Past exam of the short options paper shared by Prelims and Parts A and B of the physics course of the University of Oxford / 2012 / S25 / 1 / 3

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.

For the quadratic Planck function profile,The optically thick limit and the instantaneous doubled-abundance result are thereforeThe new coefficient is $\beta /4$, since $B(\tilde{\tau })=B_0+\beta (\tilde{\tau }/2{)}^2$ at fixed physical temperature. For positive $\beta$, the outgoing spectral irradiance decreases. The factor differs from the linear case because the temperature-dependent source function curves with depth; doubling abundance does not mean holding its old functional coefficients fixed in the new optical-depth coordinate.