3 Comparison of the model with the observations

Figure 3: [Fe/H] vs. $\log \,M_{\rm GC}$ diagram for the globular clusters studied by Rosenberg et al. (1999). The old (OC) and the younger (YC) globular clusters exhibit different distributions in the diagram relatively to the boundary foreseen by the self-enrichment model. Also shown is the tidal destruction zone for the less massive halo globular clusters located inside the solar circle.

Our self-enrichment model suggests the existence of an anti-correlation between the mass Mof a proto-cluster and the metallicity [Fe/H] reached at the end of the self-enrichment process, in the sense that the least massive proto-clusters create the most metal-rich globular clusters (see Table 1 of Paper I):

$$[{\rm Fe/H}]=4.3-{\log}~M.$$ (1)

However, this (anti-)correlation applies to the gaseous progenitors; the mass-metallicity relation observed among the studied sample of clusters, if any, should only be a relic of Eq. (1). For instance, the -1 slope will be conserved only if there is a universal and constant star formation efficiency for the second stellar generation which forms the majority of the stars from the chemically enriched gas swept up from the first generation supernovae. Since the mass M of a gaseous progenitor is an upper limit for the mass $M_{\rm GC}$ of the globular cluster formed, Eq. (1) delimits a permitted area in the ( $\log \,M_{\rm GC}$, [Fe/H]) plot: all the data should be located to the left of Eq. (1) (plain curve in Figs. 3 and 5). Figure 3 represents Eq. (1) together with the globular clusters for which the age and the mass are respectively provided in Rosenberg et al. (1999) and Pryor & Meylan (1993). Obviously, the two Rosenberg et al. (1999) groups (old clusters: open symbols; younger clusters: filled symbols) behave in a different way compared to our self-enrichment mass-metallicity relation. While the old, coeval and metal-poor GCs are all located in the permitted area of the plot, i.e. their mass-metallicity diagram is consistent with the self-enrichment of primordial gaseous progenitors, half of the young clusters, either presumed accreted or belonging to the bulge subsystem, are located in the forbidden area of the diagram, i.e. as expected, their formation cannot be accounted for by the self-enrichment model. In Fig. 3, we also represent three of the most massive globulars (filled triangles), namely $\omega$ Cen, M 54 and NGC 5824. Their location in the forbidden part of the plot points to a different star formation history. This is not surprising since, at least in the case of $\omega$ Cen and M 54, an intrinsic abundance spread is seen. M 54 is of course a member of the Sagittarius dwarf spheroidal galaxy, and is not (yet) a Galactic globular cluster.

Figure 4: [Fe/H] vs. log $M_{\rm GC}$ plot including Old Halo and Younger Halo subgroups (49 globular clusters).

Figure 3 is also clearly depleted in low-mass globular clusters ( $\log~M_{\rm GC} < 4.8$). However, at a galactocentric distance smaller than 8kpc, these low-mass clusters are not expected to survive more than a Hubble time (see the "survival triangle'' in the mass vs half-mass radius diagram defined by Gnedin & Ostriker 1996, their Fig. 20a). The vast majority of the globular clusters located at these galactocentric distances, i.e. $\log~D<8\,$kpc, exhibit a metallicity higher than $\rm [Fe/H]=-2$. The depletion zone, represented by the box in Figs. 3 and 4, is therefore not surprising and corresponds to the tidal destruction of these low-mass clusters. The globular clusters used in our Paper are therefore no more than a surviving sample. The distance a given cluster lies to lower masses from the model upper bound is, to first order, a measure of the star formation efficiency of cluster formation. A "typical'' surviving cluster lies a factor of order 5 below the bound, suggesting an efficiency factor of order 20%. As noted above, however, lower mass clusters will have preferentially failed to survive until today, so that this value is an upper limit. Star formation efficiencies in the range from a few to a few tens of percent seem appropriate for most clusters. Only the few percent of clusters which are the most massive require star formation efficiencies in excess of unity, and so are inconsistent with this formation model. Interestingly, these very massive clusters are those which show internal abundance spreads, which are themselves direct evidence for self-enrichment during cluster formation.

Figure 5: [Fe/H] vs. $\log ~M_{\rm GC}$ plot for the Old Halo subgroup (37 globular clusters). The self-enrichment model (plain curve) defines a permitted area (left part of the plot) in which most of the observational points are located. The correlation between the globular cluster masses and their metallicities is particularly striking for $\rm [Fe/H]> -1.8.$

In order to increase our sample and to look for a surviving correlation between the mass and the metallicity, we also consider the Old Halo subgroup (Zinn 1993). As for the metallicity gradient (see Paper II), an Old Halo/Younger Halo separation is fruitful. Figure 4 shows a plot of [Fe/H] versus mass for the 49 halo globular clusters whose mass has been computed by Pryor & Meylan (1993): there is no correlation between the mass and the metallicity, the linear Pearson correlation coefficient being -0.15.

Considering the Old Halo group only (Fig. 5), as stated in the previous Section, a weak correlation between the logarithm of the mass of the globular clusters and their metallicity emerges. The linear Pearson correlation coefficient improves to a value of -0.35, with a corresponding probability of correlation of 96.92%. Moreover, most of the Old Halo globulars are located in the permitted area of the plot.