Annu. Rev. Astron. Astrophys. 1982. 20: 547-85
Copyright © 1982 by . All rights reserved

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5.1 Correlations of Cluster Properties

Most cluster classification systems contain the common thread of a sequence of cluster types that span the cluster properties and divide the clusters between regular and irregular. Bahcall (1977a) summarized the correspondence between cluster classification sequences and these two broad categories. The regular clusters are usually early Bautz-Morgan types (I, I-II, and II), Rood-Sastry types with one or two dominant galaxies (cD, B), compact Zwicky clusters, and Oemler's elliptical-rich systems. The irregular clusters tend to be late Bautz-Morgan types (II-III and III), Rood-Sastry types with less symmetry (F and I), open Zwicky clusters, and Oemler's spiral-rich systems.

Several researchers have proposed that the parameters used to classify clusters could be related to their dynamical evolution (Richstone 1976, Hausman & Ostriker 1978, Oemler 1974, Gunn & Gott 1972). Hausman & Ostriker suggested that the increasing dominance of the brightest galaxy in the Bautz-Morgan "sequence of cluster types is essentially one of increasing dynamical evolution... with denser systems (more properly those with shorter internal relaxation times) progressing further in a given time." Whether S0 galaxies are formed from stripped spirals or created preferentially in dense galactic regions, both scenarios correlate lower spiral fraction with denser regions and therefore relate Oemler's (1974) cluster classifications to dynamical evolution. (Denser regions have shorter dynamical timescales.) The later phases of White's (1976a) numerical models show the reduction of substructure and the increase in central concentration, thus relating the dynamical collapse to the Zwicky characterization of clusters and the general division of clusters by Abell into regular and irregular types.

Most correlations of the X-ray and optical properties of clusters also may be interpreted as supporting the evolutionary patterns, although little progress has been made to model the gas dynamics during the cluster collapse. Increasing X-ray luminosity and temperature were found to correlate with indicators of increasing dynamical evolution. With a sample of 37 clusters (14 with z < 0.05), Bahcall (1977b, c) showed that the most X-ray-luminous clusters had the highest central galaxy densities (measured within a 0.5 Mpc radius) and the lowest spiral fractions (measured within a 3 Mpc radius). For thermal emission from clusters in virial equilibrium, Solinger & Tucker (1972) and Silk (1976) predicted that LX propto sigmaalpha, with alpha = 4 if the gas temperature is independent of the velocity dispersion and alpha = 5 if the temperature is proportional to the square of the cluster dispersion. Hintzen & Scott (1979) found values for alpha between 3.4 and 5 for different cluster subsets in their sample of 26. From observations of twenty clusters, Mushotzky et al. (1978) suggested that X-ray temperatures increase with central galaxy density and tend to correlate with the square of the velocity dispersion. Hintzen & Scott found that Tgas propto alpha2.7 (taking Abell 2319 and Abell 2029 as single clusters). These correlations were sought primarily as support for the thermal bremsstrahlung emission mechanism. However, velocity dispersion and X-ray temperature also measure the depth of the potential or the central concentration of a cluster, and should be larger for more relaxed, and evolved clusters. Other relationships between X-ray luminosity and temperature, galactic content, and cluster morphology and richness have been found (see McKee et al. 1980, Mitchell et al. 1976, 1979, Jones & Forman 1978, McHardy 1978, Melnick & Sargent 1977, Bahcall 1977c, Mushotzky et al. 1978).

Some caution should be taken in interpreting the X-ray-optical correlations, because most of the cluster samples used were not complete and the selection tended to be biased toward the more X-ray-luminous clusters. Also, luminosity limits were often ignored. The correlations also are not strict functional relations, since there are often exceptions to the trend or significant scatter. For example, from the spiral fraction/X-ray luminosity relation found by Bahcall (1977b), the predicted X-ray luminosity for Abell 194 is 50-100 times larger than observed. Also, Coma (Section 3.3) has a lower velocity dispersion than predicted from its observed X-ray temperature, based on the average in the Mushotzky et al. (1978) sample (Smith et al. 1979). In addition, for a sample of 25 clusters observed with Einstein, Helfand et al. (1980) found no strong correlation between X-ray luminosity and Bautz-Morgan class. And finally, since many clusters are probably not relaxed, the assumption of virial equilibrium is false, which also introduces scatter into some of the correlations. In the context of hierarchical clustering, a dispersion in the parameters of the initial density perturbations would further complicate the correlations. For example, stochastic variations in the densities of initial perturbations with otherwise comparable scales would introduce changes in the dynamical time-scales and the population distribution of cluster galaxies.

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