The mass of hydrogen MH retained in the surface layers of a single cool degenerate dwarf depends on the abundance of XCNO elements in its precursor. For a dwarf of mass 0.6 Msun, this mass of hydrogen is MH ≍ 1.3 × 10⁻⁴ Msun if XCNO ≍ 5 × 10⁻⁴ and MH ≍ 0.8 × 10⁻⁴ if XCNO ≍ 0.01. The corresponding hydrogen masses near the surface of the planetary nebula nucleus (PNN) precursors of the cool dwarfs are MH ≍ 2.3 × 10⁻⁴ Msun and MH ≍ 1.2 × 10⁻⁴ Msun, respectively. The amount of hydrogen converted into helium during the white dwarf phase is smaller for stars of larger initial CNO abundances primarily because of a smaller initial value of MH. The amount of burning following the PNN stage is independent of the mass ΔMBUF of the helium buffer zone initially separating the hydrogen-rich surface layers from the carbon-rich layer produced in the final helium shell flash, unless ΔM>SUB>BUF < Δ MCRIT BUF ≍ ΔMH/10. Here, ΔMH is the mass processed by hydrogen burning between shell flashes during the preceding asymptotic giant branch (AGB) phase. When ΔMBUF < Δ MCRIT BUF chemical diffusion, which causes carbon to migrate upward from the carbon-rich zone and hydrogen to migrate downward from the hydrogen-rich surface layers, can lead (if MH is large enough) to a sufficiently large CNO cycle burning rate at a sufficient depth below the surface to initiate a thermonuclear runaway. The ensuing hydrogen shell flash produces a convective zone which reaches the surface, resulting in a surface hydrogen-to-helium ratio of about 1 (by number), and forces the star to become a red giant. Subsequent behavior is speculated upon and the possible importance of this phenomenon is discussed in connection with the scenario which supposes that planetary nebula ejection from the AGB precursor occurs during a helium shell flash.
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I. et al. (1986) studied this question.