3 Modelling

3.1 LNA pulsation models

The AGB linear nonadiabatic (LNA) pulsation models used in this study are based on the code of Tuchman et al. (1978), modified as explained in Barthès & Tuchman (1994). The equation of state (EOS) includes the radiation and an ideal gas of e^-, H₂, H₂⁺, H, H^-, H⁺, He, He⁺ and He⁺⁺, as well as a few heavy elements, while abundances are determined by solving the Saha equation. Convection is treated according to the mixing-length formalism of Cox & Giuli (1968), with instantaneous adjustment to pulsation. Recent opacity tables, assuming solar composition and including molecules at low temperature, namely OPAL92 and Alexander (1992) (see Alexander & Ferguson 1994), are used.

The grid of models assumes X=0.7 and covers metallicities Z=0.02 and 0.001, masses ranging from 0.8 to 2 $M_\odot$, luminosities ranging from 1000 to possibly 50000 $L_\odot$, and three values of the mixing-length parameter: $\alpha=1$, 1.5 and 2. Then, a log-linear least-squares fit (excluding the extreme luminosities where isomass lines would turn back) gives us theoretical relations between the effective temperature or pulsation mode periods on the one hand, and the mass, metallicity, luminosity and mixing-length parameter on the other (MLZ$\alpha$T and MLZ$\alpha$P relations). Between $\alpha=2$ and 3, the extrapolated periods and effective temperatures are precise within 1-2%.

Figure 1: a) Adopted relation between V-K and effective temperature (full line), compared to dynamical models of Bessell et al. (1989a, 1996) (filled circles), static models of Bessell et al. (1998) (open circles) and the empirical non-LPV relations of van Belle et al. (1999) (dashed line) and Perrin et al. (1998) (dot-dashed line). Solar metallicity assumed. b) The same for J-K. The empirical non-LPV relation (dashed line) is from Bessell et al. (1983)

3.2 Temperature scale and bolometric correction

Comparison between the theoretical models and observational data requires one to convert effective temperatures into V-K or J-K indices.

Bessell et al. (1989a, 1996) have computed dynamical models of Mira atmospheres with various fundamental parameters (but always solar metallicity) and derived many colour indices, including V-K and J-K, at various phases, thus various effective temperatures (up to 3770 K), gravities and atmospheric extensions. Recently, Bessell et al. (1998) have published broad-band colours derived from static models of giant star atmospheres for temperatures higher than 3600 K. Comparison with the older static models of Bessell et al. (1989b) (which were the basis of their dynamical models) shows that the recent ones are systematically redder by a few 10^-1 mag for V-K and a few 10^-2 mag for J-K in the small overlapping domain of temperature and gravity. This is due to differences in the opacities and their handling.

On the other hand, van Belle et al. (1999) have published an empirical relation between V-K and $T_{\rm eff}$ (above 3030 K) derived from the interferometric angular diameters of 59 non-LPV giant stars (non-variable or with a very small V amplitude). Similar work, extending to lower temperatures, was also performed by Perrin et al. (1998). Moreover, an empirical relation involving J-K has been published by Bessell et al. (1983), again for non-LPV stars.

Considering the inconsistency of all these sources, there is no better way to derive a colour-temperature (CT) relation for each colour index than to perform a simple eyeball fit to the model data, as shown in Fig. 1. This takes into account the above-mentioned dynamical models of Bessell et al. (1989a, 1996), and the static models of Bessell et al. (1998) at gravities $\log g=-0.5$, 0 and +0.5. Roughly, one may estimate that our empirical relations are precise within 0.5 mag for V-K and 0.05 for J-K (subject to possible systematic error due to the imperfect modelling of the molecular lines).

Of course, it was necessary to correct the CT relations for intrinsic metallicity-dependence. The only information that we could find in the literature originated from static models. So, at temperatures $\leq 3350$ K, we applied a parabolic { $\Delta(\log Z)$, $\Delta$(colour)} fit and a series of linear { $T_{\rm eff}$, $\Delta$(colour)} interpolations to the models XX, X, Y and YY of Bessell et al. (1989b). They correspond respectively to [M/H]=+0.5, 0, -0.5 and -1, and to $\log g$ evolving from -1.02 to -0.43 as the temperature increases, so as to mimic the AGB. At $T_{\rm eff} \geq 3600$ K, we proceeded in the same way with the models listed in Table 5 of Bessell et al. (1998), while extrapolating the gravity sequence initiated by the models of Bessell et al. (1989b).

Around 3480 K, the { $T_{\rm eff}$, Z, V-K} relation resulting from Bessell et al. (1989b) exhibits a crossing-over that appears only around 3800 K in the Bessell et al. (1998) models. As a consequence, the V-K correction was simply interpolated between 3350 and 3600 K, without considering the intermediate model data. This does not concern J-K.

On the other hand, wherever necessary (see Sect. 6), we have converted the theoretical bolometric magnitudes into K magnitudes, by subtracting the the empirical bolometric correction $BC_K\,=\,f(V-K)$ given by Bessell & Wood (1984). The used V-K is, of course, the value derived from the model temperature.