Full evolutionary models for PG 1159 stars. Implications for the helium-rich O(He) stars

Aims. We present full evolutionary calculations appropriate to post-AGB PG 1159 stars for a wide range of stellar masses. Methods. We take into account the complete evolutionary stages of PG 1159 progenitors starting from the Zero Age Main Sequence. We consider the two kinds of Born Again Scenarios, the very late thermal pulse (VLTP) and the late thermal pulse (LTP), that give rise to hydrogen-deficient compositions. The location of our PG 1159 tracks in the effective temperature - gravity diagram and their comparison with previous calculations as well as the resulting surface compositions are discussed. Results. Our results reinforce the idea that the different abundances of N-14 observed at the surface of those PG 1159 stars with undetected hydrogen is an indication that the progenitors of these stars would have evolved through a VLTP episode, where most of the hydrogen content of the remnant is burnt, or LTP, where hydrogen is not burnt but instead diluted to very low surface abundances. We derive new values for spectroscopical masses based on these models. We discuss the correlation between the presence of planetary nebulae and the 14N abundance as another indicator that N-14-rich objects should come from a VLTP episode while N-14-deficient ones should be the result of a LTP. Finally, we discuss an evolutionary scenario that could explain the existence of PG 1159 stars with unusually high helium abundances and a possible evolutionary connection between these stars and the low mass O(He) stars.