255 Acta Polytechnica CTU Proceedings 1(1): 255–258, 2014 255 doi: 10.14311/APP.2014.01.0255 Probing Gravitational Theories with Eccentric Eclipsing Detached Binary Stars Leopoldo Milano1,2, Rosario De Rosa1,2, Mariafelicia De Laurentis1,2, Fabio Garufi1,2 1Dipartimento di Fisica Universita’ Federico II di Napoli Complesso Universitario di Monte S. Angelo, Via Cinthia, - I 80125, Napoli, Italy 2Sezione INFN di Napoli,Complesso Universitario di Monte S. Angelo, Via - I 80125, Napoli, Italy Corresponding author: milano@na.infn.it Abstract In this paper, we compare the effects of different theories of gravitation on the apsidal motion of eccentric eclipsing detached binary stars. The comparison is performed by using the formalism of the post-Newtonian parametrization to calculate the theoretical advance at periastron and compare it to the observed one, after having considered the effects of the structure and rotation of the involved stars. A variance analysis on the results of this comparison shows that no significant difference can be found due to the effect of the different theories under test with respect to the standard general relativity (GR). It will be possible to observe differences, as we would expect, by checking the observed period variation on a much larger lapse of time. Keywords: relativity - gravitation - eclipsing binaries - apsidal motion. 1 Introduction The problem of the motion of two bodies under their mutual gravitational attraction and the study of binary stellar systems has always been the ideal test bed for the theories of gravitation. Several authors in the last decades dedicated a lot of work in analyzing, both on the theoretical and the experimental point of wiew, the phenomenon of the periastron precession in binary sys- tems to test various gravitational theories [1, 2] as well as to find correction to the Newtonian and General Rel- ativistic behaviour of the systems due to stellar form factors, spin, tides and other phenomena [3]. The classical effect of General Relativity (GR) on the apsidal motion rate at periastron is well known since long time and described by Levi-Civita in a famous pa- per in 1937 [4, 5]. Another possible formulation of the problem, that allows also to test other gravitational the- ories besides GR, is the use of Parametrized Post New- tonian (PPN) formalism [6, 7]. Using this formalism, the different gravitational theories can be compared side by side on the basis of a set of Post-Newtonian (PN) parameters: the masses, the system major semiaxis and the eccentricity. Thus, using a sample of Eccentric Eclipsing Detached Binary (EEDB) systems, for which masses and orbital parameters are known with sufficient precision, it is possible to compare the apsidal motion rate at periastron ω̇T h, as expected in the different the- ories with the observations in order to verify whether the observations can select one theory or another. The choice of this class of stellar objects to test the theories is dictated by the circumstance that the both orbital and the structural parameters can be precisely determined by observation of the eclypses (the only alea being the orbit plane observation angle), and that in close stellar orbits, the gravitational field is supposed to be strong, thus enhancing eventual effects of devia- tions from the classical theory. There are so many aspects of binary star evolution and angular momentum exchange (see, e.g., [8]) that any attempt to dig out from these other effects the sub- tleties of GR could seem futile, nonetheless, we con- sider useful to ascertain whether from the observation of many binary systems, some statistical information about the prevalence of one or another theory of grav- itation can be extracted and to ground the bases of a method that can be used when other more significant data will be available. 2 Advance at Periastron The idea of considering relativistic gravitational tests in terms of a metric expansion is originally based on a work by Shiff [9] who expanded the single body met- ric in terms of the ratio between the geometrized mass mg = Gm/c 2 and the distance r: g00 = 1 − 2α mg r + 2β (mg r )2 255 http://dx.doi.org/10.14311/APP.2014.01.0255 Leopoldo Milano et al. g0k = 0 gik = − ( 1 + 2γ mg r ) δik i,k = 1, 2, 3 (1) Four new parameters α′, α′′, α′′′, ∆ were then intro- duced to account for relative velocities and accelera- tions. For General Relativity all the parameters are equal to 1 and the advance at periastron can be veri- fied to reproduce the ”classical” formula by Levi Civita [4]. We conveniently modified the ’classical’ formula by introducing a factor (KT h) to take into account the de- pendance on the theory dependent PN terms in order to test the different relativistic theories. [10]: ω̇Rel = 1.8167 × 10−4KT h ( M P ) 2 3 c2(1 −e2) (2) where: KT h = αT h(8∆T h −αT h + 2α ′′ T h −α ′′′ T h −γT h −α ′ T h) 2 (3) The PN parameters, are calculated for the different gravitational theories , i.e. the General Relativity (’classical’ term), the Nordvedt, the Brans-Dicke the- ories and the so called f(R) theories that take into account higher order terms of the Ricci scalar R, giv- ing a general expression for the relativistic term ω̇Rel, that contributes to the advance at periastron. Us- ing for the different theories the appropriate values of αT h,α ′ T h,α ′′ T h,α ′′′ T h, ∆T h and γT h the numerical values of KT h can be obtained [10]: KT h =   KGR = 3 → (General−Relativity) KBD = 19 7 → (Brans−Dicke) KND = 11 4 → (Nordvedt) Kf(R) = 13 4 → (f(R)) (4) 3 Comparison with Experimental Data To test the effects of deviations from the GR, we choose to study binary stellar systems with small radius orbits, so that the gravitational field is strong enough to evi- dence these deviations, if any. Among the various bi- nary stars catalogues available in literature, we choose a sample of Eccentric Eclipsing Detached Binary stars such that the period, the eccentricity, the masses of the components, and, possibly, the observed internal struc- ture function are known with a good precision as e.g. [11]. For these systems the passage at periastron precedes in a way that is precisely predictable from the gravita- tional theory, once given the stellar parameters such as masses, radius of the components, and orbital parame- ters. To compare the global rate of theoretical apsidal motion in a binary system with the measured one we must take into account the individual contributions of each component due to tidal and rotational distortions, and the general relativistic term ω̇Th, where the index Th indicates the theory under test (e.g. ωGR for Gen- eral Relativity). Assuming that rotation of both com- ponents of an eclipsing binary system is perpendicular to the orbital plane, the apsidal motion rate, ω̇ is given by the following simple relation [12]: ω̇Obs = ω̇cl + ω̇Rel (5) Where ω̇cl is the classical Newtonian term and ω̇Rel is the relativistic contribution of Eq. 2 and Eq. 4. The dependance of ω̇cl on the binary system parameters is expressed through the Internal Second-order Constants (ISC): k̄2T h = c21k21T h + c22k22T h c21 + c22 (6) k̄2Obs = ω̇cl c21 + c22 (7) where the parameters c2i are related to the masses and the orbital eccentricity of the binary system. It must be noticed that the individual ISC’s k2,i cannot be ob- tained from the observations although they can be in- terpolated from evolutionary codes like those used in [13, 14]. So we can evaluate a mean model dependent k̄2T h and a mean observation dependent k̄2Obs, and compare them to test the evolution stellar models from the ob- servations of apsidal motion. Taking into account that k̄2Obs is generally smaller than k̄2T h (this means that the evolution models predict stellar cores less dense than those found by observed data), remembering Eq. 2 and Eq. 5 we see that ω̇cl will vary according to ω̇T h. In this way we can test the different relativistic theo- ries of gravitation by verifying whether the agreement of the model dependent mean ISC, with those coming from observations is significantly improved by using the different ω̇T h relativistic terms. In Fig. 1 we show the trend of the apsidal mo- tion rate ω̇GR vs ω̇GR,BD,ND,f(R). It results: ω̇BD ∼= 0.92ω̇GR, ω̇ND ∼= 0.90ω̇GR, ω̇f(R) ∼= 1.10ω̇GR. Obvi- ously the numerical coefficients are the ratios KT h KGR (see Eq. 4). It is interesting to note that the f(R) theory gives a relativistic contribution that is slightly higher than GR,BD and ND. Moreover, GR is ≈ the mean between f(R) and BD and ND theories. It is also ev- ident that there is no significant difference among the theories under test within the errors. In Fig. 2 we show the trend of observed ω̇Obs vs ω̇Rel for different rela- tivistic terms (Th ≡ GR,BD,ND,f(R)). The red line is the trend of ω̇Rel = ω̇Obs. It is evident that, apart 256 Probing Gravitational Theories with Eccentric Eclipsing Detached Binary Stars from a few systems, the relativistic term is always less than the observed one. 0 0.5 1 1.5 2 2.5 3 3.5 x 10 −3 0 0.5 1 1.5 2 2.5 3 3.5 x 10 −3 dω/dtGR d ω /d t G R ,B D ,N D ,f (R ) dω/dt GR dω/dt BD dω/dt ND dω/dt f(R) Figure 1: ω̇GR vs ω̇GR,BD,ND,f(R) for the different relativistic theories (GR,BD,ND,f(R)): ω̇GR = ω̇GR, ω̇BD ∼= 0.92ω̇GR, ω̇ND ∼= 0.90ω̇GR, ω̇f(R) ∼= 1.10ω̇GR Fig. 3 shows the ISC’s comparison: Log(k̄2ObsT h ) vs Log(k̄2StellarModel) are shown for different rel- ativistic terms. The blue line is the trend of Log(k̄2StellarModel) = Log(k̄2ObsT h ). It is evident that, apart from a few systems, Log(k̄2StellarModel) is always greater than the observed one. So the stellar core den- sities derived from the observations is higher than those coming from stellar model prevision. −0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 0 0.5 1 1.5 2 2.5 3 3.5 x 10 −3 dω Obs /dt d ω re l/d t dω Rel /dt=dω Obs /dt Figure 2: Observed ω̇Obs vs ω̇Rel for different relativis- tic terms (Th ≡ GR,BD,ND,f(R)). The red line is the trend of ω̇Rel = ω̇Obs. It is evident that apart from some systems the relativistic term is always less than the observed one. 4 Discussion and Conclusion Using data coming from apsidal motion rate of EEDB we compared the variation of the relativistic term of the apsidal motion rate due to different theories of gravi- tation, that accordingly produces variation of classical Newtonian term (see Eq. 5). The results of this com- parison was that we could not find any significant differ- ence due to the effect of the different theories under test with respect to the standard General Relativity. Since the advance at periastron accumulates, the trend and the amount of this motion can be better determined by the observation of more orbits (or a larger fraction of orbit). Longer observations also improve the deter- mination of ISC’s; thus, it would be possible, perhaps, to observe more significant differences, by checking the period variation on a much larger lapse of time and verifying the assumptions of syncronous orbital and ro- tation motion of the binary star components. A lot of observing work are producing new and more accurate data and step by step it is getting a better agreement between theory and observations. −4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −2.6 −2.4 −2.2 −2 −1.8 −1.6 log k2Obs Th lo g k 2 S te ll ar M od el HP Aur BM Mon HH Car V1765 Cyg δ Ori ι Ori log k2 Obs Th =log k2 Stellar Model Figure 3: Internal second order structure constants (ISC) Log(k̄2ObsT h ) vs Log(k̄2StellarModel) for differ- ent relativistic terms (Th ≡ GR,BD,ND,f(R)). The blue line is the trend of Log(k̄2StellarModel) = Log(k̄2ObsT h ). It is evident that apart from some sys- tems Log(k̄2StellarModel) is always greater than the ob- served one. So the stellar core according to the obser- vations is more dense than the stellar model prevision. Acknowledgement This work was supported by INFN grant 2011.The au- thors gratefully acknowledge the referee Prof. Todor Stanev for his useful suggestions. References [1] Breen B., 1973, J. Phys. A: Math., Nucl. Gen., 6, 150. 257 Leopoldo Milano et al. [2] L. Lin-Sen, 2010, Astrophys. Space Sci., 327, 59. doi:10.1007/s10509-010-0267-4 [3] Giménez A., Claret A., 2010, Astron. and Astroph. 519, A57. [4] Levi Civita T., 1937, Amer. J. of Math., 59, 225. doi:10.2307/2371404 [5] Giménez A., 1985, Astroph. J., 297, 405. doi:10.1086/163539 [6] Thorne K.S., Will C.M., 1971, Astrophys. J., 163, 595. doi:10.1086/150803 [7] Nordtvedt K., 1969, Phys. Rev., 180, 1293. doi:10.1103/PhysRev.180.1293 [8] Biermann, P., et al. 1985, Astrophys. J., 293, 303 [9] Schiff L.I. , 1967, Relativity Theory and Astro- physics vol 1, ed J Ehlers 1967 (Philadelphia: American Mathematical Society) [10] M. De Laurentis,R. De Rosa,F. Garufi & L. Mi- lano,2012,MNRAS, 424, 2371-2379 [11] Bulut I., Demircan O., 2007, Mon. Not. R. Astron. Soc. 378, 179. [12] Kopal Z., Dynamics of Close Binary Systems. Rei- del,Dordrecht (1978) [13] Claret A., Giménez A., 1992, Astron. and Astroph. Suppl. 96, 255. [14] Claret, A. 2004, A&A, 424, 919 doi:10.1111/j.1365-2966.2007.11756.x DISCUSSION JIM BEALL: What systems would give a proper test? LEOPOLDO MILANO: The systems that give a proper test are about eleven. CARLOTTA PITTORI: Do you think that it could be useful to study in more datail the outliers? Which kind of obsevations could help to constrain the theory? LEOPOLDO MILANO: Many group of astronomers are doing new observations on the outliers with the aim of improving the knowledge on the rotations velocities and other parameters that can be the cause of the fail- ure of the simple model that is generally adopted. 258 http://dx.doi.org/10.1007/s10509-010-0267-4 http://dx.doi.org/10.2307/2371404 http://dx.doi.org/10.1086/163539 http://dx.doi.org/10.1086/150803 http://dx.doi.org/10.1103/PhysRev.180.1293 http://dx.doi.org/10.1111/j.1365-2966.2007.11756.x Introduction Advance at Periastron Comparison with Experimental Data Discussion and Conclusion