Annu. Rev. Astron. Astrophys. 1988. 26: 509-560
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4. THE LUMINOSITY FUNCTION OVER ALL HUBBLE TYPES

4.1. Cluster Galaxies

As mentioned in Section 1.3, Zwicky (1942) postulated the exponential nature of the LF. Although he had derived the result as a theoretical necessity of thermodynamics rather than having found it observationally, he later claimed observational confirmation from cluster data (e.g. Zwicky 1957).

The completion of the Palomar Observatory Sky Survey in the late 1950s initiated a host of cluster LF studies with Abell and his collaborators as the driving force (for reviews, see Abell 1962, 1972, 1975). The basic hypothesis was that clusters of galaxies obey a universal LF, which, in its integrated form Phi(M), can be modeled by two straight lines intersecting at a characteristic magnitude M* (sometimes referred to as the "knee"). Although Abell's shape of the LF is no longer used, it is clearly a first approximation to the Schechter (1976) function in its asymptotic behavior at both the bright and faint ends.

The modern, post-Abell cluster research began with Oemler (1974) who used large-scale plates with good morphological resolution and discussed a model-free LF. His work was followed up by many authors; a partial list in Table 3 sets out those rich clusters whose LFs have been determined since 1974. The quality and depth of these investigations vary widely. Some authors give only a crude LF for a distant cluster (e.g. Iannicola et al. 1987), while others achieve great morpological detail and/or drive for very faint (surface) magnitudes [e.g. Sandage et al. (1985) in the nearby Virgo cluster]. Most cluster studies have adopted the Schechter fitting law. An exception is the "Oxford group" (Austin & Peach 1974, Austin et al. 1975, Godwin & Peach 1977, Bucknell et al. 1979, Carter & Godwin 1979, Carter 1980, Godwin & Peach 1982), who have used model-free as well as Abell fits for their LFs.

Table 3. Rich clusters of galaxies with LFs determined since 1974
 

Cluster Investigators Cluster Investigators

Virgo Bucknell et al. 1979 A1413 Dressler 1978
  Kraan-Korteweg 1981 A1553 Bucknell et al. 1979
  Sandage et al. 1985   Yamagata et al. 1984
Fornax Jones & Jones 1980 A1656 Oemler 1974
  Caldwell 1987 (Coma) Godwin & Peach 1977
A85 Murphey 1984   Thomson & Gregory 1980
A98 Dressler 1978   Beckkman 1982
A154 Dressler 1978   Lugger 1986
A168 Dressler 1978 A1904 Oemler 1974
  Oegerle et al. 1986 A1930 Austin et al. 1975
A194 Oemler 1974 A1940 Dressler 1978
A274 Dressler 1978 A2029 Dressler 1978
A400 Oemler 1974 A2065 Bucknell et al. 1979
A401 Dressler 1978 (Corona Borealis)
A426 Bucknell et al. 1979 A2147 Bucknell et al. 1979
(Perseus) Egikyan et al. 1985   Lugger 1986
A539 Oemler 1974 A2151 Oemler 1974
A569 Lugger 1986 (Hercules) Bucknell et al. 1979
A665 Oemler 1974   Lugger 1986
  Dressler 1978 A2175 Oegerle et al. 1987
A744 Kurtz et al. 1985 A2197 Oemler 1974
A777 Iannicola et al. 1987   Lugger 1986
A779 Lugger 1986 A2199 Oemler 1974
A910 Iannicola et al. 1987   Bucknell et al. 1979
A999 Chapman et al. 1987   Lugger 1986
A1016 Chapman et al. 1987 A2218 Dressler 1978
A1146 Carter & Godwin 1979 A2256 Dressler 1978
A1228 Oemler 1974   Oegerle et al. 1987
A1314 Oemler 1974 A2384 Oegerle et al. 1987
A1367 Oemler 1974 A2634 Lugger 1986
  Godwin & Peach 1982 A2670 Oemler 1974
  Lugger 1986   Dressler 1978
A1377 Bucknell et al. 1979   Bucknell et al. 1979
(Ursa Major I)   0004.8-3450 Carter 1980
A1413 Austin & Peach 1974 CA 0340-538 Quintana & Havlen 1979
  Oemler 1974 Zw 1545.1+2104 Oemler 1974

A discussion of the individual cluster LFs is not given nor is a detailed intercomparison of the results. This has already been provided by Dressler (1984) in his review on the evolution of cluster galaxies. Instead, we concentrate on the investigations that covered several clusters in a search for systematic trends.

First indications for the nonuniformity of the cluster LFs are due to Oemler (1974), who classified clusters recording to galaxy content as being "spiral-rich," "spiral-poor," and "cD." The mean LFs for these three classes, which correlate with the kinematic properties of the clusters, appeared to be marginally, but significantly, different. However, Schechter (1976) fitted his expression to Oemler's clusters and, following Abell, found a rather high degree of uniformity with respect to the parameters alpha and M*. This came as a surprise because significant differences could be expected in view of the cluster types presumably being in different evolutionary stages. If so the result had to be interpreted in the sense that cluster evolution had little effect on the LF. From the near-agreement of the parameters alpha ~ -1.25 and MBT* ~ -21.0 for field and cluster galaxies, Schechter further concluded that the shape of the "general" LF is universal. Yet the processes of tidal stripping and dynamical friction are bound to have some bearing on the LF (cf. Dressler 1984). Dressler (1978) consequently searched for deviations from the first-order universality in 12 rich clusters and did indeed find significant differences. Several clusters showed an unusually flat faint end (alpha ~ -1). Furthermore, the data for cD clusters supported Oemler's (1974) observations of a steeper bright end than for non-cD clusters, which was also suggested by Bucknell et al. (1979) and many others (as reviewed by Dressler 1984). The latter result could be interpreted as an evolutionary effect where the central cD galaxy formed at the expense of the next brightest cluster galaxies (Miller 1983, Malumuth & Richstone 1984, Dressler 1984). The question of whether the first-ranked cluster galaxy is within the statistics of the cluster LF or whether it is a singular object has been long debated and may now have been settled in favor of the latter possibility (cf. Dressler 1984, and references therein) for cD clusters. The jury is still out, however, for non-cD clusters [see Sandage (1988) in this volume for a review of the continuing debate].

Besides the cD effect, it has remained unclear which evolutionary processes are actually responsible for the significant variance of alpha and M* among Dressler's (1978) clusters, although theoretical explanations have been suggested (Dressler 1984, Kashlinsky 1987). Lugger (1986) could not correlate the LFs of nine Abell clusters with the cluster morphology. Merritt (1984, 1985) argued on theoretical grounds that the cluster LF was determined very early during the violent relaxation phase, and that correspondingly no dependence of the LF on the present-day evolutionary stage of the clusters should be expected.

An intriguing explanation of the variance of alpha and M* among clusters has been offered by Thompson & Gregory (1980). Using large-scale plates of the Coma cluster, they established the LFs of E, S0, and S+Im galaxies separately, from which they suggested that the LFs, which are clearly different for different types, remained the same in every cluster, but that the total LF as the sum over different types varies according to the types mixture. By synthesizing clusters of different type composition they were able to reproduce the variance of alpha and M* observed by Dressler (1978). This hypothesis requires additional tests from many clusters with detailed morphological information that is presently only available for Virgo (Sandage et al. 1985) and Fornax (Sandage & Ferguson 1988). It should be noted that modern, sophisticated LF studies like those of Lugger (1986), Oegerle et al. (1986, 1987), and others are unsuitable for this purpose because they are based on small-scale plates where morphological binning gives unsatisfactory results. Thompson & Gregory's hypothesis is discussed and supported further in Section 6.

The limited data on the variance of the cluster LF can be explained satisfactorily as the effect of the first-ranked galaxies and the different type mixture. However, if the influence of the type mixture is denied, the remaining differences of the LF probably must be explained by evolution. Since this is not our preferred solution now, the reader is referred to the earlier review by Dressler (1984) for the consequences of this possibility.

The Schechter parameters are not listed in Table 2 because the results from individual investigations are too inhomogeneous. Different magnitude systems are used, as well as different fitting procedures. In some cases alpha and M* were solved for, whereas in others one of the Schechter parameters was fixed a priori. A set of parameters that could be compared was provided by Lugger (1986) for nine clusters. Fitting simultaneously for alpha, M*, and the normalization, she found for alpha a mean value of alpha = -1.24 ± 0.22 with the brightest galaxy excluded, or alpha = -1.47 ± 0.19 if it was included. As noted earlier by Schechter (1976), one obtains better fits if first-ranked galaxies (or cD clusters) are excluded. Therefore, Lugger's first-mentioned result is to be preferred. It is in good agreement with Schechter's canonical average of alpha = -1.25, which still describes the faint-end slope of cluster LFs fairly well. Yet, as mentioned before, there is no single value of alpha that applies to all clusters (Dressler 1978). Moreover, field galaxies deviate significantly from this average, having an alpha closer to -1.0 (cf. Section 4.2).

It should be stressed that what most cluster studies refer to as the "faint end" lies still at fairly high luminosities, i.e. MBT ~ -20 or -18 at best. The exponential for the faint end defined in this way is usually a mere extrapolation of the Schechter function fitted to bright galaxies. It was therefore quite surprising that the Virgo cluster LF, which was measured to a much fainter completeness limit of MBT ~ -14, has confirmed a faint-end slope of alpha ~ -1.25 (SBT). A somewhat shallower slope of alpha ~ -1.14 has been found in a preliminary study of the Fornax cluster (Caldwell 1987). This study, however, used small-plate scale material and may suffer incompleteness.

The range of M* for different clusters is given by Lugger (1986; see also Dressler 1978) as MBT* ~ -21.0 ± 0.7 (standard deviation) if cDs are excluded, or MBT* ~ -21.8 ± 0.6 if cDs are included. The former value is also recovered in the field (Section 4.2). The quoted magnitudes are reduced to H0 = 50 throughout this paper.

In addition to the variations in alpha and M*, there are clusters that seem to deny any choice of Schechter parameters (Dressler 1978). Although the significance of this effect is marginal because of the limited size of cluster samples, it is not surprising. As mentioned before, different Hubble types have quite different LFs. Depending on the type mixture, they enter into the total LF with different weight. It would be sheer coincidence if the latter could always be represented by a simple analytical formula. The expected mini-structure of the LF over all types is discussed further in section 5 (cf. Figure 1).

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