2 The theoretical background

A major requirement for population synthesis models concerns the set of stellar evolutionary tracks adopted to predict the effective temperature and the luminosity of stars contributing to the total energy flux. Here we will rely on the extensive set of stellar models presented for high (Brocato & Castellani 1993), intermediate (Cassisi et al. 1994) and low mass (Straniero & Chieffi 1991; Castellani et al. 1992) stars. This set provides an homogeneous and complete evolutionary scenario since all the computations have been performed with the same stellar evolutionary code and with similar physical assumptions. It covers a wide range of values of both stellar masses and/or chemical compositions. As an example, Fig. 1 shows the run of evolutionary tracks for Y=0.27 and Z=0.02.

  

Figure 1: The adopted set of stellar evolutionary tracks for Y=0.27 and $Z=2 \ 10^{-2}$.The plotted masses are 0.6, 0.7, 0.8, 0.9, 1.0, 1.2, 1.5, 2.0, 2.3, 2.5, 3.0, 4.0, 5.0, 7.0, 9.0, 12, 16, 20, 22, 25 $M_{\odot}$

The adopted grid of stellar evolutionary tracks has been already submitted to extensive comparisons with observations, which have shown a reasonable accuracy of the models in reproducing the behavior of real stars in terms of lifetimes, effective temperatures and luminosities for a large range of ages. Colour-Magnitude (CM) diagrams of stellar clusters as NGC 2004 (t = 8 10⁶ yr: Bencivenni et al. 1991), NGC 1866 and NGC 1850 ($t\simeq 1\ 10^8$ yr: Brocato et al. 1989 and Gilmozzi et al. 1994), M11 ($t~=~1.5\ 10^8$ yr: Brocato et al. 1993), other open galactic clusters (Castellani et al. 1992) and NGC 188 (Caputo et al. 1990) all show to be in substantial agreement with the quoted theoretical models concerning magnitudes, colours and stellar counts in the various evolutionary phases.

To compute integrated colours we updated the population synthesis code presented by Brocato et al. (1990). The major changes are related to the new set of evolutionary tracks and to the relation adopted to derive magnitudes and colours from the theoretical data log$(L/L_{\odot}$), log($T_{\rm e}$). The input parameters are the age (t), helium content (Y), metallicity (Z) and total number of stars (N) and slope $\alpha$ of the Initial Mass Function (IMF), assumed to be a power law. A value of $\alpha=2.35$ corresponds to the classical Salpeter (1955) slope The original chemical composition (Y and Z) defines the set of tracks which is then used to compute the synthetic CM diagram. In all the models, we use a Monte Carlo simulation to generate the mass of each "star'' according to the power law distribution, simulating in this way the stochastic behavior of the IMF. Finally, a simple sum of the flux of each star provides the integrated fluxes and colours. This procedure has the advantage of keeping under control the results of the simulations by showing the "theoretical'' CM distribution of the simulated population that can be compared to the observed CM diagram of real stellar clusters for a further check. Moreover, the contribution of each evolutionary phase can be easily evaluated either in terms of contribution to the integrated bolometric magnitude or to the integrated magnitude in a given filter.

The choice of the filters has been based on the three following considerations:

1. the major age indicator in CM diagrams of young and intermediate age populations is the upper and hottest Termination of the Main Sequence (TMS). The stars located at the TMS are the most significant contributors to the integrated light for those populations. This means that the most efficient region of the spectral energy distribution to investigate the age is the UV side. Moreover, a not negligible aspect is that the MS phase is also the most populated evolutionary phase (thanks to the large H-burning timescales), making this indicator the best choice also for statistical reasons. Barbero et al. (1990) presented a two UV colour diagram (C(15-31) vs. C(18-25)) based on the photometric bands explored by the ANS-satellite, showing a fair correlation with the age. More recently, Cassatella et al. (1996) proved that ages obtained with these integrated UV colours are consistent with ages derived by isochrone fittings for a sample of LMC clusters. A survey on the available HST filters discloses that F152M, F170M, F253M, F278M and F307M may represent a valid counterpart for the quoted "ANS" filters.

2. The V filter is one of the most common filters and it has been extensively used also by HST observers. Moreover Dorman et al. (1993, 1995) have already shown that it could be very useful in investigating chemical composition when used in conjunction with UV filters.

3. Standard U, B, R, I, J, K, L and HST-WFPC2 F606W filters have been also selected because of their extended use with the Wide Field Planetary Camera 2 and NICMOS on board HST as well as in ground-based observations of stellar systems. It is thus possible to compare population synthesis models with wide band observations of distant stellar objects. Integrating in such a way the evaluation already given by Chiosi et al. (1997) concerning the HST expectations.

According to these choices, our population synthesis code evaluate colours on the basis of stellar atmosphere model by Kurucz (1979a,b: K79). This choice relies on the evidence that the mixing length parameter adopted in the computations connected to K79 gives a better (but not perfect) approximation of the observed colours of cool stars than new model atmospheres do (see the discussion in Brocato et al. 1997). Adopting more recent model atmospheres (for example as given by Kurucz 1992) would only increase the discrepancy between predicted and observed magnitudes and colours.

In order to translate the theoretical isochrones from the HR diagram to the different colour-magnitude diagrams (CMDs), and to calculate the integrated colours by simply summing the contribution of all the stars, we used the standard HST synphot task running under the IRAF package. We computed the (M _i-V) colours, where the index i stands for the various filters mentioned above, as expected from the Kurucz model atmospheres for a wide range of temperatures (3750 K $\leq$  $T_{\rm e}~~\leq50000$ K) and gravities (0.75 $\leq$ log g $\leq$ 5 in cgs units), covering the excursion of these quantities during the whole stellar life.

The adopted model grid has a lower mass limit of M=0.6 $M_{\odot}$. However, one already knows that lower masses should give a negligible contribution to the cluster light. As a test, we performed a set of numerical experiments implementing the grid of Reference Frame (RF) models (see next section for definition) with the stellar models of very low mass stars by Alexander et al. (1996), extending the lower mass limit down to 0.15 $M_{\odot}$.The resulting integrated colours are, for all the ages, within the expected statistical fluctuations (see below) of the RF models. As expected, the larger difference is found for the V-K colour ($\Delta (V-K)$ $\le$ 0.1 mag), due to the very low temperature of the faint but numerous lower main sequence stars. However note that the lower portion of the MS should play a not negligible role when dealing with colours at even larger wavelengths, i.e., in the far IR.