Envelope burning over-luminosity: a challenge to synthetic TP-AGB models

Until recently synthetic AGB models had not taken into account the break-down of the core mass-luminosity (Mc L) relation due to the occurrence of envelope burning in the most massive (M > 3:5 M for Pop. II and M > 4:5 M for Pop. I) and luminous (Mbol < 6) stars. Marigo et al. (1998) made the first attempt to consistently include the related over-luminosity effect (i.e. above the Mc L relation) in synthetic TP-AGB calculations. The method couples complete envelope integrations with analytical prescriptions, these latter being presently updated with the highly detailed relations by Wagenhuber & Groenewegen (1998). In this paper the reliability of the solution scheme is tested by comparison with the results of complete evolutionary cal- culations for a 7 M AGB star undergoing envelope burning (Bl¨ ocker & Sch¨ 1991; Bl¨ ocker 1995). Indeed, the method proves to be valid as it is able to repro- duce with remarkable accuracy several evolutionary features of the 7 M star (e.g. rate of brightening, luminosity evolution as a function of the core mass and envelope mass for different mass-loss prescriptions) as predicted by full AGB models. Basing on the new solution method, we present extensive synthetic TP-AGB calculations for stars with initial masses of 3:5; 4:0; 4:5; and5:0 M, and three choices of the initial metal- licity, i.e. Z =0 :019, Z =0 :008, and Z =0 :004. Three values of the mixing-length parameter are used, i.e. =1 :68; 2:0; 2:5. We investigate the dependence of envelope burning on such stellar parameters (M , Z, and ). The comparison between different cases gives hints on the interplay between envelope burning over-luminosity and mass loss, and related effects on TP-AGB lifetimes.