5 Models calibration: Clusters and LMC stars

The sample of globular cluster stars can be divided in three sets: one with the metallicity of the LMC (Z=0.008); a second one with about the metallicity of the SMC ( $Z=0.004\pm0.001$); and the last one with $Z\approx 0.0005$. Only the two first sets (hereafter called LMC-like and SMC-like) will be used, because the adopted log-linear model fit and colour-temperature relation are no longer reliable at very low metallicities and masses. In fact, each set may be divided in two sub-sets, obviously corresponding to two different pulsation modes (they are well separated in the PL and PC planes, see Fig. 5). Each set will be represented by two points, namely the barycenters (mean periods, magnitudes and colours) of its two sub-sets. This facilitates the model fitting, reduces the observational error bars down to low levels ($\le 0.03$ on J-K), and reduces the possible effects of the scattering of $<J-K>-~(J-K)_{\rm stat}$ down to . In fact, there are only four stars in LMC-like globular clusters. We thus prefered to merge them with the sets of LMC stars (i.e. two with the Mira-like stars and two with the higher-order pulsators).

As an additional constraint, we assume that the mean mass of the Mira-like population of the LMC is 1 $M_\odot$. This ensures that, whatever the choice of the free parameters, the masses of the sample LPVs of the LMC do not exceed 1.5 $M_\odot$, consistent with the evolutionary calculations (Wood & Sebo 1996). Then, for every value of $\Delta\log P$, the theoretical MLZ$\alpha$T and MLZ$\alpha$P relations give us the single possible value of $\alpha$ and of $\Delta (J-K)$. The pulsation mode that gives the better fit to Mira-like stars is always the fundamental.

Then, keeping the three parameters unchanged, we obtain two masses for the barycenter of the SMC-like subset corresponding to the fundamental mode: one derived from the period, the other from the colour. The calibration thus consists in minimizing the difference $\delta M$ of these two masses. As shown in Fig. 3, the mass discrepancy decreases as the period (and colour) shift increases. The model fit starts being acceptable ( $\delta M \le 0.1~M_\odot$) at $\Delta\log P_0 = +0.13$. The mean colour shift derived from the dynamical models (Fig. 2), viz. $\Delta (J-K) = -0.01$ and thus $\Delta\log P_0 =+0.16$ and $\alpha=2.29$, gives a reasonable fit: $\delta M = 0.07~M_\odot$. We adopt this solution, for which the most probable value of $\Delta(V-K)$ is already known (-0.15, of course, i.e. the mean of the a priori estimates of Sect. 4). Indeed, considering all uncertainties (especially concerning <J-K> and the CTZ relation) as well as the lack of solid theoretical ground, it would make little sense to look for an exact agreement of the masses by further increasing the correction parameters (see Sect. 7 for further discussion).

 

 

Table 1: Mean theoretical masses and metallicities of the fundamental, first overtone and second overtone pulsators of the LMC, Globular Clusters with SMC metallicity and the solar neighbourhood. The mixing-length parameter is assumed to be constant (left) or to decrease by 35% every -2 bolometric magnitudes along the AGB (right)

	$\alpha=2.29$				$\alpha=f(L)$
	Mode	<Z>	<M>		Mode	<Z>	<M>
			$(M_\odot)$				$(M_\odot)$
LMC (Mira-like)	F	0.008	1.00		F	idem	idem
LMC	1ov	0.008	0.95		1ov	idem	0.95
$Z_{\rm SMC}$ GC	F	0.004	0.8		F	idem	0.75
$Z_{\rm SMC}$ GC	1ov	0.004	0.6		1ov	idem	0.5
Group 1 pop.	F	0.020	0.9		F	0.020	0.9
sample	F	0.04	1.7		F	0.024	1.85
Group 2 pop.	F	0.027	0.95		F	0.07	0.85
sample	F	0.035	1.2		F	0.05	1.1
Group 3 pop.	1ov	0.04	1.1		1ov	0.07	1.0
	2ov	0.04	0.75a		__	__	__
Group 4 pop.	F	0.009	1.1		F	0.010	1.1
sample	F	0.014	1.65		F	0.010	1.7
$\Delta\log P_0$	+0.16				+0.126
$\Delta\log P_1$	+0.056				+0.033
$\Delta\log P_2$	+0.019				+0.009
$\Delta (J-K)$	-0.01				-0.065
$\Delta(V-K)$	-0.15				-0.53
a Lower boundary.

Figure 4: Left: the LC distribution of LPVs in globular clusters with SMC-like metallicity compared to the calibrated models, assuming Z=0.004. The two barycenters are indicated by filled squares. Right: the same for clusters with LMC-like metallicity and for the barycenter of the Mira-like population of the LMC with $P \le 420$ d; Z=0.008 assumed

As a last step, we can now determine the period correction of the first overtone, which must ensure that consistent masses are obtained from the MLZ$\alpha$T and MLZ$\alpha$P relations at the barycenters of the subsets corresponding to this mode. We obtain $\Delta\log P_1= 0.056$, with negligible mass discrepancies.

The so-calibrated model grid is represented in Figs. 4 and 5 by a series of isomass lines (0.6, 0.8, 1 and 1.5 $M_\odot$). The results of this calibration are summarized in the upper-left quarter of Table 1. Let us however recall that masses as low as 0.6 $M_\odot$ are probably slightly underestimated by the log-linear fit scheme.

The likelihood of the calibrated model grid can be checked by confronting it to the K magnitudes and periods of the MACHO sample of Wood (1999). As can be seen in Fig. 6, the strip corresponding to Mira-like stars still fits the fundamental mode, with mean mass 1 $M_\odot$. The strip immediately on its left fits the first overtone. Last, a third strip obviously corresponds to the second overtone. We could not calibrate its correction parameter but, having noticed that $\Delta \log P_1 \simeq {1\over 3} \Delta \log P_0$, we adopted $\Delta \log P_2 = {1\over 3} \Delta \log P_1$. Our interpretation is consistent with that of Wood et al. (1999) and Wood (1999), who also considered the stars lying on the right of the figure as probable binaries or stars pulsating on a thermal mode coupled to the fundamental.