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

The third dredge-up phenomenon in asymptotic giant branch (AGB) stars is analyzed through evolutionary model calculations of a \mass{3}, 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 calculations, several questions are addressed regarding the occurrence of the third dredge-up in AGB star models, the laws governing that phenomenon, and some of its implications on the structural and chemical evolution of those stars. 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 dredge-up. Model calculations of a \mass{3} star using no extra-mixing confirm the failure to obtain dredge-up in those conditions. That conclusion 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 \mass{3} star, but with extra-mixing (using a decreasing bubble velocity field in the radiative layers and a diffusion algorithm for the mixing of the chemical elements), lead to efficient dredge-ups at a rate of \mass{10^{-5}-10^{-4}}/yr. Test calculations using different extra-mixing extents and efficiencies reveal that the dredge-up predictions are rather insensitive to those extra-mixing parameters. This important conclusion is understood by analyzing the physics involved in the dredge-up process. It is shown that the dredge-up rate is determined by the thermal relaxation time-scale of the envelope as C-rich matter 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 feedback of the dredge-ups on the AGB evolution is taken into account. They predict the dredge-up efficiency to level off at unity during the AGB evolution, at which point the core mass remains constant from one pulse to the next. The core 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 dredge-up laws obtained from the full AGB model calculations predict a continuous increase of the stellar luminosity L with time, contrary to the predicted behavior of M_c and lambda . This results from an adopted dependence of L on both M_c and the radius R_c of the H-depleted core of the form L~ M_c(2/R_c) . As a result of this increase of L with time, the initial-final mass relation can further be constrained if mass loss is taken into account. If, for example, a superwind is assumed to eject all the remaining envelope of the \mass{3} star at \lsun{L=15000}, then the mass of the white dwarf remnant is predicted to be \mass{0.66}, instead of \mass{0.73} predicted by models without dredge-up. Finally, the synthetic calculations predict the formation of a \mass{3} 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 the interpulse phases, such a \mass{3} carbon star could be observed at luminosities as low as \lsun{7500}.