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

2 T_{0} = Ω^{2} a_{0}^{2}

. Linearizing the

a

\frac{δ a ¨}{a _{0}} = Ω^{2} (- 2 \frac{δ a}{a _{0}} + \frac{δ T}{T _{0}}),

with cyclic analogues. The temperature scaling gives

\frac{δ T}{T _{0}} = - (γ - 1) (\frac{δ a}{a _{0}} + \frac{δ b}{a _{0}} + \frac{δc}{a _{0}}) .

For the affine breathing mode of a star, all three fractional axis changes equal

q

. Then

\overset{q}{¨} + (3 γ - 1) Ω^{2} q = 0, ω = Ω 3 γ - 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

δ T = 0

and

\overset{q}{¨} + 2 Ω^{2} q = 0, ω = 2 Ω .

These two degenerate modes deform the sphere into an ellipsoid while preserving its volume to first order.

Solved by gpt-5.6-sol high.

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