2.1. Specific Frequencies

The simplest parameter describing a GCS is the total number of clusters it contains, quantified as the specific frequency S_N or number of clusters per unit galaxy luminosity (Harris & van den Bergh 1981; Harris 1991). In numerical terms it is defined simply as S_N = N_cl . 10^0.4(M_VT+15) for a total cluster population N_cl and galaxy luminosity M_V^T. In the early years of this subject, it was a major suprise to find that individual galaxies differ in this ratio by factors of twenty! Why are some galaxies, apparently, vastly more efficient at creating - or holding - their globular clusters? This is the classic ``specific frequency problem'' which cuts across most types of galaxies, particularly the ellipticals. Understanding these differences between galaxies that are otherwise similar in structure has been a continual challenge to virtually all of our formation models.

As Figure 7 shows, the average S_N for ellipticals is 3.5, essentially constant over more than a 10⁴ range in luminosity. But large scatter exists at all levels, and interesting trends affect both the the dE,N and cD-type galaxies (solid dots in Figure 7). These will be discussed below. The mean or ``baseline'' specific frequency S_N⁰ = 3.5 can be translated into a typical formation efficiency e₀, which is essentially the number of clusters per unit stellar mass in the galaxy halo. Assuming (M/L) 8 for the old-halo stars in E galaxies, then we obtain directly from the definition of S_N above that e₀ 1 cluster per 2 x 10⁸ M of halo mass. The globular clusters evidently represent quite a tiny fraction of all the old stellar population in these galaxies, even for the higher-S_N cases.

Figure 7. Specific frequency SN against galaxy luminosity, for elliptical galaxies (with data from Harris et al. 1998 and Miller et al. 1998). For dwarfs (left side of figure), solid symbols denote nucleated dE,N types and open symbols are non-nucleated dE types. For giant ellipticals (right side of figure), solid symbols denote Brightest Cluster Galaxies (BCGs, most of which are cD galaxies at the centers of rich galaxy clusters). The horizontal line is at the ``baseline'' level of SN0 = 3.5 (see text for interpretation).

The S_N graph in Figure 7 falls conveniently into the dwarf and giant ellipticals, whose distributions are in some sense mirror images of each other (the largest specific frequencies occur at either the very lowest or very highest luminosities; as we will see below, these high-S_N cases may have ultimately very similar causes). The giant ellipticals, as we discuss in more detail below, may have been built in a variety of ways (isolated or in situ from a single protogalaxy; by mergers; or by ongoing accretions of smaller satellites), and the supergiant brightest-cluster galaxies (BCGs) with their very high specific frequencies present a special problem which we will return to below.

The dE's are likely to present a fundamentally simpler story, but even here they fall into two natural subgroups according to the presence or absence of a central nucleus. The surveys of the GCSs in these small galaxies (Durrell et al. 1996; Miller et al. 1998) demonstrate that the non-nucleated dE's have rather similar specific frequencies in the range S_N ~ 2 - 3, like disk galaxies or ellipticals in sparse groups. These same dwarfs also tend to be more elongated in structure and, in cluster environments like Virgo or Fornax, form a spatially extended subsystem of galaxies resembling the spirals and irregulars rather than the giant ellipticals. Durrell et al. and Miller et al. suggest that most of the dE's may simply be original irregulars which formed relatively few clusters at early times and were then stripped of most of their gas in the cluster potential well, preventing further star formation.

The nucleated dE,N types exhibit a different trend, with the least luminous ones having the highest specific frequencies. These galaxies are also generally rounder in structure and occupy more centrally concentrated space distributions in Virgo and Fornax resembling the giant ellipticals. The authors cited above therefore interpret them as more likely to have been ``genuine'' small elliptical galaxies from the start, with their central nuclei forming either from residual gas infall or from individual globular clusters that were drawn in by dynamical friction. The interesting trend of S_N with luminosity can be logically interpreted as the result of the main star formation stage. Presumably, in the smallest dE,N systems shown in Figure 7 (which have total stellar masses 10⁸ M), the globular clusters formed rather early, along with most of the field stars, but then the remaining gas (which would have been a high fraction of the original gas supply) was expelled from its small potential well by the first round of stellar winds and supernova (cf. the authors cited). The more massive (~ 10⁹ M) dE,N systems would have held on to more of their original gas supply and finished more star formation, leaving a lower-S_N system in the end. A key part of this scenario is that the globular clusters need to form early, out of the densest initial clumps of gas. McLaughlin (1999) has used the Dekel & Silk (1986) model for gas loss from these small galaxies to show quantitatively that the expected trend of S_N with luminosity does indeed match the observations.