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 - 100 ~ { } \textrm { d } ( Schwarzschild ) , 0.1 - 10 ~ { } \textrm { h } ( Lense-Thirring ) and 1 - 10 ^ { 3 } ~ { } \textrm { s } ( quadrupole ) for a target star with a = 300 - 800 ~ { } \textrm { AU } and e \approx 0.8 , which could be possibly measurable by the future facilities .