On the Newtonian and Spin-induced Perturbations felt by the Stars Orbiting around the Massive Black Hole in the Galactic Center

The S-stars discovered in the Galactic center (GC) are expected to provide unique dynamical tests of the Kerr metric of the massive black hole (MBH) orbited by them. In order to obtain unbiased measurements of its spin and the related relativistic effects, a comprehensive understanding of the gravitational perturbations of the stars and stellar remnants around the MBH is quite essential. Here, we study the perturbations on the observables of a typical target star, i.e., the apparent orbital motion and the redshift, due to both the spin-induced relativistic effects and the Newtonian attractions of a single or a cluster of disturbing object(s). We find that, in most cases, the Newtonian perturbations on the observables are mainly attributed to the perturbed orbital period of the target star, rather than the Newtonian orbital precessions. The Newtonian perturbations have their unique features when they peak around the pericenter passage in each revolution, which is quite different from those of the spin-induced effects. Looking at the currently detected star S2/S0-2, we find that its spin-induced effects on both the image position and redshift are very likely obscured by the gravitational perturbations from the star S0-102 alone. We also investigate and discuss the Newtonian perturbations on a hypothetical S-star located inside the orbits of the currently detected ones. By considering a number of possible stellar distributions near the central MBH, we find that the spin-induced effects on the apparent position and the redshift dominate over the stellar perturbations for target stars with orbital semimajor axis smaller than $100-400$AU if the MBH is maximally spinning. Our results suggest that, in principle, the stellar perturbations can be removed as they have distinctive morphologies comparing to those of the relativistic Kerr-type signatures.