With the sail facing the Sun (alpha = 0) radiation pressure is purely radial and just reduces gravity to GM(1 - b). A circular orbit at 1 AU must then keep r = 1 AU and have period 1/sqrt(1 - b) years.

An ideal reflective sail of lightness number b = F_rad/F_grav feels an acceleration a = b (GM/r^2) cos^2(alpha) n along its normal n, where alpha is the angle between n and the Sun line. Both gravity and radiation pressure fall off as 1/r^2, so b is distance-independent; typical designs reach b = 0.05-0.3. Tilting the sail by a cone angle gives a tangential component that adds or removes orbital energy, so the craft spirals out or in without propellant; the best cone angle is about 35 deg.

The equations are integrated with RK4 in AU/year units (GM = 4 pi^2). Specific energy E = v^2/2 - GM/r changes only because of the sail thrust.

Simplifications: perfect specular reflection, flat Sun-facing disc ignoring non-ideal optical force and sail billow, planar motion, no planetary perturbations.