Specific intensity is defined by $dE=I_\nu\cos\theta\,dA\,d\Omega\,d\nu\,dt$. A ray bundle in free space expands in area while its solid angle contracts by the same factor, so
$$
\boxed{\frac{dI_\nu}{ds}=0}.
$$
Thus <specific intensity> does not obey an inverse-square law; the flux of an unresolved source does because its apparent solid angle scales as distance${}^{-2}$.

For a cold medium with coherent, isotropic, conservative scattering, the source function is the <mean intensity> $J_\nu$. With scattering optical depth increasing along the ray,
$$
\frac{dI_\nu}{d\tau_\nu}=-I_\nu+J_\nu,
$$
and
$$
\boxed{
I_\nu(\tau)=I_\nu(0)e^{-\tau}
+\int_0^\tau J_\nu(t)e^{-(\tau-t)}dt}.
$$
The unscattered pencil beam is attenuated by $e^{-\tau}$ and reappears as a diffuse halo in other directions. Coherent scattering preserves frequency, and conservative scattering preserves total luminosity when all outgoing directions are collected.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 315 / 4 / a / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-315/4/a/solution