Effect of some modified models of gravity on the radial velocity of binary systems

For many classes of astronomical and astrophysical binary systems, long observational records of their radial velocity $V$, which is their directly observable quantity, are available. For exoplanets close to their parent stars, they cover several full orbital revolutions, while for wide binaries like, e.g., the Proxima/$\alpha$ Centauri AB system, only relatively short orbital arcs are sampled by existing radial velocity measurements. Here, the changes $\mathrm{\Delta }V$ induced on a binary’s radial velocity by some long-range modified models of gravity are analytically calculated. In particular, extra-potentials proportional to $r^{-N},N=2,3$ and $r^2$ are considered; the Cosmological Constant $\mathrm{\Lambda }$ belongs to the latter group. Both the net shift per orbit and the instantaneous one are explicitly calculated for each model. The Cosmological Constant induces a shift in the radial velocity of the Proxima/$\alpha$ Centauri AB binary as little as $\mid \mathrm{\Delta }V\mid \lesssim {10}^{-7}{\mathrm{m}\mathrm{s}}^{-1}$, while the present-day accuracy in measuring its radial velocity is ${\sigma }_V\simeq 30{\mathrm{m}\mathrm{s}}^{-1}$. The calculational scheme presented here is quite general, and can be straightforwardly extended to any other modified gravity.