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In Figure 2, I present four LFs from my earlier work [38 - 41]. Other published LFs (in addition to the ones already listed) are presented in references 12, 19, 20, 33, 34, and 47.

Figure 2

Figure 2. The R-band luminosity functions of A 262 (z = 0.016), Coma (z = 0.023), A 1795 (z = 0.063), and A 665 (z = 0.182). In converting apparent to absolute magnitudes, I assume H0 = 75 km s-1 Mpc-1 and Omega0 = 1.

For A 262 (a poor cluster at z = 0.016), signs of a turn-up at the faint-end (MR < - 14) are seen, but the LF is not well-constrained at bright magnitudes here because the galaxy density is too low. In Coma, a richer cluster at approximately the same distance as A 262, a similar turn-up at the faint-end is seen, but a far more shallow LF is observed at brighter magnitudes. In both cases, the statistics are poor at the faint-end because the field-to-field variance of the background is so large - nevertheless, in both cases the turn-up at the faint-end is statistically significant and the LF is not fit by either a power-law or a Schechter function [30] over the entire range -20 < MR < - 11). For the more distant clusters, A 1795 and A 665, a flatter LF is seen but the very faint magnitudes at which the turn-up is seen in A 262 and Coma are not probed in these clusters.

The approximate similarity of the form of the LFs in the four clusters in Figure 2 suggests that it might be productive to combine the individual cluster LFs in order to improve the statistics. This is done in Figure 3. When performing this calculation, I found that each individual cluster LF is consistent with the composite LF to a high degree of statistical significance. This is suggestive that the LF of galaxies in the cores of rich clusters is remarkably constant. Also, the Fornax LF of Ferguson [10] is also consistent with this composite function at a high level of significance down to MB = - 13 (see ref. 42 - this is particularly intriguing because the errors in the Fornax sample at the faint end come only from counting statistics and are quite small).

Figure 3

Figure 3. Composite R-band luminosity function of rich clusters (derived using the LFs from Coma, A 2199, A 1795, A 1146, A 665, and A 963). This represents the weighted average of the individual LFs, normalized to a projected galaxy density corresponding to that of a typical Abell richness 2 cluster. Local values of alpha(MR) derived by measuring the slope between MR - 1 and MR + 1, and their uncertainties, are given too. Slopes corresponding to alpha = - 1 and alpha = - 2 are also shown.

The composite LF in Figure 3 has much better statistics than the LFs shown in Figure 2, now that a sample of clusters have been combined. The composite LF shows the following properties:
1) the LF shows significant curvature at both the bright and faint-ends. Neither a power-law nor Schechter function [30] provides a good fit to the data. A similar conclusion was made for the Coma LF in Figure 2, but the smaller error bars in Figure 3 now make this true at a far higher level of confidence.
2) at no magnitude is alpha = - 1 a good fit to the data. We note however the peculiar result that the LF is flattest (alpha = - 1.2) at about MR = - 19, which is the transition magnitude between giants ellipticals and dwarfs in the Binggeli diagram (most giant galaxies in the centers of rich clusters are ellipticals). This is suggestive of a conspiracy in which the giant galaxy LF is falling by an amount almost exactly compensated for by the rise in the dwarf galaxy LF at MR = - 19).
3) the turn-up at the faint-end appears to be very steep. The new results for Virgo (ref. 27, Jones et al. these proceedings) suggest a value of alpha = - 2.1 at MR = - 12. The statistics that Phillipps, Jones and collaborators got for Virgo are significantly better than I show for A 262 or Coma in Figure 2, because in Virgo a background subtraction is not required.

On theoretical grounds, this steep LF is not predicted to extend to indefinitely faint magnitudes because we expect photoionization from the UV background to suppress the conversion of gas into stars in very low mass systems [36]. On observational grounds, we arrive at the same prediction from the requirement that the integrated light from small galaxies not overproduce the measured intracluster light (see Section 8 below). It is, however, interesting to note that between about MR = - 13 and whenever this turnover occurs, the LF has approximately the same shape (alpha ~ - 2) as the mass function predicted [49] from Press-Schechter theory (i.e. if Press-Schechter theory is valid in this context, then over this magnitude range, the combined efficiency of all the baryonic processes which convert gas to stars is independent of the mass of the galaxy).

An important caveat is that the LFs presented in Figures 2 and 3 are only valid for the cores of rich clusters, which are dense elliptical-rich environments. Perhaps the agreement between the composite LF and the Fornax LF outlined above suggests that it is the elliptical galaxy fraction, not the total galaxy density, that is important (Fornax is anomalously elliptical-rich). There is good evidence [8] that the ellipticals evolve differently from all other giant galaxies (including S0s) in clusters, so it is perhaps unexpected if the global form of the LF is the same in elliptical-rich and elliptical-poor environments.

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