Irradiation grazing angle of a disk ($\alpha \simeq z_s^{\prime }-z_s/R=Rd(z_s/R)/dR$)

For a thin illuminated surface $z=z_s(R)$ and a central point source, the radial light ray has slope $z_s/R$ whereas the surface tangent has slope $z_s^{\prime }$. Their angle difference is the grazing angle. The incident flux is $L_{\ast }\alpha /(4\pi R^2)$ to leading order. This requires a positive grazing angle and an unobstructed ray; a negative formal flux instead signals failure of direct illumination, not negative heating.