2 Self-enrichment and a mass-metallicity relation

The lack of any obvious correlation, in any globular cluster system, between the mass (or the luminosity) and the metallicity of individual globulars is often used as an argument against the self-enrichment hypothesis. Indeed, were one to assume that gravitational potential gradients dominated mass loss, the most massive objects would be better able to retain their metal-enriched supernova ejecta, so that metal abundance should increase with cluster mass in case of self-enrichment. Before adressing the discussion of a luminosity-metallicity relation, we would like to dismiss this idea that more massive clusters would be more metal-rich in the case of self-enrichment. If a more massive object is indeed better able to retain more supernova ejecta, this larger amount of metallic ejecta is mixed with a larger amount of gas. Therefore, no firm conclusion can be drawn concerning the resulting metal abundance (or metallicity), i.e. the ratio of the two increased quantities. It is the fractional efficiency of gas retention which is important. Most importantly, though, mass loss in this class of models is determined by external gas pressure and not by the pressure equivalent of the gravitational potential gradient. This means that the absence of a mass-metallicity relation, in the sense that the most massive globulars would also be the most metal-rich (e.g. McLaughlin 1997; Barmby et al. 2000), can not be considered as evidence against the self-enrichment hypothesis. In sharp opposition with these statements, the self-enrichment model we develop foresees a mass-metallicity relationship in the sense that the most metal-rich proto-globular clusters are the least massive ones.

Figure 1: Mass-luminosity relation for the 56 globular clusters of the Pryor & Meylan (1993) compilation and the corresponding least-squares fit.

Unlike globular clusters, dwarf galaxies exhibit well-defined correlations between luminosity and metallicity (e.g. Gilmore 2000; Mateo 2000) such that the dimmest ones are the most metal weak. The standard explanation for this correlation being self-enrichment in the presence of galactic winds which are limited by gravitational potential gradients (Dekel & Silk 1986), Djorgovski & Meylan (1994) conclude that globular clusters cannot be self-enriched systems. However, Dekel & Silk (1986) point out that the dwarf galaxy observed luminosity-metallicity relation can be successfully explained only if the gaseous proto-galaxies are embedded within dominant halos of dark matter. While there is indeed clear evidence of the presence of such halos around dwarf galaxies (Mateo 1996), this is not the case for globular clusters (Moore 1996; Meylan & Heggie 1997). Therefore, the Dekel & Silk model, built for dwarf galaxies, can certainly be not extrapolated to globular clusters. Moreover, dwarf galaxies and GCs have undergone very different star formation histories: their respective star formation rate and duration differ by, at least, an order of magnitude (Gilmore 2000). Dwarf galaxies also exhibit metallicity spreads, often larger than 1dex (Mateo 2000), in marked contrast with the chemical homogeneity of globular clusters. Considering these many differences, the comparison between globular clusters and dwarf galaxies therefore appears irrelevant.

Figure 2: Metallicity-luminosity diagram for the whole galactic globular cluster sytem. The dashed line at $\rm [Fe/H]=-0.8$ represents the generally assumed metallicity limit between the halo and the bulge/disk subsystems. The different types of globulars are marked by different symbols. Old Halo and Younger Halo clusters are respectively represented by open and full circles. The crosses label the Sgr globulars. The open squares ("noHBR'' group) stand for the halo globular clusters for which the horizontal brach morphology index is not given in Harris (1996). The full triangle and the asterisks respectively represent $\omega$ Cen and the bulge/disk clusters.

In searching for a luminosity-metallicity relation in the Galactic globular cluster system, it should be kept in mind that, while the observed quantity is the luminosity, the physical quantity of interest is the mass. Figure 1 represents the relation between the mass and the absolute visual magnitude for the 56 globular clusters of the Pryor & Meylan (1993) mass compilation. The M_v values come from the McMaster Catalogue (Harris 1996, updated 1999). The scatter superimposed on the correlation, of the order of $\sigma_{M_v}\simeq0.6$, is equivalently the variations of the mass-to-light ratio from cluster to cluster. This ranges from $\sim$1 to $\sim$4 (Pryor & Meylan 1993) and reflects possible differences in the initial mass function and the dynamical evolution of the clusters. Therefore, any mass-metallicity correlation will be, at least partly, washed out in the corresponding $M_v-\rm [Fe/H]$ plot. This effect is illustrated in Fig. 2, the metallicity-luminosity diagram for the whole globular cluster system (the [Fe/H] values are taken from the McMaster Catalogue). Also plotted are the corresponding error bars in [Fe/H], $\pm$0.15dex (King 1999), and in M_v, $\pm$0.6dex from Fig. 1, if the latter is considered to be a mass indicator. The size of the M_v errorbars (reflecting the different luminosities that a globular cluster with a given mass but varying mass-to-light ratios may exhibit) is clearly not negligeable compared to the size of the observed distribution, the dispersion of the best-fitting gaussian to the galactic globular cluster luminosity function being $\simeq$1.2 (Harris 1991). Unfortunately, determination of the physical quantity of interest, i.e. the relative masses of the globular clusters at their formation, is still uncertain at least by a factor 2 (Meylan 2000). For instance, the use of single-mass King models is a simplification which tends to underestimate cluster mass (Ashman & Zepf 1998; Mermilliod 2000). Therefore, one of the key points in the search for a mass-metallicity correlation is to use an homogeneous set of globular cluster masses in order to limit additional scatter in the ( $\log M_{\rm GC}$, [Fe/H]) plot. We use the globular cluster mass compilation computed by Pryor & Meylan (1993): this compilation is the most complete set of globular cluster masses computed with an internally consistent family of multi-component King-Michie models.

Another source of scatter in the luminosity(mass)-metallicity plot is introduced by the various origins of the Galactic globulars. Indeed, evidence has now accumulated that the Galactic globular cluster system does not consist of globular clusters with a single origin. While the majority of globular clusters in the halo are old, with a remarkably small age spread (Rosenberg et al. 1999), there is a small subset, particularly among the more metal-rich clusters, with inferred ages of several Gyr younger than the dominant old population. These younger globular clusters are either clusters being/having been accreted by the Galaxy recently or metal-rich clusters associated with the bulge/disk subsystem. These clusters being significantly younger, their formation is not expected to be taken into account by our self-enrichment model, which deals with globular clusters whose gaseous progenitors have a primordial composition. The age spread highlighted by Rosenberg et al. (1999) also confirms the globular cluster system subdivisions early suggested by Zinn (1985, 1993). From the point of view of the metallicity distribution, the Galactic globular cluster system is composed of two subpopulations, a metal-poor halo group and a metal-rich, centrally concentrated, bulge (or disk) group (Zinn 1985). Furthermore, the halo subsystem itself includes an Old Halo, made of globular clusters perhaps born in situ, during the collapse of the protogalactic cloud, and a Younger Halo likely made of globulars later stripped from neighbouring dwarf galaxies (see Paper II for a review of these evidences).

Since our self-enrichment model deals with globular clusters whose progenitors were embedded in the hot phase of the protogalactic cloud and whose gaseous material was pristine, it is not expected to apply to the Younger Halo group, the presumed accreted component of the halo, nor to the bulge clusters. Thus, in what follows, we focus either on the coeval and old sample of Rosenberg et al. (1999) or on the Old Halo defined by Zinn.