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Neglecting cell rotational inertia, the light and viscous torque sum to zero. Therefore the angular velocity is $\mathbf{\Omega }=\mathbf{\omega }/2+(\mathbf{p}\times \mathbf{L})/\alpha$. A material orientation evolves by $\dot{\mathbf{p}}=\mathbf{\Omega }\times \mathbf{p}$. The vector triple-product identity givesThe last term is the projection of $\mathbf{L}/\alpha$ onto the tangent plane to the unit sphere. In particular, $\mathbf{p}\cdot \dot{\mathbf{p}}=0$, as required to preserve the unit orientation. It aligns the swimmer towards the light; the vorticity rotates that orientation.

For rotational diffusion coefficient $D_r$, the phototactic orientation Fokker-Planck equation balances deterministic orientation drift with diffusion on the sphere. Its density is normalized with respect to solid angle, not $d\theta d\varphi$. When $\epsilon =0$, there is no azimuthal drift and the zero-current condition is $f_{0,\theta }=-\lambda \sin \theta f_0$. Thus $f_0=\eta e^{\lambda \cos \theta }$, and normalization givesThe limiting density at $\lambda =0$ is $1/(4\pi )$.

To justify the angular dependence of the first correction, define the operatorsThe steady Fokker-Planck equation is $\mathcal{A}f=-\epsilon \mathcal{R}f$. Its regular expansion $f=f_0+\epsilon f^{(1)}+O({\epsilon }^2)$ therefore givesThe coefficients of $\mathcal{A}$ are independent of $\varphi$, so each azimuthal Fourier harmonic is preserved. The forcing has only the sine harmonic with azimuthal number one. There can be no extra unforced harmonic in a smooth normalized first correction: writing a homogeneous solution as $f_{0}h$ gives $\mathcal{A}(f_{0}h)={\nabla }_p\cdot (f_0{\nabla }_{p}h)$. Multiplication by $h$ and integration by parts on the sphere gives $\int f_0|{\nabla }_{p}h{|}^{2}d\Omega =0$, hence $h$ is constant. Normalization $\int f^{(1)}d\Omega =0$ removes that multiple of $f_0$. It follows thatNo explicit calculation of $g$ is needed. Its dependence is on $\theta$ and the parameter $\lambda$, and its behavior at the poles must make the density smooth.

In the stated spherical coordinates, $\mathbf{p}=\cos \theta {\mathbf{e}}_x+\sin \theta \cos \varphi {\mathbf{e}}_y+\sin \theta \sin \varphi {\mathbf{e}}_z$. Azimuthal integration makes the leading $y,z$ components and the first-order $x,y$ corrections vanish. The surviving coefficients areThus the small-vorticity phototactic orientation response isBoth coefficients depend only on $\lambda$. In particular, $K_0$ is the Langevin function of $\lambda$ and is positive for positive light strength; its integral representation above suffices.