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

TL;DR

This paper analytically examines how various long-range modified gravity models, including the cosmological constant, affect the radial velocity shifts in binary systems, providing explicit calculations and assessing observational detectability.

Contribution

It introduces a general analytical framework to compute radial velocity shifts caused by different modified gravity models in binary systems.

Findings

The cosmological constant causes negligible velocity shifts compared to measurement accuracy.

Explicit formulas for velocity shifts are derived for models with potentials proportional to r^{-N} and r^2.

The method can be extended to other modified gravity theories.

Abstract

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 $\Delta V$ 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 $\Lambda$ 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 $\left|\Delta V\right|\lesssim 10^{-7}\,\mathrm{m\,s}^{-1}$, while the present-day accuracy in measuring its radial velocity is $\sigma_V\simeq 30\,\mathrm{m\,s}^{-1}$. The calculational scheme presented here is quite general, and can be straightforwardly extended to any other modified gravity.