= 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.
Back to article page
