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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 (CC BY) 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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