6 Application to the solar neighbourhood

Having fully calibrated the models, we can now investigate the results of Paper I, i.e. the four de-biased PLC distributions of the local LPVs and the calibrated and de-biased individual data (sample stars). For the sake of clarity, members of the old disk (Groups 1 and 2 as defined in Paper I), thin-disk (Group 3) and exended-disk/halo (Group 4) populations will be separately investigated. Mathematically speaking, the work to be done consists in determining the pulsation mode, mass and metallicity corresponding to the barycenter of the de-biased distribution of each group, by solving the MLZ$\alpha$T and MLZ$\alpha$P equations with the above-calibrated free parameters.

At each barycenter, the absolute K magnitude was converted into the bolometric one by applying the bolometric correction defined in Sect. 3.2, using the corresponding <(V-K)₀>. Then, the period and colour correction parameters ( $\Delta\log P=0.16$ or 0.056 and $\Delta(V-K)=-0.15$) were subtracted from the data points before solving the two equations. It must be mentioned, however, that Figs. 7 through 11 were plotted keeping the observations unchanged, i.e. using K magnitudes. We thus applied to the models bolometric corrections that were deduced from the theoretical V-Kafter adding $\Delta(V-K)$. The results are detailed in the next subsections and summarized in the left-hand part of Table 1. The isomass lines in the figures correspond to 0.8, 1 and 1.5 $M_\odot$.

The de-biased 3D distribution of each Group, shown in Fig. 7, appears as a quasi-elipsoidal volume containing 60% of the population. In the three fundamental planes, it is represented by a quasi-elliptical contour which is the projection of the elliptical $2\sigma$ iso-probability contour defined in the main symmetry plane (see Paper I). Then, the pulsation mode and the mean mass and metallicity are given by the barycenter of the de-biased distribution, i.e. the center of the "ellipsoid'' or "ellipse'': if the models have been well calibrated and if their adopted metallicity equals the actual population mean, the surface corresponding to one theoretical mode in the 3D diagram must include the barycenter. In the three 2D figures, this point must lie on the same iso-mass line, which property was used above for calibrating the models. Another advantage of working on the barycenter is that we avoid the projection effects that occur when the "ellipsoid'' crosses the single-mode single-metallicity PLC surfaces with a high incidence.

Figure 5: The PL ( top row) and PC ( bottom row) distributions of LPVs in globular and LMC clusters and the ones of LMC Mira-like stars, compared to the calibrated models: fundamental mode (solid lines) and first overtone (dashed lines); filled squares indicate the single-mode barycenters of the data. Left: SMC-like metallicity. Right: LMC-like metallicity. The superimposed linear strips correspond to the whole Mira-like population of the LMC with $P \le 420$ d: mean PL and PC relations (and the standard deviations about them) given by Feast et al. (1989) and Hughes & Wood (1990)

6.1 Old disk stars

For Group 1, the only possible pulsation mode is the fundamental. The metallicity is Z=0.02 and the mass 0.9 $M_\odot$. On the other hand, Group 2 pulsates on the same mode, but with Z=0.027 and $M=0.95~M_\odot$.

A careful look at the PL and LC diagrams on the one hand and the PC on the other, shows a contradiction if a single metallicity is assumed within Group 1. Indeed, when increasing the period and colour, the mass decreases in the latter plane, while it increases in the two former. This is the 2D translation of the fact that, in 3D, the main symmetry plane of the population is much inclined with respect to the single-Z single-mode theoretical surfaces. If $\Delta\log P$ is supposed to increase by any reasonable amount with the period, the problem is only slightly attenuated. If $\Delta(V-K)$ too is assumed to depend on the period, then it has to reach about -3 at the top of the Group 1 sample. This is at the limit of (or exceeds) the a priori estimates, which anyway exhibit no obvious correlation with the period (see Sect. 4 and Fig. 2). Therefore, complete explanation of the PLC distribution of Group 1 probably requires the metallicity to significantly increase with the period. The mass, too, must strongly increase with the period (even if only the colour shift is invoked). Actually, if the correction parameters are kept unchanged, a metallicity of 0.04 is found at the barycenter of the Group 1 sample, together with a mass of 1.7 $M_\odot$. Concerning the Group 2 sample, the metallicity (0.035) differs just a little from the population mean, and the mass is 1.2 $M_\odot$.

6.2 Thin-disk stars

Group 3 stars cannot pulsate in the fundamental mode, since this would require Z=0.44. If we assume that they are pulsating on the first overtone, the mean mass and metallicity are $1.25~M_\odot$ and Z=0.08, which seems a bit too metallic for stars of the solar neighbourhood.

Figure 6: PL distribution of red variable stars of the LMC in the MACHO data base (magnitudes taken from Wood 1999), compared to the calibrated models: fundamental mode (solid lines), first (long-dashed) and second (dashed) overtones. Also shown are the Mira-like PL relations of Feast et al. (1989) and Hughes & Wood (1990)

However, the MACHO sample (see Fig. 6) suggests that this group may be a mixture of first and second overtone pulsators. That is, the "ellipse'' representing the population would, in fact, be a global fit to two steeper, less extended ones. Unfortunately, we were unable to separate these two sub-groups (modes) when doing the luminosity calibration, probably because of the paucity of the sample in view of the observational error bars. We are thus deprived of any direct estimate of the barycenter of each overtone population. Nevertheless, we can use the MACHO sample as a guide to determine two points, typical of the first and second overtone pulsators of the Group 3 population. To this purpose, we first have to shift the MACHO sample by 0.1 magnitude (roughly corresponding to the luminosity and colour shifts involved by the metallicity difference between the LMC and our Galaxy), so that the Group 3 "ellipse'' crosses the two bulges at the bottom of the PL strips, and the top-left limit of the Group 3 sample matches that of the MACHO data. Then the intersections of the strips with the de-biased period-luminosity relation defined by the "ellipse'' can be picked out. They are found around $\log P=1.33$ and 1.85. The corresponding colours are given by the PLC relation found in Paper I. The models fit these points with Z=0.04 and $1.1~M_\odot$for the first overtone pulsators, and the same metallicity but $0.75~M_\odot$for the second overtone. The latter value is, in fact, a lower boundary of the mean mass, since the adopted point lies at the bottom edge of the second overtone population.

6.3 Extended disk and halo stars

Group 4 stars (Figs. 8 and 11) appear pulsating on the fundamental mode with Z=0.009 and 1.1 $M_\odot$ (0.014 and 1.65 $M_\odot$ for the sample). No solution is found for the first overtone. However, these results must be taken with some caution, because of the paucity of the sample in view of the data error bars (see Sect. 2.1).