Optically Thick and Optically Thin Media

Optical Depth

Dust Extinction

Light emitted from light sources like stars as it reaches to us (observers) can attenuate through dust in three major ways:

1. Absorption by the dust present along the line of sight to the observer.
2. Scattering out of the line of sight to the observer.
3. Scattering into the line of sight to the observer (also called forward scattering).

From the definition of magnitude, we can express visual extinction as

More generally,

Simplifying by substituting the expression for optical depth,

where .

Observationally, the measurement of overall amplitude of extinction (that is, what we see, implying the part of spectrum which lies in the visual band) and reddening of light are separate tasks and it is better to model them using separate quantities. Thus, we define

The latter factor represents dust extinction and accounts for grain composition and shape, but more importantly also is a function of local geometry of dust and stars along the line of sight as well the global geometry of the galaxy. There are multiple empirical derivation for values of , with one of them being the defined as the slopes of the (dust) attenuation curves. The value for this factor used in Tielens (2005), uses (see Equation 3.19).

Now, given this formulation, if one were to express visual extinction () using, say, UV flux or intensities, the final expression comes together as follows

which is the same form as described in Du & Bergin (2014).

References

1. Wild et al, 2011
2. Tielens, 2005
3. Du & Bergin, 2014