Journal of Physics and Astronomy Research
JPAR
Vol. 2(1), pp. 049-055, January, 2015. © www.premierpublishers.org, ISSN: 2123-503X
Review
Limit radius in a binary system: Cosmological and Post-Newtonian effects Jose J. Arenas Department of Physics, Monterroso Institute, Santo Tomás de Aquino S/N, 29680 Málaga (Spain) Email:
[email protected],
[email protected] Frequently, in dynamical astronomy, the quantitative effect of the large-scale cosmological expansion on local systems is studied in the light of Newtonian approach. We, however, analyze the influence of cosmological expansion on binary systems (galaxies or black holes) in the light of Post-Newtonian approximation. Furthermore, we obtain the new radius at which the acceleration due to the cosmological expansion has the same magnitude as the two-body attraction, and the classical limit radius is obtained when the Schwarzschild radius approaches zero (for example, the Solar System). Keywords: Celestial Mechanics; two-body system; Post-Newtonian approximation; Cosmological Expansion; Pioneer anomaly
1. Introduction In Celestial Mechanics, the problem of the influence of the cosmic expansion on local gravitationally bound system has been analyzed by several authors (McVittie, 1933; Einstein & Strauss, 1945; Anderson, 1995; Bonnor, 1996; Cooperstock et al., 1998; Domínguez & Gaite, 2001; Sereno & Jetzer, 2007; Carrera & Giulini, 2010; Bochicchio & Faraoni, 2012; Arenas, 2012; Iorio, 2014). The most common approach is to study a Newtonian gravitationally bound system, such as a binary stellar system embedded in an expanding Friedmann-Lemaître-Robertson-Walker (FLRW) universe and computes the effect of the cosmological expansion as a perturbation of the local dynamics (!!" ). We consider the dynamical problem of two bodies attracting each other via a force with 1/!! fall-off (cosmological framework). For simplicity we may think of one mass as being much smaller than the other one. The result for a particle of polar coordinates (!, !) and conserved angular momentum per unit mass L in the field of a mass M embedded d in a FLRW universe with scalar factor !(!) is given by the equations of motion (Carrera & Giulini, 2010; Bochicchio & Faraoni, 2012): !=−
!" ! !" + !! ! + !! = − ! + !! ! + !!" , !! ! = !, ! ! ! ! (1.1)
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with G the universal gravitational constant. The global expansion is described by the Hubble law; ! = !", which states that the relative radial velocity of two comoving objects at a mutual distance R grows proportional to that distance. H denotes the Hubble parameter, it is given in terms of the scalar factor a(t), via ! ≡ !/!. So, the acceleration that results from the Hubble law is given by ! = ! = !! + !! =
! ! = −!!! !, !
(1.2)
where !≡−
!! ! = − !!! , ! ! ! (1.3)
is the dimensionless deceleration parameter. Sereno & Jetzer (2007) and Carrera & Giulini (2010) analyzed the restricted two-body problem in an expanding universe; deviations from Keplerian orbits, evolution of the orbital radius, etc. In principle, a time varying !/! causes changes in the semi-major axis and eccentricity of Kepler orbits. Others previously (Souriau, 1975; Mizony & Lachièze-Rey, 2004) considered that on a planet in the Solar System, the relevant time scale of the problem is the period of its orbit around the Sun, and the factor !/! = −!! !!! is treated as constant during an orbit (the relative error is smaller than 10-9). Then, the classical limit radius at which the acceleration due to the cosmological expansion has the same magnitude as the two-body attraction is obtained (!!" ); !" !!! !
= !! !! ! !!" ⟹ !!" =
!
!" !! !! !
~3.59 · 10!" !~379.6!!"!, (1.4)
(!!! ≈ 73.8!!"/!/!"#!with and!!! ≈ −1/2 at the present epoch (Zheng et al., 2014). Hence, for ! > !!" , the effect of the cosmological expansion is the dominant one. Hence, the effect of the large-scale cosmological expansion on local systems is studied in the light of Newtonian approach, and is neglected the General Relativity (usually). Moreover, in a weak-field approximation ( ! /! ! ≲ 10!! , with ! the Newtonian gravitational potential and ! the speed of light), the effect of the general relativity is computed as a perturbation of the Newtonian mechanics of the Solar System (Post-Newtonian approximation). The present study analyzes the influence of cosmological expansion on binary systems in the light of Post-Newtonian approximation.
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2. Equation of motion Many authors have studied the Solar System by the first Post-Newtonian order (1PN for a system of N point masses (Portilla & Tejeiro, 2001, Poisson, 2007, Ramos, 2009) and the effect is calculated as a perturbation of the Newtonian dynamics (!!!" ) in the special case in which the system contains only two bodies, the equation of motion reduces to: !=−
! ! ! + !! ! ! !! =−
! 4 + 2! 3! − !! 3! + 1 + ! ! · ! ! 2!
!
! + 4 − 2! ! · ! ! =
! ! + !!!" , !!
(2.1) where ! =
!" ,! !!! !
= !(! + !)! and R≡mutual distance
If we also assume that ! ≫ ! (test particle model) we obtain: ! → 0, ! = !", with !=−
!" !" 4!" !" !+ ! ! − !! ! + 4 ! · ! ! = − ! ! + !!!" . ! ! ! ! ! ! (2.2)
Using the eq. (1.1) and (2.2) (absolute value of R); !=−
!" + !! ! + !!" + !!!" . !! (2.3)
The eq. (2.3) represents the equation of motion for a particle in the field of a mass M embedded in a FLRW universe with scalar factor a(t) in the light of Post-Newtonian approximation. Furthermore, assuming circular orbits (Cooperstock et al., 1998; Sereno & Jetzer, 2007; Carrera & Giulini, 2010), the eq. (2.3) becomes, !! ! =
!" ! 4! ! !! − ! − . !! ! ! ! !! (2.4)
Equating both perturbations for the Solar System (cosmological perturbation and relativistic perturbation); !! !!! !!"
=
4! ! !! ! ! !!" !
⟺ !!" =
2!" !! !
!!
≈ 4835!!".
(2.5) Limit radius in a binary system: Cosmological and Post-Newtonian effects
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So, at !!" ≈ 4835!!", the cosmological perturbation and the relativistic perturbation are equals. For distances less than !!" (all the planets in the Solar System), the relativistic perturbation is greater than the cosmological perturbation, hence, it should be considered in the influence of the cosmological expansion on local systems. Curiously, there is an approximate coincidence between the value of the anomalous acceleration of Pioneer 10/11 (Rosales & Sánchez-Gómez, 1998; Nieto, 2008; Lämmerzahl, 2009; Toth & Turyshev, 2009; Lämmerzahl & Rievers, 2011; Shojaie, 2012): [8.74 ± 1.33] · 10!!" !!! !! and the product !! ! ≈ 7.16 · 10!!" ms !! (2.5). It seems that this anomaly is due to the anisotropy of thermal radiation from the spacecraft (Turyshev et al., 2012), but Turyshev et al. have not taken into account the geometric effect of the expanding space on the propagation of light on local systems (Kopeikin, 2014) 3. New limit radius Now, using (2.3) and (2.4), the new limit radius (!! ) at which the acceleration due to the cosmological expansion has the same magnitude as the two-body attraction is given by the equation: −
!" ! 4! ! !! + ! + = 0. ! ! !!! ! ! !!! (3.1)
Put another way: −!"!! + !! !!! !!! +
4! ! !! = 0. !!
(3.2) This eq. can be defined in term of parameters of the Schwarzschild radius (!! = 2!"/! ! ) and the ! classical limit radius (!!" = !"/ !! !!! ); ! !!! + 2!! − !! !!" = 0.
(3.3) Furthermore, the general solution (real solution with physical meaning) of the eq. (3.3) is given by (Mathematica): !! =
! !
!+
!!!!" !
−! . (3.4)!
Limit radius in a binary system: Cosmological and Post-Newtonian effects
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where != !=
!
! !
18
8 +
!
! !
! !! !!"
!
,
!" ! ! ! + 9! ! . 3 27!!" − 2048!! !" !"
(3.5) 4. Results The solution (3.4) is valid for local systems (see appendix) up to the first Post-Newtonian order. If !! → 0 (for example, ~3!" in the Solar System), !! = !!" is obtained as a particular case of (3.4). Hence, this method may improve predictions about formation and evolution of galaxies and black holes (very massive systems). On the other hand, (3.3) is especially relevant for 2!! ∼ !!" : !! !∼ ∼ 1.5 · 10!! !!⊙ 4 2!! ! (3.6) Note that the most massive halo in the observable universe today (Alimi, 2012) has a mass of 15 quadrillion solar masses (!⊙ ). 5. Discussion The new limit radius (!! ) at which the acceleration due to the cosmological expansion has the same magnitude as the two-body attraction is given by the general solution of equation of motion, and it is valid for local systems. This result improve the limit radius obtained by classical methods (Souriau, 1975; Mizony & Lachièze-Rey, 2004; Carrera & Giulini, 2010). Moreover, there is an approximate coincidence between the value of this anomalous acceleration and the product !! ! ≈ 7.16 · 10!!" ms !! . That is, the product of the current value of the Hubble constant and the speed of light in vacuum. Note that this product is obtained in (2.5) and it is due to the combination of the cosmological expansion (FLRW universe) and the Post-Newtonian approximation. These Physics frameworks may be related with the anomalous acceleration of spacecraft. 6. Conclusions Frequently, the quantitative effect of the large-scale cosmological expansion on local systems is studied in the light of Newtonian approach, and the General Relativity Theory is neglected. We have obtained the equation of motion for a particle in the field of a mass M embedded in a FLRW universe with scalar factor !(!) in the light of Post-Newtonian approximation (1PN). For distances less than !!" (all the planets in the Solar System), the relativistic perturbation is greater than the cosmological perturbation, hence, it should be considered in the influence of the cosmological expansion on local systems. Hence, the Post-Newtonian approximation should be considered in the influence of the cosmological expansion on local systems, especially for very massive systems. Limit radius in a binary system: Cosmological and Post-Newtonian effects
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Acknowledgments We thank Michel Mizony of the Institute Girard Desargues (University of Lyon I) for his comments on classical limit radius. We thank Claus Lämmerzahl of the Center Applied Space Technology and Microgravity (ZARM) of the University of Bremen for his references on the Pioneer anomaly. We also acknowledge the comments on this article about the deceleration parameter made by Domenico Giulini (Institute of Theoretical Physics, Gottfried Wilhelm Leibniz Universität Hannover). We thank especially to Rosa. Appendix The four solutions are: 1. !! = −
2. !! =
!+ −
! 2!!"
!
−!
1 2!! − −! + − !" − ! 2 !
3. !! = −
4. !! =
1 2
1 2
! 1 2!!" − !+ −! 2 !
!+
! 2!!"
!
−!
Note that if !! → 0 (for example, our Solar System), we can obtain as a particular case: !=
!
! 18!!"
! ! = !!"
Hence, 1. !! = −
1 ! ! + −3!!" 2 !"
1 ! ! − −!!" + −3!!" 2 3. !! = 0 4. !! = !!" 2. !! =
Note that (4) is the only real solution with physical meaning. Conflicts of Interest The authors declare no conflict of interest. Limit radius in a binary system: Cosmological and Post-Newtonian effects
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Accepted 20 November, 2014 Citation: Arenas JJ (2015). Limit radius in a binary system: Cosmological and Post-Newtonian effects. Journal of Physics and Astronomy Research, 2(1): 049-055.
Copyright: © 2015 Arenas JJ. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are cited.
Limit radius in a binary system: Cosmological and Post-Newtonian effects
