Annu. Rev. Astron. Astrophys. 1991. 29: 543-79 Copyright © 1991 by Annual Reviews. All rights reserved

7. THEORETICAL ISSUES

The globular clusters we see today must be the hardiest survivors of a larger original population. In their pioneering discussion, Fall & Rees (57) emphasized the potential importance of dynamical erosion on clusters by their surroundings. With the advent of high-speed computer simulation, the subject has now flourished. A long list of interrelated processes operating on clusters within galaxies can be identified [see Ostriker (163) for a good summary of most of them]:

• The tidal field of the galaxy sets a physical limit to the radius of any cluster, depending on its location and orbital eccentricity (e.g. 3, 120, 129, 146, 184).
• Clusters on highly eccentric orbits may be subject to more dramatic tidal shocking if they pass through the galactic bulge or disk (2, 29, 163).
• Shocking by collisional encounters with individual objects of over 10⁶ M(giant molecular clouds, black holes) will also create significant disruption, if these latter objects are common (222, 223).
• Dynamical friction may effectively remove many of the clusters near the center of a large galaxy, preferentially working on the more massive clusters (76, 128, 197).
• Internal relaxation of an individual cluster and the drive toward core collapse will cause evaporation of stars past the tidal boundary, shrinkage of the tidal radius, and thus increased evaporation (30, 31, 32). The effects of gaseous mass loss from the clusters, driven by stellar evolution and photoionization, operate in the same sense and can have major effects - especially in the early history of the cluster if the initial mass function is weighted toward high-mass stars (4, 30, 32, 193).
• Clusters on box orbits in triaxial potentials may have enhanced destruction rates because eventually their orbits will take them right through the nucleus. Clusters on tube orbits then have systematically higher chances of long-term survival (164). The presence of a central bar potential may also increase destruction rates (143).

A thorough study of all the effects mentioned above - and perhaps others not yet fully examined - may well cause feelings of bemused that any globular clusters still exist today. Indeed, it is sometimes argued that the present clusters might represent as little as ~ 1% of their original numbers. For the galaxy as a whole, such views are likely to be unrealistically extreme. (If the total number of globular clusters existing now were as small as ~ 1% of their initial numbers, then a large fraction of the entire halo today would have to be made from disrupted clusters, which is ruled out by the metallicity-based arguments given above). However, residual numbers of order 10% or larger do remain plausible by most criteria; see also the formation efficiency arguments by Larson (126). More importantly, the effectiveness of most of the destructive physical processes depends strongly on galactocentric distance. Within R_gc 1 kpc, the destructive mechanisms have their full influence and no known globular clusters should survive a Hubble time (163, 197). But beyond R_gc 4 kpc, tidal shocking and dynamical friction are reduced to low levels, and the clusters no longer fill up their whole survival triangle in the mass-radius plane (27, 59). The comprehensive simulation by Aguilar et al. (2) demonstrates that for these outer regions the slow process of evolutionary mass loss coupled to the tidal field becomes, by default, the dominant erosive mechanism. More extensive simulations, which include a range of initial cluster LFs and follow their changes over a Hubble time would be of great interest. At present, the implications are that for the intermediate-to-outer halo, the GCLF shape is dominated by the initial conditions of formation; dynamical evolution strongly affects only the smallest and/or lowest-density clusters. Therein may lie the reason why the observed GCLFs look so similar within very different parent galaxies.

Orbital encounters between galaxies which are not close enough to cause mergers may still result in mass exchange between them, or in loss of material to intergalactic space. The possibility of ``cluster swapping'' between galaxies has been explored in a series of modelling studies by Muzzio and collaborators (63, 157, 158, 159, 160, 161, 162, 170, 218). For Virgo-like environments, large galaxies in the central regions both gain and lose the most clusters by mass exchange. Other large galaxies more distant from the center are little affected, as are dwarfs with their tiny collision cross sections; and all levels of this swapping process are reduced if a dark-matter potential dominates the cluster. However, regardless of the actual amounts of cluster exchange, this mechanism is cannot explain the large cluster population (high S_N) around M87, as has sometimes been claimed. The clusters being swapped are actually test particles, which can be either halo stars or globular clusters, and the ratio of clusters to stars will be largely immune to exchanges between galaxies since both are affected alike. Exchanges may, however, sometimes be important for diluting metallicity gradients or other such internal features of GCSs.

7.2 Formation Models and Observational Constraints

The theoretical understanding of how globular clusters might have formed overlaps with many issues in modelling both star and galaxy formation and reaches far beyond the limits of this paper; excellent reviews can be found in Fall & Rees (59) and Larson (124, 125, 126), and only brief comments are made here. Fall & Rees provide a useful classification of formation models as primary, secondary, or tertiary according to whether the epoch concerned was before, during, or after the protogalactic collapse stage. The classic primary model by Peebles & Dicke (166) identifies the post-recombination Jeans mass of 10⁵-10⁶ M with globular cluster masses. Difficulties encountered by all pregalactic models are that (a) real GCSs exhibit well-established characteristics that differ between galaxies, especially the metallicity distributions and internal gradients, and (b) observed GCSs are more spatially concentrated systems than the primordial ~ r^-2 dark matter which dominates the galactic potential wells. Rosenblatt et al. (177) circumvented some of these problems in a modified pregalactic model, though certain arbitrary assumptions were adopted to match the data.

In the secondary formation picture, clusters forming in concert with a parent protogalaxy. Fall & Rees (58, 59) developed a model that successfully predicts a mean cluster mass roughly independent of parent galaxy size. The protoglobular clusters are postulated to be cool dense clouds in pressure equilibrium with a surrounding much hotter protohalo gas. In order to imprint the appropriate 10⁶ M characteristic Jeans-type mass, the clouds must cool slowly relative with their free fall time. Detailed concerns about cloud heating and cooling mechanisms in this picture have been discussed (e.g., 58, 144, 153, 165, 187, 228). The basic Fall-Rees model requires that protoclusters have low metallicity in order to avoid too-rapid cooling and fragmentation. Plainly, this requirement presents a problem for understanding how the high-metallicity globulars in giant ellipticals formed in large numbers and with masses quite similar to the metal-poor ones. Kang et al. (121a) present a development of the Fall/Rees scheme in which the protocluster gas is ionized by the shocks from cloud-cloud collisions, and cooling below the critical 10⁴ K is delayed by ambient UV flux from a first generation of massive young stars or an active galactic nucleus.

Within the general Fall-Rees framework, Ashman (7) constructed a model in which most of the mass in dense clouds ends up in dark low-mass objects. He also derived a minimum mass for protoglobular clouds. Another theoretical analysis that predicts a full cluster initial mass spectrum is the hierarchical fragmentation model of Di Fazio (52). He obtained realistic mass histograms; though again, fairly arbitrary starting conditions were chosen to match those of the Milky Way system.

Larson (124, 125, 126, 127) has developed a somewhat different picture in which globular clusters formed within the centers of giant molecular clouds and cloud complexes, as do open clusters today. It is significant that almost no clusters of high density and characteristic masses 10^5-6 M are seen forming today. Even the ``blue populous'' clusters in the Magellanic Clouds have masses an order of magnitude or more smaller, and have a luminosity distribution like that of open clusters, i.e. with no characteristic mass scale (53). The rare ``super star clusters'' found in some starburst-type galaxies such as NGC 1140, 1569, or 1705 (e.g. 6, 70) seem to be the only clear instances of young clusters with globular-like sizes. Larson argues that such massive and compact protoclusters need to be embedded within similarly massive and dense clouds - clouds that must have been more prevalent in the early stages of protogalaxies. In this picture, the clouds themselves may be at least ~ 10⁹ M, large enough to be considered dwarf galaxies if left to evolve in isolation. These dwarfs could combine while still gaseous in larger numbers, and over longer periods of time, to build up larger systems with a higher degree of heavy-element enrichment (148a, 186). In addition, if the embedded globulars form early enough and their primeval host dwarf units are rather similar everywhere, then the resulting clusters should have sizes roughly independent of galaxy type. Two other factors reasonably fit the observations: 1. The ratio of total cloud mass to central cluster mass should be of order 10^-2 (thus explaining why globular clusters make up only this fraction of the visible halo without the need for extensive dynamical destruction later); and 2. the central dense cluster should form stars before the rest of the dwarf, and thereby could enrich the surrounding cloud, which would later disperse into the field (thus producing the metallicity offset between GCS and halo stars). Since the spheroids of galaxies probably emerged at redshifts z ~ 5 or earlier (126), the principal epoch of globular cluster formation may then approach z ~ 10.

The various tertiary scenarios for cluster formation proposed now seem at best to be minor contributors. One is that globulars form in mergers between disk galaxies, in the dense, highly shocked gaseous regions which develop during the collisions. If the cluster formation in shocks were to be very efficient [see Schweizer (185) for a recent presentation of this general approach], a large elliptical could emerge with a higher specific frequency than either of the two progenitor systems. The most telling arguments against this scheme [see van den Bergh (208, 212)] are that clusters that formed during mergers should have similar metallicity and space distributions to the bulk of the resulting galaxy; neither of which conforms to the observations. Another idea (56), relevant especially to the super-large GCSs in central giant ellipticals, is that globular clusters have formed continuously up to the present time by condensing out of the cooling flows usually present in such systems. However, the observed mean metallicities of the halo clusters are [Fe/H] ~ -1, which are 5 to 10 times lower than the X-ray halo gas metallicity (e.g. 152, 156); and the actual total cluster populations N_t seem to be quite uncorrelated with the calculated cooling flow rate. An outstanding example is NGC 6166, which has a huge deduced cooling flow but a subnormal specific frequency (168). Lastly, a third possibility discussed by Zinnecker et al. (227) and Freeman (66) is that the nuclei of nucleated-dwarf galaxies may be the progenitors of globular clusters, after the dwarfs merge to build up bigger galaxies. Many, and probably fatal, counterarguments are summarized by van den Bergh (210).

In summary, the formation epoch of the globular clusters seems to have occupied a fairly well defined place in the sequence of events that bridges primordial gas to fully formed galaxies. The epoch must have occurred early enough for clusters everywhere to form with a similar mass spectrum (GCLF), and through a common process that was insensitive to the size and type of the host galaxy that would eventually emerge. But it must also have happened late enough to allow some environmental pre-enrichment to take place and thus to build up the correlations seen in Figure 8. This epoch may thus have been quite short in the dwarf ellipticals but have lasted a few Gy in giant galaxies. We can speculate that the central supergiants (with their high specific frequencies) made a particularly early start in extraordinarily dense regions, and thus achieved a much higher than average cluster formation efficiency. The starburst galaxies that we see today, spectacular though they may be, can be nothing but pale shadows of what the giant systems like M87 looked like during those early epochs: Our imagination evokes a scene in which thousands of newly created globular clusters, shining at supernova-like luminosities, lie scattered throughout a vast assemblage of gas and dust clouds just beginning their own conversion into stars and the main body of the galaxy.