Substitute (32) into (33), m — m0 = +2.5 (loge) agngr (10.34)

and recall the definition of the extinction coefficient (5),

The extinction coefficient is thus found to be proportional to the cross section ag as well as to the density ng and distance r. The dependence on r is a restatement of (4) and (7). If the matter were not uniformly distributed, one would replace anr with k£, where the opacity k (m2 kg-1) is another version of the cross section. One could choose to substitute the grain column density Ng, after (13), for ngr, to obtain

which exhibits the linearity between Ay and ag that we set out to show.

The expressions (35) and (36) take us from the microscopic physics of the individual interactions to the macroscopic attenuation of starlight by large column densities of interstellar grains. For example, the extinctions in Fig. 3 for various frequencies are a direct indicator of the variation with frequency v of the cross section aav(v) for photon-grain interactions, averaged over the grains along the lines of sight to a number of stars.

Since Ay a EB-y (11), we have, from (36), EB-y a Ng as we examine many different stars. If further the hydrogen and grains are clumped with a constant number abundance ratio everywhere, the two column densities along the lines of sight to stars at various locations in the Galaxy will be proportional to one another, Ng a Nh from star to star. Hence we have finally

which demonstrates the expected linearity of the correlation in Fig. 4.

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