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

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4. Single-Lorentz oscillator models

Actually, in view of the widely assumed formation mechanisms of the carbonaceous component responsible for the UV bump, the highly anisotropic dielectric function of planar graphite is unlikely to be a good model of the purported dielectric function of the bump carrier. First, the exposure of the carbonaceous interstellar grains to UV radiation, though conducive to a "graphitization" of the material through dehydrogenation, is unlikely to produce perfect graphite sheets as an end-product (Sorrell 1990). There must still be considerable defects in the structure, the topology of the grains being a collection of randomly oriented graphitic crystallites in a sp3 bonding matrix (possibly containing some hydrogen). This structure is more characteristic of amorphous carbon (especially UV processed or annealed hydrogenated amorphous carbon - HAC; Fink et al. 1984; Mennella et al. 1995a, b). Furthermore, modification of the optical constants of graphite have already been shown to be necessary in order to reproduce even the most basic properties of the interstellar UV feature (Mathis 1994). An intrinsic variability in chemical composition must also be considered in order to satisfy the observational constraints.

Virtually all UV bumps observed to date can be reproduced extremely well by a "Drude" profile. This is the profile generated by a sphere whose optical properties are characterized by a single-Lorentz oscillator model. Thus, in this section, we consider a single-Lorentz oscillator model to approximate the dielectric function hypothesized for the interstellar UV feature carrier, and see how shape and clustering can affect the peak position, width, and shape of the UV bump. We assume that variability in the width is attributable to shape and clustering effects, as well as to intrinsic variations in chemical composition along different lines of sight. Variability in the peak position, which is observed to be uncorrelated with width, is attributed to variations in mean chemical composition alone.

The dielectric function of a Lorentz oscillator is given by [FORMULA], where [FORMULA], [FORMULA], and [FORMULA], are the plasma frequency, the peak position, and the damping constant, respectively, all in units of the inverse wavelength ([FORMULA]). A Drude model is characterized by [FORMULA] and usually describes metals (or semi-metals, like graphite). Through these parameters, the model can simulate the variability of the chemical composition of the grains and provide some physical insight into the mechanisms involved.

Interstellar grains cannot be expected to be perfectly smooth spheres in all environments. For example, one may expect aggregation of the primary grains, especially in denser regions. Furthermore, the grains could be characterized by surface roughness and/or by porosity, as well as chemical inhomogeneities. An interesting question is whether such shape and clustering effects (neglecting chemical inhomogeneities within a given grain) can conserve the Drude-like profile that is observed, along with the other observational constraints.

4.1. Combining a Lorentz oscillator model with shape and clustering

The effect of shape and clustering on the scattering properties of chemically homogeneous grains in the Rayleigh limit can be modelled via a spectral density, [FORMULA], of the geometric factor L, where [FORMULA] (Bohren & Huffman 1983; Fuchs 1987; Rouleau & Martin 1991). The requirements are that the zeroth and first moments of [FORMULA] are unity and [FORMULA], respectively. This approach is closely related to the "Bergman representation" which is based on effective medium theory in the case of a binary mixture (Stognienko et al. 1995). The two approaches are equivalent if one assumes that one of the components is vacuum. However, the spectral density [FORMULA] can be directly interpreted in terms of shape and clustering of small grains, whereas the Bergman representation deals with bulk material, and so gives only indirect information about the scattering properties of small grains.

Combining a single-Lorentz oscillator model of dielectric function [FORMULA] with a model of shape and clustering using [FORMULA], the extinction cross section per unit volume can be written as


where [FORMULA] denotes the imaginary part. The familiar form of [FORMULA] for spheres is obtained from [FORMULA]. Note that [FORMULA] using this model has the form of the "bump" term in Eq. (1) if [FORMULA]. In that case, the maximum [FORMULA] occurs close to


One advantage of this type of representation is that a statistical ensemble of grains of various shapes or clustering states can also be represented by an average [FORMULA]. Actually some mean [FORMULA] and mean [FORMULA] could even describe qualitatively a statistical ensemble of grains with varying shape and chemical composition, as long as there are no correlations between the two.

The irregular grains considered here arise from the agglomeration of spheres (i.e. the spheres can touch their neighbour at one point, but they are not allowed to interpenetrate). In the case of agglomerates of spheres, the spectral representation approach brings about a considerable reduction in computer burden compared to direct computations using a multiple Mie sphere code since the problem has to be solved only once. Results between the two approaches in the case of clusters of spheres using the optical constants of graphite are very similar.

To compute the spectral density [FORMULA] we use a code kindly provided by Hinsen & Felderhof (1992). This code computes ensemble-averaged electromagnetic interactions between identical spheres in the electrostatic limit. As stated in Stognienko et al. (1995), the spectral density depends on the choice of the maximum order for the multipole expansion. We present the spectral densities computed with multipole expansion order of 9 for the [FORMULA] CCA clusters and the [FORMULA] compact clusters in Fig. 3. Note that a multipole expansion order of 6 is sufficient to obtain convergence for the [FORMULA] results using Eq.  2 with the optical constants of graphite.

[FIGURE] Fig. 3. Spectral density for [FORMULA] CCA clusters (dashed line) used in Fig. 4, and spectral density for [FORMULA] compact clusters (solid line) used in Fig. 5. The vertical line represents the spectral density of spheres, a delta function at [FORMULA]

For our simple Lorentz model, we assign one of the smallest widths derived from interstellar extinction curves, [FORMULA] (for HD 93028; Fitzpatrick & Massa 1986), to spheres. This might not be completely correct, but the sphere is a useful reference shape. We assume here that broader widths arise exclusively from shape and clustering effects without any variation in chemical composition. We choose the peak position as being the mean value, [FORMULA]. This is an arbitrary choice since the narrowest profiles ([FORMULA]) span almost the entire range of [FORMULA], from [FORMULA] to [FORMULA]. Other choices of [FORMULA] would yield qualitatively similar results, but simply shifted in wavenumber. These would correspond to intrinsic variations of the assumed chemical composition of the grains.

For spheres, a pure Drude profile is obtained. Due to shape and clustering effects, however, the synthetic bump computed from [FORMULA] may differ from a pure Drude profile. We wish to find out under which conditions such clustering models can produce an increase in width of the bump without appreciably changing its peak position or its Drude-like shape.

4.2. Results for CCA clusters

Fig. 4 shows the extinction cross section per unit volume, [FORMULA], and the real and imaginary part of the dielectric function, [FORMULA] and [FORMULA], for various single-Lorentz oscillator models using the [FORMULA] computed from [FORMULA] CCA clusters. The models were parameterized using [FORMULA] and [FORMULA]. This choice of [FORMULA] gives rise to a peak at [FORMULA] for spheres ([FORMULA] in Eq. [ 3]). The damping constant is set to [FORMULA]. For spheres, this produces a Drude profile of width [FORMULA].

[FIGURE] Fig. 4. [FORMULA], [FORMULA], and [FORMULA], for various single-Lorentz oscillator models and the [FORMULA] computed from [FORMULA] CCA clusters. By order of increasing amplitude, the models are given by [FORMULA] and 0.0 (solid lines). The dotted lines are the fit using Eq. (1)

In order of increasing amplitude, results shown are for [FORMULA] and 0.0 (solid lines). Also shown for comparison are the fits using the procedure of Fitzpatrick & Massa (1990; dotted lines). Note the considerable amount of structure in the profiles, except for the weakest model shown, [FORMULA]. The corresponding [FORMULA] is shown in Fig. 3 (dashed line). A comparison between the two figures indicates that the [FORMULA] of models with the smallest [FORMULA] ("strong" Lorentz models, i.e. closest to a Drude model) are just a convolution of [FORMULA] with a broadened Drude profile. Thus, for "strong" Lorentz models, extra peaks in the profile of [FORMULA] will translate into structure in the [FORMULA] curves. Conversely, for "weak" Lorentz models ([FORMULA] in Eq. [ 2]) the profile will be close to a "Drude" profile, irrespective of the form of [FORMULA].

Table 2 is similar to Table 1, but for the one-Lorentz oscillator models using the [FORMULA] of CCA clusters. In the present case, [FORMULA] and [FORMULA] are meaningless for [FORMULA] since the curves are poorly represented by a "Drude" profile (thus being inconsistent with observations of actual interstellar extinction curves). Again, note the significant discrepancies between "PEAK" and [FORMULA], and between "FWHM" and [FORMULA] because of the spurious fitted linear rise. Only [FORMULA] values of less than 500 are listed, corresponding to models that are acceptable. Note that, though the curves are not strictly Drude-like, "PEAK" shifts to larger values compared to that of spheres (4.60 [FORMULA]) for broader profiles, precisely what is observed in denser interstellar environments. This is related to the fact that the [FORMULA] profile peaks at a value of L larger than 1/3 (the value of L for spheres, indicated by a vertical line in Fig. 3).


Table 2. Same as Table 1 but for single-Lorentz oscillator models parameterized by [FORMULA] in the case of [FORMULA] CCA clusters (PEAK  [FORMULA] and FWHM  [FORMULA] for spheres). "MAX" (cm-1) is the computed peak value [FORMULA]

The ratio of carbon locked up in interstellar grains relative to hydrogen, [FORMULA], is thought be be around [FORMULA] part-per-million (ppm) by number (Cardelli et al. 1996). Assuming the bump grains have a density of 2 g cm-3, then only the "weakest" Lorentz models ([FORMULA]) require more carbon than available (cf. the values of [FORMULA] in Table 2). As expected, putting all the carbon in a single extinction feature requires only modest amounts of the element.

Thus cluster-cluster agglomeration (i.e. fluffy interstellar grains) can reproduce the observational constraints relating to the interstellar UV feature, provided the bump grains are described by a single-Lorentz oscillator model with [FORMULA] - [FORMULA].

4.3. Results for compact clusters

We also computed [FORMULA] using Eq. (2) for a [FORMULA] corresponding to an ensemble of compact clusters containing [FORMULA] spheres. Fig. 5 is similar to Fig. 4, but for an ensemble of [FORMULA] compact cluster models. The corresponding [FORMULA] is shown in Fig. 3 (solid line). Note that the [FORMULA] of compact clusters also peaks at [FORMULA], but that it is more "compact" and contains less structure than the [FORMULA] of CCA clusters. This translates into [FORMULA] profiles that are closer to a Drude profile, except again for "stronger" Lorentz models ([FORMULA]). This emphasizes how tightly constrained the shape and clustering of grains described by the stronger single-Lorentz oscillator models are in terms of their allowable [FORMULA] profile. Table 3 lists the same parameters as Table 2 for the various compact clusters using these Lorentz models. Models with [FORMULA] satisfy most of the observational constraints. Only models with [FORMULA] violate the cosmic carbon abundance constraint. Note again the trend of larger FWHM's usually leading to larger values of "PEAK", as required observationally.

[FIGURE] Fig. 5. Same as Fig. 4, but for a [FORMULA] computed from [FORMULA] compact clusters (solid line in Fig. 3)


Table 3. Same as Table 2, but in the case of [FORMULA] compact clusters

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998