5.2. Other Evidence of Second-Order Clusters
The tendency of clouds, clusters, and groups of galaxies to form assemblages of higher order than single clusters was noted long ago by Shapley (1933, 1957). The phenomenon of superclustering was demonstrated dramatically, however, by the analysis of the counts by Shane and his associates of galaxies brighter than mpg 18 on photographs taken for the Lick Astrographic Survey. Shane and Wirtanen (1954) describe six clouds of larger dimension than normal clusters, each containing multiple condensations. Three of the Shane-Wirtanen clouds (Nos. 4, 5, and 6) correspond to apparent groups of two or more clusters, and two (Nos. 2 and 3) to single clusters in the writer's catalog of rich clusters; the remaining clusters in the Shane-Wirtanen clouds apparently are not rich enough for inclusion. More recently Shane (1956b) has called attention to additional rich aggregations of galaxies, some of which contain several centers of condensation, suggesting multiple clusters. Some of these systems are described by Dr. Shane elsewhere in this volume. Typical dimensions of these clouds (for H = 50 km s-1 Mpc-1) lie between 15 and 60 Mpc.
Even the very rich clusters in the writer's catalog (Abell 1958) show a strong tendency for second-order clustering, Of the 2712 clusters catalogued, 1682 were selected as comprising a homogeneous statistical sample. Clusters in the sample all have populations (defined in Section 3.4) of at least 50, redshifts in the range d / = 0.02 - 0.20, and lie at great enough galactic latitudes that interstellar absorption does not prevent their identification (usually at latitudes greater than about 30°). The surface distribution of these clusters is shown in figure 7. The clusters are classified according to distance, the mean redshifts of clusters in distance groups 1 through 6 being, respectively, 0.027, 0.038, 0.067, 0.090, 0.140, and 0.180.
Figure 7. The distribution in galactic coordinates (old system) of the clusters in Abell's catalog. Those clusters closer to the galactic equator than the dotted line are not part of the homogeneous statistical sample. The large empty oval region is the part of the sky not covered by the Palomar Sky Survey. The plot is on an Aitoff equal-area projection.
Superficial examination of figure 7 shows an obvious clustering tendency of the clusters themselves. To test the significance of possible superclustering, the part of the sky covered by the homogeneous sample was divided into grid cells of various sizes, and for each sized cell the frequency distribution, f (t), of cells containing t clusters each was determined. A 2 test was used to estimate the probability that f (t) would be obtained in a random sampling from a population whose frequency distribution is the binomial distribution, B(t). The f (t) and comparison with B(t) was determined separately for clusters in distance groups 5 and 6, and for clusters in distance groups 1 through 4 combined. It was found that f (t) approaches B(t) for very small cell sizes, for then every cell contains either one cluster or none. With increasing cell size, N(t) departs more and more from B(t); the probability P(2), of N(t) being a random sampling from a population with frequency distribution B(t) for the most distant clusters (for which the sample is largest), is as low as 10-30 to 10-40. For larger cell areas P(2) increases again, mainly because the sample size diml area ones) and the deviation of N(t) from B(t) is less significant. P(2) should also eventually increase with cell size if the cells become large compared to any anisotropics in the cluster distribution - that is, if superclustering is "smoothed out." The writer originally interpreted an observed inverse correlation between the cell diameters for which P(2) is a minimum with the cluster distance class as an indication that the second-order clustering occurs on the same scale at all distances surveyed. This interpretation is not strictly justified because of the smaller significance of the results for large cell sizes. However, at cell sizes smaller than those for which P(2) is at a minimum, the descent of P(2) with cell size is steepest for the most distant clusters and least steep for the nearest, as one would expect for superclusters of a common scale displaying smaller angular sizes at greater distances.
The evidence that second-order clusters may have similar linear sizes at different distances argues against their being illusions produced by interstellar or intergalactic obscuration. Simple inspection of figure 7 would also seem to rule out absorption as the cause of the clumpy cluster distribution; if apparent clumps of relatively near clusters are merely parts of a uniform or random distribution of clusters seen through holes in absorbing material, then apparent clumps of more remote clusters should be seen in the same directions, but certainly not between them, as is the case.
About 50 apparent groupings of clusters - probably second-order clusters - can be identified in figure 7. The writer has described 17 of these groupings (Abell 1961). The mean number of clusters (in the homogeneous sample) among the 17 second-order groups is 10.6±6.0 (s.d.). This number, of course, refers only to the very rich clusters in the Abell catalog; the total number of clusters and groups of all kinds in a typical second-order cluster might be greater by an order of magnitude or more. The mean linear diameter of the 17 groups of clusters is 78.0± 23.8 (s.d.) Mpc. The list of 17 systems includes two of the Shane clouds - the Corona cloud and the Serpens-Virgo cloud - described elsewhere in this volume.
The distribution of clusters in Volumes 1, 2, 3, and 5 of the Catalogue of Galaxies and Clusters of Galaxies (Zwicky et al. 1961-1968), the first four volumes of the catalog to be published, have been analyzed in the same way as were those in the Abell catalog with nearly identical results by Abell and Seligman (1965, 1967).
Numerous other investigators have attempted analyses of the distribution of rich clusters of galaxies in the published catalogs. Among them, Kiang and Saslaw (1969) computed serial correlations of Abell clusters in 50-Mpc cubic cells to determine the three-dimensional cluster distribution, and find correlations over a scale of at least 100 Mpc and possibly to 200 Mpc. Bogart and Wagoner (1973) performed nearest-neighbor tests on the Abell clusters, and found that the distribution of nearest-neighbor distances from half of the clusters (sources) to the other half (objects) has a significantly smaller mean than does the corresponding distribution when a set of random points is used for sources, indicating that the clusters are significantly clustered. Bogart and Wagoner estimated the scale of the clustering by rotating the "object" half of the clusters in galactic longitude until the distribution of nearest-neighbor distances approached the random one. The angular scale found for distance group 5 clusters is slightly greater than for the more distant group 6 clusters, suggesting a physical association of clusters; the corresponding linear scale is ~ 200 Mpc.
Statistical analyses of three catalogs of extragalactic objects have been carried out recently by the Princeton group (Peebles 1973; Hauser and Peebles 1973; Peebles and Hauser 1974; Peebles 1974). The sources are the Abell (1958) catalog, the galaxies catalogued by Zwicky and his associates (Zwicky et al. 1961-68), and the galaxies catalogued from the Lick Astrographic plates (Shane and Wirtanen 1967). Peebles and Hauser have investigated the correlation between objects in the individual catalogs and the cross correlations of objects in different catalogs. They find that the clusters and individual galaxies seem to correlate in direction, both separately and with each other over angular distances of up to 6°. The linear size of these homogeneities is of the order 100 Mpc.
Peebles has also developed a powerful statistical method for detecting variations over the surface distribution of galaxies or clusters by means of a two-dimensional power spectrum. It was first applied (Yu and Peebles 1969) to test the hypothesis of complete second-order clustering of the Abell catalog clusters. Yu and Peebles found that if second-order clusters contain an average of 10 rich clusters each, then only about 10 percent or less of the Abell clusters can be members of such superclusters, and that in a model of complete superclustering, on the average there could be at most about 2 clusters per supercluster. It should be noted that in these calculations those clusters of distance class 5 in the southern galactic hemisphere, where inspection of figure 7 suggests second-order clustering to be most pronounced, were omitted because that part of the Abell catalog seemed to Yu and Peebles to be atypical.
Peebles (1973) developed the power-spectrum approach further, and he and Hauser reanalyzed the Abell catalog (Hauser and Peebles 1973). They report "clear and direct evidence of superclusters with small angular scale" and that the structure corresponds to an average of 2 to 3 clusters per supercluster.
The early 2 tests described above are subject to misinterpretation because of the possibility of a general absorption gradient and other systematic effects, and the results of these tests alone should thus be viewed with caution. However, as we have seen, the same results are obtained with more sophisticated tests, made possible with modern computing equipment, especially the powerful power-spectrum analysis. These studies of the catalogs of observed galaxies and clusters of galaxies show very strong - perhaps overwhelming - evidence for inhomogeneities in the large-scale distribution of matter in space with a scale (for H = 50 km s-1 Mpc-1) of the order of 108 pc.
If Newton's laws are valid over dimensions of second-order clusters, and if the latter do not partake of the general expansion of the Universe, we can use the virial theorem to estimate the velocity dispersion within such a system. We denote the mean separation of its members by R', and have
where and R' are in solar masses and parsecs, respectively. If the mass of a typical rich cluster is 5 × 1014 , the entire mass of a typical supercluster probably lies in the range 1015 to 1017 . Adopting 20 Mpc for R', we find that <V2>1/2 should lie in the range 300-3000 km s-1. If the velocity field is isotropic, the observed rms dispersion in radial velocity should actually lie under 2000 km s-1.
Radial velocities are known for six clusters that are suspected of forming a second-order cluster covering an elongated region centered near = 16h14m, = + 29° (Abell 1961). The total range of these six velocities is about 3000 km s-1. There are not enough data to determine a meaningful velocity dispersion for the system, but at least the observations are not incompatible with the assumption that gravitational interactions occur between its members.
If second-order clusters are expanding, or if they do not have negative total energy, the observed dispersion in radial velocities could be higher than the value derived above. Suppose, for example, that gravitational forces within such a system are negligible, and that it expands at the general universal rate. Then the corresponding spread of radial velocities across a second-order cluster of diameter D Mpc should be V ~ H × D = 50 × 75 = 3750 km s-1. Since our estimate of the value of D is proportional to H-1, the derived value of V is independent of the value assumed for H.