The mass of helium in white dwarf stars and the formation and evolution of hydrogen-deficient post-AGB stars

We use two sets of ‘cradle-to-grave’ evolutionary calculations to investigate how the mass of the helium buffer layer between the CO core and the hydrogen-burning shell in thermally pulsing asymptotic giant branch (AGB) stars depends on the initial stellar mass and heavy element abundance. Cool star mass loss is included by augmenting the Reimers' formula with fits to mass-loss rates observed for Galactic Mira variables, obscured Large Magellanic Cloud (LMC) AGB variables and Galactic OH/IR sources. Resulting white dwarf masses are in good agreement with the semi-empirical final mass–initial mass relation. We derive lower and upper limits on the mass of helium in white dwarfs as functions of their mass and initial heavy element abundance. We find that stars that experience a very late thermal pulse (VLTP) have final helium masses that are about 25 per cent below the lower limit for stars that do not experience a VLTP. We have derived a modified form of Iben's criterion for a star to experience a post-AGB helium shell flash, and subcriteria for discriminating between the occurrence of late and VLTPs. We find that a post-AGB flash does not necessarily result in formation of a non-DA white dwarf. Furthermore, we find that the relative amount of time spent burning helium or hydrogen depends on how the mass-loss rate varies with stellar surface parameters. These two considerations complicate the relationship between the probability that a star experiences a post-AGB flash and the relative formation rates of DA and non-DA white dwarfs. However, our calculations do lead to a non-DA/DA ratio that is consistent with the observed ratio. We find that our predicted abundances for PG1159 stars agree within the error bars with the observed abundances, without the need of convective overshoot. We also find that nitrogen is produced during VLTPs but not in the other evolutionary paths to hydrogen-deficient objects. Hence, we propose this as the reason why nitrogen is observed in some PG1159 stars but not all. Our VLTP models also produce surface abundances close to those of the low-gravity hybrid PG1159 stars. An inconsistency between asteroseismological and our evolutionary model determinations of helium layer mass remains.