  Alternative theories of gravity may serve to overcame several shortcomings of
the standard cosmological model but, in their weak field limit, General
Relativity must be recovered so as to match the tight constraints at the Solar
System scale. Therefore, testing such alternative models at scales of stellar
systems could give a unique opportunity to confirm or rule them out. One of the
most straightforward modifications is represented by analytical $f(R)$-gravity
models that introduce a Yukawa-like modification to the Newtonian potential
thus modifying the dynamics of particles. Using the geodesics equations, we
have illustrated the amplitude of these modifications. First, we have
integrated numerically the equations of motion showing the orbital precession
of a particle around a massive object. Second, we have computed an analytic
expression for the periastron advance of systems having their semi-major axis
much shorter than the Yukawa-scale length. Finally, we have extended our
results to the case of a binary system composed of two massive objects. Our
analysis provides a powerful tool to obtain constraints on the underlying
theory of gravity using current and forthcoming datasets.
