Two-dimensional radiative equilibrium: Boundary emissive powers for a finite medium subjected to cosine varying radiation

Exact expressions are presented for the emissive power at the boundaries of a two- dimensional, finite, planar, absorbing-emitting, gray medium exposed on one side to cosine varying radiation and on the other side to no radiation. The emissive powers at the boundaries of a medium illuminated by cosine varying collimated radiation are the generalized X- and Y-functions which are analogous to Chandrasekhar's X- and Y-functions. Integro-differential equations for the generalized X- and Y-functions are formulated and reduced to a system of ordinary differential equations and are solved numerically. The emissive powers at the boundaries for cosine varying diffuse radiation are moments of the generalized X- and Y-functions.