8 Conclusion

The aim of this paper was to interpret the results of Barthès et al. (1999) in terms of pulsation modes and fundamental parameters, i.e. the fact that the Long-Period Variable stars of the solar neighbourhood are distributed among four groups (according to kinematic and photometric criteria), and to study the period-luminosity-colour distributions of these groups. This was done by confronting them with a grid of linear nonadiabatic pulsation models.

Preliminary discussion of the colour-temperature relations and of the existing linear and nonlinear modelling codes showed that the periods and colours predicted by LNA models may significantly differ from the observed values, even if the adopted fundamental parameters are the true ones. In order to mimic this behaviour, we added a few free parameters to the LNA models: a systematic period correction $\Delta\log P$ for each mode, and systematic colour corrections $\Delta(V-K)$ and $\Delta (J-K)$. These parameters, as well as the mixing length, were then calibrated by demanding that consistent masses be derived from the period and from the colour, for the fundamental and first-overtone pulsators of the LMC and of globular clusters with LMC or SMC metallicity. It was assumed that the mean mass of the Mira-like stars (fundamental pulsators) of the LMC is 1 $M_\odot$. Then, the mean mass of the LMC first-overtone pulsators was found to be about 0.95 $M_\odot$, while that of the fundamental and first-overtone pulsators with SMC metallicity was 0.8 and 0.6 $M_\odot$ respectively (or down to 0.75 and 0.5 $M_\odot$ if the mixing-length parameter is allowed to strongly vary with the luminosity).

We were thus able to determine the pulsating mode and the mean masses and metallicities of the neighbouring LPV populations:

• Group 1 stars (old disk LPVs, mostly Miras) are pulsating on the fundamental mode. The mean mass is $<M>\,\simeq\,0.9\,M_\odot$ and the mean metallicity $<Z>\,\simeq\,0.02$, both strongly increasing with the period;
• Group 2 stars (old disk LPVs, mostly SRb's) are pulsating on the fundamental mode also, with $<M>\,\simeq\,0.9\,M_\odot$ and . Metallicity variations of about 50% are likely;
• Group 3 stars (thin-disk LPVs, mostly SRb's) are pulsating on the first and second overtones, with [4] $<M_{\rm 1ov}>$ $\;\approx\,1.05\,M_\odot$, $<M_{\rm 2ov}>$ $\;>\,0.75\,M_\odot$ and [4] $<Z>\;\geq\,0.04$;
• Group 4 stars (extended disk and halo) appear to be pulsating on the fundamental mode with $<M>\,\simeq\,1.1\,M_\odot$ and $Z\,\simeq\,0.01$. This result must however be taken with caution.

These findings, based on the V-K index as a temperature estimator, have been checked and confirmed by using J-K. They were proven to be stable and consistent with the kinematics. The error bars of the masses are about 5%. If one shifts the reference mass of the LMC Miras by any reasonable amount, all other masses are simply similarly shifted and the metallicities remain nearly unchanged. Metallicity differences with respect to the LMC may be a little overestimated if molecular opacities have been significantly underestimated in the atmospheric models. Nevertheless, the hierarchy holds anyway.

The mixing-length parameter probably decreases along the AGB, but its variation should not exceed 15% per magnitude. This was taken into account in the abovementioned results.

This study confirms the findings in Paper I, the discrimination between Miras and semiregulars is not pertinent: Groups 1 and 2 not only have similar kinematics but also the same pulsation mode.

It has also been shown that both the linear and nonlinear models that were the basis of all previous studies of LPV pulsation are probably far from the real pulsational behaviour of these stars. While dynamical calculations including a modern equation of state predict a strong reduction of the fundamental nonlinear period with respect to the linear one, important, positive systematic corrections have to be applied to the periods of our linear models (30-45% for the fundamental mode and 8-13% for the first overtone). Improvements of the physics of the sub-photospheric envelope (phase lagged convection, turbulent pressure, horizontal opacity averaging...) appear insufficient to explain these two shifts altogether, so that the actual fundamental period should always exceed the theoretical one by at least 15%. This led us to conclude that all existing linear and nonlinear pulsation codes probably suffer from neglecting the stellar wind generated by the interaction with the circumstellar envelope.

As a consequence, the works of Barthès & Tuchman (1994) and Barthès & Mattei (1997), who confronted LNA models with the Fourier components of the lightcurves of a few nearby Miras and concluded in favour of the first overtone, should now be reconsidered by taking into account the necessary period corrections and the variations of metallicity and, possibly, mixing-length parameter.

This study may also have consequences in AGB evolutionary calculations, especially those that use the period as a substitute for a fundamental parameter (M or $T_{\rm eff}$) or as the variable in the empirical mass-loss function (e.g. Whitelock 1986; Vassiliadis & Wood 1993; Reid et al. 1995; Marigo et al. 1996). Indeed, these studies are based on pulsation models by Wood that appear to strongly overestimate the periods and their dependence on luminosity and metallicity, probably because of the equation of state. Concerning the fundamental mode, this peculiarity allows Wood's models to roughly mimic the abovementioned systematic period shifts and variation of the mixing length parameter, but with an uncertainty that remains to be assessed. Moreover, while many Long Period Variables are obviously pulsating on the first or second overtone, this possibility is usually neglected in evolutionary calculations, and Wood's models appear definitely inapropriate for these modes. These issues will be investigated in a forthcoming paper.

Acknowledgements

This work was supported by the European Space Agency (ADM-H/vp/922) and by the Hispano-French Projet International de Coopération Scientifique (PICS) No. 348. We thank the referees, Drs. Bessell and Bono, for numerous comments that helped us to improve this paper.