This manuscript is devoted to the analysis of the local controllability of non-ideal solar sails in planet-centered orbits. Classical approaches fail when considering this control problem because sails cannot generate forces with positive components toward the Sun direction. More precisely, the control set is delimited by a convex cone of revolution with axis toward the Sun and, as such, it is not defined in a neighborhood of the origin, which precludes the use of standard sufficient conditions for controllability. A novel necessary condition is introduced that certifies the noncontrollability of the system for given optical properties of the sail and orbital elements. The condition can be verified by solving a semi-infinite programming problem. Numerical solution is achieved by leveraging on the formalism of squared functional systems. Minimal optical requirements of the sail are inferred for all combinations of orbital elements. These results are independent of the planetary constant.