 Solution

For the linear Planck function profile, direct integration givesin the optically thick limit. A sudden doubling of absorber abundance doubles optical depth at every physical height while initially leaving the temperature profile fixed. With $\tilde{\tau }=2\tau$, the same profile is $B=B_0+(\alpha /2)\tilde{\tau }$, not $B_0+\alpha \tilde{\tau }$. ThusIf temperature increases downward, $\alpha >0$ and outgoing spectral irradiance decreases immediately because the effective emitting region moves upward into cooler gas. Subsequent thermal adjustment is outside this instantaneous comparison.