## 5. Two-Lorentz oscillator modelsThe 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 due to the - electronic transition (Mennella et al. 1995a; Fink et al. 1984). Moreover, any plausible material in the small particle limit will produce strong absorption at . To our knowledge, no physical mechanism can suppress the - transition while keeping intact the - 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 To investigate these effects we approximate such a dielectric function as the sum of two Lorentz oscillators, giving rise to features at about (the UV bump) and 10-15 (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 , , and
, which on its own yields a
peak at 4.60 assuming a
sphere (dotted line). The FUV oscillator is given by
, , and various widths,
which on its own yields a peak at
assuming a sphere. This could be viewed as a
very crude model of the -
peak observed for HAC in laboratory extinction
experiments (Mennella et al. 1995a). The solid, dashed, dash-dotted,
and dash-triple-dot lines are for , 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 (, and 4.573
, for , and 12.0,
respectively). An extra broadening also occurs, but only on the larger
The main effect of adding such a background to the dielectric function is to increase and, to a lesser extent, , shifting the plasmon condition () 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 ## 5.1. The effect of scattering on the FUV curvatureAnother 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
. However, this is unlikely, as shown in
Fig. 7. Plotted (in order of decreasing strength) are the
Mie curves for spheres with radius
## 5.2. Synthetic interstellar extinction curvesA 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
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
(labelled
Assuming (Mathis 1994) and using
, where is the column
density of dust and is the cross section of
component where (B-V) is the total-to-selective extinction ratio in the V band. We have , where is the mass of the hydrogen atom, and and are the density and the mean volume of the grains, respectively. The density of bump grains is again assumed to be . The computed cross section per unit volume of the bump grains using two Lorentz oscillators (i.e. seven free parameters: , , , , and ) is fitted by between 3 and 8.5 . Assuming that
is linear with respect to where , , and . 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 ) do not warrant taking the extra complication of scattering into account. The value of 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 for spheres using
and (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
( Ori D; lower solid line, modelled
by the dash-dotted lines). The values of 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 values (relative
to , not ; in units of
), the fit is excellent. Surprisingly, the fit
can actually
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 © European Southern Observatory (ESO) 1997 Online publication: June 5, 1998 |