2.2 ... But There is Some Order Amid the Chaos

The recognition that interstellar extinction is spatially highly variable is critically important for both the study of dust grains and for the purposes of dereddening. Fortunately for the latter, it has been realized that within the seemingly endless variety of observed extinction curves there are actually some constraints on the wavelength dependence and some links between extinction in the UV, optical, and IR.

This began with Savage (1975) who showed, using OAO-2 satellite data, that the shape of 2175 Å bump can be reproduced well with a Lorentzian profile. This result - which provides important clues about the nature of the 2175 Å feature - allows a functional representation of the bump, and permitted Seaton (1979) to present an estimate of the average Galactic UV extinction curve as a simple analytical formula. This curve is shown in Figure 1 by the dashed curve, extending from 2.7 to 8.7 µm^-1.

Following the lead of Savage and Seaton, Fitzpatrick & Massa (1986, 1988, 1990; collectively referred to hereafter as FM) examined IUE extinction curves for many lines-of-sight and found that all these curves could be fitted extremely well by a single analytical expression with a small number of free parameters. This expression consists of (1) a Lorentzian-like bump term (requiring three parameters, corresponding to bump width , position x0, and strength c3), (2) a far-UV curvature term (one parameter c4), and (3) a linear term underlying the bump and the far-UV (two parameters c1 and c2). This set of basis functions is shown in Figure 3. With the proper choice of the 6 parameters, essentially all UV extinction curves can be reproduced to within the uncertainties inherent in the data. For most sightlines, this number of free parameters can be reduced to 4, since it has been seen that the position of the 2175 Å bump is nearly invariant and that the intercept and slope of the linear term are tightly correlated (FM; see also Carnochan 1986 and Jenniskens & Greenberg 1993), such that the linear term appears to pivot about the point [1/, {E ( - V)} / {E ( B - V)}] [3,2]. The UV portions of the curves shown in Figure 2 ( < 2700 Å) were all constructed using these functions. There is, thus far, no convincing evidence for substructure in UV extinction curves.

Figure 3. Analytical fitting functions for UV extinction curves from Fitzpatrick & Massa 1990. A normalized UV extinction curve (thick solid curve) can be represented by a combination of three functions: (1) a linear background component (thin dashed line), (2) a UV bump component (thin solid curve), and (3) a far-UV curvature component (thin dotted line). The linear background is parameterized by two tightly correlated coefficients (slope and intercept), the bump by three coefficients (strength, width, and central position), and the far-UV curvature by a single scale factor. Given the near invariance in bump central position and the correlation between the linear coefficients, most Galactic UV extinction curves can be reproduced to within the observational uncertainties with only four free parameters.

The FM results show that some order underlies the wide spectrum of wavelength dependences seen in the UV, but do not necessarily permit a more accurate dereddening of an object along any given line of sight. Fortunately it has been observed that extinction curves do not vary randomly across the sky, but show distinct regional signatures (e.g., Meyer & Savage 1981; Morgan et al. 1982; Panek 1983; Massa & Savage 1984; Clayton & Fitzpatrick 1987) and it is clear that extinction properties reflect the physical conditions and past processing histories of the environments in which the dust grains reside.

Figure 4. Far-IR through UV extinction curves from Cardelli et al. 1989 (CCM). CCM found that extinction curves can be expressed approximately as a 1-parameter family that varies linearly with R-1, where R A (V) / E (B - V) and has a mean value in the diffuse interstellar medium of 3.1. Examples of CCM's results are shown for four representative values of R, listed on the righthand side of the figure next to the corresponding curve.

The work of Cardelli, Clayton, & Mathis (1988, 1989; hereafter CCM) provides a link between one measure of dust grain environment and the wavelength dependence of UV-through-IR extinction. CCM found that the shape of UV extinction curves correlates with the parameter R (defined in Section 2.1 above). This suggests that - although there is considerable scatter - extinction curves from the UV through the IR can be characterized as approximately a one-parameter family dependent on R. Thus, if the value of R can be determined (from optical and IR photometry), then the properties of the the entire UV-IR extinction curve can be predicted. The essence of the CCM result is illustrated in Figure 4. Four representative extinction curves are shown, each determined by the value of R listed at the righthand side of the curves. This figure shows that extinction curves which are seen to be very ``flat,'' or ``grey,'' in the UV roll over strongly in the optical and are characterized by large values of R. Steep UV curves remain steep in the optical and are characterized by small values of R. The CCM curve for R = 3.1 is shown in Figure 1 by the dotted line.

The results of CCM are important in 3 ways: (1) They indicate that some of the spatial variations seen in extinction curves behave coherently and systematically over a wide wavelength range, thus potentially allowing for the consistent dereddening of multiwavelength energy distributions. (2) The observed dependence on R allows for the definition of a meaningful average Galactic extinction curve. Existing datasets of extinction curves, particularly for the UV, are biased toward ``interesting'' regions where extinction properties are extreme. Computing a simple mean curve from these data does not necessarily yield a reasonable estimate of a Galactic mean. Since it is well-established, however, that the appropriate mean value of R for the diffuse ISM is ~ 3.1 (e.g., Schultz & Wiemer 1975; Whittet & van Breda 1980; Rieke & Lebofsky 1985), a reasonable definition of a mean Galactic extinction law is that which corresponds to the case where R = 3.1. (3) They demonstrate a general correlation between dust grain environment and the wavelength dependence of extinction, since large values of R are generally found in dense environments where dust grain growth is thought to occur. The ``greyness'' of the UV/optical extinction curves for large R is consistent with a larger than normal dust grain population. These results have an important bearing on how to correct for the effects of extinction, discussed in the following section.