The third dredge-up and the carbon star luminosity functions in the Magellanic Clouds

We investigate the formation of carbon stars as a function of the stellar m ass and parent metallicity. Theoretical modelling is based on an improved s cheme for treating the third dredge-up in synthetic calculations of thermal ly pulsing asymptotic giant branch (TP-AGB) stars. In this approach, the us ual criterion (based on a constant minimum core mass for the occurrence of dredge-up, M-c(min)) is replaced by one on the minimum temperature at the b ase of the convective envelope, T-b(dred), at the stage of the post-flash l uminosity maximum. Envelope integrations then allow determination of M-c(mi n) as a function of stellar mass, metallicity, and pulse strength (see Wood 1981), thus inferring if and when dredge-up first occurs. Moreover, the fi nal possible shut down of the process is predicted. Extensive grids of TP-AGB models were computed by Marigo (1998a,b) using th is scheme. In this paper, we present and discuss the calibration of the two dredge-up parameters (i.e. efficiency lambda and T-b(dred)) aimed at repro ducing the carbon star luminosity function (CSLF) in the LMC, using TP-AGB models with original metallicity Z = 0.008. Tn addition to this, the effect s of different input quantities on the theoretical CSLF are analyzed. It tu rns out that the faint tail`is almost insensitive to the history of star fo rmation rate (SFR) in the parent galaxy, in contrast to the bright wing whi ch may be more affected by the details of the recent history. Actually, we find that the faint end of the CSLF is essentially determined by the temper ature parameter T-b(dred). Once the faint end is reproduced, the peak locat ion is a stringent calibrator of the efficiency parameter lambda. The best fit to the observed CSLF in the LMC is obtained with lambda = 0.50, log T-b (dred) = 6.4, and a constant SFR up to an age of about 5 x 10(8) yr. This r ecent drop of the SFR is invoked to remove a slight excess of bright carbon stars otherwise predicted. A good fit to the observed CSLF in the SMC is then easily derived from the Z = 0.004 models, with a single choice of parameters lambda = 0.65, log T-b (dred) = 6.4, and a constant SFR over the entire significant age interval. The result for lambda is consistent with the theoretical expectation that t he third dredge-up is more efficient at lower metallicities.