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2 Computational procedures

As well known, theoretical predictions of cluster colours make use of suitable libraries of evolutionary tracks to produce synthetic stellar clusters, i.e., to simulate the CM distribution of stars for each given assumption about the cluster age and original chemical composition. In such a procedure, one is usually neglecting faint evolutionary phases which cannot sensitively contribute to the total cluster luminosity. However, since synthetic stellar clusters are of relevance in a much more general context than in predicting colours, the evolutionary library adopted in Paper I has been implemented to cover with suitable theoretical predictions all the evolutionary phases of cluster stars, from the Very Low Mass MS stars, at the lower mass limit for H burning ignition, to the final sequence of cooling White Dwarfs. As a result, with the present libraries the program can produce "complete" synthetic clusters in the range of ages 50 Myr to 20 Gyr and for metallicity from Z=0.0001 to Z=0.02. Figure 1 gives an example of these predictions, as given for a synthetic cluster with the given values of age and chemical composition.

\par\includegraphics[width=8.8cm,clip]{ds1846f1.eps} \end{figure} Figure 1: The predicted CM diagram location of stars in a cluster with age t=15 Gyr and Y=0.23, Z=0.001, with a suitable simulation of photometric uncertainties. $M_{\rm MS}$ is the mass of Main Sequence star at the corresponding luminosity level; $M_{\rm TO}$ is the mass of the star at the Turn Off point; $M_{\rm tip}$ is the mass at the RGB tip; $<M_{\rm HB}>$ gives the actual mean value of HB stars after mass loss, whereas the two labelled $M_{\rm pr} $ values give the progenitor mass (i.e. the mass in MS) of the stars along the WD sequence

In the present form, the bulk of the code relies on the set of homogeneous evolutionary computations presented by Straniero & Chieffi (1991), Castellani et al. (1991), Castellani et al. (1992a), Cassisi et al. (1994), covering both the H and the He burning phase for stars with original masses in the range $0.6 - 9.0~M_{\odot}$ and where the He burning phase is followed till the Carbon ignition or, alternatively, the onset of the thermal pulse phase. In the last case, the relevant one for old globulars, the tracks have been prolonged through the thermal pulse phase according to the semianalitical procedure envisaged by Groenewegen & De Jong (1993, see also Marigo et al. 1996). The above evolutionary computations have been implemented with theoretical predictions about VLM structures from Cassisi et al. (2000) whereas cooling White Dwarf sequences have been finally evaluated according to the Brocato et al. (1999b) procedure, as based on evolutionary data by Wood (1992).

A detailed description of the program can be found in Poli (1997) and Paper I. Here we notice that mass loss has been taken into account by adopting Reimers formula

{\dot {M}}= -4~10^{-13}\ \eta {\frac{L}{gR}} {\frac{M_{\odot}}{\rm yr}}
\end{displaymath} (1)

connecting the amount of mass loss to the stellar parameters L, M and $T_{\rm e}$, with $\eta $ as a free parameter governing the efficiency of the process. Since the evolution of a Red Giant structure is largely independent of the amount of mass loss (see Castellani & Castellani 1993) the above relation can be integrated over the time all along the H burning evolution of models without mass loss in order to obtain the total amount of mass lost during this phase and, in turn, the mass of the new born He burning HB star.

To orientate the reader in such a scenario, Fig. 2 gives the predicted mass and effective temperature of new born HB stars as a function of $\eta $ for selected assumption about the cluster age or metallicity.

\par\includegraphics[width=8.8cm,clip]{ds1846f2.eps} \end{figure} Figure 2: Predicted masses and temperatures of HB stars as a function of $\eta $ for the labelled assumptions about the cluster age or metallicity

As already known, one recognizes that decreasing the HB mass (i.e., increasing $\eta $ and/or the age) HB stars become hotter (and fainter). One should also notice the critical dependence of the HB temperature of Z=10-3 models on the mass, to be connected with the observational evidence for extended HB in intermediate metallicity clusters like M 3 or M 5.

For each given age and chemical composition and for each given assumption on $\eta $, we make use of the above procedure to evaluate the original mass of the star at the tip of the red giant branch and the amount of mass lost, deriving from these values, the mass of the HB star, its CM diagram location and the lifetime in the HB phase. This lifetime is used to derive the number of expected HB stars ( $N_{\rm HB}$) through a suitable proportion with the proper RGB lifetime and the already known number of RGB stars.

As well known, the HB colour distribution observed in the galactic globulars has to be interpreted as an evidence for a dispersion in HB masses, as due to a dispersion in the amount of mass loss. To simulate this occurrence, we follow the prescriptions early given by Rood (1973) by assuming the computed HB mass as the mean value of a stochastic distribution of masses with a given standard deviation, normally in the range $\sigma_{\rm M} = 0.02-0.04$. For each given HB mass, the star is put on its evolutionary track with a random fraction of its He burning lifetime, still evaluating the amount of mass loss (Reimers) during the rising along the AGB phase. As derived by Fig. 4 of Castellani et al. (1992b), if and when the mass of the H-rich envelope decreases below a critical value

$\displaystyle %
M^{\rm env}_{\rm cr}$   $\displaystyle = 0.0153\cdot [{\rm log}(L/L_{\odot})]^{2}$ (2)
    $\displaystyle - 0.0905\cdot {\rm log}(L/L_{\odot} + 0.1481)$  

the star is removed from the AGB and added to the WD sequence. Note that in this way we deliberately neglect the occurrence of luminous stars crossing the HR diagram to reach their final WD structure. Since the typical lifetime of such a crossing is of the order of only 104-105years, depending on the star mass (Schönberner 1983; Blöcker & Schönberner 1997), even in rich globulars the occurrence of similar stars is expected as a rare and stochastic phenomenon, as confirmed by observation. Therefore the contribution of these object has to be treated as an additional occurrence. One can easily quantify the expected number ratio of post-AGB to HB stars, as simply given by the ratio of their corresponding evolutionary lifetimes. According to numerical experiments we report in Table 1 the expected P-AGB number ( $N_{\rm cross}$) as a function of the cluster integrated absolute magnitude ( $M_{V}^{\rm tot}$) together with the corresponding number of the He-burning stars.

Table 1: The expected number of P-AGB stars as a function of the total absolute cluster magnitude $M_{V}^{\rm tot}$. $N_{{\rm cross}}^{1}$ refers to the model with $M_{\rm H}=0.55~M_{\odot}$ and $N_{{\rm cross}}^{2}$ to the model with $M_{\rm H}=0.525~M_{\odot}$ (Blöcker & Schönberner 1997)
$M_{V}^{\rm tot}$ $N_{\rm HB}$ $N_{{\rm cross}}^{1}$ $N_{{\rm cross}}^{2}$
-4.1 12 0.005 0.01
-5.9 74 0.03 0.08
-6.7 146 0.07 0.15
-7.5 288 0.13 0.30
-7.9 436 0.20 0.46
-8.5 730 0.33 0.77
-9.0 1166 0.53 1.22
-10.0 2904 1.32 3.08
-11.0 7304 3.32 7.74
-12.0 18326 8.33 19.45

These results, in general agreement with predictions by Renzini (1998), confirm that for galactic globulars ( $M_V^{\rm tot} > -10$ mag) post-AGB stars can only give a stochastic contribution.

Assuming cluster ages in the range 10-15 Gyr, from preliminary computations we find that the assumption $\eta \sim 0.4$ appears able to nicely reproduce the observed dependence of the HB type on metallicity, when the occurrence of a second parameter is not taken into account. This is shown in Fig. 3, where we report the predicted CM diagram for cluster with an age of 15 Gyr and for the various labelled assumptions about the metallicity, adopting Yale ( colour temperature relations and bolometric corrections.

\par\includegraphics[width=8.8cm,clip]{ds1846f3.eps} \end{figure} Figure 3: Predicted CM diagrams for clusters with age t=15 Gyr and for the labelled assumptions on the metallicity Z

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