The effect of the cosmological expansion on local systems: Post-Newtonian approximation

Abstract: 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, however, analyze the influence of cosmological expansion on small systems in the light of Post-Newtonian approximation. The equations show the product Hc=6.99*10^-10 ms^-2, and it so happens that the magnitude of this acceleration is very close the apparently anomalous acceleration of the Pioneer 10 and 11 spacecraft. 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 critical radius is obtained when the Schwarzschild radius approaches zero (for example, the Solar System).