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

Consider an element displaced upward by $\xi$ in hydrostatic equilibrium. Assume it quickly reaches the ambient pressure, but moves adiabatically and retains its composition. Put $\nabla =d\log T/d\log P$, ${\nabla }_{\mu }=d\log \mu /d\log P$, and let ${\nabla }_{\mathrm{ad}}$ be the adiabatic temperature gradient. The equation of state has differentialwhere $\delta =-(\partial \log \rho /\partial \log T{)}_{P,\mu }$ and $\varphi =(\partial \log \rho /\partial \log \mu {)}_{P,T}$.

At the new pressure, the parcel-to-environment density difference isWith positive pressure scale height $H_P=-(d\log P/dr{)}^{-1}$, $\Delta \log P=-\xi /H_P$. The buoyancy acceleration is therefore $\ddot{\xi }=-N^2\xi$, with stellar buoyancy frequencyA negative $N^2$ amplifies the displacement. For a radiative background, $\nabla ={\nabla }_{\mathrm{rad}}$, so the Ledoux criterion for instability isFor uniform composition, ${\nabla }_{\mu }=0$, giving the Schwarzschild criterionEquality is marginal stability. A mean molecular weight increasing inward has ${\nabla }_{\mu }>0$ and supplies a stabilizing composition term when $\varphi ,\delta >0$.

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.