Origin of the abundance patterns in Galactic globular clusters: constraints on dynamical and chemical properties of globular clusters

Aims. We analyse the effects of a first generation of fast rotating massive stars on the dynamical and chemical properties of globular clusters. Methods. We use stellar models of fast rotating massive stars, losing mass through a slow mechanical equatorial winds to produce material rich in H-burning products. We propose that stars with high Na and low O abundances (hereafter anomalous stars) are formed from matter made of slow winds of individual massive stars and of interstellar matter. The proportion of slow wind and of interstellar material is fixed in order to reproduce the observed Li-Na anticorrelation in NGC 6752. Results. In the case that globular clusters, during their lifetime, did not lose any stars, we found that to reproduce the observed ratio of normal to anomalous stars, a flat initial mass function (IMF) is needed, with typically a slope x=0.55 (a Salpeter's IMF has x=1.35). In the case that globular clusters suffer from an evaporation of normal stars, the IMF slope can be steeper: to have x=1.35, about 96% of the normal stars would be lost. We make predictions for the distribution of stars as a function of their [O/Na] and obtain quite reasonable agreement with that one observed for NGC 6752. Predictions for the number fraction of stars with different values of helium, of the 12C/13C and 16O/17O ratios are discussed, as well as the expected relations between values of [O/Na] and those of helium, of [C/N], of 12C/13C and of 16O/17O. Future observations might test these predictions. We also provide predictions for the present day mass of the clusters expressed in units of mass of the gas used to form stars, and for the way the present day mass is distributed between the first and second generation of stars and the stellar remnants.