We consider general metric $f(R)$ theories of gravity by solving the field equations in the presence of a spherical static mass distribution by analytical perturbative means. Expanding the field equations systematically in $cO(G)$, we solve the resul
ting set of equations and show that $f(R)$ theories which attempt to solve the dark energy problem very generally lead to $gamma_{PPN}=1/2$ in the solar system. This excludes a large class of theories as possible explanations of dark energy. We also present the first order correction to $gamma_{PPN}$ and show that it cannot have a significant effect.
We study stellar configurations and the space-time around them in metric $f(R)$ theories of gravity. In particular, we focus on the polytropic model of the Sun in the $f(R)=R-mu^4/R$ model. We show how the stellar configuration in the $f(R)$ theory c
an, by appropriate initial conditions, be selected to be equal to that described by the Lane-Emden -equation and how a simple scaling relation exists between the solutions. We also derive the correct solution analytically near the center of the star in $f(R)$ theory. Previous analytical and numerical results are confirmed, indicating that the space-time around the Sun is incompatible with Solar System constraints on the properties of gravity. Numerical work shows that stellar configurations, with a regular metric at the center, lead to $gamma_{PPN}simeq1/2$ outside the star ie. the Schwarzschild-de Sitter -space-time is not the correct vacuum solution for such configurations. Conversely, by selecting the Schwarzschild-de Sitter -metric as the outside solution, we find that the stellar configuration is unchanged but the metric is irregular at the center. The possibility of constructing a $f(R)$ theory compatible with the Solar System experiments and possible new constraints arising from the radius-mass -relation of stellar objects is discussed.
