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

Abstract

We analytically work out the cumulative, i.e. averaged over one orbital revolution, time variations of the radial velocity vρ of 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 Pb= 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 , and the diffuse material distribution, modelled with a Plummer-type mass density profile, with 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-1, the future accuracy in measuring can be evaluated to be of the order of 2.4 × 10-5 m s-2 over an observational time-span Δt= 20 yr. Currently, the available radial velocity measurements cover just 7 yr. © 2010 The Authors Monthly Notices of the Royal Astronomical Society © 2010 RAS.