# Mesoscopic superconducting memory based on bistable magnetic textures

R. Fermin  N. Scheinowitz  J. Aarts  Huygens-Kamerlingh Onnes Laboratory,
Leiden University, P.O. Box 9504, 2300 RA Leiden, The Netherlands.  K. Lahabi
Huygens-Kamerlingh Onnes Laboratory, Leiden University, P.O. Box 9504, 2300 RA
Leiden, The Netherlands.  Department of Quantum Nanoscience, Kavli Institute
of Nanoscience, Delft University of Technology, Delft, 2628 CJ, The
Netherlands

###### Abstract

With the ever-increasing energy need to process big data, the realization of
low-power computing technologies, such as superconducting logic and memories,
has become a pressing issue. Developing fast and non-volatile superconducting
memory elements, however, remains a challenge. Superconductor-ferromagnet
hybrid devices offer a promising solution, as they combine ultra-fast
manipulation of spins with dissipationless readout. Here, we present a new
type of non-volatile Josephson junction memory that utilizes the bistable
magnetic texture of a single mesoscopic ferromagnet. We use micromagnetic
simulations to design an ellipse-shaped planar junction structured from a
Nb/Co bilayer. The ellipse can be prepared as uniformly magnetized or as a
pair of vortices at zero applied field. The two states yield considerably
different critical currents, enabling reliable electrical readout of the
element. We describe the mechanism used to control the critical current by
applying numerical calculations to quantify the local stray field from the
ferromagnet, which shifts the superconducting interference pattern. By
combining micromagnetic modeling with bistable spin-textured junctions, our
approach presents a novel route towards realizing superconducting memory
applications.

###### pacs:

##  I Introduction

In recent years, energy-hungry data centers have accounted for more than 1% of
the global electricity consumption; this number is projected to increase to
3-13% by 2030.[1] As data centers are already responsible for 0.3% of the
overall carbon emissions, the development of low-power supercomputers has
become an immediate global concern.[2] Due to their non-dissipative nature,
superconducting logic and memory devices present a tantalizing route to
address this issue, promising enormous savings, even accounting for cryogenic
cooling.[3, 4, 5, 6, 7] While several classes of high-performance
superconducting processors already exist, cryogenic memories remain relatively
underdeveloped and far from meeting their applications. Non-volatility and
scalability, in particular, have been the major obstacles in the industrial
realization of superconducting memories.

$\begin{array}[]{c}\includegraphics[width=390.25534pt]{Figure1_REB1_V2.pdf}\end{array}$

Figure 1: (a) False colored scanning electron micrograph of the ellipse-shaped
SFS junction device A. The Co weak link is formed by the indicated trench
($\backsim$20 nm in size) that separates the two Nb electrodes. The scale bar
corresponds to 500 nm. (b) Shows the simulated bistable spin textures at zero
field. The cobalt is either magnetized along the long axis of the device
(M-state), or hosts two stable vortices (V-state). The vortex cores are
highlighted by orange circles and the trench location is indicated by the
orange line. The color scale indicates the out-of-plane magnetization of the
top of the cobalt layer.

The combination of superconducting (S) and ferromagnetic (F) elements can
tackle the non-volatility issue through SFS devices, in which different
(stable) magnetic configurations yield different critical supercurrents. Much
work has been done on stacked magnetic Josephson junctions with a weak
ferromagnetic barrier, where small magnetic field pulses could switch the
critical current. [8, 9, 10, 11] However, the weak ferromagnet used in such
systems has a typically low Curie temperature (around 10 K), limiting their
non-volatility. Josephson memories have also been extensively studied in so-
called pseudo-spin-valve (SFNF′S) devices, where the magnetization of two
ferromagnetic layers, separated by a normal metal, could be set parallel or
antiparallel to yield a change in critical current. [12, 13, 14, 15, 16, 17,
18] The work on pseudo-spin-valves has also been extended to multilayer
Josephson devices featuring triplet supercurrents, which not only carry charge
but also spin. [19, 20] Finally, all-oxide devices, which can operate at high
temperatures were recently examined. [21] In general, the existing non-
volatile Josephson memories utilize an intricate layer set and face a number
of obstacles, such as interlayer coupling between magnetic layers and
stochastic distribution of magnetic flux in the junction. Other problems
include, insufficient contrast between states for electrical readout, and the
requirement for applying a large writing field.

To address these challenges, we present an alternative route for developing
non-volatile Josephson memory elements, where information is stored by the
spin texture of a single mesoscopic ferromagnet. Magnetic textures in general,
and specifically ferromagnetic vortices, can be manipulated by the application
of pulsed microwave radiation, leading to the realization of ultra-fast
electronics.[22, 23, 24] Combining information inscription in spin textures
with the non-dissipative nature of supercurrents, can thus lead to the
realization of high-speed, low-power, and non-volatile memory elements. The
synergy between superconductivity and magnetic textures is amplified further
by allowing for triplet supercurrents, paving the road for superconducting
spintronics.[25, 26, 27] Textured SF-hybrid devices, therefore, open a truly
new paradigm in superconducting memory applications.

Here we show the realization of a micrometer-sized superconducting memory
element based on bistable spin textures in the F-layer of an elliptical planar
SFS Josephson junction. By combining micromagnetic simulations with
nanostructured hybrid devices, we are able to control the transport behaviour
of our devices with their magnetic texture. By applying a relatively small
magnetic field of 40 mT for the writing operation, we can reliably switch
between two stable spin textures: one is uniformly magnetized, and the other
hosts a pair of ferromagnetic vortices. These correspond to minimum and
maximum critical supercurrent of the junction, respectively. We obtain a
factor of five difference in the critical current between the two states,
which allows for a facile and reliable electrical readout of the element. By
quantifying the local stray field from the ferromagnet, we show that the
difference in critical current is caused by a shift in the superconducting
interference pattern. Moreover, the magnetic bistability enables us to use
relatively small writing fields to control a considerably larger local stray
field from the ferromagnet. We confirm that the memory is stable, non-
volatile, and non-destructive upon readout.

$\begin{array}[]{c}\includegraphics[width=390.25534pt]{Figure2_HD.pdf}\end{array}$

Figure 2: Basic transport characteristics of device A. In (a) we show the
resistance versus temperature curves for both the M- and V-state, measured
using 20 $\mu$A current bias. The critical current is suppressed below 20
$\mu$A in the M-state. The inset shows the low-temperature behavior. (b)
depicts three $IV$-characteristics of the sample in the V-state, acquired
under different RF radiation powers. The frequency is 1.6 GHz and the
temperature is 1.6 K. The measured voltage is normalized to $hf/2e$, and shows
a clear Shapiro response.

##  II Results and discussion

###  II.1 Ellipse-shaped SFS junctions

$\begin{array}[]{c}\includegraphics[width=411.93767pt]{Figrure3_REB1.pdf}\end{array}$

Figure 3: The critical current $I_{\text{c}}$, the total simulated energy
$E_{tot}$ and the spin texture of the device as a function of Hx (field along
the long axis), and $H_{y}$ (along the short axis). Results obtained on device
B at 1.7 K. (a) and (b) depict $I_{\text{c}}(H_{y})$ and
$I_{\text{c}}(H_{x})$. In (a) the field is increased along the short axis of
the ellipse. At 40 mT $I_{\text{c}}$ increases sharply, effectively
transforming the M-state to the V-state. The reverse operation is shown in
(b): by increasing the field - parallel to the long axis of the ellipse - to
15 mT, the critical current drops to 5 $\mu$A, bringing the device back to the
M-state. (c) and (d) show $E_{tot}$ corresponding to the in-plane field
sweeps. In both panels, two branches can be seen, corresponding to the V- and
M-state. A change of magnetic state coincides with a jump from one branch to
the other. We are able to correctly predict the switching fields to within a
small error. The right-hand column contains snapshots of the spin texture
during the simulated field sweeps. Here the color scale indicates the
magnetization of the top of the Co-layer.

A cobalt (65 nm) niobium (50 nm) bilayer is deposited in an ultra-high vacuum
by Ar-sputtering on a four-probe geometry lift-off pattern. Elliptic SFS
devices are structured from the Co/Nb bilayer using focused ion beam milling.
Figure 1 shows a false-colored scanning electron micrograph of such an ellipse
device, having dimensions of 1500 nm by 750 nm (long and short axes,
respectively). Micromagnetic simulations show that the zero-field spin texture
of such devices is expected to be bistable: either the cobalt layer is
magnetized along the long axis of the ellipse (we call this the M-state) or
two ferromagnetic vortices are stable near each focal point of the ellipse
(V-state). We depict simulation results from these states in Figure 1b. The
full technical details of the simulations are presented in Appendix A.
Essentially, this allows for writing a bit with the value ”0” (M-state) or ”1”
(V-state) into the spin texture of the device. We prepare the states by first
applying a 40 mT in-plane field either parallel to the long or short axis of
the devices. Next, after removing the field along the short (long) axis, the
V-state (M-state) is stable in the device.

Using ultra-low ion-beam currents, We fabricate a planar Josephson junction in
the ellipse by locally removing the Nb top-layer. This creates an
approximately 20 nm wide trench, which effectively separates the Nb contacts
and forces the supercurrent through the ferromagnet. Planar SFS-junctions
fabricated in this manner were used to study triplet supercurrents in previous
works.[28, 29] The trench is positioned near the focal point of the ellipse,
which corresponds to the approximate location of one of the vortex cores in
the V-state. As discussed later, this particular location is designed to
optimize the stray field-driven mechanism used to assign different critical
currents to the M- and V-states. In this paper, we describe two of these
elliptical devices (devices A and B), although all observations are repeated
for a large number of samples. The sample resistance is measured using a DC
nanocurrent source (Keithley 6221) and a nanovolt meter (Keithley 2182A) in a
4-probe fashion, where the voltage drop over both the bar-shaped contacts and
the ellipse is measured. Figure 2 shows the resistance ($R$) versus
temperature ($T$) of device A, measured with a current of 20 $\mu$A. If the
sample is prepared in the V-state, we find two transitions. The first
corresponds to the bulk superconducting transition temperature
($T_{\text{c}}$) of the Nb layer, the second is due to the superconducting
proximity effect in the SFS junction formed by the trench. In contrast, the
M-state does not show a second transition, indicating that the critical
current ($I_{\text{c}}$) of the junction in this state is suppressed to below
the measurement current. This difference in $I_{\text{c}}$ between the two
states allows us to electrically readout the magnetic state of the memory
element. To unambiguously demonstrate the superconducting proximity effect in
our devices, we show the appearance of Shapiro steps in the current-voltage
($IV$-)characteristic of device A under the application of RF radiation, which
is supplied by a nearby antenna. We carry out these measurements at 1.6 K in
the V-state.

$\begin{array}[]{c}\includegraphics[width=390.25534pt]{Figure4_HD.pdf}\end{array}$

Figure 4: Fidelity of the writing cycles of device A. The $IV$-characteristics
of the two states, measured at 1.6 K, are shown in (a). Based on these curves,
we define a readout current that corresponds to a finite voltage in the
M-state and zero voltage in the V-state. We cycle the device between the two
states 78 times and measure the voltage before each subsequent switch. (b)
shows a histogram of the readout voltages, measured using a 30 $\mu$A current
(the dotted line in (a)).

The step height in the $IV$-characteristic shown in Figure 2b is in units
$hf/2e$ (where $h$ is the Planck constant, $f$ is the frequency of the RF
radiation, and $e$ the electron charge), confirming the Shapiro response.

###  II.2 Controllable switching

To show the bistability of the spin textures, we examine the dependence of
$I_{\text{c}}$ on the in-plane ’writing’ fields Hx and Hy along the long and
the short axis of the ellipse, respectively. The results are shown in Figure
3a,b, accompanied by the simulated spin textures. If the sample is in the
M-state (”0”), $I_{\text{c}}$ is 5 $\mu$A. The datum ”1” can be written by
increasing the field along the short axis of the ellipse: at a field of 40 mT,
$I_{\text{c}}$ grows to above 20 $\mu$A, which is accompanied by a buckling of
the simulated spin texture. Upon decreasing the field to zero, the high
$I_{\text{c}}$ state is retained and the buckled spin texture evolves into two
vortices. Therefore, the device is converted into the V-state (”1”).
Conversely, the M-state can be written in the device by applying a field
parallel to the long axis of the ellipse. The field effectively displaces the
vortices towards the edges of the sample. At a field of 15 mT, they are fully
pushed out from the ellipse and the spins align in the M-state. This is
accompanied by a drop in $I_{\text{c}}$ back to 5 $\mu$A. The low
$I_{\text{c}}$ is stable upon decreasing the field back to zero in the
M-state.

Figure 3c,d show the total energy associated with the simulated spin texture
as a function of externally applied field (i.e., the sum of the exchange,
demagnetization, and Zeeman energies). Two branches can be discerned
corresponding to the two states, both stable at zero field. In the
simulations, the buckled state, which is required for the stabilization of the
V-state state, appears above 45 mT. During the converse switching operation, a
field of 25 mT is required to magnetize the ellipse. Remarkably, we find both
switching fields closely follow the experimental values with only a small
offset, which can be attributed to the reduced geometry of the contacts, and
the finite field steps used in the simulations.

We examine the performance of our devices by repeatably switching them between
the two states while studying the transport behavior. Figure 4a shows the
zero-field $IV$-characteristic of the two states. Based on the difference in
$I_{\text{c}}$, we can define a readout current that corresponds to a finite
voltage in the M-state, but a zero voltage in the V-state. We cycle the device
between the two states (using a 40 mT field) and acquire the voltage at the
readout current in each state. In Figure 4b we show a histogram of the
measured voltage at a readout current of 30 $\mu$A, totaling 78 subsequent
writing cycles. There is no overlap in the histogram, making the states highly
distinguishable. As the readout current density is far too low to alter the
spin texture, the readout operation is fully non-destructive. Furthermore, by
heating the ellipse to above the $T_{\text{c}}$ of the superconducting layer
between subsequent write and read operations, we confirm the non-volatility of
its memory. This was found to hold, even when the sample is warmed to room
temperature and stored for days. Additionally, to test the resilience to
perturbations of the spin texture, we try to cycle between the states using a
field of 10 mT. We find these fields insufficient to alter $I_{\text{c}}$,
demonstrating the stability of our devices.

$\begin{array}[]{c}\includegraphics[width=411.93767pt]{Figure5_PS.pdf}\end{array}$

Figure 5: Stray fields cause a shift in I${}_{\text{c}}$. (a) to (c)
respectively show the superconducting interference (SQI) patterns in the right
magnetized M-state, left magnetized M-state, and the V-state obtained on
device B. Although the shape of the SQI pattern does not change, the center
lobe of the pattern is shifted in the V-state with respect to the pattern in
the M-state (for both magnetizations). Some measurement glitches are visible
in (a) and (b). These result from sudden small magnetic configurations in the
sample. The shift is caused by stray fields from the ferromagnetic layer, as
is demonstrated in (d): the simulated field shift for different trench
locations. The vertical line in (d) corresponds to the actual location of the
trench (see Figure 1). The inset shows the geometry of the sample with the
corresponding scale distribution. The red line indicates the actual location
of the trench.

###  II.3 Stray field driven mechanism

We now discuss the mechanism used for suppressing the $I_{\text{c}}$ in the
M-state, where we utilize the local stray field from the ferromagnet itself.
To illustrate this, we perform superconducting quantum interference (SQI)
measurements, where we sweep the out-of-plane field and acquire an
$IV$-characteristic at each field. During the SQI measurements, we obtain a
direct measure of $\text{d}V/\text{d}I$, by obtaining an AC-voltage using a
Synktek MCL1-540 multichannel lock-in. For a regular junction, the
interference pattern $I_{\text{c}}(B)$ corresponds to the well-known
Fraunhofer pattern. Note that the magnitude of the out-of-plane fields is
insufficient to alter the global spin texture of the device (i.e., the out-of-
plane fields cannot transform the M-state to the V-state or vice versa). We
present such SQI measurements, obtained on sample B, for the V-state, as well
as for the left and right magnetized M-state in the color maps of Figure 5a-c.
For this sample, we observe a pattern with a single middle lobe and heavily
suppressed side lobes. 111Samples with higher $I_{\text{c}}$ values do show a
Fraunhofer interference pattern, indicating a uniform distribution of
$I_{\text{c}}$ throughout the weak link. For the V-state (Figure 5c), the
pattern is centered at zero field. By switching to either of the M-states
(Figure 5a,b), the shape of the SQI pattern is not altered, yet clearly
shifted by around 60 mT from zero applied field. Moreover, the magnetization
direction in the M-state determines the sign of this shift. Therefore, it is
evident that the suppressed $I_{\text{c}}$ of the M-state is a result of the
local stray fields from the Co layer, introducing a shift in the
$I_{\text{c}}(B)$ pattern. However, what makes our spin-textured devices truly
stand out from previous Josephson memory elements, is that we use a writing
field (40 mT) that is noticeably smaller than the resulting shift (60 mT).

In the following, we describe how the pattern shift can be quantified by
simulating the stray flux entering the junction. We sum the z-component of the
flux - evaluated in a thin sheet above the Cobalt layer - over an integration
window to obtain the local stray fields threading area of the window. The size
of this window is chosen to reflect the trench area, therefore yielding an
estimate of the stray fields penetrating the junction. Figure 5d displays the
simulated (stray field-induced) pattern shift as a function of the window
location; the vertical line indicates the location of the trench in the actual
sample. This show that the V-state corresponds to negligible stray fields at
the junction, accounting for the centered pattern in Figure 5c with no offset.
The M-state corresponds to an 80 mT shift, which closely follows the 60 mT
offset in the SQI patterns. Note that the simulated stray field is symmetric
with respect to the center of the sample. This reflects the sign change that
occurs when the magnetization direction is reversed.

Note that the spin texture snapshots of Figure 3 show the source of the stray
fields. In the M-state, the ellipse features a dipole-like field along its
long axis, which is absent in the V-state. We do observe some stray fields in
the V-state, however these average out over the junction area. Therefore the
total shift in the SQI pattern remains zero. During the field sweep along the
short axis of the ellipse (i.e., switch from M-state to V-state), the spin
texture buckles before two vortices stabilize, as can be seen in the snapshots
B and C in Figure 3. When this buckling occurs, the dipole-like field no
longer points along the long axis the ellipse and consequently the stray
fields average out over the junction area. This explains why the high
$I_{\text{c}}$ state occurs before two vortices enter in the ellipse.

###  II.4 Pairing symmetry of the supercurrents

In previous work on similar SFS junctions with a circular geometry -
containing a single ferromagnetic vortex - we found spin texture to be a
prerequisite for the superconducting proximity effect, which led to the
conclusion that the transport in these disk-shaped SFS junctions, is dominated
by long-range triplet (LRT) Cooper pairs.[28, 29] Contrary to conventional
Cooper pairs, these LRT pairs consist of spin-polarized electrons that can
penetrate a ferromagnet over far longer length scales (up to hundreds of nm in
a half-metal).[31, 32, 33, 26]

In this case however, we cannot be certain about the LRT scenario. Due to the
similarity to the circular devices, LRT correlations can be expected in the
V-state of the ellipse-shaped devices described here. However, there are a
number of inconsistencies, the most notable ones are discussed here. Firstly,
contrary to the disk-shaped devices, removing the spin texture from our
junctions does not suppress the proximity effect, i.e., the maximum of
$I_{\text{c}}(B)$ for both the M- and V-states are quite similar. Secondly,
while the SQI patterns of the disk-shaped devices are highly sensitive to the
subtle changes in the spin texture, the elliptic junctions show the same type
of pattern for widely different magnetic states. Both of these observations
seem to counter the notion of the LRT correlations, and indicate that
transport in the elliptic junctions is carried by the short-range triplet
correlations instead (i.e., Cooper pairs with no spin projection). However,
one could argue that even the M-state still features some spin texture near
the edges of the sample (due to the tapered shape of the ellipse; see Figure
1b), which possibly allows for the generation of LRT correlations. Finally,
the similar SQI patterns between the magnetic states can possibly be caused by
the relatively short junction width.222The length of the trench in the disk-
shaped devices is typically between 1 $\mu$m and 1.6 $\mu$m, whereas the
trench length of the ellipse devices presenter here is less than 600 nm. At
this stage, no conclusions can be drawn on the triplet or singlet nature of
the supercurrents in our devices, and further experiments are necessary. We
emphasize, however, that the mechanism for the shifting in the SQI patterns
(i.e., the stray fields) hold for both singlet and triplet pairing symmetries.

##  III Conclusion and outlook

We demonstrate the realization of a non-volatile superconducting memory
element based on magnetic textures in an SFS Josephson junction. The
ellipsoidal shape of the device leads to two stable magnetic states at zero
applied magnetic field: a fully magnetized state and a state containing
ferromagnetic vortices. These can be associated with the data ”0” and ”1”
respectively. By applying and removing an externally applied in-plane field,
we can reliably switch between these states, performing a write operation. We
find a strong suppression of the critical current in the magnetized state,
which allows for a non-destructive determination of the spin state, and
therefore electrical readout of the bit. By combining transport experiments
and micromagnetic simulations, we show that the difference in critical current
between the two states results from internal stray fields originating from the
ferromagnetic layer of the device, which creates a local offset-field in the
junction. Since the ferromagnetic textures can be controllably manipulated
with RF radiation, storing information in mesoscopic SF devices can provide an
exciting avenue for realizing ultra-fast electronics. Besides, strain-mediated
switching of similar elliptical devices has been recently theoretically
studied, and shown to be a good alternative to switching using externally
applied fields.[35] Combined with the possibility of long-range triplet
supercurrents in hybrid devices, the memory devices presented here can form an
essential building block for superconducting spintronics.

##  IV Acknowledgements

This work was supported by the Dutch Research Council (NWO) as part of the
Frontiers of Nanoscience (NanoFront) program, and through NWO Projectruimte
grant 680.91.128. The work was also supported by EU Cost Action CA16218
(NANOCOHYBRI) and benefited from access to the Netherlands Centre for Electron
Nanoscopy (NeCEN) at Leiden University.

##  Appendix A Micromagnetic modeling

Micromagnetic simulations are performed using the GPU-based MuMax3
program.[36] The cobalt layer is divided into a mesh of 5 nm cubic cells. The
exchange coefficient is set to 3 $\times~{}10^{-11}$ J/m and the saturation
magnetization is 1.4 $\times~{}10^{-6}$ A/m. We choose the Gilbert damping
constant to be 0.5, to allow for faster convergence and therefore increased
computation speed. As the cobalt film is polycrystalline due to the Ar-
sputtering method, we use a zero anisotropy constant. To evaluate the local
stray fields emerging from the ferromagnet, we simulate vacuum cells above the
cobalt layer. These cells feature zero spin, however, they are threaded by a
magnetic field flux. Summing the z-component of the flux in the vacuum layer
over an integration window leads to an expression of the local stray field.
The area of the integration window is determined by the local width of the
ellipse, and a fixed length of 20 nm. This reflects the size of the trench.

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# Relativistic orbits of S2 star in the presence of scalar field

Parth Bambhaniya<EMAIL_ADDRESS>International Center for Cosmology,
Charusat University, Anand 388421, Gujarat, India  Ashok B. Joshi
<EMAIL_ADDRESS>International Center for Cosmology, Charusat
University, Anand 388421, Gujarat, India  Dipanjan Dey<EMAIL_ADDRESS>International Center for Cosmology, Charusat University, Anand 388421,
Gujarat, India  Pankaj S. Joshi<EMAIL_ADDRESS>International Center for
Cosmology, Charusat University, Anand 388421, Gujarat, India  International
Centre for Space and Cosmology, Ahmedabad University, Ahmedabad, GUJ 380009,
India  Arindam Mazumdar<EMAIL_ADDRESS>Centre for Theoretical
Studies, Indian Institute of Technology, Kharagpur, West Bengal-721302, India
Tomohiro Harada<EMAIL_ADDRESS>Department of Physics, Rikkyo University,
Toshima, Tokyo 171-8501, Japan  Ken-ichi Nakao<EMAIL_ADDRESS>Department of Physics, Graduate School of Science, Osaka Metropolitan
University, Sugimoto 3-3-138, Sumiyoshi, Osaka City 558-8585, Japan  Nambu
Yoichiro Institute of Theoretical and Experimental Physics, Osaka Metropolitan
University, Sugimoto 3-3-138, Sumiyoshi, Osaka City 558-8585, Japan

###### Abstract

The general theory of relativity predicts the relativistic effect in the
orbital motions of S-stars which are orbiting around our Milky-way galactic
center. The post-Newtonian or higher-order approximated Schwarzschild black
hole models have been used by GRAVITY and UCLA galactic center groups to
carefully investigate the S2 star’s periastron precession. In this paper, we
investigate the scalar field effect on the orbital dynamics of S2 star. Hence,
we consider a spacetime, namely Janis-Newman-Winicour (JNW) spacetime which is
seeded by a minimally coupled, mass-less scalar field. The novel feature of
this spacetime is that one can retain the Schwarzschild spacetime from JNW
spacetime considering zero scalar charge. We constrain the scalar charge of
JNW spacetime by best fitting the astrometric data of S2 star using the Monte-
Carlo-Markov-Chain (MCMC) technique assuming the charge to be positive. Our
best-fitted result implies that similar to the Schwarzschild black hole
spacetime, the JNW naked singularity spacetime with an appropriate scalar
charge also offers a satisfactory fitting to the observed data for S2 star.
Therefore, the JNW naked singularity could be a contender for explaining the
nature of Sgr A* through the orbital motions of the S2 star.

$\bm{keywords}$ : Black hole, Naked singularity, Periastron precession, Milky
way, S2 star.

††preprint: RUP-22-20, AP-GR-183, NITEP 146

##  I Introduction

Recently, the Event Horizon Telescope collaboration has announced a major
breakthrough in the imaging of an ultra-compact object at the centre of our
galaxy EventHorizonTelescope:2022xnr ; EventHorizonTelescope:2022vjs ;
EventHorizonTelescope:2022wok ; EventHorizonTelescope:2022exc ;
EventHorizonTelescope:2022urf ; EventHorizonTelescope:2022xqj . A bright
emission ring around a core brightness depression in VLBI horizon-scale images
of Sgr A*, with the latter linked to the shadow of black hole. The shadow
boundary of the Sgr A* marks the visual image of the photon region and
differentiates capture orbits from scattering orbits on the plane of a distant
observer. The radius of the bright ring can be used as an approximation for
the black hole shadow radius under specific conditions and after proper
calibration, with little reliance on the details of the surrounding accretion
flux. While there is strong evidence that there is a high concentration of
mass in the center of our Milky Way galaxy, the question of whether or not it
is a black hole is still open. They have considered various alternatives such
as naked singularities and regular black holes. They favorably acknowledge
that the naked singularity with a photon sphere Joshi-Malafarina-Narayan
(JMN-1) naked singularity could be the best black hole mimicker
EventHorizonTelescope:2022xqj . The central object and its nature remain
mysterious. This is because just like a black hole case, the JMN-1 naked
singularity would create a similar shadow, and therefore it is very difficult
to distinguish between the two. Therefore, in this paper, we study the
relativistic orbits of stars that are orbiting around our own galactic center.

Near the center Sgr A* of our Milky-way galaxy, many stars are hovering around
with very high speed. Due to the presence of the central massive object of
mass around $4\times 10^{6}M_{\odot}$, these stars can move at $\frac{1}{60}$
the speed of light and they can have highly eccentric orbits. These stars are
known as ‘S’-stars. Since they are very close to the galactic center, there
exists a possibility that they can show up some general relativistic effects.
However, it is very difficult to follow the dynamics of those stars, since
they are far away from us ($\approx 25,000$ light years). The highly sensitive
infra-red instruments namely GRAVITY, SINFONI, and NACO in the European
Southern Observatory (ESO) are capable of tracking the trajectory of the
‘S’-stars. Recently, they have released the 23 years of astrometric data of
the ‘S2’ star Do:2019txf ; GRAVITY:2018ofz ; GRAVITY:2020gka  which is one of
the important star of the ‘S’ star family. As it is mentioned above, general
relativistic effects can be seen in the dynamics of the ‘S2’ star, and
therefore, its trajectory can give us information about the spacetime around
the Sgr A*. In DellaMonica:2021xcf ; DellaMonica:2021fdr ; deMartino:2021daj ,
authors investigate the nature of Sgr A* with the geodesic motion of S-stars
in Scalar-Tensor-Vector Gravity, f(R ) gravity and with ultralight bosons.

It is a general belief that the spacetime of the central supermassive object
(Sgr A*) is a vacuum solution of Einstein field equations (e.g., Schwarzschild
solution, Kerr solution, etc.). However, from the observational results, it
can be understood that the center of a galaxy is surrounded by highly
concentrated matter. Therefore, a vacuum solution is more unlikely to be
present around a galactic center. Therefore, one could consider a matter
distribution near the central region and investigate the corresponding
physical signature. There are several papers where non-vacuum spacetimes are
considered and the different physical signatures (e.g. shadow, timelike
orbits, accretion disk, etc.) of the same are investigated Bambhaniya:2019pbr
; Dey:2019fpv ; Joshi:2019rdo ; Joshi:2020tlq ; Bambhaniya:2021ybs ;
Bambhaniya:2021ugr ; Solanki:2021mkt ; Saurabh:2022jjv ; Vagnozzi:2022moj .

In this paper, we investigate the scalar field effect on the orbital dynamics
of the ‘S2’ star. A scalar field is the simplest constitute of matter and at
the beginning, one may model the matter distribution around the galactic
center using the scalar field. Here, we do not discuss anything about the
particle physics model of the scalar field. It may be interpreted by some
beyond the standard model (BSM) of particle physics which is not the scope of
the present paper. Here, we are interested in constraining the scalar charge
by best-fitting the astrometric data of ‘S2’ with the theoretical prediction
using the MCMC technique. In order to do that, we consider JNW spacetime which
is the minimally coupled, mass-less scalar field solution of Einstein
equations. The JNW spacetime possesses a naked singularity at the center. One
can retain the Schwarzschild spacetime from the JNW spacetime by considering
zero scalar charge. However, a small but non-zero scalar field charge can
drastically change the causal structure of the Schwarzschild spacetime. The
best-fitted value of the scalar charge comes out to be non-zero which may be
interpreted as the existence of non-vacuum spacetime seeded by a scalar field
around the galactic center.

We organize this paper as follows: In Sec. (II), we discuss the properties of
timelike geodesics in JNW spacetime. In Sec. (III), we discuss the orbital
parameters of the real and apparent orbits. In Sec. (IV), we constrain the
orbital parameters’ space of JNW spacetime by best-fitting the astrometric
data of the ‘S2’ star with the theoretical prediction using the MCMC
technique. In Sec. (V), we discuss our results with concluding remarks.

##  II Timelike geodesics in JNW spacetime

The JNW spacetime is a mass-less, minimally coupled scalar field solution of
Einstein field equations, which is spherically symmetric and static spacetime
and it is given by Virbhadra:1997ie ; Janis:1968zz

$\displaystyle ds^{2}$ $\displaystyle=$
$\displaystyle-\left(1-\frac{b}{r}\right)^{n}c^{2}dt^{2}+\left(1-\frac{b}{r}\right)^{-n}dr^{2}+r^{2}\left(1-\frac{b}{r}\right)^{1-n}d\Omega^{2}\,\,,$
(1)

where $d\Omega^{2}=(d\theta^{2}+\sin^{2}\theta d\phi^{2})$,
$b=2\sqrt{\left(\frac{GM}{c^{2}}\right)^{2}+q^{2}}$ and
$n=\frac{2GM}{c^{2}b}$, here $G$ and $M$ are the gravitational constant and
ADM mass of the spacetime and $q$ is the charge of the mass-less scalar field.
It can be seen from the above equation that the JNW spacetime becomes
Schwarzschild spacetime for zero scalar charge ($q$). As the JNW is a
spherically symmetric, static spacetime, the conserved energy and angular
momentum per unit rest mass can be written as,

$\displaystyle\frac{e}{c^{2}}=\left(1-\frac{b}{r}\right)^{n}\left(\frac{dt}{d\tau}\right),\,\,$
(2)

$\displaystyle
L=r^{2}\left(1-\frac{b}{r}\right)^{1-n}\left(\frac{d\phi}{d\tau}\right),\,\,$
(3)

using these conserved quantities in equation (1), we can define the total
relativistic energy as,

$E_{n}=\frac{\frac{e^{2}}{c^{4}}-1}{2}=\frac{1}{2}\left[\frac{1}{c^{2}}\left(\frac{dr}{d\tau}\right)^{2}+V_{eff}(r)\right],$
(4)

where, $V_{eff}(r)$ is the effective potential of the JNW naked singularity
spacetime. One can derive the following timelike orbit equation for the JNW
spacetime,

$\displaystyle\frac{d^{2}u}{d\phi^{2}}+u-\frac{3bu^{2}}{2}+\frac{e^{2}b}{c^{2}L^{2}}(1-n)(1-bu)^{1-2n}-\frac{c^{2}b}{2L^{2}}(2-n)(1-bu)^{1-n}=0\,\,,$
(5)

where $u=\frac{1}{r}$.

Figure 1: Real orbit projection into the sky plane Becerra-Vergara:2020xoj .
The axes begin at Sgr A* (the focus of the ellipse). The following diagram
depicts the orbital parameters: $\tilde{\phi}=\phi-\pi/2$, where $\phi$ is the
azimuth angle of the spherical coordinate system associated with the $x,y,z$
Cartesian coordinates, i.e. the true anomaly for an elliptic motion in the
$x-y$ plane, $i$ is the angle of inclination between the real orbit and the
sky plane, $\Omega$ is the ascending node angle, and $\omega$ is the
pericenter argument. It is worth mentioning that the vector going from the
solar system to the galactic centre defines the coordinate system’s Z-axis.

In Bambhaniya:2019pbr ; Dey:2019fpv , the properties of timelike orbits in the
JNW spacetime are elaborately discussed and also compared with the timelike
orbits in Schwarzschild spacetime. The important difference which is coming
out from the analysis is that the perihelion precession of bound timelike
orbits in the JNW spacetime, can be negative, i.e., the direction of particle
motion is opposite to the direction of precession. This unique characteristic
of timelike orbit is forbidden in Schwarzschild spacetime. In Igata:2022rcm ;
Igata:2022nkt , it is shown that the presence of matter may be responsible for
the negative precession. Therefore, negative precession is very much important
in the context of the trajectories of ‘S’ stars around the Sgr A*. In the next
section, we use the above orbit equation to predict the possible trajectory of
‘S2’ star, and using the astrometric data of that star, we constrain the
parameters’ space of the JNW spacetime.

Figure 2: Left: Best fitting orbit of the JNW model (black) along with the
observed astrometric position (Blue) of the S2 star from 1995 to 2018. The red
dot shows the position of the Sgr A* at (0,0). Right: R.A. (right ascension
$\alpha$ on top) and Dec. (declination $\delta$ on bottom) offset of S2 star
with the orbital period.

##  III Orbital parameters of the real and apparent orbits

It is necessary to transform the real orbit into the apparent orbit, since the
observed astrometric data given onto the plane of the sky. We define the
position and velocity components of the real orbit in Cartesian coordinates as
$x,y,z$ and $v_{x},v_{y},v_{z}$ respectively. As in our case, $\theta=\pi/2$,
we can obtain position and velocity of the real orbit by the transformation
from spherical coordinates to Cartesian coordinates as

$x=r\cos\phi,\;\;\;y=r\sin\phi,\;\;\;z=0,$ (6)

and the corresponding three velocities transform as,

$v_{x}=v_{r}\cos\phi-
rv_{\phi}\sin\phi,\;\;\;v_{y}=v_{r}\sin\phi+rv_{\phi}\cos\phi,\;\;\;v_{z}=0,$
(7)

where, $v_{r}=dr/dt$ and $v_{\phi}=d\phi/dt$ are three velocity components and
the corresponding four velocity can be written as $u^{r}=v_{r}u^{0}$ and
$u^{\phi}=v_{\phi}u^{0}$. Now, to fit the real orbit with the astrometric
observational data, we must project the real orbit on the apparent plane of
the sky as shown in fig. (1). The observed astrometric positions $X_{obs}$ and
$Y_{obs}$ of the star in Cartesian coordinates are defined by the observed
angular positions, right ascension $(\alpha)$ and declination $(\delta)$
Becerra-Vergara:2020xoj .

$X_{obs}=r_{d}(\alpha-\alpha_{SgrA^{*}}),\;\;\;Y_{obs}=r_{d}(\delta-\delta_{SgrA^{*}}),\;\;$
(8)

Parameter | JNW (95% limits)
---|---
$L^{2}$ $(pc^{2}(km/s)^{2})$ | $4.44216^{+0.00075}_{-0.00075}$
$\log{E_{n}}$ $(km/s)^{2}$ | $10.95422391^{+0.00000034}_{-0.00000033}$
$t_{\rm ini}$ (year) | $1.199^{+0.038}_{-0.040}$
$\log{M}$ $(M\odot)$ | $6.666^{+0.010}_{-0.012}$
$\log{q}$ $(M\odot)$ | $-7.46^{+0.58}_{-0.57}$
$\theta_{\rm inc}$ (radian) | $2.316^{+0.025}_{-0.025}$
$\Omega$ (radian) | $4.017^{+0.035}_{-0.033}$
$\omega$ (radian) | $1.199^{+0.029}_{-0.029}$
Distance (parsec), $r_{d}$ | $8169^{+34}_{-39}$
Time period (yr), $T$ | $16.1379$
Minimal $\chi^{2}$ | $4.71$
Table 1: Estimated best-fit values of the parameters for the JNW metric.

where $r_{d}$ is the distance between the Sgr A* and the earth. Note that the
center of the co-ordinate system represent the position of Sgr A*. The
positions $X,Y,Z$ of the apparent orbit can be obtained from the real orbit
positions $x$ and $y$ by using classic Thiele-Innes constants with the same
notation given in Becerra-Vergara:2020xoj  as

$X=xB+yG,\;\;\;Y=xA+yF,\;\;\;Z=xC+yH,\;\;$ (9)

and the corresponding velocity components of the apparent orbit are,

$V_{X}=v_{x}B+v_{y}G,\;\;\;V_{Y}=v_{x}A+v_{y}F,\;\;\;V_{Z}=v_{x}C+v_{y}H,\;\;$
(10)

where,

$A=\cos\Omega\cos\omega-\sin\Omega\sin\omega\cos i,$ (11)
$B=\sin\Omega\cos\omega+\cos\Omega\sin\omega\cos i,$ (12) $C=\sin\omega\sin
i,$ (13) $F=-\cos\Omega\sin\omega-\sin\Omega\cos\omega\cos i,$ (14)
$G=-\sin\Omega\sin\omega+\cos\Omega\cos\omega\cos i,$ (15) $H=\cos\omega\sin
i,$ (16)

where the osculating orbital elements $\Omega$, $i$, and $\omega$ are the
ascending node angle, inclination angle, and the argument of pericenter,
respectively. Now, the fully general relativistic solution of the above
equation (5) gives the $r(\phi)$, which we have to transform into Cartesian
coordinates using the relation given in (6). The apparent orbital coordinates
$X,Y$ can be obtained from the real orbital positions $x,y$ by using the
transformation (9). Now, we can fit the observed apparent orbital data with
the apparent orbital plane, which will give us the actual nature of the
orbital shape.

##  IV MCMC analysis

The MCMC analysis performed in the paper for the astrometric data of S2 star
is based upon the Metropolis-Hastings algorithm Hastings:1970aa . The
likelihood function used in the analysis for symmetric error is as follows,

$-\log{\mathcal{L}}\propto\sum_{i}\left[\left(X_{i}-\bar{X}_{i}\over\sigma^{X}_{i}\right)^{2}+\left(Y_{i}-\bar{Y}_{i}\over\sigma^{Y}_{i}\right)^{2}\right].$
(17)

Here $X_{i}$ and $Y_{i}$ represent the observed co-ordinates of the S2 star in
its orbit and $\bar{X}_{i}$ and $\bar{Y}_{i}$ are theoretically calculated
values. The errors of the observation in the $X$ and $Y$ direction are
$\sigma^{X}$ and $\sigma^{Y}$ correspondingly. To implement the MCMC analysis,
the value of $q$ is always put within the open bounded interval $(0,q_{0})$,
where $q_{0}$ is taken to some large positive value. This means that the
present analysis excludes the possibility of $q=0$, i.e., the Schwarzschild
spacetime, all along.

Details about the data set: 46 number of data points are considered here from
the astrometric positions of the S2 star. Source of the orbital data is
adopted from the supplement material of the paper Do:2019txf . For the MCMC
analysis we used Gaussian priors. The details of the priors are given in the
table (2).

Parameter | Mean | 1-$\sigma$
---|---|---
$L^{2}$ ($pc^{2}(km/s)^{2}$) | $4.44$ | $0.04$
$\log{E_{n}}$ | $10.90$ | $0.01$
$t_{\rm ini}$ (yr) | $1.22$ | $0.01$
$\log{M}$ | $6.56$ | $0.05$
$\log{q}$ | $-8.00$ | $0.005$
$\theta_{\rm inc}$ (radian) | $2.30$ | $0.02$
$\Omega$ (radian) | $4.00$ | $0.04$
$\omega$ (radian) | $1.20$ | $0.01$
Distance, $r_{d}$ (parsec) | $8200$ | $10$
Table 2: Details of the Gaussian priors of different parameters.

##  V Conclusion

In this paper, we have derived the fully relativistic orbit equation for the
JNW spacetime, which is a second-order non-linear differential equation. We
solve this equation numerically since it is difficult to solve analytically.
To obtain the best fitting orbital parameters of the JNW model, we use the
astrometric data of S2 star from Do:2019txf . We estimate the best fitting
parameters for the JNW metric using the MCMC technique under the assumption of
the positive scalar charge and obtain the lowest $\chi^{2}$ value is 4.71 (see
the figures (2) and (3)). Here, we predict the nature of the Sgr A* using the
available observed astrometric data of the S2 star. Our results show that like
Schwarzschild black hole, the JNW naked singularity could be a possible
candidate for the compact object Sgr A* at our Milky Way galaxy center.

###### Acknowledgements.

TH is very grateful to H. Saida for fruitful discussions and helpful comments.
This work was partially supported by JSPS KAKENHI Grant Nos. JP19K03876,
JP19H01895, JP20H05853 (TH) and JP21K03557 (KN).

Figure 3: 1-$\sigma$ and 2-$\sigma$ bestfit regions and the posterior
distributions of the parameters for the JNW metric and Sgr A* derived using
MCMC. Lowest $\chi^{2}$ value obtained is 4.71.

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