2017MNRAS.472.2249I

One of the directly measured quantities used in monitoring the orbits 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 DV(t) and the net shifts per revolution <DV> 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 star S2 for which RV measurements accurate to about ≃ 30 - 50 km s^–1 dating back to t₀ = 2003.271 are currently available, and whose orbital period amounts to P_b = 16 yr. Indeed, while its expected general relativistic RV net shift per orbit amounts to just <DV^GE> = -11.6 km s^–1, it should reach a peak value as large as D V_maxGE(t_max) = 551 km s^{-1} at t_max = 2018.35. Current uncertainties in the S2 and SMBH's estimated parameters yield a variation range from 505 km s^–1(2018.79) to 575 km s^–1(2018.45). The periastron shift Do^GE of S2 over the same time span will not be larger than 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 km s^–1, 0.0039 km s^–1, respectively.