The main advantage of searching for substructures in the projected distribution of galaxies, is the availability of large data-sets, reaching thousands of galaxy positions for nearby clusters. On the other hand, these methods suffer from contamination by fore/background galaxies, groups, and other clusters.
Geller & Beers (1982) were the first to systematically address the evidence of substructures in the projected distributions of cluster galaxies. Using smoothed density-contour maps in 65 clusters, they identified substructures as regions where the local density contrast was more than 3-ct above the background fluctuations (see Fig. 2.1).
West et al. (1988) developed three new statistical tests: the /? statistics measures departures from mirror symmetry in clusters; the angular separation test detects subclustering by looking for significant galaxy overdensities at similar polar angles relative to the cluster centre; the density contrast test is similar to the method of Geller & Beers (1982).
The application of the Lee-method (Lee 1979) to clusters of galaxies is described in Fitchett & Webster (1987; see also Fitchett 1988b). The method optimally splits a data-set into two or more groups using a maximum-likelihood statistics. In practice, the method is only used for the partition of a sample into two subsamples, as the detection of more than two clumps is computer-time consuming. The method measures the dumpiness, L, of the 2D data projected onto a line, with a given orientation, a. The analysis of the function L(a) allows one to define the two groups. The significance of L is established by comparison to Monte Carlo simulations, in which the simulated galaxy distributions can be drawn from several kinds of surface density profiles. While initially applied to 2D data-sets, the Lee-method method has later been used also in its 1D and 3D versions (Fitchett & Merritt 1988).
The Wavelet transform method is described by Slezak et al. (1990). The basic idea is to convolve the 2D Dirac distribution of galaxy positions with a chosen zero-mean function of position and scale (the Wavelet), on a grid of pixels. There are different kinds of Wavelet function; the so-called 2D radial "Mexican Hat" (the second derivative of a Gaussian) is often used for studies of galaxy clusters (e.g., Escalera & Mazure 1992; Escalera et al. 1992). By varying the scale of the Wavelet function, one is able to test for the presence of substructure of different sizes (a multi-scale analysis). A given substructure can only be detected if its characteristic size is of the order of the scale of the Wavelet. It is worth pointing out that, despite of being circularly symmetric, the radial Wavelet can detect non-circular substructures. As usual, Monte Carlo simulations are needed to establish the statistical significance of the detected substructures. The method also provides the likelihood of individual galaxies to belong to given substructures, thus in practice allowing a decomposition of the cluster into its component subclusters.
A variant of the classical Wavelet method has been recently discussed by Shao & Zhao (1999). An extension of the Wavelet method to 3D is discussed below (see § 1.3).
Starting from statistical techniques generally used in the analysis of the LSS of the universe, Salvador-Solé et al. (1993a,b) implemented and applied the average two point correlation function to the study of cluster substructures. This method provides an estimate of the scale length of typical substructures.
The KMM mixture-modeling algorithm for the decomposition of a given data-set in two or more groups, described by Ashman et al. (1994), has been applied to the spatial distribution of cluster galaxies by, e.g., Kriessler & Beers (1997), Maurogordato et al. (2000). Since the simpler implementation of KMM is for a 1D distribution, we describe it at length in the next section.
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