Trajectory design and station keeping of solar sails about a celestial body can be formulated as control problems with positivity constraints. Specifically, when re-emitted radiation is neglected and the sail is modeled as a flat surface, which are reasonable assumptions for control purposes, force generated by solar radiation pressure is contained in a pointed convex cone of revolution with axis towards the Sun-satellite direction. Therefore, classical approaches to infer controllability based on the Lie algebra rank condition do not apply to these problems. This study offers a novel condition to decide on controllability of control systems with positivity constraints. This condition is effective as it can be verified by solving an auxiliary convex optimization problem for which reliable numerical methods are available. A crucial ingredient of this approach is the theory of positive trigonometric polynomials. The practical interest of this condition is the assessment of a minimum requirement on the optical properties of the sail, which may be of use for mission design purposes.