Post-Keplerian effects on radial velocity in binary systems and the possibility of measuring General Relativity with the S2 star in 2018

One of the directly measured quantities which are used in monitoring the orbital motions of many of the S stars revolving around the Supermassive Black Hole (SMBH) in the Galactic Center (GC) is their radial velocity (RV) $V$ obtained with near-infrared spectroscopy. Here, we devise a general approach to calculate both the instantaneous variations $\mathrm{\Delta }V\left(t\right)$ and the net shifts per revolution $⟨\mathrm{\Delta }V⟩$ induced on such an observable by some post-Keplerian (pK) accelerations. In particular, we look at the general relativistic Schwarzschild (gravitoelectric) and Lense-Thirring (gravitomagnetic frame-dragging) effects, and the mass quadrupole. It turns out that we may be on the verge of measuring the Schwarzschild-type 1pN static component of the SMBH’s field with the S2 star for which RV measurements accurate to about $\simeq 30-50{\text{km s}}^{-1}$ dating back to $t_0 = 2003.271$arecurrentlyavailable,andwhoseorbitalperiodamountsto$P_$\text{b}$ = 16$ yr. Indeed, while its expected general relativistic RV net shift per orbit amounts to just $\left\langle\Delta V^$\text{GE}$\right\rangle = -11.6~$\text{km s}$^{-1}$,itshouldreachapeakvalueaslargeas$\Delta V_$\text{max}$^$\text{GE}$\left(t_$\text{max}$\right) = 551~$\text{km s}$^{-1}$at$t_$\text{max}$ = 2018.35$\mathrm{.}Theperiastronshift$\Delta\omega^$\text{GE}$$ofS2overthesametimespanwillnotbelargerthan$0.2$ deg, while the current accuracy in estimating such an orbital element for this star is of the order of $0.6$ deg. The frame-dragging and quadrupole-induced RV shifts are far smaller for S2, amounting to, at most, $0.19~\textrm{km s}^{-1},0.0039~$\text{km s}$^{-1}$, respectively. Further studies should be dedicated to the impact on the RV of possible diffused mass distribution in the GC and of other individual stars inside and outside the orbit of S2.