Let us first quantify the wavelength dependence of extinction. Recall from Chapter 1 that the brightness of a star near any wavelength A is measured either by mx or Mx, its apparent and absolute magnitudes, respectively. The presence of dust between the star and Earth alters the relation between these two quantities from equation (1.4) to mx = Mx + 5 logf-U + , (2.12)
where Ax, a positive quantity measured in magnitudes, is known simply as the extinction at wavelength A. Note that, even at the fiducial distance of 10 pc, the star now has mx > Mx, i. e., it is dimmer than its absolute magnitude.
The general tendency of dust to redden distant objects implies that Ax must diminish with increasing A, at least in the optical regime. Consider, then, writing equation (2.12) for the same star at two different wavelengths Ai and A2. Eliminating r between the equations yields
The first righthand quantity is C2, the intrinsic color index measured at A1 and A2. As we noted in § 1.3, this quantity can also be written in terms of the UBV filter names as (B - V)◦, (U-B)0, etc. The lefthand quantity in equation (2.13), denoted generically as C12, or as B-V, U - B, etc., is the observed color index. The difference between the intrinsic and observed color indices is a measure of reddening known as the color excess, E12:
The color excess is written in the UBV system as EB-V, Eu-b, etc. At visual wavelengths, A\1 > A\2 for Ai < A2, and Ei2 is a positive quantity.
Both the extinction and the color excess are proportional to the column density of dust grains along the line of sight.3 Consider yet a third wavelength, A3. Then the ratios A\3 /Ei2 and E32/Ei2 depend only on intrinsic grain properties. If we now fix Ai and A2, while letting the third wavelength have the arbitrary value A, either ratio provides a measure of the wavelength-variation of dust extinction. Conventionally, Ai and A2 are chosen to correspond to the B and V filters, respectively. In this case, Ex-V/EB-V is known as the normalized selective extinction at A, while AX/EB-V is the normalized total extinction. From equation (2.14b), these two quantities are simply related:
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