Passive disk

A passive astrophysical disk receives most of its heat from external irradiation rather than internal viscous dissipation. For a stellar disk, a rough criterion is $GM\dot{M}/R_{\ast }\ll L_{\ast }$. The actual local requirement compares absorbed stellar flux with viscous flux; incidence angle, shadowing and absorption can invalidate a luminosity comparison alone.

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.