Constraining the range of Yukawa gravity interaction from S2 star orbits
aa r X i v : . [ a s t r o - ph . GA ] N ov Prepared for submission to JCAP
Constraining the range of Yukawagravity interaction from S2 starorbits
D. Borka, a, P. Jovanović, b V. Borka Jovanović a and A. F.Zakharov c,d,e a Atomic Physics Laboratory (040), Vinča Institute of Nuclear Sciences,University of Belgrade, P.O. Box 522, 11001 Belgrade, Serbia b Astronomical Observatory, Volgina 7, 11060 Belgrade, Serbia c Institute of Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117259 Moscow,Russia d Bogoliubov Laboratory for Theoretical Physics, JINR, 141980 Dubna, Russia e North Carolina Central University, 1801 Fayetteville Street, Durham, NC 27707, USAE-mail: [email protected], [email protected], [email protected], [email protected]
Abstract.
We consider possible signatures for Yukawa gravity within the Galactic CentralParsec, based on our analysis of the S2 star orbital precession around the massive compactdark object at the Galactic Centre, and on the comparisons between the simulated orbits inYukawa gravity and two independent sets of observations. Our simulations resulted in strongconstraints on the range of Yukawa interaction Λ and showed that its most probable valuein the case of S2 star is around 5000 - 7000 AU. At the same time, we were not able toobtain reliable constrains on the universal constant δ of Yukawa gravity, because the currentobservations of S2 star indicated that it may be highly correlated with parameter Λ in therange (0 < δ < . For δ > they are not correlated. However, the same universal constantwhich was successfully applied to clusters of galaxies and rotation curves of spiral galaxies( δ = 1 / ) also gives a satisfactory agreement with the observed orbital precession of the S2star, and in that case the most probable value for the scale parameter is Λ ≈ ± AU.Also, the Yukawa gravity potential induces precession of S2 star orbit in the same direction asGeneral Relativity for δ > and for δ < − , and in the opposite direction for − < δ < . Thefuture observations with advanced facilities, such as GRAVITY or/and European ExtremelyLarge Telescope, are needed in order to verify these claims. Corresponding author. ontents
The modified theories of gravity have been proposed like alternative approaches to Newto-nian gravity on the ground of astrophysical and cosmological consequences of the observationsof the Solar system, binary pulsars, spiral galaxies, clusters of galaxies and the large-scalestructure of the Universe. The search for non-Newtonian gravity is part of the quest fornon-Einsteinian physics which consists of searches for deviations from Special and GeneralRelativity [1]. Different alternative gravity theories have been proposed (see e.g. [2–5] forreviews), such as: MOND [6–9], scalar-tensor [10], conformal [11, 12], Yukawa like correctedgravity theories [13–17] and extended theories of gravity [4, 5, 18–22]. One type of the ex-tended theories of gravity is characterized by power-law Lagrangians [23, 24]. An extensionof Post-Newtonian relativistic theory is presented by Kopeikin and Vlasov, who used a gen-eral class of the scalar-tensor (Brans-Dicke type) theories of gravitation [25]. Alternativeapproaches to Newtonian gravity in the framework of the weak field limit of fourth ordergravity theory have been proposed and constraints on these theories have been discussed [26–32]. Theories of "massive gravity" have also attracted some attention, and they could givean exponential decay to Newton’s potential at the large distances (see e.g. [33] for a review).Babichev et al. introduced a new limit of that theory, in the weak-field approximation, whichis able to capture both the Vainshtein recovery of general relativity and the large distanceYukawa decay [34]. For more details about massive gravity models see also the followingpapers [35–39].So called Yukawa-like fifth force is a framework for deviations from the inverse-squarelaw in which the gravitational potential deviates from the usual Newtonian form at largedistances due to a Yukawa-like term in the gravitational potential [40–42]. Also it was pro-posed that the anomalous observations of the galactic rotation curves could be explained byaddition of a Yukawa correction term to the Newtonian gravity potential [43]. In order toobtain the constraints on Yukawa gravity, studies of the planetary and stellar orbits at thelarger scales, as well as laboratory searches at the smaller scales have to be performed. Adel-berger et al. [44] reviewed experiments with very high sensitivity, placing constraints aboutnew Yukawa forces from the exchange of very light scalar, pseudoscalar or vector particles.Weaker constraints at still smaller scales are available using the Casimir effect [see e.g. 14]. Acompilation of experimental, geophysical and astronomical constraints on Yukawa violationsof the gravitational inverse square law are given in Figs. 9 and 10 from [44] for differentranges. These results show that the Yukawa term is relatively well constrained for the short– 1 –anges (especially at sub-mm scale), but for long ranges further tests are needed, and it wouldbe very important to evaluate parameters of the Yukawa law for whole range of Λ . A rangearound a few thousand AU has not been investigated yet. However, for longer distancesYukawa corrections have been successfully applied to clusters of galaxies setting δ = +1 / [45, 46]. The same value of this parameter also gives a very good agreement between thetheoretical and observational rotation curves of spiral galaxies [40].For now it seems that the only major problem with Yukawa gravity is in the case ofelliptical galaxies where the observations give a value of δ ≈ − . [47], which is inconsistentwith the one previously found for spiral galaxies. Napolitano et al. [47] found a possibleexplanation for this inconsistency, according to which δ might be correlated with the galaxyanisotropy and the scale parameter, where both elliptical and spiral galaxies follow the samepattern. In that case, δ could be interpreted in terms of physics of the gravitating systemsafter their spherical collapse [47].On the other hand, S-stars are the bright stars which move around the centre of ourGalaxy [48–53] where the compact radio source Sgr A ∗ is located. These stars, together withrecently discovered dense gas cloud falling towards the Galactic Centre [54], provide the mostconvincing evidence that Sgr A ∗ represents a massive compact object around which S-starsare orbiting [53]. For one of them, called S2, there are some observational indications thatits orbit could deviate from the Keplerian case due to relativistic precession [50, 55], but thecurrent astrometric limit is not sufficient to unambiguously confirm such a claim.The orbital precession can occur due to relativistic effects, resulting in a prograde peri-centre shift or due to a possible extended mass distribution, producing a retrograde shift [56].Both prograde relativistic and retrograde Newtonian pericentre shifts will result in rosetteshaped orbits. Adkins and McDonnell [57] calculated the precession of Keplerian orbits un-der the influence of arbitrary central force perturbations. For some examples, including theYukawa potential, they presented the results using hypergeometric functions. Weinberg etal. [58] discussed physical experiments achievable via the monitoring of stellar dynamics nearthe massive black hole at the Galactic Centre with a diffraction-limited, next-generation, ex-tremely large telescope (ELT). They demonstrated that the lowest order relativistic effects,such as the prograde precession, could be detectable if the astrometric precision would reacha few tenths of mas. The astrometric limit for S2 star orbit today reaches 0.3 mas [59], andsome very recent studies provide more and more evidence that orbit of S2 star is not closing(see e.g. Fig. 2 in [55]).Here we study a possible application of Yukawa gravity within Galactic Central Parsec,for explaining the observed precession of orbits of S-stars. We assumed that the motion ofthese stars could be described by the gravitational potential around a massive compact centralobject (without speculating about its nature, i.e. whether it is a black hole or not), in anygravity theory which predicts a Yukawa correction term. For reviews of various theoreticalframeworks yielding a Yukawa-like fifth force, such as braneworld models, scalar-tensor andscalar-tensor-vector theories of gravity, or studies of topological defects, see e.g. [60–64] andreferences therein. Other studies of long-range Yukawa-like modifications of gravity conductedwith different astronomical techniques can be found in [41, 65–69]. Such a phenomenologicalapproach is in some sense more general than studying the stellar orbits in a metric of a massivecentral black hole, since some classes of modified gravity theories (such as e.g. f ( R ) theory)predict the black hole metrics which are equivalent to those obtained in General Relativity.The present paper is organized as follows: in section 2 we describe our simulations ofstellar orbits in Yukawa gravity potential; the procedure for fitting the simulated orbits to– 2 –wo independent sets of astrometric observations of S2 star is described in section 3; our mainresults are presented in section 4, and finally, we point out the most important conclusionsof our studies in section 5. As it was already mentioned, Yukawa gravity potential represents a widely used phenomeno-logical approach to account for possible deviations from the Newtonian inverse-square law,by introducing the following exponential (or so called Yukawa-like) modification to the New-tonian gravitational potential [5, 43, 47]:
Φ ( r ) = − GM (1 + δ ) r " δe − (cid:18) r Λ (cid:19) , (2.1)where Λ is an arbitrary parameter (usually referred to as range of interaction), depending onthe typical scale of the system under consideration and δ is a universal constant. For δ = 0 the Yukawa potential reduces to the Newtonian one, as expected.We simulated orbits of S2 star in the Yukawa gravity potential (2.1) and comparedthe obtained results with two independent sets of observations of S2 star, obtained by NewTechnology Telescope/Very Large Telescope (NTT/VLT), as well as by Keck telescope (seeFig. 1 in [50]), which are publicly available as the supplementary on-line data to the electronicversion of the paper [50]. The simulated orbits of S2 star were obtained in the standard wayby numerical integration of differential equations of motion in Yukawa gravitational potential: ˙r = v , µ ¨r = − ▽ Φ ( r ) , (2.2)where µ is so called reduced mass in the two-body problem. In our calculations we assumedthat the mass of central object is M = 4.3 × M ⊙ and that the distance to the S2 star is d ⋆ = 8.3 kpc [50]. Perturbations from other members of the stellar cluster, as well as fromsome possibly existing extended structures composed from visible or dark matter [29], wereneglected due to simplicity reasons. The obtained simulated orbits in the Yukawa potentialwere compared with the two sets of observations of the S2 star. Since the integration of theequations (2.2) results with coordinates and velocity components in orbital plane (so calledtrue orbits), the first step is to project them onto the observer’s sky plane (i.e. to calculatethe corresponding apparent orbits), in order to compare them with observed positions. Fromthe theory of binary stars it is well known that any point ( x, y ) on the true orbit could beprojected into the point ( x c , y c ) on the apparent orbit according to (see e.g.[70, 71]): x c = l x + l y, y c = m x + m y, (2.3)where the expressions for l , l , m and m depend on three orbital elements ( Ω - longitudeof the ascending node, ω - longitude of pericenter and i - inclination): l = cos Ω cos ω − sin Ω sin ω cos i,l = − cos Ω sin ω − sin Ω cos ω cos i,m = sin Ω cos ω + cos Ω sin ω cos i,m = − sin Ω sin ω + cos Ω cos ω cos i. (2.4)– 3 –adial velocity can be calculated from the corresponding true position ( r, θ ) and orbitalvelocity ( ˙ r, ˙ θ ) using the well known expression in polar coordinates [70]: v rad = sin i h sin( θ + ω ) · ˙ r + r cos( θ + ω ) · ˙ θ i . (2.5)However, in our case it was more convenient to use the rectangular coordinates x = r cos θ and y = r sin θ to calculate the fitted radial velocities: v rad = sin i p x + y [sin( θ + ω ) · ( x ˙ x + y ˙ y )++ cos( θ + ω ) · ( x ˙ y − y ˙ x )] , (2.6)where θ = arctan yx .Since the present paper does not aim to study the Keplerian orbit of S2 star and since i , Ω and ω are needed here only for transforming from true to apparent coordinates, we usedthe following values obtained from the same observations [50]: i = 134 ◦ . , Ω = 226 ◦ . and ω = 64 ◦ . . One should also take into account that in the case of orbital precession ω is ingeneral a function of time, and therefore it should be treated accordingly during the fittingprocedure. However, based on some theoretical studies of precession in Yukawa gravity [72–74] one can expect very slow change of ω during the observational interval of S2 star, andsince in equations (2.4) ω is used only as an argument of sin and cos , fixing it to a constantvalue could introduce only negligible errors. Therefore, we assumed it together with l , l , m and m as constants when projecting true positions to their corresponding apparent values. In that way, for each pair of a priori given values of δ and Λ , there are only four unknownparameters which are to be obtained by fitting: two components of initial position and twocomponents of initial velocity in orbital plane, corresponding to the time of the first obser-vation. We varied both δ and Λ , and for each pair of them found the best fit values of S2star initial conditions and the corresponding value of χ . The fitting itself was performedusing LMDIF1 routine from MINPACK-1 Fortran 77 library which solves the nonlinear leastsquares problems by a modification of Marquardt-Levenberg algorithm [75]. Unfortunately, δ and Λ could not be fitted simultaneously with initial position and velocity, most likely dueto inability of the fitting routine to correctly estimate the corresponding components of Jaco-bian by a forward-difference approximation. Therefore, these two parameters were varied incertain domains, and their best fit estimates were found at the end of the fitting procedureas those for which the minimum χ was obtained.We first adopted value of the universal constant δ = 1 / and varied length scale Λ ininterval from 10 to 10 000 AU with increment of 10 AU, in order to see whether this valueof δ could provide the satisfactory fit, as well as to estimate the most probable value of Λ inthis case. For each Λ we obtained the best fit orbit using the following fitting procedure:1. initial values for true position ( x , y ) and orbital velocity ( ˙ x , ˙ y ) of S2 star at theepoch of the first observation are specified;– 4 – igure 1 . Comparisons between the orbit of S2 star in Newtonian gravity (red dashed line) andYukawa gravity during 10 orbital periods (blue solid line) for Λ = 2 . × AU. In the left panel δ = +1 / , and in the right δ = − / .
2. the positions ( x i , y i ) and velocities ( ˙ x i , ˙ y i ) of S2 star along its true orbit are calculatedfor all observed epochs by numerical integration of equations of motion (2.2) in theYukawa gravity potential;3. the corresponding positions ( x ci , y ci ) along the apparent orbit are calculated using theexpressions (2.3) and (2.4), as well as the corresponding radial velocities ( v irad ) using(2.6);4. the reduced χ of fit is estimated according the following expression: χ = 12 N − ν N X i =1 "(cid:18) x oi − x ci σ xi (cid:19) + (cid:18) y oi − y ci σ yi (cid:19) , (3.1)where ( x oi , y oi ) is the i -th observed position, ( x ci , y ci ) is the corresponding calculatedposition, N is the number of observations, ν is number of unknown parameters (in ourcase ν = 4 ), σ xi and σ yi are uncertainties of observed positions;5. the reduced χ is minimized and the final values of initial positions and velocities areobtained.Finally, we kept the value of Λ which resulted with the smallest value of minimized reduced χ . In order to obtain some more general constraints on the parameters of Yukawa gravity,we also varied both δ and Λ and studied the simulated orbits of S2 star which give at least thesame or better fits than the Keplerian orbit. For each pair of these parameters the reduced χ of the best fit is obtained and used for generating the χ maps over the Λ − δ parameterspace. These maps are then used to study the confidence regions in Λ − δ parameter space.– 5 – .0000.0250.0500.0750.1000.1250.1500.175 -0.10-0.050.000.05 ∆ δ ( " ) ∆α (") ∆ δ ( " ) ∆α (") Figure 2 . The fitted orbits in Yukawa gravity for δ = +1 / through the astrometric observations ofS2 star (denoted by circles), obtained by NTT/VLT alone (left panel) and NTT/VLT+Keck (rightpanel). The best fits are obtained for Λ = 2 . × AU and
Λ = 3 . × AU, respectively. -75-50-25 0 25 50 ∆ α ( m a s ) NTT/VLT-10-5 0 5 10 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 ( O - C ) α ( m a s ) Time (years) O-C residuals 0 50 100 150 200 ∆ δ ( m a s ) NTT/VLT-10-5 0 5 10 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 ( O - C ) δ ( m a s ) Time (years) O-C residuals
Figure 3 . The comparisons between the observed (open circles with error bars) and fitted (solidlines) coordinates of S2 star (top), as well as the corresponding O-C residuals (bottom). The leftpanel shows the results for ∆ α and right panel for ∆ δ in the case of NTT/VLT observations andYukawa gravity potential with δ = +1 / and Λ = 2 . × AU.
The simulated orbits of S2 star around the central object in Yukawa gravity (blue solid line)and in Newtonian gravity (red dashed line) for
Λ = 2 . × AU and δ = +1 / (left panel)and δ = − / (right panel) during 10 periods, are presented in Fig. 1. We can notice that for δ = − / the precession has the negative direction and when δ = +1 / the precession has the– 6 – ∆ α ( m a s ) NTT/VLTKeck-10-5 0 5 10 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 ( O - C ) α ( m a s ) Time (years) O-C residuals 0 50 100 150 200 ∆ δ ( m a s ) NTT/VLTKeck-10-5 0 5 10 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 ( O - C ) δ ( m a s ) Time (years) O-C residuals
Figure 4 . The same as in Fig. 3, but for NTT/VLT+Keck combined observations and for Yukawagravity potential with
Λ = 3 . × AU. -1500-1000-500 0 500 1000 1500 v r a d ( k m / s ) NTT/VLT-150-100-50 0 50 100 150 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 R e s i du a l s ( k m / s ) Time (years) -1500-1000-500 0 500 1000 1500 v r a d ( k m / s ) NTT/VLTKeck-150-100-50 0 50 100 150 1992 1994 1996 1998 2000 2002 2004 2006 2008 2010 R e s i du a l s ( k m / s ) Time (years)
Figure 5 . The comparisons between the observed (circles with error bars) and fitted (solid lines) radialvelocities of S2 star (top), as well as the corresponding O-C residuals (bottom). The left panel showsthe results in the case of NTT/VLT observations and Yukawa gravity potential with
Λ = 2 . × AU, while the right panel shows the results for NTT/VLT+Keck combined observations and forYukawa gravity potential with
Λ = 3 . × AU. In both cases δ = +1 / . positive direction. Our analysis shows that the Yukawa gravity potential induces precessionof S2 star orbit in the same direction as General Relativity for δ > and for δ < − , and inthe opposite direction for − < δ < as in the case of extended mass distribution or in R n gravity [26].We used these simulated orbits to fit the observed orbits of S2 star. The best fit (ac-cording to NTT/VLT data) is obtained for the scale parameter: Λ = 2 . × AU, for– 7 – (AU) NTT/VLT (AU) NTT/VLT+ Keck
Figure 6 . The reduced χ for δ =1/3 as a function of Λ in case of NTT/VLT alone (left) and combinedNTT/VLT+Keck (right) observations.
0 1000 2000 3000 4000 5000 Λ (AU)0.00.20.40.60.81.0 δ χ Λ (AU)-2-10123456 l og δ χ Figure 7 . The maps of reduced χ over the Λ − δ parameter space in case of NTT/VLT observations.The left panel corresponds to δ ∈ [0 , , and the right panel to the extended range of δ ∈ [0 . , ] .The shades of gray color represent the values of the reduced χ which are less than the correspondingvalue in the case of Keplerian orbit, and three contours (from inner to outer) enclose the confidenceregions in which the difference between the current and minimum reduced χ is less than 0.0005, 0.005and 0.05, respectively.
0 1000 2000 3000 4000 5000 Λ (AU)0.00.20.40.60.81.0 δ χ Λ (AU)-2-10123456 l og δ χ Figure 8 . The same as in Fig. 7, but for the combined NTT/VLT+Keck observations. which even a significant strength of the Yukawa interaction could be expected according tothe planetary and Lunar Laser Ranging constraints [44]. Lunar (and artificial satellite) Laserranging (LLR) is one of the most accurate techniques to test gravitational physics and Ein-stein’s theory of General Relativity [76–78] at the corresponding length scales. In particular,LLR has provided very accurate tests of the strong equivalence principle at the foundations of– 8 –eneral Relativity and of the weak equivalence principle, at the basis of any metric theory ofgravity. Also, it has provided strong limits to the values of the so-called PPN (ParametrizedPost-Newtonian) parameters [78]. The Solar System and LLR constraints on the range ofYukawa interaction are shown in Fig. 16 (see also Table 8) in Adelberger et al. [44], accord-ing to which Λ ≫ . × m and Λ ≫ × m, respectively, and thus is in accordancewith our findings.In Fig. 2 we presented two comparisons between the fitted orbits in Yukawa gravityfor δ = +1 / through the astrometric observations of S2 star by NTT/VLT alone (left) andNTT/VLT+Keck combination (right). In order to combine NTT/VLT and Keck data sets,the position of the origin of Keck observations is first shifted by ∆ x = 3 . and ∆ y = 4 . mas,following the suggestion given in [50]. In the first case the best fit is obtained for Λ = 2 . × AU, resulting with reduced χ = 1 . , and in the second case for Λ = 3 . × AU withreduced χ = 3 . . As one can see from these figures, in both cases there is a good agreementbetween the theoretical orbits and observations, although the higher value of reduced χ inthe second case indicates possibly larger positional difference between the two coordinatesystems, as also noted in [50]. These figures also show that the simulated orbits of S2 are notclosed in vicinity of apocenter, indicating a possible orbital precession.In Figs. 3 and 4 we presented the comparisons between the observed and fitted coor-dinates of S2 star and their O-C residuals in the case of NTT/VLT observations, as well asNTT/VLT+Keck combined data set, respectively. One can notice that in both cases, O-Cresiduals are higher in the first part of observing interval (up to the 12 mas) and much less inits second part (less than 2 mas). Due to adopted merit function given by Eq.(3.1), our fittingprocedure assigned greater weight to these latter, more precise observations. Also, the O-Cresiduals are larger in the case of the combined NTT/VLT+Keck observations most likelydue to the shift of the origin of the coordinate system, which was necessary in order to get areasonable fit. That is why we also presented the results for NTT/VLT measurements alone.We also made the comparisons between the observed and fitted radial velocities of S2star (see Fig. 5) for NTT/VLT data alone (left) and NTT/VLT+Keck combination (right).In the bottom parts of both panels in Fig. 5 the best fit O-C residuals for radial velocities arealso given. As it can be seen from Fig. 5, we also obtained satisfactory agreement betweenthe predicted and observed radial velocities of S2 star.Figure 6 presents the reduced χ for all fits with fixed value of δ =1/3 as a function of theother parameter of Yukawa gravity Λ which was varied from 10 to 10 000 AU. In the case ofNTT/VLT observations the minimum of reduced χ is 1.54 and is obtained for Λ = 2 . × AU, while in the case of NTT/VLT+Keck combined data set the minimal value of 3.24 isobtained for
Λ = 3 . × AU. For both cases the reduced χ for Keplerian orbits ( δ = 0 )are 1.89 and 3.53, respectively, and thus significantly higher than the corresponding minimafor δ = 1 / . This means that Yukawa gravity describes observed data even better thanNewtonian gravity and that δ = 1 / is valid value at galactic scales.Figs. 7 and 8 present the maps of the reduced χ over the Λ − δ parameter space for allsimulated orbits of S2 star which give at least the same or better fits than the Keplerian orbits.These maps are obtained by the same fitting procedure as before. The left panels of bothfigures correspond to δ ∈ [0 , and Λ[AU] ∈ [10 , , and the right panels to the extendedrange of δ ∈ [0 . , ] and Λ[AU] ∈ [2000 , . Three contours (from inner to outer) enclosethe confidence regions in which the difference between the current and minimum reduced χ is less than 0.0005, 0.005 and 0.05, respectively. As it can be seen from Fig. 7, the mostprobable value for the scale parameter Λ , in the case of NTT/VLT observations of S2 star,– 9 –s around 5000 - 6000 AU, while in the case of NTT/VLT+Keck combined data set (Fig. 8),the most probable value for Λ is around 6000 - 7000 AU. In both cases χ asymptoticallydecreases as a function of δ , and hence, it is not possible to obtain reliable constrains onthe universal constant δ of Yukawa gravity. Also, these two parameters δ and Λ are highlycorrelated in the range (0 < δ < . For δ > (the vertical strips) they are not correlated.As it could be also seen from left panels of Figs. 7 and 8, the values δ ≈ / result withvery good fits for which the reduced χ deviate from the minimal value for less than 0.005(middle contours in both figures). The corresponding values for Λ range approximately from2500 to 3000 AU. For δ = 1 / we obtained the following values: Λ = 2590 ± AU (NTT/VLTdata) and
Λ = 3030 ± AU (NTT/VLT+Keck combined data).Although both observational sets indicate that the orbit of S2 star most likely is not aKeplerian one, the current astrometric limit is not sufficient to unambiguously confirm sucha claim. However, the accuracy is constantly improving from around 10 mas during thefirst part of the observational period, currently reaching around 0.3 mas. We hope that inthe future, it will be possible to measure the stellar positions with much better accuracy of ∼ µ as [80]. In this paper orbit of S2 star has been investigated in the framework of the Yukawa grav-ity. Using the observed positions of S2 star around the Galactic Centre we constrained theparameters of Yukawa gravity. Our results show that:1. the most probable value for Yukawa gravity parameter Λ in the case of S2 star, is around5000 - 7000 AU and that the current observations do not enable us to obtain the reliableconstraints on the universal constant δ ;2. the same universal constant δ which was successfully applied to clusters of galaxies[45, 46] and rotation curves of spiral galaxies [40] also gives a good agreement in thecase of observations of S2 star orbit;3. the scale parameter of Yukawa gravity in the case of S2 star for δ = +1 / is about: Λ ≈ ± AU;4. for vanishing δ , we recover the Keplerian orbit of S2 star;5. for δ = +1 / there is orbital precession in positive direction like in General Relativity,and for δ = − / the precession has negative direction, as in the case of extended massdistribution or in R n gravity [26];6. the two parameters of Yukawa gravity are highly correlated in the range (0 < δ < .For δ > they are not correlated.Borka et al. [26] found that R n gravity may not represent a good candidate to solveboth the rotation curves problem of spiral galaxies and the orbital precession of S2 star forthe same value of the universal constant β ( β =0.817 and β ∼ δ = 1 / .The constraints on parameter Λ obtained in the present paper are in agreement withthe corresponding Solar System and LLR constraints presented by Adelberger et al. [44],– 10 –ccording to which Λ ≫ . × m and Λ ≫ × m, respectively. Also, the constraintson the Yukawa-like modifications of gravitation from Solar System planetary motions showedthat one can assume values of the parameter Λ on the same order of magnitude, or larger thanthe typical sizes of the planetary orbits in the Solar System [17]. This is common assumptionin many modified theories of gravity [17, 79] and is also in accordance with our findings.However, one should keep in mind that we considered an idealized model ignoring manyuncertain factors, such as an extended mass distribution, perturbations from nonsymmetricmass distribution, etc. Therefore, the future observations with advanced facilities, such asGRAVITY which will enable extremely accurate measurements of the positions of stars of ∼ µ as [80], or E-ELT with expected accuracy of ∼ µ as [81], are needed in order toverify these claims. Acknowledgments
This research is part of the project 176003 ”Gravitation and the large scale structure of theUniverse” supported by the Ministry of Education, Science and Technological Developmentof the Republic of Serbia. AFZ was supported in part by the NSF (HRD-0833184) andNASA (NNX09AV07A) at NCCU in Durham. The authors thank a referee for a constructivecriticism.
References [1] E. Fischbach and C. L. Talmadge,
The Search for Non-Newtonian Gravity , 305p.,Heidelberg − New York, Springer (1999).[2] T. Clifton,
Alternative Theories of Gravity , University of Cambridge (2006).[3] T. Clifton, P. G. Ferreira, A. Padilla, and C. Skordis,
Modified gravity and cosmology , PhysicsReports , vol. (2012) 1.[4] S. Capozziello, and M. de Laurentis,
Extended Theories of Gravity , Physics Reports , vol. (2011) 167.[5] S. Capozziello, V. Faraoni,
Beyond Einstein Gravity: A Survey of Gravitational Theories forCosmology and Astrophysics , Fundamental Theories of Physics vol. 170, Springer (2011).[6] M. Milgrom,
A modification of the Newtonian dynamics as a possible alternative to the hiddenmass hypothesis , Astrophys. J. (1983) 365.[7] M. Milgrom and R. H. Sanders,
Modified Newtonian dynamics and the "dearth of dark matterin ordinary elliptical galaxies" , Astrophys. J. (2003) L25.[8] J. D. Bekenstein,
Relativistic gravitation theory for the modified Newtonian dynamics paradigm , Phys. Rev.
D 70 (2004) 083509;
Phys. Rev.
D 71 (2005) 069901(E).[9] J. Bekenstein and J. Magueijo,
Modified Newtonian Dynamics habitats within the solar system , Phys. Rev.
D 73 (2006) 103513.[10] C. Brans and H. Dicke,
Mach’s principle and a relativistic theory of gravitation , Phys. Rev. (1961) 925.[11] D. Behnke, D. B. Blaschke, V. N. Pervushin, and D. Proskurin,
Description of supernova datain conformal cosmology without cosmological constant , Phys. Lett.
B 530 (2002) 20.[12] B. M. Barbashov, V. N. Pervushin, A. F. Zakharov, and V. A. Zinchuk,
Hamiltoniancosmological perturbation theory , Phys. Lett.
B 633 (2006) 458. – 11 –
13] E. Fischbach, D. Sudarsky, and A. Szafer et al.,
Reanalysis of the Eoumltvös experiment , Phys.Rev. Lett. , (1986) ; Phys. Rev. Lett. (1986) 1427.[14] E. Fischbach and C. Talmadge, Six years of the fifth force , Nature (1992) 207.[15] C. D. Hoyle, U. Schmidt, B. R. Heckel, E. G. Adelberger, J. H. Gundlach, D. J. Kapner, andH. E. Swanson,
Submillimeter test of the gravitational inverse-square law: A search for "large"extra dimensions , Phys. Rev. Lett. (2001) 1418.[16] Z. Berezhiani, F. Nesti, L. Pilo and N. Rossi, Gravity Modification with Yukawa-type Potential:Dark Matter and Mirror Gravity , arXiv:0902.0144v2 (2010).[17] L. Iorio,
Constraints on the range Λ of Yukawa-like modifications to the Newtonianinverse-square law of gravitation from Solar System planetary motions , JHEP (2007) 041.[18] S. Capozziello, Curvature Quintessence , Int. J. Mod. Phys.
D 11 (2002) 483.[19] S. Capozziello, V. F. Cardone, S. Carloni, and A. Troisi,
Curvature quintessence matched withobservational data , Int. J. Mod. Phys.
D 12 (2003) 1969.[20] S. M. Carroll, V. Duvvuri, M. Trodden, and M. S. Turner,
Is cosmic speed-up due to newgravitational physics? , Phys. Rev.
D 70 (2004) 043528.[21] G. Leon and E. N. Saridakis,
Dynamics of the anisotropic Kantowsky-Sachs geometries in R n gravity , Class. Quantum Grav. (2011) 065008.[22] T. P. Sotiriou and V. Faraoni, f(R) theories of gravity , Rev. Mod. Phys. (2010) 451.[23] S. Capozziello, V. F. Cardone, and A. Troisi, Gravitational lensing in fourth order gravity , Phys. Rev.
D 73 (2006) 104019.[24] S. Capozziello, V. F. Cardone, and A. Troisi,
Low surface brightness galaxy rotation curves inthe low energy limit of R n gravity: no need for dark matter? , Mon. Not. R. Astron. Soc. (2007) 1423.[25] S. Kopeikin, I. Vlasov,
Parametrized post-Newtonian theory of reference frames, multipolarexpansions and equations of motion in the N-body problem , Physics Reports
400 (2004) 209.[26] D. Borka, P. Jovanović, V. Borka Jovanović, and A. F. Zakharov,
Constraints on R n gravityfrom precession of orbits of S2-like stars , Phys. Rev.
D 85 (2012) 124004.[27] C. Frigerio Martins and P. Salucci,
Analysis of rotation curves in the framework of R n gravity , Mon. Not. R. Astron. Soc. (2007) 1103.[28] A. F. Zakharov, A. A. Nucita, F. De Paolis, and G. Ingrosso,
Solar system constraints on R n gravity , Phys. Rev.
D 74 (2006) 107101.[29] A. F. Zakharov, A. A. Nucita, F. De Paolis, and G. Ingrosso,
Apoastron shift constraints ondark matter distribution at the Galactic Center , Phys. Rev.
D 76 (2007) 062001.[30] A. A. Nucita, F. De Paolis, G. Ingrosso, A. Qadir, and A. F. Zakharov,
Sgr A*: A laboratory tomeasure the central black hole and stellar cluster parameters , Publ. Astron. Soc. Pac. (2007) 349.[31] S. Capozziello, A. Stabile, and A. Troisi,
A general solution in the Newtonian limit of f(R) -gravity , Mod. Phys. Lett.
A 24 (2009) 659.[32] L. Iorio,
Galactic orbital motions in the dark matter, modified Newtonian dynamics andmodified gravity scenarios , Mon. Not. R. Astron. Soc. (2010) 2012.[33] V. A. Rubakov and P. G. Tinyakov,
Infrared-modified gravities and massive gravitons , Phys.Usp.
51 (2008) 759.[34] E. Babichev, C. Deffayet, and R. Ziour,
Recovery of general relativity in massive gravity via theVainshtein mechanism, Phys. Rev.
D 82 (2010) 104008. – 12 –
35] J. B. Pitts and W. C. Schieve,
Universally coupled massive gravity , Theor. Math. Phys.
Recovering General Relativity from Massive Gravity , Phys. Rev. Lett.
103 (2009) 201102.[37] C. de Rham, G. Gabadadze, and A. J. Tolley,
Resummation of massive gravity , Phys. Rev.Lett.
106 (2011) 231101.[38] Gong Yun-Gui,
Cosmology in Massive Gravity , Commun. Theor. Phys.
59 (2013) 319.[39] E. Babichev and A. Fabbri,
Instability of black holes in massive gravity , Class. Quantum Grav.
30 (2013) 152001.[40] V. F. Cardone and S. Capozziello,
Systematic biases on galaxy haloes parameters fromYukawa-like gravitational potentials, Mon. Not. R. Astron. Soc. (2011) 1301.[41] M. Sereno and Ph. Jetzer,
Dark matter versus modifications of the gravitational inverse-squarelaw: results from planetary motion in the Solar system, Mon. Not. R. Astron. Soc. (2006)626.[42] C. Talmadge, J.-P. Berthias, R. W. Hellings, and E. M. Standish,
Model-independentconstraints on possible modifications of Newtonian gravity , Phys. Rev. Lett. , No. 10 (1988)1159.[43] R. H. Sanders, Anti-gravity and galaxy rotation curves , Astron. Astrophys. (1984) L21.[44] E. G. Adelberger, J. H. Gundlach, B. R. Heckel, S. Hoedl, and S. Schlamminger,
Torsionbalance experiments: a low-energy frontier of particle physics , Progress in Particle and NuclearPhysics (2009) 102.[45] S. Capozziello, A. Stabile, and A. Troisi, Newtonian limit of f(R) gravity , Phys. Rev.
D 76 (2007) 104019.[46] S. Capozziello, E. de Filippis, and V. Salzano,
Modelling clusters of galaxies by f(R)-gravity , Mon. Not. R. Astron. Soc. (2009) 947.[47] N. R. Napolitano, S. Capozziello, A. J. Romanowsky, M. Capaccioli, and C. Tortora,
TestingYukawa-like potentials from f(R)-gravity in elliptical galaxies , Astrophys. J. (2012) 87.[48] A. M. Ghez, M. Morris, E. E. Becklin, A. Tanner, and T. Kremenek,
The accelerations of starsorbiting the Milky Way’s central black hole , Nature (2000) 349.[49] 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. Milosavljević, and J. Naiman,
Measuringdistance and properties of the Milky Way’s central supermassive black hole with stellar orbits , Astrophys. J. (2008) 1044.[50] 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 SGR A* from very large telescope and Keck data , Astrophys. J. (2009) L114.[51] 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. (2009) 1075.[52] R. Schödel, T. Ott, R. Genzel, et al.,
Closest star seen orbiting the supermassive black hole atthe Centre of the Milky Way , Nature (2002) 694.[53] R. Genzel, F. Eisenhauer, and S. Gillessen,
The Galactic Center massive black hole and nuclearstar cluster , Rev. Mod. Phys. (2010) 3121.[54] S. Gillessen, R. Genzel, T. K. Fritz et al., A gas cloud on its way towards the supermassiveblack hole at the Galactic Centre , Nature (2012) 51. – 13 –
55] L. Meyer, A. M. Ghez, R. Schödel, et al.,
The Shortest-Known-Period Star Orbiting OurGalaxy’s Supermassive Black Hole , Science (2012) 84.[56] G. F. Rubilar and A. Eckart,
Periastron shifts of stellar orbits near the Galactic Center , Astron. Astrophys. (2001) 95.[57] G. S. Adkins and J. McDonnell,
Orbital precession due to central-force perturbations , Phys.Rev.
D 75 (2007) 082001.[58] N. N. Weinberg, M. Milosavljević, and A. M. Ghez,
Stellar dynamics at the Galactic Centerwith an extremely large telescope , Astrophys. J. (2005) 878.[59] T. Fritz, S. Gillessen, S. Trippe, T. Ott, H. Bartko, O. Pfuhl, K. Dodds-Eden, R. Davies, F.Eisenhauer, R. Genzel,
What is limiting near-infrared astrometry in the Galactic Centre? , Mon.Not. R. Astron. Soc. (2010), 1177.[60] D.E. Krause and E. Fischbach,
Searching for extra-dimensions and new string-inspired forces inthe Casimir regime, in Gyros, Clocks, Interferometers. . . : Testing Relativistic Gravity inSpace , C. Lĺammerzahl, C.W.F. Everitt and F.W. Hehl eds., Springer-Verlag, Berlin, Germany(2001) [hep-ph/9912276].[61] O. Bertolami and J. Pťaramos,
Astrophysical constraints on scalar field models , Phys. Rev.
D71 (2005) 023521.[62] O. Bertolami, J. Pťaramos and S. Turyshev,
General theory of relativity: will it survive the nextdecade?, in Lasers, clocks, and drag-free: technologies for future exploration in space and testsof gravity , H. Dittus, C. Lĺammerzahl and S. Turyshev eds., Springer (2006) [gr-qc/0602016].[63] J.W. Moffat,
Gravitational theory, galaxy rotation curves and cosmology without dark matter , JCAP (2005) 22.[64] J.W. Moffat, Scalar-tensor-vector gravity theory , JCAP (2006) 004.[65] M.J. White and C.S. Kochanek, Constraints on the long-range properties of gravity from weakgravitational lensing , Astrophys. J. (2001) 539.[66] L. Amendola and C. Quercellini,
Skewness as a test of the equivalence principle , Phys. Rev.Lett. (2004) 181102.[67] S. Reynaud and M.-T. Jaekel, Testing the newton law at long distances , Int. J. Mod. Phys.
A20 (2005) 2294.[68] C. Sealfon, L. Verde and R. Jimenez,
Limits on deviations from the inverse-square law onmegaparsec scales , Phys. Rev.
D 71 (2005) 083004.[69] A. Shirata, T. Shiromizu, N. Yoshida and Y. Suto,
Constraining deviations from NewtonŠs lawof gravity on cosmological scales: confrontation to power spectrum of SDSS galaxies , Phys. Rev.
D 71 (2005) 064030.[70] R. G. Aitken, "The binary stars" Astronomer in the Lick Observatory, University of California,New York (1918).[71] W. M. Smart,
On the derivation of the elements of a visual binary orbit by Kowalsky’s method , Mon. Not. R. Astron. Soc. (1930) 534.[72] R. H. Sanders, Solar system constraints on multifield theories of modified dynamics , Mon. Not.R. Astron. Soc. (2006) 1519.[73] L. Iorio,
Putting Yukawa-like Modified Gravity (MOG) on the test in the Solar System ,Scholarly Research Exchange, Article ID (2008) 238385; arXiv:0809.3563v4 (2008).[74] M. L. Ruggiero and L. Iorio,
Constraining Post-Newtonian f(R) Gravity in the Solar System ,arXiv:0811.3860v1 (2008). – 14 –
75] J. J. Moré, B. S. Garbow, and K. E. Hillstrom,
User Guide for MINPACK-1 , Argonne NationalLaboratory Report ANL-80-74, Argonne, Ill. (1980).[76] I. Ciufolini,
Frame-dragging, gravitomagnetism and Lunar Laser Ranging , New Astronomy
Post-Newtonian Reference Frames for Advanced Theory of the LunarMotion and for a New Generation of Lunar Laser Ranging , Acta physica slovaca vol. 60 No. 4(2010) 393.[78] I. Ciufolini, A. Paolozzi, E. C. Pavlis, J. Ries, R. Koenig, R. Matzner, G. Sindoni, and H.Neumayer,
Testing gravitational physics with satellite laser ranging , Eur. Phys. J. Plus
A modified gravity and its consequences for the solar system, astrophysics andcosmology , International Journal of Modern Physics D, 16 (12 A) (2007) 2075.[80] S. Gillessen, F. Eisenhauer, G. Perrin, et al.,
GRAVITY spectrometer: metrology laser blockingstrategy at OD=12 , Proc. of SPIE (2010) 77342.[81] An Expanded View of the Universe: Science with the European Extremely Large Telescope: