7 The illumination and limb-darkening factor

Consider a spherical CB in its rest system. It is ``illuminated'' by incoming ISM electrons only in its ``front'' hemisphere. If observed at an angle $\rm\theta_{_{CB}}\neq 0$, a fraction of the ``dark'' CB is also exposed to the observer, like the Moon in phases other than totality. For radio waves - to which the CB is not transparent - these geometrical facts play a non-trivial role.

Place the direction of the CB motion, or of its illumination, at $(\theta,\phi)=(0,0)$; at a direction $\rm\vec n_i=(0,0,1)$ in Cartesian coordinates. The normal to a sphere's surface point at $(\theta,\phi)$is $\rm\vec n_s=(\cos\theta\sin\phi,\sin\theta,\cos\theta\cos\phi)$. The observer is in the direction $\rm (0,\theta_{_{CB}})$, where we have taken the liberty to label ``$\theta$'' what in this parametrization is an azimuthal angle; the corresponding unit vector is $\rm\vec n_{_{CB}}=(\sin\theta_{_{CB}},0,\cos\theta_{_{CB}})$. The relation between $\rm\theta_{_{CB}}$ and the terrestrial observer's viewing angle is that of Eq. (7).

When attenuation plays a significant role, an element of a CB's surface reemits an amount of energy proportional to the cosine of the illumination angle: $\rm\vec n_i \cdot \vec n_s$. Because of the limb-darkening effect, the reemitted radiation depends on the cosine of the angle between the surface element and the observer: $\rm\vec n_s \cdot \vec n_{_{CB}}\equiv \mu$. A simple characterization of the functional form of the limb darkening effect (see e.g. Shu 1991) is:

$$F(\mu)=\left({2\over 5}+{3\over 5}\,\mu\right)\,\Theta[\mu].$$ (26)

The combined effect of illumination and limb darkening is an emitted radiation proportional to:

 

$$E(\cos\theta_{_{\rm CB}}) \int^1_{-1} {\rm d}\cos\theta\int^{\pi/2}_{\theta_{_{\rm CB}}-{\p... ...(\vec n_{\rm s} \cdot \vec n_{_{\rm CB}})\, \vec n_{\rm i} \cdot \vec n_{\rm s}$$ =

$${1\over 5}\,\left[2+(2+\pi-\theta_{_{\rm CB}})\cos\theta_{_{\rm CB}} +\sin\theta_{_{\rm CB}}\right].$$ = (27)

An excellent and simple approximation to Eq. (27) is:

$$E(x)=E(1)\,{1\over 10}\, [4+x][1+x],$$ (28)

with $x=\cos\theta_{_{\rm CB}}$.

For negligible self-attenuation $A[\nu]=1$, as in the optical, there is no limb darkening and illumination effect. As absorption becomes increasingly important for longer wavelengths, the effect becomes fully relevant. We interpolate between these two extremes by writing:

$$L_{_{\rm CB}}(\nu,\cos\theta_{_{\rm CB}})\simeq A[\nu]+(1-A[\nu])\, {E(\cos\theta_{_{\rm CB}})\over E(1)}$$ (29)

to obtain the overall illumination and limb-darkening correction factor to the energy flux density of Eq. (6).