Blackbody limit of an isothermal atmosphere ($I_{\lambda }⟶B_{\lambda }(T)$)

For a thermalized radiative transfer source function $S_{\lambda }=B_{\lambda }(T)$ constant along a ray, the specific intensity after optical-depth interval $\Delta \tau$ is $I_{\lambda }=B_{\lambda }+(I_{\mathrm{in}}-B_{\lambda })e^{-\Delta \tau /\mu }$ for $\mu >0$. As $\Delta \tau \to \infty$, it tends to the Planck function, independently of a bounded boundary input. A temperature gradient is unnecessary for blackbody emission. In a scattering medium, local thermodynamic equilibrium alone does not guarantee $S_{\lambda }=B_{\lambda }$; thermalization must be justified.