Past exam of the short options paper shared by Parts A and B of the physics course of the University of Oxford / 2019 / S25 / 1 / d / Solution

Use a convective layer $T(z)=T_s-\Gamma z$ capped at an isothermal minimum $T_t$ at height $z_t$. Let the initial optical depth be $\tau (z)={\tau }_s{e}^{-z/H_{\tau }}$ and ${\tau }_t={\tau }_s{e}^{-z_t/H_{\tau }}$. Within the lapse-rate layer, $d{B}_{\lambda }/d\tau =B^{\prime }\Gamma H_{\tau }/\tau$; above it the derivative vanishes. The supplied Schwarzschild equation therefore givesAssume the additional carbon dioxide multiplies the relevant opacity by $a=1.1$, leaving temperatures, the lapse rate and $H_{\tau }$ initially fixed. All material surfaces then acquire $a$ times their previous optical depth. HenceFor ${\tau }_t\ll 1$ and ${\tau }_s\gg 10$, the specified approximation givesFor illustration, $\Gamma =6.5\mathrm{K}\mathrm{k}{\mathrm{m}}^{-1}$ and $H_{\tau }=7\mathrm{km}$ give $-4.34\mathrm{K}$; these are illustrative, not supplied data. The inequalities put the effective emitting level $\tau \sim 1$ inside the decreasing-temperature layer, well above an opaque surface and below the thin isothermal cap. Increasing opacity raises that emitting level by $H_{\tau }\ln 1.1$, so it is colder. In a spectral window or above a temperature inversion the approximation and possibly the sign change. Opacity proportional to mixing ratio is also an assumption, not exact for every absorption mechanism.