Sphericity effects

It is important to assess the relevance of the effects of spherical geometry and radiative transfer on the synthetic spectra. For hot stars, have shown that a combination of geometric effects and line blanketing can resolve a long standing problem of discrepancies between synthetic and observed spectra. The effects of spherical symmetry on cool giant atmospheres has been investigated previously by for giant M and C stars, by for M giants, and by for Carbon stars. Our model grid extends this work toward warmer stars and reduces the gap to the stars considered in . In addition, improved molecular line lists are now available and since found a close connection between line blanketing and sphericity for hot stars, the improved molecular line data could affect the results.

In Fig.  we compare normalized (to the same area) low-resolution ($20\hbox{\AA}$) spectra for 3 sets of models. Each panel displays a NG-giant synthetic spectrum (full lines) and a synthetic spectrum calculated with the same parameters and input physics, but assuming plane parallel geometry. In this figure, only the model with $\hbox{$\,T_{\rm eff}$}=4000\,{\rm K}$shows visible differences between the plane parallel and spherical cases. However, plotting the relative differences of the spectra in Fig. 4 reveals systematic differences even for low resolution spectra. For $\hbox{$\,T_{\rm eff}$}=3000\,{\rm K}$ (top panel), we find that the spherical model emits more flux than the plane parallel model in the short wavelength regime. There are also some changes in the slopes of the spectra between 0.45 and $0.55\,\mu$m, which are hardly noticeable in the low resolution spectra. The middle panel shows the results for $\hbox{$\,T_{\rm eff}$}=4000\,{\rm K}$. This model displays a different systematic change in the blue spectral region, the long wavelength part of the displayed wavelength interval does not reveal large systematic effects. The highest effective temperature shown, $\hbox{$\,T_{\rm eff}$}=5600\,{\rm K}$, shows the smallest systematic changes.

Figure 5 is similar to the previous plot but it shows the changes for a resolution of $2\hbox{\AA}$ (10 times better). From this figure it becomes clear that the changes are due to the effect of line blanketing: line overlap and individual line strengths change between spherical and plane parallel models. In low resolution spectra this produces a change in the pseudo-continuum that is formed by the overlapping spectral lines. In the $\hbox{$\,T_{\rm eff}$}=3000\,{\rm K}$ model the TiO lines are significantly affected. This shows that spectral analyses based on high-resolution spectra need to account for spherical effects.

For longer wavelengths we find that the differences between spherical and plane parallel models are smaller. Figures 6 and 7 demonstrate this for low resolution and medium resolution spectra, respectively. The changes are mostly due to the sensitivity of the CO lines to changes in the structure of the atmosphere, therefore, they are larger at lower effective temperatures. The figure with the higher resolution shows that there are significant changes in individual water lines for low $\hbox{$\,T_{\rm eff}$}$. At higher effective temperatures, individual atomic lines are up to 15% stronger in the plane parallel models (less emitted flux, since these are absorption lines). The conclusion is that high-resolution spectral analysis requires spherical models, whereas low resolution spectra or colors can be interpreted using simple plane parallel models.

In order to demonstrate the reason for these differences, we display in Fig. 8 the electron temperatures as functions of our standard optical depth $\ifmmode{\tau_{\rm std}}\else\hbox{$\tau_{\rm std}$}\fi$, which is simply the optical depth in the continuum (b-f and f-f processes) at a reference wavelength of $1.2\,\mu$m. The temperature structures for the 2 models with $\hbox{$\,T_{\rm eff}$}=5600\,{\rm K}$ are nearly identical, the differences are less than about $80\,{\rm K}$ everywhere. In the model with the lowest effective temperature ($3000\,{\rm K}$) the differences are larger, close to $200\,{\rm K}$at $\ifmmode{\tau_{\rm std}}\else\hbox{$\tau_{\rm std}$}\fi\approx 10^{-4}$. This amounts to nearly 10% change in the absolute temperature, which is quite significant for the formation of the spectrum. Note that the line optical depth in the blue spectral region can be orders of magnitude larger than $\ifmmode{\tau_{\rm std}}\else\hbox{$\tau_{\rm std}$}\fi$ and many ``blue lines'' actually form around $\ifmmode{\tau_{\rm std}}\else\hbox{$\tau_{\rm std}$}\fi\approx 10^{-4}$. The model with $\hbox{$\,T_{\rm eff}$}=4000\,{\rm K}$ lies in between the two extremes and is not shown in the figure. The temperature structure is only a part of the full structure of the atmosphere, gas pressures and partial pressures change accordingly. For larger gravities, the differences between plane parallel and spherical models become smaller and for gravities log(g)>3.0 the differences are small enough to be negligible for most applications.