Model calculations

We have calculated the models presented in this paper using our multipurpose model atmosphere code PHOENIX, version 10.5. Details of the code and the general input physics setup are discussed in and and references therein. The model atmospheres for Cepheids presented here were calculated with the same general input physics. However, they use spherical geometry (including spherically symmetric radiative transfer) rather than plane parallel geometry. For giant models with low gravities ($\log(g)\le 3.5$), this can be important for the correct calculation of the structure of the model atmosphere and the synthetic spectrum .

Our combined molecular line list includes about 500 million molecular lines. These lines are treated with a direct opacity sampling technique where each line has its individual Voigt (for strong lines) or Gauss (weak lines) line profile. The lines are selected for every model from the master line list at the beginning of each model iteration to account for changes in the model structure . This procedure selects about 190 million molecular lines for a typical giant model atmosphere with $\hbox{$\,T_{\rm eff}$}\approx 3000\,{\rm K}$.Accordingly, we generally use the parallelized version of PHOENIX to perform calculations efficiently on parallel supercomputers. Details of the calculations are given in the above references and are not repeated here.

The parameterization of giant models requires an additional parameter compared to plane parallel model atmospheres. This complicates the task of relating theoretical models to observed data, see for a discussion of these issues. In the models presented here, we set the stellar radius R by the condition $g_{\rm grav}=GM/R^2$, where we define $g_{\rm grav}$ as the gravitational acceleration at $\ifmmode{\tau_{\rm std}}\else\hbox{$\tau_{\rm std}$}\fi=1$, G is the constant of gravity, and $\ifmmode{\tau_{\rm std}}\else\hbox{$\tau_{\rm std}$}\fi$ is the optical depth in the continuum at $1.2\,\mu$. The luminosity L of the model is then given by $L=4\pi R^2 \sigma \hbox{$\,T_{\rm eff}$}^4$. For convenience, our model grid is based on the set of parameters $(\hbox{$\,T_{\rm eff}$},\log(g),M,[{\rm M/H}])$. The above formulae and the structures of the model atmospheres can be used to transform them to any target set of parameters (e.g., for a different definition of $\ifmmode{\tau_{\rm std}}\else\hbox{$\tau_{\rm std}$}\fi$).

for a discussion of these issues. In the models presented here, we set the stellar radius R by the condition $g_{\rm grav}=GM/R^2$, where we define $g_{\rm grav}$ as the gravitational acceleration at $\ifmmode{\tau_{\rm std}}\else\hbox{$\tau_{\rm std}$}\fi=1$, G is the constant of gravity, and $\ifmmode{\tau_{\rm std}}\else\hbox{$\tau_{\rm std}$}\fi$ is the optical depth in the continuum at $1.2\,\mu$. The luminosity L of the model is then given by $L=4\pi R^2 \sigma \hbox{$\,T_{\rm eff}$}^4$. For convenience, our model grid is based on the set of parameters $(\hbox{$\,T_{\rm eff}$},\log(g),M,[{\rm M/H}])$. The above formulae and the structures of the model atmospheres can be used to transform them to any target set of parameters (e.g., for a different definition of $\ifmmode{\tau_{\rm std}}\else\hbox{$\tau_{\rm std}$}\fi$).