Evolution of long-lived globular cluster stars II. Sodium abundance variations on the asymptotic giant branch as a function of globular cluster age and metallicity

Context. Long-lived stars in globular clusters exhibit chemical peculiarities with respect to their halo counterparts. In particular, sodium-enriched stars are identified as belonging to a second stellar population born from cluster material contaminated by the hydrogen-burning ashes of a first stellar population. Their presence and numbers in different locations of the colour-magnitude diagram provide important constraints on the self-enrichment scenarios. In particular, the ratio of Na-poor to Na-rich stars on the asymptotic giant branch (AGB) has recently been found to vary strongly from cluster to cluster (NGC 6752, 47 Tuc, and NGC 2808), while it is relatively constant on the red giant branch (RGB). Aims. We investigate the impact of both age and metallicity on the theoretical sodium spread along the AGB within the framework of the fast rotating massive star (FRMS) scenario for globular cluster self-enrichment. Methods. We computed evolution models of low-mass stars for four different metallicities ([Fe/H] = -2.2, -1.75, -1.15, -0.5) assuming the initial helium-sodium abundance correlation for second population stars derived from the FRMS models and using mass loss prescriptions on the RGB with two realistic values of the free parameter in the Reimers formula. Results. Based on this grid of models we derive the theoretical critical initial mass for a star born with a given helium, sodium, and metal content that determines whether that star will climb or not the AGB. This allows us to predict the maximum sodium content expected on the AGB for globular clusters as a function of both their metallicity and age. We find that (1) at a given metallicity, younger clusters are expected to host AGB stars exhibiting a larger sodium spread than older clusters and (2) at a given age, higher sodium dispersion along the AGB is predicted in the most metal-poor globular clusters than in the metal-rich ones. We also confirm the strong impact of the mass loss rate in the earlier evolution phases on the Na cut on the AGB: the higher the mass loss, the stronger the trends with age and metallicity. Conclusions. The theoretical trends we obtain provide, in principle, an elegant qualitative explanation to the different sodium spreads that are observed along the AGB in the Galactic globular clusters of different ages and [Fe/H] values. Although it is real, the slope with both age and metallicity is relatively flat, although it steepens when accounting for mass loss variations. Therefore, additional parameters may play a role in inducing cluster to cluster variations, that are difficult to disentangle from existing data.