All Figures

$\begin{figure} \par\includegraphics[width=12cm,clip]{1965fg1.eps}\end{figure}$ Figure 1: Hertzsprung-Russell diagram for the complete evolution of our $2.7 ~M_{\odot}$ stellar model from the ZAMS to the white dwarf domain. The star experiences a very last thermal pulse (VLTP) on its early cooling phase after hydrogen burning has almost ceased. The following evolutionary stages correspond to the born-again phase and are depicted, together with the VLTP stage with dotted lines. The numbers close to the filled dots along the evolutionary track correspond to the age (in 10³ yr) counted from the occurrence of the peak of the last thermal pulse. The domain of the PG 1159, DO, DB and DQ stars are indicated, as well as the thermal pulse and born-again stages. As a result of mass loss episodes, the stellar mass decreases from 2.7 to $0.5885 ~ M_{\odot}$ . After the born-again episode, the hydrogen-deficient post-AGB remnant experiences a second excursion towards lower temperatures before reaching eventually its terminal white dwarf track.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1965fg2.eps}\end{figure}$ Figure 2: The temporal evolution of surface luminosity and hydrogen- and helium-burning luminosities in solar units for an initially $2.7 ~M_{\odot}$ star during the thermally pulsing phase on the AGB. After experiencing 10 thermal pulses, the progenitor star departs from the AGB with a thermal-pulse cycle phase of 0.89 (denoted by a black dot). The last thermal pulse shown here takes place already on the white dwarf cooling track, where the hydrogen-burning luminosity exceeds $10^{8}~ L_{\odot}$ . The right panel shows the luminosity evolution after the occurrence of the tenth pulse. Age is given in 10⁶ yr counted from the main sequence.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1965fg3.eps} % \end{figure}$ Figure 3: Internal ⁴He, ¹²C and ¹⁶O profiles for the $2.7 ~M_{\odot}$ DB white dwarf progenitor shortly before the occurrence of the first thermal pulse on the AGB ( upper panel) and at the onset of the last thermal pulse at high $T_{\rm eff}$ ( bottom panel). Overshoot episodes leave recognizable features both in the inner carbon-oxygen profile and in the intershell region that are particularly rich in oxygen. See text for details.
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In the text

$\begin{figure} \par\includegraphics[width=12cm,clip]{1965fg4.eps}\end{figure}$ Figure 4: Hertzsprung-Russell diagram for the evolutionary stages following the onset of the last helium thermal pulse that takes place at high effective temperatures (born-again episode) for our post-AGB remnant of $0.5885 ~ M_{\odot}$ . Note that the remnant experiences a second excursion (double loop) to the red giant region before evolving into its final white dwarf state. The left inset displays the time-dependence of hydrogen- and helium-burning luminosities (solid and dotted lines, respectively) expressed in solar units. Selected stages are labeled by letters along the curves. In particular, points A and C correspond to the moment of maximum helium- and hydrogen-burning luminosities, respectively. At point B, protons begin to be engulfed and burnt by the outward-growing, helium-flash convection zone. Between points E and F, hydrogen burning becomes almost extinguished. The hydrogen content remaining in the star amounts to $1.3 \times 10^{-8}~M_{\odot}$ . Note the extremely short evolutionary time-scales characterizing this phase.
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In the text

$\begin{figure} \par\includegraphics[width=14cm,clip]{1965fg5.eps}\end{figure}$ Figure 5: Internal abundance distribution of ¹H, ⁴He, ¹²C, ¹³C, ¹⁴N and ¹⁶O as a function of the outer mass fraction q for the $0.5885 ~ M_{\odot}$ remnant at various selected epochs during the course of the last helium thermal pulse. Panel a) corresponds to the moment just before the occurrence of the peak helium-burning luminosity (point A in Fig. 4). The situation when the outward-moving, helium-flash convection zone has reached the base of the hydrogen-rich envelope (between points B and C in Fig. 4) is illustrated by panel b). Note that protons are ingested into deeper and hotter layers. As a result of the ongoing hydrogen burning a small radiative and salt-finger zone establishes at log $q \approx -1.6$ , which splits the convection zone into two. The situation two weeks later is visualized in panel c). During this time, the convection zone powered by hydrogen burning propagates outwards. Finally, panel d) illustrates the abundance profiles near point F in Fig. 4. Only traces of hydrogen remain after hydrogen burning. Nuclear energy release due to hydrogen and helium burning are also shown in panel b) and c).
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1965fg6.eps}\end{figure}$ Figure 6: Evolution of the internal abundances of ¹³C and ¹⁶O (dashed and solid lines respectively) across the PG 1159 domain in terms of the outer mass fraction q for the $0.5885 ~ M_{\odot}$ remnant. Thin and thick lines correspond, respectively, to a model at $T_{\rm eff}$ = 71 000 K before the point of maximun effective temperature in the HR diagram and $T_{\rm eff}$ = 125 000 K below that point. Note the formation of a bump in the oxygen distribution as a result of $\alpha$ captures by ¹³C.
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In the text

$\begin{figure} \par\includegraphics[width=12cm,clip]{1965fg7.eps}\end{figure}$ Figure 7: Time-dependence of the different luminosity contributions (in solar units) for the post born again $0.5885 ~ M_{\odot}$ remnant of the evolution of the $2.7 ~M_{\odot}$ star: surface luminosity, $L_{\rm sur}$ , helium burning luminosity, $L_{\rm He}$ , neutrino losses, $L_{\rm neu}$ and rate of gravothermal (compressional plus thermal) energy release $L_{\rm g}$ . The evolution of the effective temperature (right scale) is also plotted. Time is in years counted from the moment at which the remnant reaches $T_{\rm eff}$ = 70 000 K. The vertical lines bracket the domain where neutrino losses exceed photon luminosity.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1965fg8.eps}\end{figure}$ Figure 8: Evolution of the surface gravity (in cgs units) as a function of the effective temperature for the post born-again $0.5885 ~ M_{\odot}$ remnant and its comparison with some observed hot hydrogen-deficient stars. Filled (open) circles denote pulsating (non-pulsating) PG 1159 stars and asterisks observed WC stars (Werner et al. 1997; Werner 2001).
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In the text

$\begin{figure} \par\includegraphics[width=14cm,clip]{1965fg9.eps} % \end{figure}$ Figure 9: Abundance by mass of ⁴He, ¹²C, ¹⁶O, ¹³C, ¹⁴N and ²²Ne in terms of the outer mass fraction for the $0.5885 ~ M_{\odot}$ remnant at four selected white-dwarf evolutionary stages, characterized by values of $\log L/L_{\odot}$ and $\log T_{\rm eff}$ (the corresponding values are given in parentheses). Panel a) corresponds to the start of the cooling branch, panel c) to an evolutionary stage during the DB instability strip and panel d) to the domain of the carbon-enriched DQ white dwarfs. Panel d) illustrates the significant carbon enrichment in the surface layers as a result of convective dredge-up of the carbon diffusive tail by the superficial helium convection zone. It is clear that element diffusion strongly modifies the internal chemical profiles of the hydrogen-deficient white dwarf.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1965fg10.eps}\end{figure}$ Figure 10: Abundances by mass of ⁴He, ¹²C and ¹⁶O as a function of the outer mass fraction for the $0.5885 ~ M_{\odot}$ white-dwarf remnant having a final helium content of $5.2 \times 10^{-3} ~ M_{\odot}$ . Initial profiles at the start of the cooling branch (log $T_{\rm eff}$ = 5.0) are plotted with thin lines. Later stages, denoted with normal and heavy lines, correspond to log $T_{\rm eff}$ = 4.41 and 4.28 respectively.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1965fg11.eps}\end{figure}$ Figure 11: Logarithm of the number density of surface ¹²C, ¹⁶O, ¹³C and ¹⁴N relative to that of ⁴He as a function of the effective temperature.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1965fg12.eps}\end{figure}$ Figure 12: Same as Fig. 10 but for a final helium content of $1.4 \times 10^{-3} ~M_{\odot}$ . The initial profiles at the start of the cooling track (log $T_{\rm eff}$ = 4.89) are plotted with thin lines. Later stages, denoted with normal and heavy lines, correspond to log $T_{\rm eff}$ = 4.407 and 4.29 respectively.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1965fg13.eps}\end{figure}$ Figure 13: Same as Fig. 10 but for a final helium content of $9.4 \times 10^{-4} ~ M_{\odot}$ . The initial profiles at the start of the cooling branch (log $T_{\rm eff}$ = 4.93) are plotted with thin lines. Later stages, denoted with normal and heavy lines, correspond to log $T_{\rm eff}$ = 4.42 and 4.31 respectively.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1965fg14.eps}\end{figure}$ Figure 14: Number density of surface ¹²C, relative to that of ⁴He as a function of the effective temperature. Solid and dotted lines correspond to the white dwarf remnant with helium content of $5.2 \times 10^{-3}$ and $1.4 \times 10^{-3} ~M_{\odot}$ , respectively. Filled circles refer to observed carbon abundance as taken from MacDonald et al. (1998).
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1965fg15.eps}\end{figure}$ Figure 15: Abundance by mass of ¹H, ⁴He, ¹²C and ¹⁶O as a function of the outer mass fraction for the $0.5885 ~ M_{\odot}$ white dwarf remnant at $T_{\rm eff}$ = 27 000 K. Here, post-born again mass-loss episodes have been neglected. Thus, traces of hydrogen remain in the star at the beginning of white dwarf evolution. Note that element diffusion leads to the formation of a DA white dwarf with a pure hydrogen envelope mass of about $5 \times 10^{-9} ~M_{\odot}$ . In this particular simulation, we have neglected the diffusion of minor species such as ¹³C and ¹⁴N.
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In the text

$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1965fg8.eps}\end{figure}$	Figure 8: Evolution of the surface gravity (in cgs units) as a function of the effective temperature for the post born-again $0.5885 ~ M_{\odot}$ remnant and its comparison with some observed hot hydrogen-deficient stars. Filled (open) circles denote pulsating (non-pulsating) PG 1159 stars and asterisks observed WC stars (Werner et al. 1997; Werner 2001).
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1965fg11.eps}\end{figure}$	Figure 11: Logarithm of the number density of surface ¹²C, ¹⁶O, ¹³C and ¹⁴N relative to that of ⁴He as a function of the effective temperature.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1965fg12.eps}\end{figure}$	Figure 12: Same as Fig. 10 but for a final helium content of $1.4 \times 10^{-3} ~M_{\odot}$ . The initial profiles at the start of the cooling track (log $T_{\rm eff}$ = 4.89) are plotted with thin lines. Later stages, denoted with normal and heavy lines, correspond to log $T_{\rm eff}$ = 4.407 and 4.29 respectively.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1965fg13.eps}\end{figure}$	Figure 13: Same as Fig. 10 but for a final helium content of $9.4 \times 10^{-4} ~ M_{\odot}$ . The initial profiles at the start of the cooling branch (log $T_{\rm eff}$ = 4.93) are plotted with thin lines. Later stages, denoted with normal and heavy lines, correspond to log $T_{\rm eff}$ = 4.42 and 4.31 respectively.
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$\begin{figure} \par\includegraphics[width=8.8cm,clip]{1965fg14.eps}\end{figure}$	Figure 14: Number density of surface ¹²C, relative to that of ⁴He as a function of the effective temperature. Solid and dotted lines correspond to the white dwarf remnant with helium content of $5.2 \times 10^{-3}$ and $1.4 \times 10^{-3} ~M_{\odot}$ , respectively. Filled circles refer to observed carbon abundance as taken from MacDonald et al. (1998).
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