Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 314 / 1 / d / Solution

Let the equilibrium radius be $a_0=b_0=c_0$ and $2T_0={\Omega }^2{a}_0^2$. Linearizing the $a$ equation giveswith cyclic analogues. The temperature scaling gives

For the affine breathing mode of a star, all three fractional axis changes equal $q$. ThenThis is the homologous compressional mode, which changes volume, density, and temperature.

For either independent affine quadrupole mode of a star, the three fractional changes sum to zero. Then $\delta T=0$ andThese two degenerate modes deform the sphere into an ellipsoid while preserving its volume to first order.