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In this paper we constrain the sizes of hypothetical new weak forces bymaking use the data coming from the precession of Planets. We consider the weakfield approximation of Scalar-Tensor Fourth Order Gravity (STFOG), whichinclude several models of modified gravity. The form of the corrections to theNewtonian potential if of the form of Yukawa-like potential (5th force), i.e.$V(r) = {\alpha} \dfrac{e^{-\beta r}}{r}$, where ${\alpha}$ is the parameterrelated to the strength of the potential, and ${\beta}$ to the range of theforce. The present data on periastron advance allow to infer a constraint onthe free parameter of the gravitational models. Moreover, the Non-CommutativeSpectral Gravity (NCSG) is also studied, being a particular case STFOG. Here weshow that the precession shift of Planet allows to improve the bounds onparameter ${\beta}$ by several orders of magnitude. Finally such an analysis isstudied to the case of power-like potential, referring in particular todeformation of the Schwarzschild geometry induced by a quintessence field,responsible of the present accelerated phase of the Universe.

Precession shift in curvature based Extended Theories of Gravity and Quintessence fields