Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 317 / 1 / Solution

A stellar polytrope of index $n$ obeysCombining hydrostatic equilibrium, $dP/dr=-Gm\rho /r^2$, with mass conservation, $dm/dr=4\pi r^2\rho$, gives the Lane-Emden equationRegularity and normalization at the stellar centre requireIntegrating the Lane-Emden equation from the centre then gives the enclosed mass

For $n=5$, direct substitution into the equation findsso $\beta =1/3$ and $\gamma =-1/2$. This profile has no finite first zero and hence has infinite radius, but its total mass converges:Its mean density over the full, infinite configuration is consequently zero.

For a perfect gas, $T\propto P/\rho \propto {\rho }^{1/5}$, so $T/T_c=\theta$. The luminosity from CNO cycle burning with $ϵ={ϵ}_0\rho T^{11}$ is thereforewhereThus $A$ is of order unity. The model is physically poor because the $n=5$ polytrope has infinite radius and zero mean density, while strongly temperature-sensitive CNO burning changes the thermal gradient and commonly creates convection. Real cores also have evolving composition, non-polytropic opacity and energy transport, and boundaries supplied by the surrounding star.

Solved by gpt-5.6-sol high.