Formation of Multiple Populations in Globular Clusters: constraints on
  the dilution by pristine gas
  The star-to-star differences in the abundance of light elements observed in
the globular clusters (GCs) can be explained assuming that a second generation
(SG) of stars form in the gas ejected by the asymptotic giant branch (AGB)
stars belonging to a first stellar generation. However, while Na and O appear
to be anticorrelated in the cluster stars, from the stellar models they turn
out to be correlated into the AGB ejecta. In order to reconcile the stellar
theory with the observational findings, all the GC models invoke an early
dilution of AGB ejecta with pristine gas occurring during the SG formation.
Despite a vast consensus about the occurrence of such dilution, the physical
process behind it is still unknown. In the present paper we set some general
constraints on the pristine gas dynamics and on the possible amount of pristine
gas involved in the SG formation, making use of a one zone chemical model. We
find that such a dilution is a necessary ingredient in the SG star formation to
explain the observed abundance patterns. We confirm the conclusion of our
previous works showing that clusters must have been initially much more
massive. We also show that models assuming that clusters had an initial mass
similar to their current one, and adopting a large fraction of pristine gas to
form SG stars, fail to reproduce the observed Na-O anticorrelation and are not
viable. We finally show that the dilution event should be restricted in time,
rather than extended for the all duration of the SG formation.
