6 Conclusions

Thanks to an up-to-date maximum-likelihood method, using parallaxes, proper motions, radial velocities and other independent data (periods and colours), we have calibrated the luminosity of about 350 Oxygen-rich Long-Period Variable stars (Miras, SRa and SRb) observed by HIPPARCOS. Meanwhile, the stars got classified in several groups differing by their distributions of kinematical parameters, magnitude, period and colour. Four groups were found: Group 1, mainly composed of Miras and having the kinematics of old disk stars; Group 2, mainly composed of SRb, with the same kinematics as Group 1; Group 3, also mainly composed of SRb, but with a much younger kinematics; Group 4, corresponding to the extended disk and halo, and containing no SRb. For each of them, we obtained de-biased PLC, PL, PC and LC relations.
We thus confirm the existence of two SRb populations, already suggested on other grounds by Kerschbaum & Hron ([1992,1994]). SRa stars do not seem to constitute a separate class. They can be considered as a small-amplitude subset of the Mira and SRb classes. The presence of a small but significant number of Miras in Group 2, and of SRb's in Group 1, may be due to the probabilistic character of our classification method. However, it also tends to confirm that the usual (GCVS) classification criteria are not fully pertinent, as shown by Lebzelter et al. ([1995]).

As expected, since they all belong to the AGB, the stars seem to obey a global luminosity-colour relation, both in (V-K)₀ and (J-K)₀. More precisely, each group has its own relation, nearly parallel to the others, with a slight shift. Though belonging to the same Galactic population, Group 2 stars (SRb's) are fainter, bluer, and have shorter period and shallower period-colour relation than Group 1 (Miras). They probably pulsate on a higher-order mode. The higher luminosity and shorter period of Group 3 with respect to Group 2 is probably due to higher mass and metallicity. The slope of Groups 2 and 3 in a period-luminosity diagram, as well as a close look at the distribution of the three variability types within these groups, indicate that the two sequences of Semiregulars correspond to evolutionary sequences towards the Mira instability strip (i.e. SRb's are, generally, a little younger than Miras of the same population).

All these findings will be confronted in detail to theoretical models of pulsation and evolution in Paper II.
Another important result of our study is that, contrary to a usual assumption (e.g. in van Leeuwen et al. [1997]), but consistently with the work of Menzies & Whitelock ([1985]) on a few Globular Cluster stars, the PL and PC relations of Oxygen-rich Miras found in the LMC may not be trivially transposed to other galaxies by simply shifting the zero-point: their slopes are inconsistent with the ones found for O-rich Miras in the solar neighbourhood.

The first explanation is a misclassification of many LMC stars, since the observers simply discriminated the SRa stars on grounds of their small amplitude, and thus the remaining so-called "Miras'' included a significant proportion of (younger) SRb stars. Concerning the PL relations, additional discrepancy may be generated by a metal-deficient, probably older LMC Mira population, the existence of which was already suspected by Wood et al. ([1985]) and Hughes et al. ([1991]). All this, together with the fact that the stars distribution within the LMC "Mira'' strip (at least below 500 days) seems rather uniform, suggests that these LPVs derive from a quite smooth star formation history, rather than well-separated bursts.

Concluding, the fact that the global PL relation of LMC "Miras'' approximately matches the global one of Galactic Miras (in the solar neighbourhood) does not guarantee that it holds for every galaxy: everything depends on the relative number of misclassified SRb's and on the respective proportion of the different populations of stars, i.e. on the star formation history.

Nevertheless, consistently with the LMC and globular clusters data (see e.g. Hughes & Wood [1990] and Menzies & Whitelock [1985]), our calibrations show that an LPV M-giant (Mira, SRa or SRb) pulsating with a period of 300-330 days is expected to have a mean absolute K magnitude of $-7.5 \pm 0.5$, whatever the stellar population. This may be used as a distance estimator.

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. R.A. benefits from an EU TMR "Marie Curie'' Fellowship. We also gratefully thank J.A. Mattei and the AAVSO staff for their help in evaluating the GCVS data.