5 Discussion

The physical structure of AGB stars are certainly variable to different extents. In order to model the atmospheres of these variable stars in a proper physical manner one needs to take the time dependence into account; this is increasingly important in the infrared wavelength range where the light originates from the outermost layers. The resulting dynamic effects could affect the density and temperature structures of the outer photosphere and therefore affect the line formation. For dynamic models of variable stars see, among others, Höfner & Dorfi (1997) and Bessell et al. (1996). The partial pressures of molecules in the outer photosphere may vary greatly over a pulsation period. However, the fact that R Dor shows only a modest variability of $\delta V \sim 1.5~\mbox{mag}$ (the AAVSO International Database; J. A. Mattei 2001, private communication) suggests the feasibility of an attempt to model the stellar photosphere of this red giant with a hydrostatic model, as opposed to a dynamic model which takes into account time variations of the photospheric structure.

Figure 5: The observed (upper, full line) and synthetic spectra (upper, dotted line) are compared directly. In order to outline all the observed, medium-resolution spectra (SWS06) the low-resolution ISO observation (SWS01) is also shown. This observation is convolved to a resolution of $\Delta \lambda = 0.05~\mbox{$\mu$ m}$. The lower spectrum shows the ratio of the observed and the modeled spectra. The ordinata is given in logarithmic flux units arbitrarily scaled.

5.1 The hydrostatic modeling

In the hydrostatic modeling of M giants, there are several concerns to be addressed, which could potentially be important and explain discrepancies between observations and their modeling.

For example, the main source of continuous opacity in red giants in the IR spectral region is due to H^- free-free processes, a source of opacity that increases with wavelength. Therefore, IR lines are formed far out in the photosphere, where the assumptions made in the calculation of the model photosphere may be invalid. The outer atmospheres of M giants are very tenuous and the densities are low in the line-forming regions. One should therefore also be aware of the possibility of uncertainties due to NLTE effects affecting the lines.

Furthermore, the input data regarding molecular lines (line positions, excitation energies, and line strengths) and the completeness of the data, i.e. whether all relevant molecular bands are taken into account or not, are a concern. This is especially important for water vapour since it dominates the spectrum. Even bands far from the wavelength region considered, can contribute significantly to the synthetic spectrum through a large number of weak lines. Therefore, the inclusion of a relevant number of lines, including weak ones, is important (see also Jørgensen et al. 2001). Jones et al. (2002) discuss the completeness of the line list of water vapour by Partridge & Schwenke (1997), which is used here. They find a good match between their observations of M dwarfs and synthetic spectra based on the Partridge & Schwenke line list, indicating its relevance for studies of M stars.

Finally, the relative extension ( $d=\Delta r_{\rm atm}/R_*$) of the photosphere, which is typically approximately $10\%$, means that sphericity effects could be important both in the calculation of the photospheric structure and in the radiative transfer, which generates a synthetic spectrum. Indeed, we calculate our model photospheres and synthetic spectra in spherical geometry as opposed to plane-parallel geometry normally used for dwarf-star models.

In spite of all the approximations made, the synthetic spectrum presented here reproduces the ISO-SWS observations reasonably well in the region of 2.8- $3.66~\mbox{$\mu$ m}$, also keeping in mind the uncertainties in the observational data and in the reductions of the data. A direct comparison is presented in Fig. 5. The low-resolution SWS01 observation used to align the different medium-resolution SWS06 observations (see page 875) is also shown in the figure, as is the ratio of the observed and the modeled spectra. The negative spike in the ratio close to $3~\mbox{$\mu$ m}$ probably shows an inaccuracy in the line list used. Thus, the use of a hydrostatic model photosphere for the synthesis of the photospheric spectrum of this spectral region, seems to be quite adequate for the red, semi-regular variable R Doradus. This is certainly not the case for the more variable Mira stars (see for example Aringer et al. 2001).

5.2 The discrepancy at 2.6-2.8 $\mu$m

However, at the beginning of our interval, at 2.6- $2.8~\mbox{$\mu$ m}$, we find a discrepancy of the over-all spectral shape, between our model and the ISO observations, see Fig. 5. We have not identified the cause of this, but we will discuss a few points in connection with this discrepancy, which reflect the approximations and shortcomings in the modeling.

First, the relevant absorption band of water vapour in the wavelength region we are considering here has its maximum at $2.65~\mbox{$\mu$ m}$ (Jørgensen et al. 2001; Jones et al. 2002). Therefore, one would expect the lines at these wavelengths to be formed far out in the photosphere and one might be concerned as to whether the extension of the model photosphere is large enough to encompass this line-forming region properly. In our models the water vapour lines in the 2.6- $2.8~\mbox{$\mu$ m}$ region are indeed formed furthest out, with the strongest lines formed at an optical depth of approximately $\log\tau_{\rm {Ross}}=-4.8$( $\log\tau_{\rm {Ross}}$ is the standard optical depth calculated by using the Rosseland mean extinction coefficient). However, our photospheric model is calculated out to $\log\tau_{\rm {Ross}}=-5.6$ which means that the strongest of the water vapour lines are formed within our photosphere and not on the outer boundary of the model. As a check, we also calculated a model extending to $\log\tau_{\rm {Ross}}=-6.2$, showing no noticeable differences in the calculated spectra. Still, the further out in the photosphere, the less likely are the assumptions made in the calculations valid. Thus, the discrepancy could have its origin in a too warm temperature structure in the outermost layers. Furthermore, the model is calculated in LTE and no line scattering is considered. Line scattering could yield a larger absorption in the water lines. Neither should one rule out dynamic effects here.

Second, some problems may exist concerning the input data; the line list calculations could be wrong just here, for instance due to missing bands, but this is not likely (see the discussion in Jones et al. 2002).

Third, there is certainly an extended atmosphere beyond the photosphere of AGB stars, which could affect their spectra in the infrared. The mass loss from the star creates a circumstellar envelope which can extend far out. Also, it is seen in dynamic model atmospheres of, for example, carbon-rich Mira stars, that the photospheres could extend far out due to pulsations and shocks (Höfner & Dorfi 1997). (These basic features of dynamic model photospheres are certainly valid also for oxygen-rich Mira stars.) Densities at a few stellar radii from the star can be an order of magnitude higher than the ones given by the equation of continuity in the stellar wind. Thus, the discrepancy could perhaps be explained by an additional absorption from an extended atmosphere. A discrepancy similar to the one we find between our modeled and observed spectra at 2.6- $2.8~\mbox{$\mu$ m}$, was also found by Tsuji et al. (1997, 1998) in their analysis of the spectrum of the M 7 giant SW Vir. They attributed this to an excess absorption of warm water-vapour of non-photospheric origin. In order to explain several differences between models and infrared observations, Tsuji and collaborators (see for example Tsuji et al. 1997, 1998; Tsuji 2000a,b; Yamamura et al. 1999) have introduced the idea of a new, non-expanding, warm envelope situated further out than the photosphere but distinct from the cool, expanding circumstellar-envelope, at a distance of a few stellar radii from the star. The origin of the envelope is neither theoretically expected nor has it as yet received a theoretical explanation, but seems to be a common feature of M supergiants and M giants in general (Tsuji et al. 1998; Matsuura et al. 1999). This new envelope is shown to contain water vapour at temperatures of 1000- $2000~\mbox{K}$ (Tsuji et al. 1997), resulting in non-photospheric signatures in IR spectra of M giants. This view was recently corroborated by ISO observations of the $6.3~\mbox{$\mu$ m}$ bands of water vapour in early M giants, stars not expected to show signatures of water vapour (Tsuji 2001). Thus, the discrepancy between our modeled and observed spectra at 2.6- $2.8~\mbox{$\mu$ m}$ could be a result of an extra absorption component due to a non-expanding, warm envelope. We are in the process of modeling such a region (Ryde et al. in preparation). It is tempting to suggest in this scenario, that a star with a sufficiently low mass-loss rate will not be able to accelerate a wind from the stellar surface to velocities above the velocity of escape, but instead accumulates material "close'' to the star where it could form a relatively dense, warm layer including $\rm {H_2O}$.

5.3 The derived effective temperature of R Dor

Our derived effective temperature of $3000\pm 100~\mbox{K}$, which is based on the ISO spectrum of the 2.60- $3.66~\mbox{$\mu$ m}$ region, is somewhat higher than values found in the literature. For example, Bedding et al. (1997) estimate an effective temperature of $2740\pm190~\mbox{K}$ from measurements of the angular diameter and the apparent flux of R Doradus. As is discussed by these authors, other indirect estimates of the effective temperature in the literature yield even lower temperatures. We note that, given the inferred radius of $R=370~\pm~50~\mbox{$R_\odot$ }$ of Bedding et al. (1997), our temperature would yield a luminosity more characteristic of a supergiant. However, we also note that Fluks et al. (1994), who derive effective temperatures for all M-spectral sub-types of the MK classification based on spectroscopic observations of M giants in the solar neighbourhood, find an $T_{\rm {eff}}= 3126~\mbox{K}$ for the M7 class. They classify R Dor as an M7 giant and they derive an effective temperature of $2890~\mbox{K}$ for the M8 class.

Several possible reasons exist for the slightly higher temperature deduced in this paper as compared to most of the literature values. First, as was pointed out by Bedding et al. (1997), there may be inadequacies in indirect methods of determining effective temperatures from the colours of red stars. Second, Jones et al. (2002), in their fit of ISO spectra of M dwarfs with synthetic spectra based on the line list of Partridge & Schwenke (1997), also find effective temperatures systematically higher than those found in the literature, which are obviously based on several different methods. They suggest that an explanation of this could be the non-physical line splittings of the Partridge & Schwenke (1997) line list, which lead to too strong water vapour transitions for a given temperature. This would lead to a slightly higher effective temperature for a synthetic fit of a stellar spectrum. Third, R Dor is intrinsically variable, albeit with a small amplitude. As has been discussed in, for example, Aringer (2000), low-resolution ISO-SWS spectra of oxygen-rich, semi-regular variables on the AGB can be modeled with a sequence of hydrostatic models of different effective temperatures over the period of the star. Thus, one would indeed expect slightly different effective temperatures to be determined at different phases of the pulsation.