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# Two-Body Orbit Expansion Due to Time-Dependent Relative Acceleration Rate of the Cosmological Scale Factor

Italian Ministry of Education, University and Research (M.I.U.R.), Viale Unità di Italia 68, Bari (BA) 70125, Italy

Received: 10 November 2013 / Accepted: 27 December 2013 / Published: 3 January 2014

(This article belongs to the Special Issue Beyond Standard Gravity and Cosmology)

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# Abstract

By phenomenologically assuming a slow temporal variation of the percent acceleration rate S̈S^{-1}of the cosmic scale factor

*S*(

*t*), it is shown that the orbit of a local binary undergoes a secular expansion. To first order in the power expansion of S̈S

^{-1 }around the present epoch t

_{0}, a non-vanishing shift per orbit (Δr) of the two-body relative distance

*r*occurs for eccentric trajectories. A general relativistic expression, which turns out to be cubic in the Hubble parameter

*H*at the present epoch, is explicitly calculated for it in the case of matter-dominated epochs with Dark Energy. For a highly eccentric Oort comet orbit with period

_{0}*P*≈ 31 Myr, the general relativistic distance shift per orbit turns out to be of the order of (Δr) ≈ 70 km. For the Large Magellanic Cloud, assumed on a bound elliptic orbit around the Milky Way, the shift per orbit is of the order of (Δr) ≈ 2–4 pc. Our result has a general validity since it holds in any cosmological model admitting the Hubble law and a slowly varying S̈S

_{b}^{-1}

*(t)*. More generally, it is valid for an arbitrary Hooke-like extra-acceleration whose “elastic” parameter κ is slowly time-dependent, irrespectively of the physical mechanism which may lead to it. The coefficient κ

_{1}of the first-order term of the power expansion of κ

*(t)*can be preliminarily constrained in a model-independent way down to a κ

_{1}≲ 2 x 10

^{-13}year

^{-3 }level from latest Solar System’s planetary observations. The radial velocities of the double lined spectroscopic binary ALPHA Cen AB yield κ1 ≲ 10

^{-8 }year

^{-3}.

*Keywords:*classical general relativity; cosmology

*This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.*

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**MDPI and ACS Style**

Iorio, L. Two-Body Orbit Expansion Due to Time-Dependent Relative Acceleration Rate of the Cosmological Scale Factor. *Galaxies* **2014**, *2*, 13-21.

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