6 Model results

Figure 10: Examples of the modelling results using measure MgS optical constants in a CDE shape distribution. We show the observed spectra (black), the modelled continua (dashed), the spectra with the MgS contribution subtracted (gray) and the composite of the continuum and the MgS contributions (thin black line).

In Fig. 10 we show a few typical examples of how our model results compare to the observed spectra. In Fig. .1 we show the observed spectra, the composite of the continuum and the synthetic MgS feature and the residuals after subtracting the MgS feature for the complete sample. The fits are very satisfactory in 50 out of 63 cases. In $\sim$25 sources the synthetic spectra obtained with this very simple model are able to explain the detailed profile of the "30'' $\mu$m feature very well. The onset and range of the feature and even the slight depression between 26-30 $\mu$m are reproduced by the model. We show a zoomed view of the 30 $\mu$m region of a few sources that are very well fitted by this simple model in Fig. 11. Notice the different apparent shapes of the feature that the model is able to explain. Volk et al. (2002) discuss the "30'' $\mu$m feature in IRAS 23304 and find that they need 2 separate unidentified components in order to understand the shape of the feature. Figure 11 illustrates that this is not necessarily required.

Examining the complete sample of observed "30'' $\mu$m features and the synthetic spectra, we find there are some systematic deviations. In the sample of C-stars and post-AGB objects there are numerous examples where the major part of the "30'' $\mu$m feature is explained well by our model, but the observed spectra show excess emission in the 26 $\mu$m region. The excess is not accounted for using our CDE fits. The most extreme case is IRAS 19584 but several sources exhibit the same behaviour. In Sect. 6.1, we discuss the origin of this discrepancy.

In some cases the synthetic spectra over-predict the flux at the longest wavelengths. This can be due to the very simplistic method we have used to estimate the continuum level. As the dust optical depth decreases with increasing wavelengths, the continuum level estimated from shorter wavelengths might over-predict the true continuum level. Note, however, that the discrepancies between the modelled and the observed spectra in the 26 $\mu$m region and the 40 $\mu$m region cannot be considered completely independently. If we were to weaken the strength of the MgS feature this would yield a better fit around 45 $\mu$m but would increase the discrepancy around 26 $\mu$m. We also note that MgS produces a weak continuum contribution at 45 $\mu$m (see for example Fig. 9). This continuum contribution is already taken into account when fitting the overall continuum, but it is still present in the calculated MgS contribution. Therefore, our model may slightly over-predict the fluxes near 45 $\mu$m.

As a class, the spectra of most PNe show another systematic difference. The peak position of the "30'' $\mu$m feature lies in general at longer wavelengths than in the post-AGB sources. This is in accordance with the picture of a slowly expanding and cooling dusty envelope. We can simulate the same shift in peak position using MgS grains. However the fits we obtain fail to reproduce the relatively narrow width of the observed profile. We discuss this deviation of the profiles in Sect. 6.3.

There are 4 sources in the sample that have a broader "30'' $\mu$m feature than can be fit by our simple model. Of these sources, IRAS 13416 and CD-49 11554 show a slightly flattened and broadened feature while in the cases of RAFGL 618 and RAFGL 2688 the feature is very broad with a depression around 30 $\mu$m. The latter sources are known to have a very large dust column along the line of sight. Most likely the feature shape is due to optical depth effects. We discuss these sources further in Sect. 6.2.

Despite these systematic deviations it is clear that our simple model is able to explain the profile of the "30'' $\mu$m feature in good detail in a very wide range of objects. We conclude that the carrier of the "30'' $\mu$m feature in the C-stars and post-AGB objects is solidly identified with MgS and that the variations in peak position reflect differences in grain temperature.

Figure 11: Examples of spectra that are very well fitted with MgS in a single temperature, CDE shape distribution. The black lines represent the data and the grey line the model. The different sources have been offset for clarity. The excess around 23 $\mu$m in IRAS 18240 is due to FeS (Keller et al. 2002; Hony et al. 2002). Notice how the model is able to explain the profile of the "30'' $\mu$m feature found in the full range of objects in our sample.

   
6.1 26 $\mu$m excess

Figure 12: Some examples of sources with a 26 $\mu$m excess compared to the model spectra. We show the data (full black line) the model using the CDE calculation (full grey line) and the composite model using both spherical MgS grains and the CDE calculation (dashed line). We show below the emission of spherical MgS grains at a temperature of 250 K. The continuum in RAFGL 2155 runs above the observed spectrum at the shortest wavelengths, this may be due to molecular absorptions (see Sect. 2.2)

As discussed above we find about 25 sources that are very well fitted by our simple model. Using similar parameters we also find about 25 sources which show an excess near 26 $\mu$m. Evidently, there is a contribution from an additional dust component in the latter sources. In considering this additional dust component we find that MgS itself is the best candidate. The wavelength region where this excess occurs (26 $\mu$m) is also the wavelength where the generic main resonance of MgS occurs. Spherical MgS grains exhibit a resonance at 26 $\mu$m. In our model, the MgS profile is broader and peaks at a longer wavelength because we use the calculated absorption cross-sections in a CDE shape distribution. There is no a priori reason why the shape distribution should be close to this distribution. In this respect, it is rather surprising that our simple model works so well for so many sources. We can simulate a different shape distribution by adding the contributions of spherical grains to the CDE profile. This approach is permitted since we assume optically thin emission and thus the contribution of different components add linearly. In Fig. 12, we show the result of such a composite model for one C-star and for one post-AGB object. The relative amounts of spherical grains added to the model is $\sim$35 per cent. We keep the temperature of the spherical and the CDE grains the same as in the initial model. As can be seen the spherical grains contribute at the position where our initial model fails. We conclude that variations in the distribution over grain shapes can explain the variations we observe in the profile of the "30'' $\mu$m feature.

   
6.2 Optically thick shells

The feature found in CD-49 11554, IRAS 13416, RAFGL 618 and RAFGL 2688 is different from the others in the sense that it is broader and flatter. This is especially true for RAFGL 618 and RAFGL 2688 where the profile even shows a central depression. We investigate the effect of optical depth on the emission profile. We model the effect of optical depth with two simple limiting cases.

  

$$I_{1}(\lambda) B(\lambda,T_{\rm MgS}) \times (1-{\rm e}^{-\rho\kappa_{\lambda}l}) ~{\rm and}$$ (2)

$$I_{2}(\lambda) I_{0}(\lambda,T_{\rm MgS}) \times {\rm e}^{-\rho\kappa_{\lambda}l},$$ (3)

where $T_{\rm MgS}$ is the temperature of the MgS, $B(\lambda,T)$is the Planck function of temperature T, $\rho$ is the mass density of MgS, l is the column length and $I_{0}(\lambda,T)$ is MgS emission of temperature T. $I_{1}(\lambda)$ is the limiting case of a column of MgS with a single temperature of $T_{\rm MgS}$. $I_{2}(\lambda)$ represents the case of foreground absorption only; i.e., MgS emission of temperature $T_{\rm MgS}$ obscured by a column of MgS with negligible emission. In both cases the resulting profile will be broader than the optically thin case. For a long column $I_{1}(\lambda)$ approaches $B(\lambda,T_{\rm MgS})$ while $I_{2}(\lambda)$ becomes double peaked with a depression where $\kappa_{\lambda}$ peaks. The resulting $I_{1}(\lambda)$ profile will never show a central depression. We show the effects of incorporating the optical depth in Fig. 13. Indeed, these methods yield a broadened profile closer to what is observed. The second method reproduces the central depression found in RAFGL 2688. It is important to stress that the curves shown in Fig. 13 are not the result of a proper radiative transfer modelling of the CS shell or an attempt to fit the observed spectrum of the source. Nevertheless, they are able to explain the general characteristics of the "30'' $\mu$m profile in the deviant sources well.

Figure 13: The effects of optical depth of the profile of the MgS emission. We show the absorbed MgS emission following Eqs. (2) and (3) (grey dashed and solid lines), the profile of the "30'' $\mu$m feature of RAFGL 2688 (solid black line) and the optically thin MgS emission (dashed black line).

   
6.3 PNe profiles

Figure 14: The absorptivity of MgS grains coated with various types of materials. The modelled grains are composed of 80 per cent MgS and 20 per cent coating by volume. The models have been obtained by adding 5 ellipsoidal core-mantle grain of axes ratios (10:1:1), (3:1:1), (1:1:1), (3:3:1) and (10:10:1) of equal volume. For comparison we show the absorptivity of MgS in a CDE distribution (7) and the excess in NGC 7027 (8). We show the absorptivity of FeS (1) and amorphous carbon (5) grains with a MgS coating, both with core volume of 20 per cent of the grain. None of the modelled composite grains is able to explain the profile of the emission found in many PNe.

For 9 out of 13 PNe the observed "30'' $\mu$m feature is much narrower than in our model. Of the other four cases we discussed the profile of RAFGL 618 in Sect. 6.2. It is important to note that the remaining three cases (NGC 6790, IRAS 18240 and K 2-16) are the PNe with the highest continuum temperature among the PNe in our sample.

In Fig. 8, we compare the shape of the "30'' $\mu$m profile of NGC 7027 with the profiles due to differently shaped MgS grains. As can be seen an oblate MgS grain with an axes ratio of 10:10:1 exhibits a "30'' $\mu$m feature which peaks at the right position. At present we don't know of a physical reason for a preferred oblate grain shape in PNe, and a broader CDE shape distribution in the C-stars and post-AGB objects (see also Sect. 8.4).

The shape of a resonance is also influenced by the presence of a coating. MgS is very hygroscopic. Under conditions where oxygen is available in the gas phase MgS can be oxidised and transformed into MgO (Nuth et al. 1985; Begemann et al. 1994). It is possible that the MgS is transformed as the central star of the PN heats up and the UV radiation progressively dissociates the CO molecules yielding gas phase oxygen. This could lead to MgS grains which are coated by a thin layer of MgO. We have modelled such grains using the electrostatic approximation following Bohren & Huffman (1983, Chap. 5). The result is shown in Fig. 14, curve 6. As can be seen the "30'' $\mu$m resonance is split into two features due to the MgO coating. The feature at the red wavelength is shifted to longer wavelengths compared to the pure MgS resonance. However the main feature is on the blue side of 25 $\mu$m towards the strong resonance at 18 $\mu$m in the pure MgO material, in clear contrast with the observations.

We explore other possible coatings on MgS grains to test their ability to explain the narrow feature observed in the PNe and the lack of emission at 26 $\mu$m. We find that of the composite grains we tested none give a satisfactory explanation. Mixtures of MgS and FeS have been discussed in the literature to investigate the nature of the "30'' $\mu$m feature (Begemann et al. 1994; Men'shchikov et al. 2001; Henning 2000). Curves 1 and 2 in Fig. 14 show the result of embedding an FeS core in a mantle of MgS and embedding a MgS core in a mantle of FeS, respectively. The latter compares most favourably with the position of the feature in the PNe. However, the substructure found in the spectrum of the composite grain around 33-37 $\mu$m is not found in the PNe spectra.

Szczerba et al. (1999) examine grains of amorphous carbon with a mantle of MgS to compare with the 30 $\mu$m feature in two post-AGB objects. We show simulated spectra of such grains and MgS grains coated with amorphous carbon in Fig. 14, curves 3 and 4 respectively. Curve 3 clearly does not match the observed feature in the PNe. As can been seen in curve 4 the MgS grains coated with amorphous carbon absorb less at 26 $\mu$m than pure MgS and are therefore a better spectral match to the "30'' $\mu$m feature of the PNe. However, the feature to continuum ratio in these grains is about a factor 2.5 lower than in the pure MgS grains requiring a factor 2.5 more mass in the MgS component in order to explain the observed band strength. Note also that such grains will still produce a weak feature at 26 while in some PNe spectra we find no excess at that wavelength at all.

Lastly, in curve 5 (Fig. 14) we show the effect of water ice on the MgS grains. The effects on the optical properties of a water ice coating are marginal and the profile cannot explain the PNe observations. We conclude that of the composite materials we have experimented with MgS grains coated with amorphous carbon give the best spectral match. However we find no composite grains that match satisfactorily.

We stress that although our model does not reproduce the "30'' $\mu$m profile in the PNe in its width it is safe to assume that its carrier is MgS based. These PNe are believed to be the evolutionary descendants of the sources which exhibit the MgS feature. The shift in peak position compared to the post-AGB objects follows naturally from an expanding and cooling shell. Also, the feature strength for the PNe is similar to these found in the post-AGBs further strengthening the physical link between the MgS in the C-stars and the post-AGBs on one hand and the "30'' $\mu$m feature in the PNe on the other (see also Sect. 7).