Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 317 / 4

LetA displaced fluid element remains in pressure balance, changes temperature adiabatically, and retains its composition. Comparing its density with the environment after an upward displacement gives the Ledoux criterion for stability,The reverse inequality causes convection. For uniform composition, or if composition is ignored, ${\nabla }_{\mu }=0$ and this reduces to the Schwarzschild criterionfor stability. A molecular weight increasing inward has ${\nabla }_{\mu }>0$ and stabilizes the stratification.

Write $\mathcal{R}=k_B/(\mu m_u)$. At fixed density,The specific internal energy isApplying the first law of thermodynamics adiabatically givesCombining these logarithmic derivatives yieldsAs gas pressure dominates,the result for a monatomic perfect gas. In the radiation-pressure limit it tends to $1/4$.

Uniform composition and adiabatic convection give $T\propto {\rho }^{2/3}$ for a monatomic perfect gas. Hence near the centreThe luminosity equation isSince ${\rho }^2{T}^{\eta }\propto 1-(2+2\eta /3)\lambda r^2$, integration givesSimilarly,

The radiative temperature gradient isExpanding $\kappa ={\kappa }_0{\rho }^a{T}^{-b}$ and all the preceding factors giveswhereThe Schwarzschild criterion requiresfor the centre to be convectively unstable. If $A>0$, the radiative gradient decreases outward and can cross $2/5$, producing a finite convective core.