5 Comparison with previous models and to LMC clusters

As a first step, Fig. 12 compares the UV two-colours diagram (C(15-31) vs. C(18-28)) presented in the already quoted paper by Barbero et al. (1990) with analogous results, but obtained from the present computations over the relevant range of ages (t $\leq$ 1 Gyr). The time evolution of the diagrams appear in rather good agreement, supporting the scenario discussed in that paper. However, a not negligible difference can be found in the absolute calibration of the age which is now revised according to Table 5. In the same figure we plot observational data for the sample of LMC stellar clusters presented by Barbero et al. (1990), implemented with more recent data by Cassatella et al. (1996).

  

Figure 12: Two UV colours diagram as in Barbero et al. (1990). (dashed line) compared to the present work (solid line). Open square represent observational data of LMC clusters

  

Table 5: Two colours calibration for the RF model

Further comparisons can be made only for broad band colours, since HST filters, as presented in this work, up to now have been presented only by Yi et al. (1995) but for older populations ($\ge 12$ Gyr). One finds a general agreement with the behavior of UBVK colours given by Bruzual & Charlot (1993) for simple stellar populations. These authors adopted stellar evolution tracks by Maeder & Meynet (1989, 1991) in which a rather efficient core overshooting is assumed, thus predicting different (larger) evolutionary times. Our models predict the occurrence of red supergiants in moderately metal poor massive stars, not found by Maeder & Meynet because of the adoption of the Schwarzschild convective criterion (see, for a discussion on that matter Stothers & Chin 1992; Brocato & Castellani 1993 and references therein). Correspondingly, we predict redder colour for very young, moderately metal poor clusters.

To enter in more details, let us compare our results with the colours more recently presented by Bressan et al. (1994: B94) and by Bruzual & Charlot (GISSEL95 and GISSEL96: Leitherer et al. 1996), again on the basis of stellar evolutionary models allowing for an efficient core overshooting. The comparison, as given in Fig. 13 for clusters with solar metallicity, discloses a remarkable agreement in both U-B and B-V colours. As expected, for a given U-B colour B94 and GISSEL95/96 give larger ages. As a matter of fact, this colour is dominated by the luminous termination of the cluster MS, and overshooting gives (roughly) a similar termination but for larger ages than canonical computations do. This difference vanishes for the larger ages, perhaps - at least in part - because for less massive stars in the range 1.0 to 1.5 $M_{\odot}$ B94 adopted a reduced amount of overshooting.

  

Figure 13: Present integrated colours (solid line) compared to similar models by Bressan et al. (1994) (dashed line), Bruzual & Charlot GISSEL95 (dot-dashed line) and GISSEL96 (dotted line)

The comparison of B-V colours deserves a bit more discussion. One finds a close similarity of results for log t $\leq$ 9, whereas for larger ages B94 predicts redder colours. This last occurrence can be taken as an evidence that for old cluster dominated by Red Giants B94 gives cooler Giant Branches than we do. Both computations use a mixing length l $\simeq$ 1.6 $H_{\rm p}$, and the above occurrence should be likely ascribed to the use by B94 of improved model atmosphere by Kurucz (1992) which give slightly redder colour for red giants. However, we have already discussed in the introduction the evidence that even our branches appear too red in comparison with actual clusters. Thus we can only conclude that predictions about red colour indexes should wait for evolutionary computations calibrated on the cluster giant branch rather than on the Sun. GISSEL95/96 are slightly bluer than B94 probably due to the different assumption on the model atmospheres, however more details can be found in Charlot et al. (1996).

Finally, one finds that our, B94 and GISSEL95/96 V-K colours appear reasonably similar only in a restricted range of ages, namely for 7 < log t < 8. For smaller age we predict more red giants and, thus, redder colours than B94 does. A possible explanation could be related to the fact that B94 models in this range of masses have been possibly interpolated between the 12 $M_{\odot}$ (which has a He-burning loop in the red side of the CMD) and the 30 $M_{\odot}$ (which has the He-burning phase at high temperature, i.e. low emission in the K band). Consequently their V-K colours move to the blue following this interpolation.

The discrepancy at the larger age can be understood bearing in mind that this colour largely follows the occurrence of AGB stars (see, e.g., Ferraro et al. 1995). The results by B94 show a jump in the V-K colour at $\log t~=~7.8$due to the Phase Transition (Renzini & Buzzoni 1986) at $t(M_{\rm UP})$. In our models we included the AGB phase when $M \le t(M_{\rm UP})$ but neglecting the thermal pulses phase (TP-AGB). GISSEL95/96 models present a trend similar to our model even if they are sistematically redder than our of about 0.5 mag. This discrepancy is again related to the different treatment of the TP-AGB phase which foresee a different number of expected TP-AGB stars (see also Charlot et al. 1996). However, we recall that the actual time extension of the TP-AGB is still a debated question (Blöcker & Schönberner 1991; Renzini 1992).

  

Figure 14: Present integrated colours (Z = 0.02 solid line, Z = $6 \ 10^{-3}$ dashed line) compared to observational data for MC clusters. Error bars on cluster age determination are from the literature

To shed light on this problem and, more generally, to test theoretical predictions Fig. 14 compares theoretical prediction with observational U-B, B-V and V-K colours vs. age relations for MC clusters with known age from isochrone fitting. The population synthesis models for each colour are plotted for both Z=0.02 and Z=0.006. The agreement is good for U-B and B-V, and the somewhat large uncertainties on age do not allow any discrimination about the efficiency of overshooting. The V-K colour is much less clear, and - in particular - one finds clusters distributed on both the alternative predictions of curves in Fig. 14. The possible statistical fluctuations, which for V-K colour can be larger than $\sigma \simeq 0.9$ mag for clusters with $M_V \ge -7$, and the uncertainties in the age evaluation of MC stellar clusters could largely take into account the scatter of observational data, forbidding any conclusion on that matter.

Beyond such an uncertainty, let us notice that the fair agreement between our broad band U-B and B-V colours and those by B94 and by Bruzual & Charlot (GISSEL95/96) suggest that the results of population synthesis appear rather solid, given current uncertainties in the stellar evolution theories.