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Volume 559, November 2013
Article Number A66
Number of page(s) 13
Section Stellar structure and evolution
Published online 15 November 2013

Online material

Appendix A: Constraining CE ejection efficiency

Our choice of the CE ejection efficiency parameter αCE is based on the following population synthesis estimate for the Galactic nova rate (per year) which is constrained by observations (Shafter 2002). In the population synthesis algorithm for counting novae, the total frequency (νtot) of nova-explosions (per year) for Z = 0.02 and a constant star formation rate, S = 7.086     yr-1 (following the prescription of Hurley et al. 2002), is estimated from, (A.1)with the nova recurrence time for a binary system calculated for every timestep δtnova of its evolution in a nova-phase from, (A.2)using the expression of Yungelson et al. (1995) for the critical ignition mass , (A.3)The WD radius RWD is determined according to the formula of Nauenberg (1972), (A.4)where MCh = 1.43   M is the Chandrasekhar mass.

Appendix A.1: Galactic nova rate

With the above prescription, the total rate of novae in the Galaxy (Z = 0.02) is calculated using binary_c/nucsyn as, (A.5)assuming a constant binary fraction fbin = 0.5 (Duquennoy & Mayor 1991). As shown in Table A.1, our estimate of the Galactic nova rate is consistent with existing estimates of 30 ± 10   yr-1 (Shafter 2002) within a factor of ~2–3, being closer to the observed rate for values of αCE < 0.5, and decreasing with higher values of αCE.

Table A.1

Estimated values of the Galactic nova rate, Rnova, average WD-mass in nova systems, ⟨ MWD ⟩, and fraction of WD binaries leading to novae, α (according to Eq. (A.6)), as a function of the CE ejection efficiency αCE.

thumbnail Fig. A.1

Histograms of predicted Galactic nova rate as function of WD mass, MWD per 0.1   M bin, for αCE = 0.2 (upper panel) and 0.5 (lower panel). The total rate of novae in the Galaxy Rnova is the sum of the contributions from all MWD bins.

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Figure A.1 further illustrates the sensitivity of the nova rate to the choice of αCE. The nova rate decreases sharply at higher WD masses and for more efficient CE ejection. This is expected because the final separation after the CE phase increases with αCE, so that for a higher αCE, the post-CE binary is too wide for the secondary to evolve to fill its Roche lobe. Because of shorter recurrence times for massive WDs which lead to more frequent outbursts, the total nova rate drops sharply with increasing αCE   (0.2to0.5). Nelson et al. (2004) also find an increase in the nova rate by a factor of ~2–3 for lower CE ejection efficiency. For αCE < 0.2, the nova rate decreases as more systems merge following the CE phase that leads to the WD-MS binary.

Appendix A.2: Additional tests for nova binaries

The fraction, α, of binary systems hosting WDs that lead to novae is also estimated as, (A.6)where is the birth rate of WDs in binary systems. Also, the frequency averaged WD mass (⟨MWD⟩ in M) is calculated for different choices of αCE (<1). As shown in Table A.1, our estimates for this fraction α (defined by Eq. (A.6)) is close to the value of 0.02 used by Romano & Matteucci (2005) to match the observed nucleosynthesis yields for Galactic novae. The averaged (by nova frequency) WD mass ⟨MWD⟩ is also in agreement with the values in the range of 0.8−1.1   M as obtained by Nelson et al. (2004).

Appendix B: HBB J-stars: single stars vs. binary mergers

If mass ratio q exceeds a critical value qcrit (as defined in Hurley et al. 2002), RLOF from a GB star to a lower mass MS companion is dynamically unstable and mass transfer leads to a CE phase in which the system can merge depending on the initial separation. For such a Case-B RLOF merger, the MS star is absorbed into the envelope and the core-mass of the merger is determined by the core of the GB donor star at the onset of the CE phase. Consequently, following the CE phase, the merged star has a lower core mass on the GB compared to a single star of same total mass. Because the core mass at the base of the AGB phase in our synthetic models depends on the core mass at the base of the GB (Hurley et al. 2002), such mergers ascend the AGB with lower core masses than single AGB stars of similar total masses. Consequently they live longer on the AGB and, for total masses higher than about 4   M for which HBB occurs along with TDU, we predict such mergers also behave as J-stars for a longer phase on the AGB as compared to single HBB AGB stars. Figure B.1 shows an example of the J-star phase for such a binary merger that lasts for about 1  Myr compared to the much shorter span of about 0.1  Myr for an equivalent single star. The binary system initially consists of a 3.1   M primary star which overflows its Roche lobe on the GB with a core mass of 0.45   M that subsequently forms the core of the merged star following a CE phase with the 1.3   M (MS) secondary absorbed into the giant envelope. Thus, the merger core (and envelope) mass is significantly different from a single star of similar total mass, and consequently the core mass can only grow to 0.47   M at the end of the GB – significantly lower than the expected core mass (~0.73   M) of a corresponding single star. This in turn leads to a lower core mass at the start of the TPAGB for the binary merger and consequently it evolves for a longer phase with TDU because of which its surface C/O exceeds 1, and HBB that decreases the isotopic ratios of on the stellar surface classifying it as a J-star.

thumbnail Fig. B.1

Evolutionary properties on the TPAGB for a single star (upper panel) and a GB-MS (Case-B RLOF) binary merger (lower panel) of similar total mass. The core mass Mc and surface C/O are plotted on the left scale, along with surface isotopic ratios, (multiplied by a factor of 10) on the right scale. The dotted horizontal lines mark C/O = 1 and while the dashed vertical lines mark the duration of the J-star phase.

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© ESO, 2013

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