1. The difference between a group and a cluster of galaxies is not rigorously defined. The division is often based on the Abell catalog [Abell G. O., Corwin H. G., Olowin R. P., Astrophys. J. Suppl. 70, 1 (1989)]. Systems of galaxies with richness class <0 (that is, <30 member galaxies) are usually termed groups, whereas their richer counterparts are called clusters. As defined by Abell et al., richness is a measure of the number of galaxies within a radius of 2h 75 −1 Mpc of the cluster center (whereh 75 =H 0/75, andH 0 is the Hubble constant).
2. The density parameter Ω 0 is defined as ρ 0 /ρ c = 8π G ρ 0 /(3 H 0 2 ) where G is the gravitational constant H 0 is the Hubble constant at the present epoch (75 km s −1 Mpc −1 is assumed throughout this paper) ρ 0 is the true mass density of the universe and ρ c is the critical density (48). For ρ 0 > ρ c (Ω 0 > 1) the universe will collapse into a big crunch whereas for ρ 0 < ρ c (Ω 0 < 1) the universe will continue monotonic expansion.
3. The evolution of clusters is driven by mass accretion from large-scale structures. The accretion rate depends on Ω0because clusters must form earlier [that is, at larger redshifts or look-back times (5)] in a low density universe. Estimates of the density parameter from cluster properties can be found, for example, in D. Richstone, A. Loeb, E. L. Turner, Astrophys. J. 393, 477 (1992); A. Evrard, J. Mohr, D. Fabricant, M. Geller, Astrophys. J. Lett. 419, L9 (1993); N. A. Bahcall, X. Fan, R. Cen, ibid 485, L53 (1997); J. P. Henry, ibid. 489, L1 (1997); Evrard A. E., Mon. Not. R. Astron. Soc. 292, 289 (1997).
4. For Ω 0 = 1 (flat universe) the universal expansion rate declines exponentially but for Ω 0 < 1 the universe is open and will expand forever (48).
5. Look-back time, Δt, is defined as the time measured back from the present and is a nonlinear function of the redshift (z) [in (48), p. 313]. For Δt = 6 Gy (corresponding to z = 0.9 forH 0 = 75 km s−1 Mpc−1and Ω0 = 0.2), the fraction of blue galaxies is measured to be ∼80% of the observed galaxies in each cluster [K. Rakos, J. Schombert, Astrophys. J. 439, 47 (1995)]. This diminishes to ∼20% for Δt = 3 Gy (z = 0.4) and ∼4% at present [Butcher H., Oemler A., ibid 285, 426 (1984). . The trend is generally known as the Butcher-Oemler effect].