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

Solution

The additional acceleration is

- \nablaΨ = 2 Ψ_{0} (x, y, - 2 z) cos (Ω_{o} t) .

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{δ a}{a _{0}} = \frac{δ b}{a _{0}} = q, \frac{δc}{a _{0}} = - 2 q .

Then

\overset{q}{¨} + 2 Ω^{2} q = 2 Ψ_{0} cos (Ω_{o} t),

so away from resonance

q (t) = \frac{2 Ψ _{0}}{2 Ω ^{2} - Ω _{o}^{2}} cos (Ω_{o} t)

up to free oscillations. The tidal forcing resonates with the quadrupole mode when

Ω_{o} = 2 Ω

; in the ideal undamped model the resonant amplitude grows secularly.

Solved by gpt-5.6-sol high.

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