Show that for a plane-parallel gray atmosphere in LTE, the (constant) value of the radiative flux is equal to π times the source function evaluated at an optical depth of 2/3:
Frad=πS(τv=2/3).
This function, called the Eddington–Barbier relation, says that the radiative flux received from the surface of the star is determined by the value of the source function at τv=2/3.