Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 317 / 2 / ii / Solution

The Eddington closure approximation with the standard surface boundary condition gives the grey atmosphere profileTogether with the specified pressure law, logarithmic differentiation givesConsequently the radiative temperature gradient isFor a monatomic perfect gas, $\gamma =5/3$ and the adiabatic temperature gradient is $(\gamma -1)/\gamma =2/5$. Uniform composition makes the Schwarzschild criterion applicable. The first marginal layer therefore satisfies $\log (1+3{\tau }_c/2)=4/5$, giving the convection onset in a logarithmic-pressure grey atmosphereThe assumed profile becomes convectively unstable for $\tau >{\tau }_c$; further inward, a self-consistent model must account for convective transport.