= Solution
The <specific intensity> $I_\nu$ is <energy> transported per unit time, projected area, <solid angle> and <frequency>. The path coordinate $s$ increases along the ray. The <emission coefficient> $j_\nu$ is emitted <energy> per unit time, volume, <solid angle> and <frequency>, whereas the <absorption coefficient> $\alpha_\nu$ has units of inverse length. Thus $j_\nu ds$ adds <specific intensity> and $\alpha_\nu I_\nu ds$ removes it. <Stimulated emission> can be included as negative <absorption>.
Define the <optical depth> between two points by $\tau_\nu(s_1,s_2)=\int_{s_1}^{s_2}\alpha_\nu(u)\,du$. Multiplying the <radiative transfer equation> by the <integrating factor> $\exp[\tau_\nu(s_0,s)]$ and integrating gives
$$
\boxed{I_\nu(s)=I_\nu(s_0)e^{-\tau_\nu(s_0,s)}+\int_{s_0}^{s}j_\nu(u)e^{-\tau_\nu(u,s)}\,du.}
$$
Each local contribution is attenuated only by the material between its emission point and the observer. Where $\alpha_\nu\ne0$, the <source function> is $S_\nu=j_\nu/\alpha_\nu$ and the same <formal solution of the radiative transfer equation> is $I_\nu(\tau)=I_\nu(0)e^{-\tau}+\int_0^\tau S_\nu(t)e^{-(\tau-t)}dt$. A constant <source function> gives $I_\nu=I_\nu(0)e^{-\tau}+S_\nu(1-e^{-\tau})$.
Back to article page
