Extinction and Optical Depth

At this point, we have two nondimensional measures of extinction, the optical depth and the quantity A\ introduced in equation (2.12). To see the relation between them, we first use the equation of transfer to obtain the specific intensity at a point P located a distance r from the center of a star (Figure 2.10). The plan is to use this result to obtain the star's apparent magnitude at P. Suppose we measure the specific intensity at a frequency where the intervening dust has negligible emission, so that jv = 0 in equation (2.20). Reverting from frequency to wavelength, we integrate the equation along any ray from the stellar surface to P, obtaining

Here Atx denotes the optical depth from the stellar radius to P. Referring to Figure 2.10, we see that this depth depends on the precise location of the emitting point within the cone converging on P. However, we suppose that r > R*, so that this variation can safely be ignored. Similarly, Ix(R*) is independent of propagation direction within the cone if the stellar surface is assumed to radiate like a blackbody (see below).

4 Here we follow the imprecise, but accepted, procedure of employing the term "optical" depth even for frequencies outside the visual regime.

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