A general analytic procedure is developed to deal with the Newtonian limit of $f(R)$ gravity. A discussion comparing the Newtonian and the post-Newtonian limit of these models is proposed in order to point out the differences between the two approaches. We calculate the post-Newtonian parameters of such theories without any redefinition of the degrees of freedom, in particular, without adopting some scalar fields and without any change from Jordan to Einstein frame. Considering the Taylor expansion of a generic $f(R)$ theory, it is possible to obtain general solutions in terms of the metric coefficients up to the third order of approximation. In particular, the solution relative to the $g_{tt}$ component gives a gravitational potential always corrected with respect to the Newtonian one of the linear theory $f(R)=R$. Furthermore, we show that the Birkhoff theorem is not a general result for $f(R)$ gravity since time-dependent evolution for spherically symmetric solutions can be achieved depending on the order of perturbations. Finally, we discuss the post-Minkowskian limit and the emergence of massive gravitational wave solutions.

We investigate the hydrostatic equilibrium of stellar structure by taking into account the modified Lan\'e-Emden equation coming out from $f(R)$ gravity. Such an equation is obtained in a metric approach by considering the Newtonian limit of $f(R)$ gravity, which gives rise to a modified Poisson equation, and then introducing a relation between pressure and density with polytropic index $n$. The modified equation results an integro-differential equation, which, in the limit $f(R)$\rightarrow${}R$, becomes the standard Lan\'e-Emden equation. We find the radial profiles of the gravitational potential by solving for some values of $n$. The comparison of solutions with those coming from general relativity shows that they are compatible and physically relevant.

We investigate the possibility of explaining theoretically the galaxy rotation curves by a gravitational potential in total absence of dark matter. To this aim an analytic fourth-order theory of gravity, nonminimally coupled with a massive scalar field, is considered. Specifically, the interaction term is given by an analytic function $f(R,$\phi${})$, where $R$ is the Ricci scalar and $$\phi${}$ is the scalar field. The gravitational potential is generated by a pointlike source and compared with the so-called Sanders's potential that can be exactly reproduced in this case. This result means that the problem of dark matter in spiral galaxies could be fully addressed by revising general relativity at galactic scales and requiring further gravitational degrees of freedom instead of new material components that have not been found up to now.