Measuring extinction curves of lensing galaxies

We critique the method of constructing extinction curves of lensing galaxies, using multiply imaged QSOs. If one of the two QSO images is lightly reddened, or if the dust along both sight lines has the same properties, then the method works well and produces an extinction curve for the lensing galaxy. These cases are likely rare and hard to confirm. However, if the dust along each sight line has different properties, then the resulting curve is no longer a measurement of extinction. Instead, it is a measurement of the difference between two extinction curves. This lens difference curve'' does contain information about the dust properties, but extracting a meaningful extinction curve is not possible without additional, currently unknown information. As a quantitative example, we show that the combination of two Cardelli, Clayton, and Mathis (CCM)-type extinction curves with different values of R-V produces a CCM extinction curve with a value of R-V that is dependent on the individual R-V values and the ratio of V-band extinctions. The resulting lens difference curve is not an average of the dust along the two sight lines. We find that lens difference curves with any value of R-V, even negative values, can be produced by a combination of two reddened sight lines with different CCM extinction curves with R-V values consistent with Milky Way dust (2.1 <= R-V <= 5.6). This may explain extreme values of R-V inferred by this method in previous studies. However, lens difference curves with more normal values of R-V are just as likely to be composed of two dust-extinction curves with R-V values different from that of the lens difference curve. While it is not possible to determine the individual extinction curves making up a lens difference curve, there is information about a galaxy's dust contained in the lens difference curves. If the lens difference curve can be fit with the CCM relationship ( regardless of the fitted R-V value), this implies that the dust along the two sight lines can be described by CCM. In addition, the presence of the 2175 angstrom feature in the lens difference curve means that this feature is present along at least one of the two lensed sight lines.