4 Procedure and results

In order to model the observations and to find a good fit, we have calculated a grid of synthetic spectra, allowing the effective temperature ( $T_{\rm eff}$) and the surface gravity ($\log g$) of the models they are based on, to vary by increments of 50 K and 0.5 dex, respectively.

First, the over-all spectral shape was matched. This was done by convolving the observations and the models to a resolution of $R\sim 60$. The wavelength coverage of 2.6- $3.7~\mbox{$\mu$ m}$is quite large and we find that especially the "low-frequency'' spectral shape, in particular the slope, at 3.1- $3.7~\mbox{$\mu$ m}$ is sensitive to the effective temperature. This fact confines the possible range of effective temperatures. Subsequently, the molecular features, both their relative strengths and their amplitude are fitted visually. We find that we are able to fit our observations well, both the over-all shape and the molecular features, of the $3~\mbox{$\mu$ m}$ region, for the phase at which the observations were made, with a synthetic spectrum based on a model photosphere of a temperature of $3000~\mbox{K}$ and a surface gravity of $\log g = 0 \rm {~(cgs)}$, see Figs. 1 and 2. We point out that since the spectral region covered is sufficiently large we can fit both the general shape, or slope, of the spectrum as well as the molecular features themselves. However, we are not able to fit the flux level at the beginning of our observed region, at 2.6- $3.8~\mbox{$\mu$ m}$. There is a discrepancy here and this will be discussed in Sect. 5. We also performed a more formal $\chi^2$ fitting with the same result. We determined the correct multiplicative factor to multiply the modeled fluxes by (in order to account for the distance to the star), to be able to compare them with the observations, by minimising the $\chi^2$ for every model in our grid. The parameters of the model giving the lowest overall $\chi^2$ are also $3000~\mbox{K}$ and $\log g=0$.

Figure 3: The observed spectrum is shown uppermost, and below are shown three synthetic spectra generated from model atmospheres of temperatures of (from top to bottom) $3100~\mbox{K}$, $3000~\mbox{K}$, and $2900~\mbox{K}$ and of a surface gravity of $\log g=0$. The SWS01 observation is shown to outline the observations and is repeated for the model spectra. These are arbitrarily normalised with the synthetic spectra at $3.5~\mbox{$\mu$ m}$.

Actually, a synthetic spectrum calculated on the basis of a model photosphere with an effective temperature of $3100~\mbox{K}$ fits some of the molecular features of the observed range better (though only marginally) than does the $3000~\mbox{K}$ model, but this synthetic spectrum is then too steep in the long-wavelength region and the overall strength of the water bands is too weak, as is shown in Fig. 3. The uppermost spectrum in the figure shows the observations with the SWS01 observation as a back bone to outline the spectrum. The three spectra below are synthetic spectra based on model atmospheres of temperatures of $3100~\mbox{K}$, $3000~\mbox{K}$, and $2900~\mbox{K}$. The low-resolution SWS01 observation is repeated for the three synthetic spectra. The overall shape of the hottest model is too steep, reflecting weaker water bands. The reverse is true for the coldest model. These facts may provide us with an estimate of the uncertainties in the temperature. Thus, with a conservative estimate of the uncertainties we arrive at a determination of the temperature of $3000\pm 100~\mbox{K}$ for R Doradus.

The surface gravity ($\log g$) is also a fundamental parameter, and is varied together with the effective temperature in the search for the best model, but the spectra are not as sensitive to this parameter as to the effective temperature. For the grid of synthetic spectra that we have generated, we chose an increment of $\Delta \log g = 0.5 \rm {~(cgs)}$. A change in surface gravity changes the relative strengths and details of the molecular features. We have chosen $\log g=0$ as our best model but the uncertainty is quite large due to the low sensitivity of this parameter. Based on the fits, we estimate the limits of the surface gravity to be $\delta \log g = \pm 1$ (cgs). For a given surface gravity the stellar mass does not greatly change the spectral features or the spectral shape (typically by a few percent for a change in mass by a factor of 3).

Also, the metallicity of the star has an influence on the spectrum. This was also found by Decin et al. (2000) in their study of the red giant $\alpha$ Tau. We find that a decrease of the metallicity by $0.15~\mbox{dex}$ affects the absolute flux level and the relative slope of the spectrum but only beyond $3.3~\mbox{$\mu$ m}$. The molecular features hardly change. The slope of the spectrum is changed in such away that for a given effective temperature the slope decreases slightly for a lower metallicity.

In addition to the observed spectrum, Figs. 1 and 2 show our synthetic ones calculated with a hydrostatic model photosphere of an effective temperature of 3000 K and a surface gravity of $\log g=0$. The stellar mass is assumed to be 1 solar mass and the chemical composition is assumed to be solar (Feast et al. 1999). For a metallicity of [Fe/H] =-0.15 the temperature of the model that best fits the data is $T_{\rm eff}=3050~\mbox{K}$. In the figures of the spectra, the fluxes are plotted on a logarithmic scale. The uncertainties in the absolute ISO fluxes originate from uncertainties in the gain factors of the different detectors (i.e. observed bands) and not from the levels of dark current of the bands. Allowing for these uncertainties and the distance to the star, which has to be taken into account when comparing stellar and model fluxes, the fluxes will differ only by a multiplicative factor. A plot of the logarithm of the fluxes allows a direct comparison with only an additive shift between them. The amplitude of the spectral features can be compared directly.

Figure 4: The total spectrum and the spectra of the contributing molecules are shown convolved with a Gaussian profile of a width corresponding to $300~\mbox{km~s$^{-1}$ }$. This is done to give a better overview. The SiO and OH spectra are shifted up for clarity. The dashed, straight line represents the continuum. The continuum is also shown for the shifted SiO and OH spectra.

In Fig. 4, spectra including the contributions of only one molecule at a time, are shown. The relative importance of the H₂O, OH, CO, and SiO molecules contributing to the spectral features in the observed region is displayed. The total synthetic spectrum is also shown in the figure, and the SiO and the OH spectra are shifted for clarity. CO₂ does not contribute significantly; the two vibration-rotational bands in this region are centered at $2.69~\mbox{$\mu$ m}$ and $2.77~\mbox{$\mu$ m}$ (Herzberg 1966), but these are only 2% deep as a maximum in our synthetic spectrum. The other molecules included in the spectral synthesis, i.e. CH, CN, and C₂, do not contribute to the spectrum at all. All spectra are convolved with a Gaussian profile with a width of $300~\mbox{km~s$^{-1}$ }$, in order to provide a good overview of the spectrum. As can be seen, the main features are due to water vapour. It is the signatures of the $\Delta \nu_1=1$ and $\Delta \nu_3=1$ modes of the rotational-vibrational bands of H₂O. These modes are due to symmetric and antisymmetric stretching of the water molecule. The first overtone vibration-rotational band of the CO molecule starts contributing at $3.0~\mbox{$\mu$ m}$ and increases toward shorter wavelengths. Several strong OH bands show their signatures in the synthesised spectrum including all molecules, especially at longer wavelengths (3.3- $3.7~\mbox{$\mu$ m}$). These OH signatures can also be seen in the observed spectrum. The dashed line shows the continuum of the spectrum.

The partial pressure of water vapour in the model photosphere follows the gas pressure from the surface in to regions where the temperature has reached approximately 4000 K, with values of P(H₂O)/ $P_{\rm g} \sim 3\times 10^{-3}$. Further in, the partial pressure of water vapour decreases rapidly due to the dissociation of the molecule.