2011MNRAS.411..453I

We analytically work out the cumulative, i.e. averaged over one orbital revolution, time variations ({dot}v_ρ) of the radial velocity v_ρof a typical S star orbiting the supermassive (M_• ≈ 10⁶ 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 Q₂ 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 {dot}v^(GE)_ρ = 8.2x10^–5m/s2, and the diffuse material distribution, modelled with a Plummer-type mass density profile, with {dot}v^(dm)_ρ = 3.8x10^–6m/s2. The effects caused by the general relativistic Kerr-type gravitomagnetic (GM) field and by Q₂ are smaller by orders of magnitude. By assuming an uncertainty in measuring the radial velocities of about 15 km/s, the future accuracy in measuring {img} can be evaluated to be of the order of 2.4 x 10^–5 m/s2 over an observational time-span Δt= 20 yr. Currently, the available radial velocity measurements cover just 7 yr.