2 Details of computations

Our selected ZZ Ceti model on which the pulsational results are based has been calculated by means of an evolutionary code developed by us at La Plata Observatory. The code, which is based on a detailed and up-to-date physical description, has enabled us to compute the white dwarf evolution in a self-consistent way with the predictions of time dependent element diffusion, nuclear burning and the history of the white dwarf progenitor. The constitutive physics includes: new OPAL radiative opacities for different metallicities, conductive opacities, neutrino emission rates and a detailed equation of state. In addition, a network of 30 thermonuclear reaction rates for hydrogen burning (proton-proton chain and CNO bi-cycle) and helium burning has been considered. Nuclear reaction rates are taken from Caughlan & Fowler (1988) and Angulo et al. (1999) for the $^{12}{\rm C}(\alpha, \gamma)^{16}{\rm O}$ reaction rate. This rate is about twice as large as that of Caughlan & Fowler (1988). Abundance changes resulting from nuclear burning are computed by means of a standard implicit method of integration. In particular, we follow the evolution of the chemical species ¹H, ³He, ⁴He, ⁷Li, ⁷Be, ¹²C, ¹³C, ¹⁴N, ¹⁵N, ¹⁶O, ¹⁷O, ¹⁸O and ¹⁹F. Convection has been treated following the standard mixing length theory (Böhm-Vitense 1958) with the mixing-length to pressure scale height parameter of $\alpha=1.5$. The Schwarzschild criterium was used to determine the boundaries of convective regions. Overshooting and semi-convection were not considered. Finally, the various processes relevant for element diffusion have also been taken into account. Specifically, we considered the gravitational settling, and the chemical and thermal diffusion of nuclear species ¹H, ³He, ⁴He, ¹²C, ¹⁴N and ¹⁶O. Element diffusion is based on the treatment for multicomponent gases developed by Burgers (1969). It is important to note that by using this treatment of diffusion we are avoiding the widely used trace element approximation (see Tassoul et al. 1990). After computing the change of abundances by effect of diffusion, they are evolved according to the requirements of nuclear reactions and convective mixing. Radiative opacities are calculated for metallicities consistent with the diffusion predictions. This is done during the white dwarf regime in which gravitational settling leads to metal-depleted outer layers. In particular, the metallicity is taken as two times the abundance of CNO elements. For more details about this and other computational details we refer the reader to Althaus et al. (2002) and Althaus et al. (2001).

We started the evolutionary calculations from a 3 $M_{\odot }$ stellar model at the zero-age main sequence. The adopted initial metallicity is Z= 0.02 and the initial abundance by mass of hydrogen and helium are, respectively, $X_{\rm H}= 0.705$ and $X_{\rm He}= 0.275$. Evolution has been computed at constant stellar mass all the way from the stages 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 the white dwarf state by invoking strong mass loss episodes. The adopted mass loss rate was $\approx$10^-4 $M_{\odot }$ yr^-1 and it was applied to each stellar model as evolution proceeded. After the convergence of each new stellar model, the total stellar mass is reduced according to the time step used and the mesh points are appropriately adjusted. As a result of mass loss episodes, a white dwarf remnant of 0.563 $M_{\odot }$ is obtained. The evolution of this remnant is pursued through the stage of planetary nebulae nucleus till the domain of the ZZ Ceti stars on the white dwarf cooling branch.

Figure 1: Panel a) the chemical abundance distribution of our stellar model for hydrogen (dotted line), helium (dashed line), carbon (solid line) and oxygen (dot-dashed line). Panel b) the Ledoux term, B. Panel c) the logarithm of the squared of the Brunt-Väisälä frequency (N2). In the inset of this panel, the logarithm of the squared of the Brunt-Väisälä frequency computed neglecting the term B is depicted. Note that the imprints of the chemical transition zones in the functional form of N2 are not completely eliminated, in particular at the helium-carbon-oxygen and hydrogen-helium interfaces. The stellar mass of the white dwarf model is 0.563 $M_{\odot }$ and the effective temperature is $\approx$12 000 K. Quantities are shown in terms of the outer mass fraction.

As well known, the shape of the composition transition zones plays an important role in the pulsational properties of DAV white dwarfs. In this sense, an important aspect of these calculations concerns the evolution of the chemical abundance during the white dwarf regime. In particular, element diffusion makes near discontinuities in the chemical profile at the start of the cooling branch be considerably smoothed out by the time the ZZ Ceti domain is reached (see Althaus et al. 2002). The chemical profile throughout the interior of our selected white dwarf model is depicted in the upper panel of Fig. 1. Only the most abundant isotopes are shown. In particular, the inner carbon-oxygen core emerges from the convective helium core burning and from the subsequent stages in which the helium-burning shell propagates outwards. Note also the flat profile of the carbon and oxygen distribution towards the centre. This is a result of the chemical rehomogenization of the innermost zone of the star due to Rayleigh-Taylor instability (see Althaus et al. 2002). Above the carbon-oxygen interior there is a shell rich in both carbon ($\approx$35%) and helium ($\approx$60%), and an overlying layer consisting of nearly pure helium of mass 0.003 $M_{\odot }$. The presence of carbon in the helium-rich region below the pure helium layer is a result of the short-lived convective mixing which 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. Finally, we note that the inner carbon-oxygen profile of our models is somewhat different from that of Salaris et al. (1997). In particular, we find that the size of the carbon-oxygen core is smaller than that found by Salaris et al. (1997) with the consequent result that the drop in the oxygen abundance above the core is not so pronounced as in the case found by these authors. We suspect that this different behaviour could be a result of a different treatment of the convective boundary during the core helium burning.

For the pulsation analysis we have employed the code described in Córsico & Benvenuto (2002). We refer the reader to that paper for details. Here we shall describe briefly our strategy of calculation and mention the pulsation quantities computed that are relevant in this study. The pulsational code is based on the general Newton-Raphson technique (like the Henyey method employed in stellar evolution studies). The code solves the differential equations governing the linear, non-radial stellar pulsations in the adiabatic approximation (see Unno et al. 1989 for details of their derivation). The boundary conditions at the stellar centre and surface are those given by Osaki & Hansen (1973) (see Unno et al. 1989 for details). Following previous studies of white dwarf pulsations, the normalization condition adopted is $\delta r / r= 1$ at the stellar surface. After selecting a starting stellar model we choose a convenient period window and the interval of interest in $T_{\rm eff}$. The evolutionary code computes the white dwarf cooling until the hot edge of the $T_{\rm eff}$-interval is reached. Then, the program calls the set of pulsation routines to begin the scan for modes. In order to obtain the first approximation to the eigenfunctions and the eigenvalue of a mode we have applied the method of the discriminant (Unno et al. 1989). Specifically, we adopt the potential boundary condition (at the surface) as the discriminant function (see Córsico & Benvenuto 2002). When a mode is found, the code generates an approximate solution which is iteratively improved to convergence (of the eigenvalue and the eigenfunctions simultaneously) and then stored. This procedure is repeated until the period interval is covered. Then, the evolutionary code generates the next stellar model and calls pulsation routines again. Now, the previously stored modes are taken as initial approximation to the modes of the present stellar model and iterated to convergence.

For each computed mode we obtain the eigenperiod P_k ( $P_k= 2 \pi / \sigma_k$, being $\sigma_k$ the eigenfrequency) and the dimensionless eigenfunctions $y_1,\cdots, y_4$ (see Unno et al. 1989 for their definition). With these eigenfunctions and the dimensionless eigenvalue $\omega_k^2 = \sigma^2_k (G M_* / R_*^3)^{-1}$ we compute for each mode considered the oscillation kinetic energy, $(E_{\rm kin})_k$, given by:

 

$$(E_{\rm kin})_k \frac{1}{2} (G M_* R_*^2) \omega_k^2 \int_{0}^{1} x^2 \rho$$ =

$$\times\left[ x^2 y_1^2 + x^2 \frac{\ell (\ell +1)}{(C_1 \omega_k^2)^2} y_2^2\right] {\rm d}x,$$ (1)

and the first order rotation splitting coefficient, $C_{\ell ,k}$,

$$C_{\ell,k}= \frac{(G M_* R_*^2)}{2 (E_{\rm kin})_k} \int_{0}... ...^2 y_1 y_2 + \frac{x^2}{C_1 \omega_k^2} y_2^2\right] {\rm d}x,$$ (2)

where M_* and R_* are the stellar mass and the stellar radius respectively, G is the gravitation constant, C₁= (r/R_*)³ (M_*/M_r) and x = r / R_*. In addition, we compute the weight functions, WF, and the variational period, $P_k^{\rm V}$, as given by Kawaler et al. (1985). Finally, for each model computed we derive the asymptotic spacing of periods, $\Delta P_{\rm A}$, given by (Tassoul 1980; Tassoul et al. 1990):

$$\Delta P_{\rm A} = \frac{P_0}{\sqrt{\ell(\ell+1)}},$$ (3)

and P₀ is defined as

$$P_0 = 2 \pi^2\ \left[\int_{0}^{x_2} \frac{N}{x} {\rm d}x \right]^{-1},$$ (4)

where x₂ corresponds to the location of the base of the outer convection zone. The Brunt-Väisälä frequency (N), a fundamental quantity of white dwarf pulsations, is computed employing the "modified Ledoux'' treatment. This treatment explicitly accounts for the contribution to N² from any change in composition in the interior of model (the zones of chemical transition) by means of the Ledoux term B (see Brassard et al. 1991). We want to mention that we have also employed a numerical differentiation scheme for computing N² directly from its definition. We found that this scheme yields the same results as those derived from the modified Ledoux treatment.

As mentioned, for the pulsation analysis in this study we have selected a white dwarf model representative of the ZZ Ceti instability band. Specifically, we have picked out a 0.563 $M_{\odot }$model at $T_{\rm eff}\approx$ 12 000 K. In Table 1 we show the main characteristics of our template model. In the interests of comparison, the same quantities corresponding to a similar model of P. Bradley (2002) (private communication) are shown. The Brunt-Väisälä frequency and the Ledoux term (B) corresponding to our template model are shown in the middle and bottom panels of Fig. 1 in terms of the outer mass fraction. Note the particular shape of B, which is a

 

Table 1: Our template model and Bradley's model.

	Our template	Bradley's model
	model
$M_*/{M_{\odot}}$	0.563	0.560
$T_{\rm eff}$ [K]	11996	12050
$\log(L_*$/ $L_{\odot})$	-2.458	-2.462
$\log(R_*$/ $R_{\odot})$	-1.864	-1.866
$\log(M_{\rm H}/M_*)$	-3.905	-3.824
$\log(M_{\rm He}/M_*)$	-1.604	-1.824
$\log(\rho_{\rm c})$ [g cm-3]	6.469	6.466
$\log(T_{\rm c})$ [106 K]	7.086	7.087

direct consequence of the chemical profile. In turn, the features of B are reflected in the Brunt-Väisälä frequency. As a result of element diffusion, the chemical profiles of our evolutionary models are very smooth in the interfaces (see upper panel of Fig. 1), and this explains the presence of extended tails in the shape of B. Also, note that our model is characterized by a chemical interface in which three ionic species in appreciable abundances coexists: helium, carbon and oxygen (see early in this section). This transition region gives two contributions to B, one of them of relatively great magnitude, placed at $\log q \sim -1.4$, and the other, more external and of very low height at $\log q \sim -2.2$. As a last remark, the contribution of the hydrogen-helium interface to B is less than that corresponding to the helium-carbon-oxygen transition. Note that the contributions from the Ledoux term are translated into smooth bumps on N². The characteristics of B and N² as predicted by our models are markedly different from those found in previous studies in which the white dwarf evolution is treated in a simplified way, particularly regarding the chemical abundance distribution in the outer layers (e.g., Tassoul et al. 1990; Brassard et al. 1991, 1992a,b; Bradley 1996). For more details see Althaus et al. (2002).