By phenomenologically assuming a slow temporal variation of the percent acceleration rate \ddot { S } S ^ { -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 \ddot { S } S ^ { -1 } around the present epoch t _ { 0 } , a non-vanishing shift per orbit \left \langle \Delta r \right \rangle 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 } \approx 31 Myr , the general relativistic distance shift per orbit turns out to be of the order of \left \langle \Delta r \right \rangle \approx 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 \left \langle \Delta r \right \rangle \approx 2 - 4 pc .
Our result has a general validity since it holds in any cosmological model admitting the Hubble law and a slowly varying \ddot { S } S ^ { -1 } ( t ) .
More generally , it is valid for an arbitrary Hooke-like extra-acceleration whose “ elastic ” parameter \mathcal { K } is slowly time-dependent , irrespectively of the physical mechanism which may lead to it .
The coefficient \mathcal { K } _ { 1 } of the first-order term of the power expansion of \mathcal { K } ( t ) can be preliminarily constrained in a model-independent way down to a \mathcal { K } _ { 1 } \lesssim 2 \times 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 \mathcal { K } _ { 1 } \lesssim 10 ^ { -8 } year ^ { -3 } .