Compton Getting Effect

When an observer moves with velocity $w$ through an isotropic cloud , she will notice an anisotropy in the incident particle flux. The anisotropy is measured by the ratio of the difference of the extreme fluxes to the sum of the extremes

$\xi = \frac{F_{\max} - F_{\min}}{F_{\max} + F_{\min}}$

If the velocity of the observer is much smaller than the typical velocity of the particles in the cloud $v$ , the anisotropy will be linear in the ratio of the velocities

$\xi \propto \frac{w}{v}$

To understand this result, let us consider a one dimensional ensemble with some velocity distribution $f\left(v\right)$. We further assume that the distribution is isotropic so $f\left(v\right) = f\left(-v\right)$ and $f'\left(-v\right) = -f'\left(v\right)$. If some observer moves with velocity $w$ she would see a slightly different velocity distribution

$\tilde{f}\left( v \right) = f\left(v+w\right) \approx f\left(v\right) + w f'\left(v\right)$

The flux is just the product of the distribution function and the velocity. The anisotropy is therefore

$\xi = \frac{v \left( f\left(v\right) + w f'\left(v\right) \right) - v \left( f\left(-v\right) + w f'\left(-v\right) \right)}{v \left( f\left(v\right) + w f'\left(v\right) \right) + v \left( f\left(-v\right) + w f'\left(-v\right) \right)} = \frac{w}{2} \frac{v f'\left( v \right)}{f\left(v \right)} = \frac{1}{2} \frac{w}{v} \frac{d \ln f}{d \ln v}$

Assuming $\frac{d \ln f}{d \ln v}$ is constant (or at least a weak function of the velocity), we reproduce the qualitative behaviour of the Compton - Getting effect.