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$^{-3}$.