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

Spectral irradiance $F_{\lambda }$ is radiant power crossing unit area per unit wavelength, integrated over the relevant hemisphere with the projected-area factor: $F_{\lambda }=\int I_{\lambda }\cos \theta d\Omega$. Its units are $\mathrm{W}{\mathrm{m}}^{-2}$ per unit wavelength; spectral radiance $I_{\lambda }$ additionally refers to unit solid angle. For an isotropic outgoing hemisphere with Planck function $B_{\lambda }$, the integral is $F_{\lambda }=\pi B_{\lambda }$.

The vertical optical depth above height $z$ is the dimensionless extinction integralHere $z$ increases upward, $n_a$ is absorber number density, ${\sigma }_{\lambda }$ its absorption cross section, ${\rho }_a$ its mass density, and ${\kappa }_{\lambda }$ the absorption coefficient per unit absorber mass. Equivalently, using bulk air density $\rho$ and absorber mass fraction $q_a$, $d{\tau }_{\lambda }/dz=-{\kappa }_{\lambda }{q}_a\rho$. The coefficient may depend on temperature, pressure and wavelength. For purely absorbing gas, transmission along a ray is $e^{-{\tau }_{\mathrm{ray}}}$; scattering, if present, contributes to extinction and requires its own source term.

Use the optical-depth weighting stipulated in the paper. A jump in the Planck function at ${\tau }_1$ gives $dB/d\tau =(B_1-B_0)\delta (\tau -{\tau }_1)$, soThe upper isothermal layer emits $\pi B_0(1-e^{-{\tau }_1})$ and transmits the lower region's blackbody spectral irradiance with factor $e^{-{\tau }_1}$. The lower layer and black surface have the same $B_1$, so their combined radiation entering the upper layer is $\pi B_1$, independent of how its emission is divided between the lower gas and surface.

More generally, integration by parts rewrites the supplied expression asThe terms are attenuated surface emission and the atmosphere's own emission. In the original boundary-plus-gradient form, $\pi B_0$ is the isothermal reference flux and the integral corrects for temperature variation below the top. Optical depths of order unity contribute most strongly. These formulas follow the paper's specified exponential weighting; a full angular radiative transfer calculation uses ray-angle-dependent optical paths rather than identifying every ray with a vertical path.

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.