7 Correlations

Figure 15: "30'' $\mu$m feature properties versus the peak wavelength of the continuum. The symbols are the same as in Fig. 1. We show in panel  a) the ratio of the integrated flux in the "30'' $\mu$m feature to the integrated flux in the SWS spectrum. In panel b), we show the peak over continuum values. We also show the average values for the C-stars, the post-AGBs and the PNe. The non-detections are not taken into account in determining the mean values. The centroid wavelength of the "30'' $\mu$m feature is shown in panel c).

Figure 16: The centroid position of the "30'' $\mu$m feature versus the MgS temperature in the model. The symbols are the same as in Fig. 1. We show a power-law function fitted to the data with the dashed line.

Using the large database of sources available we can study some of the properties of the "30'' $\mu$m feature statistically. We have found it most convenient to characterise the sources by the temperature of the fitted continuum. We use the wavelength where the derived continuum peaks ( $\lambda_{\rm max,cont}$) as an indicator of the continuum temperature. The derived continuum temperature itself is less well suited because of the systematic difference in the power law index we find between classes of sources. The sources are rather uniformly distributed over $\lambda_{\rm max,cont}$ as well.

First, we show in Fig. 15a the relation between the $\lambda_{\rm max,cont}$ and the ratio of the integrated flux in the "30'' $\mu$m feature to the total flux in the SWS spectrum (I₃₀/ $I_{\rm SWS}$). The C-stars demonstrate a clear increase of I₃₀/ $I_{\rm SWS}$ with decreasing continuum temperature. The post-AGB objects emit systematically a larger fraction, of up to 25 per cent, in their "30'' $\mu$m feature. The PNe emit a similar fraction in the "30'' $\mu$m feature as the post-AGB objects although with a larger scatter. Notice that the sample contains a number of PNe with warm dust indicative of young PNe. There are a few sources which do not follow the general trend. The C-stars, R Scl, IRAS 19584 and RAFGL 2256, exhibit an atypically strong "30'' $\mu$m feature. These latter two sources are further typified by very weak molecular absorptions near 14 $\mu$m (see also Fig. 3). These observed anomalies are indicative of deviating conditions in the outflows of these sources, possibly a recently halted period of efficient dust formation. The post-AGB object IRAS 19454 has a very weak and cold "30'' $\mu$m feature. RAFGL 618 has a weak feature due to self-absorption (see Sect. 6.2).

The increasing strength of the "30'' $\mu$m feature in the AGB stars in not surprising. Since the emission is optically thin I₃₀ is proportional to the amount of MgS. The low values of $I_{30}/I_{\rm SWS}$ for the warmest C-stars reflects the fact that there is little dust around these sources and most of the IR radiation comes from the stellar photosphere. Cooler C-stars have more dust and thus more MgS. The difference between the coolest C-stars and the post-AGBs is more surprising. The fact that post-AGBs emit a larger fraction in the "30'' $\mu$m feature is due to two effects. First since the dust shell becomes optically thin in the visible some fraction of the light is emitted at shorter wavelengths. Second, the temperature of the MgS decreases less rapidly than the temperature of the other dust components (see below).

It is clear that any dust component which produces 30 per cent of the IR light has to be abundant. In order to quantify the (relative) amounts of MgS present in the CS shells of these objects will require radiative transfer modelling which is beyond the scope of this paper. We can however in first approximation study the relative amounts of MgS compared to the other cold dust components by studying the peak to continuum ratio (P/C). In Fig. 15b, we show the P/C versus the $\lambda_{\rm max,cont}$. The majority of the sources lies within the 0.3-1.0 range in P/C. We indicate a few clear outliers. R Scl, IRAS 19584 and RAFGL 2256 have a very strong "30'' $\mu$m feature indicating again that these sources have "too much'' MgS for a normal C-star. The PNe NGC 6790 and NGC 6826 have an exceptionally strong MgS feature. Note that NGC 6790 also has a very warm continuum, much like a post-AGB source or a very young PN. The strong SiC band at 11 $\mu$m is consistent with this. We also show the averages for each of the classes of sources. The average P/C for C-stars is 0.5, for post-AGB objects 1.0 and for the PNe it is 0.9. The similar ratios for the post-AGB objects and the PNe suggests that the carrier of the "30'' $\mu$m feature in the PNe is indeed directly related to the MgS feature in the post-AGBs. Furthermore, the similar ranges found for the post-AGB objects and the PNe argues against any process which results in a destruction of the MgS grains during the PN phase.

In Fig. 16, we show the derived MgS temperature versus the centroid position of the "30'' $\mu$m feature. The two are well correlated. For convenience, we have fitted a power-law function (without physical meaning) to the relation.

$$T_{\rm MgS} = 5.1\times 10^{14}~\left(\lambda_{\rm c,30}\right)^{-8.34}.$$ (4)

It is not surprising that $T_{\rm MgS}$ and $\lambda _{\rm c,30}$are correlated since we have used the feature profile to estimate the $T_{\rm MgS}$. However, using Fig. 16 or Eq. (4) one can easily derive the MgS temperature from a given observation. Also indicated in the figure are IRAS 13416 and RAFGL 2688; as can be seen they fall outside the correlation. This is due to the optical depth effects discussed above.

Figure 17: The derived MgS temperature versus the continuum temperature. The symbols are the same as in Fig. 1. We show in the box in the lower right the continuum temperature of the sources without a "30'' $\mu$m feature detected. The inset shows a blow up on a logarithmic scale up of the sources with a continuum temperature below 1000 K.

Lastly, we show in Fig. 17 the relation we find between the temperature of the continuum ( $T_{\rm cont}$) and the temperature of the MgS ( $T_{\rm MgS}$). We find, tracing the evolution from hot dust sources (C-stars) to the PNe, that the MgS temperature decreases correspondingly. Surprisingly, we find for warmest sources in the sample, with $T_{\rm cont}$ > 1000 K, very cold MgS. We propose two explanations for this phenomenon. First, it may be due to the absorption properties of MgS. If MgS cannot efficiently absorb the stellar light in the optical or near-IR part of the spectrum the grains remain cold. Alternatively, the MgS grains may be located further away from the star (see below). In the inset, we show a blow-up of the left side of the figure. There is a clear correlation between $T_{\rm cont}$ and $T_{\rm MgS}$. The MgS in the post-AGB sources is systematically warmer than expected on the basis of the C-stars. Apparently, in the process of becoming a post-AGB object, when the dust shell becomes detached and moves away from the star, the general dust cools down more rapidly than does the MgS. Note that the difference in dust temperature between the coldest C-stars and the warmest post-AGBs is seen most clearly in the [12]-[25] colour or $\lambda_{\rm max,cont}$ but is also found in the [25]-[60] colour (e.g. Fig. 1). The different behaviour of the continuum and the MgS is nicely illustrated in Fig. 15c where we show that while the continuum becomes redder the position of the "30'' $\mu$m feature does not change. This effect is the cause for the discontinuity between the C-stars and the post-AGB object seen in Figs. 15a and b. The physical cause for this must be due to the fact that MgS is actually (partially) heated by mid-IR radiation. The mid-IR radiation is much slower to respond to the termination of the AGB. Conversely, this implies that mid-IR radiation is a less important heating agent for the other dust components present.