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## Optical Depth

Solving for specific intensity $I_{\nu }$ in the Radiative Transport Equation

${{dI_{\nu } \over ds}=j_{\nu }-\alpha _{\nu }I_{\nu }}\,\!$
for zero emission ($j_{\nu }=0$) gives a solution of the form:

${\begin{aligned}I_{\nu }(s)&=I_{\nu }(0)e^{-n\sigma _{\nu }s}\\&=I_{\nu }(0)e^{-\tau _{\nu }}\\\end{aligned}}\,\!$
where $\tau _{\nu }$ is the *optical depth* at $\nu$. Optical depth is often computed as:

$\tau _{\nu }=n\sigma _{\nu }s=N\sigma _{\nu }\,\!$
where $N$, the *column density*, is in $cm^{-2}$ and is the # of extinguishers per unit area. Similarly,

$\tau _{\nu }=\rho \kappa _{\nu }s=\Sigma \kappa _{\nu }\,\!$
where $\Sigma$ is the mass surface density.

$\tau _{\nu }{\begin{cases}\ll 1&optically\ thin\\\gg 1&optically\ thick\end{cases}}\,\!$
The *Mean Free Path* is given by: $\lambda _{mfp,\nu }=\alpha _{\nu }^{-1}={\frac {1}{n\sigma _{n}u}}={\frac {1}{\rho K_{\nu }}}$. Thus:

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

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

$I_{\nu }(s)=I_{\nu }(0)e^{-\alpha _{\nu }s}\,\!$
In fact, the radiative transport equation can be expressed in terms of optical depth. Dividing by $\alpha _{\nu }$ and recognizing $d\tau _{\nu }=ds\alpha _{\nu }$:

${\begin{aligned}{dI\nu \over d\tau _{\nu }}&={j_{\nu } \over \alpha _{\nu }}-I_{\nu }\\&=S_{\nu }-I_{\nu }\\\end{aligned}}\,\!$
where $S_{\nu }$ is a “source function”. In general,

$S_{\nu }{\big |}_{scattering}\propto \int {I_{\nu }d\Omega }\,\!$
There is a formal solution for $I_{\nu }$. Let’s define ${\tilde {I}}\equiv Ie^{\tau \nu }$ and ${\tilde {S}}\equiv Se^{\tau \nu }$. Then:

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

$I_{\nu }(\tau _{\nu })=I_{\nu }(0)e^{-\tau _{\nu }}+S_{\nu }(1-e^{-\tau _{\nu }})\,\!$
That second term on the righthand side can be approximated as $S_{\nu }\tau _{\nu }$ for $\tau _{\nu }\ll 1$, since self-absorption is negligible. Similarly, for $\tau _{\nu }\gg 1$, it may be approximated as $S_{\nu }$. The source function $S_{\nu }$ is everything. It has both the absorption and emission coefficients embedded in it.

* An example of the Mona Lisa at optical depth of $\tau =0.1$, for obscuring particles of various radii. To achieve the same optical depth, particles with a smaller cross-sectional area need to have a higher column density. *

* The Mona Lisa at various optical depths, illustrating how the transition from optically thin to optically thick erases the background picture. *