Limb darkening and contrast

In Fig. 8 we show the limb darkening and contrast for a continuum test case with different values of $\epsilon$. To compute the limb darkening, we calculate the intensity average $<I>$ over the visible surface for different values of $\cos(\theta)$ where $\theta$ is the angle between the observer and the normal to the surface. We similarly calculate the contrast as $\sqrt{<(I-<I>)^2>)}/<I>$ over the visible surface for different $\theta$. The absolute values of the limb darkening and the contrast depend strongly on $\epsilon$, scattering dramatically reduces the contrast and 'flattens' the limb darkening law. Overall, the limb darkening is nearly linear in $\cos(\theta)$, as would be expected from a plane parallel atmosphere with grey temperature structure.