2}+\frac{30m\mathcal{E}^{2}}{\sqrt{-2\mathcal{E}\mathcal{J}^{2}}}\Big{)}~{}.$

In the formulations, $a_{r}$, $e_{r}$ and $u$ denote the semi-major axis, the
eccentricity, the eccentric anomaly of the quasi-Keplerian motion in the PN
approximation, respectively. $f$ and $\upsilon$ are two slightly different
definitions of the true anomaly. ${\rm T}_{u}$ denotes the orbital period.
Notice that the effects of the magnetic charge on the test particle’s motion
are characterized by the terms containing $g^{2}$ in the above formulas.

The test particle’s orbital perihelion precession can be calculated by

$\displaystyle\Delta\phi\\!\equiv\\!\Phi-2\pi=6\pi\frac{m^{2}}{\mathcal{J}^{2}}\\!+\\!15\frac{\pi
m^{2}\mathcal{E}}{\mathcal{J}^{2}}\\!+\\!\frac{105}{2}\frac{\pi
m^{4}}{\mathcal{J}^{4}}\Big{(}1\\!-\\!\frac{6g^{2}}{35m^{2}}\Big{)}~{}.$ (53)

When the magnetic charge of the Bardeen black hole vanishes ($g=0$), Eq. (53)
can be simplified to the 2PN Schwarzschild periastron precession Damour and
Schäfer (1988); Memmesheimer  _et al._ (2004).

Finally, it can be observed that the test particle’s orbital period is
independent on the the magnetic charge up to the 2PN order.

## ACKNOWLEDGEMENT

This work was supported in part by the National Natural Science Foundation of
China (Grant Nos. 11973025 and 11947404).

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