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5.2. Clusters vs. field

Comparisons of the luminosity function in high and low-density regions are of great interest, both as a test of biased galaxy formation (see Sect. 7.2), and as a measure of environmental influences on galaxy evolution. One of the great difficulties here is in establishing the connection between observations and theory. On the theoretical side, there are many different variants of biasing (e.g. Rees 1985). While statistical bias in the distribution of fluctuation peaks (Kaiser 1984; Bardeen et al. 1986) may plausibly bias the spatial distribution of dark-matter halos, the connection between that distribution and the LF is not well understood (in large part because it involves star formation). On the observational side, the techniques used to probe the LF in clusters and outside of clusters are quite different, leading to different potential biases in the results (see above). Finally, there is the difficulty of relating the variations with position or local density within a virialized or partly virialized cluster (e.g. the morphology-density relation) to the expectations from linear theories of structure formation.

With these difficulties in mind, we shall briefly summarize the observations from the richest (but not necessarily the densest) environments, to the poorest, concentrating on the dE contribution.

The richest region that has been surveyed to faint limiting magnitudes is the Coma cluster. Thompson and Gregory (1993) catalogued dwarf galaxies in the Coma cluster, identifying faint cluster members on the basis of photographic b - r colors. They measured a faint-end slope alpha = -1.4, and a dwarf/giant ratio slightly lower than that of Virgo, when converted to the same luminosity limit. There is a pronounced lack of low surface-brightness dE galaxies (classified dSph by Thompson & Gregory) within the central 0.3 degrees. They interpret this as the result of tidal disruption in the core of the cluster of galaxies with velocity dispersions less than 35 km s-1. While not explicitly shown, the lack of LSB dwarfs in the core of the cluster presumably leads to a significantly flatter luminosity function. Outside the core of the cluster, the early-type dwarfs follow the spatial distribution of E and S0 galaxies.

The Virgo cluster was extensively studied by Sandage et al. (1985b). They found an overall faint-end slope of alpha = -1.30 and a dE slope alpha = -1.35 to an absolute magnitude of -11.7. Comparison to the Coma cluster survey is a bit uncertain as the morphological classifications are different and the galaxies are obviously not as well resolved at the Coma distance. In contrast to Coma, there is no strong deficit in the number of faint dE's near the center of Virgo, but there is perhaps a slight decline (see Ferguson and Sandage 1991). Bothun et al. (1991) re-examined the Virgo cluster LF, adding in a sample of 24 extremely low surface-brightness galaxies. They examine only the subset of galaxies for which there is quantitative surface photometry, and apply a rather indirect method of calculating the luminosity function, assuming all galaxies have exponential profiles and deriving the distribution of scale-lengths and central surface-brightnesses in a region of parameter space essentially free from selection effects. Modeling these distributions as power laws, they arrive at an LF slope alpha = -1.6. However, the difference between this result and that of Sandage et al. (1985b) is largely a result of the analysis technique, rather than the inclusion of the additional LSB galaxies.

Ferguson and Sandage (1991) compiled luminosity functions for seven nearby groups (including the previously-published Fornax and Virgo clusters), and found faint end slopes -1.6 < alpha < -1.3, and turnover magnitudes -23.2 < MB* < -21.2. A Schechter function was typically not a very good fit to the data, but the systematic departures from the data were different for different clusters and difficult to separate from uncertainties in the magnitude estimates and completeness. The most striking result from the survey is that the dwarf/giant ratio for early-type (E, S0, dE, and dS0) galaxies varies by a factor of ~ 5 from the richest to the poorest groups (see Sect. 6.2). This result has been extended to still poorer groups by Vader and Sandage (1991).

Surveys of the field typically find flatter luminosity functions, with alpha approx -1 (Efstathiou et al. 1988; Loveday et al. 1992; Schmidt and Boller 1994), and similar MB*. Ferguson (1992a) and Lacey et al. (1993) considered the effect of isophotal selection on the field-galaxy luminosity function, and concluded that surface-brightness selection alone could not account for the difference in the faint end slope between the Virgo cluster (Sandage et al. 1985b) and the field (Loveday et al. 1992). The Loveday et al. survey was deep enough that relatively high surface-brightness dE galaxies with -18 < MB < -16 would have been detected. However, if the dwarfs in the general field follow the Virgo cluster dE luminosity function, it is quite possible that the total luminosity function turns up below the limits of the Loveday et al. survey. The loose groups observed by Ferguson and Sandage (1991) show such an upturn.

Although they did not explicitly compute a luminosity function, the sample of Eder et al. (1989) provides some indication that the LF in the field, even when probed to fainter surface-brightness limits, is still dominated by HI rich galaxies, with a luminosity function similar to that of Virgo. 122 galaxies from a well-defined diameter-limited sample were observed at 21-cm and only 28 (22%) were not detected. For those detected, the HI properties are similar to the Virgo cluster irregular galaxy sample. However, in the Virgo cluster dE's outnumber Sd+Im galaxies by at least factor of 3 (the exact value depends on luminosity). The flat LF of the field therefore reflects the flat LF of the Sd+Im types. Jerjen et al. (1992) arrived at a similar conclusion. For a sample of groups within 10 Mpc, they found that the faint population is dominated by irregulars (only 11% being dE+dS0) types, and the resulting LF to MB = -15 was flat with alpha = -0.98.

The Local Group provides information on the very faint tail of the field-galaxy luminosity function. Van den Bergh (1992) finds alpha = -1.1, although the number of galaxies is probably too small to argue that this is significantly different from the steeper slope found in clusters. Ferguson (1992a) finds a much steeper slope alpha = -1.9 for the M81 group, although once again the statistics are very poor.

Compact groups in principle provide an interesting comparison to the above studies. Such groups can have densities as high as the centers of rich clusters, but have velocity dispersions and total numbers of galaxies consistent with ``loose groups.'' De Oliveira & Hickson (1991) found a luminosity function significantly different from that of field, loose-group, and cluster galaxies, with alpha approx -0.2. More recently, Ribeiro et al. (1994) found alpha = -0.82 ± 0.09 from a deeper survey of a large number of groups. The Ribeiro et al. luminosity function appears to be consistent with that seen in the field and in loose groups, but significantly flatter than that of the Virgo cluster. The result is in accord with the general variation of dwarf/giant ratio with richness (as opposed to density), but is rendered somewhat uncertain by the rather bright isophotal threshold of the Ribeiro et al. survey.

While there are still large uncertainties in the luminosity function and its environmental dependence at absolute magnitudes fainter than MB = -16, we believe that surveys to date have taught us the following:

1. Selection effects are important, but not dominant. Dwarf galaxies are probably not a significant contributor to the mass-density of the universe.
2. The overall luminosity function changes with environment in a way that is consistent with a simple change in the relative proportions of dE and irregular galaxies.
3. The dE luminosity function is largely independent of environment, with the exception of a possible depletion at the faint end in the centers of rich clusters (Thompson and Gregory 1993). The slope in the range -16 < MB < -13 is alpha approx -1.3. The observations for clusters and groups, even when selection effects are taken into account, do not support slopes as flat as alpha = -1 or as steep as alpha = -1.8.
4. LF variations are not consistent with simple biasing schemes. The relative abundance of low-luminosity galaxies is, if anything, lower in the field than in clusters.

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