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Wavelength dependence

Interpretation of the interstellar extinction involves two tasks: explanation of the wavelength dependence and fitting the absolute extinction. The first task assumes the searching for chemical composition and sizes of particles which give the wavelength dependence of the extinction efficiencies close to the observed dependence of $A^{(n)}(\lambda)$.

Figure 7: Wavelength dependence of the extinction efficiency factors for homogeneous spherical particles of different sizes consisting of astronomical silicate and amorphous carbon. The dashed segment shows the approximate wavelength dependence of the mean galactic extinction curve at optical wavelengths.

The average interstellar extinction curve in the visible-near UV ( $1 \,{\mu}\rm {m}\leq\lambda^{-1} \leq 3 \,{\mu}\rm {m}$) presented in Fig. 13 in Voshchinnikov [2002] can be approximated by the power law $A(\lambda) \propto \lambda^{-1.33}$. This dependence is plotted in Figs. 7-9 as a dashed segment. The Figures allow one to judge the influence of the size, chemical composition, structure and shape of particles on the wavelength dependence of extinction. In all cases, the initial growth of extinction reflects the increase of factors $Q_{\rm ext}(m,x)$ at the interval from zero up to the first maximum. As follows from Fig. 7, spheres of astrosil with $r_{\rm s} \approx 0.1 \, {\mu}\rm {m}$ or slightly smaller spheres of AC1 can produce the dependence $A(\lambda)$ resembling the observed one. Evidently, the fraction of particles with these radii in the size distribution must be considerable. As was noted many times (e.g., Greenberg, [1978]), a similar extinction occurs if the product of the typical particle size $\langle r \rangle$ on the particle refractive index is constant, i.e.

$$\langle r \rangle \, \vert m-1\vert \approx {\rm const.}$$ (5)

Using the values of $m$ from Table 3 in Voshchinnikov [2002] and the results shown in Fig. 7, it is possible to conclude that the wavelength dependence of extinction in the visible can be approximately reproduced if one choose the particles of astrosil with $r_{\rm s} \approx 0.1 \, {\mu}\rm {m}$, particles of AC1 with $r_{\rm s} \approx 0.08 \, {\mu}\rm {m}$, particles of iron with $r_{\rm s} \approx 0.04 \, {\mu}\rm {m}$, etc. This illustrates that from the wavelength dependence of extinction one can determine only the product of the typical particle size on refractive index but not the size or chemical composition of dust grains separately.

Figure 8: Wavelength dependence of the extinction efficiency factors for spherical particles of astronomical silicate with radius $r_{\rm s}=0.1\,{\mu}\rm{m}$. Calculations were made for homogeneous particles with the refractive indices found from the Bruggeman mixing rule with a different fraction of vacuum (upper panel) and for hollow particles with different core size (lower panel). The dashed segment shows the approximate wavelength dependence of the mean galactic extinction curve at optical wavelengths.

The conclusion on the impossibility to identify exactly the structure of particles can be done from Fig. 8 where the extinction for spheres with a fraction of vacuum is presented. The voids were included in two ways: using the Bruggeman mixing rule (see Table 4 in Voshchinnikov [2002]) and in the form of the core in core-mantle spheres. In both cases, the ``effective'' refractive index of particles reduces, and according to Eq. (5) to produce the observed extinction, the particle radius over $r_{\rm s}=0.1\,{\mu}\rm {m}$ must be increased. It is interesting to note the similarity in behaviour of extinction for particles with different internal structure. This shows that the EMT is not a totally hopeless matter.

Figure 9: Wavelength dependence of the normalized extinction cross-sections for spherical and spheroidal equivolume particles of astronomical silicate with radius $r_V=0.1\,{\mu}\rm{m}$. Calculations were made for homogeneous spheroids with $a/b=2$ in a fixed orientation ( $\alpha = 0\hbox{$^\circ$}$ and $90\hbox{$^\circ$}$) and for 3D-orientation. The dashed segment shows the approximate wavelength dependence of the mean galactic extinction curve at optical wavelengths.

In Fig. 9, the normalized extinction cross-sections $C_{\rm ext}/\pi r^2_{\rm V}$ for spheroids (see Eqs. (2.43) and (2.44) in Voshchinnikov [2002]) and spheres are plotted. The results for spheroids are shown for two orientations of non-rotating particles (``picket fence'' alignment) and for the case of the arbitrary orientation in space (3D alignment) when the average cross-section is:

$$\langle C_{\rm ext} \rangle^{\rm 3D} = \int^{\pi /2}_0 \, \f... ...m TE}(\alpha) \right] G(\alpha) \,\sin\alpha \, {\rm d}\alpha.$$ (6)

Here $G$ is the geometrical cross-section of a spheroid (Eqs. (2.41) and (2.42) in Voshchinnikov [2002]) and the incident radiation is assumed to be non-polarized. Figure 9 shows that the shape of particles has a small influence on the extinction at different wavelengths. Certainly, the curves for spheroids in a fixed orientation differ from those for spheres, and the difference increases with the growth of $a/b$. However, the wavelength dependence of extinction close to the observed one can be obtained if the difference in the paths of the rays inside particles with different orientation is taken into account. It means that the particles with $r_V$ smaller than 0.1 ${\mu}\rm {m}$ for $\alpha=0\hbox{$^\circ$}\,(90\hbox{$^\circ$})$ and larger than 0.1 ${\mu}\rm {m}$ for $\alpha=90\hbox{$^\circ$}\,(0\hbox{$^\circ$})$ for prolate (oblate) particles should be chosen.

Thus, neither chemical composition, nor structure, and shape of dust particles can be uniquely deduced from the wavelength dependence of the interstellar extinction.