8th MATHEMATICAL PHYSICS MEETING: Summer School and Conference on Modern Mathematical Physics August 2014, Belgrade, Serbia Orbits of S2 star in Yukawa gravity: simulations vs observations D. Borka, 1 P. Jovanović, 2 V. Borka Jovanović, 1 and A. F. Zakharov 3 1 Atomic Physics Laboratory (040), Vinča Institute of Nuclear Sciences, University of Belgrade, P.O. Box 522, Belgrade, Serbia 2 Astronomical Observatory, Volgina 7, Belgrade, Serbia 3 Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, Moscow, Russia

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

Comparison between theoretical rotation curves and the observations 1. Motivation

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: /science 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).

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 2. INTRODUCTION

Galactic center: SGR A

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. 3. Extended Theories of Gravity

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).

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 M BH =4.3 ·10 6 M S (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). 4. Simulated orbits of S2 star

For that purpose we used the following Keplerian orbital elements from i= deg, Ω = deg, and ω = 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 = 2.59·10 3 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· 10 3 AU and Λ = 3.03· 10 3 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· 10 3 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· 10 3 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· 10 3 AU, while the right panel shows the results for NTT/VLT+Keck combined observations and for Yukawa gravity potential with Λ = 3.03· 10 3 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, 10 6 ]. 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 , 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 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 R n gravity ; 6.the two parameters of Yukawa gravity are highly correlated in the range (0 2 they are not correlated. 7.Borka et al. found that R n (f(R)=f 0 R n ) 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· m and Λ >> 4· 10 8 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 R n gravity from precession of orbits of S2-like stars, Physical Review D 85, (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 R n 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, (2014).