All Figures

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f01a.eps}\pa... ...ce*{2mm} \includegraphics[angle=-90,width=8.5cm,clip]{5066f01b.eps} \end{figure}$ Figure 1: Core-mass - radius relations for the "basic'' models, as described in the text. The lines show the logarithm of the radius of the stars as a function of the helium-core mass. Upper panel a): low-mass grid models with masses of 0.91, 1.01, 1.14, 1.30, 1.48, 1.63, 1.81 and 2.00 $M_\odot$ . Lower panel b): high-mass grid models with masses of 2.00, 2.46, 2.79, 3.17, 3.60, 4.09, 4.65, 5.28 and 6.00 $M_\odot$ . The 2 $M_\odot$ model is plotted in both panels throughout as a solid line for easier comparison. The other models are shown as solid lines on the first giant branch (FGB), where they could cause a common envelope resulting in a spiral-in and creating a helium white dwarf. The dashed lines show the asymptotic giant branch (AGB), where a common envelope would lead to the formation of a carbon-oxygen white dwarf. Dotted lines are parts of the evolution where the stars either are smaller than at the tip of the FGB (at lower radii) or where their envelope binding energies become positive on the AGB (at large radii).
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f02.eps} \end{figure}$ Figure 2: Helium-core-mass - orbital period tracks for the "basic'' models. The lines show the logarithm of the orbital period at which the Roche lobe is filled for grid models with masses of 1.01, 1.27, 1.59, 2.00, 2.52, 3.17, 3.99, 5.02 and 6.32 $M_\odot$ . The period was obtained from the radius of the model star, under the assumption that it fills its Roche lobe and the companion has a mass equal to the helium-core mass of the model. This way, the system would undergo a spiral-in that would lead to a binary with mass ratio q = 1. The line styles have the same meaning as in Fig. 1.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f03.eps} \end{figure}$ Figure 3: The logarithm of the binding energy of the "basic'' model stars as a function of the helium-core mass. The grid models with masses of 0.91, 1.01, 1.14, 1.30, 1.48, 1.63, 1.81, 2.00, 2.46, 2.79, 3.17, 3.70, 4.09, 4.65, 5.28, 6.00 and 6.82 $M_\odot$ are shown. The 2.00 $M_\odot$ model is drawn as a solid line, the line styles for the other models have the same meaning as in Fig. 1. The parts where the envelope binding energy is zero (before a helium core develops) or positive are not shown.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f04.eps} \end{figure}$ Figure 4: The envelope-structure parameter $\lambda _{\rm env}$ for the "basic'' models, as a function of the helium-core mass. The same grid models are shown as in Fig. 3. The meaning of the line styles is explained in the caption of Fig. 1.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f05.eps} \end{figure}$ Figure 5: The fraction of mass lost at two moments in the evolution of a star as a function of its initial mass, for the three different wind strengths ( $C_{\rm rml} = 0.2$ , 0.5 and 1.0) used in the grid. This fraction is shown for the tip of the FGB (dashed lines and crosses), and the "tip of the AGB'' (dotted lines and plusses). See the text for details.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f06.eps} \end{figure}$ Figure 6: Comparison of a selection from the small grid of models with a stellar wind. The models displayed have masses of 1.0, 1.6, 2.5, 3.2, 4.0, 5.0 and 6.3 $M_\odot$ . The wind strength parameters are $C_{\rm rml} = 0.0$ (dotted lines) and $C_{\rm rml} = 1.0$ (solid lines, the strongest mass loss in the grid). Stars with mass loss are usually larger, but for models of 4.0 $M_\odot$ or more this effect becomes negligible. The 1.0 $M_\odot$ model loses so much mass that it never ignites helium in the core.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f07a.eps}\pa... ...2mm} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f07b.eps} \end{figure}$ Figure 7: Comparison of a selection of grid models with $C_{\rm rml} = 1.0$ with initial masses of 1.3, 1.6, 2.0, 2.5 and 3.2 $M_\odot$ to the basic models ( $C_{\rm rml} = 0.0$ ). Upper panel a): comparison of the radius of the models with a stellar wind (solid lines) and the radius of a basic model with the same mass and core mass (plusses). Lower panel b): the fraction of the radius of the wind model $R_{\rm w}$ over the radius of the basic model $R_{\rm b}$ with the same total and core mass. Each data point corresponds to a point in the upper panel. Of the data points in the upper panel, 7 out of 143 (5%) lie outside the plot boundaries in the lower panel. The dashed horizontal lines show the region where agreement is better than 10%, where 83% of the data points lie. The 1.0 $M_\odot$ model was left out because there are only a few basic models with lower mass, the higher-mass models were left out because they lose very little mass (see Fig. 5).
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f08.eps} \end{figure}$ Figure 8: Results of the spiral-in calculations, each individual symbol is a solution of the calculations and thus represents one pre-CE binary. The figure shows the logarithm of the orbital period of the intermediate binary $P_{\rm m}$ as a function of the secondary mass $M_{\rm 2m}$ . Different symbols represent different observed systems, as explained in the legend. System 1704a is the system listed in Table 1, 1704b is the same system, but with the reverse mass ratio. For solutions with $M_{\rm 2m} < 2.5~M_\odot$ , only every third solution is plotted for clarity. Around $M_{\rm 2m} = 1.2$ and $\log P_{\rm m} = 2.8$ the symbols of WD 0135-052, WD 0136+768 and WD 1204+450 overlap due to the fact that they have similar white-dwarf masses. For comparison we show the lines of the solutions for (top to bottom) WD 0136+768, WD 0957-666 and WD 1101+364 taken from Nelemans et al. (2000), as described in the text.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f09.eps} \end{figure}$ Figure 9: The logarithm of the common-envelope parameter $\alpha _{\rm ce}$ for the solutions of the spiral-in calculations shown in Fig. 8. Different symbols represent different observed systems. For $M_{\rm 2m} < 2.5~M_\odot$ every third solution is plotted only.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f10.eps} \end{figure}$ Figure 10: Results of the spiral-in calculations. This figure is similar to Fig. 8 and shows the orbital period of the pre-CE system as a function of the secondary mass. The solutions for each system are plotted in a separate panel, as labelled in the upper-right corner. All solutions with acceptable $\alpha _{\rm ce}$ are plotted. The number of solutions for each system is shown in the lower-right corner. We assumed an uncertainty in $M_{\rm 2f}$ of $0.05~M_\odot$ and calculated $M_{\rm 1f}$ using q₂.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f11.eps} \end{figure}$ Figure 11: Results of the spiral-in calculations for WD 0957-666 with period limits for a conservative first mass transfer. This figure contains the same data as the third panel in Fig. 10 (symbols) plus the period limits $P_{\rm min}$ and $P_{\rm max}$ (dashed lines). The solutions that lie between these limits are shown in black, the others in grey. See the main text for details.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f12.eps} \end{figure}$ Figure 12: Results of the spiral-in calculations with period limits for conservative mass transfer as in Fig. 11, but for all systems. The number in the upper left corner of each panel is the number of systems that lie between the period limits.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f13.eps} \end{figure}$ Figure 13: Results of the spiral-in calculations (grey symbols), obtained as in Fig. 11, and the solutions of calculations of conservative evolution (black symbols). Only the six systems shown have spiral-in solutions within the period limits (see Fig. 12). The numbers in the lower left and lower right corners are the numbers of plotted spiral-in solutions and conservative solutions respectively.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f14.eps} \end{figure}$ Figure 14: The mass ratio of model double white dwarfs formed by a conservative first mass transfer and a common envelope with spiral-in, against the age difference between the two components. The dashed horizontal lines show the observed range of possible mass ratios as shown in Table 1. The dotted vertical lines are the estimated cooling-age differences $\pm$ $50\%$ (see Table 1).
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In the text

$\begin{figure*} \par\mbox{\includegraphics[angle=-90,width=8.8cm,clip]{5066f15a.eps}\includegraphics[angle=-90,width=8.8cm,clip]{5066f15b.eps} }\end{figure*}$ Figure 15: The distribution of the logarithm of the envelope-ejection parameters for solutions of the double dynamical mass-loss scenario. Each dot represents a system that evolves through an episode of dynamical mass loss with $\gamma _{\rm s1}$ and then a common envelope with spiral-in with $\alpha _{\rm ce2}$ to form one of the observed white dwarfs. Only solutions for which $0.1\!\leq\!\gamma_{\rm s1}\!\leq\!10$ and $0.1\!\leq\!\alpha_{\rm ce2}\!\leq\!10$ are shown. Hence, each dot in one panel has a counterpart in the other panel. The dashed vertical lines show $\gamma_{\rm s1}\!=\!1.45, 1.58$ in the left panel and $\alpha_{\rm ce2}\!=\!0.61, 0.72$ in the right panel. Black dots in one panel indicate solutions that lie between the dashed lines in the other panel; black dots between black lines therefore lie in the preferred range in both panels. The vertical position of each dot within the line for its system shows the deviation from the observed secondary mass M₂. From the lower to the upper dots in a line these masses are: $M_{\rm 2f} = M_2-0.05, M_2-0.04, \ldots, M_2+0.05~M_\odot$ . The total number of solutions in the figure is nearly 150 000. The figure is made after Nelemans & Tout (2005).
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.8cm,clip]{5066f16a.eps}\includegraphics[angle=-90,width=8.8cm,clip]{5066f16b.eps} \end{figure}$ Figure 16: Solutions for the $\gamma _{\rm d}\gamma _{\rm a}$ scenario for each system. The representation is the same as in Fig. 15. The vertical dashed lines are at $\gamma = 0.9$ and $\gamma = 1.1$ in both panels. Black dots are solutions that lie between these lines in the other panel. Black dots that lie between the lines thus have $0.9 \leq \gamma _{\rm d1} \leq 1.1$ and $0.9 \leq \gamma _{\rm a2} \leq 1.1$ . The total number of solutions exceeds 200 000.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f17a.eps}\pa... ...e*{2mm} \includegraphics[angle=-90,width=8.5cm,clip]{5066f17b.eps} \end{figure}$ Figure 17: Upper panel a): the radius of a helium star as a function of its CO-core mass, for a selection of 15 models with total masses between 0.41 and $1.43~M_\odot$ . The dots show where the maximum radii are obtained and are used for the lower panel. The dashed line is the Roche-lobe radius for the intermediate primary of PG 1115+116 according to our solution 57. Lower panel b): the maximum radius of a low-mass helium star as a function of its total mass, for a selection of 33 models with masses between 0.33 and $1.4~M_\odot$ . The dots are the data points, the solid line connects them to guide the eye.
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In the text

$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f04.eps} \end{figure}$	Figure 4: The envelope-structure parameter $\lambda _{\rm env}$ for the "basic'' models, as a function of the helium-core mass. The same grid models are shown as in Fig. 3. The meaning of the line styles is explained in the caption of Fig. 1.
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$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f05.eps} \end{figure}$	Figure 5: The fraction of mass lost at two moments in the evolution of a star as a function of its initial mass, for the three different wind strengths ( $C_{\rm rml} = 0.2$ , 0.5 and 1.0) used in the grid. This fraction is shown for the tip of the FGB (dashed lines and crosses), and the "tip of the AGB'' (dotted lines and plusses). See the text for details.
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$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f09.eps} \end{figure}$	Figure 9: The logarithm of the common-envelope parameter $\alpha _{\rm ce}$ for the solutions of the spiral-in calculations shown in Fig. 8. Different symbols represent different observed systems. For $M_{\rm 2m} < 2.5~M_\odot$ every third solution is plotted only.
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$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f11.eps} \end{figure}$	Figure 11: Results of the spiral-in calculations for WD 0957-666 with period limits for a conservative first mass transfer. This figure contains the same data as the third panel in Fig. 10 (symbols) plus the period limits $P_{\rm min}$ and $P_{\rm max}$ (dashed lines). The solutions that lie between these limits are shown in black, the others in grey. See the main text for details.
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$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f12.eps} \end{figure}$	Figure 12: Results of the spiral-in calculations with period limits for conservative mass transfer as in Fig. 11, but for all systems. The number in the upper left corner of each panel is the number of systems that lie between the period limits.
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$\begin{figure} \par\includegraphics[angle=-90,width=8.5cm,clip]{5066f14.eps} \end{figure}$	Figure 14: The mass ratio of model double white dwarfs formed by a conservative first mass transfer and a common envelope with spiral-in, against the age difference between the two components. The dashed horizontal lines show the observed range of possible mass ratios as shown in Table 1. The dotted vertical lines are the estimated cooling-age differences $\pm$ $50\%$ (see Table 1).
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