On the post-Keplerian corrections to the orbital periods of a two-body system and their application to the Galactic Center

Detailed numerical analyses of the orbital motion of a test particle around a spinning primary are performed. They aim to investigate the possibility of using the post-Keplerian (pK) corrections to the orbiter's periods (draconitic, anomalistic and sidereal) as a further opportunity to perform new tests of post-Newtonian (pN) gravity. As a specific scenario, the S-stars orbiting the Massive Black Hole (MBH) supposedly lurking in Sgr A$^\ast$ at the center of the Galaxy is adopted. We, first, study the effects of the pK Schwarzchild, Lense-Thirring and quadrupole moment accelerations experienced by a target star for various possible initial orbital configurations. It turns out that the results of the numerical simulations are consistent with the analytical ones in the small eccentricity approximation for which almost all the latter ones were derived. For highly elliptical orbits, the size of all the three pK corrections considered turn out to increase remarkably. The periods of the observed S2 and S0-102 stars as functions of the MBH's spin axis orientation are considered as well. The pK accelerations considered lead to corrections of the orbital periods of the order of 1-100d (Schwarzschild), 0.1-10h (Lense-Thirring) and 1-10^3s (quadrupole) for a target star with a=300-800~AU and e ~ 0.8, which could be possibly measurable by the future facilities.