# Bound orbits around charged black strings

A S. Habibina  H. S. Ramadhan  Departemen Fisika, FMIPA, Universitas
Indonesia, Depok, 16424, Indonesia.

###### Abstract

We study the geodesics of $5d$ Reissner-Nordstrom and nonsingular black
strings, and establish a rational bound orbit taxonomy for both massive as
well as null test particles. For the timelike case, test particles with high
energy (that would have made them plunge into or scatter off a black hole)
could still form bound orbits around the black strings. We calculate the
accumulated angles of the corresponding radial periods and show that they are
higher than their $4d$ counterparts. For the null case, we found the existence
of stable null orbits outside their respective horizons, which do not exist in
the four dimensions except at their extremal limit.

##  I Introduction

Black holes in higher dimensions have long been an interesting as well as
intriguing subject. Mathematically, the additions of extra dimensions pose a
considerable challenge to obtaining exact solutions. Astrophysically, their
signature might shed light on the unification of fundamental forces. Although
the possibility of extra dimension has been considered by Kaluza and Klein as
early as the 1920s Kaluza:1921tu , it was Tangherlini who generalized the
Reissner-Nordstrom (RN) black hole to higher dimensions Tangherlini:1963bw .
Myers and Perry then followed in obtaining solutions for higher-dimensional
Kerr-Newman Myers:1986un . The spherical event horizon is not the only
possible higher-dimensional BH solution. A trivial extension of a
Schwarzschild black hole to a higher dimension would be a charged black string
whose metric does not depend on its compact extra dimension. Such solution has
been shown to be stable by Gregory and Laflamme Gregory:1987nb . More
nontrivial black strings are, of course, abundant in the literature
Gibbons:1987ps ; Gregory:1995qh ; Emparan:2001wn .

Even in $4d$ due to its very definition, a black hole cannot be observed
directly. Thus, so far the probe for their existence relies on: (i) the
gravitational wave LIGOScientific:2016aoc  and (ii) orbital dynamics around it
(e.g., perihelion precession GRAVITY:2020gka , gravitational lensing and
shadow EventHorizonTelescope:2019dse ; Akiyama2022 , or photon orbits
GRAVITY:2018 ). It was Hagihara in 1930 that first obtained the exact solution
tof the Schwarzschild geodesic Hagihara(1930)  (see also Hackmann
Hackmann:2010tqa ). The analytic solution is in the form of the Weierstrass
elliptic function. It was later extended to the case of (static and rotating)
black strings by Grunau Grunau:2013oca . The behavior of massive neutral and
charged particles around weekly magnetized Schwarzschild string was studied
Rezvanjou:2017hox .

In a series of interesting papers, Levin and collaborators propose a taxonomy
based on isomorphism between periodic orbits around (or of pairs of) $4d$
black holes and rational numbers Levin:2008mq ; Levin:2008ci ; Perez-
Giz:2008ajn ; Healy:2009zm . They found that there exists a set of three
integers $(z,w,v)$ that can be combined to form a rational number

$q\equiv w+{v\over z},$ (1)

that completely parametrizes every (timelike or null) bound orbit. Here $w$
defines the number of whirls, $z$ expresses the number of leaves, and $v$
denotes the order the leaves are traced out. Within this formalism, precession
of Mercury’s perihelion can be perceived as periodic orbit with very large
$z$, around $z\sim 1.296\times 10^{7}$ Levin:2008mq . This rational orbit
formalism has been applied to Schwarzchild, RN Misra:2010pu , and Kerr
solutions Levin:2008mq , as well to the black hole binary system.

Even though its unchraged geodesic has extensively been studied Grunau:2013oca
, to the best of our knowledge no one yet elaborates the rational orbit
formalism on the charged black string. It is therefore the purpose of this
paper to accomplish this task. Since the orbital motion is confined to $4d$,
the hope is that the effect of one extra spatial dimension will give distinct
observational signatures that can distinguish black hole from black string.
The “anomaly” orbital motion around a black object can thus serve as an
astrophysical probe for the existence of extra dimension. To do so, we
specifically study two types of charged strings: the charged Gregory-Laflamme
Gregory:1987nb  and the nonsingular black strings. The latter is the $5d$
extension of the well-known Bardeen black hole bardeen . These two are simple
toy models but at the same time are rich enough to prove our point. The
organization is as follows. We examine the general black string geodesic and
effective potential in Section II. Section III is devoted to the all possible
types of orbits and the formalism of rational orbit taxonomy. The orbital zoo
of RN and Bardeen black strings are presented in Sections IV and V,
respectively. Finally, we conclude the result of this study in VI.

##  II Black String Geodesics

A trivial $5d$ extension of a static and spherically-symmetric black hole can
always be written as Gregory:1987nb

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

where $\omega$ is an extra compact spatial dimension and $f(r)$ is any metric
solution that solves the $5d$ Einstein equations. Since the string preserves
the $4d$ spherical symmetry, under the condition of equatorial plane
($\theta=\pi/2$) the invariance condition
$g_{\mu\nu}\dot{x}^{\mu}\dot{x}^{\nu}=-\epsilon$ gives the constraint equation
as

$-\epsilon=-f(r)\dot{t}^{2}+f(r)^{-1}\dot{r}^{2}+r^{2}\dot{\phi}^{2}+\dot{\omega}^{2},$
(3)

where $\epsilon=\\{1,0\\}$ corresponds to timelike and null geodesics,
respectively. Using Euler-Lagrange equation, we obtain the equation of motion
for each coordinate:

$\displaystyle\dot{t}$ $\displaystyle=$
$\displaystyle\frac{dt}{d\tau}=\frac{\mathbb{E}}{f(r)},$
$\displaystyle\dot{\phi}$ $\displaystyle=$
$\displaystyle\frac{d\phi}{d\tau}=\frac{\mathbb{L}}{r^{2}},$
$\displaystyle\dot{\omega}$ $\displaystyle=$
$\displaystyle\frac{d\omega}{d\tau}=\mathbb{J}.$ (4)

Inserting (II) into (3) and rescaling $\mathbb{L}^{2}\rightarrow 1/L$,
$\mathbb{J}^{2}\rightarrow J$, we have the geodesic equation in the form of

$\dot{r}^{2}=\mathbb{E}^{2}-V_{eff},$ (5)

where the effective potential $V_{eff}$ can be written as

$\displaystyle V_{eff}(r)$ $\displaystyle=$ $\displaystyle
f(r)\left(\epsilon+J+\frac{1}{Lr^{2}}\right).$ (6)

##  III Black String Orbits

###  III.1 Types of Orbit

|
---|---
|
|
Figure 1: Typicals of terminating orbit, escape orbit, and bound orbit around
black hole (left column) and black ring (right columnn).

Particle trajectories in black string can be classified in terms of orbital
types and regions. Denoting $r_{EH}$ as the radius of outer event horizon, the
three possible orbit types and their definitions are listed as below, in which
their visualizations are shown in Fig. 1. Here we use similar convention as in
Grunau Grunau:2013oca .

  1. 1.

Terminating orbit (TO): particles arrive at a periapsis $r_{p}$ and falls into
singularity.

  2. 2.

Escape orbit (EO): particles come from $\infty$, approach a periapsis $r_{p}$,
and return to $\infty$.

  3. 3.

Bound orbit (BO): particles oscillate between its periapsis and apoapsis under
the condition of $r_{EH}<r_{p}<r_{a}<\infty$.

---
Figure 2: All possible regions (R I to R IV) according to various energy level. The blue line refers to energy level $\mathbb{E}^{2}$. Region |  Positive Real Zeros |  Orbit Types
---|---|---
I |  2 |  TO, EO
II |  0 |  EO
III |  1 |  TO
IV |  3 |  TO, BO
Table 1: The classification of regions based on its effective $V_{eff}$
potential and particle energy level $E$ along with their corresponding
possible orbit types.

Based on the effective potential, the regions can be classified using the
Hackman formalism Hackmann:2010tqa . Solving the Eq. (5) results in various
scenarios based on the numbers of its positive real solutions, which are shown
in Table 1. To help understanding the regions better, the dependence of the
orbits on the energy levels is shown in Fig. 2.

A typical $V_{eff}$ we study here generically have three turning points, as we
shall see later, only two of which are outside the horizon. The local maxima
represent the unstable circular orbit (UCO)111For timelike case, this local
maxima is called the marginally bound orbit (MBO) Ono:2015jqa ., while the
local minima identify the stable circular orbit (SCO). Both circular orbits
(CO) satisfy

$\dot{r}^{2}=0,\ \ \ \partial_{r}V_{eff}=0,$ (7)

which translates into

$\displaystyle f(r_{CO})\left(J+\epsilon+{1\over Lr_{CO}^{2}}\right)$
$\displaystyle=$ $\displaystyle\mathbb{E}^{2},$ $\displaystyle
f^{\prime}(r_{CO})\left(J+\epsilon+{1\over
Lr_{CO}^{2}}\right)-{2f(r_{CO})\over Lr_{CO}^{3}}$ $\displaystyle=$
$\displaystyle 0.$

Both CO depend on the angular momenta and black hole’s charge. For a given
black hole there exists a critical circular radius beyond which it no longer
supports bound orbits. For timelike case this critical value is called the
innermost stable circular orbit (ISCO), and this is achieved when UCO and SCO
radii merge. Mathematically, this is the inflection point of the effective
potential,

$V_{eff}=\mathbb{E}^{2},\ \ \ \partial_{r}V_{eff}=0,\ \ \
\partial^{2}_{r}V_{eff}=0.$ (9)

This adds the following condition,

$r\left[rf^{\prime\prime}(r)\left(Lr^{2}(J+\epsilon)+1\right)-4f^{\prime}(r)\right]+6f(r)=0,$
(10)

into (III.1).

###  III.2 Overview of Rational Orbits Taxonomy

Here we briefly review the scheme of Levin and Perez-Giz in assigning each
orbital solution to a distinct rational number Levin:2008mq . Any (positive)
rational number can always be cast as

$q=s+{m\over n},$ (11)

where $s\geq 0$ is an integer and $1\leq m\leq n-1$ are (relatively) prime
numbers. This taxonomy draws correspondence between $q$ and the topological
features of orbit. The claim is that each periodic orbit is completely
characterized by three integers $(z,w,v)$ that can be expressed as rational
number, Eq. 1. Each of them corresponds to a specific topological trait of a
periodic orbit. The first and most tangible number is $z$, which is the number
of leaves (or ”zooms”) of the orbit. The second integer, $w$, is the number of
whirls which the particle performs in its path from apastron to periastron to
the next apastron. Note that every particle performs at least a full $2\pi$
trip. The number of whirls is the additional integer number of full $2\pi$
accomplished beyond this. The last number $v$, the vertex number,
differentiate between orbits that have equal $z$ and $w$ but are geometrically
distinct. Particle can skip leaves in its motion from apastron to apastron.
One important thing to address is the degeneracy that arises when the quotient
$v=z$ is a reducible fraction. This is solved by requiring $v$ and $z$ to be
relatively prime. The bounds on v then can be written as

$\displaystyle 1\leq v\leq z-1\,\,\,\,$
$\displaystyle\text{if}\,\,\,\,\,\,\,\,z>1\,\,\text{and}\,\,z,v\,\,\text{are
relatively prime}$ (12) $\displaystyle v=0\,\,\,\,$
$\displaystyle\text{if}\,\,\,\,\,\,\,\,z=1.$

The proof of such claim is based on the fact that for a periodic orbit, the
accumulated angle between one apastron to another apastron can be expressed as

$\Delta\varphi_{r}=2\pi\left(1+q\right)={\Delta\varphi\over z},$ (13)

where $\Delta\varphi\equiv z\Delta\varphi_{r}$ is the total accumulated angle
in a complete orbit. We may also define the radial period as the (affine) time
taken by particle to return to the same radius upon starting at the apsis
$r_{a}$ (where $\mathbb{E}^{2}=V_{eff}$) Levin:2009sk ; Babar:2017gsg . If
within this period, the evolution of $\varphi$ is an integer multiple of
$2\pi$, we have a periodic orbit. Therefore the accumulated radial angle can
be written as

$\Delta\varphi_{r}=2\int_{r_{p}}^{r_{a}}\frac{\dot{\phi}}{\dot{r}}dr=2\int_{r_{p}}^{r_{a}}\frac{dr}{Lr^{2}\sqrt{\mathbb{E}^{2}-f(r)\left(\epsilon+J+\frac{1}{Lr^{2}}\right)}},$
(14)

where $r_{p}$ is the periastron radius. The rational number in Eq.(13) can be
related to the orbital angular frequency by the following arguments. Every
eccentric equatorial orbit has two types of frequencies: the radial
($\omega_{r}$) and angular ($\omega_{\varphi}$) frequencies. They are given by

$\displaystyle\omega_{r}$ $\displaystyle=$ $\displaystyle{2\pi\over T_{r}},$
$\displaystyle\omega_{\varphi}$ $\displaystyle=$ $\displaystyle{1\over
T_{r}}\int_{0}^{T_{r}}\frac{d\varphi}{dt}dt={\Delta\varphi_{r}\over T_{r}},$
(15)

where $T_{r}$ is the period of one radial cycle. Any periodic orbit must then
satisfy

${\omega_{\varphi}\over\omega_{r}}={\Delta\varphi_{r}\over 2\pi}=1+q.$ (16)

##  IV RN Black String

We consider the RN black string. It turns out that

$f(r)=1-{2M\over r}+{Q^{2}\over r^{2}},$ (17)

is still the solution of the corresponding $5d$ Einstein’s equations.
Alternatively, we can perceive this metric as a solution arising from the
Einstein-Maxwell compactification along the framework of Quasitopological
Electromagnetism (QTE) theory222We thank Adolfo Cisterna for bringing this
formalism into our attention. Cisterna:2020rkc ; Cisterna:2021ckn . The
horizons are still located at $r_{\pm}=M\pm\sqrt{M^{2}-Q^{2}}$, stretching to
extra dimension and creating a hypercylindrical topology. Despite the
remarkable simplicity in the metric solution, its geodesic phenomenology is
not quite that trivial. There is a rich family of orbital solutions, both for
timelike and null particles.

The typical $V_{eff}$ for massive particles is shown on Fig. 3. Notice that
the local maximum rises along with the increase of $J$ value. Since energy and
angular momentum are inversely proportional (see Eq. (5)), this means that the
existence of bound orbit with bigger value of the constant $J$ requires test
particles to have smaller $L$. For massless particles, the $V_{eff}$ is shown
on Fig. 4. We observe the same trend as revealed in the timelike one.
Nonetheless, we notice that in higher angular momentum case, the existence of
additional dimension ($J>0$) elevate the potential and create a condition that
allows photon to form bound orbits.

###  IV.1 Timelike CO and ISCO

The existence of CO and ISCO for timelike geodesic in $4d$ RN black hole has
been widely discussed in Pugliese:2010ps ; Schroven:2020ltb  and the
references therein. As stated earlier, the condition for CO, given by Eq.
(III.1), for the RN string translates into

$\displaystyle\left(J-\mathbb{E}^{2}+1\right)Lr_{CO}^{4}-2(J+1)MLr_{CO}^{3}+\left[\left(J+1\right)LQ^{2}+1\right]r_{CO}^{2}-2Mr_{CO}+Q^{2}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle
2(J+1)MLr_{CO}^{3}-2\left[1-\left(J+1\right)LQ^{2}\right]r_{CO}^{2}+6Mr_{CO}-4Q^{2}$
$\displaystyle=$ $\displaystyle 0,$

where $r_{CO}$ is the circular radii. Simple algebra shows that those
equations can be solved simultaneously to obtain one-parameter class of
solutions parametrized by $r_{CO}$,

$\displaystyle L$ $\displaystyle=$
$\displaystyle\frac{3Mr_{CO}-2Q^{2}-r_{CO}^{2}}{(J+1)r_{CO}^{2}\left(Q^{2}-Mr_{CO}\right)},$
$\displaystyle\mathbb{E}^{2}$ $\displaystyle=$
$\displaystyle\frac{(J+1)\left(-2Mr_{CO}+Q^{2}+r_{CO}^{2}\right)^{2}}{r_{CO}^{2}\left(-3Mr_{CO}+2Q^{2}+r_{CO}^{2}\right)}.$

In the extremal case we have

$L={r-2M\over\left(1+J\right)Mr^{2}},\ \ \
\mathbb{E}^{2}={\left(1+J\right)\left(r-M\right)^{3}\over\left(r-2M\right)r^{2}}.$
(20)

When $J\rightarrow 0$ they reduce to the circular motion conditions for
neutral test particles around $4d$ RN black hole Pugliese:2010ps .

For any given $L$ the ISCO is the smallest radius of circular orbit before the
particle plunges into the black hole. It is the inflection point of $V_{eff}$.
From condition (9), the additional equation to be solved is

$4(J+1)LMr_{ISCO}^{3}-6\left[(J+1)LQ^{2}+1\right]r_{ISCO}^{2}+24Mr_{ISCO}-20Q^{2}=0.$
(21)

Since the extra-dimensional signature always appears multiplying $L$, i.e.,
$(1+J)L$, then from the expression of $L$ in (IV.1) it is obvious that the
ISCO radii satisfy

$Mr_{ISCO}^{3}-6M^{2}r_{ISCO}^{2}+9MQ^{2}r_{ISCO}-4Q^{4}=0.$ (22)

The same thing happens for the extremal black strings. The ISCO radii becomes

$r_{ISCO}^{2}-5Mr_{ISCO}+4M^{2}=0,$ (23)

with solutions

$r_{ISCO}=\\{M,\ 4M\\},$ (24)

the same as the extremal $4d$ RN ISCO; the timelike ISCO is oblivion to the
existence of extra dimension (i.e., independent of $J$). Notice that this is
precisely the same condition for ISCO in $4d$ RN black hole Pugliese:2010ps .

|
---|---
Figure 3: [Left] Plot of effective potential for massive particles in RN two-
horizon case with $Q=0.8$ and $J=1$ and various values of $L$ and [Right]
similar plot with $L=0.04$ and various values of $J$. The dashed red lines
refer to the inner and outer horizons.

###  IV.2 Null Circular Orbit

For null geodesic ($\epsilon=0$) it is customary to define impact parameter
$b$, where $b^{2}\equiv 1/\mathbb{E}^{2}L$. The geodesic equation (5) becomes

$\dot{r}^{2}+V_{eff}=b^{2},\ \ \ V_{eff}\equiv f(r)\left(j+{1\over
r^{2}}\right),$ (25)

where $j\equiv JL$ and we subsequently rescale the proper time
$\tau\rightarrow\tau/L$. It is well-known that $4d$ RN spacetime does not
support stable CO, except in the extremal limit. To be precise, for any given
charge $Q/M$ the $\partial_{r}V_{eff}=0$ condition for null gives us a
quadratic equation whose roots are

$r^{\pm}_{CO}={3M\pm\sqrt{9M^{2}-8Q^{2}}\over 2}.$ (26)

One can easily observe that $r^{-}_{CO}$ is at the local minima of $V_{eff}$
and is between the two horizons. Thus, the only observable CO, called the
photon sphere, $r^{+}_{CO}$ is unstable Pradhan:2010ws ; Khoo:2016xqv . At
$M=Q$, the CO radii shifts further into

$r^{\pm}_{CO}=\\{2M,\ M\\};$ (27)

the stable $r_{CO}$ coincides with the extremal horizon $r_{extr}$.

The situation is rather different in this $5d$ black string. The additional
term that multiplies $j$ in $V_{eff}$ gives qubic equation for the
$\partial_{r}V_{eff}=0$,

$jMr_{CO}^{3}-\left(jQ^{2}+1\right)r_{CO}^{2}+3Mr_{CO}-2Q^{2}=0,$ (28)

i.e., in general we have an additional local minima of $V_{eff}$. By adjusting
$j$ we can have a stable photon sphere outside the outer horizon,
$r_{ps}>r_{+}$. For extremal case, Eq. (28) has three roots

$r_{CO}^{e,\pm}=\bigg{\\{}M,\ {1\pm\sqrt{1-8jM^{2}}\over 2jM}\bigg{\\}},$ (29)

one of which lies on the extremal horizon, $r^{e}_{CO}=r_{extr}=M$. The other
two ($r_{\pm}$) are real and located outside $r_{extr}$ provided the following
condition is satisfied,

$0<j<{1\over 8M^{2}}.$ (30)

The $r_{CO}^{+}$ ($r_{CO}^{-}$) act as the stable (unstable) radii. As
$j\rightarrow 0$ we have $r_{CO}^{\pm}\rightarrow\\{2M,\infty\\}$ and we
recover the condition (27).

|
---|---
Figure 4: [Left] Plot of effective potential for light particles in RN two-
horizon case with $Q=0.8$ and $J=1$ and various values of $L$ and [Right]
similar plot with $L=0.07$ and various values of $J$. The dashed red lines
refer to the inner and outer horizons.

###  IV.3 Exact Solutions of the Geodesic

Studying the shape of $V_{eff}$ can only tell us to so much about the
qualitative types of orbits. To know their specific shapes we must solve the
geodesic equation. Using the black string metric above and setting
$r=r(\phi)$, the geodesic equation (5) can be cast into

$\bigg{(}\frac{dr}{d\phi}\bigg{)}^{2}=\mathbb{E}^{2}Lr^{4}-f(r)\left(r^{2}+Lr^{4}(J+\epsilon)\right)\equiv\mathcal{R}(r).$
(31)

This can be simplified by rescaling $r\rightarrow r/M$, $Q\rightarrow Q/M$,
$L\rightarrow L/M^{2}$, and defining:
$v\equiv L(\mathbb{E}^{2}-(J+\epsilon)),\ w\equiv 2L(J+\epsilon),\ x\equiv-
LQ^{2}(J+\epsilon)-1,\ y\equiv 2,$ and $z\equiv-Q^{2},$ such that Eq. (31) now
reads

$\bigg{(}\frac{dr}{d\phi}\bigg{)}^{2}=vr^{4}+wr^{3}+xr^{2}+yr+z.$ (32)

Further simplification by expanding $r$ around (one of its zeros) $r=s+r_{0}$
and defining: $a_{0}\equiv v,\ a_{1}\equiv 4r_{0}v+w,\ a_{2}\equiv
6r_{0}^{2}v+3r_{0}w+x,$ and $a_{3}\equiv 4r_{0}^{3}v+3r_{0}^{2}w+2r_{0}x+y$,
transform Eq. (32) into

$\bigg{(}\frac{ds}{d\phi}\bigg{)}^{2}=a_{0}s^{4}+a_{1}s^{3}+a_{2}s^{2}+a_{3}s.$
(33)

The degree of the corresponding polynomial can be reduced by setting $r\equiv
1/u$, and further by $u\equiv 4y/a_{3}-a_{2}/3a_{3}$. We then obtain the
Weierstrass form

$\displaystyle\bigg{(}\frac{dy}{d\phi}\bigg{)}^{2}=4y^{3}-g_{2}y-g_{3},$ (34)

where $g_{2}$ and $g_{3}$ are given as

$\displaystyle g_{2}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\left(\frac{4a_{2}^{2}}{3}-4a_{1}a_{3}\right),$ (35)
$\displaystyle g_{3}$ $\displaystyle=$
$\displaystyle\frac{1}{16}\left(\frac{1}{3}a_{1}a_{3}a_{2}-\frac{1}{27}2a_{2}^{3}-a_{0}a_{3}^{2}\right).$
(36)

Eq. (31) admits analytical solution in terms of Weierstrass $\wp$-function
Hackmann:2010tqa :

$r(\phi)=r_{0}+\frac{3a_{3}}{12\wp(\phi-\phi_{in})-a_{2}}$ (37)

where the initial angle $\phi_{in}$ is only dependent on $\phi_{0}$ and
$y_{0}$:

$\phi_{in}=\phi_{0}+\int_{y_{0}}^{\infty}\frac{dy}{\sqrt{4y^{3}-g_{2}y-g_{3}}},\,\,\
\ \ \
y_{0}\equiv\frac{1}{4}\bigg{(}\frac{a_{3}}{r_{in}-r_{0}}+\frac{a_{2}}{3}\bigg{)}.$
(38)

Making use of (II) we have the equation for $\omega$ as

$\bigg{(}\frac{d\omega}{d\phi}\bigg{)}=J\sqrt{L}r(\phi)^{2},$ (39)

which gives the solution of $\omega$ in terms of $\phi$ as

$\omega=J\sqrt{L}\int_{\phi_{0}}^{\infty}r(\phi)^{2}d\phi.$ (40)

Eq. (40) is later to play a significant role in enabling null bound orbit
outside the string’s horizons.

|
---|---
Figure 5: The $z=1,2,3$ orbits in RN null condition: extremal case with $w=0$ for the first column and $w=1$ for the second column; two-horizon case with $w=0$ for the third column and $w=1$ for the fourth column. Note that the first entries in the first and third column are blank because the $q=0+\frac{0}{1}$ orbits are inaccessible. |
---|---
Figure 6: The $z=1,2,3$ orbits in RN timelike condition: extremal case with
$w=0$ for the first column and $w=1$ for the second column; two-horizon case
with $w=0$ for the third column and $w=1$ for the fourth column. Note that the
first entries in the first and third column are blank because the
$q=0+\frac{0}{1}$ orbits are inaccessible.

###  IV.4 RN Black String’s Bound Orbit Taxonomy

As discussed above, the black string’s $V_{eff}$ is quartic in $r$, so in
general it gives three local extrema; an additional local minimum outside the
horizon. In this work we limit ourselves only to the case of physical orbits;
that is, which lies outside the horizon and can thus be observable. We avoid
discussing the tricky case of horizon-crossing orbits despite the premise of
many-world orbits by Grunau and Kagramanova, in which the orbit is viewed as
infinite continuity of the patches of the spacetime Grunau:2010gd . Armed with
the aforementioned taxonomy in III.2, we carefully examine the bounded domain
of region IV (Fig. 2) in RN black string potential and present the periodic
orbits up to 3 zooms and 2 whirls. The whole existence of stable null orbits
in our case is genuine. The $J$-factor leads to the existence of periodic
orbits, as opposed to the $4d$ case where no such stable orbits present
Misra:2010pu ; Pradhan:2010ws . The plots are shown in Fig. 5.

| $q$
---
SCO
1/2
2/3
1
1 1/2
1 2/3
UCO
$\mathbb{E}$ (Extremal) | Position
---|---
1.791566220 | 0.00%
1.802000000 | 5.62%
1.885900000 | 50.79%
1.951300000 | 86.00%
1.973065000 | 97.72%
1.974968000 | 98.75%
1.977293000 | 100.00%
$\mathbb{E}$ (Two-horizon) | Position
---|---
1.841640600 | 0.00%
1.872700000 | 23.38%
1.928400000 | 65.31%
1.964250000 | 92.30%
1.973330000 | 99.14%
1.973920000 | 99.58%
1.974477000 | 100.00%
Deviation
---
-
12.56%
10.27%
4.45%
1.00%
0.59%
-
Table 2: Comparison of energy level and the position of particular orbit inside the energy range as percentage for timelike case in extremal and two-horizon RN black string. SCO refers to the bottom level of energy range (no whirl), while UCO refers to the top one (maximum whirl). | $q$
---
SCO
1/2
2/3
1
1 1/2
1 2/3
UCO
$\mathbb{E}$ (Extremal) | Position
---|---
0.891967985 | 0.00%
0.894300000 | 2.96%
0.925500000 | 42.51%
0.957750000 | 83.39%
0.968662000 | 97.22%
0.969635000 | 98.46%
0.970851807 | 100.00%
$\mathbb{E}$ (Two-horizon) | Position
---|---
0.919894000 | 0.00%
0.932500000 | 20.07%
0.959600000 | 63.20%
0.977550000 | 91.77%
0.982130000 | 99.06%
0.982430000 | 99.54%
0.982718600 | 100.00%
Deviation
---
-
12.10%
14.63%
5.93%
1.30%
0.77%
-
Table 3: Comparison of energy level and the position of particular orbit
inside the energy range as percentage for null case in extremal and two-
horizon RN black string. SCO refers to the bottom level of energy range (no
whirl), while UCO refers to the top one (maximum whirl).

Levin in Levin:2009sk  shows that, since rational numbers are discrete so are
the energy levels that produce the orbits. The existence of bound orbits are
bounded by UCO and SCO, so we can quantify the level energy by the deviation
from them. We define the corresponding deviation $\eta$ as

$\eta\equiv{\left(\mathbb{E}-\mathbb{E}_{SCO}\right)\over\left(\mathbb{E}_{UCO}-\mathbb{E}_{SCO}\right)}\bigg{|}_{timelike/null}.$
(41)

In Table 2 we list the energy level of various periodic orbits according to
their energy level, for massive particles, in both extremal and two-horizon RN
black string. Similar table for light particles is shown in Table 3. The table
shows increasing number of rational number as it goes down the list. One
behavior that is observed in both cases is the standard deviations, while
fluctuate, tends to decrease as energy level rises to UCO. If we calculate the
average of deviation for timelike and null condition, we obtain the value
$6.36\%$ and $6.94\%$, respectively. The reasoning of why we separate the
table by horizon case becomes clear when we pay attention to each deviation
average: $2.03\%$ for extremal case and $0.86\%$ for two horizon one, which is
way smaller than the numbers earlier. This finding means that the distance
between energy levels is more consistent in each specific horizon case.

q |  $\Delta\varphi_{r-4d}$ |  $\Delta\varphi_{r-5d}$
---|---|---
1/2 |  31.1018 |  44.0076
2/3 |  34.5575 |  48.8576
1  |  41.4694 |  58.634
1 1/2 |  51.8373 |  73.2913
1 2/3 |  48.085 |  78.1832
Table 4: Accumulated radial angle for extremal RN black holes (second column) and black strings (third column) strings as a function of $q$. q |  $\Delta\varphi_{r-4d}$ |  $\Delta\varphi_{r-5d}$
---|---|---
1/2 |  35.8142 |  51.5273
2/3 |  39.7936 |  57.2777
1 |  47.7531 |  68.7163
1 1/2 |  59.6837 |  85.9175
1 2/3 |  63.6605 |  91.6488
Table 5: Accumulated radial angle for non-extremal RN black holes (second
column) and black strings (third column) strings as a function of $q$.

For the timelike case we obtain a qualitatively similar pattern of orbits
found in RN black hole ”zoo” (see Misra:2010pu ), as shown in Fig. 6. One
might also notice that the whirl is stronger near the radius of UCO
($r_{UCO}$), which is the same behaviour observed in the $4d$ RN. The
significant difference is that the test particle in RN black string requires
higher energy for the orbit to exist ($\mathbb{E}\approx 2$). We can also
calculate the accumulated radial angle of black strings’ rational orbit for
any given $q$. The Eq. (14) reads

$\Delta\varphi_{r}=2\int_{r_{p}}^{r_{a}}\frac{dr}{Lr^{2}\sqrt{\mathbb{E}^{2}-\left(1-{2M\over
r}+{Q^{2}\over r^{2}}\right)\left(\epsilon+J+\frac{1}{Lr^{2}}\right)}}.$ (42)

This equation can be integrated numerically. In Table 4 and 5 we show the
accumulated angle for extremal and two-horizon RN black strings, respectively.
For each case we compare their values with their corresponding $4d$
counterpart from Misra:2010pu . The accumulated radial angle for black strings
is higher than for black holes.

##  V Nonsingular Black String

In 1968 in his seminal paper, Bardeen proposed an analytic solution of a
charged BH with no singularity everywhere; a regular333Many nonsingular black
hole models have been developed ever since. For example, the static cases may
be found in Hayward:2005gi ; Bogojevic:1998ma , while the rotating cases are
in Bambi:2013ufa ; Ghosh:2014hea ; Toshmatov:2014nya ; Dymnikova:2015hka ;
Abdujabbarov:2016hnw . BH bardeen . The Bardeen solution is given by

$f(r)=1-\frac{2Mr^{2}}{(r^{2}+Q^{2})^{3/2}},$ (43)

where $Q$ can be identified as a charge. Surely this solution is different
from RN, but also reduces to Schwarzschild in the limit of vanishing $Q$. The
Kretschmann scalar is finite everywhere, including at the center

$\lim_{r\rightarrow
0}R^{\alpha\beta\gamma\delta}R_{\alpha\beta\gamma\delta}={96m^{2}\over
Q^{8/3}}.$ (44)

At $r\rightarrow 0$ the spacetime behaves de Sitter-like, $f(r)\approx
1-2mr^{2}/Q^{3}$. The horizons $r_{h}$ are given as the roots of

$\left(r_{h}^{2}+Q^{2}\right)^{3}-4M^{2}r_{h}^{4}=0.$ (45)

In general there are at most two horizons. The typical values of $Q$ for each
horizon condition are shown in Fig. 7. The extremal Bardeen is obtained when
Ayon-Beato:2000mjt :

$Q^{2}={16\over 27}M^{2}\ \longrightarrow r_{extr}=\sqrt{32\over 27}M,$ (46)

beyond which the black hole becomes naked. This no-horizon case does not
violate the cosmic censorship theorem Penrose:1964wq  since no singularity is
present, and indeed the formation of naked Bardeen spacetime as a result of
destroying its event horizon is strongly supported theoretically Li:2013sea .
As in the RN case, Bardeen lacks stable null bound orbit outside the horizons
and the SCO exists only in the extremal case, where $r_{SCO}=r_{extr}$. This
can be seen in Fig. 8.

Figure 7: The three conditions of metric function of Bardeen spacetime. |
---|---
Figure 8: Plot of effective potential in timelike (left) and null geodesics
(right) in the extremal $4d$ Bardeen black hole. The dashed red line refers to
their respective event horizon.

When extended to $5d$ the metric (43) describes a nonsingular black string.
The appearance of $J$ creates a new dynamics different from its $4d$
counterpart, as can be seen in Fig. 9 and 10. As in the RN case, we shall
analyze the circular (timelike and null) orbits before obtaining the
geodesic’s exact solutions.

|
---|---
Figure 9: [Left] Plot of effective potential for massive particles in $5d$ two-horizon case with $Q=0.62$ and $J=1$ and various values of $L$ and [Right] similar plot with $L=0.035$ and various values of $J$. The dashed red lines refer to the inner and outer horizons. |
---|---
Figure 10: [Left] Plot of effective potential for null particles in $5d$ two-
horizon case with $Q=0.62$ and $J=1$ and various values of $L$ and [Right]
similar plot with $L=0.063$ and various values of $J$. The dashed red lines
refer to the inner and outer horizons.

###  V.1 Timelike Circular Orbit

Applying the condition (III.1) for the black string yields

$\displaystyle\left(J+1-\mathbb{E}^{2}\right)Lr_{CO}^{2}\left(r_{CO}^{2}+Q^{2}\right)^{3/2}-2M\left(J+1\right)Lr_{CO}^{4}+\left(r_{CO}^{2}+Q^{2}\right)^{3/2}-2Mr_{CO}^{2}$
$\displaystyle=$ $\displaystyle 0,$
$\displaystyle\left(J+1\right)MLr_{CO}^{6}-\left(r_{CO}^{2}+Q^{2}\right)^{5/2}+\left[3+2\left(J+1\right)LQ\right]Mr_{CO}^{4}$
$\displaystyle=$ $\displaystyle 0.$

One can easily verify that upon $J\rightarrow 0$ they reduce to the MBO
conditions for $4d$ Bardeen black hole Gao:2020wjz . Solving them for $L$ and
$\mathbb{E}^{2}$ yield

$\displaystyle L$ $\displaystyle=$
$\displaystyle\frac{\left(Q^{2}+r_{CO}^{2}\right)^{5/2}-3Mr_{CO}^{4}}{(J+1)Mr_{CO}^{4}\left(r_{CO}^{2}-2Q^{2}\right)},$
$\displaystyle\mathbb{E}^{2}$ $\displaystyle=$
$\displaystyle\frac{(J+1)\left(Q^{2}\left(3r_{CO}^{4}-4Mr_{CO}^{2}A\right)+4Mr_{CO}^{4}\left(M-A\right)+Q^{6}+3Q^{4}r_{CO}^{2}+r_{CO}^{6}\right)}{-3Mr_{CO}^{4}A+Q^{6}+3Q^{4}r_{CO}^{2}+3Q^{2}r_{CO}^{4}+r_{CO}^{6}},$

where $A\equiv\sqrt{Q^{2}+r_{CO}^{2}}$.

The ISCO condition gives the additional condition:

$\displaystyle
3\left(Q^{2}+r_{ISCO}^{2}\right)^{7/2}+Mr_{ISCO}^{4}\bigg{[}-2(J+1)LQ^{4}+Q^{2}\left(11(J+1)Lr_{ISCO}^{2}+3\right)$
$\displaystyle-2r_{ISCO}^{2}\left((J+1)Lr_{ISCO}^{2}+6\right)\bigg{]}=0.$

In the extremal case, $Q^{2}=16/27$, the condition becomes

$\displaystyle
2(J+1)LMr_{ISCO}^{8}+2M\left[3\frac{88}{27}(J+1)LM^{2}\right]r_{ISCO}^{6}+{16M^{3}\over
9}\left[\frac{32}{81}(J+1)LM^{2}-5\right]r_{ISCO}^{4}$
$\displaystyle-\frac{512M^{5}r_{ISCO}^{2}}{243}+3\left(\frac{16M^{2}}{27}+r_{ISCO}^{2}\right)^{2}\left[2Mr_{ISCO}^{2}-\left(\frac{16M^{2}}{27}+r_{ISCO}^{2}\right)^{3/2}\right]=0,$

which can be determined numerically.

###  V.2 Null Circular Orbit

Null geodesic in $4d$ Bardeen spacetime has been discussed widely, for example
in Eiroa:2010wm ; He:2021htq . In $5d$ by defining $b^{2}$ and $j$, and by
rescaling the proper time the effective potential can be written as

$V_{eff}=\left(1-\frac{2Mr^{2}}{(r^{2}+Q^{2})^{3/2}}\right)\left(j+{1\over
r^{2}}\right).$ (51)

One can see that the appearance of $j$-term implies the existence of another
turning point in $V_{eff}$ outside the horizon, as can be seen in Fig. 10. The
CO radii are determined by the roots of the following equation

$2Mr_{CO}^{4}\left(j\left(r_{CO}^{2}-2Q^{2}\right)+3\right)-2\left(Q^{2}+r_{CO}^{2}\right)^{5/2}=0.$
(52)

For the extremal case, the condition becomes

$\displaystyle-81\sqrt{3}Mr_{CO}^{4}\left(JL\left(32M^{2}-27r_{CO}^{2}\right)-81\right)-\left(16M^{2}+27r_{CO}^{2}\right)^{5/2}=0.$

Both conditions are polynomial equations of degree six, and the roots can best
be found numerically.

###  V.3 Exact Solutions and the Taxonomy of Bound Orbits

|
---|---
Figure 11: The $z=1,2,3$ orbits in the timelike condition: extremal case with
$w=0$ for the first column and $w=1$ for the second column; two-horizon case
with $w=0$ for the third column and $w=1$ for the fourth column. Note that the
first entries in the first and third column are blank because the
$q=0+\frac{0}{1}$ orbits are inaccessible.

Unlike the RN case, we have been unable to cast Eq. (5) into a form that can
be solved analytically,

$\bigg{(}\frac{dr}{d\phi}\bigg{)}^{2}=\frac{2\left(r^{6}L(J+\epsilon)+r^{4}\right)}{\left(Q^{2}+r^{2}\right)^{3/2}}+r^{4}L\left(\mathbb{E}^{2}+J+\epsilon\right)-r^{2}.$
(54)

To solve it numerically in Mathematica, it is better to rewrite it in terms of
the $2^{nd}$-order ODE:

$\displaystyle\frac{d^{2}r}{d\phi^{2}}$ $\displaystyle=$
$\displaystyle\frac{3Lr^{7}(J+\epsilon)}{\left(Q^{2}+r^{2}\right)^{5/2}}+\frac{3r^{5}\left(2Q^{2}L(J+\epsilon)-1\right)}{\left(Q^{2}+r^{2}\right)^{5/2}}$
(55)
$\displaystyle+r^{3}\left(\frac{4}{\left(Q^{2}+r^{2}\right)^{3/2}}-2L\left(-\mathbb{E}^{2}+J+\epsilon\right)\right)-r.$

|
---|---
Figure 12: The $z=1,2,3$ orbits in the null condition: extremal case with
$w=0$ for the first column and $w=1$ for the second column; two-horizon case
with $w=0$ for the third column and $w=1$ for the fourth column. Note that the
first entries in the first and third column are blank because the
$q=0+\frac{0}{1}$ orbits are inaccessible.

We obtain the numerical solutions of periodic orbits, both for timelike as
well as null conditions. They are shown up to $q=1{2\over 3}$ in Figs. 11-12,
respectively. For each case we are able to obtain the extremal and two-horizon
solutions. What is interesting is that the orbits found in extremal and two-
horizon case have almost identical form for each equivalent counterpart.

| $q$
---
SCO
1/2
2/3
1
1 1/2
1 2/3
UCO
$\mathbb{E}$ (Extremal) | Position
---|---
1.815650000 | 0.00%
1.816250000 | 0.44%
1.870600000 | 39.88%
1.929300000 | 82.48%
1.949485900 | 97.13%
1.951050000 | 98.27%
1.953438690 | 100.00%
$\mathbb{E}$ (Two-Horizon) | Position
---|---
1.827332000 | 0.00%
1.829750000 | 1.95%
1.889250000 | 49.86%
1.935600000 | 87.18%
1.949349000 | 98.25%
1.950209151 | 98.94%
1.951519700 | 100.00%
Deviation
---
-
1.07%
7.06%
3.32%
0.79%
0.48%
-
Table 6: Comparison of energy level and the position of particular orbit
inside the energy range as percentage for timelike case in extremal and two-
horizon nonsingular black string. SCO refers to the bottom level of energy
range (no whirl), while UCO refers to the top one (maximum whirl).

To create a more comprehensive view of periodic orbits in the nonsingular
string, we provide similar tables as in IV.4. Tables 6 and 7 show various
energy levels and each of their corresponding position in extremal and two-
horizon cases, respectively. A quick calculation of average of standard
deviation of each case would give $0.09\%$ for extremal and $0.07\%$ for two
horizon one. This is way smaller than the ones found in RN black string, which
means the prediction for the position of each periodic orbit will be more
precise in the nonsingular black string.

| $q$
---
SCO
1/2
2/3
1
1 1/2
1 2/3
UCO
$\mathbb{E}$ (Extremal) | Position
---|---
0.908371200 | 0.00%
0.908659500 | 0.41%
0.936700000 | 39.96%
0.967200000 | 82.98%
0.977244258 | 97.14%
0.978060029 | 98.29%
0.979268855 | 100.00%
$\mathbb{E}$ (Two-Horizon) | Position
---|---
0.913804300 | 0.00%
0.915000000 | 1.91%
0.944900000 | 49.68%
0.968400000 | 87.23%
0.975259983 | 98.19%
0.975842810 | 99.12%
0.976394418 | 100.00%
Deviation
---
-
1.06%
6.88%
3.01%
0.74%
0.58%
-
Table 7: Comparison of energy level and the position of particular orbit
inside the energy range as percentage for null case in extremal and two-
horizon Bardeen black string. SCO refers to the bottom level of energy range
(no whirl), while UCO refers to the top one (maximum whirl).

Another point to be noted is that the average of deviation for timelike and
null condition is $2.54\%$ and $2.45\%$, respectively. This is a similar
manner found in RN black string: the distance between energy levels is more
consistent when we compare each specific horizon case, which means this
behavior exists generally on charged black holes. By carefully inspecting the
numbers from both table, we also find that the orbits in two-horizon condition
are more dense toward the local maximum (point of UCO). This behavior is
observed in both timelike and null orbits. Nevertheless, one might notice that
in the two-horizon nonsingular black string the shift is less drastic compared
to similar behavior in two-horizon RN IV.4 where the orbits become highly
condensed near the maximum.

As in the timelike RN, the accumulated radial angle can be calculated as
follows:

$\Delta\varphi_{r}=2\int_{r_{p}}^{r_{a}}\frac{\dot{\phi}}{\dot{r}}dr=2\int_{r_{p}}^{r_{a}}\frac{dr}{Lr^{2}\sqrt{\mathbb{E}^{2}-\left(1-\frac{2Mr^{2}}{(r^{2}+Q^{2})^{3/2}}\right)\left(\epsilon+J+\frac{1}{Lr^{2}}\right)}}.$
(56)

We calculate numerically the accumulated angle and show it in Table 8. We
compare the angles we calculate with the corresponding angle from the $4d$
counterpart. As in the RN case, the nonsingular black strings exhibits higher
values for the accumulated angle. From the Table 8 the $4d$ angles reach
maximum at $q=1{1\over 2}$, then decreases. the $5d$ counterpart, however,
keeps increasing as $q$ increasing.

q |  $\Delta\varphi_{r-2\ horizon}$ |  $\Delta\varphi_{r-extremal}$ |  $\Delta\varphi_{r-4dextremal}$
---|---|---|---
1/2 |  50.3815 |  49.8285 |  -
2/3 |  55.5692 |  54.0185 |  37.9724
1 |  66.6127 |  64.715 |  44.4775
1 1/2 |  83.3911 |  81.0931 |  50.7586
1 2/3 |  87.6954 |  85.7237 |  48.5055
Table 8: Accumulated angles for the nonsingular black strings. The second and
third columns are the angles for 2-horizon and extremal string, respectively.
The fourth column is the corresponding angles in the $4d$ extremal Bardeen BH
from Gao:2020wjz . The angle for $q=1/2$ in $4d$ extremal Bardeen is empty
because the authors do not show solutions for $(2,0,1)$.

##  VI Conclusion

This works deals with the bound orbits of massive and massless particles
around charged RN and nonsingular black strings. These toy models are quite
simple, but the geodesics are rich enough to have genuine observables that can
be distinguished from their $4d$ counterparts. Following Hackmann
Hackmann:2010tqa  the RN bound orbits can be expressed analytically in terms
of the Weierstrass function, while the nonsingular string must be solved
numerically. We present solutions for timelike and null bound orbits in both
models, characterized by the rational number $q$. We investigate bond orbit
solutions for several $q$ up to $q=1{2\over 3}$. Higher values of $q$ can, in
principle, be obtained easily.

What is novel here are twofold. First, the existence of stable bound orbits
for photon which does not exist in its 4d counterpart. The addition of
(trivial) extra dimension modifies the geodesic equations and thus the shapes
of the corresponding $V_{eff}$. As can be seen from Figs. 4 and 10, the non-
zero $J$ raises the tail of $V_{eff}$ to create a local maxima outside the
corresponding horizons. This is confirmed, for example in the case of RN, by
Eq. (28) where the existence of $j\equiv JL$ gives a cubic term in the
circular orbit condition. This another extremum turns out to be minimum and
lies outside the corresponding horizon(s). Thus this extra bound orbit is
stable and observable. For the extremal RN the roots for the cubic equation
can be found analytically, as shown in Eq. (29). All $r_{CO}$s are located
outside or on the extremal horizon, provided $0<j<1/8M^{2}$.

For the massive case, we show that the circular orbits require higher
particle’s energy and angular momentum (remember that the angular momentum
$\mathbb{L}\equiv 1/\sqrt{L}$). The ISCO radii are indistinguishable from its
$4d$ counterpart. However, as pointed out by authors of Babar:2017gsg , the
accumulated angle of the timelike orbit might provide observational signature
that can distinguish these black strings from their corresponding black hole
counterparts. In Tables 4-5 and 8 we show comparison of radial accumulated
angles for black strings and black holes, both for the RN and the nonsingular,
respectively. In all cases, the black string accumulated angles are higher
compared to the black holes. Thus, measuring such angle for bound orbits of
stars around some black hole might give a hint on whether our universe is
higher-dimensional or not.

It is probably worthwhile to comment on the size of extra dimension. While the
existence of black string solution itself is oblivion to the extra-dimensional
size, stability prefers it to be compact, not infinite Gregory:1987nb ;
Harmark:2007md . The bounds comes from particle physics as well as cosmology.
From particle physics, recent report by CMS collaboration suggests that the
size for one “large” extra dimension based on the Randall-Sundrum scenario
Randall:1999ee ; Randall:1999vf  is of order $\sim 10^{-4}$ fm CMS:2022pjv . A
much more optimistic bounds comes from cosmology. Upper bounds value for the
emission ring and the angular shadow diameter from the EHT observation on
SgrA* indicates that the extra dimension might be as large as $\sim 0.4$ mm
Tang:2022hsu .

In this work we only consider static charged black strings in the simplest
extension. It would be interesting to extend our investigation to black
strings endowed with NLED charged, to regular black strings beyond the Bardeen
solutions, or to black strings possessing Lorentz boost as in Gregory:1995qh .
We are addressing those issues in the forthcoming works.

###### Acknowledgements.

We thank Heribertus Hartanto and Byon Jayawiguna for the enlightening
discussions, Adolfo Cisterna for informing us about Refs. Cisterna:2020rkc ;
Cisterna:2021ckn , Reinard Primulando for the Ref. CMS:2022pjv , and Jason
Kristiano for helping us with the Ref. Gao:2020wjz . This work is supported by
grants from Universitas Indonesia through Hibah Riset PPI Q1 No. NKB-583/UN2.
RST/HKP.05.00/2021.

## Data Availability Statement

Data sharing is not applicable to this article as no data sets were generated
or analyzed during the current study.

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# Serendipitous Science from the K2 Mission

Derek L. Buzasi1 Lindsey Carboneau1 Carly Hessler1 Andy Lezcano1 Heather
Preston1,2 1Dept. of Chemistry & Physics, Florida Gulf Coast University,
10501 FGCU Blvd. South, Fort Myers, FL 33919 USA
email<EMAIL_ADDRESS>
2 Calusa Nature Center & Planetarium, 3450 Ortiz Ave., Fort Myers, FL, 33905
USA

(2015)

###### Abstract

The K2 mission is a repurposed use of the Kepler spacecraft to perform high-
precision photometry of selected fields in the ecliptic. We have developed an
aperture photometry pipeline for K2 data which performs dynamic automated
aperture mask selection, background estimation and subtraction, and positional
decorrelation to minimize the effects of spacecraft pointing jitter. We also
identify secondary targets in the K2 “postage stamps” and produce light curves
for those targets as well. Pipeline results will be made available to the
community. Here we describe our pipeline and the photometric precision we are
capable of achieving with K2, and illustrate its utility with asteroseismic
results from the serendipitous secondary targets.

###### keywords:

techniques:photometric, stars: oscillations

††journal: Astronomy in Focus, Volume 1††editors: Piero Benvenuti, ed.

##  1 Introduction

The impressive rescue and repurposing of the K2 mission ([Howell et al.
(2014), Howell et al. (2014)]) has the promise of high-precision photometry
for numerous ecliptic fields of view, with access to numbers and types of
targets not envisioned in the original Kepler mission. However, the operating
mode for K2 differs significantly from that of the original Kepler mission, so
the data present significantly greater analysis complexities. This situation
is exacerbated by the fact that at present the K2 Guest Observer office does
not plan to provide a vetted and standardized pipeline product, but instead
simply access to the relatively raw observing frames.

The lack of a pipeline and data product endorsed by the mission has led a
number of teams to develop their own pipelines ([Aigrain et al. (2015),
Aigrain et al. (2015)], [Armstrong et al. (2015), Armstrong et al. (2015)],
[Lund et al. (2015), Lund et al. (2015)], [Van15, Vanderburg & Johnson
(2015)]), and the work presented here is broadly similar. However, since there
is no “ground truth” for K2 data products, and different approaches to K2 data
reduction are independent of one another, the existence of products from
multiple pipelines is complementary rather than competitive, and the differing
approaches are likely to be more or less suited to different kinds of targets.
In addition, we are further motivated by the desire to perform serendipitous
science using the background sources in each K2 frame, and by the desire to
include undergraduate students in the development and analysis work. In some
cases, we compare our results to those of [Van15, Vanderburg & Johnson
(2015)], as this is currently the most widely-used pipeline.

##  2 Pipeline

Here we briefly describe the operation of the pipeline and its products for
long-cadence K2 data. While operation and output shown are generally for the
pipeline operating in batch mode, it can also be run in single-frame mode to
obtain more refined results, particularly for targets displaying unusual
photometric characteristics.

Preprocessing & Target Identification

Long-cadence K2 time series, like the Kepler time series, have 30-minute
cadence, and the spacecraft returns “postage stamps” surrounding primary
targets. These small images have varying sizes, typically approximately 25
pixels square in Campaign 0, but with decreasing sizes in following Campaigns.
We begin by estimating and removing the background level in each image, using
the median of the faintest 20% of the pixels, and then construct a mean image
from the time series as shown in Figure 1 for the Campaign 0 primary target
EPIC 202066811. Due to the $\approx 6$ hourly spacecraft pointing corrections,
stellar images show varying degrees of elongation depending primarily on the
distance of the postage stamp from boresight.

Figure 1: The mean image from a K2 time series for EPIC 202066811 from
Campaign 0. Multiple targets are clearly visible, as is the asymmetry typical
of K2 stellar images, due to the 6 hourly pointing corrections of the
spacecraft.

Targets are identified in the mean image through a brute-force search for
local maxima in the frame, avoiding edge pixels. This is a relatively
inefficient process which is nonetheless practical due to the small frame
size. To avoid accidentally identifying background noise peaks as sources, as
a default we require that each peak be more than $0.5\sigma$ above its
immediate neighbors, though this level can be manually modified in single-
frame mode. The example shown in Figure 2 imposed a $1\sigma$ requirement,
leading to 3 targets in the frame rather than the 5 which would result in
default mode.

Figure 2: Targets identified in the frame shown in Figure 1.

Mask Generation

We next generate a mask to use for aperture photometry. The initial mask for a
target is simply the one-pixel local maximum previously identified as the
target location, and a time series is calculated based on that pixel. The
resulting time series is then twice-differenced to improve stationarity ([Box
& Jenkins (1970), Box & Jenkins (1970)]), and $\sigma$ is taken as a figure of
merit. Additional mask are then generated algorithmically by adding the
largest contiguous pixel. A new time series and figure of merit are then
calculated, and the mask resulting in the time series with the lowest twice-
differenced $\sigma$ is adopted. The process can be terminated early if adding
an additional pixel would result in the addition to the mask of a pixel with a
larger mean image value than any previously included in the mask. This latter
constraint helps to avoid “bridging” between nearby targets.

Figure 3: The figure of merit used by the pipeline ($\sigma$ of the twice-
differenced time series), shown as a function of mask size for a K2 Campaign 1
target. The best mask size in this case was 103 pixels, which is typical of
targets in the uncrowded Campaign 1 frames; C0 mask sizes are generally
smaller (see Figure 4).

No attempt is made to ensure that masks for individual targets in a frame do
not overlap. However, the mask construction procedure makes this uncommon and
in practice it only occurs for truly overlapping targets. When masks do
overlap, the number of overlapping pixels is noted and used as a proxy for
crowding. Once the masks are constructed, simple aperture photometry is used
to calculate fluxes along with $(x,y)$ centroid and background estimates.
Figure 4 illustrates the resulting masks for the frame in Figure 1.

Figure 4: Aperture photometry masks for the targets identified in Figure 2.
Note that the derived mask sizes in this more-crowded field are significantly
smaller ($\approx 20$ pixels) than those typical of the uncrowded C1 field
seen in Figure 3.

Detrending

Detrending is normally required due to the presence of spacecraft pointing
resets, along with additional correlated noise. We have the capability to
detrend against $(x,y)$ centroid, background level, and FWHM, but normally
only positional detrending is used. Points corresponding to spacecraft
pointing resets are identified through examination of the differenced centroid
time series and removed, and the entire time series is then detrended using a
low-order (typically $n=3$) polynomial fit. The time series is then broken
into overlapping segments of length $\Delta t\geq 1~{}\rm d$ (variations are
possible depending on the behavior of the time series DFT), and further
detrended using low-order ($n=1-3$) polynomial fits, with the exact order a
function of the number of points in each segment. Detrending is repeated in an
iterative fashion until the improvement in successive time series second-
differenced $\sigma$ fails to exceed 1%; typically $3-10$ times. Figure 5
shows the typical improvement from raw to detrending time series, while Figure
6 illustrates the effect of detrending on the flux-centroid correlation.

Figure 5: Improvement in time series quality between raw, stage 1 detrending
(entire time series), and stage 2 detrending (segmented time series). The
overall offsets between the time series are added to aid visibility. Figure 6:
Decorrelation effectively removes the relation between flux and centroid
position which is present in the raw time series. The two data sets are
arbitrarily offset to aid visualization.

##  3 Results

The resulting time series have noise characteristics (particularly CDPP,
[Gilliland et al. (2011), Gilliland et al. (2011)]) which are similar to those
shown by Vanderberg & Johnson (2014), though typically our low-frequency noise
is modestly better and our high-frequency noise is slightly worse. One aspect
of this behavior, shown in Figure 7 for the $K_{mag}=12.8$ Campaign 1 red
giant star EPIC 201757842, is that the offsets between data before and after
time series gaps which frequently occur in the Vanderberg & Johnson (2014)
pipeline are absent from ours. In addition, we tend to produce cleaner time
series for fainter targets, but Vanderberg & Johnson’s results are generally
superior for bright targets and those with larger-amplitude periodic
variability.

Figure 7: A comparison of time series from our pipeline (black) and Vanderberg
& Johnson (2014, in red) for the same red giant target (EPIC 201757842). The
power spectrum for this target is shown in Figure 8.

Figure 8 illustrates the DFT for the time series shown in Figure 7, which
should be compared with Figure 2 in Stello et al. (2015). A simple analysis
broadly following the procedure in Stello et al. (2015) leads to
$\nu_{max}=148\mu Hz$ and $\Delta\nu=14\mu Hz$, comparable (within errors) to
the results in Stello et al.

Figure 8: The power spectrum for the red giant whose time series is shown in
Figure 7. Data were processed similarly to those in Stello et al. (2015). A
simple quick-look analysis yields $\nu_{max}=148\mu Hz$ and $\Delta\nu=14\mu
Hz$, comparable (within errors) to the results in Stello et al. (2015).

##  4 Further Directions

We plan to begin an undergraduate “student-sourced” search for transiting
exoplanet candidates, concentrating on the secondary sources in the K2 frames,
and making use of students taking an “Exoplanets” course at Florida Gulf Coast
University. In support of this project, we have used Python to construct some
simple GUI-based tools which allow visualization and period-searching (using
DFT, BLS, etc.) of the K2 time series (Figure 9). The tools run under both
Windows and Macintosh operating systems, allowing students to use their
personal computers to contribute to the project.

Figure 9: A screenshot of one of our simplified Python-based tools for
visualization and period-finding in the K2 time series. These support
undergraduate student exoplanet candidate searches among the K2 secondary
targets.

In addition, the pipeline supports additional ongoing K2 projects, including
rotation/activity correlations among solar analogs, red giant oscillation
detection and characterization, and eclipsing binary identification.

## References

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