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Alternative theories of gravity may serve to overcame several shortcomings ofthe standard cosmological model but, in their weak field limit, GeneralRelativity must be recovered so as to match the tight constraints at the SolarSystem scale. Therefore, testing such alternative models at scales of stellarsystems could give a unique opportunity to confirm or rule them out. One of themost straightforward modifications is represented by analytical $f(R)$-gravitymodels that introduce a Yukawa-like modification to the Newtonian potentialthus modifying the dynamics of particles. Using the geodesics equations, wehave illustrated the amplitude of these modifications. First, we haveintegrated numerically the equations of motion showing the orbital precessionof a particle around a massive object. Second, we have computed an analyticexpression for the periastron advance of systems having their semi-major axismuch shorter than the Yukawa-scale length. Finally, we have extended ourresults to the case of a binary system composed of two massive objects. Ouranalysis provides a powerful tool to obtain constraints on the underlyingtheory of gravity using current and forthcoming datasets.

Analysis of the Yukawa gravitational potential in $f(R)$ gravity II: relativistic periastron advance