2.3. Luminosity and Mass Distributions

By far the most robust and predictable feature of the GCS in different galaxies is the luminosity distribution of the clusters (LDF), which is the visible trace of the cluster mass spectrum. In its classic form plotted as number of clusters per unit magnitude M_V, the LDF has a roughly Gaussian-like shape with a characteristic ``turnover'' or peak point at MV⁰ -7.4, differing by little more than ± 0.2 magnitudes from any one large galaxy to another. The symmetric Gaussian-like shape has been consistently verified in every galaxy with sufficiently deep photometry (e.g., Harris et al. 1991; Whitmore et al. 1995; Kundu & Whitmore 1998), and the turnover luminosity is consistent enough to be an entirely respectable standard candle for distance determination (see H99; Whitmore 1997; Harris 1997 for comprehensive recent discussions).

The potential use of the LDF for standard-candle purposes was, in fact, the original stimulus for studying globular clusters in distant galaxies, beginning with the Virgo ellipticals (see Hanes 1977 and H99). But as more data have come in, and as the remarkable uniformity of the LDF has emerged more clearly, it has become more interesting for its strong astrophysical constraints on cluster formation and evolution. The LDF in its entirety must be the product of the initial mass spectrum of the clusters at formation, combined with the subsequent ~ 10¹⁰ years of dynamical evolution in the tidal field of the parent galaxy. Most of the relevant destruction mechanisms (tidal shocking due to passage through the disk or bulge, evaporation coupled to the tidal field) are much more effective at lower cluster mass, and current models (e.g. Murali & Weinberg 1997; Gnedin & Ostriker 1997; Vesperini 1997, 1998; Vesperini & Heggie 1997; and Elson's lectures in this volume) indicate that the transition mass above which these effects are not critically important is the LDF turnover near 10⁵ M. Above this point, the LDF we observed today is therefore likely to be much closer to the original mass spectrum at the time of formation. Although a consensus is gradually growing in this direction, more comprehensive dynamical simulations incorporating the full range of dynamical effects, starting from a realistic formation mass spectrum, still need to be done.

Over the upper ~ 90% of the globular cluster mass range (M 10⁵ M, where the very most massive clusters reach as high as ~ 10⁷ M), the cluster mass spectrum is better plotted as number of clusters per unit mass or unit luminosity, rather than per unit magnitude. In this form, it is well described by a simple power-law form dN / dM ~ M^{-1.8 ± 0.2} in all galaxies (Harris & Pudritz 1994), gradually steepening to higher masses. This distribution is the form which needs to be reproduced by a quantitative formation model; it also needs to be virtually independent of other factors such as metallicity, total cluster population (S_N), or size and type of host galaxy! Whatever mechanism we settle on must be extremely robust.

McLaughlin & Pudritz (1996) have developed a quantitative theory for the LDF in which protocluster gas clouds build up within very large ``supergiant'' molecular clouds (SGMCs) by collisional agglomeration. The SGMC is visualized as supplying a large number of initial small-mass cloud ``particles'' (physically, these particles can be visualized as probably resembling the ~ 100 M cloud cores found in Galactic GMCs). These cloud particles then collide and amalgamate to form larger ones; after several crossing times, a power-law distribution of cloud masses results (e.g., Field & Saslaw 1965; Kwan 1979). The larger clouds become the ``protoclusters'' which eventually turn into full-fledged star clusters once their internal pressure support (due to turbulence and weak magnetic field) leaks away. Since the small clouds always vastly outnumber the large ones, the reservoir of gas contained in the entire SGMC needs to be very much larger than the masses of the individual protoclusters that build up inside it (observationally, the typical star cluster mass is ~ 10^-3 of the host GMC mass; see Harris & Pudritz 1994).

The emergent mass spectrum from this collisional growth process has the expected power-law form, but its detailed shape is controlled by two key input parameters: (a) the ratio of cloud lifetime against star formation relative to the cloud-cloud collision time; and (b) the dependence of cloud lifetime on mass. More massive clouds have higher mean densities and are expected to have shorter lifetimes. Thus, cloud growth is a stochastic race against time: large clouds continue to grow by absorbing smaller ones, but as they do, their survival time before turning into stars becomes shorter and shorter. Thus at the high-mass end, the slope of the mass spectrum dN / dM gradually steepens as it becomes more and more improbable that such massive clouds can survive before turning into stars. Encouragingly, this very feature is matched extremely well by the observations of LDFs in populous cluster systems such as in giant ellipticals (see McLaughlin & Pudritz 1996). Even more encouraging is the fact that this theory is also able to match the entire LDF of the newly formed star clusters in recent mergers such as NGC 4038 / 4039 (Whitmore & Schweizer 1995) and NGC 7252 (Miller et al. 1997). In these cases, we should be looking at something much closer to the initial mass spectrum, relatively unaffected even at low masses by dynamical evolution (see Figure 8).

Figure 8. Luminosity distribution function (LDF, or number of clusters per unit luminosity) for the ``young'' globular clusters formed in the merger remnants NGC 4038 / 39 and NGC 7252. The model lines (from McLaughlin 1998, private communication) are computed from the collisional growth model of McLaughlin & Pudritz (1996). The original gas ``particles'' have masses m0 = 100 M, the cloud lifetime varies as ~ m-0.4, and the parameter defines the ratio of fiducial cloud lifetime to collisional crossing time. Larger values of generate shallower mass functions that extend to higher mass. In this case a relatively small value 20 fits the data.

The related new observations of globular clusters in young and merging systems, some of which were used above, represent a major stride forward in our understanding of globular cluster formation. Clusters are now seen to form in an amazingly wide range of situations: in protogalactic halos, in starburst dwarf galaxies with or without obvious external ``triggers'', in merging disk galaxies, or at the centers of giant ellipticals that are accreting gas. We see from the same observations that cluster formation is a highly inefficient process: the typical star cluster mass is ~ 10<sup>-3</sup> of the host GMC mass, and any one GMC appears to produce only a handful of star clusters that remain bound over many Gyr. In other words, the star clusters end up using typically less than 1% of the host GMC gas supply regardless of environment (see H99; Kissler-Patig et al. 1998; McLaughlin 1999). Although most stars may well form in ``clustered'' mode, the majority of these clumps and associations dissolve quickly away into the field, leaving the bound clusters as those rare sites in which the star formation efficiency was ~ 50% or more.

These observations show that it is no longer tenable to regard globular cluster formation as a ``special'' event which happened only in the early universe; though it is a rare mode of star formation, it is also clearly a robust process which can happen at any metallicity and at any time that a sufficient supply of gas is collected together. Exactly how the gas is accumulated into sufficiently large SGMCs during the protogalactic era is of interest on its own merit but is, apparently, not a critical issue for the mass spectrum of the globular clusters it produces.

At the present time, the collisional-growth model provides the only fully quantitative theoretical fit to the cluster mass spectrum at formation that we have available. Though it is obviously successful as far as it goes, many steps remain to be taken: for example, we would like to be able to predict from the gas dynamics of the host GMCs what the cloud lifetimes should be and exactly how the lifetime depends on mass. We also need to understand why the ratio of cluster mass to host GMC mass is typically ~ 10^-3, and what fraction of the total GMC mass can be expected to turn into clusters. (NB: In Elmegreen & Efremov (1997), a somewhat different scenario is adopted in which the mass distribution of clouds within a GMC is stated to resemble a fractal structure generated by turbulence. In this scheme, however, the expected slope dN / dM ~ M^-2 is slightly too steep to match most real galaxies, and does not clearly predict the progressive change in slope with mass that is embedded in the collisional growth model.)