Apply the <formal solution of the radiative transfer equation> successively to the two isothermal layers at normal incidence. In <local thermodynamic equilibrium>,
$$
\boxed{
I_{\nu,2}=I_{\nu,0}e^{-(\tau_1+\tau_2)}
+B_\nu(T_1)(1-e^{-\tau_1})e^{-\tau_2}
+B_\nu(T_2)(1-e^{-\tau_2})}.
$$
For <optically thin> layers this becomes
$$
\boxed{
I_{\nu,2}=I_{\nu,0}
+\tau_1[B_\nu(T_1)-I_{\nu,0}]
+\tau_2[B_\nu(T_2)-I_{\nu,0}]
+O(\tau^2)}.
$$
This assumes no scattering and negligible reflection between layers.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 315 / 2 / b / i / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/iii/paper-315/2/b/i/solution