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Astron. Astrophys. 322, 633-645 (1997)

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5. Two-Lorentz oscillator models

The simple approach taken in the previous section - to model the optical properties of the interstellar UV feature carrier with a single Lorentz oscillator - may appear somewhat unrealistic. It assumes that the bump carrier does not contribute at all to the FUV rise. Actually, along some lines of sight (many passing through Orion), the FUV rise is observed to be very small or virtually absent, even though the interstellar UV feature is still present, albeit weaker than usual. But all carbonaceous materials (including diamond) are expected to give rise to a peak at roughly 10-15 [FORMULA] due to the [FORMULA] - [FORMULA] electronic transition (Mennella et al. 1995a; Fink et al. 1984). Moreover, any plausible material in the small particle limit will produce strong absorption at [FORMULA].

To our knowledge, no physical mechanism can suppress the [FORMULA] - [FORMULA] transition while keeping intact the [FORMULA] - [FORMULA] transition responsible for the UV feature. Thus, the assumption that the UV bump carrier is carbonaceous in nature appears at first glance to be difficult to reconcile with the very small FUV rise observed in a number of cases. In this section, we ignore this difficulty and assume that the dielectric function of the interstellar UV feature carrier is the sum of two Lorentz oscillators - one producing the UV bump, and the other producing the FUV curvature- to see how the conclusions of the preceding section are modified.

One important distinction must be made here. If two grain populations are unrelated, then one can add their cross sections, like is implicit in the decomposition of Eq.(1). For example, adding an extra linear background or an extra FUV curvature will affect only their corresponding fitting parameters in Eq. (1), but will leave the other fitting parameters unchanged. However, if the UV bump and the FUV curvature arise from the same parent material, then their respective contributions to the total dielectric function must be added instead, and then only can the cross section be computed assuming a certain grain shape. These various contributions to the dielectric function do not necessarily add up linearly when translated into cross sections, unless [FORMULA] is small. Therefore, adding an extra Lorentz oscillator at larger x (in [FORMULA] space) to an initial one that produces a UV bump well characterized by a "Drude" profile (for example, assuming a spherical shape) might actually result into a bump that is no longer Drude-like. As a result of this process, all fitting parameters may be affected in a non-trivial way.

To investigate these effects we approximate such a dielectric function as the sum of two Lorentz oscillators, giving rise to features at about [FORMULA] (the UV bump) and 10-15 [FORMULA] (the FUV bump), respectively. This guarantees that the total dielectric function of the material satisfies Kramer-Kronig relations (Bohren & Huffman 1983). Of course, more complicated dielectric functions are also possible, but then the number of free parameters becomes rapidly excessive, and the physical interpretation becomes less straightforward.

The effects of this extra Lorentz oscillator on the UV bump are illustrated in Fig. 6. The UV oscillator is given by the parameters [FORMULA], [FORMULA], and [FORMULA], which on its own yields a [FORMULA] peak at 4.60 [FORMULA] assuming a sphere (dotted line). The FUV oscillator is given by [FORMULA], [FORMULA], and various widths, which on its own yields a [FORMULA] peak at [FORMULA] assuming a sphere. This could be viewed as a very crude model of the [FORMULA] - [FORMULA] peak observed for HAC in laboratory extinction experiments (Mennella et al. 1995a). The solid, dashed, dash-dotted, and dash-triple-dot lines are for [FORMULA], and 12.0, respectively (also assuming spheres). The main effect of the FUV oscillator is to decrease the strength of the UV bump, more or less independently of the width of the FUV bump (keeping the FUV oscillator strength constant). Another effect is to shift the peak position of the UV bump to smaller wavenumbers ([FORMULA], and 4.573 [FORMULA], for [FORMULA], and 12.0, respectively). An extra broadening also occurs, but only on the larger x wing of the bump. Changing the width of the FUV oscillator without changing its strength affects the FUV curvature dramatically. Actually, this suggests an explanation for the quasi-absence of FUV curvature observed along some lines of sight: if the FUV bump is extremely narrow (or shifted to large enough wavenumbers), then the FUV rise may be virtually absent (solid line). This is also obtained for very wide FUV bumps, but then there is a "jump" in the linear component of the extinction across the UV bump (dash-triple-dot line). However, Fitzpatrick & Massa (1988) have shown that the linear background of interstellar extinction curves, once the fitted bump has been subtracted out, is very smooth across the bump, with no evidence of jumps or changes of slope. Therefore extremely broad FUV bumps are more or less ruled out as an explanation of the lack of a FUV curvature concurrent with the presence of a UV bump.

[FIGURE] Fig. 6. Same format as Figs. 4 and 5, but for various Lorentz oscillator models and for spheres. Results shown are for a single-UV-Lorentz oscillator model with [FORMULA], [FORMULA], [FORMULA] (dotted lines), and for models involving an extra FUV Lorentz oscillator with [FORMULA], [FORMULA], and [FORMULA] (solid lines), 3.0 (dashed lines), 6.0 (dash-dotted lines), and 12.0 (dash-triple-dot lines)

The main effect of adding such a background to the dielectric function is to increase [FORMULA] and, to a lesser extent, [FORMULA], shifting the plasmon condition ([FORMULA]) to smaller wavenumbers. Note how little the dielectric function of the UV oscillator is changed for smaller wavenumbers when changing the width of the FUV oscillator (keeping the strength constant).

When a shape different from a sphere is assumed (either a CCA or a compact cluster), as a result of this extra background, the peak position is shifted to smaller wavenumbers, contrary to what was obtained in the previous section. A moderate increase in FWHM is also obtained. These results are similar to what was obtained in the case of graphite (Table 1).

5.1. The effect of scattering on the FUV curvature

Another mechanism that can flatten the FUV bump is scattering by larger grains. But scattering is likely to affect the UV feature as well (mainly shifting it to smaller wavenumbers - which is not observed in all interstellar cases), unless a size distribution could be such that it would affect only wavenumbers larger than [FORMULA]. However, this is unlikely, as shown in Fig. 7. Plotted (in order of decreasing strength) are the [FORMULA] Mie curves for spheres with radius a in the Rayleigh limit, and [FORMULA] and 0.1  [FORMULA] m, respectively, using the dielectric function with [FORMULA] in Fig. 6 (dashed lines). Note that the UV bump is strongly affected in peak position and shape for [FORMULA] m, even though the FUV peak is still quite large (with much FUV curvature). The size which produces a FUV bump that is essentially flat, [FORMULA] m (lowest curve), also yields a UV bump that is virtually unrecognizable. Here, the UV peak is more strongly affected than the FUV peak because the dielectric function in the UV range is closer to a plasmon condition (see Fig. 6). Therefore, a size distribution of grains (weighting these curves by the volume of individual grains) is unlikely to flatten the FUV rise without severely affecting the shape and peak position of the UV bump as well.

[FIGURE] Fig. 7. [FORMULA] of Mie spheres for various radii a using the dielectric function with [FORMULA] in Fig. 6 (dashed lines). In order of decreasing strength, [FORMULA] for radius a in the Rayleigh limit, and [FORMULA] and 0.1  [FORMULA] m, respectively

5.2. Synthetic interstellar extinction curves

A direct comparison between the fitting parameters of interstellar curves and some synthetic ones can be useful to assess the qualities of the Drude profile for the UV bump and the term [FORMULA] for the FUV curvature when two-Lorentz oscillator models are considered. To make such a comparison, a few simplifying assumptions are required. The total optical depth is assumed to arise from two contributions, one from grains producing most of the linear rise between 3 and 8.5 [FORMULA] (labelled l) and one from the grains producing the bump, the rest of the linear rise, and the FUV curvature (labelled b). The component l is assigned to silicate grains. The lack of correlation between the linear rise parameters and the bump parameters suggests that the grains responsible for the UV bump do not contribute appreciably to the linear rise (Jenniskens & Greenberg 1993). Thus assigning most of the linear rise to silicate grains is reasonable. The different slopes observed in the interstellar extinction curves (see, e.g., Fig. 8) could be assumed to arise from different size distributions of silicate grains (with larger mean grain sizes producing flatter curves). For the sake of simplicity, we shall not be concerned here with the detailed modelling of this component, but will simply assume that its [FORMULA] can be described as a simple linear rise in the range 3-8.5 [FORMULA]. For the carbonaceous component b, a key normalization parameter is the carbon to hydrogen ratio (by number), [FORMULA].

[FIGURE] Fig. 8. [FORMULA], [FORMULA], and [FORMULA], for various Lorentz oscillator models assuming spheres. The parameters of the models, listed in Table 4, were chosen to fit actual interstellar curves (solid lines). Models are for HD 204827 (dotted lines), HD 229196 (dashed lines), and HD 37023 ([FORMULA]  Ori D; dash-dotted lines: thick- [FORMULA]  ppm, thin-45 ppm)

Assuming [FORMULA] (Mathis 1994) and using [FORMULA], where [FORMULA] is the column density of dust and [FORMULA] is the cross section of component j (where [FORMULA]), we have from the definition of [FORMULA],


where [FORMULA] (B-V) is the total-to-selective extinction ratio in the V band. We have [FORMULA], where [FORMULA] is the mass of the hydrogen atom, and [FORMULA] and [FORMULA] are the density and the mean volume of the grains, respectively. The density of bump grains is again assumed to be [FORMULA]. The computed cross section per unit volume of the bump grains using two Lorentz oscillators (i.e. seven free parameters: [FORMULA], [FORMULA], [FORMULA], [FORMULA], and [FORMULA]) is fitted by


between 3 and 8.5 [FORMULA]. Assuming that [FORMULA] is linear with respect to x, Eq. (4) can be rewritten as


where [FORMULA], [FORMULA], and [FORMULA].

Interstellar extinction curves along specific lines of sight were modelled using this two-Lorentz oscillator model assuming Rayleigh spheres over the whole range. Using the same optical constants and the same normalization, results for CCA clusters and compact clusters were also computed to see how the bump parameters changed. Uncertainties related to the extrapolation beyond the observed range (3-8.5 [FORMULA]) do not warrant taking the extra complication of scattering into account. The value of [FORMULA] was set more or less arbitrarily to 90 ppm (i.e. slightly below the current derived limit) and to 45 for the line of sight where a weak UV bump is observed (case of HD 37023). Smaller values than 90 ppm for the other two lines of sight would require more extreme Lorentz oscillator parameters and extinction curves may exhibit some substructure (see Fig. 4), and so it might be more difficult to obtain a good fit.

Fig. 8 shows [FORMULA] for spheres using [FORMULA] and [FORMULA] (shown in the middle and bottom panels, respectively). The dielectric function was computed from the parameters of two-Lorentz oscillator models. These parameters, listed in Table 4, were chosen to reproduce (assuming spheres) the extinction along three lines of sight spanning a wide range in FUV curvature (Fitzpatrick & Massa 1988; 1990). The lines of sight are in the direction of HD 204827 (top solid line, modelled by the dotted lines), HD 229196 (middle solid line, modelled by the dashed lines), and HD 37023 ([FORMULA]  Ori D; lower solid line, modelled by the dash-dotted lines). The values of [FORMULA] were taken from Cardelli et al. (1989). In Table 5, the resulting fitting parameters obtained for spheres, CCA clusters and compact clusters are compared to those of these interstellar extinction curves. As can be shown by the [FORMULA] values (relative to [FORMULA], not [FORMULA] ; in units of [FORMULA]), the fit is excellent. Surprisingly, the fit can actually improve in going from spheres to clusters (case of HD 204827). This is mainly due to a better fit of the FUV curvature, not the UV bump. In case of HD 37023 the fit becomes slightly worse (due to a slight FUV curvature mismatch) in reducing [FORMULA] from 90 to 45 ppm. Both oscillators are shifted to smaller wavenumbers in case of [FORMULA]  ppm, and, as expected, the fitted UV oscillator appears to be much stronger (larger [FORMULA]) than for the case of 90 ppm. Interestingly, [FORMULA] is even more reduced in case of 45 ppm. At any rate, the [FORMULA] values are usually smaller than in the single-Lorentz oscillator case (see entries [FORMULA] in Tables 2 and 3). Thus it appears that an extra background under the UV bump (due to the FUV oscillator contribution to the UV oscillator) makes the bump profile less sensitive to shape and clustering.


Table 4. Parameters of two-Lorentz oscillator models to reproduce interstellar extinction curves along selected lines of sight


Table 5. Fitting parameters of synthetic interstellar extinction curves compared to actual ones

Unfortunately, given the already large number of free parameters (7), the models are not unique. But they give a qualitative idea of what the optical constants of actual bump grains might look like in various environments. Note that for curves exhibiting less FUV curvature, the second Lorentz oscillator must be shifted to larger x, otherwise the fit can be very poor in the FUV portion of the observed range. The FUV curvature observed in the direction of HD 204827 is so large ([FORMULA]) that no FUV Lorentz oscillator with [FORMULA] can reproduce it (these tend to give [FORMULA]). These large curvatures can only be generated by bringing the FUV oscillator closer to the UV oscillator ([FORMULA]). The fit is slightly poorer in that case for spheres since the tail of the FUV Lorentz oscillator significantly differs from the shape of this curvature. The bump peak position, [FORMULA], is sensitive mainly to [FORMULA] (a [FORMULA] variation in [FORMULA] produces a [FORMULA] variation in [FORMULA] - which is a substantial part of the observed range of variation of [FORMULA]), and to a lesser extent, [FORMULA]. The bump strength [FORMULA] is sensitive to [FORMULA] and [FORMULA], whereas the curvature [FORMULA] is sensitive to [FORMULA]. Table 5 also shows that clustering, though still broadening the UV bump compared to spheres, systematically shifts the peak position of the bump to smaller wavenumbers, which is contrary to what was found for single-Lorentz oscillator models (and contrary to observations). Thus, the inclusion of a background dielectric function (arising from the oscillator associated with the FUV peak) inhibits the conditions under which a shift to larger wavenumbers is possible. Therefore, if the optical constants of Fig. 8 are representative of the interstellar UV feature carrier, then intrinsic variations in chemical composition, rather than clustering, may be responsible for this shift.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998