  We study some possible observational signatures of $R^n$ gravity at Galactic
scales and how these signatures could be used for constraining this type of
$f(R)$ gravity. For that purpose, we performed two-body simulations in $R^n$
gravity potential and analyzed the obtained trajectories of S2-like stars
around Galactic center, as well as resulting parameter space of $R^n$ gravity
potential. Here, we discuss the constraints on the $R^n$ gravity which can be
obtained from the observations of orbits of S2-like stars with the present and
next generations of large telescopes. We make comparison between the
theoretical results and observations. Our results show that the most probable
value for the parameter $r_c$ in $R^n$ gravity potential in the case of S2-like
stars is $\sim$100 AU, while the universal parameter $\beta$ is close to 0.01.
Also, the $R^n$ gravity potential induces the precession of S2-like stars orbit
in opposite direction with respect to General Relativity, therefore, such a
behavior of orbits qualitatively is similar to a behavior of Newtonian orbits
with a bulk distribution of matter (including a stellar cluster and dark matter
distributions).
