Empirical estimates of the Na-O anti-correlation in 95 Galactic globular clusters

Large star-to-star abundance variations are direct evidence of multiple stellar populations in Galactic globular clusters (GCs). The main and most widespread chemical signature is the anti-correlation of the stellar Na and O abundances. The interquartile range (IQR) of the [O/Na] ratio is well suited to quantifying the extent of the anti-correlation and to probe its links to global cluster parameters. However, since it is quite time consuming to obtain precise abundances from spectroscopy for large samples of stars in GCs, here we show empirical calibrations of IQR[O/Na] based on the O, Na abundances homogeneously derived from more than 2000 red giants in 22 GCs in our FLAMES survey. We find a statistically robust bivariate correlation of IQR as a function of the total luminosity (a proxy for mass) and cluster concentration c. Calibrated and observed values lie along the identity line when a term accounting for the horizontal branch (HB) morphology is added to the calibration, from which we obtained empirical values for 95 GCs. Spreads in proton-capture elements O and Na are found for all GCs in the luminosity range from Mv=-3.76 to Mv=-9.98. This calibration reproduces in a self-consistent picture the link of abundance variations in light elements with the He enhancements and its effect on the stellar distribution on the HB. We show that the spreads in light elements seem already to be dependent on the initial GC masses. The dependence of IQR on structural parameters stems from the well known correlation between c and Mv, which is likely to be of primordial origin. Empirical estimates can be used to extend our investigation of multiple stellar populations to GCs in external galaxies, up to M31, where even integrated light spectroscopy may currently provide only a hint of such a phenomenon.