{
  "Abstract ": [
    "Here we investigate possible applications of observed stellar orbits around Galactic Center for constraining the Rn gravity at Galactic scales. For that purpose, we simulated orbits of S2-like stars around the massive black hole at Galactic Center, and study the constraints on the Rn gravity which could be obtained by the present and next generations of large telescopes. Our results show that Rn gravity a ects the simulated orbits in the qualitatively similar way as a bulk distribution of matter (including a stellar cluster and dark matter distributions) in Newton’s gravity. In the cases where the density of extended mass is higher, the maximum allowed value of parameter in Rn gravity is noticeably smaller, due to the fact that the both extended mass and Rn gravity cause the retrograde orbital precession. "
  ],
  "keywords": [
    "Keywords: Black hole physics; Galactic Center; Astrometry; Alternative theories of gravity "
  ],
  "1. Introduction ": [
    "Dark matter (DM) ((<>)Zwicky, (<>)1933) and Dark Energy ((<>)Turner, (<>)1999) problems are fundamental and diÿcult for the conventional General Relativity approach for gravity, ((<>)Zakharov et al., (<>)2009) see also the monograph by (<>)Weinberg ((<>)2008) for a more comprehensive review. ",
    "There is an opinion that an introduction of alternative theories of gravity (including so-called f(R) theories ((<>)Capozziello and Fang, (<>)2002; Capozziello et al., (<>)2003, (<>)2006; (<>)Carroll et al., (<>)2004; (<>)Capozziello et al., (<>)2007; Capozziello and Faraoni, (<>)2012; (<>)Mazumdar and Nadathur, (<>)2012)) could after all give explanation of observational data without DM and DE problems. However, a proposed gravity theory has to explain not only cosmological problems but many other observational data because sometimes these theories do not have Newtonian limit for a weak gravitational field case, so parameters of these theories have to be very close to values which correspond to general relativity ((<>)Zakharov et al., (<>)2006). Earlier, constraints on f(R) = Rn have been obtained from an analysis of trajectories of bright stars near the Galactic Center ((<>)Borka et al., (<>)2012, (<>)2013), assuming a potential of bulk distribution of matter is negligible in comparison with a potential of a point like mass. In this paper we consider modifications of results due to a potential of a bulk distribution of matter assuming that this potential is small in comparison with point like mass one. For the standard GR approach these calculations have been done ((<>)Zakharov et al., (<>)2007; (<>)Nucita et al., (<>)2007). ",
    "We would like to mention that not only trajectories of bright stars but also probes such as LAGEOS and LARES ((<>)Ciufolini and Pavlis, (<>)2004; (<>)Ciufolini, (<>)2007) could provide important test for alternative theories of gravity, see for instance constraints on the Chern–Simons gravity ((<>)Smith et al., (<>)2008). "
  ],
  "2. Method ": [
    "We simulated orbits of S2 star around Galactic Center in the Rn gravity potential, assuming a bulk distribution of mass in the central regions of our Galaxy. The Rn gravity potential is given by (<>)Capozziello et al. ((<>)2006, (<>)2007): ",
    "(1) ",
    "where rc is an arbitrary parameter, depending on the typical scale of the considered system and is a universal constant depending on n (Capozziello ",
    "et al., (<>)2006; (<>)Zakharov et al., (<>)2006) ",
    "(2) ",
    "We use two component model for a potential in the central region of our Galaxy which is constituted by the central black hole of mass MBH = 4.3 ×106M⊙((<>)Gillessen et al., (<>)2009) and an extended distribution of matter with total mass Mext(r) (including a stellar cluster and dark matter) contained within some radius r. For the density distribution of extended matter we adopted the following broken power law proposed by (<>)Genzel et al. ((<>)2003): ",
    "(3) ",
    "where ˆ0 = 1.2 ×106 M⊙·pc−3 and r0 = 10 ′′. This leads to the following expression for the extended mass distribution: ",
    "(4) ",
    "Figure 1: Parameter space for Rn gravity with di erent contributions of extended mass under the constraint that, during one orbital period, S2-star orbits in Rn gravity di er less than 10 mas (\" = 0 ′′.01) from its Keplerian orbit. The assumed values for mass density constant ˆ0 from Eq. ((<>)3) are denoted in the title of each panel. ",
    "The corresponding potential for extended distribution of matter is then: ",
    "(5) ",
    "Figure 2: The same as in Fig. (<>)1, but for 10 times higher astrometric precision of 1 mas (\" = 0′′.001). ",
    "where r∞is the outer radius for extended distribution of matter which is enclosed within the orbit of S2 star. The total gravitational potential for the two component model can be evaluated as a sum of Rn potential for central object with mass MBH and potential for extended matter with mass Mext(r): ",
    "(6) ",
    "Thus, the simulated orbits of S2 star could be obtained by numerical integration of the following di erential equations of motion in the total gravitational potential: ",
    "(7) "
  ],
  "3. Results ": [
    "In Figs. (<>)1–(<>)3 we present the parameter space for Rn gravity with di erent contributions of extended mass for which the discrepancies between the orbits ",
    "Figure 3: The same as in Fig. (<>)1, but for 100 times higher astrometric precision of 0.1 mas (\" = 0′′.0001). ",
    "of S2-star in Rn gravity and its Keplerian orbit during one orbital period are less than an assumed astrometric precision. As it can be seen from Fig. (<>)1, it is very diÿcult to detect the contribution of extended mass with the astrometric precision of 10 mas, which was the actual limit during the first part of the observational period of S2 star, some 10–15 years ago. The blue area of parameter space changing very little with variations of ˆ0. However, with the current astrometric limit reaching less than 1 mas, this contribution significantly constrains the maximum allowed value (see Fig. (<>)2). In the cases where the density of extended mass is higher, the maximum allowed value of is noticeably smaller, due to the fact that the both extended mass and Rn gravity have the similar e ect on S2 star orbit, i.e. they both cause the retrograde orbital precession. However, the astrometric limit is constantly improving and in the future it will be possible to measure the stellar positions with much better accuracy of ∼10 µas ((<>)Gillessen et al., (<>)2010). The parameter space for currently unreachable accuracy of 0.1 mas ",
    "Figure 4: Comparison between the simulated orbits of S2 star in the Rn gravity potential with (solid line) and without (dashed line) contribution of extended mass, and NTT/VLT (left) and Keck (right) astrometric observations. The parameters of Rn gravity are obtained by fitting the simulated orbits in total potential to the observations, assuming astrometric accuracy of 10 mas and the density constant ˆ0 = 12 ×106 M⊙·pc−3 . ",
    "is presented in Fig. (<>)3, from which one can see that even a small amount of extended mass would practically exclude the Rn term from the total gravity potential. Even more, we found that in such a case expression ((<>)3) would result with overestimated amount of extended mass at Galactic center, and therefore, we assumed 2 and 10 times smaller densities ˆ0. Besides, from Figs. (<>)1–(<>)3 it is obvious that both astrometric precision and amount of extended mass have significant influence on the value of rc for which maximum is expected. For example, in the top right panel of Fig. (<>)1 the maximum ≈0.027 is expected for rc ≈250 AU, while in the top right panel of Fig. (<>)3 the maximum ≈0.00023 is expected for rc ≈450 AU. ",
    "Several comparisons of the simulated S2 star orbits in the Rn gravity potential with and without contribution of extended mass with NTT/VLT and Keck astrometric observations are given in Fig. (<>)4, assuming the as-trometric accuracy of 10 mas. In this case, influence of extended mass for ˆ0 = 1.2 ×106 M⊙·pc−3 ((<>)Genzel et al., (<>)2003) is almost negligible, and hence it cannot explain the observed precession of S2 star orbit. Therefore, in order to explain the observed precession, either a higher value of in Rn gravity ",
    "potential (see e.g. (<>)Borka et al., (<>)2012), or much higher density of extended mass are necessary (as it is the case in Fig. (<>)4). "
  ],
  "4. Conclusions ": [
    "In this paper we analyze stellar orbits around Galactic Center in the Rn gravity with extended mass distribution. Our results show that Rn gravity with extended mass distribution could significantly a ect the simulated orbits. Both Rn gravity and extended mass distribution give retrograde direction of the precession of the S2 orbit. In the cases if the density of extended mass is higher, the maximum value of which is consistent with observations in Rn gravity is noticeably smaller. We confirmed that the Rn gravity parameter must be very close to those corresponding to the Newtonian limit of the theory. When parameter is vanishing, we recover the value of the Keplerian orbit for S2 star. When we take into account extended mass distribution, parameter is less than in case without extended mass distribution and faster approaching to zero. Even more, one can see that relatively small amount of extended mass would practically exclude the Rn term from the total gravity potential. ",
    "We can conclude that both e ects, additional term in Rn gravity and extended mass distribution, produce a retrograde shift, that results in rosette shaped orbits. Also, we can conclude that both astrometric precision and extended mass distribution have significant influence on the value of rc for which maximum could be expected. ",
    "Although both observational sets (NTT/VLT and Keck) indicate that the orbit of S2 star might not be closed, the current astrometric limit is not suÿcient to unambiguously confirm such a claim. However, the astrometric accuracy is constantly improving from around 10 mas during the first part of the observational period, currently reaching less than 1 mas. ",
    "Acknowledgments. D. B., V. B. J. and P. J. acknowledge support of the Ministry of Education, Science and Technological Development of the Republic of Serbia through the project 176003 ”Gravitation and the large scale structure of the Universe”. A. F. Z. acknowledges a partial support of the NSF (HRD-0833184) and NASA (NNX09AV07A) grants at NCCU (Durham, NC, USA) and RFBR 14-02-00754a at ICAD of RAS (Moscow). Authors thank anonymous referees for their useful critical remarks. "
  ]
}