**ABSTRACT**

The role of f(R) gravity, as well as the other modifications of standard Einstein’s gravity, is to explain the accelerated expansion, structure formation of the Universe, and some other phenomena at extragalactic scales (such as e.g. flat rotation curves of spiral galaxies) without adding unknown forms of dark energy or dark matter. In f(R) model, the Ricci scalar in the Einstein-Hilbert action is replaced by a general function of it. Its power-law form R^n, here is analyzed using observed orbits of S-stars around Galactic Center and also their computer simulations. We review the various consequences of the R^n gravity parameters on stellar dynamics and investigate their constraints from the observed S-star orbits. The presented results show that these observations could put reliable constraints on the parameters of R^n gravity. Also, other aim of this investigation is to show the connection between the half-light radius, central velocity dispersion, and mean surface brightness of elliptical galaxies and the parameters of R^n gravity potential. We also want to show that R^n gravity fit the observations of fundamental plane of elliptical galaxies very well, without adding unknown forms of dark energy or dark matter.