The first obvious manifestation of the differences between AGB and supergiant spectra is the difference in their broad features. However, the various "silicate'' features are basically identical for both AGB stars and supergiants (see Fig. 15). Therefore, assuming that the observed spectral features represent evolution of the dust from the most refractory species, responsible for the broad features, to magnesium-rich silicates, it is clear that the AGB star condensation sequence is different from that of supergiants but both condensation sequences eventually leading to similar dust types, i.e. magnesium-rich silicates. The suggested evolutionary paths for the dust around O-rich AGB stars and supergiants are shown in Table 8. To explain these we need to have distinct differences between the dust-forming regions of AGB stars and supergiants in terms of elemental abundances, oxidation properties, temperatures or densities of the dust-forming atmospheres.

The difference between the broad features exhibited by O-rich AGB stars and supergiants implies a different pathway to the magnesium-rich silicates. The short-wavelength rise of the supergiant broad feature may be indicative of calcium-aluminium-rich silicates, e.g. Ca₂Al₂SiO₇, which match this part of the spectrum well (Fig. 19a). However, thus far no Ca-Al-rich silicate spectrum has a broad enough feature to account for the entire broad band. A second spectral feature peaking at about 11.5-12.0 $\mu$ m and extending to beyond the 13.5 $\mu$ m limit of our spectra is needed to account for the long wavelength part of the broad feature. This feature (but not the 12.5 - 13.0 $\mu$ m feature) could be attributable to alumina (Al₂O₃). Likewise the short wavelength onset of the AGB star broad feature may be due to Mg-Fe-rich silicates, but again this does not account for the long wavelength part of the feature. However, the shape of the long wavelength feature needed to complete the broad feature in both the AGB star and supergiant cases is very similar. We have found that this extra feature is best fit using alumina (Al₂O₃), although the exact form of alumina is not identical in each case to account for the flux longwards of $\sim$ 13 $\mu$ m. Figure 19a shows that the supergiant broad feature can be fit using a combination of Ca-Al-rich silicate (Ca₂Al₂SiO₇) and compact amorphous alumina, while the AGB star broad feature (Fig. 19b) can be fitted using a combination of olivine (MgFeSiO₄) and porous amorphous alumina.

**Table 8:** The different evolutionary paths of dust around AGB stars and around supergiants
$\begin{displaymath}\begin{tabular}{c\vert c} {\rm AGB~stars} & {\rm Supergiants}... ...rm feature}\\ {\rm in~some~spectra} & \\ & \\ \end{tabular}\end{displaymath}$

Therefore the main difference between the supergiant and AGB dust features seems to be that the supergiants have a distinct evolutionary phase in which calcium-aluminium-rich silicate condensation takes place to an extent that is observable, whereas for AGB stars this phase is not seen. The first distinct spectral features to be seen in AGB star spectra can be attributed to a combination of Mg-Fe-rich silicates and alumina. The AGB stars go through a phase in which the Mg-Fe-rich silicate feature strengthens and eventually overwhelms the alumina feature. The supergiant spectra exhibit either Ca-Al silicate and alumina features or strong Mg-Fe silicate features. No transitional states have been observed.

The 12.5 - 13.0 $\mu$ m feature seen in the spectra of some AGB stars is almost completely absent from the supergiant spectra, appearing only in the spectrum of S Per, a supergiant with a particularly thin dust shell similar to that of SRs (see Sect. 2.3.2; Richards 1997; Richards et al. 1999). Therefore, whatever dust species is responsible for this feature is not included in the usual condensation sequence for supergiants. As discussed in Sect. 3, the 12.5 - 13.0 $\mu$ m feature is predominantly seen in the spectra of semiregular variables, so there must also be a difference between dust-forming properties of Miras and semiregular variables. We suggest that this feature is not due to Al₂O₃, MgAl₂O₄, or core-mantle alumina-silicate grains, as has previously been proposed (e.g. Vardya et al. 1986; Posch et al. 1999; Kozasa & Sogawa 1997, 1998 respectively), but rather that it may be associated with silicon dioxide (see Morioka et al. 1998; Speck 1998), which also accounts for the 9.25 $\mu$ m peak seen in the spectra of some AGB stars with strong silicate features (see Speck 1998). Another possible carrier may by highly polymerized silicates which bear more structural resemblance to SiO₂ than to isolated silicates like olivine.

$\begin{figure}\includegraphics[width=9cm,clip]{H1934F19.ps}\end{figure}$

Figure 19: Fits to broad features. a) The supergiant broad feature is fit by Ca₂Al₂SiO₇ (dotted line) and compact amorphous Al₂O₃ (dashed line). b) The AGB star broad feature is fit by MgFeSiO₄ (dotted line) and porous amorphous Al₂O₃ (dashed line)

The difference between the chemistries of Miras, semiregular variables and supergiants needs to be addressed. There are two proposed dust sequences (for want of a better term): the classic condensation sequence discussed in Sect. 4 (see also Tielens 1990); and the new chemical evolution sequence based on C/O ratios (SP98). The SP98 model was constructed using a dataset that had no supergiant spectra exhibiting the broad feature. With this supergiant broad feature included, we need to re-examine the possibilities.

AGB stars experience the so-called "third dredge-up'' which brings the products of the helium-burning shell to the surface of the star (see e.g. Iben & Renzini 1981). In particular, this process enriches the atmospheres of these stars with ¹²C. With successive dredge-ups the atmospheres of AGB stars get progressively more carbon-rich. Therefore oxygen-rich AGB stars can have carbon-to-oxygen ratios (C/O) ranging from cosmic, $\sim$ 0.4, to approximately unity. Supergiants, on the other hand, are not expected to experience a "third dredge-up'' and are therefore expected to remain oxygen-rich, with approximately cosmic C/O. Furthermore, Sylvester et al. (1994) found UIR bands in the spectra of some M-supergiants. They suggested that UV photodissociation of CO molecules provided the free carbon atoms required to form the organic materials responsible for the UIR bands. Obviously, this dissociation mechanism would also release extra oxygen atoms, and if the carbon atoms become trapped in organic molecules, this could lead to an even more oxygen-rich environment (C/O < 0.4). Therefore, supergiants exhibiting the UIR bands could have even more oxygen available for dust formation and, following the SP98 scheme, we might expect these stars to exhibit the "classic'' strong silicate spectral feature, since they should have more than enough oxygen to make substantial amounts of Mg-silicates. However, many of the supergiants with UIR bands exhibit "broad'' spectral features. These supergiant "broad'' spectral features are alone enough to call into question a condensation mechanism based on the amount of available oxygen, since the C/O ratio for supergiants is always low. Combining this with the UIR band evidence, it seems unlikely that the type of dust forming around these stars is determined solely by the amount of oxygen available.

Contrary to SP98, we have found that the "broad'' feature in AGB star spectra cannot solely be attributed to Al₂O₃. The feature appears to be due to a combination of Al₂O₃ and ``classic'' silicate (e.g. amorphous olivine). A fit to the average "broad'' feature spectrum using Al₂O₃ and amorphous olivine is shown in Fig. 19b. This does not contradict the SP98 dust evolution scheme, but implies that oxygen availability is never so low as to preclude some silicate formation. It does, however, cause a problem for the supergiant sequence. The SP98 scheme cannot account for the formation of Ca-Al-rich silicates rather than Mg-Fe-rich silicates in the very oxygen-rich supergiant environments, where there should be more than enough oxygen to form the ``classic'' silicates.

It is clear that the C/O ratio is an important factor in determining which dust species will form around which stars, however, other factors (e.g. dust shell temperature and density) are also important and need to be taken into account. In trying to accommodate as much of the available information, and previous interpretations, as possible, the current work suggests the following: the type of dust species formed around a given star is controlled by the C/O ratio and by the density and temperature of the dust-forming region, but that one of these factors will dominate, depending on the exact balance between the C/O ratio, the density and the temperature. The work of Hron et al. (1997) suggested that the classical condensation sequence is valid for Miras, since they found that the "classic'' narrow silicate feature is associated with the most optically thick dust shells (i.e. more dust = more evolved dust). This is not the case for the semiregular variables. In fact, for these the optical depth is lowest for the narrowest silicate features and is higher for the broader features (Ivezic & Elitzur 1995). However, the most optically thick SR dust shells are only as optically thick as the most optically thin Mira dust shells. This may be the key to the problem. In the case of the Miras, with relatively high optical depths, and therefore relatively high density dust shells, it is the density of the dust shells that controls the dust formation, and these environments undergo dust formation which follows the classic condensation sequence. For Miras the evolutionary stage of the dust reflects the evolutionary stage of the star (i.e. Miras with broad features are less evolved than those with the "classic'' silicate features). In the case of SRs, with much less dense shells, it is the C/O ratio that controls the dust formation and therefore the dust "evolution'' follows the opposite path to the classic condensation sequence, as suggested by SP98. This fits with the findings of Hron et al. (1997). In the case of the SRs, the stars with the "classic'' silicate feature are less evolved, with less dense dust and more free oxygen, while the "broad'' feature is associated with more evolved SRs, which have denser dust shells and fewer available oxygen atoms.

We now return to the enigmatic 13 $\mu$ m feature. This feature appears to be associated with SR variables and therefore with less dense dust-forming regions. Following the above argument, this in turn implies that species formation is dominated by the C/O ratio. However, it is not clear that the 13 $\mu$ m feature is associated with any other dust feature. While Begemann et al. (1997) found a close correlation between this feature and the classic silicate feature, both Sloan et al. (1996) and the current work found that the 13 $\mu$ m feature is not particularly associated with any of the other dust features - appearing in examples from all classes and implying that the 13 $\mu$ m feature is not associated with the evolutionary stages of the dust around SRs, or the prevailing C/O ratio. Furthermore, it is rarely found in supergiant or Mira spectra. It has been suggested that the 13 $\mu$ m feature is carried by silicon dioxide (SiO₂; Speck 1998). However, some objections to this attribution have been raised. Firstly, there is the exact position of the feature at $\sim$ 12.9 $\mu$ m. Posch et al. (1999) state that the feature in amorphous SiO₂ peaks at 12.3 $\mu$ m, which is too far removed from the observed feature. However, infrared spectra of different polytypes of SiO₂ presented by Speck (1998) show that the "13 $\mu$ m'' feature peak position varies from one polytype to another from $\sim$ 12.5 $\mu$ m to $\sim$ 12.8 $\mu$ m, with the feature width (FWHM) varying from 0.33 to 0.73 $\mu$ m. While this is still not identical to the observed feature, it is much closer than any other suggested carrier. Second, there are other spectral features in the infrared spectrum of SiO₂ which need to be accounted for. As shown in Fig. 18, SiO₂ has two features in the 7-14 $\mu$ m range relevant to our observed spectra, the 12.5 - 13 $\mu$ m feature and a stronger 9.2 $\mu$ m feature. There is also a feature at $\sim$ 20 $\mu$ m outside the range of our observations. The relative strength of the 13 $\mu$ m feature to the other features depends on the polytype (i.e. crystal structure; see Speck 1998), the level of disorder of the crystal structure, and the relative optical depths (Hofmeister et al. 2000). While the 13 $\mu$ m feature is usually weaker than the 9.2 $\mu$ m feature, optical depth effects can hugely increase the relative strength of the 13 $\mu$ m feature (see Hofmeister et al. 2000). The observed 13 $\mu$ m feature is never found in isolation - there is always some contribution in the 8 - 10 $\mu$ m region, probably from Si-O bond stretching in amorphous forsterite/enstatite. It is possible to bury the 9.2 $\mu$ m feature in this general Si-O stretch region, so that the only evidence for SiO₂ above the general silicate contributions to the spectra is the 13 $\mu$ m feature. In fact, there is some evidence for an emerging 9.2 $\mu$ m feature, over and above the "classic'' 9.7 $\mu$ m silicate feature, which may be attributable to SiO₂ (Speck 1998). It is possible that the $\sim$ 20 $\mu$ m SiO₂ feature could be buried in the amorphous silicate Si-O bending feature nearby, but this cannot be assessed with the current observations. However, it was noted by LML90 and Goebel et al. (1989) that the 13 $\mu$ m feature seen in IRAS spectra was often associated with an unidentified feature at about 19 - 20 $\mu$ m, and this may be the other SiO₂ spectral feature. Another objection raised to SiO₂ is that equilibrium calculations have shown that it is not expected to form around AGB stars (Lodders & Fegley 1999). Any SiO₂ that does form is expected to be rapidly converted to forsterite and then enstatite. The key point, however, is that objection requires that equilibrium is reached. In order to form forsterite and enstatite, the condensation sequence is likely to go through SiO₂ formation. Current thinking has been that SiO₂forms at most only a minor constituent of the dust, but this is based on the absence of observed features near 9.2 and 12.5 $\mu$ m and a shoulder near 8.4 $\mu$ m (J. Nuth, pers. comm), which, as stated above, may be buried in the strong Si-O stretching feature. If equilibrium is not reached, we would expect there to be some residual SiO₂ in these regions. In Miras, the denser dust shells are more likely to reach equilibrium than the less dense SR dust shells because each atom/molecule in a denser environment has more chance of interacting with another atom/molecule. This might explain the formation of a disequilibrium product (SiO₂) in the less dense SR dust shells and not in the denser Mira dust shells.

In Sect. 3, we discussed the difference between SRs and Miras. It is not clear whether SRs are the progenitors of Miras or vice versa, whether these stars go through successive SR and Mira phases, or whether they are completely separate objects. However, assuming that they are related and that SRs may evolve into Miras and then back into SRs, how can we account for the appearance of a species in the dust-forming region of one of these variability types and not in the other? The easiest explanation is to assume that the dust around SRs, which we assume includes some SiO₂ or highly polymerized silicate, is ejected by the stellar wind prior to the Mira phase and then does not form again until the star returns to the SR phase. Alternatively, the dense dust regions around Miras may cause the SiO₂-rich (parts of) grains to be transformed into olivines and pyroxene as the progression towards equilibrium is enhanced in the denser environment. It is known that the implantation of impurities, such as metal ions, tends to transform SiO₂ and highly polymerized silicates into less polymerized silicates like olivines and pyroxenes (Zachariasen 1932; Hess 1995). The less dense dust forming regions around SRs would allow more of the purer SiO₂/highly polymerized silicate grains to form and survive. All other features in the spectra of Miras and SRs are identical. The fact that SiO₂/highly polymerized silicate is plausible from an abundance point of view and that the disappearance of the species in Mira spectra can be explained makes the attribution of the 13 $\mu$ m feature to SiO₂ or highly polymerized silicates even more attractive. While laboratory spectra of SiO₂ are commonly available, as discussed above, the position of the 12.5 - 13.0 $\mu$ m feature is not ideal. Therefore, laboratory studies of highly polymerized silicates are needed to find a better match to the observed feature.

How do we relate these ideas to the supergiants? Most supergiant spectra do not exhibit the 13 $\mu$ m feature. They seem to follow a different chemical evolution to either Miras or SRs. While many supergiant spectra show "classic'' silicate features identical to those in AGB stars, it is the difference in the broad feature that is the mystery. We have to ask: what differences between supergiants and AGB stars can shed light on this problem? One obvious difference is that some supergiants exhibit UIR emission bands, attributed by Sylvester et al. (1994, 1998) to the presence of chromospheres around such supergiants, whereas AGB stars do not have chromospheres. The presence of a strong UV radiation field in supergiant outflows, as well as dissociating CO and permitting the formation of the carbon-rich UIR-band carriers, may also significantly influence the early dust formation sequence in the remaining oxygen-rich gas once the UIR-band carriers have formed.

Another avenue for future investigation is the relation of asymmetry factors for AGB stars to their dust spectra. It has been suggested that the asymmetry of the light curve of an AGB star is related to its evolutionary stage (Vardya et al. 1986). Unfortunately it appears that useful published data is only available for the light curves of Miras, so that comparison of the light curves of Miras to those of SRs is not currently possible. Other factors to take into account in future work are dust feature strength and profile variations as a function of pulsation cycle phase, as studied by Monnier et al. (1998) and earlier workers, and the secular long-term variation of the observed dust features of some sources that was discovered by Monnier et al. (1999).

5 Discussion