3 Results

Next, we shall discuss the pulsational properties of a selected WD model at $T_{\rm eff} \sim 12\,000$ K. We should remark that, although the chemical profiles evolve as the WD cools down through the instability strip (see Althaus et al. 2001c), the conclusions of the present paper remain valid for any model belonging to the ZZ Ceti instability strip. We begin by showing in Fig. 2 the square of the Brunt-Väisälä frequency N (computed as in Brassard et al. 1991) and the Ledoux term B of such a model. The results for the diffusive equilibrium approximation are also plotted as thin lines. Note the smooth shape of B, which is a direct consequence of the chemical abundance distribution. The contributions from the Ledoux term are characterized by extended tails, and translate into smooth bumps on N².

Figure 3: Oscillation kinetic energy (upper panel) and period spacing (lower panel) values for $\ell = 1$ in terms of the computed periods, Pk. Filled dots correspond to pulsational computations for the non-equilibrium diffusion model, and empty dots for the diffusive equilibrium one. In the interests of clarity, the scale for the kinetic energy in the case of diffusive equilibrium is displaced upwards by 1 dex. The kinetic energy values correspond to the normalization $y_1= \delta r / r = 1$ at the surface of the non-perturbed models for each mode.

The characteristic of B and N² as predicted by our models is markedly different from those found in previous studies in which the WD evolution is treated in a simplified way, particularly regarding the chemical abundance distribution (e.g. Tassoul et al. 1990; Brassard et al. 1991, 1992a,b; Bradley 1996). Clearly, non-equilibrium chemical profiles lead to B values with markedly less pronounced peaks as compared with the diffusive equilibrium treatment. Accordingly, the Brunt-Väisälä frequency turns out to be smoother as a result of non-equilibrium diffusion.

For the pulsation analysis we have employed the general Newton-Raphson code described in Córsico & Benvenuto (2001). We have computed g-modes with $\ell= 1, 2$ and 3 with periods in the range of 50 s $\mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$<$ }}}P_k \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$<$ }}}$ 1300 s (k being the radial order of modes). The upper panels of Figs. 3-5 show, respectively, the values of oscillation kinetic energy for modes with $\ell= 1, 2$ and 3 in terms of computed periods. Lower panels depict the corresponding values for the forward period spacing $\Delta P_k$ ( $\equiv P_{k+1} - P_k$). Filled dots depict the results corresponding to our model with non-equilibrium diffusion, whereas empty dots indicate the results predicted by the diffusive equilibrium approximation for the hydrogen-helium interface. In the interests of clarity, the scale for the kinetic energy in the case of diffusive equilibrium is displaced upwards by 1 dex.

For the non-equilibrium diffusion model the quantities plotted (especially the $E_{\rm kin}$ values) exhibit two clearly different trends. Indeed, for $P_k \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$>$ }}}600$ s and irrespective of the value of $\ell$, the kinetic energy of adjacent modes is quite similar, which is in contrast with the situation found for lower periods. Interestingly, the $\Delta P_k$ minima are commonly associated with $E_{\rm kin}$ maxima, but that modes with  $E_{\rm kin}$ maxima are adjacent to the modes with  $E_{\rm kin}$ minima. On the other hand, the period spacing diagrams show appreciable variations of  $\Delta P_k$ for $P_k \mathrel{\hbox{\rlap{\hbox{\lower4pt\hbox{$\sim$ }}}\hbox{$<$ }}}600$ s. This is due mostly to the presence of chemical abundance transitions in DA WD models as explained by Brassard et al. (1992a,b). In contrast, for higher periods the  $\Delta P_k$ of the modes tend to a constant, asymptotic value (Tassoul 1980).

Figure 4: Same as Fig. 3 but for $\ell = 2$.

Figure 5: Same as Fig. 3 but for $\ell = 3$.

The assumption of diffusive equilibrium in the trace element approximation in WD modeling gives rise to a kinetic energy spectrum and period spacing distribution in which the presence of the well known mode trapping phenomenon is clearly visible, as previously reported by numerous investigators (see Brassard et al. 1992b, particularly their Figs. 20a and 21a for the case of $M_{\rm H} = 10^{-4}~M_*$). The trapped modes correspond to modes with local minima in $\log (E_{\rm kin })$ and $\Delta P_k$. Here, we find that these trapping properties virtually vanish when account is made of WD models with diffusively evolving stratifications. This is particularly true for large periods, though for low periods trapping is also substantially affected. We attribute the differences found between both treatments to the markedly different shapes of the Ledoux term at the hydrogen-helium interface as predicted by non- and equilibrium diffusion.

From the results presented in this letter we judge that, for high periods, trapping mechanism in massive envelopes of stratified WDs is not an appropriate one to explain the fact that all the modes expected from theoretical models are not observed in ZZ Ceti stars. It is worth mentioning that Gautschy & Althaus (2001) have recently found, on the basis of a consistent diffusion modeling, a weaker trapping effect on the periodicities in DB WDs. Our results give strong theoretical support to recent evidence against the claimed correlation between the observed luminosity variations amplitude and trapping of modes. Finally, to place these assertions on a firmer basis, a non-adiabatic stability analysis of the pulsational properties of non-equilibrium diffusion models is required. A more extensive exploration of the results presented in this letter will be presented in a future work.

Acknowledgements

We warmly acknowledge to our referee, Paul Bradley, for the effort he invested in the revision of our article. His comments and suggestions strongly improved the original version of this work.