On the third dredge-up phenomenon in asymptotic giant branch stars

The third dredge-up phenomenon in asymptotic giant branch (AGB) stars is an alyzed through evolutionary model calculations of a 3 M. solar metallicity star. The Schwarzschild criterion is used to test the stability of a given layer against convection, and the calculations are performed either with or without extra-mixing below the convective envelope. Based on these calcula tions, several questions are addressed regarding the occurrence of the thir d dredge-up in AGB star models, the laws governing that phenomenon, and som e of its implications on the structural and chemical evolution of those sta rs. The use of the Schwarzschild criterion without extra-mixing of any sort is shown to lead to unphysical afterpulse models which prevent the occurrence of third dodge-up. Model calculations of a 3 M. star using no extra-mixing confirm the fail ure to obtain dredge-up in those conditions. That conclusi on is found to be independent of the mixing length parameter, stellar mass, or numerical accuracy of the models. Model calculations performed on selected afterpulses of the 3 M. star, but with extra-mixing (using a decreasing bubble velocity field in the radiativ e layers and a diffusion algorithm for the mixing of the chemical elements) , lead to efficient dredge-ups at a rate of 10(-5)-10(-)4 Mo/yr. Test calcu lations using different extra-mixing extents and efficiencies reveal that t he dredge-up predictions are rather insensitive to those extra-mixing param eters. This important conclusion is understood by analyzing the physics inv olved in the dredge-up process. It is shown that the dredge-up rate is dete rmined by the thermal relaxation time-scale of the envelope as C-rich matte r is added from the core into the envelope. The dredge-up predictions are, however, expected to depend on the convection prescription in the envelope. Linear relations both between the dredge-up rate and the core mass M-c and between the dredge-up efficiency lambda and M-c are predicted by the model calculations. Those linear relations are expected to still hold when the fe edback of the dredge-ups on the AGB evolution is taken into account. They p redict the dredge-up efficiency to level off at unity during the AGB evolut ion, at which point the core mass remains constant from one pulse to the ne xt. The con mass is concomitantly predicted to evolve towards an asymptotic value. The existence of such an asymptotic core mass naturally provides an upper limit to the mass of the white dwarf remnant, and helps to constrain the initial-final mass of white dwarfs. Synthetic calculations taking into account the Jredge-up laws obtained from the full AGB model calculations predict a continuous increase of the stell ar luminosity L with time, contrary to the predicted behavior of M-c and la mbda. This results from an adopted dependence of L on both M-c and the radi us R-c of the H-depleted core of the form L proportional to M-c(2)/Rc. As a result of this increase of L with time, the initial-final mass relation ca n further be constrained if mass loss is taken into account. If, for exampl e, a superwind is assumed to eject all the remaining envelope of the 3 M. s tar at L = 15000 M., then the mass of the white dwarf remnant is predicted to be 0.66 M., instead of 0.73 M. predicted by models without dredge-up. Finally, the synthetic calculations predict the formation of a 3 M. carbon star after about 20 pulses experiencing dredge-up. Taking into account the fact that the luminosity decreases by a factor of two during about 20% of t he interpulse phases, such a 3 M. carbon star could be observed at luminosi ties as low as 7500 L..