1 The Fundamental Equation of Radiative Transfer

The fundamental equation of radiative transfer is governed by emission and extinction. Extinction is brought about by absorption (which changes photon energy) or by scattering (which does not). Examples of scattering are Thomson scattering of light off of cold electrons, Rayleigh scattering in the atmosphere, and Line scattering (reemission in a different direction). An example of absorption is photoionization (where a photon ionizes an atom, say by knocking off an electron).

1.1 Absorption

Let’s say radiation ${\displaystyle I_{\nu }}$ passes through a region ${\displaystyle ds}$ of absorption/scattering on its way to us. Then:

${\displaystyle dI_{\nu }=-\alpha _{\nu }I_{\nu }ds\,\!}$

where ${\displaystyle \alpha _{\nu }}$ is the extinction coefficient (units of ${\displaystyle cm^{-1}}$). We may compute ${\displaystyle \alpha _{\nu }}$ a couple different ways:

${\displaystyle \alpha _{\nu }=\overbrace {n} ^{\#\ density}\overbrace {\sigma _{\nu }} ^{cross\ section}=\overbrace {\rho } ^{mass\ density}\overbrace {\kappa _{\nu }} ^{opacity}\,\!}$

Solving for intensity:

{\displaystyle {\begin{aligned}I_{\nu }(s)&=I_{\nu }(0)e^{-n\sigma _{\nu }s}\\&=I_{\nu }(0)e^{-\tau _{\nu }}\\\end{aligned}}\,\!}

where ${\displaystyle \tau _{\nu }}$ is the optical depth at ${\displaystyle \nu }$.

Optical depth is often computed as:

${\displaystyle \tau _{\nu }=n\sigma _{\nu }s=N\sigma _{\nu }\,\!}$

where ${\displaystyle N}$, the column density, is in ${\displaystyle cm^{-2}}$ and is the # of extinguishers per unit area. Similarly,

${\displaystyle \tau _{\nu }=\rho \kappa _{\nu }s=\Sigma \kappa _{\nu }\,\!}$

where ${\displaystyle \Sigma }$ is the mass surface density and ${\displaystyle \kappa _{\nu }}$ is the density-weighted extinction coefficient.

${\displaystyle \tau _{\nu }{\begin{cases}\ll 1&optically\ thin\\\gg 1&optically\ thick\end{cases}}\,\!}$

The Mean Free Path is given by: ${\displaystyle \lambda _{mfp,\nu }=\alpha _{\nu }^{-1}={\frac {1}{n\sigma _{\nu }}}={\frac {1}{\rho \kappa _{\nu }}}}$. Thus:

${\displaystyle \tau _{\nu }={s \over \lambda _{mfp,\nu }}\,\!}$

That is, the optical depth is the number of mean-free-paths deep a medium is. For Poisson processes, the probability of absorption is given by:

${\displaystyle P(n)={e^{-{s \over \lambda _{mfp,\nu }}}\left({s \over \lambda _{mfp,\nu }}\right)^{n} \over n!}\,\!}$

Therefore:

${\displaystyle I_{\nu }(s)=I_{\nu }(0)e^{-\alpha _{\nu }s}\,\!}$

1.2 Emission

If ${\displaystyle j_{\nu }}$ is the emissivity, then the contribution of the emissivity of a medium to the flux is:

${\displaystyle dI_{\nu }=j_{\nu }ds\,\!}$

Emission and Extinction together:

${\displaystyle {{dI_{\nu } \over ds}=j_{\nu }-\alpha _{\nu }I_{\nu }}\,\!}$

(Fundamental Equation of Transfer)

It is often convenient to express this in terms of optical depth. Dividing by ${\displaystyle \alpha _{\nu }}$ and recognizing ${\displaystyle d\tau _{\nu }=ds\alpha _{\nu }}$:

{\displaystyle {\begin{aligned}{dI\nu \over d\tau _{\nu }}&={j_{\nu } \over \alpha _{\nu }}-I_{\nu }\\&=S_{\nu }-I_{\nu }\\\end{aligned}}\,\!}

where ${\displaystyle S_{\nu }}$ is a “source function”. In general,

${\displaystyle S_{\nu }{\big |}_{scattering}\propto \int {I_{\nu }d\Omega }\,\!}$

There is a formal solution for ${\displaystyle I_{\nu }}$. Let’s define ${\displaystyle {\tilde {I}}\equiv Ie^{\tau \nu }}$ and ${\displaystyle {\tilde {S}}\equiv Se^{\tau \nu }}$. Then:

${\displaystyle {d{\tilde {I}} \over d\tau _{\nu }}={\tilde {S}}\,\!}$
${\displaystyle {\tilde {I}}(\tau _{\nu })={\tilde {I}}(0)+\int _{0}^{\tau _{\nu }}{{\tilde {S}}d{\tilde {\tau }}_{\nu }}\,\!}$
${\displaystyle {I_{\nu }(\tau _{\nu })=\overbrace {I_{\nu }(0)e^{-\tau _{\nu }}} ^{atten\ bg\ light}+\overbrace {\int _{0}^{\tau _{\nu }}{S_{\nu }(\tau _{\nu }^{\prime })\underbrace {e^{-(\tau _{\nu }-\tau _{\nu }^{\prime })}} _{self-absorption}d\tau _{\nu }^{\prime }}} ^{glowing\ medium}}\,\!}$

If ${\displaystyle S_{\nu }}$ is constant with ${\displaystyle \tau _{\nu }}$, then:

${\displaystyle I_{\nu }(\tau _{\nu })=I_{\nu }(0)e^{-\tau _{\nu }}+S_{\nu }(1-e^{-\tau _{\nu }})\,\!}$

That second term on the righthand side can be approximated as ${\displaystyle S_{\nu }\tau _{\nu }}$ for ${\displaystyle \tau _{\nu }\ll 1}$, since self-absorption is negligible. Similarly, for ${\displaystyle \tau _{\nu }\gg 1}$, it may be approximated as ${\displaystyle S_{\nu }}$. The source function ${\displaystyle S_{\nu }}$ is everything. It has both the absorption and emission coefficients embedded in it.