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 (M_c - L) relation due to the occurrence of envelope burning in the most massive (M >~ 3.5 Msun for Pop. II and M >~ 4.5 Msun for Pop. I) and luminous (M_bol <~ -6) stars. Marigo et al. (1998) made the first attempt to consistently include the related over-luminosity effect (i.e. above the M_c-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 calculations for a 7 Msun AGB star undergoing envelope burning (Bloecker & Schoenberner 1991; Bloecker 1995). Indeed, the method proves to be valid as it is able to reproduce with remarkable accuracy several evolutionary features of the 7 Msun 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, and 5.0 Msun, and three choices of the initial metallicity, i.e. Z=0.019, Z=0.008, and Z=0.004. Three values of the mixing-length parameter are used, i.e. alpha =1.68, 2.0, 2.5. We investigate the dependence of envelope burning on such stellar parameters (M, Z, and alpha ). 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.