A possible criterion for envelope ejection in asymptotic giant branch or first giant branch stars

Following Paczyńiski & Ziółkowski, we investigate the relationship between the zero-age mass of a star and the mass of its degenerate core at that point in the star's evolution when the binding energy of its envelope changes from negative to positive. This relationship is fairly consistent, to within observational error, with the relationship by Weidemann & Koester between the initial mass and the mass of the white dwarf (WD) remnant, and with the observed distribution of masses of planetary nebula nuclei. We derive the equivalent relation for Population II (Pop II) stars (Z = 0.001), finding white dwarf masses ~ 10–20 per cent greater for the same initial masses. For both Population I (Pop I) and Pop II, however, we do not obtain C/O WD masses above ~1.1 M⊙ – the putative massive progenitors (≳ 8 and 6 M⊙ respectively) have negative envelope binding energies even at fairly extreme red supergiant radii. For low-mass Pop I stars (≲ 1 M⊙), we obtain He WD remnants by the same process on the first giant branch. We discuss the implications of our calculations for C/O WD masses, thermonuclear Type II supernovae, and the use of planetary nebulae as distance candles.