2 Evolutionary models

The WD models on which the present results are based have been calculated by means of a detailed evolutionary code developed by us at La Plata Observatory. The code has been employed in previous studies on WD evolution (Althaus et al. 2001a,b) and it has recently been modified to study the evolutionary stages prior to the WD formation (see Althaus et al. 2001c). The constitutive physics include: up-to-date OPAL radiative opacities for different metallicities, conductive opacities, neutrino emission rates, a detailed equation of state and a complete network of thermonuclear reaction rates for hydrogen and helium burning (see Althaus et al. 2001c). For a proper treatment of the diffusively evolving chemical stratification, gravitational settling and the thermal and chemical diffusion of nuclear species have been considered.

Figure 2: The logarithm of the squared Brunt-Väisälä frequency and the Ledoux term, B, for the non-equilibrium diffusion model. The results for the diffusive equilibrium approximation are shown in thin lines.

The evolutionary stages prior to the WD formation have been fully taken into account. Specifically, we started our calculations from a 3 $M_{\odot}$ star at the zero-age main sequence and we follow its further evolution all the way from the stage of hydrogen and helium burning in the core up to the tip of the asymptotic giant branch where helium thermal pulses occur. After experiencing 11 thermal pulses, the model is forced to evolve towards its WD configuration by invoking strong mass loss episodes. As a result, a WD remnant of 0.563 $M_{\odot}$ is obtained. The evolution of this remnant is pursued through the stage of planetary nebulae nucleus to the domain of the ZZ Ceti stars on the WD cooling branch. An important aspect of these calculations is related to the evolution of the chemical abundance during the WD cooling. In particular, the shape of the composition transition zones is of the utmost importance regarding the pulsational properties of the ZZ Ceti models. In this respect, diffusion processes cause near discontinuities in the abundance distribution at the start of the cooling branch to be considerably smoothed out by the time the ZZ Ceti domain is reached. This can be appreciated in Fig. 1, which also illustrates the profile of the hydrogen-helium interface resulting from the predictions of diffusive equilibrium in the trace element approximation (thin dotted line). The shape of the innermost carbon and oxygen distribution emerges from the chemical rehomogenization process due to the Rayleigh-Taylor instability occurring at early stages of the WD evolution (see Althaus et al. 2001c and also Salaris et al. 1997). Surrounding the carbon-oxygen interior there is a shell rich in both carbon ($\approx$35%) and helium ($\approx$60%), and a overlying layer consisting of nearly pure helium of mass 0.003 $M_{\odot}$. The presence of carbon in the helium-rich region below the helium buffer stems from the short-lived convective mixing episode that has driven the carbon-rich zone upwards during the peak of the last helium pulse on the asymptotic giant branch. We want to mention that the total helium content within the star once helium shell burning is eventually extinguished amounts to 0.014 $M_{\odot}$ and that the mass of hydrogen that is left at the start of the cooling branch is about $1.5 \times 10^{-4}$ $M_{\odot}$, which is reduced to $7 \times 10^{-5}$ $M_{\odot}$ due to the interplay of residual nuclear burning and element diffusion by the time the ZZ Ceti domain is reached.