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+ AV ~ ——- = 0.18 csc b mag (Extinction through (10.9)

This is sometimes referred to as the cosecant law. It is applicable for observations of sources outside the (MW) Galaxy, i.e., of other galaxies.

The expression (9) is approximately correct over a wide range of galactic latitudes away from the galactic equator, approximately 10° < |b| < 90°. At lower latitudes the extinction becomes very large and uncertain.

Measurements of extragalactic objects at numerous galactic latitudes can be used to confirm that the flat-slab model of the dust is approximately correct. The csc b dependence of the path length in our model remains valid even if the dust density decreases with height from the galactic plane as long as it does not vary in the lateral direction. This is approximately true in the vicinity of the sun from where we do our observing.

Color excess (reddening) The reddening of the starlight due to scattering by grains may be described as a color excess, EB-V, where

Eb-v = Ab — AV (Color excess for B and V bands) (10.10)

Recall that the difference in magnitudes, mB — mV = B — V, is a color to the human eye because the difference in magnitudes translates to a ratio of the B and V fluxes. With the introduction of dust grains (in a thought experiment), both the blue light (B filter) and the yellow light (V filter) will become fainter. However, the blue light will decrease more than the yellow light because of the frequency dependence of the scattering. Thus both the B and V magnitudes increase, but B increases more than V. This represents the change in the perceived color. The difference in the two colors (B — V) — (B — V )0 is described as a color excess Eb-v. It may be written as (B — B0) — (V — V0) = AB — AV, from (5), which justifies (10).

The extinction AV and the reddening or color excess, EB—V, go hand in hand; if the one increases because of more dust grains along the line of sight, the other also increases. From measurements, one finds

Thus a visual extinction of AV = 0.6 mag per 1000 LY in the galactic plane (4) corresponds to a color excess of EB—V ~ 0.6/3 = 0.2 mag per 1000 LY. That is, for a star at 1000 LY, the V magnitude would be increased 0.6 mag by extinction, but the B magnitude would be increased an additional 0.2 mag for a total of 0.8 mag. This latter result may be obtained from the expression which follows from (10) and (11).

Frequency dependence

The measured dependence of the extinction A(v) upon frequency over a wide range of frequency is given in Fig. 3. The extinction is due to the integrated effect of grains of many different sizes, each with its own efficiency for scattering the light. The ordinate is the ratio of A(v)/Av . The effective frequencies of the V and B bands are shown. The curve passes through unity at the V band as it should. In the optical region, the rapid increase of extinction with frequency is apparent.

This increase and the broad peak in the ultraviolet at X ~ 220 nm provide information about the sizes and constituents of the grains. For example, the 220-nm peak could be due to small uncoated grains of graphite. The (approximate) straight-line character of the data in the visible region demonstrates that A(v) a ~v1. This is roughly in accord with the increase of A B over AV discussed above.

Astronomers must take care to properly account for extinction when deducing the spectral properties of an object, because the frequency-dependent extinction will modify the spectrum. The measured extinction of an object in the Galaxy, through its spectrum or colors, can be used to deduce the distance to the object if one also measures the extinction of other stars in the same direction in order to build up a model of extinction vs. distance. In this manner one takes into account the clumpiness of the absorbing medium.

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