A stellar fly-by close to the Galactic centre: Can we detect stars on highly-relativistic orbits?

The Galactic centre Nuclear Star Cluster is one of the densest stellar clusters in the Galaxy. The stars in its inner portions orbit the supermassive black hole associated with compact radio source Sgr~A* at the orbital speeds of several thousand km/s. The B-type star S2 is currently the best case to test the general relativity as well as other theories of gravity based on its stellar orbit. Yet its orbital period of $\sim 16\,{\rm yr}$ and the eccentricity of $\sim 0.88$ yields the relativistic pericentre shift of $\sim 11'$, which is observationally still difficult to reliably measure due to possible Newtonian perturbations as well as reference-frame uncertainties. A naive way to solve this problem is to find stars with smaller pericentre distances, $r_{\rm p}\lesssim 1529$ Schwarzschild radii ($120\,{\rm AU}$), and thus more prominent relativistic effects. In this contribution, we show that to detect stars on relativistic orbits is progressively less likely given the volume shrinkage and the expected stellar density distributions. Finally, one arrives to a sparse region, where the total number of bright stars is expected to fall below one. One can, however, still potentially detect stars crossing this region. In this contribution, we provide a simple formula for the detection probability of a star crossing a sparse region. We also examine an approximate time-scale on which the star reappears in the sparse region, i.e. a "waiting" time-scale for observers.