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).

1. Limit radius in a binary system: Cosmological and Post-Newtonian effects
JPAR
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: jjarenasfisica@iesmonterroso.org, arenasferrer@hotmail.com
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
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 and Strauss, 1945; Anderson, 1995; Bonnor, 1996;
Cooperstock et al., 1998; Domínguez and Gaite, 2001; Sereno and Jetzer, 2007; Carrera and Giulini, 2010;
Bochicchio and 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 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 in a FLRW universe with scalar factor is given by the equations of motion (Carrera and Giulini, 2010;
BochicchioandFaraoni, 2012):
(1.1)
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)
Journal of Physics and Astronomy Research
Vol. 2(1), pp. 049-053, March, 2015. © www.premierpublishers.org, ISSN: 2123-503X
Review

2. Limit radius in a binary system: Cosmological and Post-Newtonian effects
Arenas JJ. 049
where
(1.3)
is the dimensionless deceleration parameter.
Sereno and Jetzer (2007) and Carrera and 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 and
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 ( );
(1.4)
( with 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 ( ,
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.
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 and 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:
(2.1)
where and R mutual distance
If we also assume that (test particle model) we obtain: , with
(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 and Jetzer, 2007; Carrera and Giulini,
2010), the eq. (2.3) becomes,
2.4)

3. Limit radius in a binary system: Cosmological and Post-Newtonian effects
J. Phys. Astron. Res. 050
Equating both perturbations for the Solar System (cosmological perturbation and relativistic perturbation);
(2.5)
So, at , 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 and Sánchez-Gómez, 1998; Nieto, 2008; Lämmerzahl, 2009; Toth and Turyshev, 2009; Lämmerzahl and
Rievers, 2011; Shojaie, 2012):
and the product (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)
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:
(3.1)
Put another way:
(3.2)
This eq. can be defined in term of parameters of the Schwarzschild radius and the classical limit
radius ( );
(3.3)
Furthermore, the general solution (real solution with physical meaning) of the eq. (3.3) is given by (Mathematica):
(3.4)
where
(3.5)
RESULTS
The solution (3.4) is valid for local systems (see appendix) up to the first Post-Newtonian order.
If (for example, 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 :

4. Limit radius in a binary system: Cosmological and Post-Newtonian effects
Arenas JJ. 051
(3.6)
Note that the most massive halo in the observable universe today (Alimi, 2012) has a mass of 15 quadrillion solar
masses ( ).
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 improves the limit radius obtained by classical methods (Souriau, 1975; Mizony and Lachièze-Rey, 2004;
Carrera and Giulini, 2010).
Moreover, there is an approximate coincidence between the value of this anomalous acceleration and the product
. 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.
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 forvery massive systems.
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:

5. Limit radius in a binary system: Cosmological and Post-Newtonian effects
J. Phys. Astron. Res. 052
Note that if (for example, our Solar System), we can obtain as a particular case:
Hence,
Note that (4) is the only real solution with physical meaning.
CONFLICTS OF INTEREST
The authors declare no conflict of interest.
REFERENCES
Alimi JM (2012). DEUS Full Observable ΛCDM Universe Simulation: the numerical challenge, (arXiv: 1206.2838)
Anderson JL (1995). Multiparticle Dynamics in an Expanding Universe. Phys. Rev. Lett. 75, 3602
Arenas JJ (2012). Estabilidad y Caos en el Sistema Solar, Master Thesis. UNED/University of Granada.
Bochicchio I, Faraoni V (2012). Cosmological expansion and local systems: a Lemaître–Tolman–. Bondi model.
General Relativity and Gravitation, 44(6): 1479-1487
Bonnor WB (1996). The cosmic expansion and local dynamics. MNRAS, 282(4): 1467-1469
Carrera M, Giulini D (2010). Influence of global cosmological expansion on local dynamics and kinematics. Rev.
Mod. Phys., 82: 169-208
Cooperstock FI, Faraoni V, Vollick N (1998). The influence of the cosmological expansión on local system,
Astrophysical Journal, 503: 61-66.
Domínguez A, Gaite J (2001). Influence of the cosmological expansion on small system, Europhysics Letters, 55: 4
Einstein A, Strauss EG (1945). The Influence of the Expansion of Space on the Gravitation Fields Surrounding the
Individual Stars. Rev. Mod. Phys., 17: 120
Iorio L (2014). The Lingering Anomalous Secular Increase of the Eccentricity of the Orbit of the Moon: Further
Attempts of Explanations of Cosmological Origin. Galaxies, 2: 259-262
Iorio L (2014). Two-Body Orbit Expansion Due to Time-Dependent Relative Acceleration Rate of the Cosmological
Scale Factor. Galaxies, 2: 13-21
Kopeikin SM (2014). Local gravitational physics of the Hubble expansion, (arXiv: 1407.6667)
Lämmerzahl C, Rievers B (2011). High precision thermal modeling of complex systems with application to the flyby
and Pioneer anomaly. Ann. Phys, 523: 439-449
Lämmerzahl C (2009) The Pioneer Anomaly and the Solar System in the Expanding Universe, presentation at the
workshop “Intertwining Theory and Observational Evidence in Contemporary Cosmology”. Wuppertal University,
Germany.
McVittie GC (1933). The mass-particle in an expanding universe. MNRAS, 93: 325
Mizony M, Lachièze-Rey M (2004). Cosmological effects in the local static frame. (arXiv: 0412084)
Nieto MM (2008). New Horizons and the onset of the Pioneer anomaly. Phys. Lett. B, 659: 483
Poisson E (2007). Post-Newtonian theory for the common reader, Lecture Notes. University of Guelph.
Portilla JG, Tejeiro J (2001). The bitales or post-Newtonian elements of the analytical solution of Damour and
Deruelle for the relativistic problem of the two Army Nurse Corps. Rev. Acad. Colomb. Cienc., 256: 537-55
Ramos JF (2009). Study of the effect of the post-Newtonian corrections to the evolution of systems of many particles
autogravitantes, Doctoral Thesis. Industrial University of Santander. Faculty of Sciences. School of Physics.
Rosales JL, Sánchez-Gómez JL (1998). The “Pioneer effect” as a manifestation of the cosmic expansion in the Solar
System (arXiv: 9810085)
Sereno M, Jetzer P (2007). Evolution of gravitational orbits in the expanding universe. Physical Review D, 75(6):
064031

6. Limit radius in a binary system: Cosmological and Post-Newtonian effects
Arenas JJ. 053
Shojaie H (2012). Pioneer Anomaly in Perturbed FRW Metric, (arXiv: 1201.3889)
Souriau JM, (1975). Géométrieet Thermodynamique en cosmologie, in “Géométriesymplectique et Physique
mathématique” CNRS, Paris.
Toth VT, Turyshev SG (2009). The Pioneer Anomaly in the Light of New Data. Phys. Rev. D, 79: 043011
Turyshev SG, Toth VT, Kinsella G, Lee SC, Lok SM, Ellis J (2012). Support for the thermal origin of the Pioneer
anomaly. Physical Review Letters, 108(24), 241101
Zheng W, Li H, Xia JQ, Wan YP, Li SY, Li M (2014). Constraints on cosmological models from Hubble parameters
measurements. International Journal of Modern Physics D, 23 (05).
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): 049-053.
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.