TP-AGB evolution with overshoot for low-mass stars as a function of metallicity

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We give a brief review on the properties of asymptotic giant branch models with overshoot. Then we describe new model calculations with overshoot. Initial masses are ranging from 1 to 3Msun and metallicities are Z=0.02, 0.01 and 0.001. Third dredge-up occurs efficiently for low masses and carbon stars are formed, with some at core masses as low as 0.58Msun. After the thermal pulse at which stars become C-rich the luminosities are in the range of the observed C-star luminosity function during the whole interpulse phase and for all C-star models. The dredge-up evolution depends mainly on the core mass at the first thermal pulse and on the metallicity. The Z=0.001 models of the 2 and 3Msun sequence become C-rich almost instantaneously after the onset of the first thermal pulses. For the 2Msun case the C/O ratio initially exceeds 4. During following dredge-up episodes the C/O ratio decreases.