Two-Body Orbit Expansion Due toTime-Dependent Relative Acceleration Rate of the Cosmological Scale Factor

By phenomenologically assuming a slow temporal variation of the percent acceleration rate SS-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 SS-1 around the present epoch t(0) , a non-vanishing shift per orbit 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-0 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 P-b approximate to 31 Myr, the general relativistic distance shift per orbit turns out to be of the order of approximate to 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 approximate to 2-4 pc. Our result has a general validity since it holds in any cosmological model admitting the Hubble law and a slowly varying SS-1(t) . More generally, it is valid for an arbitrary Hooke-like extra-acceleration whose elastic parameter K is slowly time-dependent, irrespectively of the physical mechanism which may lead to it. The coefficient K-1 of the first-order term of the power expansion of K(t) can be preliminarily constrained in a model-independent way down to a K-1 less than or similar to 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 K-1 less than or similar to 10(-8) year(-3).