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AGB stars of the intermediate-age LMC cluster NGC 1846 II. Dredge up along the AGB

Aims: We investigate the change in the surface abundance of $^{12}$C during the evolution along the AGB, aiming to constrain third dredge-up models. Methods: High-resolution, near-infrared spectra of a sample of AGB stars in the LMC cluster NGC 1846 were obtained. A cluster sample ensures a high level of homogeneity with respect to age, metallicity, and distance. The C/O ratio and the ratio of $^{12}$C/$^{13}$C were measured and compared with our evolutionary models. Results: For the first time, we show the evolution of the C/O and $^{12}$C/$^{13}$C ratios along a cluster AGB. Our findings allow us to check the reliability of the evolutionary models and, in particular, the efficiency of the third dredge up. The increase in both C/O and $^{12}$C/$^{13}$C in the observed O-rich stars is reproduced by the models well. However, the low carbon isotopic ratios of the two C-stars in our sample indicate the late occurrence of moderate extra mixing. The extra mixing affects the most luminous AGB stars and is capable of increasing the abundance of $^{13}$C, while leaving unchanged the C/O ratio, which has been fixed by the cumulative action of several third dredge-up episodes. We find indications that the F abundance also increases along the AGB, supporting an in situ production of this element.