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Orbits of S2 star in Yukawa gravity:
simulations vs observations
D. Borka,1 P. Jovanović,2 V. Borka Jovanović,1 and A. F. Zakharov3
1Atomic
Physics Laboratory (040), Vinča Institute of Nuclear Sciences,
University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia
2Astronomical Observatory, Volgina 7, 11060 Belgrade, Serbia
3Institute of Theoretical and Experimental Physics,
B. Cheremushkinskaya 25, 117259 Moscow, Russia
8th MATHEMATICAL PHYSICS MEETING:
Summer School and Conference on Modern Mathematical Physics
24 - 31 August 2014, Belgrade, Serbia
Our aim is to constrain parameters of the gravitational potentials derived
from modified gravity models in absence of dark matter.
We simulate the orbit of S2 star around the Galactic Centre in f(R)
(Yukawa-like) gravity potentials and compare it with NTT/VLT and Keck
observations.
The approach we are proposing can be used to constrain the different
modified gravity models from stellar orbits around Galactic Centre.
D. Borka, P. Jovanović, V. Borka Jovanović and, A. F. Zakharov,
Constraining the range of Yukawa gravity interaction from S2 star orbits,
Journal of Cosmology and Astroparticle Physics 11, 050, (1-15) (2013).
Outline of the talk
1. Motivation
2. Introduction (S2 star orbit)
3. Extended Theories of Gravity
4. Simulated orbits of S2 star
5. Results: comparison between the theoretical results
and observations
6. Conclusions
1. Motivation
Comparison between theoretical rotation curves and the observations
1.
Excellent agreement is obtained using Yukawa gravity between
theoretical and observed rotation curves by Capozziello and
coworkers.
V. F. Cardone and S. Capozziello, Systematic biases on galaxy
haloes parameters from Yukawa-like gravitational potentials, Mon.
Not. R. Astron. Soc. 414, 1301 (2011).
2.
For S2 star, there are some observational indications that this orbit
deviates from the Keplerian case.
S. Gillessen, F. Eisenhauer, T.K. Fritz, H. Bartko, K. Dodds-Eden,
O. Pfuhl, T. Ott, and R. Genzel, The orbit of the star S2 around
SgrA* from VLT and Keck data, Astrophys. J. 707, L114 (2009).
Astrometric data for the star S2
L. Meyer et al.
DOI: 10.1126/science.1225506
Science 338, 84 (2012).
NTT/VLT
The New Technology Telescope/Very Large Telescope (NTT/VLT) is a telescope
operated by the European Southern Observatory on Cerro Paranal in the Atacama
Desert of northern Chile. The VLT consists of four individual telescopes, each with a
primary mirror 8.2 m across, which are generally used separately but can be used
together to achieve very high angular resolution.
S. Gillessen, F. Eisenhauer, S. Trippe, T. Alexander, R. Genzel, F. Martins and T.
Ott, Monitoring stellar orbits around the Massive Black Hole in the Galactic
Center, Astrophys. J. 692, 1075 (2009).
Keck
The Keck Observatory is a two-telescope astronomical observatory at an elevation of
4,145 meters near the summit of Mauna Kea in the U.S. state of Hawaii. Both
telescopes feature 10 m primary mirrors, currently among the largest astronomical
telescopes in use. The combination of an excellent site, large optics and innovative
instruments has created the two most scientifically productive telescopes on Earth.
A. M. Ghez, S. Salim, N. N. Weinberg, J. R. Lu, T. Do, J. K. Dunn, K. Matthews,
M. R. Morris, S. Yelda, E. E. Becklin, T. Kremenek, M. Milosavljevic and J.
Naiman, Measuring Distance and Properties of the Milky Way's Central
Supermassive Black Hole with Stellar Orbits, Astrophys. J. 689, 1044 (2008).
2. INTRODUCTION
S2 star
=======
- S2 is a massive young star that closely orbits the supermassive black hole at the
center of the Milky Way Galaxy
- it is one of the brightest members of the so-called S-star cluster (Sgr A—
Cluster) orbiting the central black hole
- it has about 15 times Sol's mass and seven times its diameter, with orbital period
about 15.8 yr
Galactic center: SGR A
3. Extended Theories of Gravity
Extended Theories of Gravity have been proposed like alternative approaches to
Newtonian gravity in order to explain galactic and extragalactic dynamics without
introducing dark matter.
In the case of f(R) gravity, one assumes a generic function f of the Ricci scalar R (in
particular, analytic functions) and searches for a theory of gravity having suitable
behavior at small and large scale lengths.
Yukawa-like corrections have been obtained in the framework of f(R) gravity. The
physical meaning of such corrections needs to be confirmed at different scales: for
short distances, Solar system, spiral galaxies and galaxy clusters.
1.
E. Fischbach and C.L. Talmadge, The Search for Non-Newtonian
Gravity, 305p., Heidelberg-New York, Springer (1999).
2.
T. P. Sotiriou and V. Faraoni, f(R) Theories Of Gravity, Rev. Mod.
Phys. 82, 451 (2010) .
3.
S. Capozziello and M. De Laurentis, Extended Theories of Gravity,
Phys. Rept. 509, 167 (2011).
4.
S. Capozziello and V. Faraoni, Beyond Einstein Gravity: A Survey
of Gravitational Theories for Cosmology and Astrophysics, in
Fundamental Theories of Physics vol. 170, Springer (2011).
5.
S. Nojiri and S. D. Odintsov, Unified cosmic history in modified
gravity: From F(R) theory to Lorentz non-invariant models, Phys.
Rept. 505, 59 (2011).
6.
T. Clifton, P.G. Ferreira, A. Padilla and C. Skordis, Modified
Gravity and Cosmology, Phys. Rept. 513, 1 (2012).
4. Simulated orbits of S2 star
We performed two-body simulations in Yukawa gravity potential
The positions of the S2 star along its true orbit are calculated at the observed
epochs using two-body simulations in the Yukawa gravity potential, assuming
that distance to the S2 star is d = 8.3 kpc and mass of central black hole
MBH=4.3 ·106 MS (S. Gillessen et al., Astrophys. J. 707, L114, 2009).
All the above two-body simulations in Yukawa gravity potential resulted with the
true orbits (ξ, η) of S2-like stars, i.e. the simulated positions of S2-like stars. In
order to compare them with observed positions, the first step is to project them to
the observer's sky plane, i.e. to calculate the corresponding apparent orbits (x, y).
For that purpose we used the following Keplerian orbital elements from
i= 134.87deg, Ω = 226.53deg, and ω = 64.98 deg (S. Gillessen et al.,
Astrophys. J. 707, L114, 2009).
RESULTS: Comparison between
the theoretical results and observations
Figure 1. Comparisons between the orbit of S2 star in Newtonian gravity (red
dashed line) and Yukawa gravity during 10 orbital periods (blue solid line) for Λ =
2.59·103 AU. In the left panel δ = +1/3, and in the right panel δ = -1/3.
Figure 2. The fitted orbits in Yukawa gravity for δ = +1/3 through the astrometric
observations of S2 star (denoted by circles), obtained by NTT/VLT alone (left
panel) and NTT/VLT+Keck (right panel). The best fits are obtained for Λ = 2.59·
103 AU and Λ = 3.03· 103 AU, respectively.
Figure 3. The comparisons between the observed (open circles with error bars) and
fitted (solid lines) coordinates of S2 star (top), as well as the corresponding O-C
residuals (bottom). The left panel shows the results for Δα and right panel for Δδ in
the case of NTT/VLT observations and Yukawa gravity potential with δ = +1/3 and
Λ = 2.59· 103 AU.
Figure 4. The same as in Figure 3, but for NTT/VLT+Keck combined observations
and for Yukawa gravity potential with Λ = 3.03· 103 AU.
Figure 5. The comparisons between the observed (circles with error bars) and fitted
(solid lines) radial velocities of S2 star (top), as well as the corresponding O-C
residuals (bottom). The left panel shows the results in the case of NTT/VLT
observations and Yukawa gravity potential with Λ = 2.59· 103 AU, while the right
panel shows the results for NTT/VLT+Keck combined observations and for Yukawa
gravity potential with Λ = 3.03· 103 AU. In both cases δ = +1/3.
Figure 6. The reduced χ² for δ =1/3 as a function of Λ in case of NTT/VLT alone
(left) and combined NTT/VLT+Keck (right) observations.
Figure 7. The maps of reduced χ² over the Λ-δ parameter space in case of NTT/VLT
observations. The left panel corresponds to δ in [0, 1], and the right panel to the
extended range of δ in [0.01, 106]. The shades of gray color represent the values of
the reduced χ² which are less than the corresponding value in the case of Keplerian
orbit, and three contours (from inner to outer) enclose the confidence regions in
which the difference between the current and minimum reduced χ² is less than
0.0005, 0.005 and 0.05, respectively.
Figure 8. The same as in Figure 7, but for the combined NTT/VLT+Keck
observations.
CONCLUSIONS
In this paper orbit of S2 star has been investigated in the framework of the
Yukawa gravity. Using the observed positions of S2 star around the Galactic
Centre we constrained the parameters of Yukawa gravity. Our results show
that:
1.
the most probable value for Yukawa gravity parameter Λ in the case of S2 star,
is around 5000 - 7000 AU and that the current observations do not enable us to
obtain the reliable constraints on the universal constant δ;
2.
the same universal constant δ which was successfully applied to clusters of
galaxies and rotation curves of spiral galaxies also gives a good agreement in
the case of observations of S2 star orbit;
3.
the scale parameter of Yukawa gravity in the case of S2 star for δ = +1/3 is
about: Λ ≈ 3000 ± 1500 AU, for vanishing δ, we recover the Keplerian orbit of
S2 star;
5.
for δ = +1/3 there is orbital precession in positive direction like in General
Relativity, and for δ = -1/3 the precession has negative direction, as in the case of
Newtonian precession due to extended mass distribution or in Rn gravity ;
6.
the two parameters of Yukawa gravity are highly correlated in the range (0 < δ <
1). For δ > 2 they are not correlated.
7.
Borka et al. found that Rn (f(R)=f0Rn) gravity may not represent a good candidate
to solve both the rotation curves problem of spiral galaxies and the orbital
precession of S2 star for the same value of the universal constant β (β =0.817
and β ≈ 0.01, respectively). According to the above results, the opposite
conclusion is not eliminated in the case of Yukawa gravity with δ =1/3.
8.
The constraints on parameter Λ obtained in the present paper are in agreement
with the corresponding Solar System and LLR constraints presented by
Adelberger et al., according to which Λ >> 1.5· 1011 m and Λ >> 4· 108 m,
respectively.
9.
The approach we are proposing can be used to constrain the different modified
gravity models from stellar orbits around Galactic Centre.
REFERENCES
1.
D. Borka, P. Jovanović, V. Borka Jovanović and A. F. Zakharov, Constraints
on Rn gravity from precession of orbits of S2-like stars, Physical Review D
85, 124004-1-11 (2012).
2.
D. Borka, P. Jovanović, V. Borka Jovanović and A. F. Zakharov, Constraining
the range of Yukawa gravity interaction from S2 star orbits, Journal of
Cosmology and Astroparticle Physics 11, 050, (1-15) (2013).
3.
A. F. Zakharov, D. Borka, V. Borka Jovanović and P. Jovanović, Constraints
on Rn gravity from precession of orbits of S2-like stars: case of bulk
distribution of mass, Advances in Space Research 54,1108–1112, (2014).
4.
S. Capozziello, D. Borka, P. Jovanović and V. Borka Jovanović, Constraining
Extended Gravity Models by S2 star orbits around the Galactic Centre,
Physical Review D 90, 044052-1-8 (2014).
THANK YOU FOR YOUR ATTENTION!