= Small-vorticity phototactic orientation response
{title2=$\langle\mathbf p\rangle=K_0\mathbf e_x+\varepsilon K_1\mathbf e_z+O(\varepsilon^2)$}
For light along $\mathbf e_x$, <vorticity> along $\mathbf e_y$ and $\varepsilon=\omega/(2D_r)$, spherical coordinates $\mathbf p=(\cos\theta,\sin\theta\cos\phi,\sin\theta\sin\phi)$ give a first orientation-density correction proportional to $\sin\phi\,g(\theta)$. The linearized alignment-diffusion operator preserves azimuthal harmonics and the <rotation> of the axisymmetric equilibrium forces only this sine harmonic. Angular integration makes the first correction to the mean purely vertical. Uniqueness follows from the weighted identity $\mathcal A(f_0h)=\nabla_p\cdot(f_0\nabla_ph)$: <integration by parts> shows the only homogeneous smooth solution is a multiple of $f_0$, removed by <normalization>.
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