3. Dust optical properties
3.1. Infrared emission bands and the underlying infrared continuum
To account for the presence of the infrared emission features in the spectrum of IRAS 22272, we have included in our code PAH grains, assuming that PAHs have the same enthalpy as graphite grains (see Siebenmorgen & Krügel 1992) - knowledge of the enthalpy is required during non-equilibrium temperature calculations. The cross-sections for the PAH grains are based upon the analytical formulae given by Desert et al. (1990). Some changes have been made in the discrete PAH feature strengths following the values in Schutte et al. (1993) because their values for the bands near 11 µm seem to be better. Some plateau shapes have also been added to the cross-section values based upon the observations of Bregman et al. (1989). In addition the equations from Desert et al., which were only intended to apply for wavelengths greater than the Lyman limit, were not used at wavelengths less than 912 Å as they give values that continue to increase rapidly as the wavelength decreases. Instead a linear extrapolation of the values near the Lyman limit was used for the shorter wavelength cross-sections, producing values that do not rise very rapidly as the wavelength decreases. In addition the sharp cut-off of the optical cross-section at 8000 Å in the paper of Desert et al. has been replaced by an exponential decline with increasing wavelength that starts at 2500 Å and which is set to produce a feature to continuum ratio of 100 at 3.3 µm. This was done in part to avoid discontinuities in the opacity values and in part to allow the possibility of modeling the unusual red emission seen in a few reflection nebulae that have strong PAHs emission. It should be noted that the long wavelength opacity function for the PAH grains is poorly known, and therefore the infrared emission from the PAH grains is suspect aside from the discrete features. This is not a problem in practical situations however, because the PAH grains are not observed to occur without other forms of dust which dominate the emission except at the discrete PAH features.
The PAH grain cross-sections were calculated for 12 logarithmically spaced grain radii from 3.5 Å up to 50 Å and, as an example, the adopted mass absorption coefficient () for PAHs of 10 Å radius ( 120 C atoms) in the wavelength range from about 0.25 to 125 µm (in total, 256 's from 100 Å up to 3000 µm were used in the radiative transfer calculations) is shown in Fig. 1.
As the other form of carbon-based dust appearing together with the PAHs we have chosen amorphous carbon of type AC (soot produced by striking an arc between two amorphous carbon electrodes in a controlled argon atmosphere - see Bussoletti et al. 1987 for details). Using real and imaginary parts of the complex refractive index computed for AC dust by Rouleau & Martin (1991) we have calculated absorption and scattering efficiencies on the basis of Mie theory for 30 logarithmically spaced grain radii from 10 Å up to 1 µm. Finally, we have assumed that the dust responsible for the infrared emission bands and the infrared continuum emission in post-AGB objects is made of: PAHs for 5 Å a 10 Å, amorphous carbon grains for a 50 Å and dust with an opacity obtained from averaging of the absorption efficiences for PAH and AC grains according to the formula
for grain sizes between 10 and 50 Å. Here f = 1 for a = 10 Å and f = 0 for a = 50 Å. The last form of dust was introduced to keep continuous distribution of dust grain sizes and to fill the gap between properties of large carbon-bearing molecules and small grains.
We performed a few numerical trials trying to incorporate graphite grains into our model (see Laor & Draine 1993 for the most recent source of the optical opacity data for graphite). However, we found that the absorption efficiency of graphite is too steep in the far-infrared (FIR) and also in the mid-infrared (MIR). In the FIR range of the spectrum our model is not able to predict enough flux at 100 µm. Only increasing the number of cold graphite grains (e.g. by assumption that decreased significantly during the formation of the shell) could explain the observed far-infrared emission. On the other hand, if we try to match an observed continuum level at around 18-19 µm the predicted emission at near-infrared wavelengths is too high. This seems to suggest that the dust around the C-rich post-AGB objects is amorphous rather than graphitic in form.
3.2. Features around 21 and 30 µm
IRAS 22272 5435 is one of the four post-AGB sources where the 21 µm feature was discovered by analysis of the IRAS LRS database (Kwok et al. 1989). Subsequent measurements made on the KAO have detected a prominent emission band around 30 µm in four 21 µm sources including IRAS 22272 (Omont et al. 1995b). It has been proposed that the 30 µm emission band, which is also detected in C-rich AGB stars and post-AGB stars including planetary nebulae (see Omont et al. 1995b and references therein), could be due to solid magnesium sulfide, MgS (Goebel & Moseley 1985). Recently, Begemann et al. (1994) tabulated the optical constants of Mg-Fe sulfides as derived from laboratory measurements in the wavelength range from 10 to 500 µm. In Fig. 2 we display the mass absorption coefficient for a mixture composed of 90 % MgS and 10 % FeS as derived by us from the optical constants of Begemann et al. (1994). The dotted line shows the results based on Mie theory and the short dashed line the mass absorption coefficients for a continuous distribution of ellipsoids (hereafter CDE: see Bohren & Huffman 1983). As compared to the Mie theory, the CDE computations predict a feature which is much broader with a peak shifted towards longer wavelengths at about 35 µm. We note that the lack of the refractive index data for sulfides in the short wavelength range prevents detailed quantitative modeling. An empirical approach to estimate their FIR emission will be given in Sect. 4.
As far as 21 µm band is concerned the problem of the identity of the emitter remains unknown. Currently, the most accepted materials responsible for this feature in C-rich PPNe seem to be PAHs or PAH clusters (a much wider discussion of the possible carriers of the 21 µm emission band as well as physical and chemical conditions necessary for their production can be found in recent papers by Kwok et al. 1995, Omont et al. 1995b and Henning et al. 1996). While Kwok et al. and Omont et al. suggested that 21 µm feature is present (excited) only during a short PPNe phase of evolution, Hening et al. found that this band is observed in some young stellar objects as well. It could mean that the physical and chemical conditions which produce the carrier of the 21 µm feature in the circumstellar environment (e.g. high speed molecular stellar winds and/or high velocity streaming of grains through gas) are similar during these two different stages of stellar evolution.
Since the real carriers of 21 and 30 µm bands are still not identified (and therefore their optical properties are unknown), we adopted an empirical opacity function (EOF) which takes into account both these features to perform a quantitative modeling of the energy distribution. This empirical opacity was defined by adding features onto the mass absorption coefficient for AC amorphous carbon grains with radius of 0.1 µm. In principle, is constant in the mid- and far-infrared (i.e. is independent of a), so the exact choice of the AC grain size is unimportant. Also, scattering efficiencies are small at these wavelengths, so changes were made only to the absorption opacity. For the 21 µm emission band we approximated its shape by a gaussian fit with a centre wavelength of 20.6 µm and a width at half maximum of 1.5 µm as determined from the source IRAS 07134+1005 which has the strongest known 21 µm feature (Kwok et al. 1989). For the 30 µm emission band shape a simple gaussian shape is inadequate because the feature is asymmetric with the short-wavelength side of the peak significantly sharper than the long-wavelength side. Therefore we fitted the 30 µm band with two half-gaussians instead, having the same central wavelength and peak strength but different widths. The values of these parameters as determined from modelling of IRAS 22272 are: 27.2 µm for the central wavelength and 4.0 and 9.0 µm for the widths at half maximum. It was necessary to perform several radiative transfer calculations to find out the values of the feature parameters that gave a reasonable fit.
The resulting opacity function including the 21 and 30 µm features and the mass absorption coefficient of AC amorphous carbon is displayed in Fig. 2 as a solid line. For comparison purposes, Fig. 2 also shows the underlying continuum due to pure AC grains (long-dashed line). We note that for the EOF is much smaller at 30 µm as compared to that for a mixture of Mg and Fe sulfides. However, this is not crucial since the empirical opacity was added to the opacities of the amorphous carbon grains implying that the strength of the EOF is determined by the required amount of the AC providing the best fit to the infrared continuum emission. The shape of the opacity function is much more important. It is clear from Fig. 2, that long-wavelength side of the EOF lies in between for the mixture of sulfides calculated from the Mie theory (spherical grains) and from CDE approximation (distribution of ellipsoids).
The strong dependence of the MgS absorption properties on the shape of the grains is a consequence of the very large values for the refractive index (see Begemann et al. 1994). Because our opacity function gives an excellent fit to the observed energy distribution of IRAS 22272 (see below) we can conclude that if sulfides are indeed responsible for the observed 30 µm feature then the shape of these grains is quite important. A simple variation in the distribution of the dust grain shapes could change the long-wavelength shape of this feature. Finally, it is worth noting that the short-wavelength parts are quite similar for all of the opacities used for modelling of the 30 µm emission band.
3.3. Carbon grains coated by sulfide particles
In order to identify the carrier of the 30 µm feature, we have tested the possibility of MgS condensing on the surface of AC grains. The material responsible for the feature in IRAS 22272 must absorb as much as 20 % of the heating ultraviolet (UV) and visual radiation which is absorbed by dust. Therefore, the material must either have sufficient absorption cross section or exist as a thin mantle on grains of sufficient cross section (Nuth et al. 1985). To calculate the optical properties of spherical AC (core) grains coated by MgS (coat), we used the method and the computer code described in Bohren and Huffman (1983). The thickness of the coating layers was computed assuming that all available sulphur (, e.g. Aller & Czyzak 1983) is tied up in Mg-Fe sulfides composed of 90 % of MgS and 10 % of FeS and that all the coating material is distributed proportionally to the surface areas of the AC grains. One of the consequence of such an assumption is that smaller AC grains will have thicker mantles (relative to their radius). For the parameters describing the dust size distribution of pure AC grains in our model of IRAS 22272 (i.e., Å, µm, 1 and dust-to-gas ratio ), we obtained radii of the coated grains up to 11 % larger than the radii of the core grains in the case of the smallest (50 Å) AC particles and only 0.3 % greater in the case of the largest (0.25 µm) ones (for larger the ratio of the radii would be even smaller).
In Fig. 3 we present the mass absorption coefficient of the homogeneous AC spheres of different radii coated with a homogeneous layer of Mg-Fe sulfides as computed under the above assumptions. The scale and the range of the and wavelengths are the same as in Fig. 2 to allow a direct comparison between the expected features. As one can see for thin sulfide mantles ( / ), which are required because of the sulphur abundance constraints, we obtain two peaks at about 24 and 38 µm with a broad minimum around 28-32 µm. Since all the conditions (described in Appendix B in Bohren & Huffman 1983) which are required for the code to work properly are fulfilled these features are believed to be real. Similar results were obtained by B. Glaccum (private communication). Glaccum also tested the case of the core-mantle spheroidal grains, in the Rayleigh limit, and found that for thin mantles the peaks do not really shift in frequency, but are approximately independent of the shape of the grains. The two peaks can be interpreted as the interface modes at the AC/sulfide and sulfide/vacuum interfaces. This interpretation is confirmed by analysis of the polarizability of small coated spheres, which shows that the calculated peaks occur at the positions where conditions for singularities of the polarizability are the best fulfilled (Ossenkopf, private communication).
On the other hand, for thick MgS-FeS mantles on the AC core ( / ) the feature is close to the observed one, but one would have a relatively small number of grains with thick mantles and most grains would be uncoated due to the limited amount of sulphur available. We did not perform radiative transfer calculations for the case of thick mantles for two further reasons: first it is not clear how to define populations of coated and uncoated grains, and, second it is not exactly known how sulfides condense. During condensation on carbon grains other materials (such as SiC) could condense at the same time so that the mantles formed need not necessarily be homogeneous. On the other hand, when condensation on AC grains is likely a separate MgS dust component seems to be possible as well.
© European Southern Observatory (ESO) 1997
Online publication: July 8, 1998