Two-body orbit expansion due to time-dependent relative acceleration
  rate of the cosmological scale factor
  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 $P_{\rm b}\approx 31$ 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 \virg{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
$\mathcal{K}_1\lesssim 2\times 10^{-13}$ yr$^{-3}$ level from latest Solar
System's planetary observations. The radial velocities of the double lined
spectroscopic binary $\alpha$ Cen AB yield $\mathcal{K}_1\lesssim 10^{-8}$
yr$^{-3}$.
