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

4. SPECIFIC FREQUENCY AND SPECIFIC LUMINOSITY

The simplest statistic describing a GCS is the total number of clusters N_t. Hanes (77) and Harris & Racine (108) demonstrated to first order that for a given type of galaxy, N_t scaled directly with galaxy luminosity. The specific frequency (111) is defined as the cluster population normalized to M_V^T = -15,

S_N N_t · 10^{-0.4 (M_VT + 15)} . 3.

Table 1 lists values for S_N. These have all been recalculated from the published cluster counts as described in Section 2, and make up a new and homogeneous catalog of specific frequencies. The quoted internal uncertainties on S_N include the given errors in both the raw counts N_obs and the ± 0.2-mag uncertainty in (GCLF), but not the potential errors in distance modulus or limiting magnitude. As a rule of thumb, the specific frequency for any particular galaxy should be regarded as valid to roughly a factor of two. Except for the few best-studied systems, the combined uncertainties in the cluster counts, galaxy luminosities, and the extrapolations necessary to estimate N_t over all magnitudes and radii, prevent any higher precision.

In spite of its approximate nature, very real differences in S_N from galaxy to galaxy do exist. The prototype high-S_N system M87 has repeatedly been shown to have two to three times more clusters per unit luminosity than do the other Virgo ellipticals; these, in turn, are quite a bit more cluster-rich than most of the field ellipticals. No obvious correlations of S_N with parameters such as M_V^T (see Figure 4) or galaxy ellipticity have been found. However, significant mean differences appear with environment, as is summarized in Table 3. By and large, the ellipticals in smaller groups and sparser environments contain approximately two times fewer clusters for their size than those in rich environments (Virgo, Fornax). The few dwarf ellipticals in the list have specific frequencies that are no different in the mean from the giants, indicating that they were at least as efficient per unit mass at forming clusters as were the bigger galaxies. An outstanding and still quite puzzling exception is the Fornax dwarf in the Local Group (68); its S_N 70 is by far the highest in the list, and too anomalous to be easily ascribed just to small-number statistics.

Figure 4. Specific frequency SN of globular cluster systems, as listed in Table 1. In the upper panel, SN is plotted versus the luminosity of the parent galaxy; E/S0 types are denoted by the filled symbols, and spiral/irregular types by the crosses. The five giant ellipticals at the centers of rich clusters (Virgo, Fornax, Hydra, Coma, A2199) are denoted by the circled dots. In the lower panel, SN is plotted against morphological type. Here ES and ER refer to ellipticals in sparse and rich clusters, and the last bin (cD) again denotes the central-giant ellipticals.

For disk galaxies, S_N is harder to interpret directly since M_V^T includes the disk and Population I light, which is generically less related to the halo clusters. For these, an adjusted quantity S_N^* is usually used (cf 91, 214), which is the ratio of N_t to only the spheroid light specifically excluding the disk. For Sa/Sb galaxies, reasonably accurate estimates of S_N^* can be made since the spheroid makes up a large fraction of the total light (49a). For Sc/Irr types, however, the fraction of light belonging to the old spheroid (if any!) is too small and uncertain to permit any sensible conversion from S_N to S_N^*, at least in optical bandpasses. It is nevertheless remarkable that, even with vanishingly small amounts of the oldest stellar populations, these late-type galaxies have still managed to produce very old clusters in noticeable numbers (184a). This property of the very late-type galaxies such as M33 and the LMC suggests, as does much other evidence to be discussed in the following sections, that the GCS and halo formation processes were at least partly decoupled.

Table 3. Mean specific frequencies of various types of galaxies

Galaxy Type	<SN>	N	Comment
Sc/Irr	0.5 ± 0.2	4
Sa/Sb	1.2 ± 0.2	9	<SN*> 2.1 ± 0.4
E/S0 (Small Groups)	2.6 ± 0.5	13
E/S0 (Virgo, Fornax)	5.4 ± 0.6	15	excludes M87, N1399
dE	4.8 ± 1.0	4	excludes Fornax, M32

The specific frequency was introduced as a way to remove the first-order proportionality of N_t to galaxy size and thus to compare systems more easily. But it turned out also to be an interesting discriminator for ideas by Toomre (196) and others about the formation of E galaxies by mergers. As a ratio of clusters to field halo stars, S_N is relatively invariant to interactions between galaxies, because both stars and clusters alike behave essentially as massless test particles. Thus in a collision between two galaxies, S_N (or S_N^* for disk systems) for the merged product will be a luminosity-weighted average between the two (or somewhat smaller if the remnant is stripped of gas and the age-dimmed disk light eventually joins the spheroid). The typical S_N^* for Sa/Sb galaxies (see Table 3) is similar to S_N for E galaxies in smaller groups, but significantly smaller than in the Virgo Es or the dwarfs. On this basis, Harris (91) suggested that spirals were fundamentally less efficient at forming globular clusters than were the Virgo ellipticals, and that the lower specific frequencies for field and small-group ellipticals might be explained by their higher expected merger rates; that is, many of the large ellipticals with the lowest S_N values might indeed be remnants of long-past disk mergers.

The high S_N values found in the rich Virgo and Fornax ellipticals have several implications repeatedly emphasized by van den Bergh (202, 203, 204, 205, 207, 208, 212). It appears highly improbable that galaxies, with low specific frequencies(such as present-day disk galaxies), could have merged to form these cluster-rich ellipticals. Finally, it is especially difficult to understand the existence of the huge cluster populations around the central giant ellipticals M87, NGC 1399, NGC 3311, and (possibly) NGC 4874 in any way involving mergers of normal galaxies that had already fully formed. These rare high-S_N supergiant systems all sit very close to the dynamical centers of rich clusters (99), and almost certainly were unusual from the start. However, whether they represent just the upper end of a continuum of S_N values (Figure 4) or if they are truly distinct remains unclear. In M87, the characteristics of the globular clusters themselves (colors, metallicities, space distribution, and luminosity distribution) are no different from those within the more normal big ellipticals. In other words, the only distinguishing characteristic of clusters in the high-S_N systems appears to be their sheer numbers.

With the salient exceptions noted above, there remains a remarkable first-order uniformity from galaxy to galaxy in the number of globular clusters per unit halo (spheroidal-component) luminosity over a vast range of sizes. An average S_N 4 corresponds to one cluster per M_V(halo) = -13.5, or in terms of mass, of order 1 M in globular clusters per ~ 10³ M in field-star mass. This ratio is very close to that estimated from formation efficiency arguments (e.g. 126; see Section 7).

Though useful for rough comparisons, S_N is unavoidably an imprecise parameter. A potentially better index of the contribution of the GCS to the total galaxy population is one that I will define here as the specific luminosity S_L,

S_L 100 · L_cl / L_T = 10^{0.4 (M_VT - M_Vcl + 5)} 4.

Here, L_cl denotes the summed visual luminosity of all the globular clusters in the galaxy, and L_T the luminosity of the galaxy itself; M_V^cl and M_V^T are the corresponding integrated magnitudes. (S_L can of course be defined equally well in any bandpass; V is adopted here only for convenience.) The ratio S_L is simply the percentage of the total galaxy light contributed by the globular clusters. Although it requires more complete photometric information to measure than does S_N, note that S_L has several advantages, listed below.

1.	As a luminosity ratio, SL is strictly independent of the assumed distance.
2.	It is utterly insensitive to the details of the faint half of the cluster luminosity function, thus is easier than SN to calculate accurately for more distant galaxies. (2)
3.	SL can readily be defined locally as well as globally within the galaxy; for example, the spatial variation in cluster population could be traced out by calculating SL(r) within radial annuli around the galaxy center.

Wagner et al. (219) have used a similar quantity to trace the radial structure of the GCS in NGC 1399. Probably the best galaxy to act as a reference standard for S_L is M49; with both a well sampled GCLF and full radial profile available, it is the most well understood ``normal'' GCS among the large ellipticals. Its S_L(r) profile is plotted in Figure 5: Globally, S_L = 2 r S_L(r) dr equals 1%, but locally it increases drastically from center to outer halo. This effect is a manifestation of the more extended spatial structure of the GCS compared with the halo light, discussed again in the next section. Because the clusters are bluer than the integrated galaxy light (Section 6), S_L will vary somewhat with adopted wavelength. For example, if S_L equals 1% in B, then it will be roughly 1.2% in U, 0.8% in V, and 0.6% in I.

Figure 5. The local specific luminosity profile for the giant elliptical M49, in B light. Here SL(r) is the ratio (in units of percentage) of the total light from the globular clusters to the light from the halo, at projected radius r from the galactic center. The solid dots are six radial points sampled using CCD photometry of the clusters (103). SL(r) increases with radius because the clusters follow a spatially more extended distribution than the halo, thus are relatively more common at larger radii. The solid line is derived from the difference between the r1/4 profile curves describing the halo and the GCS (94); it is used here to show schematically the radial increase of SL(r). Integrating this profile over all radii gives the global specific luminosity SL, which for M49 is close to 1% in B.