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Astron. Astrophys. 323, 202-210 (1997)

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3. Results

3.1. Influence of SiO on stellar atmospheres

As mentioned in the previous section, we have calculated all model atmospheres with and without taking into account the opacity of SiO. By comparing the results it became possible to estimate, how important this molecule is for the atmospheric structure. Fig. 3 displays a temperature versus pressure plot for an object with [FORMULA], [FORMULA], solar mass and chemical composition. In addition to the models with and without SiO we also show the curve for an atmosphere that has been calculated neglecting the opacities of TiO as well as of SiO. This allows to compare the effects of those two molecules. As one can clearly see, the presence of TiO causes a substantial heating over a large region of the atmosphere. This is related to the fact that its strong bands are generally located at wavelengths shorter than the spectral maximum of the stellar radiation field (Gustafsson & Jorgensen 1994). On the other hand SiO with its main rotation-vibration transitions in the mid infrared tends to cool the surface layers and gives rise to a corresponding small back-warming effect, which is seen in Fig. 3 for regions deeper than about [FORMULA].

[FIGURE] Fig. 3. The atmospheric temperature-pressure structure for [FORMULA], [FORMULA], solar mass and chemical composition. The three different curves have been computed including all available molecular opacities (full line), neglecting the opacities of only SiO (short dashes) and neglecting the opacities of TiO and SiO (long dashes). The largest changes are seen to be due to TiO.

Another obvious result from Fig. 3 is that the influence of SiO remains small in all layers, especially if one compares it with the enormous changes produced by TiO (shown) or H2 O, the latter being important in cooler stars. As one can see, the corresponding temperature differences never exceed 50 K, although this is one of the models with the strongest effects from SiO. If all other atmospheric parameters are kept constant, the temperature differences always decrease monotonously with growing values of log (g), which is due to a general weakening of the SiO bands (see next subsection). As a consequence the SiO molecule has practically no influence on objects with [FORMULA]. The same is also true for stars with [FORMULA] or [FORMULA]. If the temperatures become too high, the SiO bands will be too weak to cause any significant changes of the atmosphere. On the other hand it turns out that in very cool objects the opacity is completely dominated by TiO and H2 O, while SiO only plays an unimportant role. The strongest effects from SiO appear in stars with [FORMULA] and [FORMULA]. But even in these models we have never found temperature changes exceeding 50 K, which allows us to conclude that the influence of SiO on the atmospheric structure of cool stars remains always small as long as hydrostatic and chemical equilibrium can be assumed.

3.2. The intensity of SiO bands as a function of temperature and atmospheric extension

In order to investigate the behavior of the intensity of the SiO absorption we have calculated equivalent widths for several of the corresponding molecular features using our synthetic spectra. The following bandheads have been measured from our high resolution data: [FORMULA] of 28 SiO ([FORMULA] m), [FORMULA] of 28 SiO ([FORMULA] m), [FORMULA] of 28 SiO ([FORMULA] m), [FORMULA] of 29 SiO ([FORMULA] m) and [FORMULA] of 29 SiO ([FORMULA] m). As one can see in Tab. 1, the wavelength ranges defined to study the [FORMULA] and [FORMULA] bands of 28 SiO also include 30 SiO features. But since the latter are comparatively weak, they show no significant influence on the results. This is not true for the absorption caused by the main isotope and extending into the spectral windows used to measure the 29 SiO bandheads. As a consequence the corresponding equivalent widths will always depend on both of these two features. In addition we have determined the total intensity of the whole [FORMULA] ([FORMULA] m) and 2 ([FORMULA] m) bands from our low resolution spectra.

In Figs. 4 and 5 we present the intensity of the SiO absorption as a function of stellar temperature for different values of log (g). In Fig. 4, which displays the total equivalent width of the whole fundamental band ([FORMULA]), one can see that the latter decreases monotonously with [FORMULA], if the gravitational acceleration remains constant. This trend is almost linear. Only for the coolest objects and the smallest log (g) values there seems to be some saturation effect. The situation is a little bit different for the sum of the equivalent widths corresponding to the [FORMULA], [FORMULA] and [FORMULA] bandheads of the main isotope, which is shown in Fig. 5 and will be called [FORMULA] in the following text (L for the photometric L-band). As for the fundamental band, the SiO absorption always goes monotonously down with the temperature. But for the more extended atmospheres there appears a very strong decrease between [FORMULA] and 4200 K, while the gradient is only weak for cooler and hotter stars. This behavior causes the curves to show some kind of "s-shape". It is not so well pronounced in objects with [FORMULA], where the decrease is much more linear. For the most compact atmospheres the largest gradient even seems to appear at [FORMULA].

[FIGURE] Fig. 4. The total equivalent width of the SiO fundamental band ([FORMULA]) as a function of effective temperature and gravitational acceleration for stars with solar mass and abundances. log (g) is in units of [FORMULA].

[FIGURE] Fig. 5. The sum of the equivalent widths of the [FORMULA], [FORMULA] and [FORMULA] bandheads of the main SiO isotope ([FORMULA]) as a function of effective temperature and gravitational acceleration for stars with solar mass and abundances. log (g) is in units of [FORMULA].

In Figs. 4 and 5 it is also obvious that at every temperature the SiO absorption becomes smaller, if the gravitational acceleration grows. As it turned out in all of the investigated cases, the relation between the equivalent width of the SiO bands and log (g) is almost linear. Thus, the strongest SiO features are expected in very cool giants and supergiants, while they always will be relatively weak in dwarfs.

From our observations of AGB stars (Aringer et al. 1995) we found that the intensity ratios of the SiO bandheads show a large scatter, if different objects are compared. For example, in our sample the value of [FORMULA] changes by a factor of more than 2. This is partly related to the total amount of the SiO absorption. But even if the latter remains constant, there is still a considerable scatter left, which may be caused by a different [FORMULA] or log (g). In principle the ratio between the intensities of any band originating from transitions to the [FORMULA] level and its hot bands (transitions involving only vibrationally excited levels) will reflect the temperature at the depth(s) of formation in the stellar atmosphere (primarily via the Boltzmann factor). Since the latter may be a function of [FORMULA] and log (g), those parameters could also affect the intensity ratios. In order to see, if this is true and if the SiO features can be used for an independent determination of effective temperature and gravitational acceleration, we compared the equivalent widths (EW) of some bandheads. First we concentrated on the 28 SiO transitions in the photometric L-band. Fig. 6 displays a plot of the ratio [FORMULA] as a function of the sum [FORMULA], which represents a good measure for the SiO absorption in this range. It is obvious that [FORMULA] decreases with growing [FORMULA] in objects showing weak features, while it remains approximately constant at [FORMULA]. The symbols and curves correspond to different gravitational accelerations. As one can see, the latter do not produce any systematic trends or larger deviations from the overall relation. The same also applies to the effects caused by temperature variations, and the other ratios like [FORMULA] show a similar behavior. Thus, we must conclude that the 28 SiO bandheads in the photometric L-band alone are not a good tool for an independent measurement of [FORMULA] and log (g), as long as one works with low or medium resolution spectroscopy. Their ratios are almost only a function of the total intensity of the SiO absorption in this wavelength range, especially if one takes into account that there are always uncertainties in the observations and atmospheric models.

[FIGURE] Fig. 6. The ratio [FORMULA] as a function of the SiO absorption in the photometric L-band ([FORMULA]) for different gravitational accelerations. Solar mass and abundances are assumed. log (g) is in units of [FORMULA].

The last two statements are also valid, if the study is extended to the bandheads of the isotope 29 SiO. For example, the ratio [FORMULA] for the [FORMULA] transition increases with [FORMULA], and the scatter around the corresponding relation, which is almost linear, remains very small. On the other hand this may offer an advantage, because it could make it easier to estimate the isotopic abundances.

Finally we have compared the total intensities of the fundamental and the first overtone bands. It turned out that for [FORMULA] the ratio is almost independent of temperature and only determined by variations of the gravitational acceleration: For higher values of log (g) the [FORMULA] transitions become weaker relative to the [FORMULA] features. On the other hand, if the atmospheres become hotter than 3600 K, there exists no clear trend being caused by different values of either [FORMULA] or log (g). Of course, the ratio of the equivalent widths is also correlated with the total intensity of the SiO absorption (in all bands). But the scatter around the corresponding relation is very large. However, there are again no systematic trends caused by temperature or gravitational acceleration.

3.3. Sphericity effects

Since we have calculated our synthetic atmospheres using spherical radiation transport routines, we are able to investigate deviations from plane parallel geometry. The latter are related to the fact that the structure of spherical models is a function of stellar mass (as oppose to the plane parallel models). As one would expect, we found the strongest sphericity effects in the SiO features of the most extended and least massive stars. On the other hand, for objects with high mass and/or gravity the spherical models approach the plane parallel ones. To give some typical numbers, for [FORMULA] and solar chemical abundances the ratio between [FORMULA] for a [FORMULA] and for a [FORMULA] star (which is almost plane parallel) varies from 1.02 to 1.6, the largest values corresponding to the highest effective temperatures. In objects with [FORMULA] the sphericity effects never produce significant changes of the SiO opacities. This demonstrates that the intensity of the SiO features decreases as a function of stellar mass, although the effect does not become very important, since larger differences appear only in hotter models, where the bands are already weak. And in high gravity stars deviations from plane parallel geometry can be completely neglected.

3.4. Chemical abundances

It is evident that changes of the chemical abundances may have a large impact on the formation of SiO and consequently on the appearance of the SiO features. Therefore we studied such effects starting with the metallicity. In Fig. 7, which presents [FORMULA] as a function of [FORMULA], the different curves and symbols correspond to selected temperatures and gravitational accelerations. One can see that for all parameters the intensity of the bandheads increases steadily with log (Z) showing almost a linear growth. This means that stars belonging to the galactic population II will always have much weaker SiO features than those with solar metallicity.

[FIGURE] Fig. 7. The equivalent width of the [FORMULA], [FORMULA] and [FORMULA] bandheads of the main SiO isotope ([FORMULA]) as a function of metallicity for different effective temperatures, gravitational accelerations and for one solar mass. log (g) is in units of [FORMULA].

A similar and also very strong effect can be obtained, if one changes only the silicon abundance, which is shown in Fig. 8. As a consequence in principle it should be possible to estimate this quantity from low or medium resolution spectroscopy in the photometric L-band. But in practice it is not that simple, because the intensity of the SiO bands also depends on [FORMULA], log (g) and log (Z), which are in general not precisely known, especially for the coolest giants.

[FIGURE] Fig. 8. The equivalent width of the [FORMULA], [FORMULA] and [FORMULA] bandheads of the main SiO isotope ([FORMULA]) as a function of the silicon abundance for [FORMULA] 2800 and 3600 K, [FORMULA] and one solar mass.

Finally, we have also investigated the effects of variations in the C/O ratios. Such changes may occur in AGB objects, when they transform into carbon stars (e.g. Lambert 1994). We studied the range between [FORMULA] (solar) and 0.9. We did not go to higher values, since it is not likely that the used chemical equilibrium routine still gives correct results, if one comes too close to [FORMULA]. This is due to the neglect of typical S-type molecules like VO, YO, ZrO and LaO. For [FORMULA] there will be a carbon rich environment, where almost no SiO forms (Jorgensen 1994b). Within the limits of our investigation, which covered only values of [FORMULA] and log (g) that are typical for AGB stars, we did not find any systematic or strong variations of the SiO bands, reflecting that the SiO density is determined by the silicon abundance only. For some models there was a slight decrease with C/O and for others we even observed an increase. The latter may be due to changes of the atmospheric structure.

3.5. Microturbulence

As it has already been mentioned in the previous section, the appearance of the SiO bands at a low or medium resolution is also influenced by the microturbulence. In general their intensity increases, if the value of [FORMULA] becomes larger, since many of the lines are saturated. As one would expect, the size of this effect depends on the original intensity of the SiO absorption and the most striking changes can be found for the very cool and extended atmospheres. For example, in an object, which is characterized by [FORMULA], [FORMULA], solar mass and chemical abundances, [FORMULA] grows by a factor of approximately 2, if [FORMULA] increases from 1.0 to 3.5 km/s. Of course, this is one of the models with the strongest variations. The corresponding value for a similar star with [FORMULA] is still almost 2, but it drops quickly at higher temperatures. On the other hand, if [FORMULA] is smaller than 20 Å, the influence of the microturbulence remains always negligible. Thus, one has only to worry about the [FORMULA] values of very cool giants and supergiants.

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© European Southern Observatory (ESO) 1997

Online publication: June 5, 1998

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