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.
