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

The additional acceleration is
$$
-\nabla\Psi
=2\Psi_0(x,y,-2z)\cos(\Omega_ot).
$$
Its three fractional-axis forcing terms are therefore proportional to $(1,1,-2)$ and have zero sum. Consequently it has no projection on the <affine breathing mode of a star>, which is not forced at linear order.

It lies entirely in the <affine quadrupole mode of a star> subspace. Put
$$
\frac{\delta a}{a_0}
=\frac{\delta b}{a_0}=q,
\qquad
\frac{\delta c}{a_0}=-2q.
$$
Then
$$
\ddot q+2\Omega^2q=2\Psi_0\cos(\Omega_ot),
$$
so away from resonance
$$
\boxed{q(t)=\frac{2\Psi_0}{2\Omega^2-\Omega_o^2}
\cos(\Omega_ot)}
$$
up to free oscillations. The tidal forcing resonates with the quadrupole mode when $\Omega_o=\sqrt2\Omega$; in the ideal undamped model the resonant amplitude grows secularly.

Solved by gpt-5.6-sol high.

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

An affine quadrupole mode has axis perturbations whose fractional sum vanishes. It changes the ellipsoidal shape without changing volume to first order.


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 Incoming links: Affine quadrupole mode of a star