Low-mass, helium-enriched PG 1159 stars: a possible evolutionary origin and implications for their pulsational stability properties

Aims. We examine a recently-proposed evolutionary scenario that could explain the existence of the low-mass, helium-enriched PG 1159 stars. We focus in particular on studying the pulsational stability properties of the evolutionary models predicted by such a scenario. Methods. We assess the overstability of pulsation g-modes of stellar models as evolution proceeds in the PG 1159 domain. Stellar models are extracted from the full evolution of a 1-$M_{\odot}$ model star that experiences its first thermal pulse as a late thermal pulse (LTP) after leaving the AGB. The evolutionary stages corresponding to the born-again episode and the subsequent helium sub-flashes are taken into account in detail. Results. Under reasonable assumptions of mass-loss rate, the evolutionary scenario reproduces the high helium abundances observed in some PG 1159 stars. We find that, despite the high helium abundance in the driving layers, a narrow region exists in the $\log T_{\rm eff}-\log~g$ diagram for which the helium-enriched PG 1159 sequence exhibits unstable pulsation modes with periods in the range 500 to 1600 s. In particular, the nonpulsating helium-enriched PG 1159 star, MCT 0130-193