# Synthesis of infrared Stokes spectra in an evolving solar chromospheric jet

T. Matsumoto,1,2 Y. Kawabata,2 Y. Katsukawa2 H. Iijima1 and C. Quintero
Noda3,4
1Centre for Integrated Data Science, Institute for Space-Earth Environmental
Research, Nagoya University, Furocho, Chikusa-ku, Nagoya, Aichi 464-8601,
Japan
2National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo
181-8588, Japan
3Instituto de Astrofísica de Canarias, E-38200, La Laguna, Tenerife, Spain
4Departamento de Astrofísica, Univ. de La Laguna, La Laguna, Tenerife,
E-38205, Spain E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Chromospheric jets are plausible agents of energy and mass transport in the
solar chromosphere, although their driving mechanisms have not yet been
elucidated. Magnetic field measurements are key for distinguishing the driving
mechanisms of chromospheric jets. We performed a full Stokes synthesis in the
infrared range with a realistic radiative magnetohydrodynamics simulation that
generated a chromospheric jet to predict spectro-polarimetric observations
from the Sunrise Chromospheric Infrared spectro-Polarimeter (SCIP) onboard the
SUNRISE III balloon telescope. The jet was launched by the collision between
the transition region and an upflow driven by the ascending motion of the
twisted magnetic field at the envelope of the flux tube. This motion is
consistent with upwardly propagating non-linear Alfvénic waves. The upflow
could be detected as continuous Doppler signals in the Ca ii 849.8 nm line at
the envelope where the dark line core intensity and strong linear polarisation
coexist. The axis of the flux tube was bright in both Fe i 846.8 nm and Ca ii
849.8 nm lines with down- flowing plasma inside it. The structure, time
evolution, and Stokes signals predicted in our study will improve the physical
interpretation of future spectro-polarimetric observations with SUNRISE
III/SCIP.

###### keywords:

Sun: photosphere – Sun: chromosphere – Sun: infrared – Sun: magnetic fields –
MHD – radiative transfer.

††pubyear: 2015††pagerange: Synthesis of infrared Stokes spectra in an
evolving solar chromospheric jet–Synthesis of infrared Stokes spectra in an
evolving solar chromospheric jet

##  1 Introduction

Chromospheric jets are collimated eruptions that exist ubiquitously in the
chromosphere.Various types of jets, such as spicules (Beckers, 1968), surges
(Roy, 1973), and anemone jets (Shibata et al., 2007) have been reported so
far. Significant amounts of energy and mass are considered to be transported
via chromospheric jets, however, the driving mechanisms have not yet been
elucidated. Because the magnetic field plays a crucial role in dynamics,
spectro-polarimetry could be the key observation to determine the driving
mechanism.

The collision between the magnetohydrodynamics (MHD) shock and transition
region is a promising mechanism for accelerating chromospheric jets
(Osterbrock, 1961). When a certain amount of energy is released in the deep
layer, it produces an acoustic wave that propagates upward. As the acoustic
wave propagates upward, it evolves into a shock owing to the atmospheric
stratification and hits the transition region (Shibata et al., 1982). When the
shock hits the transition region, it breaks into shocks and a contact
discontinuity, below which the chromospheric materials are lifted.

There are several candidates for the origin of the MHD shock. First, the
magnetic pressure and centrifugal force associated with Alfvén waves generate
magneto-acoustic waves via nonlinear mode conversion (Hollweg et al., 1982;
Kudoh & Shibata, 1999). Horizontal magnetic signals followed by propagating
shocks are expected in this process. Second, sudden downflow is caused by
processes, such as magnetic pumping (Kato et al., 2011, 2016) or convective
collapse (Parker, 1978; Spruit, 1979) that pull down the plasma in the flux
tubes to produce rebounding slow shocks. When the flux tubes are twisted owing
to a photospheric vortex (Brandt et al., 1988; Bonet et al., 2008), the
perturbed twist propagates with slow shock. In such a case, it might be
difficult to distinguish between the Alfvén wave model and the rebound shock
model because both processes may contribute to feeding the magneto-acoustic
waves. Third, magnetic reconnection between pre-existing and newly emerged
fields will be the agent for energy release in the chromosphere (Shibata et
al., 2007). The photospheric magnetic features, as well as the chromospheric
current structure, could provide key evidence for reconnection.

To determine the origin of chromospheric jets, we need to infer the properties
of the magnetised atmosphere at multiple layers, which requires observing
spectral lines that form at different heights. Unfortunately, these
transitions also fall far apart in the wavelength spectrum, strictly
simultaneously requiring the observation of the Sun at multiple wavelengths.
Polarimetric observations will provide magnetic field information, such as
mixed polarity at the foot point, current concentrations near the magnetic
inversion lines, and helical field, including rich information in the driving
process. Line-of-sight structures also help in understanding the height of the
energy source that can be deduced from the localised temperature increase
(Robustini et al., 2018) and bidirectional flow (Nelson et al., 2019), and
propagating wave signals (Tziotziou et al., 2020). The dynamical nature of
polarimetric signals in the chromosphere has been revealed in several contexts
(Joshi & de la Cruz Rodríguez, 2018; Siu-Tapia et al., 2020; Morosin et al.,
2022); the demands for theoretical predictions for polarimetric signals in the
chromospheric jets are increasing to properly interpret the complex
observations. The theoretical predictions will help interpret the upcoming
polarimetric observations with a wide spectral range, as well as high
polarimetric sensitivity, such as SUNRISE III111Official website of the
SUNRISE III project can be found in https://www.mps.mpg.de/solar-
physics/sunrise, Daniel K. Inouye Solar Telescope (Rimmele et al., 2020,
DKIST), and European Solar Telescope (Quintero Noda et al., 2022, EST), which
provides smoking-gun evidence for the origin of the chromospheric jets.

This study aims to predict the Stokes signals that will be observed by SUNRISE
III/SCIP (Katsukawa et al., 2020, Sunrise Chromospheric Infrared spectro-
Polarimeter). SCIP will be deployed aboard the third flight of SUNRISE
stratospheric balloon-borne solar telescope (Barthol et al., 2011). The
instrument is a slit scanning spectro-polarimeter capable of simultaneously
observing two wide spectral bands, covering 770 nm and 850 nm bands. These
bands include both photospheric lines, such as Fe i (846.8 nm) and K i (766.5
nm & 769.9 nm), as well as lower chromospheric lines, such as Ca ii (849.8 nm
& 854.2 nm). Additionally, SCIP features a high degree of polarization
accuracy of $3\times 10^{-4}$ (1 sigma) and a high spatial resolution of 0.2
arcsec, owing to negligible effect of atmospheric seeing, providing a powerful
tool for studying the fine-scale magnetic fields in the solar chromosphere. By
utilizing the unique capabilities of SCIP, this study has the potential to
deepen our understanding of the magnetic activities in the solar chromosphere.

Forward modeling using full Stokes synthesis of radiative MHD simulations is a
robust tool for interpreting complex observed spectra and simulating the
capabilities of a given instrument or mission. A series of studies have been
devoted to predicting the potential of the spectral lines to be observed by
SCIP to infer the properties of, e.g., magnetic pumping (Quintero Noda et al.,
2017) and chromospheric jets (Quintero Noda et al., 2019). Although
polarisation signals from the helical magnetic field associated with
chromospheric jets are expected to be detectable with SUNRISE III/SCIP, their
origin and relationship with the jet have not been investigated yet. Hence we
expand on the previous work using a time-series analysis to investigate
further the driving processes and the evolution of the chromospheric jet.

Our analysis suggests that Alfvénic motion in the flux tube generates an
upflow that drives the chromospheric jet. Consequently, strong linear
polarisation with dark, blue-shifted, and arc-like structures around the flux-
tube axis will be detectable with Ca ii infrared line. Along the axis of the
flux tube, bright and red-shifted cores are continuously observed in both Fe i
(846.8 nm) and Ca ii (849.8 nm) lines. Careful consideration of observation
configurations, including the field of views and integration time in SUNRISE
III/SCIP can capture all these features and help distinguish the origin of the
jets.

This paper is organised as follows: Section 2 presents methods and dataset,
and Section 3 details our data analysis and results with discussions. Finally,
the conclusions of the study can be found in Section 4.

##  2 Methods and Dataset

Figure 1: Summary of the time evolution of the analysed jet. (a) Height of the
transition region whose temperature was defined to be 4$\times 10^{4}$ K (The
maximum value was taken in x = 2.7 Mm plane) as a function of time. (b)–(d)
vertical velocity, $v_{\rm z}$ at y = 7.8 Mm. The dashed lines in (b)–(d)
indicate the height of the transition region. Temporal evolution ($\Delta$t =
-1, 0, 1 min) is displayed from left to right. Blue represents upflows while
red colour indicates downflowing materials.

In this study, the full Stokes profiles were derived using a radiative MHD
simulation combined with a Stokes synthesis code. Infrared spectra emerging
from a simulated chromospheric jet were modeled to predict future observations
from SUNRISE III/SCIP. Time-series analysis of the synthesised spectra enabled
us to distinguish the driving mechanisms of the jet that could not be
clarified by spectropolarimetry with a single snapshot.

###  2.1 Numerical simulation

As the reference solar atmosphere, we used a dataset from a radiative MHD
simulation identical to that used in Iijima & Yokoyama (2017). The simulation
was conducted using RAMENS code, and the full details of the simulation can be
found in (Iijima & Yokoyama, 2015; Iijima, 2016). In brief, the RAMENS code is
dedicated to simulating a realistic solar atmosphere in which various types of
physics are coupled. In addition to the ideal MHD equations, gravity, Spitzer-
type thermal conduction, equation of state under the assumption of LTE,
including the latent heat of partial ionisation, and radiative cooling are
included in the model. Radiative cooling is computed as a combination of
optically thick radiative cooling with gray approximation and optically thin
cooling. Based on these assumptions, the RAMENS code enables us to provide a
realistic solar atmosphere.

The MHD dataset represents a unipolar quiet region with an average magnetic
flux density of 10 G. The total duration analysed was approximately 22 min (t
$\in$ [396, 418] min) with a temporal cadence of 5 s. The horizontal domain is
9 Mm for in both the x- and y-directions with a periodic boundary. The
vertical domain ranges from -2 to 14 Mm and includes both the surface
convection zone and corona. The height z = 0 Mm roughly corresponds to the
surface where the optical depth is unity at the continuum wavelengths at 500
nm.

Among the chromospheric jets appearing in the dataset, we focused on the jet
investigated in Iijima & Yokoyama (2017). The jet started to erupt from t =
404 min to t = 413 min, showing a parabolic trajectory (Fig. 1a). Hereafter,
we use a different time coordinate, $\Delta{\rm t}~{}\equiv{\rm t}-404$ min,
to set $\Delta$t = 0 when the jet started for convenience. Starting from
$\Delta$t = 0, the jet reached its peak height of approximately 11 Mm at
$\Delta$t = 5 min and ended at $\Delta$t = 9 min. A photospheric magnetic
sheet existed below the jet to form a flux tube before the formation of the
jet. At $\Delta$t = -1 min, there appears to be a blob of upflow (the upflow A
in Fig. 1b) that will collide with the transition region to drive the jet at
$\Delta$t = 0 min (Fig. 1c). After upflow A hits the transition region,
chromospheric materials are lifted to form the jet.

###  2.2 Synthesis of the Stokes profiles

We synthesised full-Stokes profiles in the infrared range, including the Ca ii
849.8 nm and Fe i 846.8 nm lines. We focused on the Ca ii 849.8 nm line rather
than the more commonly used 854.2 nm line in this study because the
polarisation signals turned out to be stronger in the 849.8 nm line, although
SCIP can observe both spectral lines simultaneously. The RH code was used to
synthesise these lines (Uitenbroek, 2001). The equations of radiative transfer
and statistical equilibrium under non-LTE conditions are solved in the RH
code. Among the four modules specific for different geometries, rhf1d was used
to calculate the emergent spectra assuming a 1D plane-parallel geometry. This
geometry module allows us to calculate the Stokes profiles on a column-by-
column basis, which can be appropriate for the Ca ii lines where horizontal
scattering does not have significant effects (Leenaarts et al., 2009). We used
the RH code version 2, last modified on 2020 May 1.

Several modifications have been made to the input files that are necessary for
the RH code. First, complete redistribution was assumed for all spectral
lines. Second, the LTE hydrogen population was calculated using an abundance
identical to that used in the RAMENS code. The rest of configurable parameters
were identical to their default settings.

A spatial and spectral degradation was applied to the synthetic Stokes
profiles to reproduce the spatial and spectral resolution of SUNRISE III/SCIP.
For the spatial and spectral convolution, a Gaussian kernel with an FWHM of
0.2 $\times~{}(\lambda/854~{}{\rm nm})$ arcsec and 4 pm were applied,
respectively. Although the spectral resolution has a small dependence on
wavelength, we used a constant value of 4 pm for simplicity. Because there
were several pixels with poor convergence in Stokes synthesis (less than 1%),
we ignored these pixels during the convolution process.

To simplify the analysis, six representative variables were mainly used rather
than showing all the Stokes profiles: the intensity at the line core at the
rest wavelength, $I_{\rm core}$, Doppler velocity $V_{\rm LOS}$, maximum
amplitude of circular (MCP) and linear (MLP) polarisation, total linear
polarisation (TLP), and linear polarisation azimuth (LPA). The centre-of-mass
wavelength of $I_{\rm cont}-I$ at 11 spectral points near the line centre is
used to determine the line-of-sight velocity for each line. MCP and MLP are
defined as follows:

$\displaystyle\mathrm{MCP}$ $\displaystyle\equiv$
$\displaystyle\underset{|\lambda-\lambda_{0}|<\Delta\lambda}{\rm
max}(|V(\lambda)|),$ (1) $\displaystyle\mathrm{MLP}$ $\displaystyle\equiv$
$\displaystyle\underset{|\lambda-\lambda_{0}|<\Delta\lambda}{\rm
max}\left(\sqrt{Q(\lambda)^{2}+U(\lambda)^{2}}\right)$ (2)

where $\Delta\lambda$ = 20 pm (five spectral sampling sizes) for each line and
$\lambda_{0}$ is the central wavelengths in the rest frame. The TLP is defined
as

$\displaystyle{\rm TLP}\equiv\sqrt{\langle Q\rangle^{2}+\langle
U\rangle^{2}},$ (3)

where

$\displaystyle\langle
f\rangle\equiv\frac{1}{2\Delta\lambda}\int^{\lambda_{0}+\Delta\lambda}_{\lambda_{0}-\Delta\lambda}f(\lambda)d\lambda.$
(4)

Following the equations given in Landi Degl’Innocenti & Landolfi (2004), LPA,
which provides an estimation of the magnetic field azimuth, is defined as

$\displaystyle\phi=\frac{1}{2}\arctan\left(\frac{U}{Q}\right),$ (5)

where spectral points with maximum amplitudes of $Q$ and $U$ in
$|\lambda-\lambda_{0}|<\Delta\lambda$ were used.

One of the main advantages of our approach is that it analyses the time series
of Stokes profiles throughout the entire evolution of the jet. With this
approach, we attempted to specify the spectral properties from the driving
process of a jet that were not obtained in the previous study of Quintero Noda
et al. (2019).

##  3 Results and Discussion

Overall, the results presented below show that the jet was driven by a blob of
upflow connected to the twisted field lines above a flux tube. A bright down-
flowing core and dark up-flowing envelope structure with strong linear
polarisation signals can be detected using spectro-polalimetric observations
from SUNRISE III/SCIP.

Figure 2: Pearson’s product moment correlations between synthetic data in (a)
Fe i and (b) Ca ii line and MHD variables as a function of height. The black,
orange, blue, and green lines represent the correlation between temperature &
$I_{\rm core}$, horizontal field strength ($B_{\rm h}\equiv\sqrt{B_{\rm
x}^{2}+B_{\rm y}^{2}}$) & MLP, vertical field strength & MCP, and vertical
velocity & $V_{\rm LOS}$, respectively. Symbols designate the height where
each correlation is maximum.

###  3.1 Estimation of formation layers

To roughly estimate the formation layers of each line, Pearson’s product
moment correlations were obtained between temperature & $I_{\rm core}$,
horizontal field strength & MLP, vertical field strength & MCP, and vertical
velocity & $V_{\rm LOS}$ at each height. The peaks in the correlations from Fe
i line ranged in 0–0.3 Mm (Fig. 2a), while those from the Ca ii line ranged in
0.5–1.0 Mm (Fig. 2b). These analyses will help us understand the relationship
between vertical structures and synthetic data, although further analysis is
necessary to retrieve exact height information

###  3.2 Evolution of the synthesized observables during the jet

Figure 3: Time evolution of the synthetic images (MCP & MLP) and the related
MHD variables during the jet (- 6 min $<$ $\Delta$t $<$ \+ 9 min). From left
to right column, (a) $B_{\rm z}$ at z = 0.27 Mm, (b) MCP for the Fe i line,
(c) $B_{\rm z}$ at z = 0.62 Mm, (d) MCP for the Ca ii line, (e) $B_{\rm h}$ at
z = 0.06 Mm, (f) MLP for the Fe i line, (g) $B_{\rm h}$ at z = 0.56 Mm, and
(h) MLP for the Ca ii line were shown. The red arrows in columns (e) and (g)
indicate the horizontal component of magnetic field vector. The black line
segments in columns (f) and (h) represent the LPA where polarization signals
were larger than 3$\times$10-4 of $I_{\rm cont}$.

The synthetic MCP signals capture the evolution of the axial field of the jet
(columns a–d in Fig. 3). From $\Delta{\rm t}=-6{\rm~{}min}$, the two flux
sheets (B & C in column a in 3) in the granular lanes approached the
intersection of the granular lanes. The lower half of sheet B merged with
sheet C at $\Delta$t = - 4 min to create a flux tube where the foot point of
the jet is located. The upper half of sheet B was moved up to create another
flux tube. These flux sheets produced strong polarisation signals well above
10-3 of $I_{\rm cont}$ on Fe i line during the lifetime of the jet. A cadence
of 1 min is sufficient to capture the flux-sheet merger ($\Delta$t = - 4 min).
At a height of z = 0.6 Mm, flux sheet B expanded to create a round-shaped
axial field of the jet. The axial field was surrounded by a negative spiral-
like structure after the flux-sheet merger (column c in Fig. 3). We found that
the negative structure was a part of the twisted field, whose elevation angle
was locally negative. Although the axial-field components have strong
polarisation signals in the Ca ii line, a twisted field with a negative LOS
component exhibited weak signals (column d in Fig. 3).

The twisted field structure around the axis of the jet was spatially resolved
with reasonable linear polarisation signals (columns e–h in Fig. 3) that are
above the noise level expected for SCIP. At the photospheric height, the
twisted field around the flux tube appears after the flux merger ($\Delta$t =
- 4 min) and gradually disappears after the jet reaches its maximum height
($\Delta$t = 5 min). The photospheric twist revealed strong linear
polarisation signals around the envelope of the axial field. A tornado-like
structure is continuously observed at the chromospheric heights (column g in
Fig. 3). The corresponding structure was found in MLP signals, although the
twist direction was not available owing to the 180° ambiguity. The magnetic
twist became increasingly prominent until $\Delta$t = 5 min, and gradually
disappeared by $\Delta$t = 9 min (column h in Fig. 3).

Figure 4: Time evolution of the synthetic images ($I_{\rm core}$ & $V_{\rm
LOS}$ ) and the related MHD variables during the jet (-6 min $<\Delta$t$<$ \+
9 min). From left to right column, (a) Te at z = 0.27 Mm, (b) $I_{\rm core}$
for the Fe i line, (c) Te at z = 0.77 Mm, (d) $I_{\rm core}$ for the Ca ii
line, (e) $V_{\rm z}$ at z = 0.09 Mm, (f) $V_{\rm LOS}$ for the Fe i line, (g)
$V_{\rm z}$ at z = 1.01 Mm, and (h) $V_{\rm LOS}$ for the Ca ii line were
shown.

From the line-core images, we can infer that the jet had a hot core and cool
envelopes (columns a–d in Fig. 4). A bright spot of approximately 0.2 Mm
appeared in the line core intensity of Fe i line above the flux tube axis
after the flux merger. This enhancement continued until at least $\Delta$t = 5
min, and disappeared at $\Delta$t = 9 min. The temperature at the photospheric
heights also exhibited similar behaviour. At the chromospheric heights, hot
core and cool envelope appeared well before and after the jet initiation
($\Delta$t = -6,-4, and 5 min). The corresponding structure was found in the
line-core images of Ca ii as a bright core and dark envelope. This feature
sometimes can be observed as magnetic tornadoes or chromospheric swirls found
in past observations (Wedemeyer-Böhm & Rouppe van der Voort, 2009; Wedemeyer-
Böhm et al., 2012).

The Doppler signals help to find the descending core and ascending envelope of
the jet (columns e–h in Fig. 4). The granulation pattern was observed in the
Doppler signals of the Fe i line. There was a small downflow near the axis of
the flux tube, particularly after the flux sheet merger, which is consistent
with the magnetic vortex appearing in the previous simulation (Kitiashvili et
al., 2013). At the chromospheric heights, the flux tube axis tended to have a
downward flow, while the envelope tended to have an upflow in the Ca ii line.
However, it was challenging to distinguish the upflow A in the x-y plane at
the chromospheric heights from a single snapshot. This was because the
fluctuations in the vertical velocity were dominated by acoustic waves that
hindered the determination of the velocity in the jet. A blue shift was
sometimes obtained at the same location as the jet-related upflow, although a
time-series analysis, such as sit and stare observation, was necessary to
confirm that the blue shift was related to the jet-upflow.

###  3.3 Sit-and-stare observation

A set of time distance diagrams for the Fe i 846.8 nm and Ca ii 849.8 nm lines
were derived to simulate a sit-and-stare observation (Fig. 5). The slit was
located at the centre of the photospheric magnetic concentration (x = 2.1 Mm)
along the y-direction. At the photospheric heights, we found that the
localised enhancement of the line core intensity continued throughout the
lifetime of the jet (Fig. 5a). At the same location, the continuous MCP
signals overlapped with the line-core enhancement with strong MLP signals at
the envelope (Fig. 5b). Moreover, the down-flow signals with 2–3 km s-1 were
associated with the hot and magnetised core of the flux tube (Fig 5c). At the
chromospheric heights, both the dark envelope and bright core can be well
distinguished, especially in the later phase of the jet ($\Delta$t > 5 min)
(Fig 5d). The axial field was detectable using MCP because the signal of
approximately 2 % of $I_{\rm cont}$ was well above the polarimetric
sensitivity of SUNRISE III/SCIP ($1\sigma=3\times 10^{-4}$ of $I_{\rm cont}$)
(Fig. 5e). The amplitude of MLP signals in the arc-like structure was weaker,
so it would be difficult to distinguish between noise and signals using MLP
signals for this case. Because the arc-like structure had a width of 0.5 Mm
and a lifetime of more than 5 min, spatio-temporal binning will increase the
S/N ratio. The arc structure in MLP originated from the field lines at the
envelope of the flux tube is writhing counterclockwise. An upward Doppler
velocity of approximately 1 km s-1 above the arc indicates the upward motion
of the writing flux tube (Fig. 5f). Because the fluctuations of the Doppler
velocity in the ambient region were comparable to those of the flux tube, it
was necessary to combine the Doppler and polarimetric signals to infer the
velocity of the flux tube.

Figure 5: Time-distance diagram of (a) $I_{\rm core}$, (b) MCP, and (c)
$V_{\rm LOS}$ derived from the synthetic spectra of the Fe i 846.8 nm line,
and (d) $I_{\rm core}$, (e) MCP, and (f) $V_{\rm LOS}$ derived from the
synthetic spectra of the Ca ii 849.8 nm line. Contours of 2, 4, and 6 (1, 2,
and 3) sigma levels in MLP were over-plotted in the panels from (a) to (c) (
(d) to (f) ) . The slit position was taken along x = 2.1 Mm.

###  3.4 Detection of linear polarization signals

Analysing the linear polarisation signals in chromospheric lines is
challenging because they are generally less sensitive to the magnetic field
due to their lower effective Landé factors. So, in our case, it may be
advisable to use the reference variable TLP instead of MLP to analyse the Ca
ii linear polarisation signals. This is because the averaging operation over
wavelength reduces the noise level, whereas the maximum operation increases
the noise level. For instance, when applying Gaussian noise with 4 $\times$
10-4 Icont, both MLP and TLP signals became comparable to the noise level
(Fig. 6a & b). Performing a spatial binning (the spatial sampling increasing
from 0.1 to 0.2 arcsec) increased the signal-to-noise ratio up to 6 (Fig. 6c &
d). The arc structure was still visible owing to the spatial scale of
approximately 1 Mm, although further binning destroyed the structure.
Performing an additional binning on the time domain ( temporal sampling
increases from 10 to 20 s) further increases the signal-to-noise ratio to more
than 6 (Fig. 6e & f) in the most prominent phase.

Figure 6: Linear polarization signals from the Ca ii 849.8 nm line at
$\Delta$t = 5 min: (a) TLP with noise, (b) MLP with noise, (c) TLP with noise,
2 binned in the spatial domain, (d) MLP with noise, 2 binned in spatial domain
(e) TLP with noise, 2 binned both in the spatial and time domain, (f) MLP with
noise, 2 binned in the spatial and time domain. The colour corresponds to S/N
ratio. For temporal binning, data in $\Delta$t = 5 min - 10 sec was also used.

To optimize the observations with SCIP, it is important to consider the trade-
off between temporal cadence and polarimetric accuracy, given that SCIP is a
slit scanning spectro-polarimeter. In this study, the magnetic arc had a
spatial scale of approximately 1 Mm, and its evolution could be effectively
resolved with a cadence of less than 2 minutes. By selecting an integration
time of 10.24 seconds, the slit scan cover an area of 1 arcsec times the slit
size in 1.9 minutes. In addition, that integration times will produce a
polarization accuracy of $4\times 10^{-4}$ of $I_{\rm cont}$ (1 sigma) without
spatial nor temporal summing in the Ca ii 849.8 nm line. Although the FOV is
narrower than the magnetic arc, a S/N ratio greater than 6 can be obtained by
spatially summing up to two pixels (Fig. 6 c). A narrower FOV can further
improve the S/N ratio by temporal binning.

###  3.5 Driving process of the jet

Figure 7: Evolution of the jet-related field lines. The magnetic field lines
at $\Delta t$ = -1 min were temporally traced assuming ideal MHD conditions.
The colours in x-y and x-z slices indicate $B_{\rm z}$ and $V_{\rm z}$,
respectively. The colours on the field lines represent the local temperature.

During the jet, the foot point of the twisted field sank and swirled
counterclockwise from the edge to the centre of the magnetic concentration in
the photosphere. Simultaneously, the twisted part moved upward and became a
vertical axial field (Fig. 7). At $\Delta$t = -1 min, upflow A, which was
about to collide with the transition region, was connected to the twisted
field (Fig 7a). We temporally traced a group of field lines that were
connected to the upflowing material assuming an ideal MHD process, and found
that the twisted field lines evolved into the axial field with time (Fig 7b &
c). As the twisted part moved upward, part of the cool material inside the
flux was lifted to the corona to form the jet, while the other half was
flowing down along the axial field. After the field lines became the flux
tube’s axial field, the ambient plasma continuously supplied another twisted
field, which maintained the tornado-like structures, as observed in the
horizontal field evolution (column g & h in Fig. 3).

There are several candidates for the triggering process of upflow A. The first
candidate is the flux merger at $\Delta$t = -4 min. By temporally tracing
field lines in sheets B & C, we found that the field lines were swirling
counterclockwise when approaching the intersection of the granular lanes.
Because B${}_{\rm z}>0$ in the flux sheets, and the current density in the
z-direction was negative (inferred from column g in Fig. 3), the additional
counterclockwise vorticity drove the upward propagating Alfvénic motion. This
Alfvénic motion produces the upflow A through nonlinear wave pressure (Hollweg
et al., 1982). The second candidate is the downflow along the axis of the flux
tube. This downflow contributes to the upflow via the rebounding process
(Sterling & Hollweg, 1988; Kato et al., 2016). The third candidate is magnetic
reconnections between the twisted field and its overlying field. This
reconnection may contribute to the upflow via sling shot effects, similar to
the reconnection above the emerging flux (Yokoyama & Shibata, 1996). The
combinations of the three candidates could contribute to the upflow, although
the further theoretical investigation is needed to quantitatively measure the
degree of contribution of each process.

###  3.6 Summary of the evolving jet

Herein, we briefly summarise the important physics and observables during the
evolution of the chromospheric jet.

  * •

Before $\Delta$t = - 6 min, two flux sheets B and C were approaching the
intersection of granular lanes. This motion can be detectable by tracing
enhanced MCP signals or bright Icore in Fe i line (Fig. 3 & 4).

  * •

At $\Delta$t = - 4 min, the two flux sheets merged. After the flux merger, the
down-flowing hot core was continuously observed in the Fe i line at the foot
point during the lifetime of the jet (Fig. 4 & 5).

  * •

At $\Delta$t = 0 min, the twisted field line moved up to collide with the
transition region (Fig. 1 & 7). The twisted field lines can be observed as the
arc-like linear polarization signals with strong enough amplitude in Ca ii
line (Fig. 3 & 6). The arc-like structure was dark in the line core, showing
blueshifted velocities (Fig. 5). Moreover, a bright core and a dark envelope
in Ca ii would be useful to specify the jet structure.

  * •

The magnetic-tornado-like structure with a hot core and cool envelope
continued to exist during the lifetime of the jet (0$<\Delta$t$<$9 min). This
feature would be detectable from both Icore, MLP, and TLP signals in Ca ii
(Fig. 3, 4, 5, and 6).

##  4 Conclusion

Although there are several candidate mechanisms for generating chromospheric
jets, it has been difficult to conclusively determine the driving mechanism
directly from observations. Because the magnetic field plays a key role, high
spatial, spectral, and spectro-polarimetric observations are essential for
distinguishing between competing mechanisms. We found that flux sheet merger,
bright and down-flowing core, and arc-like linear polarisation signals with
dark and blue-shifted envelope were expected from chromospheric jets, although
we analysed only one chromospheric jet appearing in the RMHD simulation. These
properties are the manifestation of the transition process from the ambient
twisted field to the axial field, which is consistent with the upwardly
propagating nonlinear Alfvénic waves. The triggering process is still under
investigation, although we considered that the key mechanisms would the flux
merger, downflow along the core, or magnetic reconnection above the twisted
field.

Our prediction will be a useful tool for distinguishing the driving mechanisms
of chromospheric jets when combined with future observations, such as SUNRISE
III, DKIST, or EST. An appropriate time cadence, integration time, and field
of view should be selected to fully capture the characteristic magnetic field
associated with the chromospheric jets.

## Acknowledgements

We would like to express our sincere gratitude to the reviewer for the
invaluable feedback and constructive suggestions on our manuscript. Numerical
computations for the radiation MHD simulation were performed on a Cray XC30
supercomputer at the Centre for Computational Astrophysics, National
Astronomical Observatory of Japan. Numerical analyses and Stokes synthesis
were performed on the analysis servers at the Centre for Computational
Astrophysics at the National Astronomical Observatory of Japan. A part of this
study was conducted using computational resources from the Centre for
Integrated Data Science at the Institute for Space-Earth Environmental
Research, Nagoya University, through a joint research program. This work was
supported by JSPS KAKENHI Grant Number JP18H05234 (PI: Y. Katsukawa).

## Data Availability

The data underlying this article will be shared upon reasonable request by the
corresponding author.

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# Bound States of Central Force System in Special Relativity

Iwan Setiawan1, Ryan Sugihakim2, and Bobby Eka Gunara2 1 Department of Physics
Education, University of Bengkulu, Kandang Limun, Bengkulu 38371, Indonesia  2
Department of Physics, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung
40132, Indonesia<EMAIL_ADDRESS>
###### Abstract

In this paper we consider the central force problem in the special theory of
relativity. We derive the special relativistic version of the Binet equation
describing the orbit of a body. Then, the motion of a planet in a solar-like
system where the gravitational potential is modified by adding a term that is
proportional to the inverse of the square of the radial coordinate is
discussed. Using perturbative method, we obtain the explicit orbital function.
At the end, the orbital precession of the planet and its relation to the
result from general relativity are also discussed.

##  1 INTRODUCTION

One of the common situations in physics is the motion of a body under
influence of the central force in which the magnitude of the force depends
only on the radial coordinate. The problem is relevant to realistic conditions
such as the motion under either gravitational force or electrostatic force.
Many standard textbooks in classical mechanics discuss the central force
problem such as two-body system to solve the motion of interacting objects,
see for example, [1, 2]. We can say that this problem is essential to physics
students improving their competence.

In the non-relativistic mechanics, the central force problem can be reduced to
second order ordinary differential equation called Binet equation whose
solutions are called orbit in which the position can be viewed as a function
of angle in the polar coordinate. Several papers have been published to study
this equation. For example, in [3, 4] some authors considered a macroscopic
case where the potential energy has polynomial form which covers the standard
Newton gravitational potential energy that is proportional to the inverse of
the radial coordinate. They find that the orbit is an implicit function
related to a hypergeometric function. At the microscopic level, the authors in
[5] have studied the Sommerfeld atomic model of Hydrogen-like model. However,
we know that this is unrealistic since according to the astronomical
observation the relativistic effect cannot be ignored especially in the motion
of a planet near a massive object. Therefore, the Binet equation has to be
extended to the relativistic domain.

In the extension to the relativistic mechanics, although general relativity is
the most favourable at the level of macroscopic system, it is still of
interest to consider special relativity since it is much simpler than the
general theory. In other words, we can study the dynamics of a particle
without curving the spacetime. There are several attempt in this direction.
For example, in [6] the author extend the concepts the central and
conservative forces to special relativity. The other authors in [7] have tried
to extend Bertrand’s theorem to the central force problem in special
relativity. They also find that there is a special form of potentials which
admits a stable circular orbit. There are also other authors that have
considered this problem at the microscopic level to produce Sommerfeld-Dirac
fine structure formula [8, 9].

According to the above developments, we also attempt to consider the central
force problem in special relativity. First, we review the derivation of the
relativistic Binet equation describing the orbit of a body [7, 10]. We shortly
discuss the stability criterion of the circular orbit. Then, a simple case is
considered where the potential is proportional to the inverse of the radial
coordinate. We obtain an explicit orbital function in which the orbital
precession of a body appears naturally for elliptical orbits which can be
regraded as the relativistic effect [10], see also the discussion in [11].
This precession vanishes if the orbit is circular.

Next, at the microscopic level the above results are applied to the case of
the hydrogen-like atom. Using the Wilson-Sommerfeld quantization rule, we
obtain the energy level of the system so called Sommerfeld-Dirac fine
structure formula [8, 9] which can also be derived from Dirac equation in
relativistic quantum mechanics [12, 13].

At the final part, we discuss a planetary motion in which the potential energy
is fine-tuned by adding the term which is proportional to the cubic of of the
radial coordinate with arbitrary pre-coefficient. This potential energy has
been appeared in the case of a free falling body in Schwarzschild spacetime,
see for example, [14, 15] and Randall-Sundrum model of gravity [16]. Using
perturbative method, we get an explicit orbital function that admits the
orbital precession of the planet and then, we compare it with the result from
general relativity. It is worth mentioning that this explicit orbital function
has never been considered in the literature above. Also, we pointed out the
second Kepler’s laws which differs to the case of Newtonian mechanics.

In this paper, it is convenient to use Lagrangian formulation as a starting
point. We use the relativistic Lagrangian that can be found from textbooks
such as by Landau and Lifshitz [14]. We discuss the motion of a body under the
influence of the central force using this formulation in section 2. In section
3, we derive the energy levels of hydrogen-like atom [8, 9, 12, 13]. Then, in
section 4 we discuss the orbit precession and the fine-tuning of model to
explain a planet’s orbital precession. We add some appendices to support our
arguments in the paper.

##  2 Special Relativistic Central Force Problem

In this section we discuss our general setup of a central force system in
special theory of relativity. Then, our discussion will be focus on an
attractive force whose scalar potential is proportional to the inverse of the
distance.

Let us first consider a relativistic particle with the rest mass $m_{0}$
moving with speed $v$ under the influence of a central force that depends only
on the radial coordinate $r$ whose potential energy function is given by
$V\equiv V(r)$. This implies that we could reduce the case to two dimensional
problem such that using the polar coordinate $(r,\theta)$ for space with
coordinate time $t$. We define

$\displaystyle\gamma(v)\equiv\quantity(1-\frac{v^{2}}{c^{2}})^{-1/2}~{}.$ (1)

The Lagrangian of this system is

$\displaystyle
L(r,\theta,\dot{r},\dot{\theta})=-\frac{m_{0}c^{2}}{\gamma(v)}-V(r)~{}.$ (2)

The dot over a quantity denotes the derivative with respect to $t$. After some
computation we could show that the constants of motion are the angular
momentum $p_{\theta}$ and the total ’mechanical’ energy $E$ (see A for
details)

$\displaystyle p_{\theta}=\gamma m_{0}r^{2}\dot{\theta}~{},$ (3)
$\displaystyle E=\gamma m_{0}c^{2}+V(r)~{},$ (4)

whereas the radial momentum

$\displaystyle p_{r}=\gamma m_{0}\dot{r}=-p_{\theta}u^{\prime}~{},$ (5)

is not constant where $u\equiv 1/r,u^{\prime}\equiv\derivative*{u}{\theta}$.
We note that

$\displaystyle\gamma=\quantity[1+\frac{p_{\theta}^{2}}{m_{0}^{2}c^{2}}\quantity(u^{\prime
2}+u^{2})]^{1/2}=\frac{E-V}{m_{0}c^{2}}~{}.$ (6)

From (4) and (6), we obtain then

$\displaystyle m_{0}^{2}c^{4}=(E-V)^{2}-u^{\prime
2}p_{\theta}^{2}c^{2}-u^{2}p_{\theta}^{2}c^{2}~{},$ (7)

such that if we differentiate (7) with respect to $\theta$, it leads to the
relativistic Binet equation

$\displaystyle\derivative[2]{u}{\theta}+u+\frac{(E-V)}{p_{\theta}^{2}c^{2}}\derivative{V}{u}=0~{}.$
(8)

Some comments are in order. If $u$ is a positive constant, then we may have a
circular orbit. This is the simplest closed orbit which is possible if the
following condition fulfills

$\displaystyle
u_{0}=\frac{(V(u_{0})-E)}{p_{\theta}^{2}c^{2}}\frac{dV}{du}(u_{0})~{},$ (9)

where $r_{0}=1/u_{0}$ is the radius of the circular orbit implying
$V(u_{0})\neq E$ and $u_{0}$ is not the extremal point of $V(u)$, namely,
$\frac{dV}{du}(u_{0})\neq 0$. These features distinguish to the case in
Newtonian mechanics as discussed in [1]. Moreover, this orbit is stable under
small perturbations if it satisfies [7]

$\displaystyle\xi\equiv 1-\frac{dJ}{du}(u_{0})>0~{},$ (10)

that should be a rational number where

$\displaystyle J\equiv\frac{(V-E)}{p_{\theta}^{2}c^{2}}\frac{dV}{du}~{}.$ (11)

In the case of attractive potentials such as Newtonian gravitational potential
and Coulomb potential, the condition (10) follows

$\displaystyle K^{2}/p_{\theta}^{2}<c^{2}~{}.$ (12)

###  2.1 Orbital Motion in a Special Class of Central Force

Now, we particularly discuss the potential energy function whose form is given
by

$\displaystyle V(r)=-\frac{K}{r},\quad K>0~{},$ (13)

satisfying (12) which appears only in relativistic cases. The potential (13)
describes the attractive force between two particles which is proportional to
the inverse of $r^{2}$. In the case at hand, we may have a solution with
closed orbit. Substituting to the equation (8), we have

$\displaystyle\derivative[2]{u}{\theta}+\quantity(1-\frac{K^{2}}{p_{\theta}^{2}c^{2}})u=\frac{KE}{p_{\theta}^{2}c^{2}}~{}.$
(14)

Using the condition $u^{\prime}(0)=0$, we have the solution, see the
discussion in [11],

$\displaystyle u(\theta)=\frac{1+e\cos(\kappa\theta)}{a(1-e^{2})}~{},$ (15)

where $e$ and $a$ are eccentricity and semi-major axis of the ellipse,
respectively, with

$\displaystyle e=\sqrt{\beta}~{},$ (16) $\displaystyle
a=\frac{K}{E}\quantity[\quantity(\frac{p_{\theta}c}{K})^{2}-1]\frac{1}{1-e^{2}}~{},$
(17)
$\displaystyle\kappa\equiv\quantity[1-\quantity(\frac{K}{p_{\theta}c})^{2}]^{1/2}~{},$
(18)
$\displaystyle\beta\equiv\quantity(\frac{m_{0}c^{2}}{E})^{2}+\quantity(\frac{p_{\theta}c}{K})^{2}-\quantity(\frac{m_{0}c^{2}}{E}\frac{p_{\theta}c}{K})^{2}~{}.$
(19)

We could express the angular momentum and the energy in terms of $a$ and $e$,
namely,

$\displaystyle\frac{p_{\theta}^{2}c^{2}}{K^{2}}=\frac{1+e^{2}}{2}\quantity[1+\frac{1-e^{2}}{1+e^{2}}\sqrt{1+4\eta^{2}}]~{},$
(20) $\displaystyle
E=\frac{K}{a(1-e^{2})}\quantity[\frac{p^{2}_{\theta}c^{2}}{K^{2}}-1]~{},$ (21)

respectively, with $\eta\equiv\frac{m_{0}c^{2}a}{K}$. It is also important to
state that in this relativistic case we have the orbital precession

$\displaystyle
2\pi\delta=\frac{2\pi}{1-\frac{K^{2}}{2p_{\theta}^{2}c^{2}}}-2\pi~{}.$ (22)

As a numerical simulation, we set $m=K=1$. For $e=1/4,a=1$, the orbit of mass
is shown in the figures 3, 3, and 3.

Figure 1: Case $e=1/4$, $a=1$

Figure 2: Case $e=1/3$, $a=1$

Figure 3: Case $e=1/2$, $a=1$

In the case of $K/p_{\theta}c\ll 1$, we could have $\kappa\approx
1-\frac{K^{2}}{2p_{\theta}^{2}c^{2}}$ so that

$\displaystyle\delta=\quantity(\frac{2p_{\theta}^{2}c^{2}}{K^{2}}-1)^{-1}\approx\frac{K^{2}}{2p_{\theta}^{2}c^{2}}~{}.$
(23)

We would like to write some comments as follows. Although the orbital
precession can be regarded as relativistic effect, it is of less interest to
consider it at the level of microscopic physics since it is almost impossible
to observe this effect. Instead, we have relativistic quantum mechanics for
this level. Therefore, it is more interesting to discuss it at the level of
macroscopic physics, for example, involving a case of gravitational
interaction in the special theory of relativity.

##  3  Hydrogen-like Atomic Model

In this section we apply our study in the preceding case to the model of
hydrogen-like atom where an electron with mass $m_{e}$ orbits a proton as
nucleus with mass $m_{N}$. They interact by the Coulomb force with
$K=q^{2}/4\pi\epsilon_{0}$, where $q$ is the magnitude of electron and proton
charges. This case at hand can be viewed as a single particle model moving
under the influence of the Coulomb force with reduced mass
$\mu=\quantity(1/m_{e}+1/m_{N})^{-1}$. Using Wilson-Sommerfeld rule of
quantization

$\displaystyle\oint p_{\theta}\differential\theta=n_{\theta}h,\rightarrow
p_{\theta}\int_{0}^{2\pi}\differential\theta=n_{\theta}h~{},$ (24)
$\displaystyle\oint p_{r}\differential r=n_{r}h\rightarrow
p_{\theta}\oint\frac{u^{\prime
2}}{u^{2}}\differential\theta=p_{\theta}\oint\frac{\kappa^{2}\sin[2](\kappa\theta)}{\quantity(1/e+\cos(\kappa\theta))^{2}}\differential\theta=n_{r}h~{},$
(25)

with $n_{\theta},n_{r}\in\mathbb{Z}$. From the first integral, we have

$\displaystyle p_{\theta}=n_{\theta}\hbar~{},$ (26)

whereas for the second integral, it gives

$\displaystyle\int_{0}^{2\pi}\frac{\sin[2](\tilde{\theta})}{\quantity(1/e+\cos(\tilde{\theta}))^{2}}\differential\tilde{\theta}=\frac{2\pi}{\sqrt{1-e^{2}}}+2\pi=\frac{2\pi
n_{r}}{\sqrt{n_{\theta}^{2}-\alpha^{2}}}~{},$ (27)

where $\tilde{\theta}=\kappa\theta$,
$\kappa=\sqrt{1-(\alpha^{2}/n_{\theta}^{2})}$, and
$\alpha=q^{2}/(4\pi\epsilon_{0}\hbar c)$ implying

$\displaystyle
e=\quantity[1-\quantity(\frac{\sqrt{n_{\theta}^{2}-\alpha^{2}}}{n_{r}+\sqrt{n_{\theta}^{2}-\alpha^{2}}})^{2}]^{1/2}.$
(28)

Since $e\in[0,1]$, $n_{\theta}$ cannot be zero since it produces complex
number for $e$, then $n_{\theta}\geq 1$. While $n_{r}$ can be chosen
$n_{r}\geq 0$. Defining $\tilde{n}\equiv\sqrt{n_{\theta}^{2}-\alpha^{2}}$ and
$n^{\prime}\equiv n_{r}+\sqrt{n_{\theta}^{2}-\alpha^{2}}$, we have

$\displaystyle e=\sqrt{1-\frac{\tilde{n}^{2}}{n^{\prime 2}}}~{}.$ (29)

The values of $n^{\prime}$ is given in Figure 4.

Figure 4: The values of $n^{\prime}$. Each branch of clusters value is
approximately equal to the parent branch.

From equation (16), the eccentricity is given by

$\displaystyle e=\pm\quantity[\quantity(\frac{\mu
c^{2}}{E})^{2}+\quantity(\frac{n_{\theta}}{\alpha})^{2}-\quantity(\frac{n_{\theta}}{\alpha}\frac{\mu
c^{2}}{E})^{2}]^{1/2}~{}.$ (30)

The energy of the system is

$\displaystyle E$ $\displaystyle=\mu
c^{2}\sqrt{\frac{n_{\theta}^{2}-\alpha^{2}}{n_{\theta}^{2}-e^{2}\alpha^{2}}}=\frac{\mu
c^{2}}{\sqrt{1+\dfrac{\alpha^{2}}{n^{\prime 2}}}}=\mu
c^{2}\quantity(1+\dfrac{\alpha^{2}}{\quantity(n_{r}+\sqrt{n_{\theta}^{2}-\alpha^{2}})^{2}})^{-1/2}.$
(31)

The above energy formula resembles the Sommerfeld’s fine structure [8, 13, 9]
(see also,[12] p. 231)

$\displaystyle
E=mc^{2}\quantity(1+\frac{\alpha^{2}}{\quantity[n-j-\frac{1}{2}+\sqrt{(j+\frac{1}{2})^{2}-\alpha^{2}}]^{2}})^{-1/2},$
$\displaystyle n=1,2,3,\dots,\quad
j=\frac{1}{2},\frac{3}{2},\frac{5}{2},\dots$ (32)

where $n$ is the principal quantum number, and $j$ is the total angular
momentum number. We conclude
$j=n_{\theta}-\frac{1}{2},n=n_{r}+j+\frac{1}{2}=n_{r}+n_{\theta}$.

##  4 The Orbital Precession in Modified Gravity

This section is devoted to consider a solar system where a planet with mass
$m$ moves around a massive star with mass $M$. We assume that the orbit is
closed. In this section, we take $K=GMm$ where $G$ is the universal
gravitational constant.

First, let us focus on a modified gravitational theory whose potential energy
has the form

$\displaystyle V(r)=-\frac{K}{r}+\frac{\epsilon K_{1}}{r^{3}}~{},$ (33)

where $K_{1}$ is a real constant and $\epsilon$ is a small dimensionless
quantity. The above form of potential energy is motivated by generalized Binet
equation derived using Schwarzschild geometry, see for example [15] and a
modified gravitational theory in the context of Randall-Sundrum Model [16]. It
is important to notice that since $K_{1}$ is an arbitrary real constant in
(33), it also covers the case in [10] where they have fixed the constant
coming from perturbative modified gravitational force.

Setting $\epsilon\ll 1$, the Binet equation (8) simplifies to

$\displaystyle
u^{\prime\prime}+\kappa^{2}u-\frac{EK}{p_{\theta}^{2}c^{2}}=-\epsilon
K_{1}\quantity(\frac{4K}{p_{\theta}^{2}c^{2}}u^{3}+\frac{3E}{p_{\theta}^{2}c^{2}}u^{2})+\mathcal{O}(\epsilon^{2})~{},$
(34)

such that we can write the solution of (34) as

$\displaystyle u(\theta)=u_{0}(\theta)+\epsilon u_{1}(\theta)~{}.$ (35)

The function $u_{0}$ is the unperturbed solution given in (15), whereas the
function $u_{1}$ is the first order of perturbation with the boundary value
$u_{1}(0)=u_{1}^{\prime}(0)=0$. Inserting the ansatz (35) into (34), it
results

$\displaystyle u_{0}^{\prime\prime}+\epsilon
u_{1}^{\prime\prime}+\kappa^{2}(u_{0}+\epsilon
u_{1})-\frac{EK}{p_{\theta}^{2}c^{2}}=-\epsilon
K_{1}\quantity(\frac{4K}{p_{\theta}^{2}c^{2}}u_{0}^{3}+\frac{3E}{p_{\theta}^{2}c^{2}}u_{0}^{2})+\mathcal{O}(\epsilon^{2})~{}.$
(36)

Then, equalizing the first order terms, we have

$\displaystyle
u_{1}^{\prime\prime}+\kappa^{2}u_{1}=-u_{0}^{2}\quantity(\frac{4KK_{1}}{p_{\theta}^{2}c^{2}}u_{0}+\frac{3EK_{1}}{p_{\theta}^{2}c^{2}})~{},$
(37)

whose solution has the form

$\displaystyle
u_{1}=c_{1}\cos(\kappa\theta)+c_{2}\sin(\kappa\theta)+v_{1}(\theta)\cos(\kappa\theta)+v_{2}(\theta)\sin(\kappa\theta)~{},$
(38)

where $c_{1},c_{2}$ are real constants, and

$\displaystyle v_{1}(\theta)$ $\displaystyle=\frac{K_{1}}{\kappa
p_{\theta}^{2}c^{2}}\int_{0}^{\theta}\differential\theta^{\prime}\
u_{0}^{2}\quantity(4Ku_{0}+3E)\sin(\kappa\theta^{\prime})$
$\displaystyle=-\frac{K_{1}}{ea\kappa^{2}p_{\theta}^{2}c^{2}\quantity(1-e^{2})^{2}}\bigg{[}\frac{K\quantity[1+e\cos(\kappa\theta)]^{4}}{ea\quantity(1-e^{2})^{3}}+E\quantity[1+e\cos(\kappa\theta)]^{3}\bigg{]}~{},$
(39) $\displaystyle v_{2}(\theta)$ $\displaystyle=-\frac{K_{1}}{\kappa
p_{\theta}^{2}c^{2}}\int_{0}^{\theta}\differential\theta^{\prime}\
u_{0}^{2}\quantity(4Ku_{0}+3E)\cos(\kappa\theta^{\prime})$
$\displaystyle=-\frac{K_{1}}{8a\kappa^{2}p_{\theta}^{2}c^{2}\quantity(1-e^{2})^{3}}\bigg{[}Ke^{3}\sin(4\kappa\theta)+\quantity{12Eea(1-e^{2})+8Ke^{3}+24Ke}\sin(2\kappa\theta)$
$\displaystyle\quad+\quantity{-8Ee^{2}a(1-e^{2})-32Ke^{2}}\sin[3](\kappa\theta)+\quantity{(24Ee^{2}+24E)a(1-e^{2})+96Ke^{2}+32K}\sin(\kappa\theta)$
$\displaystyle\quad+\quantity{24Eea(1-e^{2})+12Ke^{3}+48Ke}\kappa\theta\bigg{]},$
(40)

where $e$ and $a$ are given in (16) and (17) respectively. According to the
initial value, $u_{1}(0)=u_{1}^{\prime}(0)=0$, we should have
$c_{1}=-v_{1}(0)$ and $c_{2}=-v_{2}(0)=0$. Therefore, the complete orbit
function is

$\displaystyle
u(\theta)=\frac{1+e\cos(\kappa\theta)}{a(1-e^{2})}-v_{1}(0)\epsilon\cos(\kappa\theta)+\epsilon
v_{1}(\theta)\cos(\kappa\theta)+\epsilon v_{2}(\theta)\sin(\kappa\theta)~{}.$
(41)

From the equations (41), (4), and (4), it is easy to see that $v_{1}$ is
bounded from above and below since it consists of cosinus functions, thus we
can ignore the term $\epsilon v_{1}\cos(\kappa\theta)$. We can also ignore the
term $v_{1}(\theta)\epsilon\cos(\kappa\theta)$ by the same reason. Meanwhile,
$v_{2}$, is a non bounded function on $\theta$, so that we can approximate the
last term in (41) to

$-\frac{\epsilon
K_{1}\quantity{24Eea(1-e^{2})+12Ke^{3}+48Ke}\theta\sin(\kappa\theta)}{8a\kappa
p_{\theta}^{2}c^{2}(1-e^{2})^{3}}~{}.$

Thus, we have

$\displaystyle
u(\theta)\approx\frac{1+e\cos(\kappa\theta)}{a(1-e^{2})}-\frac{\epsilon
K_{1}}{\kappa
p_{\theta}^{2}c^{2}}\frac{6Eea(1-e^{2})+3Ke^{3}+12Ke}{2a(1-e^{2})^{3}}\theta\sin(\kappa\theta)~{}.$
(42)

Let us define

$\displaystyle\sigma\equiv-\frac{\epsilon K_{1}}{\kappa
p_{\theta}^{2}c^{2}}\frac{6Eea(1-e^{2})+3Ke^{3}+12Ke}{2a(1-e^{2})^{3}}~{}.$
(43)

Since $\sigma$ is small, then

$\displaystyle\alpha\cos(\kappa\theta)+\sigma\theta\sin(\kappa\theta)\approx\alpha\cos(\quantity(\kappa-\frac{\sigma}{\alpha})\theta)~{},$
(44)

where

$\displaystyle\alpha=\frac{e}{a(1-e^{2})}~{}.$ (45)

So we can simplify (42) to

$\displaystyle u(\theta)=\frac{1+e\cos(\quantity[\kappa-\frac{\sigma
a(1-e^{2})}{e}]\theta)}{a(1-e^{2})}~{}.$ (46)

The orbital precession when the body revolves in one period

$\displaystyle 2\pi\delta=\frac{2\pi}{\kappa-\frac{\sigma
a(1-e^{2})}{e}}-2\pi~{}.$ (47)

Taking the approximation

$\displaystyle\frac{K^{2}}{p_{\theta}^{2}c^{2}}\ll 1~{},$ (48)

we finally have the orbital precession

$\displaystyle\delta=\frac{K^{2}}{2p_{\theta}^{2}c^{2}}\quantity[1-\frac{3\epsilon
K_{1}}{K(1-e^{2})^{2}}\quantity(\frac{2p_{\theta}^{2}c^{2}}{K^{2}}+e^{2}+2)]~{}.$
(49)

###  4.1 Comparison with Other Theories

We consider an example of Mercury’s orbit. In this case $K=GMm$, where $G$ is
Newton gravity constant, $M=1.989\times 10^{30}\ \mathrm{kg}$ is mass of Sun,
$m=3.301\times 10^{23}\ \mathrm{kg}$ is mass of Mercury. Its semi-major axis
$a=57.91\times 10^{9}\mathrm{m}$ and eccentricity $e=0.2056$. Since

$\displaystyle\frac{mc^{2}}{GMm}\approx 10^{-3}>\frac{e}{a(1-e^{2})}\approx
10^{-8}~{},$ (50)

and $\eta\gg e$, then eq. (20) becomes approximately

$\displaystyle\frac{p_{\theta}^{2}c^{2}}{(GMm)^{2}}=\frac{c^{2}a(1-e^{2})}{GM}~{}.$
(51)

Inserting (51) into (49), we obtain

$\displaystyle\delta$
$\displaystyle=\frac{GM}{2c^{2}a(1-e^{2})}-\frac{3\epsilon
K_{1}}{2mc^{2}a(1-e^{2})^{3}}\quantity(\frac{2c^{2}a(1-e^{2})}{GM}+e^{2}+2)~{}.$
(52)

Meanwhile, in general relativity the orbit precession of Mercury is given by
[15]

$\displaystyle\delta_{GR}=\frac{3GM}{c^{2}a(1-e^{2})}=\epsilon\approx
10^{-7}.$ (53)

In order to make a contact with observation, we have to set
$\delta=\delta_{GR}$ implying

$\displaystyle
K_{1}=-\frac{5a(1-e^{2})^{3}GMmc^{2}}{9\quantity(GM(2+e^{2})+2ac^{2}(1-e^{2}))}~{}.$
(54)

Another interesting gravitational theory is Randall-Sundrum model [16]. In
this theory we have a five dimensional bulk spacetime with negative
cosmological constant in which we embed two 3-branes (3+1 dimensional
spacetimes) lying on the fifth coordinate with distance $r_{c}$ where $r_{c}$
is called compactification radius. After integrating out all massive Kaluza-
Klein particles by taking $r_{c}$ to be very large (compared to Planck length
$\sim 10^{-35}$ m), we finally obtain the modified Newtonian gravitational
potential energy

$\displaystyle V(r)=-\frac{K}{r}-\frac{\epsilon_{RS}K}{r^{3}}~{},$ (55)

where $\epsilon_{RS}\propto 1/k^{2}$ is a dimensionless quantity and $k$ is a
warped factor on the bulk spacetime. After employing similar computation as
above, we obtain the orbit precession of Mercury

$\displaystyle\delta_{RS}$
$\displaystyle=\frac{GM}{2c^{2}a(1-e^{2})}+\frac{3\epsilon_{RS}GM}{2c^{2}a(1-e^{2})^{3}}\quantity(\frac{2c^{2}a(1-e^{2})}{GM}+e^{2}+2)~{}.$
(56)

However, since the warped factor $k$ is propotional to Planck energy in SI
unit, that is, $k\approx 10^{8}$, so the last term in (56) is suppressed to
zero. Thus, this Randall-Sundrum model is ruled out in this case.

At the end, it is worth mentioning that in Newtonian theory we could also use
the potential energy (33). Using the same reason as above, we have to take

$\displaystyle K_{1}=-\frac{a^{2}}{3}(1-e^{2})^{2}GMm~{},$ (57)

in order to have consistent model with observation.

###  4.2 Relevance with second Kepler’s law

It is well known that in the problem of planetary orbit, the motion should
satisfy the second Kepler’s law. Let us consider a section area in the orbit
plane $\differential A=\frac{1}{2}r^{2}\differential\theta$. Then, it can be
shown that the rate of swept area is constant since the $\gamma$ factor and
the angular momentum $p_{\theta}$ are constant. In other words, $\dot{A}$ is
constant, namely,

$\displaystyle\dot{A}=\frac{p_{\theta}}{2\gamma
m_{0}}=\frac{a(1-e)}{2m_{0}\sqrt{m_{0}^{2}c^{2}+p_{\theta}^{2}a^{2}(1-e)^{2}}}~{}.$
(58)

The area of ellipse is $\pi\sqrt{1-e^{2}}a^{2}$. From (58), the Kepler’s law
in this case has the form

$\displaystyle\int_{0}^{T}\differential t\ \dot{A}=\frac{p_{\theta}T}{2\gamma
m_{0}}=n\pi\sqrt{1-e^{2}}a^{2}~{},$ (59)

where $T$ is the period such that the orbit of a planet is closed and $n$ is
the number revolution of a planet forming closed orbit which has to be a
positive integer as illustrated by the Figure 5.

Figure 5: The orbit of a planet shifted by precession.

##  5 Conclusion

We have discussed the motion of a body under the influence of central force in
the special theory of relativity. We particularly reviewed the motion under
the force that is proportional to $1/r^{2}$ where $r$ is the radial coordinate
in which it admits the precessing elliptical orbit. This orbital precession
depends on the eccentricity and semi-major axis of the orbit. If the orbit is
circular, then the orbital precession does not appear.

Then, we applied the above results to the case of the hydrogen-like atom. We
used the Wilson-Sommerfeld quantization rule to obtain the energy level of the
system so called Sommerfeld-Dirac fine structure formula which can also be
derived from Dirac equation in relativistic quantum mechanics.

At the end, we have considered the planetary motion in which the potential
energy is fine-tuned by adding the term which is proportional to $1/r^{3}$. We
obtained the explicit orbital function and then, derived the precession of the
planetary orbit. Since the pre-coefficient of the additional term in the
potential energy is arbitrary, we could make a contact with the result from
general relativity. However, in the Randall-Sundrum model this pre-coefficient
is suppressed to zero implying that the model can be ruled out. Also, we
pointed out the second Kepler’s laws which differs to the case of Newtonian
mechanics.

We are grateful to Fiki Taufik Akbar, Ardian Nata Atmaja, Andy Octavian
Latief, and Rudy Kusdiantara for useful discussion and valuable suggestions.
The work in this paper is supported by World Class Professor program of
Kemendikbud Ristek 2021.

##  Appendix A  Derivation of Constant of Motion

For a relativistic particle motion under the influence of central potential,
the Lagrangian is given by

$\displaystyle
L=-\frac{m_{0}c^{2}}{\gamma(v)}-V(r),\quad\gamma=\quantity(1-\frac{\dot{r}^{2}}{c^{2}}-\frac{r^{2}\dot{\theta}^{2}}{c^{2}})^{-1/2}.$
(60)

The conjugate momenta correspond to $r,\theta$ are

$\displaystyle p_{r}=\gamma m_{0}\dot{r}~{},$ (61) $\displaystyle
p_{\theta}=\gamma m_{0}r^{2}\dot{\theta}~{},$ (62)

respectively. From Euler-Lagrange equation of motions for $\theta$, we have

$\displaystyle\derivative{t}\quantity(\gamma m_{0}r^{2}\dot{\theta})=0~{},$
(63)

where the quantity $\gamma m_{0}r^{2}\dot{\theta}$ is the angular momentum
$p_{\theta}$. From the above equation, it implies that $p_{\theta}$ is
constant.

Now we consider the total time derivative of Lagrangian,

$\displaystyle\dot{L}=\partialderivative{L}{r}\dot{r}+\partialderivative{L}{\theta}\dot{\theta}+\partialderivative{L}{\dot{r}}\ddot{r}+\partialderivative{L}{\dot{\theta}}\ddot{\theta}+\partialderivative{L}{t}.$
(64)

The last term vanish since the Lagrangian does not depend explicitly on time.
Meanwhile,

$\displaystyle\partialderivative{L}{\dot{q}_{i}}\ddot{q}_{i}=\derivative{t}\quantity(\partialderivative{L}{\dot{q}_{i}}\dot{q}_{i})-\derivative{t}\quantity(\partialderivative{L}{\dot{q}_{i}})\dot{q}_{i},\quad\text{where}\quad
q_{i}=r,\theta~{}.$ (65)

Then,

$\displaystyle\dot{L}=\quantity[\partialderivative{L}{r}-\derivative{t}\quantity(\partialderivative{L}{\dot{r}})]\dot{r}+\quantity[\partialderivative{L}{\theta}-\derivative{t}\quantity(\partialderivative{L}{\dot{\theta}})]\dot{\theta}+\derivative{t}\quantity(\partialderivative{L}{\dot{r}}\dot{r})+\derivative{t}\quantity(\partialderivative{L}{\dot{\theta}}\dot{\theta})~{}.$
(66)

From Euler-Lagrange equation of motions, the first and second terms vanish
implying

$\displaystyle\derivative{t}\quantity(\partialderivative{L}{\dot{r}}\dot{r}+\partialderivative{L}{\dot{\theta}}\dot{\theta}-L)=0~{},$
(67)

which is nothing but the energy function of the system. Thus, we conclude that
the total energy of the system

$\displaystyle
E=\partialderivative{L}{\dot{r}}\dot{r}+\partialderivative{L}{\dot{\theta}}\dot{\theta}-L=\gamma(v)m_{0}c^{2}+V(r)~{},$
(68)

is also the constant of motion.

##  Appendix B  Orbit Equation

We begin by writing the Binet equation

$\displaystyle\quantity(\derivative[2]{\theta}+1-\frac{K^{2}}{p_{\theta}^{2}c^{2}})u=\frac{KE}{p_{\theta}^{2}c^{2}}~{}.$
(69)

The characteristic equation of the above differential equation for homogeneous
case is

$\displaystyle\rho^{2}+1-\frac{K^{2}}{p_{\theta}^{2}c^{2}}=0~{},$ (70)

whose solution is given by

$\displaystyle\rho=\pm
i\Omega,\quad\Omega=\sqrt{1-\frac{K^{2}}{p_{\theta}^{2}c^{2}}}~{}.$ (71)

Let $u_{0}$ be the solution to the homogeneous equation

$\displaystyle u_{0}=A\exp(i\Omega\theta)+B\exp(-i\Omega\theta)~{},$ (72)

whereas $u_{1}$ is the particular solution. Since the rhs of the Binet
equation is constant, then $u_{1}^{\prime\prime}=0$ such that we have

$\displaystyle u_{1}=\frac{KE}{p_{\theta}^{2}c^{2}-K^{2}}~{}.$ (73)

Finally, we can write the solution as

$\displaystyle
u(\theta)=u_{0}(\theta)+u_{1}=A\exp(i\Omega\theta)+B\exp(-i\Omega\theta)+\frac{KE}{p_{\theta}^{2}c^{2}-K^{2}}~{}.$
(74)

##  Appendix C Derivation of Eccentricity

Let us first consider

$\displaystyle\quantity(m_{0}c^{2})^{2}=\quantity(E+Ku)^{2}-u^{\prime
2}p_{\theta}^{2}c^{2}-u^{2}p_{\theta}^{2}c^{2}~{},$ (75)

for $\theta=0$ which implies

$\displaystyle\quantity(m_{0}c^{2}a(1-e))^{2}=\quantity(Ea(1-e)+K)^{2}-\quantity(p_{\theta}c)^{2}~{}.$
(76)

Defining $\alpha=a(1-e)$, we get

$\displaystyle\quantity(m_{0}^{2}c^{4}-E^{2})\alpha^{2}-2KE\alpha-\quantity(K^{2}-p_{\theta}^{2}c^{2})=0~{},$
(77)

resulting

$\displaystyle\alpha=\frac{KE}{m_{0}^{2}c^{4}-E^{2}}\quantity[1\pm\sqrt{\quantity(\frac{m_{0}c^{2}}{E})^{2}+\quantity(\frac{p_{\theta}c}{K})^{2}-\quantity(\frac{m_{0}c^{2}}{E}\frac{p_{\theta}c}{K})^{2}}]~{}.$
(78)

Since

$\displaystyle
a=\frac{K}{E}\quantity(\frac{p_{\theta}^{2}c^{2}}{K^{2}}-1)\frac{1}{1-e^{2}}~{},$
(79)

it gives then

$\displaystyle\frac{K}{E}\quantity(\frac{p_{\theta}^{2}c^{2}}{K^{2}}-1)\frac{1}{1+e}=\frac{KE}{m_{0}^{2}c^{4}-E^{2}}\quantity[1\pm\sqrt{\quantity(\frac{m_{0}c^{2}}{E})^{2}+\quantity(\frac{p_{\theta}c}{K})^{2}-\quantity(\frac{m_{0}c^{2}}{E}\frac{p_{\theta}c}{K})^{2}}]~{},$
(80)

whose solution is

$\displaystyle e=\sqrt{\beta}~{},$ (81)

where

$\displaystyle\beta=\quantity(\frac{m_{0}c^{2}}{E})^{2}+\quantity(\frac{p_{\theta}c}{K})^{2}-\quantity(\frac{m_{0}c^{2}}{E}\frac{p_{\theta}c}{K})^{2}~{}.$
(82)

##  Appendix D Solving the angular momentum

Let $x=p_{\theta}c/K$, then we substitute (79) into (82) we have

$\displaystyle(x^{2}-1)\quantity[(x^{2}-e^{2})(x^{2}-1)-\eta^{2}(1-e^{2})^{2}]=0~{},$
(83)

whose solutions are given by

$\displaystyle
x^{2}\in\quantity{1,\frac{1+e^{2}}{2}\quantity[1+\frac{1-e^{2}}{1+e^{2}}\sqrt{1+4\eta^{2}}],\frac{1+e^{2}}{2}\quantity[1-\frac{1-e^{2}}{1+e^{2}}\sqrt{1+4\eta^{2}}]}~{},$
(84)

where $\eta\equiv\frac{m_{0}c^{2}a}{K}$. Note that in the third solution,
$\eta\geq\frac{e}{1-e^{2}}$ must hold.

##  Appendix E Solving The Radial Momentum Quantization

We start from

$\displaystyle n\int_{0}^{2\pi}\frac{u^{\prime
2}}{u^{2}}\differential\theta=2\pi k~{},$ (85)

such that

$\displaystyle\int_{0}^{2\pi}\quantity(n\frac{u^{\prime
2}}{u^{2}}-k)\differential\theta=0~{}.$ (86)

Since $u^{\prime
2}/u^{2}=\kappa^{2}e\sin[2](\kappa\theta)/(1+e\cos(\kappa\theta))^{2}$,
$\kappa^{2}=1-(\alpha^{2}/n^{2})$, then

$\displaystyle\frac{\quantity(n-(\alpha^{2}/n))e\sin[2](\kappa\theta)}{(1+e\cos(\kappa\theta))^{2}}=k~{}.$
(87)

We use the above equation to solve the relation between $e$, $k$, and $n$. For
$\theta=\pi/2$, we have

$\displaystyle\quantity(n-\frac{\alpha^{2}}{n})e=k~{},$ (88)

and for $\theta=\pi/4$, we have

$\displaystyle\frac{\quantity(n-\frac{\alpha^{2}}{n})e}{2(1+\quantity(e/\sqrt{2}))^{2}}=k~{}.$
(89)

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