To calculate the sizes of voids between clusters and superclusters
we used the empty sphere method. In this method we divide
the cubic sample volume into 
cubic cells, where n is a
resolution parameter. For each cell centre we determine the
distance to the nearest cluster. Cells having the largest
distances to the nearest clusters are located in centres of voids.
The distances to nearest clusters correspond to the radii of voids.
Therefore we obtain the void centre coordinates and radii. For
details of the method see Einasto et al. (1989, EEG)
and EETDA.
We determined the diameters of voids, delineated by all clusters, by supercluster members, and by members of rich superclusters (Table 1 (click here)). The number of clusters in these samples is 1304, 900, and 580 clusters, respectively, and samples are denoted as Acl, Asc2, and Asc4 (A stands for Abell). In order to see the influence of the change of the number of clusters on the void sizes we used randomly diluted cluster samples, i.e. from the observed sample (Acl) we removed clusters in a random way so that in the resulting sample the number of clusters was 900 and 580 (correspondingly Ard900 and Ard 580, rd stands for random dilution).
Additionally, we calculated void sizes for random supercluster catalogues. Here again we used samples of all clusters, all supercluster members and members of rich superclusters (correspondingly the samples Rcl, Rsc2, and Rsc4). We used ten realizations of random catalogues. Although this number is rather small, the results for random catalogues are seen quite well. The median diameters of voids for these catalogues are also given in the Table 1 (click here). Since the results of void analysis for both sets of random supercluster catalogues were essentially the same, we give in this table the diameters of voids for only one set, the censored random catalogues.
Table 1 (click here) shows that the median void sizes in the case of observed cluster and supercluster samples are very close to each other. Also the scatter of void diameter values is rather small (see Fig. 8 in EETDA). We see only a slight increase of void sizes as we move from the sample of all clusters to the supercluster members and to the members of rich superclusters. The reason for the increase of void sizes is clear: although isolated clusters and poor superclusters are located close to void walls, some of them enter into voids determined by rich superclusters, and thus voids determined by all clusters are smaller - the sizes of voids are determined by the location of clusters in the periphery of voids. If we remove clusters in a random way then of course we remove part of the clusters from the central regions of void walls that have no effect to void sizes. Thus the increase of void sizes in this case is smaller than in the first case. Real rich superclusters form a quasi-regular lattice which is almost identical for supercluster samples of all richness classes; much stronger random dilution is needed to destroy this lattice.
Figure 8: The distribution of distances between centres of
superclusters. Upper panel shows the distributions for poor and
medium rich superclusters, lower panel - for the very rich
superclusters. Curves correspond to the first (line with short
dashes), second (line with long dashes) and third (solid line)
neighbour
Comparison with the random catalogues shows that if the clusters and superclusters are located randomly then the removal of part of the clusters increases the void sizes much more than in the observed case.
Table 2: Median distances between high-density regions