All Figures

$\begin{figure} \par\includegraphics[width=12.6cm,clip]{4199f-01.eps} \end{figure}$ Figure 1: The observed location of PG 1159 stars in the $\log T_{\rm eff}\!-\!\log g$ plane. Filled squares represent GW Vir pulsators, hollow squares represent non-pulsators PG 1159 stars, diamonds correspond to "hybrid-PG 1159'' stars, and triangles represent stars with no photometric data available. Data are taken from Kawaler et al. (2004), Werner et al. (1997), Dreizler & Heber (1998), Nagel & Werner (2004), and Vauclair et al. (2005). PG 1159 evolutionary tracks - this work; see also Althaus et al. (2005) - are indicated with solid curves; stellar masses are from right to left $0.53, 0.54, \ldots, 0.64~ M_{\odot}$ . The dashed curve corresponds to an AGB-manquè evolutionary track of a model star with $M_*= 0.49~ M_{\odot}$ (Scóccola et al. 2005). In this and the remainig figures displaying results from our additional evolutionary sequences with different stellar masses, only those results for evolutionary stages after the transitory phase has long vanished are shown (see text toward the end of Sect. 2).
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In the text

$\begin{figure} \par\includegraphics[width=7.4cm,clip]{4199f-02.eps} \end{figure}$ Figure 2: The surface (thick line) and helium-burning (thin line) luminosities ( $\log L_*/L_{\odot}$ and $\log L_{\rm He}/L_{\odot}$ , respectively) in terms of the effective temperature. The upper panel depicts the situation for our consistent 0.5895- $M_{\odot }$ sequence. The bottom panel corresponds to the 0.64- $M_{\odot }$ artificially generated sequence, in which the the unphysical evolutionary stages are indicated with dotted lines. For details, see text.
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In the text

$\begin{figure} \par\includegraphics[clip,width=8.1cm,clip]{4199f-03.eps} \end{figure}$ Figure 3: Evolutionary tracks for sequences with stellar masses, from right to left, of $0.53, 0.54, \ldots, 0.64~ M_{\odot}$ . The sequence with $0.5895 ~M_{\odot}$ is indicated with a thicker line. For this sequence, the black dots mark the location of five specific models to be analyzed in Sect. 3.2.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{4199f-04.eps} \end{figure}$ Figure 4: The run of the squared Brunt-Väisälä frequency in terms of the mass coordinate, corresponding to a 0.5895- $M_{\odot }$ PG 1159 model at $T_{\rm eff}= 139~000$ K and $\log(L_*/L_{\odot})= 2.31$ . The mass of the helium content is 0.008821 M_* ( $0.0052~ M_{\odot}$ ). Dark regions denote the contributions of the Ledoux term B (shown in the inset) to the Brunt-Väisälä frequency. The chemical profile is displayed in the upper zone of the plot.
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In the text

$\begin{figure} \par\includegraphics[width=8.1cm,clip]{4199f-05.eps} \end{figure}$ Figure 5: The evolution of the $\ell = 1$ pulsation periods $\Pi _k$ in terms of the effective temperature, corresponding to the sequence of $0.5895 ~M_{\odot}$ -models. Panel a) depicts the situation before the models reach the maximum value of effective temperature (Max. $T_{\rm eff}$ ), and panel b) shows the situation when the models have already passed through this stage. Arrows indicate the effective temperature corresponding to two elected models that are analyzed in Figs. 6 and 7. Note that the orientation of the $T_{\rm eff}$ axis in panel a) is opposite that of panel b). See text for details.
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In the text

$\begin{figure} \par\includegraphics[width=8.2cm,clip]{4199f-06.eps} \end{figure}$ Figure 6: Propagation diagram corresponding to a PG 1159 evolutionary model characterized by $[M_*/M_{\odot}, \log(L_*/L_{\odot}), \log T_{\rm eff}]= [0.5895, 3.68, 5.168]$ for $\ell = 1$ modes. The effective temperature of this model is marked with an arrow in panel (a) of Fig. 5. White (gray) areas correspond to propagation (evanescent) zones according the local analysis described in the text. The square of the Brunt-Väisälä frequency (N) and the Lamb frequency ( $L_{\ell }$ ) are depicted with dashed and solid lines, respectively. Horizontal thin lines show the square of the eigenfrequencies, and small symbols show the loci of the nodes of the radial eigenfunction, y₁ (filled circles: g-modes; filled squares: p-modes). Open circles correspond to nodes of modes with mixed character. These modes propagate in both P- and G-regions.
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In the text

$\begin{figure} \par\includegraphics[width=8.2cm,clip]{4199f-07.eps} \end{figure}$ Figure 7: Same of Fig. 6, but for a PG 1159 model characterized by $[M_*/M_{\odot}, \log(L_*/L_{\odot}), \log T_{\rm eff}]= [0.5895, 2.31, 5.143]$ . The effective temperature of the model is that marked with an arrow in panel (b) of Fig. 5.
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In the text

$\begin{figure} \par\includegraphics[width=8.2cm,clip]{4199f-08.eps} \end{figure}$ Figure 8: The normalized weight function of the mode g₄₀ vs. the stellar radius corresponding to $0.5895 ~M_{\odot}$ -models at different effective temperatures. The location of the stellar models in the H-R diagram is shown in Fig. 3.
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In the text

$\begin{figure} \par\includegraphics[width=8.2cm,clip]{4199f-09.eps} \end{figure}$ Figure 9: The evolution of $\Pi _0$ in terms of stellar luminosity for evolutionary models of different stellar masses ( $0.53,...,0.64~M_{\odot}$ from top tobottom).
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In the text

$\begin{figure} \par\includegraphics[width=8.3cm,clip]{4199f-10.eps} \end{figure}$ Figure 10: The forward period spacing $\Delta \Pi _{k}$ vs. period $\Pi _k$ for dipole ( $\ell = 1$ ) modes. The asymptotic period spacing $\Delta \Pi _{\ell }^{\rm a}$ as given by Eq. (4) is depicted with a thin horizontal line. Panel a) corresponds to the PG 1159 model considered in Fig. 4. The remaining panels correspond to the same stellar model, but for the situation in which the Ledoux term has been artificially suppressed in specific regions of the model: b) B= 0 at $M_r/M_* \approx 0.96$ , c) B= 0 at $M_r/M_* \approx 0.4{-}0.6$ , and d) B=0 everywhere in the star.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{4199f-11.eps} \end{figure}$ Figure 11: Period-spacing diagram and kinetic energy distribution ( upper and lower panels, respectively) for the same PG 1159 model considered in panel (a) of Fig. 10. Vertical dotted lines link local minima in $\Delta \Pi _{k}$ with the corresponding values of K_k. Maximum values of kinetic energy are indicated with arrows. Also labeled are the modes with k= 56 and k= 58, to be analyzed in Fig. 12.
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In the text

$\begin{figure} \par\includegraphics[width=8.2cm,clip]{4199f-12.eps} \end{figure}$ Figure 12: The spatial run of the modulus of the eigenfunctions y₁ and y₂, shown in panels b) and c), respectively, and the weight function W_k depicted in panels d) and e), for dipole pulsation modes with k= 56 (thick line) and k= 58 (thin line) corresponding to the same PG 1159 model analyzed in Fig. 11. We also include the Ledoux term B and the abundance (by mass) of ¹⁶O, ¹²C, and ⁴He in panel a).
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In the text

$\begin{figure} \par\includegraphics[width=8.2cm,clip]{4199f-13.eps} \end{figure}$ Figure 13: The period-spacing distributions for PG 1159 models with stellar masses, from bottom to top: $0.53, 0.54, \ldots, 0.64~ M_{\odot}$ at $T_{\rm eff} \approx 140~000$ K. In the interests of clarity, each curve has been arbitrarily shifted upwards, starting from the lowest one, corresponding to the model with $M_*= 0.53 ~M_{\odot}$ .
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In the text

$\begin{figure} \par\includegraphics[clip,width=8.4cm,clip]{4199f-14.eps} \end{figure}$ Figure 14: Period-spacing distributions for 0.5895- $M_{\odot }$ PG 1159 models at different effective temperatures. In the interest of clarity, each curve has been shifted upwards, except for the lowest one. In a few cases we have included numbers on the right side of the plot, corresponding to the $T_{\rm eff}$ value of the models (in units of kK). In particular, the thick curve corresponds to the highest value of effective temperature reached by the star, $T_{\rm eff} \sim 176$ kK.
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In the text

$\begin{figure} \par\includegraphics[width=8.4cm,clip]{4199f-15.eps} \end{figure}$ Figure 15: The asymptotic period spacing for PG 1159 models with stellar masses of $0.53,...,0.64~M_{\odot}$ in terms of the effective temperature. This figure is based in Fig. 9. The observed mean period spacing $\overline {\Delta \Pi }$ of the high-gravity GW Vir stars PG 1159-035, PG 2131+066, PG 1707+427, and PG 0122+200 ( $21.5 \pm 0.1$ s, $21.6 \pm 0.4$ s, $23.0 \pm 0.3$ s, and $21.1 \pm 0.4$ s, respectively) is also shown (taken from Kawaler et al. 2004).
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In the text

$\begin{figure} \par\includegraphics[clip,width=8.4cm,clip]{4199f-16.eps} \end{figure}$ Figure 16: The average of the computed period spacings for PG 1159 models with different stellar masses in terms of the effective temperature. Each panel corresponds to a specific GW Vir star. The observed mean period spacing $\overline {\Delta \Pi }$ is also shown (taken from Kawaler et al. 2004).
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In the text

$\begin{figure} \par\includegraphics[width=8.2cm,clip]{4199f-17.eps} \end{figure}$ Figure 17: The $\chi ^2$ values as defined by Eq. (5) corresponding to PG 1159-035 versus the effective temperature for $\ell = 1$ modes. Each panel depicts the results for a specific mass value. The gray strip corresponds to the allowed values of $T_{\rm eff}$ according to spectroscopic measurements. The arrow indicate the minimum value of $\chi ^2$ associated with the effective temperature of the best-fit model.
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In the text

$\begin{figure} \par\includegraphics[width=8.3cm,clip]{4199f-18.eps} \end{figure}$ Figure 18: Same as Fig. 17, but for PG 2131+066.
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In the text

$\begin{figure} \par\includegraphics[clip,width=8.1cm,clip]{4199f-03.eps} \end{figure}$	Figure 3: Evolutionary tracks for sequences with stellar masses, from right to left, of $0.53, 0.54, \ldots, 0.64~ M_{\odot}$ . The sequence with $0.5895 ~M_{\odot}$ is indicated with a thicker line. For this sequence, the black dots mark the location of five specific models to be analyzed in Sect. 3.2.
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$\begin{figure} \par\includegraphics[width=8.2cm,clip]{4199f-07.eps} \end{figure}$	Figure 7: Same of Fig. 6, but for a PG 1159 model characterized by $[M_/M_{\odot}, \log(L_/L_{\odot}), \log T_{\rm eff}]= [0.5895, 2.31, 5.143]$ . The effective temperature of the model is that marked with an arrow in panel (b) of Fig. 5.
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$\begin{figure} \par\includegraphics[width=8.2cm,clip]{4199f-08.eps} \end{figure}$	Figure 8: The normalized weight function of the mode g₄₀ vs. the stellar radius corresponding to $0.5895 ~M_{\odot}$ -models at different effective temperatures. The location of the stellar models in the H-R diagram is shown in Fig. 3.
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$\begin{figure} \par\includegraphics[width=8.2cm,clip]{4199f-09.eps} \end{figure}$	Figure 9: The evolution of $\Pi _0$ in terms of stellar luminosity for evolutionary models of different stellar masses ( $0.53,...,0.64~M_{\odot}$ from top tobottom).
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$\begin{figure} \par\includegraphics[width=8.2cm,clip]{4199f-13.eps} \end{figure}$	Figure 13: The period-spacing distributions for PG 1159 models with stellar masses, from bottom to top: $0.53, 0.54, \ldots, 0.64~ M_{\odot}$ at $T_{\rm eff} \approx 140~000$ K. In the interests of clarity, each curve has been arbitrarily shifted upwards, starting from the lowest one, corresponding to the model with $M_*= 0.53 ~M_{\odot}$ .
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$\begin{figure} \par\includegraphics[clip,width=8.4cm,clip]{4199f-16.eps} \end{figure}$	Figure 16: The average of the computed period spacings for PG 1159 models with different stellar masses in terms of the effective temperature. Each panel corresponds to a specific GW Vir star. The observed mean period spacing $\overline {\Delta \Pi }$ is also shown (taken from Kawaler et al. 2004).
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$\begin{figure} \par\includegraphics[width=8.3cm,clip]{4199f-18.eps} \end{figure}$	Figure 18: Same as Fig. 17, but for PG 2131+066.
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