= Blackbody limit of an isothermal atmosphere
{title2=$I_\lambda\longrightarrow B_\lambda(T)$}
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 $I_\lambda=B_\lambda+(I_{\rm in}-B_\lambda)e^{-\Delta\tau/\mu}$ 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.
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