Why the globular cluster NGC 6752 contains no sodium-rich second-generation AGB stars

Context. Globular clusters host multiple stellar populations showing different sodium enrichments. These various populations can be observed along the main sequence, red giant and horizontal branch phases. Recently it was shown, however, that at least in the globular cluster NGC 6752, no sodium-rich stars are observed along the early asymptotic giant branch (AGB), posing an apparent problem for stellar evolution. Aims. We present an explanation for this lack of sodium-rich stars in this region of the colour–magnitude diagram. Methods. We computed models for low-mass stars following the prediction of the so-called fast rotating massive stars model for the initial composition of second-generation stars. We studied the impact of different initial helium contents on the stellar lifetimes and the evolutionary path in the Hertzsprung-Russell diagram. Results. We propose that the lack of sodium-rich stars along the early-AGB arises because sodium-rich stars were born with a high initial helium abundance, as predicted by the fast rotating massive stars scenario. Helium-rich stars have much shorter lifetimes for a given initial mass than stars with a normal helium abundance, and above a cutoff initial helium abundance that slightly depends on the mass-loss rate on the RGB they do not go through the AGB phase and evolve directly into a white dwarf stage. Within the fast rotating massive stars framework we obtained a cutoff in [Na/Fe] between the second-generation models evolving into the AGB phase and those skipping that phase between 0.18 and 0.4 dex, depending on the mass loss rate used during the red giant phase. In view of the uncertainties in abundance determinations, the cutoff obtained by the present model agrees well with the one inferred from recent observations of the cluster NGC 6752. Conclusions. The helium-sodium correlation needed to explain the lack of sodium-rich stars along the early-AGB of NGC 6752 corresponds to the one predicted by the fast rotating massive stars models. A crucial additional test of the model is the distribution of stars with various helium abundances among main-sequence stars. Our model predicts that two magnitudes below the turnoff a very large percentage of stars, about 82%, probably has a helium content lower than 0.275 in mass fraction, while only 5% of stars are expected to have helium abundances greater than 0.4