Spectral flux density
Energy passing through a unit surface in unit of time and unit of wavelength/frequency \[
F_\lambda = \frac{dE}{dS \, dt \, d\lambda}
\]
- It represents the spatial, temporal, and spectral distribution of the energy flow
- We have to add propagation direction
Specific monochromatic intensity
Energy passing through a unit surface in unit of time and unit of wavelength/frequency and unit of spatial angle \[
I_\lambda = \frac{dE}{dS \, dt \, d\lambda \, d\Omega}
\] with units \[
[ I_\lambda ] = \frac{W}{m^2 \, srad \, nm}.
\]
- Generally, it depends on 7 parameters (\(x,y,z,\varphi,\theta,\lambda,t\)).
- Later, we refer it only as ‘intensity’.
Radiative transfer equation
- It describes how intensity \(I_\lambda\) evolves after
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Propagation slab (nso.edu)
\[
\frac{dI_\lambda}{dl} = j_\lambda - \chi_\lambda I_\lambda
\]
where:
- \(\lambda\) is the considered wavelength
- \(j_\lambda\) is the emissivity
- \(\chi_\lambda\) is the absorption coefficient / opacity
Emissivity and absorption
Emissivity can be also expressed as \[
j_\lambda = \frac{dE_{emitted}}{dS \, dl \, dt \, d\Omega \, d\lambda} = \frac{dn_{emitted} \frac{hc}{\lambda} }{dt \, d\Omega \, d\lambda}
\]
That leads to a volume emission coefficient \(n_{emitted}\).
The opacity can be expressed as
\[
\chi_\lambda [m^{-1}] = n_{absorbers} [m^{-3}] \times \sigma [m^2]
\] where \(\sigma\) is the effective cross-section along the path \(l\).
Mean free path of a photon
- The higher the opacity is, the shorter distance the photon can travel without interaction.
- Optically thick and thin media.
- The interaction is usually absorption followed by an emission (which can be at a different wavelengths/frequency, emission angle, and intensity).
- More opaque means more photons are trapped.
- For enough photon interactions along the path, the medium gets into an equilibrium state.
- Wavelength dependent
\[
\bar{l_\lambda} = \frac{1}{\chi_\lambda}.
\]
Notes
- Radiative transfer cover all parts of the electromagnetic spectrum.
- Equation of radiative transfer is numerically solved as a set of equations for various wavelengths/frequencies.
- Though, various frequencies can have their specifics.
- Whole zoo of absorption and emission mechanisms.
