Optical Depth

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Optical Depth

Solving for specific intensity in the Radiative Transport Equation

for zero emission () gives a solution of the form:

where is the optical depth at . Optical depth is often computed as:

where , the column density, is in and is the # of extinguishers per unit area. Similarly,

where is the mass surface density.

The Mean Free Path is given by: . Thus:

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:

Therefore:

In fact, the radiative transport equation can be expressed in terms of optical depth. Dividing by and recognizing :

where is a “source function”. In general,

There is a formal solution for . Let’s define and . Then:

If is constant with , then:

That second term on the righthand side can be approximated as for , since self-absorption is negligible. Similarly, for , it may be approximated as . The source function is everything. It has both the absorption and emission coefficients embedded in it.

Mona lisa vs radius.png
An example of the Mona Lisa at optical depth of , 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.
Mona lisa vs opt depth.png
The Mona Lisa at various optical depths, illustrating how the transition from optically thin to optically thick erases the background picture.