Constraining Extended Gravity Models by S2 star orbits around the
  Galactic Centre
  We investigate the possibility to explain theoretically the observed
deviations of S2 star orbit around the Galactic Centre using gravitational
potentials derived from modified gravity models in absence of dark matter. To
this aim, an analytic fourth-order theory of gravity, non-minimally coupled
with a massive scalar field is considered. Specifically, the interaction term
is given by analytic functions $f(R)$ and $f(R,\phi)$ where $R$ is the Ricci
scalar and $\phi$ is a scalar field whose meaning can be related to further
gravitational degrees of freedom. We simulate the orbit of S2 star around the
Galactic Centre in $f(R)$ (Yukawa-like) and $f(R,\phi)$ (Sanders-like) gravity
potentials and compare it with NTT/VLT observations. Our simulations result in
strong constraints on the range of gravity interaction. In the case of analytic
functions $f(R)$, we are not able to obtain reliable constraints on the
derivative constants $f_1$ and $f_2$, because the current observations of S2
star indicated that they may be highly mutually correlated. In the case of
analytic functions $f(R,\phi)$, we are able to obtain reliable constraints on
the derivative constants $f_0$, $f_R$, $f_{RR}$, $f_{\phi}$, $f_{\phi\phi}$ and
$f_{\phi R}$. The approach we are proposing seems to be sufficiently reliable
to constrain the modified gravity models from stellar orbits around Galactic
Centre.
