5 Comparison to evolutionary tracks

It is of course of great interest to confront the predictions of evolution calculations with the M-L values which we have extracted from the OGLE data. Recently a number of such calculations, all performed with the OPAL opacities, have become available: Alibert et al. (1999), Girardi et al. (2000), Bono et al. (). Alibert et al. have also compared their results to observational EROS P-magnitude-color values and have obtained reasonable agreement. Of course our results, by construction, satisfy these observational constraints.

The evolution calculations have not been compared to the SMC and LMC OGLE-derived stellar parameters, neither in a HR diagram nor in a M-L diagram, both of which we now present in Fig. 5. We plot the evolutionary tracks of Girardi with (X=0.756, Z=0.004) and (X=0.742, Z=0.008), respectively, which are the closest to our chosen compositions. The Girardi et al. M-Lrelations for the 2nd/3rd crossing (taken as the points of slowest evolution near the blue edge) are shown as solid lines, those of Alibert et al. as long dashes and those of Bono et al. with (Y=0.226, Z=0.004) and (Y=0.216, Z=0.004) as short dashes.

We note that none of the three sets of evolutionary calculations is fully in agreement with our OGLE-derived LMC and SMC M-L data (nor are the older calculations), even when we adopt the most favorable choice of distance modulus and reddening. At fixed mass, the evolved stellar models are not luminous enough. The results of Girardi et al. are closest to our derived M-L relations, and they also seem to have the right curvature (Alibert et al. and Bono et al. used straight line M-L fits). Indeed, if the M-L of Girardi et al. are shifted by $\sim$0.35 in $\log L$ for SMC (respectively by $\sim$0.25 in $\log L$ for LMC) metallicities, a reasonable agreement obtains at low and high luminosities. This could be achieved, at least partially, with overshooting or an increase thereof (Baraffe, priv. comm.).

We have not shown the M-L relations for the faster, first crossing to avoid cluttering the figures. It can be seen from the left-hand sub-figures that the luminosities are about 0.2 lower for the same mass on these crossings.

The density of stars is definitely lower at the low luminosity end. A natural explanation is that the low luminosity stars are first crossers. The Girardi et al. tracks for the LMC are compatible with this interpretation, but it would be useful to do the statistics on the basis of the evolution speed along the tracks. However, for the SMC, even the Girardi M-L is much too low for the first crossers.

Finally, the Girardi low L tracks do not loop sufficiently far for the SMC. The problem is slightly worse for both the LMC and SMC tracks of Alibert et al..

The reader may wonder why both the evolutionary calculations and our pulsational calculations claim to give agreement with the observational data, yet they are based on substantially different stellar masses. To discuss the origin of this discrepancy we first need to compare the two procedures.

In our calculations, we rely only on observed periods, colors and magnitudes, transformed to $T_{\rm {ef f}}$, L with Kurucz models, and from which we construct stellar models that exactly satisfy these observational constraints. The computed periods are essentially independent of any physical and numerical uncertainties, as pointed out in Sect. 3.2. Furthermore, we do not rely on the stability of the models. The latter are quite sensitive to physical uncertainties, especially the turbulent convective parameters. (We note though that, with our "standard'' parameters, the vast majority of our models are linearly unstable.)

In order to determine the $T_{\rm {ef f}}$ range of Cepheid behavior along the evolutionary tracks, Alibert et al., for example, performed a linear stability analysis along these tracks and then compared the properties of their unstable models to the observational EROS P-magnitude and P-color data sets. The use of a stability analysis with the inherent uncertainties coming mostly from turbulent convection necessarily introduces an uncertainty in the temperature range of the Cepheid models (width of the instability strip).

Equation (7) gives a clue to the apparent discrepancy: It is possible to absorb rather large changes in M, say $\delta {{\rm Log}}~M = 0.1$ (the largest difference between the evolutionary calculations, or with our average M-L), with a tiny change in ${{\rm Log}}~{T_{\rm {ef f}} }$of $\delta{{\rm Log}}~{T_{\rm {ef f}} } = 0.02$. This shows that our procedure of extracting masses directly from the observational data can therefore impose a novel, external and stronger constraint on the evolutionary calculations than is available from the previous comparison to observed P-magnitudes and colors.