4 Mass loss on the RGB, and synthetic TP-AGB evolution

Before presenting the isochrones derived from our evolutionary tracks, we briefly describe the way we have considered the effect of mass loss on the RGB. Also, we describe the extension of our tracks of low- and intermediate-mass stars through the complete TP-AGB phase. The latter point is an important one, since this evolutionary phase constitutes a significant fraction of the bolometric luminosity of stellar populations.

Mass loss by stellar winds during the RGB of low-mass stars is considered only at the stage of isochrone construction. We use the empirical formulation by Reimers (1975), but with mass-loss rates multiplied by a parameter $\eta$ which is set equal to 0.4 (see Renzini & Fusi Pecci 1988). The procedure is basically the following: In passing from the RGB-tip to the ZAHB, we first integrate the mass loss rate along the RGB of every single track, in order to estimate the total amount of mass that has to be removed. Then the mass of the evolutionary models (that were computed at constant mass) is simply scaled down to the value suited to the ZAHB stars. This approximation is a good one since the mass loss does not affect significantly the internal structure of models at the tip of the RGB.

In addition, before constructing the isochrones, we need to complete the stellar evolution along the TP-AGB. This phase is followed in a synthetic way (see Iben & Truran 1978; Groenewegen & de Jong 1993; Bertelli et al. 1994; Marigo et al. 1996, 1998; Girardi & Bertelli 1998). In a few words, we evolve the core mass, total mass, effective temperature and luminosity of each star, from the first thermal pulse on the AGB up to the stage of complete envelope ejection. The following relations are used to this aim:

• The core mass-luminosity relation described in Eqs. (10) and (11) of Groenewegen & de Jong (1993, and references therein);
• The rate of core growth from Eq. (18) of Groenewegen & de Jong (1993);
• The luminosity-effective temperature relations described in Eq. (12) of Girardi & Bertelli (1998). Suffice it to recall that these relations are obtained by extrapolating the slope of the early-AGB phase of our models to higher luminosities and lower effective temperatures. This is performed separately for each value of the metallicity;
• We adopt the Vassiliadis & Wood's (1993) prescription for the mass-loss along the TP-AGB, since it proved to provide a reasonable description for the initial-final relation of low- and intermediate-mass stars of solar metallicity (see e.g. Marigo 1998a), as well as to the maximum luminosity of AGB stars of different ages (and hence initial masses) observed in the LMC (see Marigo et al. 1996).

It is worth remarking that this prescription for the TP-AGB evolution can be considered only as a crude first approximation to the real evolution. We do not consider, for instance, key processes as the third dredge-up and hot-bottom burning in TP-AGB stars. We intend to replace soon the present prescription, for the detailed TP-AGB models of Marigo et al. (1998).

For the moment, we just compare the initial-final mass relation, as derived from the present tracks, with the empirical one of Herwig (1996). This is done in Fig. 4. It is important to recall some aspects of the empirical relation. First, the Herwig (1996) relation is very different from the (largely used) Weidemann (1987) one, at least in the range of initial masses  $M_{\odot }$; this is due, essentially, to the dramatic re-evaluation of the mass of the white dwarfs in the Hyades (cf. Weidemann 1996). Second, the Herwig (1996) relation extends up to a initial mass of 8 $M_{\odot }$, value which represents the white dwarfs in the open cluster NGC 2516. Recently, the initial mass of the white dwarfs in this cluster have been re-evaluated, to new values of 5-6 $M_{\odot }$, by Jeffries (1997). Therefore, the upper mass limit of stars which produce white dwarfs, is roughly consistent with the values of $\mbox{$M\mbox{$_{\rm up}$ }$ }\sim5$ $M_{\odot }$ we find in our stellar models.

Figure 4 evidences that our theoretical initial-final mass relation for the solar metallicity, reproduces in a satisfactory way the empirical one from Herwig (1996) and Jeffries (1997). This is an important point for a set of isochrones aimed to be used in evolutionary population synthesis calculations.