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Astron. Astrophys. 342, 799-808 (1999)

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4. Synthetic spectra

4.1. Hydrostatic atmospheres

One of the main conclusions in Paper I was that our synthetic spectra based on hydrostatic MARCS atmospheres are able to reproduce the SiO bands of K and early M giants, but they fail to explain the relatively weak features seen in very cool and extended stars. Instead, they predict a monotonic increase of the equivalent width with lower temperature and surface gravity. This problem is clearly illustrated by the objects in our sample. If one compares the [FORMULA]- and [FORMULA]-values listed in Table 1 with the theoretical results, one finds that even the most intense of the observed bands are at least a factor of two weaker than expected from the synthetic spectra for [FORMULA]. The situation only becomes worse for the Mira variables with weak or no SiO absorption and for more extended models. The discrepancy is demonstrated in Fig. 5 where the observations (stars) as well as the original theoretical [FORMULA]-values derived for atmospheres with [FORMULA] (uppermost curve with open circles) are shown. For the stars being too red for the [FORMULA] versus (J-K) relation from Ng et al. (in preparation) we have adopted [FORMULA], which is between the coolest hydrostatic models (2600 K) and the smallest [FORMULA] value found in Table 1 (2414 K). In this context it should be noted that the discrepancies cannot be caused by the uncertainty of the effective temperatures discussed in the previous section. In order to follow the theoretical predictions all objects in our sample must be at least hotter than 3600 to 3800 K, depending on their surface gravity. For the stars with weak or no SiO absorption one would even need [FORMULA]-values much larger than 4000 K. All of this is not realistic. Thus, we have to conclude that the synthetic spectra published in Paper I really fail to reproduce the SiO bands of the observed objects presented in this work. However, Paper I was not dedicated to AGB stars and if one wants to study very extended sources showing strong SiO absorption the theoretical treatment should be improved for two reasons:

[FIGURE] Fig. 5. [FORMULA] as a function of the effective temperature. We show four sequences based on hydrostatic spherical MARCS models with [FORMULA], solar mass and chemical abundances: a) plane-parallel radiative transfer (open circles), b) plane-parallel radiative transfer and corrected for the influence of the SiO absorption on the observed continuum (full circles), c) spherical radiative transfer (open squares), d) spherical radiative transfer and corrected for the influence of the SiO absorption on the observed continuum (full squares); The observed [FORMULA]-values are displayed as stars. The corresponding effective temperatures are derived from the (J-K) color. We have also included the ranges of equivalent widths covered by the two dynamical atmospheres p5t297solu2 and p5t288solu2, which are shown as vertical lines at the temperatures of the hydrostatic initial models.

First, as it is described in Sect. 2, the observed spectra have been normalized using a polynomial fit to wavelength regions which do not contain any prominent SiO or OH features. In contrast, the equivalent widths published in Paper I are based on the theoretical stellar continuum corresponding to a radiative transfer calculation without molecular or atomic lines. Both methods will give similar results as long as the SiO bands remain relatively weak ([FORMULA]Å). But if the SiO absorption becomes more intense, it affects all those spectral intervals used for the polynomial fit, which are situated at [FORMULA]m. This effect can already be seen in the synthetic spectrum of an object with [FORMULA], solar mass and [FORMULA] as it is shown in the first figure of Paper I, although it gets much stronger for cooler stars. It depends on the resolution and causes the local continuum derived from the observations to be steeper and at a lower level than the theoretical one. As a consequence the measured equivalent widths will decrease considerably. In order to take this effect into account when comparing the theoretical predictions with the objects in our sample we have rebinned the synthetic spectra to the same resolution as the observed ones and then we normalized them to the same kind of pseudo-continuum (polynomial fit). The results can be seen in Fig. 5 where we show the temperature sequence of the original (open circles) and the continuum corrected (full circles) [FORMULA]-values for models with [FORMULA], solar mass and chemical abundances. As it is demonstrated in the diagram the total equivalent width of the SiO bands decreases in the coolest stars by approximately 15%.

The second improvement that has to be applied to the predicted equivalent widths is connected to sphericity. In principle, the results published in Paper I are based on atmospheric structures derived from spherical calculations (MARCS code with spherical routines from Nordlund 1984). However, for the subsequent computation of the synthetic spectra a plane-parallel radiative transfer was used. As it turned out, this approach introduces errors, which are much larger than we had expected. Especially for the very extended objects the equivalent widths will decrease significantly, if the spectra are calculated using a spherical code. This is demonstrated in Fig. 5 where we present [FORMULA]-values for our standard sequence of stars with [FORMULA], which are derived from the results of a spherical radiative transfer program (Windsteig et al. 1997, open squares). As one would expect the differences become larger with higher temperature and, of course, with lower surface gravity. For the very extended objects they are even big enough to invert the relation found in Paper I that [FORMULA] monotonically grows with decreasing values of log (g).

In Fig. 5 we also show the final temperature sequence, which results after correcting for both of the two discussed effects (full squares). It is obvious that the total equivalent widths of the SiO bands have decreased significantly, and consequently they are much closer to the highest values derived from our observations. The situation becomes even better, if one takes the uncertainties concerning the effective temperatures into account. In addition, as it has just been mentioned, we found for the very extended objects that the SiO features get weaker at lower surface gravities. For example, in a model with [FORMULA] the value of [FORMULA] is 5 to 15 Å smaller than at [FORMULA], depending on temperature. Thus, we can conclude that after correcting for the effect of the SiO absorption on the observed continuum and using a spherical radiative transfer for the spectrum synthesis the hydrostatic MARCS atmospheres are in principle able to explain those stars in our sample, which have the most intense SiO bands ([FORMULA]Å). Of course, as it was already shown in Paper I, these models are very successful in reproducing the observed band intensities of giants with spectral types earlier than about M5 III and M2 II. Nevertheless, they still fail to reproduce the SiO spectra of all AGB stars with weaker or no features, most of which are Miras.

4.2. Dynamical atmospheres

Up to now we have calculated our synthetic spectra for AGB stars based only on hydrostatic atmospheres. This is problematic, since these objects are dominated by such phenomena as strong pulsations creating shock waves, dust formation and mass loss. All of those processes can be successfully simulated by dynamical models as they have been developed by Fleischer et al. (1992), Höfner & Dorfi (1997) or Höfner et al. (1998). The atmospheric structure there is obtained by solving the equations of grey radiation hydrodynamics together with a time-dependent description of the dust formation. The stellar pulsation is introduced as a variable boundary located beneath the photosphere moving sinusoidally with a velocity amplitude [FORMULA] and a period P (piston). The calculation is started with a hydrostatic initial model characterized by its effective temperature (T[FORMULA]), luminosity (L[FORMULA]) and mass (M[FORMULA]). Since the original models were mainly designed to understand the behaviour of circumstellar shells and mass loss caused by dust driven winds of carbon stars, we had to modify them slightly for our purpose:

First, for the earlier computations (e.g. Höfner & Dorfi 1997, Fleischer et al. 1992) a constant value of the gas opacity independent of thermodynamical conditions and chemical composition has been used as it was introduced in the work of Bowen (1988). However, it turns out that this assumption generally produces completely unrealistic molecular spectra (Höfner et al. 1998), since the density at a given temperature may be orders of magnitude too large, which will also change the derived mass loss rates significantly. As a consequence, we have calculated our atmospheres using Planck mean absorption coefficients for the gas opacity in an environment with solar chemical abundances. Although this is still a somewhat crude approximation, the agreement with the results from detailed frequency-dependent computations in the hydrostatic limit case is already much better. This method, its advantages and problems are discussed in Höfner et al. (1998).

The second modification is connected to the circumstellar dust shell. In contrast to the carbon stars, no applicable self-consistent description of the formation and growth of grains in an oxygen-rich environment is available at the moment. As a consequence, we did not include any dust in our models. However, we do not expect this to be a large problem concerning the atmospheric structure, since it will only have strong effects in the outer regions, where the temperatures are already low enough that solid particles can be created. On the other hand, almost all of the SiO absorption comes from layers situated much closer to the star. Exactly the same argument applies to the depletion of gaseous SiO caused by the dust formation: it happens in regions of the atmosphere which do not contribute significantly to the observed 4 µm-bands. We have checked this by assuming that all SiO condenses into solid particles at [FORMULA]. Although this is already a quite high temperature (e.g. Sedlmayr 1997), we found only very small changes of the resulting synthetic spectra. However, a thick dust shell might also influence the observed 4 µm-spectra through its continuous opacity. Thus, one should be careful, if one applies our results to objects with high mass loss rates, although we did not find any correlation of the SiO bands with quantities probing the optical depth of the circumstellar envelope (Sect. 3).

In order to obtain synthetic spectra based on the given dynamical atmospheric structures we have computed chemical abundances as well as continuous and SiO opacities in each layer. This is done by a program named COMA (Copenhagen Opacities for Model Atmospheres). The routines for the equilibrium chemistry and the continuum absorption are taken from a version of the MARCS code that has already been used for the construction of the hydrostatic models in Paper I (Gustafsson et al. 1975, Jorgensen et al. 1992). The SiO opacities are derived from the linelist prepared by Langhoff & Bauschlicher (1993) assuming Doppler profiles and a microturbulence of 2.5 km s-1 (for details see Paper I). The results of COMA are then used as an input for a spherical radiative transfer program (Windsteig et al. 1997) producing the synthetic spectra.

Up to now we have only calculated spectra for a very small number of dynamical models. In addition, one should not forget that there are still problems concerning their construction like the grey radiative transfer or the neglecting of dust in an oxygen-rich environment. Nevertheless, the first results look promising as it can be seen in Fig. 5, where we present the ranges of [FORMULA]-values covered by the two atmospheres p5t297solu2 and p5t288solu2 during their pulsation cycle. They are shown as vertical lines situated at the effective temperatures of the corresponding hydrostatic initial models. The parameters for p5t297solu2 are [FORMULA], [FORMULA], [FORMULA], P = 295 d and [FORMULA]. p5t288solu2 differs only by its [FORMULA], which is 2880 K. It is obvious that the observed and predicted band intensities are in good agreement, and it becomes possible to explain the spectra of Mira variables with weak or no SiO absorption.

As an example we compare in Fig. 6 the observation of S Gru with a synthetic spectrum calculated from p5t288solu2 at phase 0.4 (bolometric). Taking into account that the dynamical atmosphere was not computed to fit the stellar parameters of S Gru and that only SiO is included into the molecular opacity, the agreement is quite good. Fig. 6 also shows a spectrum derived from a MARCS model with [FORMULA], [FORMULA], solar mass and chemical abundances. Although it is calculated using a spherical radiative transfer and normalized to the same pseudo-continuum as the observations, the SiO bands are much too intense. The main reason, why the dynamical models generally predict weaker SiO features than the hydrostatic ones, is their much larger extension producing emission components in the lines. As a function of the radius, the density, and thus the partial pressure of SiO, does not decrease exponentially, and in the presence of strong pulsations even not monotonically.

[FIGURE] Fig. 6. The spectrum of S Gru (full line) is compared with two synthetic spectra based on the dynamical atmosphere p5t288solu2 at phase 0.4 (bolometric, dotted line) and a MARCS model with [FORMULA], [FORMULA], solar mass and chemical abundances (dashed line), respectively. The V = 2 [FORMULA] 0 and V = 3 [FORMULA] 1 bandheads of the main SiO isotope are shown. The synthetic spectra are calculated using a spherical radiative transfer and normalized to the same pseudo-continuum as the observations.

In Fig. 7 we present [FORMULA] calculated from the dynamical atmospheres as a function of the bolometric phase. In addition we show the values for the Miras listed in Table 1. Especially p5t288solu2 seems to reproduce the observed trend that the SiO bands become weak or disappear around the time of the light maximum very well, while for p5t297solu2 the minimum of the SiO absorption happens later. However, the SiO spectra derived from the dynamical models do not show a strictly periodic behaviour (roughly periodic with perturbations) and we expect small shifts between the visual and the bolometric phase (up to 0.1).

It should be noted that the behaviour of the SiO bands depends very much on the value of [FORMULA]. We have calculated spectra from models characterized by the same parameters as those in Fig. 5 except for [FORMULA] and 4 km s-1. It turned out that these atmospheres produce a considerable SiO emission during some phases of their pulsation cycle, which does not agree with the observations.

[FIGURE] Fig. 7. [FORMULA] as a function of the bolometric phase for the two dynamical atmospheres p5t297solu2 (dotted) and p5t288solu2 (solid) and as a function of the visual phase for the Miras from Table 1 (squares).

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© European Southern Observatory (ESO) 1999

Online publication: February 23, 1999