Perturbative solutions of the $f(R)$-theory of gravity in a central
  gravitational field and some applications
  Exact solutions of an $ f(R) $-theory (of gravity) in a static central
(gravitational) field have been studied in the literature quite well, but, to
find and study exact solutions in the case of a non-static central field are
not easy at all. There are, however, approximation methods of finding a
solution in a central field which is not necessarily static. It is shown in
this article that an approximate solution of an $f(R)$-theory in a general
central field, which is not necessary to be static, can be found perturbatively
around a solution of the Einstein equation in the general theory of relativity.
In particular, vacuum solutions are found for $f(R)$ of general and some
special forms. Further, applications to the investigation of a planetary motion
and light's propagation in a central field are presented. An effect of an
$f(R)$-gravity is also estimated for the SgrA*--S2 system. The latter
gravitational system is much stronger than the Sun--Mercury system, thus the
effect could be much stronger and, thus, much more measurable.
