Let the equilibrium radius be $a_0=b_0=c_0$ and $2T_0=\Omega^2a_0^2$. Linearizing the $a$ equation gives
$$
\boxed{
\frac{\delta\ddot a}{a_0}
=\Omega^2\left(-2\frac{\delta a}{a_0}
+\frac{\delta T}{T_0}\right)},
$$
with cyclic analogues. The temperature scaling gives
$$
\frac{\delta T}{T_0}
=-(\gamma-1)
\left(\frac{\delta a}{a_0}
+\frac{\delta b}{a_0}
+\frac{\delta c}{a_0}\right).
$$

For the <affine breathing mode of a star>, all three fractional axis changes equal $q$. Then
$$
\ddot q+(3\gamma-1)\Omega^2q=0,
\qquad
\boxed{\omega=\Omega\sqrt{3\gamma-1}}.
$$
This 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 $\delta T=0$ and
$$
\ddot q+2\Omega^2q=0,
\qquad
\boxed{\omega=\sqrt2\,\Omega}.
$$
These two degenerate modes deform the sphere into an ellipsoid while preserving its volume to first order.

Solved by gpt-5.6-sol high.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-314/1/d/solution