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.

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