2 Evolutionary synthesis models

We have computed models for single stellar populations (instantaneous burst), at four different metallicities (Z= 0.2 $Z_{\odot}$, 0.4 $Z_{\odot}$, $Z_{\odot}$ and 2.5 $Z_{\odot}$, and ranging in age between 1 Myr (logt = 6.00) and 13 Gyr (logt = 10.12), with a logarithmic step in age (given in years) of 0.1. The model at 17 Gyr was also computed to compare with other authors (see Sect. 4.2 and Fig. 6 ).

The total mass of the SSP is $1 \; 10^{6}$ ${M}_{\odot}$ with a Salpeter-type IMF (Salpeter 1955), $\phi(m)=m^{-\alpha}$, $\alpha$ = 2.35, from the lower limit $m_{\rm low}$ = 0.8 ${M}_{\odot}$ to the upper limit $m_{\rm up}$ = 100 ${M}_{\odot}$.Taking into account the Padova evolutionary tracks (see Sect. 2.1), a fine grid of isochrones has been computed following the method outlined by Bertelli et al. (1994). We also synthesized a complete grid of isochrones with $m_{\rm low}$ = 0.6 ${M}_{\odot}$ to check the effect of considering a lower limit of the IMF on CaT index, finding a maximum discrepancy of 10% and only for ages older than 4 Gyr.

Once the HR Diagram is calculated and the SED for the SSP computed (see Sects. 2.2, 2.3), we are able to calculate the EW(CaT) in the integrated populations by taking the EW(CaT) of individual stars from theoretical models (JCJ92), or observed stellar libraries (DTT89, Z91) as will be outlined in Sect. 3.

2.1 Stellar evolution

Isochrones were constructed at several ages by interpolating between the evolutionary sequences calculated by Bressan et al. (1993), and Fagotto et al. (1994a,b). These tracks were computed using the radiative opacities of Iglesias et al. (1992) for the initial chemical compositions Z=0.004, Y=0.24; Z=0.008, Y=0.25; Z=0.02 and Z=0.05, Y=0.352 (Padova models).

Recent reviews on stellar evolution can be found in Maeder & Conti (1995) and Chiosi et al. (1992). Here we will briefly summarize the main properties of the adopted models, with particular emphasis to the red giant and red supergiant phases which are the most relevant to the CaT synthesis.

Red giant stars appear suddenly after hydrogen in the center has been exhausted. Two remarkable exceptions are constituted by the most massive stars if mass-loss is strong enough to peal-off the envelope of the star thus avoiding the expansion phase, and by stars around $20\; M_\odot$ if the mixing criterion in the intermediate convective shell and in the previous H-burning core is such that the model ignites and burns He in the center as a yellow supergiant star (case A evolution, usually associated with the Schwarzschild criterion for the convective instability, Deng et al. 1996). For all the other initial masses the models possess a red giant phase of significant duration. Old clusters with turn-off mass, $M_{\rm Toff}$, lower than 2 ${M}_{\odot}$, have a well populated red giant branch (RGB). For a sufficient high metallicity and/or relatively young age these clusters also show a red clump of He-burning stars tied to the RGB. On the contrary, in intermediate-age and young clusters, only the red clump of He-burning stars is populous and luminous enough to have observable effects. Usually for a sufficiently high initial mass and low metal content part of the central He is burnt in a blue loop toward higher effective temperatures. Finally old and intermediate age clusters, $M_{\rm Toff} < 5{-}6~M_\odot$, also display the asymptotic giant branch (AGB) phase. The fuel consumed in this phase is relatively high so that the contribution to the integrated light is not negligible.

As already anticipated the evolution of the most massive stars is still unclear because of our poor knowledge of the efficiency of internal mixing processes and of the mass-loss phenomenon. The Padova models account for mild overshoot from the convective core, and mass loss by stellar winds has been accounted for according to the rates given by de Jager et al. (1988) from the main sequence up to the so-called de Jager limit in the HRD. Beyond the de Jager limit the most massive stars enter the region where Luminous Blue Variables (LBV) are observed and, accordingly, the mass-loss rate has been increased to 10^-3 ${M}_{\odot}$ yr^-1. As the evolution proceeds, the surface hydrogen abundance by mass in the most massive stars eventually falls below the value of 0.3. In this case the model is supposed to become a Wolf-Rayet (WR) star and the mass-loss rate is derived according to Langer (1989).

As a matter of fact there are several unsolved questions in the HR diagram of the most massive stars, among which we recall the existence of the so called blue Hertzsprung gap, a region where, contrary to what is observed, theory predicts a negligible number of stars; the observational evidence of the de Jager limit at the highest luminosities, which is reproduced by the models only by adopting an arbitrarily high mass-loss rate (of the order of 10^-3 ${M}_{\odot}$/yr) in the corresponding region of the HR diagram; and finally the problem of the Wolf-Rayet stars, which are either much cooler or less luminous than predicted by the models. Nevertheless the theory predicts that massive stars with initial mass between 10 and 30 solar masses, spend a significant fraction ($\simeq$50%) of their He burning phase as red supergiant but the effective temperature of these stars is a matter of debate, and this must be reminded when assigning the spectral type during the synthesis process. In general the effective temperature predicted by the theory is higher than what is observed, but one must bear in mind that the majority of the models adopt a static gray atmosphere as a boundary condition, while that of RSG stars is an extended and expanding atmosphere. Moreover the suppression of the density inversion or the adoption of a density scale-height in the convective envelope, both result into a higher effective temperature (Bressan et al. 1993). Finally RSG are losing mass at a rate of about 10^-5 ${M}_{\odot}$/yr and dust processes in the circumstellar envelope can also affect their color and then their apparent location in the HR diagram.

Another important question is whether a young SSP may contain RSG and WR stars at the same time. Bressan (1994) and García-Vargas et al. (1995a) have shown that this is marginally possible for an age of 6 Myr and Z=0.02. In fact in our standard view, WR stars evolve in the HR diagram from the highest luminosities almost vertically downward and thus their presence is associated with very young ages. RSG on the contrary, only appear after a few Myr have elapsed from the burst onset. However Bressan (1994) showed that by adopting the mass-loss parameterization of de Jager et al. (1988), the predicted mass-loss rate of a typical RSG model of 20 ${M}_{\odot}$ of solar composition is significantly lower than that derived by means of the Feast formulation (1992), which empirically links the mass-loss rate to the period of pulsation of the RSG stars. Recent models of massive stars of solar composition, in which one adopts the mass-loss formulation by Feast (1992), show that 20 ${M}_{\odot}$ and 18 ${M}_{\odot}$ stars leave the RSG phase and enter the main sequence band with a surface hydrogen abundance of 0.43, which is comparable to the one selected by Maeder for the BSG-WNL transition (Salasnich et al. 1997). This 3horizontal'' evolution into the channel of the low luminosity WR stars allows the presence of WR and RSG stars simultaneously in an instantaneous burst.

Clusters of intermediate age (between 0.1 and 1 Gyr) are characterized by the presence of the very luminous Asymptotic Giant Branch stars (AGB). While their life-time is quite short (around 1 Myr), they are among the brightest stars in the cluster, their fuel consumption is large and their contribution to the integrated light is significant. The appearance of the AGB phase as the SSP evolves is quite sudden at an age of 100 Myr and causes a jump in the colors, in particular when near infrared pass-bands are considered (see e.g. Bressan et al. 1994). The same happens to the EW(CaT) in clusters of about 0.1 Gyr (see Fig. 4).

At older ages the contribution of the AGB phase declines while that of the red giant branch (RGB) becomes more and more pronounced. Above 10 Gyr, red giants mainly belong to the RGB phase and the integrated light from the AGB phase has become negligible.

2.2 The stellar energy distributions

We have synthesized the emergent spectrum of an evolving star cluster by calculating the number of stars in each element of the isochrone and assigning to it the most adequate stellar atmosphere model, i.e. the closest one in effective temperature and surface gravity. The stellar spectrum has then been scaled to the luminosity of the corresponding theoretical star in the HRD.

To build our stellar spectral library we assembled the stellar atmospheres of Clegg & Middlemass (1987) for stars with $T_{\rm eff}$$\geq$ 50000 K and those of Kurucz (1992) for stars with 5000 K $\leq$ $T_{\rm eff}$< 50000 K. The later models are available at different metallicities. Since the precise shape of the spectrum of the hottest stars does not have any influence in the CaT models presented here, we will not discuss the selected atmosphere models for them (a detailed discussion can be found in García-Vargas 1996 and references therein). For the coolest stars, we have used a blackbody distribution since it can model the level of the continuum at 8600 Å better than Kurucz models (of course the SEDs are not used in any case to synthesize the features, but to locate the continuum level). As an example, Fig. 1 shows observed stellar spectra together with the corresponding Kurucz model and blackbody (BB) distribution for some representative spectral types. We have checked quantitatively the differences in the continuum level at 8600 Å between these three representations (BB, Kurucz, and observed) finding a maximun discrepancy of 15% for the coolest RSG, and only 1% for giants.

  

Figure 1: Comparison between different near-IR spectra of cool stars and stellar atmosphere models. Left panel shows a sequence of giants with effective temperature decreasing from top to bottom. Right panel shows the Red Supergiant sequence. Data (lines with higher spectral resolution) are true stars for the labelled spectral type and luminosity class (Danks & Dennefeld 1994). The degraded spectra correspond to a Kurucz's model of a $T_{\rm eff}$ and logg appropriate for each given star. Finally, the featureless line is the spectral energy distribution of a blackbody whose $T_{\rm eff}$ has been chosen to be equivalent to the assigned Kurucz's model. All the spectra are normalized at 8800 Å

2.3 The nebular continuum

Because our aim is to build models that can be applied to star-forming regions, we have computed the continuum nebular emission under the following hypothesis.

  

Figure 2: Ratio between the nebular and the total luminosity as a function of the burst age. Panels a), b), c), and d) show the ratio at different wavelengths: 4850 Å, 8600 Å, 2.17 $\rm \mu m$ and 2.30 $\rm \mu m$ respectively. Different line-types are used to show the effect at different metallicities: Z=0.004 (0.2 $Z_{\odot}$, dash-dotted line), Z=0.008 (0.4 $Z_{\odot}$, dashed line), Z=0.02 ($Z_{\odot}$, solid line) and Z=0.05 (2.5 $Z_{\odot}$, dotted line)

The gas is assumed to have an electron temperature, $T_{\rm e}$, which is metallicity dependent. The values for $T_{\rm e}= 11000$ K (Z= 0.2 $Z_{\odot}$), 9000 K (Z= 0.4 $Z_{\odot}$), 6500 K (Z= $Z_{\odot}$) and 4000 K (Z= 2.5 $Z_{\odot}$) have been chosen according to the observational determination of $T_{\rm e}$ in star-forming regions (for the lowest metallicities), and the average value, in the age-range 1.5- 5.4 Myr, given by photoionization models (García-Vargas et al. 1995b) for the highest Z values. The assumed helium abundance by number is 10%. The free-free, free-bound emission by hydrogen and neutral helium, as well as the two photon hydrogen-continuum have been included. The atomic data were compiled from Aller (1984) and Ferland (1980) according to the selected value of $T_{\rm e}$.

Tables 1 and 2 list the integrated luminosity of the SSPs with different metallicity at some characteristic wavelengths in the UV (2000 Å), optical (4850 Å, representative of the continuum near H$_{\beta}$), and infrared (at 2.17 $\rm \mu m$, near Br$_{\gamma}$). A complete set of tables, including the nebular and stellar contributions separately as well as the total luminosity for our grid of models at wavelengths of 1400, 2000, 4850, 8600 Å, 2.17 and 2.30 $\rm \mu m$ and the synthetic SEDs are available upon request.

Figure 2 shows the ratio between the nebular and the total luminosity as a function of the age at four selected wavelengths. At $Z_{\odot}$ the nebular contribution in the earliest stages of the burst changes between 20% in the optical (H$_{\beta}$) to almost 90% in the infrared (2.3 $\rm \mu m$). This effect becomes negligible for evolved SSP (older than 5.5 Myr) when the production of ionizing photons is negligible.

However, if a very young (ionizing) burst coexist with a slightly older population (around 10 Myr), RSG rich, such as in the case of some star-forming regions (García-Vargas et al. 1997), the effect of the nebular continuum competes with that of the older stellar component, and some stellar infrared features can be partially diluted. This could be the case of the CO absorption bands at 2.3 $\rm \mu m$, where the contribution of the nebular to the total luminosity can be as high as 90%. The same effect also applies to the near-IR colors. For example, if we assume two coexisting populations (one around 2-4 Myr and the other one around 9-12 Myr) contributing with similar percentage in mass, the resulting V-K color would be affected both by the nebular continuum of the young burst and by the stellar continuum from RSG present in the intermediate-age burst. Thus detailed evolutionary synthesis models, using other constraints, would be required in order to correctly interpret the photometric observations.