Constraining the Yukawa Gravity with Post Newtonian Approximation using S-star Orbits around the Supermassive Black Hole in our Galactic Center

A number of modified gravity theories (e.g., $f(R)$-gravity) lead to a Yukawa-like metric in the weak field limit which can be described by two Yukawa parameters, i.e., the strength $\kappa$ and the length scale $\lambda$. The S-stars, orbiting around the supermassive black hole in the Galactic Center, are unique probes to test these gravity theories in relatively strong gravitational field. The Newtonian Yukawa gravity potential or a simple approximation to the Yukawa metric was usually adopted in previous studies when using the orbital motion of S-stars to constrain such modified gravity theories, which may be not sufficiently accurate considering recent and future high resolution observations. In this paper, we first derive the Post-Newtonian (PN) Yukawa motion equation at the 2PN order, and then investigate the high order effects on the orbital motions by comparison with those from the Newtonian Yukawa gravity potential. We further obtain constraints on $\kappa$ by using the observations on the orbital motions of several S-stars (i.e., S2, S38, and S55). Our results show that the current observations of these stars are compatible with the General Relativity and $\kappa$ can be constrained to $|\kappa|<0.01$ with $95\%$ confidence if $\lambda\in(100,250)$\,AU. We also estimate the possible improvements (about an order of magnitude or more) to the constraints by future higher resolution observations and the inclusion of closer S-stars, such as S4716.