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

The boundary is material precisely when its material derivative vanishes on $f=0$. Direct differentiation givesFor this to vanish at every point of the ellipsoidal free surface, each coefficient must vanish:With these relations the expression vanishes for every interior point as well, sothroughout the affine stellar model. Thus $f$ is a Lagrangian label carried by each fluid element.

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Since $Df/Dt=0$, the density ansatz hasThe velocity divergence isThe mass conservation equation therefore givesSimilarly,and hence

The $x$ component of the material acceleration isBecause $d\hat{p}/df=\hat{\rho }$,Combining this with $-{\partial }_x\Phi =-{\Omega }^{2}x$ givesThe $y$ and $z$ components give the analogous equations for $b$ and $c$. Finally, $\hat{p}(0)=0$ makes $p=0$ on the material free surface, so both dynamic and kinematic boundary condition requirements are satisfied.

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Let $V=abc$ and write $T=K{V}^{-(\gamma -1)}$ for a constant $K$. The axis equations are Newton equations in the effective potentialbecauseand similarly for $b,c$. Thereforeis conserved. Multiplying by the fixed profile-dependent mass moment converts $\mathcal{E}$ into the physical kinetic, trapping-potential, and internal energy of the star, so it is proportional to total energy.

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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.

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The additional acceleration isIts 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. PutThenso away from resonanceup to free oscillations. The tidal forcing resonates with the quadrupole mode when ${\Omega }_o=\sqrt{2}\Omega$; in the ideal undamped model the resonant amplitude grows secularly.

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