Long-term classical and general relativistic effects on the radial velocities of the stars orbiting Sgr A*

We analytically work out the cumulative, i.e. averaged over one orbital revolution, time variations v of the radial velocity v of a typical S star orbiting the supermassive (M 10 6 M ) black hole (SBH) hosted by the Galactic Centre (GC) in Sgr A * caused by several dynamical effects. They are the general relativistic gravitoelectromagnetic (GEM) fields of the SBH, its quadrupole mass moment Q 2 and a diffuse dark matter distribution around the SBH. All of them induce non-zero long-term radial accelerations proportional to the eccentricity e of the orbit. By taking the S2 star, orbiting the SBH along a highly eccentric (e = 0.8831) ellipse with a period P b = 15.9 yr and semimajor axis a = 1031.69 au, we numerically compute the magnitudes of its radial accelerations. The largest effects are due to the general relativistic Schwarzschild-like gravitoelectric (GE) field, with v(GE) = 8.2 10 -5 m s -2 , and the diffuse material distribution, modelled with a Plummer-type mass density profile, with v(dm) = 3.8 10 -6 m s -2 . The effects caused by the general relativistic Kerr-type gravitomagnetic (GM) field and by Q 2 are smaller by orders of magnitude. By assuming an uncertainty in measuring the radial velocities of about 15 km s -1 , the future accuracy in measuring v can be evaluated to be of the order of 2.4 10 -5 m s -2 over an observational time-span t = 20yr. Currently, the available radial velocity measurements cover just 7 yr.