The Orbital Structure and Selection Effects of the Galactic Center S-Star Cluster

The orbital distribution of the S-star cluster surrounding the supermassive black hole in the center of the Milky Way is analyzed. A tight, roughly exponential dependence of the pericenter distance r$_{p}$ on orbital eccentricity e$_{\star}$ is found, $\log ($r$_p)\sim$(1-e$_{\star}$), which cannot be explained simply by a random distribution of semi-major axes and eccentricities. No stars are found in the region with high e$_{\star}$ and large log r$_{p}$ or in the region with low e$_{\star}$ and small log r$_{p}$. G-clouds follow the same correlation. The likelihood P(log r$_p$,(1-e$_{\star}$)) to determine the orbital parameters of S-stars is determined. P is very small for stars with large e$_{\star}$ and large log r$_{p}$. S-stars might exist in this region. To determine their orbital parameters, one however needs observations over a longer time period. On the other hand, if stars would exist in the region of low log r$_{p}$ and small e$_{\star}$, their orbital parameters should by now have been determined. That this region is unpopulated therefore indicates that no S-stars exist with these orbital characteristics, providing constraints for their formation. We call this region, defined by $\log$ (r$_p$/AU) $<$ 1.57+2.6(1-e$_{\star})$, the zone of avoidance. Finally, it is shown that the observed frequency of eccentricities and pericenter distances is consistent with a random sampling of log r$_{p}$ and e$_{\star}$. However, only if one takes into account that no stars exist in the zone of avoidance and that orbital parameters cannot yet be determined for stars with large r$_{p}$ and large e$_{\star}$.