The effects of the spin and quadrupole moment of SgrA* on the orbits of S stars

Context. Measuring the astrometric and spectroscopic data of stars orbiting the central black hole in our galaxy (Sgr A*) offers a promising way to detect relativistic effects. In principle, the “no-hair” theorem can be tested at the Galactic Center by monitoring the orbital precession of S stars caused by the angular momentum (spin) and quadrupole moment of Sgr A*. Closer-in stars, which tend to be more strongly affected by the black hole’s rotation, might be required for such studies. While such stars are currently too faint for GRAVITY, it would be possible to detect them with GRAVITY+. Aims. We aim to analytically and numerically characterize orbital reorientations induced by spin-related effects of Sgr A* up to the second post-Newtonian (2PN) order. Methods. To study the interaction between the orbital and spin orientations, we introduced observer-independent quantities that offer insight into the Kerr geometry. We also used the PN code OOGRE to simulate hypothetical stars orbiting closer to Sgr A*, where the spin and quadrupole effects are stronger. This enabled comparison with our analytical predictions. Results. We exhibit three orbital-timescale precession rates that encode the in-plane pericenter shift and the out-of-plane redirection of the osculating ellipse. We provide the 2PN expressions of these precession rates and express the orbit-integrated associated angular shifts of the pericenter and of the ellipse axes. We relate these orbital-timescale precession rates to the secular-timescale precession of the orbital angular momentum around the black hole spin axis. We consider the theoretical insight provided in this article to be useful in constraining the spin effect of Sgr A* with GRAVITY+ observations.