The Morphological Classification of Galaxy Clusters: Algorithms for Applying the Numerical Criteria

Abstract

We summarize the experience of studying 2D features in the galaxy distribution of galaxy cluster fields. For the detailed study of the inner structure of galaxy clusters, algorithms were developed for detecting various types of regular substructures inside such objects automatically. Substructures in galaxy clusters arise from interactions as well as the evolution of the cosmic web, but cannot be described according to the schemes of morphological classification, both classical and modern, because some regular substructures are not present. Our algorithms are based on numerical criteria that permit the determination of classical morphological types, connected with parameters such as the degree of concentration to the cluster center and/or to a straight line, on a statistically significant level. Other types of substructures can also be detected with corresponding algorithms. As a result, we can analyze intracluster features, such as crosses, semi-crosses, complex crosses, and compact dense chains. All algorithms are realized in the “Cluster Cartography” tool and can be used with data taken from different catalogs. The algorithms and their realization in program code must simplify, standardize, and speed up the analysis of 2D distributions of galaxies in clusters. It is possible in future to adapt the algorithms for the 3D case. The results of statistically valid morphological classification are useful for studies of the evolution of galaxy clusters.

1. Introduction

Modern systematic surveys of extragalactic objects are excellent material for the study of large-scale structures in the Universe: the galaxy’s distribution is organized in a complex network, with filaments surrounding underdense regions and crossing at overdensities, which host galaxy clusters.

Classification is an essential stage of any scientific study, and galaxy cluster morphological classification is not an exception to the rule. The features of galaxy morphology for clusters at optical and X-ray wavelengths permit one to highlight subtypes that occur in interactions between cluster members, in interactions with neighboring elements of the Large-Scale Structure of the Universe, or at their evolutionary stage. The three main components of a galaxy cluster are best established using different instruments and methods of study. Galaxies are presently the main optical markers of structure in other, more massive, cluster components—dark matter and intracluster gas. Common features in various, modern, numerical simulations must agree with the distribution of true cluster members.

The basis of morphological classification in the optical range is the distribution of galaxies in a cluster field. The first catalogs by Abell [1] and maps by Zwicky et al. [2] of galaxy clusters were created on Palomar Sky Survey plates. Simultaneously, the first attempts at the classification of galaxy clusters were executed: Abell [1] proposed an initial approach and divided clusters into rich and poor, as well as into regular and irregular morphologies; Zwicky et al. [2] proposed another separation into compact, medium-compact, and open clusters. Prevalent during the era of Bautz–Morgan (BM) classification [3] was the relative contrast (dominance in extent and brightness) of the brightest galaxy relative to other cluster members, ranging from type I to III in decreasing order of dominance. Rood and Sastry, [4] and later Strubble and Rood [5], used the geometry in their system of the distribution of the ten brightest members, and divided clusters into cD, with the brightest giant central galaxy, binary B, core C, line L, and F, irregular I. Oemler [6] recognized three types of clusters that depended on the prevalent type of galaxies: “spiral rich”, “spiral poor”, and “cD”. The last type describes clusters with a dominant giant elliptical cD galaxy at the center. López-Cruz et al. [7] and López-Cruz [8] introduced the same type as special, proposing the definition of cD clusters; the complement to that class was called non-cD clusters. Bahcall [9] summarized the properties of clusters and groups of galaxies. Panko [10] proposed an integrated approach based on the generalization of the listed classical schemes and the quality of the observational material, PF catalog [11], for the morphology classification of galaxy clusters.

The correspondence of these schemes is demonstrated in Table 1.

Table 1. The comparison of the schemes of the morphological classification of galaxy clusters.

The Scheme and Property	Regular Clusters	Intermediate Clusters	Irregular Clusters
Zwicky type	Compact	Medium-Compact	Open
Bautz–Morgan type	I, I–II, II	(II), II–III	(II–III), III
Rood–Sastry type	cD, B, (L,C)	(L), (F), (C)	(F), I
Lòpez-Cruz	cD	non-cD	non-cD
Symmetry	Spherical	Intermediate	No
Central concentration	High	Moderate	Very little
Central profile	Steep	Intermediate	Flat
Panko type	C, (CF), CcD, CBG	I, IBG, IL7 1, IL11, IP	O, OBG, OL9 1, OP

¹ The Arabic numeral can be 5, 7, 9, or 11.

At first, the Panko [10] scheme was based on numerical criteria. That permitted the detection of differences in concentration towards the cluster center (C—compact, I—intermediate, and O—open clusters), and/or to some preferential line (L or F types, depending on the degree of concentration). The role of bright cluster members is also indicated in that scheme as cD or BG, with an Arabic numeral corresponding to the number of significant bright galaxies. Other features are marked as P. Later [12], the symbol F for concentration to a preferential line was excluded, and the symbol L was modified to L with an Arabic numeral (5, 7, 9, or 11) according to the degree of concentration to the line. The statistical approach became the next step in the study: it permits the detection of other regular peculiarities, such as X- and Y-types (crosses and semi-crosses), curved bands, and short dense chains ([12], as quoted there). Simultaneously, the criteria for feature detection were changed from numerical to statistical. The statistical criteria were based on the difference in the value of the normalized surface density to the overdensity region and its average value calculated for other parts of the cluster field, at the level 2σ and more. The overdense region can be either round or elongated in shape. During the testing of the scheme, it was found that crosses and semi-crosses are not rare features of the inner structure of galaxy clusters, and an algorithm for the detection of cruciform substructures was constructed.

Here, we describe computer algorithms for the analysis of such features. All algorithms were tested both on real and simulated clusters. The data set containing the data for real clusters was the list of the 247 ACO clusters with the corresponding PF clusters. The algorithms were then applied to the detection of substructures in about 500 PF clusters, and the results are used as illustrations in the corresponding topics.

2. Observational Basis for the Study

The classification scheme proposed by Panko [10] was initially intended for studying the morphology of rich galaxy clusters extracted from “A Catalogue of Galaxy Clusters and Groups” (PF hereafter) [11]. The PF catalog was created from the Münster Red Sky Survey galaxy list [13], the MRSS hereafter. The MRSS is the last photographic sky survey, and of a high quality, but, unfortunately, it is 2D. It includes 217 ESO Southern Sky Atlas R Schmidt plates, obtained at La Silla Observatory. The survey covers an area of 5000 square degrees in the region of Galactic latitudes b < –45°. The plates were digitized using two PDS 2020GMplus microdensitometers of the Astronomisches Institut at Münster. The classification of objects into stars, galaxies, and perturbed objects was performed via an automatic procedure, with a posterior visual check of the automatic classification. The external calibration of the photographic magnitudes was carried out using CCD sequences obtained with three telescopes in Chile and South Africa [13]. The MRSS contains positions, red magnitudes r_F, radii, ellipticities, and position angles in the best fitted ellipse approximation for about 5.5 million galaxies to r_F = 24^m. The MRSS galaxy list is complete to magnitude r_F = 18.3^m, i.e., the MRSS contains all galaxies to that magnitude. About 1.2 million galaxies in the completeness limit were used as input data for creating the PF catalog [11]. Each PF cluster has several parameters including the following: Right Ascension and Declination (2000.0), equivalent radius in arc seconds for full area of structure, the number of galaxies, major and minor semiaxes of the best fitted ellipse, ellipticity of the structure (E = 1 − b/a, where a, b are ellipse semiaxes), and the position angle of the major axis of structure (counted clockwise from that direction to the North Celestial Pole, as for the position angle of galaxies in the MRSS). We also have the full list of galaxies in each cluster field. That provides a qualitative observational basis for the study of the 2D distribution of galaxies in the cluster fields.

From 460 PF galaxy clusters of a richness of 100 or more and with no boundary effects, only 247 have counterparts in the ACO [14] catalog. Accordingly, only those clusters have BM and Abell morphological types. There is a connection between the magnitude limit for the Palomar Observatory Sky Survey and the ESO Southern Sky Atlas R Schmidt plates. Nevertheless, even the short list of PF galaxy clusters leads to significant results [15,16]. Yet, the determination of morphological types for other PF clusters becomes necessary, with our main goal here being the detection of substructures in the clusters.

Based on the adopted classification scheme, the derived algorithms determine the requirements for cluster mapping, the method for establishing excess concentration to the cluster centers and/or linear concentrations, as well as for detecting other features in the positions and orientations of galaxies. In the next section, we demonstrate the algorithms for the MRSS and PF data. They can be readily adapted to other lists of galaxies.

3. The Algorithms for the Feature Search

3.1. Cluster Mapping

As a first step, we generate a cluster map suitable for visual control. The equatorial coordinates are recalculated as rectangular coordinates in the usual manner, with point (0, 0) corresponding to the cluster center. The size of the cluster map, 4000 × 4000 arc seconds, is selected in order to show the positions of galaxies at a convenient scale.

The best way to visualize galaxies in the map is by using symbols with the same shapes and orientations as galaxies in the MRSS. The brightness (magnitude) of each galaxy is an important integral parameter, so the symbol size must be associated with it. We adopted the size of each circle symbol m′ as follows:

m′ = a × b^c(d−m) + f,

(1)

where a, b, c, d, and f are constants, and m is the magnitude of the galaxy cited in the catalog.

For the MRSS data for the selected size of the map, the equation is as follows:

m′ = 3 × 2^{0.6(18.5−m)} + 6.

(2)

For other catalogs, the conversion constants must be calculated separately. As the next step, we must determine the symbol shape and orientation corresponding to the galaxy parameters. We assume that an ellipse is the appropriate figure for the symbol and calculate the axes A and B of each ellipse using the ellipticity E from the MRSS. We assume that the areas of the ellipse and a circle with diameter m′ are equal. The appropriate equations are as follows:

$$A={\displaystyle \frac{m^{\prime }}{\sqrt{1-E}}},B={\displaystyle \frac{{(m^{\prime })}^2}{A}},$$

(3)

where A and B are the axes of the elliptical symbol for the galaxy at the scale of the map, and E is the ellipticity of the image of the galaxy, according to the value cited in the MRSS.

The sizes of galaxies in arcseconds are small for the map scale. The computer code must also indicate each galaxy according to its angular size by a specific magnification.

The positional angle of the major axis of the symbol must also correspond to the MRSS data. The brightest cluster members can be noted by darker symbols: the brightest galaxy can be shown in black, the second-ranked in dark gray, etc.

The realization of such a cluster mapping was presented in [12], and examples of maps for the PF 0020-4224 and PF 2199-2391 clusters are shown in Figure 1. The PF 0020-4224 cluster (Figure 1a) has crossed filaments (X-type), and the brightest cluster members are aligned across the dominant linear substructure. The PF 2199-2391 cluster (Figure 1b) has the brightest galaxy in the cluster center, with no significant concentration toward the cluster center. As a difference from PF 0020-4224 in Figure 1a, for PF 2199-2391, the cruciform feature (Y-type) is not seen, but was detected by the corresponding algorithm.

Figure 1. The maps for two PF galaxy clusters. The axes units are arcseconds; the size symbol corresponds to the r_F magnitude of the galaxy, as it’s shown in the additional panel in (a). The brightest galaxy is shown as black, the next-ranked galaxies are gray. The alignment of the brightest galaxy along its mother substructure is seen in PF 0020-4224 (a), but the three-beam substructure in PF 2199-2391 (b) can be confirmed by the LightHouse Beam (LHB) procedure.

3.2. The Detection of the Degree of Concentration Toward the Cluster Center

A standard morphological feature, beginning with Abell [1], Zwicky et al. [2], and Bautz and Morgan [3], is the degree of concentration toward the cluster center. Abell’s regular clusters correspond to Zwicky et al.’s compact ones, BM I type, and C in the Panko [10] scheme. We show typical histograms for concentrated C and open O clusters in Figure 2. To search for such features, the cluster field is divided into concentric ring zones, and the normalized density of galaxies f′ in each zone is calculated, taking into account the difference in the areas of the zones. Histograms of the normalized density f′ permit one to establish a statistically valid degree of concentration.

Figure 2. The distributions of the normalized densities f′ of galaxies in the ring zones: (a) C-type clusters PF 2187-1958 with significant overdense central zone; (b) O-type cluster PF 0031-2046 displaying a relatively smooth distribution. R is the cluster radius.

For C-type clusters (Figure 2a), the overdense region is well defined, with a typical size of 0.1 of the radius of the cluster. For PF 2187-1958, the calculations indicate a mean normalized density value for all zones—excluding the central zone—of 0.0109, with a Standard Deviation of 0.0075. The normalized density f′ of the central zone is 0.105, and we assume that the peak in Figure 2a is statistically significant. For I-type, the overdense region has blurred borders and its radius is larger than in the C-type. It is typically about 0.15 of the radius of such clusters. Open galaxy clusters have no significant concentration toward the cluster center (Figure 2b). In the program code, the position of the maximum of the normalized density and the normalized densities of galaxies in each circle zone for the histogram must be calculated automatically. Additionally, if any cluster has two overdense regions, the positions of the secondary center (or some other such feature) must be set separately with corresponding calculations and histograms. The number of zones is typically 10, but it can be changed if necessary.

3.3. The Detection of the Linear Substructures

One more standard morphological feature in galaxy clusters is the linear substructure. It can also be detected via corresponding calculations and histograms. The cluster field in such cases is divided into N parallel zones, and the normalized density of galaxies f′ in each zone is calculated. Our experience indicates that N can be 5, 7, 9, or 11, etc., because in the typical case, the overdense bands pass through the central part of the cluster, i.e., the number of parallel zones must be odd. We have checked value 3, but significant detection of the band is possible starting from value 5. The width of the zone in the map is calculated according to the diameter of the cluster and noted in the histograms as part of the diameter. The parallel grid rotates relative to the cluster with a step of 1°. The maximal value of the normalized density f′ corresponds to the direction of the linear substructure. If the linear substructure is real, variations in the studied region or number of zones give the difference in the derived value of the position angle at the limit of 3°. Thus, step 1° is optimal. For the detection of the linear substructure, the overdensity of its band must be statistically different from the mean density of other bands (Figure 3a,b). The number of parallel zones is transformed into an Arabic numeral in the L-classification.

Figure 3. The distributions of densities of galaxies in parallel zones: (a) OL7-type clusters PF 0121-3001, (b) OL11-type cluster PF 0443-4308. In both cases, the filamentary feature is well defined, with sharp borders. D is the cluster’s diameter.

The case L7 (OL7-type clusters PF 0121-3001, Figure 3a) is clearly different from the L11 case (OL11-type cluster PF 0443-4308, Figure 3b) by the width of the overdense band. We did not detect a linear substructure like L13. In the program code, the direction and densities of galaxies in each band zone for the histogram are determined automatically. Additionally, if any cluster has two overdense bands, the positions of the secondary one (or some other direction) must be set separately with corresponding calculations and histograms. For C or I clusters, the central zone must be excluded from the analysis because the central condensation distorts the distribution in the zone. For example, for some C- or I-type clusters that have no linear substructure, the low overdensity in any band with the contribution of central cluster members becomes significant. As a result, the L-type for such clusters will be assumed falsely.

These two steps allow us to classify the clusters as C, I, or O morphological types and add the L-type with the Arabic numeral (5, 7, 9, or 11) according to the overdense band width, if it is detected.

3.4. The Detection of Complex Regular Substructures

About 20% of the clusters in our data set (PF catalog) have regular nonstandard features in their 2D distributions of galaxies in the cluster field. We detected crossed or merged linear substructures (X, Y, and complex cruciform types) and compact chains ([12], as quoted there). We did not find the classification of such clusters in the literature. Nevertheless, the detected peculiarities are statistically significant; they reflect the evolutionary moments in cluster forming, as was shown in numerous numerical simulations by the famous Millennium Simulation [17] to modern Illustric [18], EAGLE [19], and Uchuu simulations [20], for example. The manner of detecting cruciform peculiarities is the “lighthouse beam” analysis. The cross in the cruciform corresponds to the position of the origin of the “lighthouse beam”. The width of the beam is set as N (5, 7, 9, or 11), as for the detection of linear substructures, but the number of galaxies must be calculated only for the upper half of the zone, much like the real beam of a lighthouse. The steps for beam rotation are also 1°, and we obtain the galaxy counts for each angle. Visual control can be executed in the cluster map. The example of “lighthouse beam” analysis for clusters is shown in Figure 4. For the PF 0020-4224 histogram, for the width of the beam as 1/7 of the diameter, we detect the four maxima (in Figure 4a they are marked with arrows) corresponding to the directions of the portions of the cruciform substructures. The map of this cluster is shown in Figure 1. The X-type substructure is seen and underlined by the positions of the six brightest galaxies. The Y-type substructure in PF 2199-2391 is detected in the histogram, and is smoothed out when looking at the map.

Figure 4. The “lighthouse beam” analysis for the X-type cluster PF 0020-4224 (a) and Y-type cluster PF 2199-2391 (b). Unsmoothed values are shown. α is the positional angle of the substructure element; for clarity, the angle is shown within the range 0–365°. The peaks in the distributions correspond to the elements of cruciform substructures; the spatial maps of the same clusters are shown in Figure 1a,b.

3.5. Additional Modes in the Program Code

All described algorithms are realized in the “Cluster Cartography” tool (CC) [12]. At present, the final version of the CC code is being tested. The CC tool consists of two parts. The first part, “CC_Input”, recalculates the catalog data to the input CC format: rectangular coordinates X and Y, the symbol size m′, ellipticity from catalog E, positional angle PA, and sign of the brightest galaxy—0 by default, 1 for brightest galaxy, 2 for the second-rank galaxy, etc. The next part executes the other steps, including the recalculation of the major and minor axes of the ellipse from m′ and E. Additionally, sometimes there is a need to export information about galaxies forming the features. CC carries this out, as well as exporting all images, namely the cluster map, cluster map with zones, and histograms for analysis.

One more mode is realized for checking the presence of the Binggeli effect [21] in the detected substructures. It must be presented as a histogram for the values of the acute angle between the direction to the nearest neighbor and the positional angle of the major axis of the galaxy image. The alignment of galaxies in the substructure must also be studied in computer code.

In the next mode, “Hubble mix,” the ratio of the counts of elliptical and spiral galaxies in the cluster can be estimated as an excess in the distribution of the ellipticities of galaxies in the case when we do not know their morphological types. Panko and Flin [22] compare the individual distribution of ellipticities for two PF clusters with the same morphological type I and estimated redshifts. They detected two kinds of ellipticity distributions: bimodal and single-mode. The maximum frequency in the single-mode clusters corresponds to the bin 0.2–0.3; for bimodal clusters, the main peak occurs in the bin 0.1–0.2, with a secondary maximum in the bins 0.4–0.6, corresponding to the spiral/disk-rich type. CC can also build the distribution of ellipticities.

All histograms and maps of the cluster can be exported as pictures for future use with the cluster name.

4. Discussion

The proposed algorithms allow us to detect interesting features in 2D galaxy distributions, such as linear bands, crosses, and compact or curved chains. The rapid and statistically valid detection of substructures in galaxy clusters is one way to compare the galaxies (optical), intracluster gas (X-ray), and DM distributions. The main goal of the application of the proposed algorithms shifted from purely morphological classification to the search for features associated with the evolution of galaxy clusters, according to [4]. Interaction between such clusters manifests itself in the presence of “bridges” between clusters, as shown in Dietrich et al. [23], Gu et al. [24], or HyeongHan et al. [25]. The “bridge” manifests in hot gas and DM distributions [23]. For one data set, Tugay et al. [26] detected the coincidence of directions for the linear substructure in the PF 2187-1958 galaxy cluster and the corresponding X-ray image. Our cluster maps and radio or X-ray images can be compared using the positions of the compass substructures and the positions and orientations for the elongated substructures, as in the noted paper. Cluster X-Ray Morphological Classes by Jones and Forman [27] have a good correspondence with Panko types: type S (single symmetric peak) corresponds to CcD; O (offset center) and E (elliptical) to the L-type, etc. We plan to obtain one more data set for the comparison of optical and X-ray data, and also of simulated galaxy cluster mergers.

The directions of linear and/or cruciform substructures are connected with the positions of neighbors, as in the Binggeli effect [21]. The alignment of galaxies on the substructures must correspond to Joachimi et al. [28]. They demonstrated that elliptical galaxies tend to align their major axes with the linear substructure direction, while disk galaxies tend to align their spin perpendicular to the linear substructure direction (Figure 1a). The confirmation of such alignments provides evidence for the physical nature of the detected substructures.

5. Conclusions

The proposed algorithms permit the determination of morphological types for galaxy clusters using the scheme by Panko [10]. Additional possibilities permit discussion of the evolutionary status of galaxy clusters. A complex inner structure is likely for young clusters. The excess of spiral/disk galaxies is also noted for young clusters. The Strubble and Rood [5] ideas about the evolution of galaxy clusters from open ones without any substructures to relaxed, concentrated cD clusters are consistent with previous results obtained with PF and MRSS data. The main goal of the present paper is a detailed description of the algorithms, as the basis for future studies.

There is no fundamental difference between morphological features in the 2D and 3D data. The concentration to a point must be conserved; the concentration to a line can be cruciform or transformed from the concentration to the plane, and cruciform features and compact chains must also be conserved. Two-dimensional analysis requires much less time. It can be used in the first step of a study: open clusters without features can be excluded from future consideration. Moreover, even SDSS data are cut off at specific redshifts by magnitude, and only the brightest galaxies outline the clusters, for example [29]. From that point of view, the PF and MRSS data with completeness to r_F = 18.3^m remain important and promising observational data for studying the morphology of galaxy clusters. The proposed algorithms standardize the process, while “Cluster Cartography” improves its efficiency. The approach can be adapted to input data from other catalogs, but the main task remains the morphology of PF galaxy clusters and the comparison of the results with numerical simulations. The updated computer code for Cluster Cartography in online mode [30] has important prospects for galaxy cluster studies.