3.1 Using a Global Mean Curve
If there is no specific information available about the wavelength dependence of the extinction along a sightline of interest then the only alternative is to adopt some globally defined mean curve and perform a realistic error analysis. This is the least attractive of the three cases discussed here, but is by far the most commonly used. The average Galactic extinction curves from either Seaton (1979; see Figure 1) or Savage & Mathis (1979) are often adopted for dereddening UV data. These should now be superseded by the use of an R-dependent curve computed for the case R = 3.1 (see Section 2.2). Figure 1 shows the R = 3.1 curve from CCM (dotted line) and a new determination of the R = 3.1 case (solid line) which is described in the Appendix to this paper. This new curve aims to reproduce the detailed wavelength dependence of the R = 3.1 extinction law and has been constructed to account for bandpass effects properly in optical/IR extinction data and to reproduce the observed broad-, intermediate-, and narrow- band extinction measurements. The curve is thus suitable for dereddening multiwavelength spectrophotometric observations and can be used to derive the average extinction relationships in any photometric system.
A complete evaluation of the likely error in a dereddened relative energy distribution m ( - V) requires the propagation of the uncertainty in E (B - V) and the often-neglected but often-dominant uncertainty in the adopted mean curve. This is given by
where k ( - V) represents the normalized extinction curve E ( - V) / E (B - V). If absolute fluxes are desired, then the uncertainty in the assumed value of R must also be incorporated (two additional terms on the righthand side of eq. 1, similar to the current terms but with R substituted for k ( - V)).
The large database of ANS satellite extinction measurements from Savage et al. (1985) shows that the 1- scatter at 1500 Å is _{k(15-V)} = 0.74, based on ~ 400 sightlines with E (B - V) 0.5. (Only relatively large values of E (B - V) were considered in order to minimize the effects of spectral mismatch error and random noise on the measurement.) To extend this analysis to other wavelengths, we computed the standard deviation at each wavelength point for the 80 curves shown in Figure 2, and then scaled the result to match the ANS value at 1500 Å. (The actual standard deviation at 1500 Å for the 80 IUE curves is somewhat higher than the ANS result because of the bias in Figure 2 toward extreme extinction curves). The resultant values of _{k(-V)} are shown by the thick dotted curve near the bottom of Figure 2 (labeled ``'') and are listed at selected wavelengths in the third column of Table 1. This estimate of _{k(-V)} should be adopted whenever the average Galactic extinction curve is used for dereddening an observed energy distribution. The uncertainties approach zero for 1/ < 3 µm^{-1} due to the curve normalization. The quantity E (B - V) is usually derived directly from photometry and often has an easily quantifiable uncertainty; thus the computation of ^{2}_{m(-V)} from eq. 1 is straightforward.
^{a} k (-V) E ( - V) / E (B - V). These values of _{k (-V)} should be inserted into Eq. 1 to compute the uncertainty in a corrected relative energy distribution m ( - V) when there is no information available about the true shape of the extinction law along the sightline and the average Galactic curve is adopted by default. See Section 3 in the text. | |||
^{b} These values (multiplied by the derived E (B - V)) give the uncertainty in the final relative energy distribution m ( - V) when an object is dereddened by ``ironing out'' the 2175 Å bump using the average Galactic extinction curve. See Section 3.1 in the text. |
Sometimes E (B - V) is not known a priori and is estimated by ``ironing out'' the 2175 Å extinction bump using an assumed extinction curve shape (e.g., see Massa et al. 1983). In many cases, uncertainties in E (B - V) as small as 0.01-0.02 mag for moderately reddened objects have been quoted from this process. This is incorrect! The normalized ``height'' of the 2175 Å extinction bump - which is the quantity that is important in the ironing-out process - has a 1- scatter of about ± 20% around its mean value (from the data in Figure 2 and Savage et al. 1985). Thus the uncertainty in an E (B - V) measurement derived from the bump must be considered to have a similar relative uncertainty. Since bump strength does not correlate well with other aspects of UV extinction, such as the slope of the linear component or the strength of the far-UV rise (FM), the uncertainty in an energy distribution dereddened this way can be estimated by using eq. 1 with _{E(B-V)} 20% and the values of _{k(-V)} from Table 1. The result of this calculation needs to be modified somewhat by removing the signature of the 2175 Å bump. The final estimate of ^{2}_{m(-V)} computed this way is listed in the last column of Table 1. Multiplying these values by E (B - V) gives the uncertainty in a dereddened relative energy distribution. The values in Table 1 agree well with estimates ^{2}_{m(-V)} at 1500, 1800, 2200, 2500, and 3300 Å derived from the ANS data of Savage et al (1985), and are consistent with the analysis of Massa (1987).