distribution of ${\ensuremath{\hat{{ {\mathbf g}}}}_T\xspace}$ by [Eq. ]{}\[eq:snrt\]. We know ${\ensuremath{{ {\mathbf r}}}\xspace}(t)\cdot{\ensuremath{{ {\mathbf r}}}\xspace}(0) = |{\ensuremath{{ {\mathbf r}}}\xspace}(0)|^2\cos \Omega t$, and hence $C_{rr}(t) = \cos \Omega t$ and its Fourier transform $C_{rr}(\omega) = \pi \left[\delta(\omega-\Omega) + \delta(\omega+\Omega)\right]$, and thus ${\ensuremath{\sigma_{\mathbf{g},T}}}^2 = {\ensuremath{\sigma_{\mathbf{g},0}}}^2 \times |K_T(\Omega)|^2 = {\ensuremath{\sigma_{\mathbf{g},0}}}^2/(1+\Omega^2T^2)$.

By rescaling to unitless parameters, we then find that $${\ensuremath{\left\langle v_x \right\rangle}}/v_\textrm{max} = \sqrt{1 - \frac{1}{2} \omega^2} \times C(\left[{\ensuremath{\textrm{SNR}}}_0 (1+\omega^2\widetilde{T}^2)\right]^{-1/2})$$ where $\omega = \Omega R / v_{\textrm{max}}$ is the unitless angular velocity, ${\ensuremath{\textrm{SNR}}}_0 = {\ensuremath{\sigma_{\Delta}}}^{-2} g^2 \chi$ is the usual SNR with no averaging, and $\widetilde{T} = T v_{\textrm{max}}/R$ is the ratio of the averaging time to the characteristic rotational time $R/v_{\textrm{max}}$, and $C(\sigma)$ is the function given by [Eq. ]{}\[eq:csigma\]. When ${\ensuremath{\textrm{SNR}}}_0$ is sufficiently small, and $\widetilde{T}$ sufficiently large, ${\ensuremath{\left\langle v_x \right\rangle}}/v_\textrm{max}$ has a maximum at finite $\omega$ ([Fig. ]{}\[fig:rot\]).

In the limit of low [$\textrm{SNR}$]{}, $C(\sigma) \approx \sqrt{\pi/8} \sigma^{-1}$, and we find ${\ensuremath{\left\langle v_x \right\rangle}}/v_\textrm{max}$ is maximized by $\omega = \pm\sqrt{1-\frac{1}{2}\widetilde{T}^{-2}}$ when $\widetilde{T} > 1/\sqrt{2}$ and $\omega = 0$ otherwise. For the large ${\ensuremath{\textrm{SNR}}}$ limit, $C(\sigma) \approx 1-\sigma^2/2$ and rotation will increase the mean velocity in the direction of the gradient when $\widetilde{T}^2 > {\ensuremath{\textrm{SNR}}}_0/2-1/4$. More generally, it is possible to 