7 Discussion

7.1 A problem of mixing length?

A single value of the mixing-length parameter has been used throughout this work. We did not possess enough data on LMC and globular cluster stars to securely calibrate variations of $\alpha$ and of the correction parameters, but some discussion of this deserves attention and is given below.

Comparison between the mixing-length theory and 2D numerical simulations of stellar convection indicate that, at the bottom of the RGB, the mixing-length parameter should exceed the solar value by at least 5% - or even 10% according to some 3D calculations. Low-mass K sub-giant models have $\alpha=\alpha_\odot+10$-15%, and this parameter is a decreasing function of the effective temperature while it increases with the surface gravity (Freytag & Salaris 1999; Ludwig et al. 1999).

Empirical results are also available: Chieffi & Straniero (1989), Castellani et al. (1991), Bergbusch & Vandenberg (1997), Vandenberg et al. (2000) were able to fit the Red Giant Branch (RGB) of Globular Clusters (masses $\le 1~M_\odot$) with the same mixing length parameter as for solar-like stars. However, Stothers & Chin (1995) as well as Keller (1999), showed that the mixing length parameter must exceed $\alpha_\odot$ by roughly 35% at 3 $M_\odot$ and strongly decrease at larger masses (and luminosities) if the red giants and supergiants of Galactic open clusters and of young clusters of the Magellanic Clouds are to be fitted. This non-monotonic behaviour illustrates the competition between the mass and the luminosity in determining the gravity and temperature. No clear metallicity-dependence was found (Stothers & Chin 1996; Keller 1999). Concerning the Horizontal Branch and the Early-AGB, Castellani et al. (1991) found the solar mixing length to be suitable for Globular Cluster stars with masses $\le 0.8~M_\odot$.

Figure 7: The de-biased PLC distribution of LPVs in the solar neighbourhood ( top row: Groups 1 and 2; bottom row: 3 and 4), compared to calibrated models (Z=0.02, 0.02, 0.04 and 0.01 respectively). Each group appears as a quasi-ellipsoid containing 60% of the population. The PLC relation (main symmetry plane) and the PC distribution (relief on the xy plane) are also represented. The curved surfaces represent the models (fundamental mode for Groups 1, 2 and 4; first overtone for Group 3)

Figure 8: The LC distribution of local LPVs (sample points and projected, de-biased $2\sigma$ iso-probability contours of the populations), compared to calibrated models (solid lines: Z=0.02; dashed: Z=0.01; dot-dashed: Z=0.04; masses are 0.8, 1 and 1.5 $M_\odot$ from the bottom to the top)

Although the LPVs are TP-AGB stars, with much higher and diverse luminosities, these previous works give us the order of magnitude of the possible variations of $\alpha$ between the mean mass or luminosity that were the basis of the calibration (barycenter of the LMC Mira-like stars) and 3 $M_\odot$ or the maximum luminosity of the sample. We have thus derived a mass-mixing-length relation from the RGB results, by performing a spline interpolation and scaling so as to match the $\alpha$ and mass that were obtained at the barycenter of the LMC Mira-like stars. On the other hand, when investigating the possibility of a luminosity-dependence, we have assumed that $\alpha$ varies by 35% every 2 magnitudes.

Figure 9: The PL a) and PC b) distributions of old-disk LPVs, compared to calibrated models with Z=0.02: fundamental mode (solid lines) and first overtone (dashed lines)

Figure 10: The PL a) and PC b) distributions of thin-disk LPVs, compared to calibrated models with Z=0.04: fundamental mode (solid lines), first overtone (long-dashed lines) and second overtone (dashed lines). Also shown are the de-biased distributions of old-disk stars and, shifted by +0.1 mag, the MACHO sample (dots). The two squares show the adopted typical first and second overtone pulsators in the thin-disk population

Figure 11: The PL a) and PC b) distributions of extended-disk/halo LPVs, compared to calibrated models with Z=0.01

If $\alpha$ depends on the mass, the isomass lines in this paper must move away from each other in the PL and LC diagrams below $3\,M_\odot$ and get closer again at larger masses. This means that, in the previous sections, the masses that appeared larger than 1 $M_\odot$ and the corresponding metallicities were respectively over- and underestimated. As a consequence, a totally unlikely metallicity (Z > 0.10) is now required in a large part of Group 1. The Group 4 sample, too, gets unlikely high mass and metallicity in view of its kinematics. Moreover, the period corrections also have to be increased, which makes the model fit more unlikely in view of the theoretical background. The mixing-length parameter is thus probably not or is little dependent on the mass. The same arguments also allow us to rule out the hypothesis that it increase with the luminosity.

On the other hand, let us assume that $\alpha$ is a decreasing function of L. Then, the slopes of the isomass lines get smaller in the PL and LC diagrams. The same combination of $\Delta\log P_0$, $\Delta (J-K)$ and $\alpha_{\rm LMC~mira}$ as in the preceding sections gives a better fit, with a mass discrepancy $\delta M=0.04~M_\odot$ for the SMC-like GC fundamental pulsators. However, we get $\delta M=0.08~M_\odot$ for the first overtone pulsators. As a compromise, we may adopt $\Delta\log P_0=+0.126$, $\Delta(J-K)=-0.03$ and $\alpha_{\rm LMC~mira}=2.22$, yielding $\Delta\log P_1=0.033$ and $\delta M = 0.07~M_\odot$ for both modes. Then, the deviation of $\Delta (J-K)$ from the mean of its a priori estimates (see Sect. 4.4), expressed in units of standard deviation, defines a scaling to apply to $\Delta(V-K)$, which yields -0.53.

As a first consequence, this hypothesis helps to solve a puzzling contradiction that we found in Figs. 4 and 5: while the theoretical mass increases along the Mira-like strip of the LMC in the PL and PC planes it decreases in the PC diagram if a constant mixing length is assumed. A bit of metallicity dispersion and of period-dependence of the correction parameters might also help.

The barycenter masses and metallicities of the local, LMC and GC populations are shown in Table 1, together with the ones obtained with a constant $\alpha$. One can see that <M> and <Z> are nearly constant for Groups 1 and 4. For Groups 2 and 3 (1st ov.), the mean masses decrease by 10%, but the metallicities reach high values that are unlikely for old stars of the solar neighbourhood. Moreover, no solution is found for the Group 3 second-overtone pulsators, even when varying $\Delta\log P_2$ by a factor 3. All of this, together with the very low mass found for GC 1st overtone pulsators, suggests that $\alpha$ is actually less luminosity-dependent than the adopted rate.

If we further increase the luminosity-dependence of the mixing length, smaller or even null period corrections of the fundamental and first overtone modes can be adopted. For example, if $\alpha$ varies by 70% every 2 magnitudes, the compromise is reached at $\Delta\log P_0=0.063$, $\Delta(J-K)=-0.065$ and $\alpha_{\rm LMC~mira}=2.09$, yielding $\Delta\log P_1=-0.004$. However, the fit is not better ( $\delta M = 0.07~M_\odot$ for both modes) and the mean mass of the SMC-like GC first-overtone pulsators becomes definitely unlikely ( $0.4~M_\odot$).

Concluding, the mixing-length parameter probably decreases along the AGB, but its variation should not exceed 15% per magnitude. Clearly positive period corrections are anyway required, especially for the fundamental mode (>30%) but also for the first overtone (>8%) and the latter relative shift is always smaller than a third of the former.

7.2 Stability, consistency and error bars

Our results form a consistent set including seven or eight populations with five or six different mean metallicities, three pulsation modes and two colour indices. If we vary by 0.1 $M_\odot$ the assumed mean mass of the LMC Mira-like stars, all other masses simply get shifted by a similar amount. The metallicities of the local stars remain unchanged, and the period and colour correction parameters vary by only about 0.05 and 0.015 (J-K) respectively.

In the single-$\alpha$ case, the period and colour correction parameters were taken a minima, i.e. a better fit (smaller $\delta M$) would have been obtained with larger corrections. As a test, let us increase $\Delta (J-K)$ and $\Delta(V-K)$ by one standard deviation of the a priori estimates (Fig. 2), this corresponds to $\Delta\log P=0.24$ for the fundamental mode and 0.101 for the first overtone. Then, the mean masses of Groups 1 through 3 increase by less than 2% and the metallicities by only 0.002. This is of course negligible. Thus, our results are stable, which is confirmed by their similarity when a luminosity-dependent mixing length is adopted (see Table 1).

On the other hand, the error bars of the periods, magnitudes and colours at the barycenter of each population (LMC, Globular Clusters and solar neighbourhood) are small (see Sects. 2 and 5) and yield uncertainties of a few percent on masses and, for the solar neighbourhood, 5-15% on metallicities (except perhaps for Group 4).

Since any change of the adopted temperature scale would automatically translate into the colour correction parameters (the calibrated values as well as the a priori estimates shown in Fig. 2) and into the luminosity dependence of $\alpha$, the uncertainty of this relation does not significantly affect our results at constant Z. However, metallicity differences with respect to the LMC might be a little overestimated, if molecular opacities were significantly underestimated in the atmospheric models.

Support for this model may be seen in the consistency of the mean masses and metallicities of the local populations of LPVs with their respective kinematics, determined in Paper I, and their similar evolutionary stages:

Group 1, which has the kinematics of old disk stars and is mostly composed of Miras, is actually found having about the solar metallicity, just as usually assumed.

Concerning Group 3, first and second overtone pulsators have the same, higher mean metallicity, consistent with the thin-disk kinematics.

Also consistently with its kinematics (extended disk and halo), Group 4 has a lower metallicity than the others.

Last, compared to Group 1, Group 2 is found to be much more metallic and a bit less massive (L-dependent $\alpha$) or a little more metallic and massive (single $\alpha$). Since these stars are slightly less evolved and may lose $0.05~M_\odot$ in roughly 10⁵ years, the mean initial mass must be similar to or smaller than that of Group 1. So, in both cases, Group 2 stars must be a little older (Vassiliadis & Wood 1993). This is consistent with the larger velocity dispersion and scale height found in Paper I. The evolutionary aspects of this work will be further investigated in a forthcoming paper.

7.3 Theoretical interpretation

The significant, positive period correction that has to be applied to linear models contradicts the hydrodynamical calculations. Indeed, we have seen in Sect. 4.2 that our $\Delta\log P_0$ would range from -0.19 to -0.03 if the models of Ya'ari & Tuchman (1996, 1998) did represent real stars. This would lead to large mass discrepancies ( $0.1 < \delta M \le 0.35~M_\odot$) for the fundamental pulsators of the SMC-like clusters, even with a luminosity-dependent mixing length. Moreover, the corresponding colour corrections would be extreme values among the a priori estimates calculated in Sect. 4: $-0.15\le\Delta (J-K)\le-0.10$ and $-2.7\le\Delta(V-K)\le-1.9$. This is quite a bit for an average behaviour!

Another interesting - though weaker - argument is derived from evolutionary considerations. For the predicted period shifts, the mean metallicity and mass ratios of Group 1 to Group 2 are $8\,\ge\,{Z_1/Z_2}\,>\,0.8$ and $1.5\,\ge\,{M_1/M_2}\,\ge\,1.1$. Detailed evolutionary calculation is beyond the scope of this paper but if the periods predicted by Ya'ari & Tuchman were right, Group 2 would probably be a little younger than Group 1, in contradiction with the kinematics.

Summarizing, the nonlinear behaviour calculated by Ya'ari & Tuchman (1996) is unlikely in view of the available data. Let us try to explain this:

As stated in Sect. 4.1, the core mass-luminosity relation assumed in our calculations holds over a limited part of the thermal pulsation cycle, but the effect on the period and temperature is very small and the opposite sign of the calibrated period and colour correction parameters.

The same section also shows that phase-lagged convection, horizontal opacity averaging and turbulent pressure all together might yield a period increase by as much as 40% for the fundamental and 25% for the first overtone, the relative shift of the latter always being larger than a third of the former. In case of negligible nonlinear effects, this could explain the correction parameters of the fundamental or the first overtone but not both, since the ratio would be wrong. Furthermore, the nonlinear effects predicted by Ya'ari & Tuchman would strongly reduce the final period of the fundamental while not significantly changing the first overtone, so that both modes would finally exhibit similar shifts with respect to our LNA models.

In other terms, if an improved physics of the sub-photospheric regions manages to explain the observed first overtone, then there must remain a significant, positive shift of the actual fundamental mode ( %, possibly much more) with respect to its theoretical period. The only explanation seems to be the coupling of the stellar envelope with the circumstellar layers and the subsequent wind, evoked in Sect. 4.3 above.