Stochastic self-enrichment, pre-enrichment, and the formation of globular clusters

We develop a model for stochastic pre-enrichment and self-enrichment in globular clusters (GCs) during their formation process. GCs beginning their formation have an initial metallicity determined by the pre-enrichment of their surrounding protocloud, but can also undergo internal self-enrichment during formation. Stochastic variations in metallicity arise because of the finite numbers of supernova. We construct an analytic formulation of the combined effects of pre-enrichment and self-enrichment and use Monte Carlo models to verify that the model accurately encapsulates the mean metallicity and metallicity spread among real GCs. The predicted metallicity spread due to self-enrichment alone, a robust prediction of the model, is much smaller than the observed spread among real GCs. This result rules out self-enrichment as a significant contributor to the metal content in most GCs, leaving pre-enrichment as the viable alternative. Self-enrichment can, however, be important for clusters with masses well above 10^6 Msun, which are massive enough to hold in a significant fraction of their SN ejecta even without any external pressure confinement. This transition point corresponds well to the mass at which a mass-metallicity relationship ("blue tilt") appears in the metal-poor cluster sequence in many large galaxies. We therefore suggest that self-enrichment is the primary driver for the mass-metallicity relation. Other predictions from our model are that the cluster-to-cluster metallicity spread decreases amongst the highest mass clusters; and that the red GC sequence should also display a more modest mass-metallicity trend if it can be traced to similarly high mass.