Hot bottom burning in asymptotic giant branch stars and the turbulent convection model

We investigate the effect of two different local turbulent convection models on the structure of intermediate-mass stars (IMSs, 3.5 M(.) less than or equal to M less than or equal to 7 M(.)) in the asymptotic giant branch (AGB) phase where, according to observations, they should experience hot bottom burning (HBB). Evolutionary models adopting either the mixing length theory (MLT) or the Canuto & Mazzitelli (CM) description of stellar convection are discussed. It is found that, while the MLT structures require some degree of tuning to achieve, at the bottom of the convective envelope, the large temperatures required for HBB, the CM structures spontaneously achieve these conditions. Since the observational evidence for HBB (existence of a class of very luminous, lithium-rich AGB stars in the Magellanic Clouds showing low C-12/C-13 ratios) is quite compelling, the above result provides a further, successful test for the CM convective model, in stellar conditions far from solar. With the aid of the CM model, we then explore a number of problems related to the late evolution of this class of objects, and give first results for (1) the luminosity evolution of IMSs in the AGB phase (core mass-luminosity relation and luminosity range in which HBB occurs) for Population I and Population II structures, (2) the minimum core mass for semidegenerate carbon ignition (similar to 1.05 M(.)), (3) the relation between initial mass and final white dwarf (WD) mass (also based on some observational evidences about the upper AGB stars), and (4) the expected mass function of massive WDs. Confirmation of the theoretical framework could arise from an observational test: the luminosity function of AGB stars is expected to show a gap at M(bol) similar to -6, which would distinguish between the low-luminosity regime, in which AGBs become carbon stars, and the upper luminosities, at which they undergo HBB, have very low C-12/C-13 ratios, and become lithium rich.