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After giving a short overview of previous results on constraining of Extended Gravity (EG) by stellar orbits, we discuss the Schwarzschild orbital precession of S2 star assuming the congruence with predictions of General Relativity (GR). At the moment, the S2 star trajectory is remarkably fitted with the first post-Newtonian (pN) approximation of GR. In particular, both Keck and VLT (GRAVITY) teams declared that the gravitational redshift near its pericenter passage for the S2 star orbit corresponds to theoretical estimates found with the first pN approximation. In 2020, the GRAVITY Collaboration detected the orbital precession of the S2 star around the supermassive black hole (SMBH) at the Galactic Center (GC) and showed that it is close to the GR prediction. Based on this observational fact, we evaluated parameters of the EG theories with the Schwarzschild precession of the S2 star. Using the mentioned method, we estimate the orbital precession angles for some EG models including power-law f(R), general Yukawa-like corrections, scalar-tensor gravity (STG), and non-local gravity (NLG) theories formulated in both metric and Palatini formalism. In this consideration we assume that a gravitational field is spherically symmetric, therefore, alternative theories of gravity could be described only with a few parameters. Specifically, considering the orbital precession, we estimate the range of parameters of these EG models for which the orbital precession is like in GR. We then compare these results with our previous results, which were obtained by fitting the simulated orbits of S2 star to its observed astrometric positions. In case of power-law f(R), generic Yukawa-like correction, STG and NLG theories, we were able to obtain a prograde orbital precession, like in GR. According to these results, the method is a useful tool to evaluate parameters of the gravitational potential at the GC., Comment: 19 pages, 7 figures. Accepted for publication in Universe

The global properties of elliptical galaxies are connected through the so-called fundamental plane of ellipticals, which is an empirical relation between their parameters: effective radius, central velocity dispersion and mean surface brightness within the effective radius. We investigated the relation between the parameters of the fundamental plane equation and the parameters of modified gravity potential $f(R)$. With that aim, we compared theoretical predictions for circular velocity in $f(R)$ gravity with the corresponding values from a large sample of observed elliptical galaxies. Besides, we consistently reproduced the values of coefficients of the fundamental plane equation as deduced from observations, showing that the photometric quantities like mean surface brightness are related to gravitational parameters. We show that this type of modified gravity, especially its power-law version - $R^n$, is able to reproduce the stellar dynamics in elliptical galaxies. Also, it is shown that $R^n$ gravity fits the observations very well, without need for a dark matter., Comment: 9 pages, 2 figures. Accepted for publication in Atoms