(Optical depth, (10.29) dimensionless)

This form preserves the exponential dependence even if k and p vary with x, which is often the case. An optical depth of 1.0 means that the photon beam is attenuated by a factor of e-1. An optical depth much less than unity means negligible absorption, and a large value means substantial absorption.

These several ways of describing distances traveled, mean free path, and the exponential absorption are summarized in Table 1.

Cross section and extinction coefficient

Since a cross section ag gives rise to extinction of starlight by interstellar grains, one can develop a relation between cross section ag and the extinction coefficient AV .It turns out that the two quantities are proportional to one another as we now demonstrate.

Consider starlight traversing a thickness r of uniform grain density ng. The photon flux (r) (m-2 s-1) decreases with distance similarly to N(x) in (20). Take the natural log of both sides of (20) and express the numbers N in terms of photon fluxes,

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