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Astron. Astrophys. 322, 633-645 (1997)
3. Clustering effects on the graphite UV feature
In this section we investigate whether clustering of grains
composed of graphite can lead to an increase of the width of the UV
feature without appreciably changing its peak position.
Draine (1988) and Draine & Malhotra (1993) have confirmed the
good agreement between the so called "1/3-2/3" approximation and DDA
computations taking explicitly the anisotropy of graphite into
account. In this approximation extinction cross sections
and are obtained
separately for isotropic particles having the components of the
dielectric tensor parallel and perpendicular to the c-axis of
graphite, respectively. They are then combined by taking 1/3 of the
first contribution, and 2/3 of the second, respectively. This
procedure is exact in the electrostatic limit for spheres, and is an
excellent approximation in the Rayleigh limit ( ,
where is the radius of the equal-volume
sphere). It is also expected to be a good approximation for more
complicated shapes in the Rayleigh limit provided one assumes that the
c-axis of graphite is randomly oriented with respect to any symmetry
axis that might be present within the particles.
There is no consistent report of the interstellar UV feature being
polarized (i.e. the particles are not partially oriented elongated
particles). Therefore, in this paper, we assume that the particles are
randomly oriented.
It has been confirmed (Witt 1989; Witt et al. 1992; Calzetti et al.
1995) that the extinction in the vicinity of the interstellar UV
feature is consistent with pure absorption (i.e., carrier with size in
the Rayleigh limit). This fact is exploited to simplify the
orientational average by taking only three mutually perpendicular
orientations of the incident electric field with respect to the
particles. This averaging procedure is also exact in the electrostatic
limit and is a good approximation in the Rayleigh limit. An added
advantage of the Rayleigh limit is that one needs not worry about size
distributions of grains since extinction is independent of size
(except perhaps indirectly through grains of different sizes possibly
having different shapes).
It is of interest to see how the width, shape, and peak position of
the graphite UV feature change in the case of ensembles of
agglomerated spheres to simulate clustered or irregularly shaped
interstellar grains. Such a configuration can be handled by a multiple
Mie sphere code (Mackowski 1991; Rouleau 1996), provided one assumes
that the spheres, though touching, do no interpenetrate each other,
and are isotropic. Therefore, we have performed multiple Mie sphere
computations to calculate extinction cross sections per unit volume,
, for various simple arrangements of spheres and
plausible clustering models.
The two types of clusters considered are a cluster-cluster
agglomerate (CCA), and a compact cluster. A cluster-cluster
agglomerate is constructed from hierarchically joining together
clusters containing the same number of spheres, N, in an
progression (see, e.g., Stognienko et al.
1995). These are rather fluffy and irregular in shape (in the case
considered here, the filling factor with
respect to a sphere enclosing the cluster is about 0.04). Another
alternative is to use a particle-cluster agglomerate (PCA) in which
individual spheres are added to the cluster with a sticking
probability of unity. For the small number of spheres used here, both
CCA and PCA clusters yield similar results and so PCA clusters are not
included. A compact cluster represents the tightest arrangement of
spheres possible, with each sphere touching many neighbours (Rouleau
& Martin 1993; Rouleau 1996). Again, the spheres are not allowed
to interpenetrate. The filling factor of a compact cluster is thus
much larger than either CCA or PCA clusters (about 0.26 for an
cluster). This type of cluster could be the
result of a low sticking probability of the grains or a compaction of
an initially looser agglomerate after some disruptive event (like
interstellar shocks).
The clusters are assumed to be in the Rayleigh limit. The optical
constants of Draine (1985) are used for the computations.
Qualitatively similar results are obtained for other choices of
optical constants of graphite. A multipole expansion order up to 6 was
necessary to obtain convergence for the results
with the multiple Mie sphere calculations.
There are three distinct ways to deal with the anisotropy of
graphite in the multiple Mie sphere code. The first way is to use the
"1/3-2/3" rule, i.e. to average the extinction cross sections of the
agglomerates with either or
. The second way is to randomly assign
or with probabilities
1/3 and 2/3 to the individual spheres forming the aggregates. The
third way is to average the components of the dielectric tensor and to
use the resulting dielectric function in the multiple Mie sphere
computations. These different approaches should be good approximations
if we assume that there is no correlation between the orientations of
the anisotropic spheres within the clusters and the direction of their
individual c-axes. This is true for the regions outside of the
resonance. However, it turns out that the results of the three methods
differ from each other markedly in the bump region (Fig. 1). E.g.
for the CCA cluster the averaged dielectric function (third way) leads
to a much broader feature ( ) at smaller
wavenumbers ( ) compared to the results obtained
with the "1/3-2/3" rule (results for other arrangements of spheres are
discussed in detail below) and the random assignment (second way).
Furthermore, the feature obtained with the "1/3-2/3" rule shows a
small shoulder in the long-wavelength wing. The appearance of the
shoulder is explained by the rather long linear chains of the strong
"oscillators". In case of the random assignment
the chains are interspersed by the weak
"oscillators" which makes the feature less sensitive to the detailed
cluster structure (see Sect. 4.2). The averaged dielectric function
(third way) is in any case also more or less moderate compared to
regardless of the used averaging procedure. The
feature position and width, however, certainly depend on the averaging
procedure (the numbers given above are for the Bruggeman mixing rule;
Bohren & Huffman 1983). Which way and which averaging procedure
have to be used to get the best approximation of the
of clusters of graphite spheres may be tested
with extensive DDA calculations in the future, but this subject goes
beyond the goals of this paper. In any case, one should note that more
detailed information about the arrangement of basic structured units
(in respect to their individual c-axes) from laboratory data are
required to get a more reliable basis for the calculations (see
Rouzaud & Oberlin 1989). The remaining discussion in this section
is restricted to the results obtained with the "1/3-2/3" rule for the
various sphere configurations.
![[FIGURE]](img34.gif) |
Fig. 1.
Cross section per unit volume, , of randomly oriented CCA graphite clusters. Three different ways (see text) are used to treat the anisotropy of graphite: "1/3-2/3" approximation (long dashed line), random assignment (solid line), and averaged dielectric function (short dashed line). The vertical line marks the interstellar UV feature peak position
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Fig. 2 shows for a single sphere, a
randomly oriented bisphere, a randomly oriented linear chain of three
touching spheres, a CCA cluster containing 16 spheres, and a compact
cluster also containing 16 spheres. Other random realizations of these
clusters, also containing 16 spheres, gave very similar results. In
particular the shoulder observed around is
reproduced in all CCA cluster realizations. The appearance of the UV
feature looks very similar if only is plotted.
Note that in going from a single sphere to chains of increasing
N, the peak decreases in amplitude, while the extinction rises
preferentially in the long-wavelength wing of the feature. The feature
of a CCA cluster is similar to that of a chain of three spheres,
except that this extinction enhancement has developed into a shoulder.
Also shown for comparison (short dashed line) is
for a continuous distribution of ellipsoids
(CDE; Bohren & Huffman 1983), a widely used approximate clustering
model.
![[FIGURE]](img38.gif) |
Fig. 2.
of a Rayleigh sphere (dotted line), a bisphere (dash-dotted line), a chain of three spheres (dash-triple-dot line), an CCA cluster (long dashed line), an compact cluster (solid line), and CDE (short dashed line). All arrangements of spheres are randomly oriented, using the "1/3-2/3" approximation to treat the anisotropy of graphite
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Table 1 lists the peak position ("PEAK"), full-width at
half-maximum ("FWHM") of the computed , as well
as and derived from a fit
similar in form to Eq. (1), with fitting coefficients
and , using a slightly
modified version of the routine given in Fitzpatrick & Massa
(1990). To indicate the degree of quality of the fit, the
value (in units of ) is
also given, which is defined in the usual way with the weight for each
data point (sampled at uniform intervals in
wavenumber) set to the (arbitrary) value 1 except for the bump region
- where the weight is
set to 2. Apart from the CDE case (which turns out to be a bad
clustering model for the bump), the parameter is
quite stable ( relative variation), whereas
is substantially increased (by
) in going from a single sphere to clusters.
These features satisfy qualitatively the observational constraints,
except for the fact that the peak position is wrong
( - compared to
). Note that there is a significant difference
between "PEAK" and , and between "FWHM" and
. This is due to the fact that a Drude profile
fit is applied to a profile that is not purely Drude-like, thus
introducing a spurious linear background (coefficients
, ) which modifies both
the peak position and the width of the feature. The fit in the case of
the compact cluster is reasonable, which is indicated by the small
value in Table 1, whereas the fit is
somewhat poorer in the case of the CCA cluster (due to the presence of
the shoulder), yielding a larger .
![[TABLE]](img55.gif)
Table 1.
Computed and fitted peak position ("PEAK" and , respectively) and width ("FWHM" and , respectively) for arrangements of graphite spheres (in )
The peak position is at a larger wavenumber than that observed for
the interstellar UV feature. This is why one has to resort to either
shape or size effects to shift the feature at the position of the
interstellar UV feature (see, e.g., Draine 1988; Mathis 1994).
Discounting the disagreement in peak position and concentrating only
on the qualitative aspect of the feature, one can surmise that some
irregularly shaped or clustered particles can in fact broaden
the feature without appreciably shifting its peak position, as
required for the purported interstellar UV feature carrier. However,
"fluffy" or elongated structures tend to enhance the long wavelength
wing of the feature, even introducing additional structure in the
feature (like the CCA cluster does). Assuming this is a general
feature of CCA clusters (see Sect. 4), it then appears that
fluffy particles are severely constrained as a possible "topology" of
the interstellar UV feature carrier. A compact cluster is an appealing
alternative, since its overall spherical shape and compactness
suppresses the appearance of additional structure in the feature, but
still allows the feature to be broadened substantially compared to
isolated spheres. The broadening, however, appears to be insufficient
to explain all the range of FWHM observed for the interstellar
UV feature. A further contributor to the broadening could be an
intrinsic variability in chemical composition. This interpretation is
consistent with the fact that the narrowest interstellar UV features
appear in all sorts of environments, diffuse and dense alike, with an
accompanying variation in peak position over the whole observed range.
We expect a clustering of grains to occur primarily in denser regions.
The ensuing broadening, however, appears to contradict the observation
that the broadest interstellar UV features (observed in dense regions
and thus presumably arising from clustering) are accompanied by a
shift of to larger values, whereas here,
it shifts to smaller values (see Table 1). But this does
not necessarily need to be the case, as shown in the following
section.
© European Southern Observatory (ESO) 1997
Online publication: June 5, 1998
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