Radiative transfer equation

In plane-parallel optical-depth coordinates without scattering,with source function $S_{\nu }$.

In local thermodynamic equilibrium, matter populations are thermal at the local temperature and the source function is the Planck function, $S_{\nu }=B_{\nu }(T)$.

For an outward ray through a nonscattering atmosphere,

When the source function varies approximately linearly with optical depth, the emergent intensity obeys $I_{\nu }(0,\mu )\simeq S_{\nu }({\tau }_{\nu }=\mu )$.