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

3. LUMINOSITY FUNCTIONS

In a given galaxy, the number of globular clusters per unit magnitude interval, (m) (or (M) in absolute magnitude), is the luminosity function or GCLF of the cluster system. The GCLF is the visible manifestation of the cluster mass spectrum, and must result from both the initial mass function of cluster formation and its subsequent long-term erosion by various dynamical processes (see Section 7).

The GCLF data now available for several galaxies show that (m) can be simply and accurately described by a Gaussian distribution,

(m) dm = A exp[-(m - m₀)² / 2 ²] dm 1.

where A is the simple normalization factor representing the total population N_t, m₀ is the mean or peak (turnover) magnitude of the distribution, and is the dispersion. This log-normal function is introduced purely as an empirical match to the real GCLF and as a convenient way to compare different galaxies. It is, however, an extremely successful match to the actual data; no significant departures from a strictly Gaussian form have been found even in samples corresponding to N_t 2000 clusters (see 18, 71, 78, 88, 103, 108, 206). Whatever the true underlying form of (M) is, it must closely resemble Equation 1.

To first order, GCLFs in different galaxies can then be compared through the two parameters M₀ (the absolute magnitude of the turnover) and . Estimates of these for all galaxies to date in which photometric surveys have reached clearly to the turnover point are summarized in Table 2 and Figure 1. In Table 2, <M_V> is the turnover level in absolute visual magnitude, (M₀) is the estimated GCLF dispersion, and <[Fe/H]> is the mean metallicity of the clusters. For the smaller Local Group members, M_V⁰ is simply the mean over all clusters and (M_V) is the rms dispersion around the mean; for the larger galaxies, both M_V⁰ and are obtained by fitting Equation 1 to the observed GCLF (see 103).

Table 2. Mean cluster luminosities and metallicities

Galaxy	<MV>a	(MV)	<[Fe/H]>	Sources
Fornax dE	-7.06 ± 0.42	0.95	-1.85 ± 0.14	21, 22, 226a
N147	-5.99 ± 0.92	1.8:	-2.05 ± 0.4	45, 108
N185	-6.49 ± 0.71	1.42	-1.65 ± 0.25	45, 108
N205	-7.27 ± 0.27	0.72	-1.45 ± 0.10	45, 108
LMC	-7.51 ± 0.32	0.96	-1.66 ± 0.1:	72a, 108, 201a
M33	-7.01 ± 0.17	1.17	-1.40 ± 0.2	21, 37
Milky Way	-7.36 ± 0.17	1.28	-1.35 ± 0.05	5, 103, 225
M31	-7.4 ± 0.5:	1.2:	-1.21 ± 0.05	21, 54
N1399	-7.3 ± 0.3	1.4	-0.85 ± 0.1	18, 71
N3031			-1.46 ± 0.31	21
N3377	-7.05 ± 0.6	1.2:		100
N3379	-7.59 ± 0.4	1.2:		100, 169
N4365	-6.95 ± 0.18	1.58		103
N4406	-6.9 ± 1:	1::		38, 98
N4472	-7.40 ± 0.11	1.50	-0.8 ± 0.3	103, 152
N4486	-7.32 ± 0.13	1.70	-1.0 ± 0.2	21, 39, 103, 152, 217
N4649	-7.45 ± 0.14	1.29	-1.1 ± 0.2	40, 103
N5128		1.35	-0.84 ± 0.1	85, 88

^a Observations in B for the giant E galaxies have been converted to V assuming (B - V) = 0.8. Distance moduli are as in Table 1.

For the more distant galaxies in Table 2, the turnover levels are faint (B₀ 24.7 at Virgo) and require limiting CCD photometry to establish the levels unambiguously. For the Local Group members, different but equally challenging problems have plagued this subject. In the dwarf ellipticals, the cluster samples are complete but unavoidably tiny. In the LMC and SMC, the samples are small but may also still be slightly biassed toward brighter objects, while in M33, both faint-end incompleteness and accidental inclusion of red intermediate-age clusters are likely problems. With M31, we encounter a uniquely frustrating morass of observational and interpretive difficulties involving combinations of sample contamination, large and uncertain reddening corrections for the many clusters projected on the M31 disk, and a highly nonuniform photometric database. These have eventually led to unquantified biasses in all attempts to extract the true GCLF: compare the discussions by Racine & Shara (174), Crampton et al. (41), van den Bergh (206), Sharov & Lyutyi (90), Racine (171a), as well as the error analyses by Battistini et al. (8), Elson & Walterbos (54), and van den Bergh (211). Nevertheless, new radial velocity measurements and precise multicolor photometric indices from CCD imaging are finally putting a complete, accurate, and nearly uncontaminated database within reach for this important galaxy.

Figure 1 demonstrates that over a broad range of systems, the turnover luminosity M_V⁰ is evidently nearly independent of parent galaxy size, environment, or (to within ± 0.3 mag), reasonable changes in the adopted distance scale H₀ (see 96). An unweighted mean for all the galaxies in Table 2 yields < M_V⁰ > = -7.1 with an rms scatter of ± 0.43 mag. For the 10 largest galaxies, we obtain < M_V⁰ > = -7.27 with a ± 0.23-magnitude scatter. For M / L_V 2, this luminosity corresponds to a characteristic mass 1.4 x 10⁵ M. The intrinsic dispersion may be systematically a bit larger for the giant ellipticals (for which the best functional fits are reached consistently at 1.4) than for the other systems (for which 1.2 seems preferable), though the difference has been hard to establish and is not yet definitive. Figure 2 shows some specific illustrations. In essence, globular cluster luminosities follow basically similar distributions everywhere we have looked: The data in Table 2 now represent every type of host galaxy and span almost the full known range of galaxy size (10 magnitudes in M_V^T). This uniformity is all the more remarkable when we consider that Figure 1 may represent at least three distinguishably different processes of galaxy formation (dwarf ellipticals, giant ellipticals, and the spheroids of disk galaxies) as well as a large range of dynamical erosion mechanisms.

Figure 1. Mean globular cluster luminosity as a function of parent galaxy size, with data from Table 2. MV0 is the average or turnover luminosity of the clusters in each galaxy. Smaller symbol sizes denote more uncertain mean values, as indicated by the error bars.

Historically, strong interest in GCLFs has accompanied their use as distance indicators. Hanes (79, 80) reviewd their early development as standard candles, and thorough recent discussions can be found in Hanes & Whittaker (84), Harris (98), and Harris et al. (103). In brief, for a distant galaxy one needs to obtain photometry of the globular clusters to a level approaching the turnover. Various numerical techniques can then be used to fit a model like Equation 1 to the observed GCLF and thus to deduce m₀ and . Putting in the true luminosity M₀ then immediately gives the distance modulus of the system. A practical problem is often that the data do not reach clearly past the turnover, in which case the solutions for m₀ and become correlated. In such cases one must assume a standard value for and solve only for m₀. For any reasonable choice of distance scale H₀, the first-order universality of M₀ is already clear [see Figure 1 above and Harris (96)]. However, it is not yet known at the second-order ( 0.3-mag) level just how much M₀ may vary randomly, or change systematically with the type of the parent galaxy (in particular, from the large disk galaxies to the large ellipticals). Settling this issue is crucial to calibrating GCLFs self-consistently.

With the best current ground-based imaging capabilities (subarcsecond seeing and V_lim 25), the fiducial level M₀ will be directly detectable for galaxies as distant as ~ 30 Mpc; with space-based imaging at 0."1 resolution, this limit can be extended to ~ 100 Mpc, i.e. about four magnitudes more distant than the Virgo Cluster. Some suggestions (16, 84) are that the very top end of the GCLF may itself be used as a more approximate standard candle to reach even further out: In a typical giant elliptical galaxy, the well-filled bright end of the (m) curve (see the right panel of Figure 2) falls off so steeply that the magnitude level of the brightest clusters is insensitive to the total population N_t. Preliminary consistency tests of this idea have been made at the ± 0.5-magnitude level for the distant GCSs in Coma (95, 194) and NGC 6166 (168). Because the brightest globulars lie at M_V -11, the ultimate limits of GCS detectability may approach ~ 500 Mpc with large telescopes from space.

Figure 2. Two different representations of GCLFs. (Left) The number of globular clusters per 0.4-magnitude bin is plotted for 118 clusters in the Milky Way with known luminosities (lower panel), and for a total of 48 clusters in 7 small Local Group galaxies that have been rather arbitrarily lumped together (upper panel). In the upper graph, the shaded region shows the sum of the four dE systems NGC 147, 185, 205, and Fornax, while the unshaded region adds the LMC, SMC, and M33. (Right) GCLFs for three giant E galaxies. Note that (log ) is now plotted, and the scale is arbitrary since the three curves have been shifted vertically for clarity.

Note that m₁ represents the magnitude level at which the clusters contribute the most total light as the result of their relative numbers and individual luminosities. ⁽¹⁾ The (m) curve can be fitted to the data in just the same way as the number distribution (m), but with the advantages that its turnover is quite a bit brighter and that it is insensitive to the details of the faint end of the cluster distribution because the clusters there contribute little total light. Numerical tests on the Virgo galaxy sample show that for the same data, both m₀ and m₁ can be estimated with similar precision as long as the bright end of the LF is well populated. Thus for large E galaxies, more distant than Virgo where the GCLF turnover may be too faint to reach clearly, it can be more convenient to work with the LWLF instead. Figure 3 illustrates the two functions.

Figure 3. Luminosity distributions for the globular cluster systems around the giant Virgo ellipticals, from Harris et al. (103). The four galaxies NGC 4365, 4472, 4486, and 4649 are directly summed to form a single composite distribution. In the left panel, the GCLF is plotted as the number of globular clusters found per 0.2-magnitude interval. In the right panel, the LWLF (luminosity-weighted luminosity function, as defined in the text) is plotted. Here Beq (= constant - 2.5 log ; see Section 3) is the integrated magnitude of all clusters in each 0.2-magnitude bin. For each function, a best-fitting Gaussian curve with dispersion = 1.4 magnitudes is drawn through the points. Note that the LWLF peaks at a point almost exactly 2 magnitudes brighter than the GCLF does.