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Appendix A: Observational data and time evolution in the (T_eff, log g)-diagram

	Fig. A.1 Selected tracks (Fig. 1) with a point marked for a time interval of 1 Myr (triangle), 50 Myr (square), 100 Myr (circle), and 1 Gyr (star).
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In Fig. A.1 we have plotted points for fixed time intervals of evolution along a number of selected tracks from Fig. 1. The density of points along these curves combined with the (proto) WD luminosities at these epochs can be used to evaluate the probability of detecting them. For a direct comparison with data population synthesis needs to be included to probe the distribution of WD masses. The observational data plotted in Fig. 1 were taken partly from the sources given in Table A.1 (primarily He WDs with MSP companions, main-sequence A-star companions, or He WDs that have been detected to show pulsations).Additional data for the plotted symbols can be found in Silvotti et al. (2012), Hermes et al. (2013), Brown et al. (2013).

Appendix B: The (proto) WD contraction phase

Figure C.1 shows the time Δt_proto it takes from Roche-lobe detachment until the proto-He WD reaches its highest value of T_eff and settles on the cooling track. Shown in this plot are all our calculated models for progenitor stars of 1.2 and 1.4 M_⊙ (i.e. a subset of the models plotted in Fig. 2). The black line (Eq. (1)) is an analytical result obtained from a somewhat steep core mass-luminosity function () combined with the assumption (for simplicity) that in all cases 0.01 M_⊙ of hydrogen is burned before reaching the highest T_eff. The figure shows that this line also serves as a good approximate fit to our calculated models. For a given He WD mass, the fit to Δt_proto calculated from our models is accurate to within 50%.

Appendix C: Nuclear burning during flashes

	Fig. C.1 Calculated models of proto-He WDs from Fig. 2 for M2 = 1.2 M⊙ (orange) and M2 = 1.4 M⊙ (blue). The black line is a fit to the data. It can also be derived analytically using a modified core mass-luminosity relation for low-mass evolved stars, combined with an assumed fixed amount of residual hydrogen (0.01 M⊙) to be burned. The red line separates models with and without flashes.
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	Fig. C.2 Evolutionary tracks in the HR-diagram for a 0.221 M⊙ proto-He WD with flashes (brown) and for a 0.212 M⊙ proto-He WD without flashes (blue). See Table C.1 for data.
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To compare the burning of residual envelope hydrogen for a case with and without large thermal instabilities (hydrogen shell flashes), we have plotted tracks in the HR-diagram shown in Fig. C.2. The age of the stars and the total amount of hydrogen remaining in their envelopes are given in Table C.1 for the points marked in the figure. These models were chosen very close to (but on each side of) M_flash ≃ 0.21 M_⊙, in both cases for a 1.3 M_⊙ progenitor star. As discussed in the main text, although the peak luminosity is high during a flash (and thereby the rate at which hydrogen is burned), the star only spends a relatively short time (~10⁶ yr) in this epoch. (For more massive He WDs it is even less time – for example, it only lasts ~10³ yr for a 0.27 M_⊙ He WD.) Therefore, the amount of additional hydrogen burned as a result of flashes is relatively small. In the example shown in Fig. C.2 it amounts to about 12% of the total amount of hydrogen at the point of Roche-lobe detachment. Hence, the flashes may appear to reduce Δt_proto by ~100 Myr. However, one must bear in mind that the proto-WDs that experience

flashes are also the WDs with the least amount of hydrogen in their envelopes after RLO.

For a star that experiences flashes, the residual hydrogen present in the envelope following the LMXB-phase is processed roughly as follows: 70% during the epoch from Roche-lobe detachment until reaching highest T_eff, 10% during the flashes, and 20% after finally settling on the WD cooling track.

© ESO, 2014