LIBRARY Michigan State University 5 PLACE ll RETURN BOX to move this chock” 1mm your ncotd. To AVOID FINES Mum on or More data duo. DATE DUE DATE DUE DATE DUE _—ll ll I [_j::J IT—IWF—j MSU I. An mauve Mean-l Opportunity Imam Studies of Substructure in Clusters of Galaxies: A Two-Dimensional Analysis By Jeffrey R. Kriessler A DISSERTATION submitted to Michigan State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY Department of Physics and Astronomy 1997 ABSTRACT STUDIES OF SUBSTRUCTURE IN GALAXY CLUSTERS: A TWO DIMENSIONAL ANALYSIS By Jeffrey R. Kriessler ABSTRACT In this thesis I explore a procedure for the detection and quantification of sub- structure in the projected positions of galaxies in clusters. The method is first tested by application to the 56 well-studied galaxy clusters that make up the morphological sample of Dressler (1980). This method is then applied to a much larger, volume- limited sample of 119 Abell clusters originally identified of Hoessel, Gunn, & Thuan (1980). This sample includes all Abell clusters with distance class 3 4 and richness class 2 0 with |b| > 30. Two tests for substructure, one parametric and one nonpara- metric, are applied to the galaxy positions and the results are compared. The KMM algorithm partitions the data into Gaussian sub-populations and estimates their sta- tistical significance via a hypothesis test. The DEDICA algorithm is a nonparametric technique that identifies peaks in the projected galaxy density and determines their significance with respect to the background. After a K-S test is employed on the magnitude distributions to remove background/foreground groups, 64% :t 15% of the large cluster sample is found to contain significant substructure. Nonparametric methods of density estimation are explored and applied to the construction of contour plots and the calculation of radial number-density profiles for each of the sample clusters. An average core radius of 150 :1: 100 kpc (H0 = 75 km s“) is obtained. This is however, likely to be an upper limit due to mis-specification of the cluster centers. Inside of l Mpc, the space density is found to vary as p or r‘wio'3 after a correction is made for background galaxies. The large fraction of clusters with presently-detectable substructure, as well as the shallow space—density profiles, are used to argue that rich clusters of galaxies are still in the process of formation during the present epoch and are not well described by equilibrium models. If clusters are currently accreting large amounts of material, this implies a high-density Universe, with Q 2 0.4. To my family. ACKNOWLEDGEMENTS I would like to acknowledge all those who have contributed to the successful com- pletion of this project. In particular the guidance of my advisors Tim Beers and Suzanne Hawley was very helpful. I would also like to thank the members of my committee Jim Linnemann, Ed Loh, Gerald Pollack, and Horace Smith. I further wish to acknowledge the friends I have made here at Michigan State and all the fond memories I will take with me where ever I go. Thanks to Bill Abbott, Dave Bercik, Daniel Casavant, Jen Discenna, Normand Mousseau Dennis Kuhl, Rod Lambert, Vickie Plano, Jeff Schubert, and Steve Snyder, the years I have spent here passed quickly. iv Table of Contents LIST OF TABLES LIST OF FIGURES 1 3 INTRODUCTION 1.1 The Standard Model ........................... 1.2 Observational Properties of Clusters ................... 1.3 Previous Studies ............................. 1.4 Goals of the Thesis ............................ 1.5 Chapter Overview ............................. DATASET 2.1 The Cluster Samples ........................... 2.1.1 Dressler’s Data .......................... 2.1.2 HGT Sample ........................... 2.2 Digital Sky Surveys ............................ 2.3 The Minnesota Automated Plate Scanner ................ 2.4 Neural Network Star/ Galaxy Classification ............... 2.4.1 Contamination from Stars .................... 2.4.2 Completeness ........................... 2.5 Estimation of Background Contamination ............... PROBABILITY DENSITY ESTIMATION 3.1 Introduction ................................ 3.2 The Histogram .............................. 3.3 Generalizations of the Histogram .................... 3.4 The Kernel Estimator .......................... 3.5 Adaptive Smoothing Methods ...................... V vii viii CDKIv-RNl—l 10 12 12 12 13 18 18 19 20 22 24 31 31 3.6 Application: Galaxy Number-Density Plots ............... 46 4 TESTS FOR SUBSTRUCTURE 94 4.1 Introduction ................................ 94 4.2 The KMM Algorithm ........................... 95 4.2.1 Monte Carlo Simulations ..................... 99 4.2.2 Application of KMM to the Dressler Sample .......... 105 4.2.3 Application of KMM to the HGT Sample ............ 109 4.3 The DEDICA Algorithm ......................... 120 4.3.1 Application of DEDICA to the Dressler Sample ........ 124 4.3.2 Application of DEDICA to the HGT Sample .......... 129 4.4 Background/ Foreground Cluster Identification ............. 138 4.5 Comparison of Results .......................... 140 4.6 Comparison to other Studies ....................... 140 4.7 Conclusions ................................ 142 5 ESTIMATION OF THE COSMIC DENSITY PARAMETER $20 145 5.1 Introduction ................................ 145 5.2 The Theory ................................ 145 5.3 Discussion ................................. 151 6 RADIAL NUMBER-DENSITY PROFILES 154 6.1 Introduction ................................ 154 6.2 Maximum Penalized Likelihood Estimator ............... 156 6.3 Application to the Cluster Sample .................... 191 6.3.1 Core Radius ............................ 191 6.3.2 Power Law Profiles and Estimation of S20 ............ 198 6.4 Conclusions ................................ 199 7 CONCLUSIONS 200 7.1 ....................................... 200 7.2 Future Work ................................ 201 LIST OF REFERENCES 203 vi 1.1 2.1 2.2 2.3 3.1 3.2 4.1 4.2 4.3 4.4 6.1 List of Tables Abell Richness and Distance Classes .................. 5 HGT Cluster Parameters ......................... 15 Estimated Background — Dressler Sample ................ 26 Estimated Background — HGT Sample ................. 28 Map Parameters - Dressler Sample ................... 48 Map parameters — HGT Clusters .................... 65 Mixture Model Parameters — Dressler Sample ............. 107 Mixture Model Parameters — HGT Sample ............... 113 DEDICA Cluster Parameters — Dressler Sample ............ 125 DEDICA Cluster Parameters — HGT Sample .............. 130 Core Radius and Power-Law Profiles .................. 193 vii 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 4.1 4.2 5.1 6.1 List of Figures Histogram of A400 velocity measurements. ............... Histogram of A400 velocity measurements. ............... ASH for A400 velocity data. ....................... Naive estimator for A400 data. ..................... Adaptive-kernel plot for A400 velocity data. .............. Adaptive-Kernel Maps for Dressler Sample ............... Comparison of Adaptive-Kernel Maps and GB Maps ......... Comparison of Adaptive-Kernel Maps and GB Maps ......... Adaptive-Kernel Maps for HGT Sample ................ Core region of A1656 ............................ Core region of A400. ........................... KMM success rate vs. group separation — homoscedastic case ..... KMM success rate vs. group separation - heteroscedastic case . . . . Fraction of clusters with substructure vs. (20 .............. HGT Cluster Number-Density Profiles ................. viii 33 34 36 38 41 50 152 Chapter 1 INTRODUCTION The human spirit is characterized, among other things, by an intense desire to explain the world around us. Of all the physical sciences, cosmology is perhaps the most ambitions, for it seeks to explain how and why the Universe we see today came into existence. From the fertile minds of scientists have sprung forth an incredible array of theories, from the Earth-centered, non-evolving universe of Ptolemy to the Inflationary Big Bang cosmologies which are popular today. The role of observational cosmology is to constrain these theories and try to find the model which best explains the real Universe. As a science, cosmology is still in its infancy. Only within this century have the theories been taken from the realm of pure conjecture about how the Universe ought to be, to testable theories which describe the Universe as it, at present, appears. The reasons for this transition are: 1) The existence of the mathematical frame work of General Relativity which describes the interaction of matter with space and time, 2) advances in high energy physics which have extended our understanding of the forces which act between particles at very high temperature and pressure, and 3) extension of the astronomical observations to include nearly the full range of the electromagnetic spectrum. 1.1 The Standard Model Recent progress in observational cosmology has led to the general acceptance of the Hot Big Bang theory or, borrowing a term from high energy physics, the “standard model.” The success of the standard model of the Universe rests on its ability to ex- plain, in a simple manner, three important observations. These are: 1) the blackbody spectrum of the microwave background radiation, 2) the abundance of the light ele- ments (D, He, and Li) in the Universe, and 3) the expansion of the Universe. Though there remain a number of problems with the standard model, the current evidence indicates that it is at least a useful approximation to the real Universe. The standard model indicates that the Universe did not exist in its present state forever, but was created a finite time ago in an event commonly referred to as the Big Bang. At this time the Universe existed in a singular state of unimaginable temperature and pressure where current theories of physics can not be applied. From this hot, radiation-dominated state, the Universe started to expand and cool. This process continues today as observed in the redshifts of galaxies and the, now relatively cool, 2.7 K cosmic background radiation. While there are many interesting transitions which took place as the Universe cooled, this thesis is primarily concerned with events which occurred after the decoupling of the radiation and the matter, about 3 x 105 years after the Big Bang. This is commonly referred to as the “era of recombination” , because only then is it possible for electrons to bind with protons to form atoms. This era is of interest because it signals the earliest possible formation time of structures such as galaxies and clusters of galaxies. Although it is believed that the density fluctuations which collapsed into these structures must have already existed before the era of recombination, they could not have begun to collapse due to the radiation pressure that dominated the Universe at these times. For a density fluctuation to grow due to gravitational instability it must have a mass greater than the Jeans mass M J. At the time of recombination the Universe becomes transparent to light and particles of matter can slow to non-relativistic speeds. This transition causes the Jeans mass to change abruptly from M J a: 1016M® (about ten times the mass of the Coma cluster) to M J z 106MQ. The time scales for collapse of cluster-sized, Gaussian perturbations will be discussed in detail in Chapter 5. Despite the successes of the standard model, there are a number of fundamental questions that remain unanswered. These questions include: What is the mass density of the Universe, 00? Will the Universe expand forever, or collapse sometime in the future? What is the present value of the Hubble constant, H0? How old is the Universe? What is the value of the cosmological constant, A? In a rapidly expanding, uniform Universe, how can structures such as galaxies, clusters of galaxies and superclusters of galaxies form? Does structure form on large scales and then fragment into smaller units, or do smaller units form first and join together to create larger objects? What is the nature of dark matter and how does it affect the evolution of the Universe? Clusters of galaxies play an important role in addressing these questions. In a low- density Universe clusters at the present epoch are expected to be in free expansion. Therefore, they should not be accreting new material. On the other hand, in a high-Q Universe clusters can continue to grow in the present epoch. The inflow of material into clusters should be observed as substructure (Richstone et al. 1992, Kauffmann & White 1993) and flattened density profiles (Crone et at. 1994, Jing et at. 1995, Crone et al. 1997). 1.2 Observational Properties of Clusters One of the biggest challenges faced by the researcher in the field of galaxy clusters is defining just what constitutes a cluster, with different researchers adopting different criteria. The problem is to identify galaxies which are gravitationally bound to one another, often through the use of only their projected positions on the sky and their apparent magnitudes, parameters easily estimated from photographic surveys. Abell’s (1958) solution was to visually examine the red plates of the Palomar Optical Sky Survey (POSS) and define a cluster as a region where there existed at least 30 galaxies within two magnitudes of the third-brightest galaxy within a projected radius of 1.512“1 Mpc (one Abell radius). (Throughout this thesis H0 = 100h km s‘1 Mpc"l with h=0.75.) The centers of the clusters were determined by eye and the distance to the cluster was estimated using the apparent magnitude of the tenth-ranked galaxy. The resulting catalog of clusters (after extending it to the southern sky [Abel], Corwin, & Olowin 1989]) contains 4076 systems. Abell divided the sample into “richness classes” and “distance classes,” as defined in Table 1.1. Column (1) provides the richness class R. Column (2) lists the number N of galaxies within two magnitudes of the third-ranked galaxy for each richness class. The distance class D is given in column (3). The magnitude range of the tenth-ranked galaxy mm in the V band for each distance class is listed in column (4). The Abell catalog has received a number of criticisms over the years. First, strict use of a radius within which to look for cluster members tends to favor inclusion of only those clusters which are concentrated and roughly circular, often referred to as “regular.” Large, spread-out clusters or elongated clusters having a large fraction of their members outside the Abell radius would be missed. Second, since the clusters were chosen only on the basis of concentrations in the projected galaxy distribution, Table 1.1. Abell Richness and Distance Classes R N D mm (1) (2) (3) (4) 0 30-49 0 < 13.3 1 50-79 1 13.3-14.0 2 80-129 2 14.1-14.8 3 130-199 3 14.9-15.6 4 200-299 4 15.7-16.4 5 > 300 5 16.5-17.2 the possibility arises that a large number of the clusters may simply be due to the superposition of background and foreground groups. In fact a number of numerical simulations indicate that identifying clusters using Abell’s method may lead to a catalog within which as much as 30% of the richness class 1 clusters are due simply to projection effects and that a similar percentage of real clusters have been missed (van Haarlem 1996). On the other hand, a study by Briel & Henry (1993) of Abell clusters with detectable X-ray emission by ROSAT (indicating the presence of a real potential well) found that only 10% of the Abell richness class 1 clusters are likely to have been mis-identified due to foreground/ background projection. In the past decade, the somewhat subjective nature of the Abell catalog has been addressed by the development of machine-generated catalogs (Dalton et al. 1994), but these do not as yet exist for the entire sky. Furthermore, advances in X—ray astronomy have led to the hope that a cluster catalog can be produced using X—ray- derived temperatures. If the hot intercluster gas is assumed to be in hydrostatic equilibrium, then the temperature of the gas will be a direct measure of the depth of the potential well, eliminating all possibility of mis-identified clusters. Although catalogs of clusters have been made from the ROSAT All Sky Survey (Giacconi & Burg 1993, Ebeling et al. 1996), at present they are still incomplete. Furthermore, a higher-resolution survey (with 5-10 arcsec resolution) is required for easy separation of point and extended sources. Thus, despite the potential problems, the Abell catalog is still the most complete catalog of rich clusters available for the entire sky. A typical line-of-sight velocity dispersion for a rich Abell cluster is a, z 103 km S“. If clusters of galaxies are assumed to be bound, the viral theorem can be used to determine the mass of the cluster. Typically this mass for rich Abell clusters is on the order of a few x1015M®. One important dynamical time scale is the crossing time. The crossing time is the time it takes the average galaxy to get from one end of the cluster to the other. In convenient units it is given by: R 103km 3‘1 tcfoss z 1 9 __ — , 1.1 0 yr (Mpc) ( 0,. ) ( ) where R is the radius of the cluster and a, is the line-of—sight velocity dispersion. This provides a lower limit for the time it takes substructure to be erased. For rich clusters this is about a billion years, or one-tenth the age of the Universe. Analytical work suggests that the smallest groups on radial orbits will be disrupted by tidal forces in a single crossing time, while a merger between two equal-sized clusters may take as long as four crossing times to be erased (Gonzalez—Casado et al. 1994). On the other hand, numerical simulations span the full spectrum of possible relaxation times, from a single crossing time to several Hubble times, depending on the initial conditions assumed (West, Oemler, & Dekel 1988; Cavaliere et al. 1992; Nakamura et al. 1995). Ultimately, it appears as though observations of clusters will be necessary to constrain these initial conditions. 1.3 Previous Studies The modern study of clusters of galaxies was initiated with Zwicky’s (1933) study of the Coma cluster (A1656). Using positions projected on the plane of the sky and line of sight velocities obtained from redshifts of spectral lines in the galaxies, he concluded that the amount of matter needed to keep the Coma cluster from flying apart on a time scale of a billion years was many times larger than the matter visible in the galaxies. This was the first indication of the existence of large amounts of dark matter in the Universe. In a follow-up study, Zwicky (1937) concluded that the distribution of bright galaxies was very similar to the distribution of mass density in an isothermal gas sphere. For the following five decades the Coma Cluster has been considered the prototype of a relaxed, rich galaxy cluster. With advances in computer speed and availability over the past 20 years has come enormous strides in statistical techniques which can be used to analyze data in new ways. With this and the advent of X-ray astronomy, the idea that clusters of galaxies could be described as relaxed systems in isothermal equilibrium has been challenged. The evidence used to argue against equilibrium cluster models includes: a) “clumpy” distributions of galaxies seen in projection on the sky, b) apparent structure in the distribution of radial velocities for cluster members, and c) multiple centers of X- ray emission in the cluster, and other complexities in the X—ray-derived temperature profiles, suggestive of ongoing subcluster collisions. Studies using solely the projected positions of galaxies in clusters have concluded that between 20% and 80% of clusters have statistically-significant substructure. Geller & Beers (1982, hereafter GB) made contour maps of the projected galaxy density for 65 clusters with data from Dressler (1976) and Dressler (1980). These authors concluded that 40% of the clusters in the combined Dressler samples have substructure based on multiple peaks in the contour maps. Baier (1983), on the ba- sis of secondary peaks in radial number-density distributions for some 100 clusters, concluded that as much as 80% of the clusters in his sample had substructure. More recent investigations are discussed in Baier et al. (1996). The image analysis tech- niques of West & Bothun (1990) led them to conclude that some 30% of the Dressler sample has substructure. Rhee, van Haarlem & Katgert (1991) applied six tests for substructure to the projected positions of galaxies in 104 Abell clusters obtained from digital scans of copies of the Palomar Sky Survey plates, and found that 26% had significant substructure. Salvador-Sole, Sanroma & Gonzalez-Casdado (1993, here- after SSG) looked for deviations in the density profiles of 15 clusters (after applying a redshift filter to remove obvious foreground/ background galaxies), and found that 50% of their sample showed evidence of substructure. Dressler & Shectman (1988) obtained the first sample of galaxies in clusters with a sufficient number of measured redshifts to include velocity information in the search for substructure, and concluded that 30% to 40% of their sample (of 15 clusters) exhibited deviations in the local vs. global kinematic properties, consistent with the existence of dynamically-significant substructure. Bird (1993) applied a number of statistical tests using both spatial and velocity data to demonstrate that between 30% and 80% of clusters could have substructure, depending on the test employed. Escalera et al. (1994) applied the wavelet analysis technique to projected galaxy positions and velocities in 16 clusters and found that only three clusters could be classified as unimodal. The simple kinematic test of Dressler & Shectman (the A- test) was applied to a sample of 73 clusters in the ESO Nearby Abell Cluster Survey (ENACS) by den Hartog (1995), who found that 50% showed evidence of substructure. Jones & Forman (1992) found that of the 208 clusters observed by the EINSTEIN satellite with X—ray emission bright enough to classify, some 22% showed clear sub- structure. Mohr, Frabricant & Geller (1992) examined X—ray surface brightness mo— ments of 40 EINSTEIN cluster observations, and found that 68% showed evidence of substructure. Buote & Tsai (1996) used a power-ratio technique (essentially a ratio of higher-order moments of a two-dimensional potential to the monopole moment) to examine 59 clusters with X-ray maps available from ROSAT which had substructure obvious to the eye, in order to specify the dynamical states of the clusters in their sample. These authors conclude that most clusters have some level of substructure and that the evolutionary state of the cluster can be specified by its influence on the gravitational potential. 1.4 Goals of the Thesis Most of the above studies have used relatively small numbers of clusters which suffer more or less from selection effects. Although Rhee et al. applied a battery of sub- structure tests to their sample clusters, they concluded that only 26% of them had evidence for substructure. Furthermore, no single test resulted in more than 10% of the sample being classified as containing significant substructure. Their conclusion is clearly at odds with the growing evidence that suggests most clusters do indeed have substructure. It may well be that the statistical tests applied by Rhee were simply not sensitive enough to the substructure which they were designed to detect. On the other hand, Jones & Forman studied a large sample of clusters with EINSTEIN pointed observations with which they went as far as classifying the morphologies of the clusters based on the appearance of the images. While such a catalog is poten- tially very useful, especially for comparison to optical maps such as the ones presented in this thesis, this catalog is not readily available. Also, in his thesis, Beers (1983) argued that estimates of the true fraction of X—ray clusters which exhibit substruc- ture obtained with the EINSTEIN survey are only lower limits due to selection biases 10 in the observations. In particular, because the unvignetted field of view is only 40 arcminutes, subclusters with separations greater than about 20 arcminutes from the cluster center will be missed. Therefore, armed with new and potentially very powerful techniques for the de- tection of substructure in clusters, the time is ripe to re-examine the question of substructure in the projected galaxy distributions of a statistically-complete sample of nearby Abell clusters. The goal of the thesis is to identify a subset of Abell clus- ters which are likely to contain dynamically-significant substructure. The fraction of clusters with substructure and the radial density profiles of clusters are used to place constrains on the cosmological density parameter, $20. 1 .5 Chapter Overview In Chapter 2 the motivation and selection criteria for the cluster sample is explained. The use of the Automated Plate Scanner (APS) catalogs of star and galaxy positions is discussed. In particular, the accuracy and completeness of the galaxy catalogs is addressed. The background contamination within an Abell radius of each cluster is estimated to be between 10% and 30%. Chapter 3 presents the basic concepts behind nonparametric density estimation. The motivation behind the use of the adaptive-kernel technique is explained. These concepts are applied in the construction of contour maps for each of the sample clusters. These maps can be used to identify peaks in the projected galaxy positions for detailed comparison with X-ray surface brightness maps. The tests employed for the detection of substructure are presented in Chapter 4. Two tests are applied to the sample clusters and the results discussed and compared. Comparisons are made with other techniques and the advantages and disadvantages, 11 are explored. In Chapter 5, the theory behind the estimation of $20 from the fraction of clusters with detectable substructure is reviewed. The results of Chapter 4 are used to argue that if the density perturbations are Gaussian at the time of recombination and if substructure is erased on the order of four crossing times, S20 is likely to be greater than 0.4. This is about twice the amount of matter currently inferred from the dynamics of galaxy clusters. Chapter 6 presents non-parametric density profiles for the sample clusters. The question of the existence of constant-density cores in clusters is addressed. The steep- ness of the radial-density profiles is compared to numerical simulations which suggest a high-Q Universe. Conclusions and suggestions for future work are presented in Chapter 7. Chapter 2 DATASET 2.1 The Cluster Samples In this thesis two samples of clusters will be examined for the presence of substructure in the projected galaxy positions. These are the 56 clusters included in Dressler’s morphological study (1980) and the 119 clusters in the sample identified by Hoessel, Gunn, & Thuan (1980, hereafter HGT). There is an overlap of 25 clusters between the two samples which will be used to compare the APS (Automated Plate Scanner project at Minnesota) data with that obtained by Dressler. The combined samples contain a total of 150 clusters. 2.1.1 Dressler’s Data Theisample of clusters selected by Dressler includes clusters with 2 S 0.06 (oz 3 18000 km S“), and N Z 50 with magnitude my _<_ 16.5 contained in an area of few square degrees on the sky. Unlike Abell’s definition of a cluster, the area definition was left purposefully vague in order to avoid selecting only circularly-symmetric clusters. This nevertheless includes 38 Abell clusters from the northern catalog, though some (6.9. A14) have richness class 0 due to the different area definitions. In addition, 18 southern clusters (those with prefix DC and Centaurus) satisfying these criteria 12 13 were identified in the southern sky using the plate copies of the ESO Quick Blue Sky Survey. Though many of the these clusters have since been given Abell numbers in the expanded ACO (Abell, Corwin, & Olowin 1989) catalog, the older designation will be retained here for easy comparison with previous work. The cluster redshift was obtained by Dressler from the literature when available. When not available he obtained redshifts for at least two cluster members. Dressler (1980) lists positions, estimated magnitudes rounded to the nearest mag- nitude, bulge sizes, ellipticities, and morphological type for each galaxy in the survey. The fact that this information was published and that these are among the closest clusters has made members of the Dressler sample some of the most well-studied clus- ters in the sky, hence a good test bed for the evaluation of new statistical techniques. The background in the Dressler sample was estimated by taking an additional 15 plates at random areas of the sky and repeating the same procedure of galaxy identification carried out for the program plates. A median value of 8 galaxies deg‘2 is quoted by Dressler or 0.0022 galaxies arcmin‘z. Follow-up studies which included the gathering of redshift data confirmed that in most cases this is a good estimate (Dressler & Shectman 1988). 2.1.2 HGT Sample In addition to a re-examination of the question of substructure in the 56 clusters of Dressler’s morphological sample, this study also includes the 119 Abell clusters in the sample of HGT (1980). This sample was an attempt to be a volume-limited sample, in that it consists of all northern clusters in the Abell catalog with distance class less than or equal to 4 and richness class greater than 0, with galactic latitude |b| Z 30 (107 clusters). In addition it contains 12 clusters with richness class 0 and distance class 3 or less at high galactic latitude. 14 Although the APS project offers the unique possibility of testing (1112714 northern Abell clusters, the reason for choosing to study these 119 clusters first is that these clusters are the richest and closest clusters to us. As such, they have attracted the most attention from the astronomical community and are likely to continue to do so in the coming years. They are among the most likely targets for new redshift surveys. At least 54 of these clusters have detectable X—ray emission indicating that they are real systems and not simply due to the projection of physically different foreground and background groups. (At the time of this writing 36 have pointed ROSATobservations available from the public archive.) Each cluster has a measured redshift, so that their distances do not need to be approximated. Lastly, avoiding clusters close to the plane of the Galaxy helps to minimize the effect of obscuring dust which would need to be estimated and corrected for in the calculation of a limiting magnitude, as well as helping to keep down the number of misclassified stars in the sample. Table 2.1 is a listing of cluster parameters for the HGT sample. Column (1) lists the cluster name. Columns (2) and (3) list the center of the cluster as specified by Abell in 1950 coordinates. The galactic latitude is given in column (4). Distance classes and richness classes are given in columns (7) and (8), respectively. Column (7) lists the Bautz-Morgan (BM) type (Bautz & Morgan 1970). The revised Rood- Sastry (RS) type given by Struble & Rood (1982) is listed in column (8). Column (9) lists the cluster redshift from Struble & Rood (1991). 15 TABLE 2.1. HGT Cluster Parameters Cluster RA (1950) DEC (1950) b D R BM RS 2 (1) (2) (3) (4) (5) (6) (7) (8) (9) A 21 00 07.9 +28 22 —33.74 4 l I B 0.0948 A 76 00 07.2 +06 30 —55.97 3 0 II-III L 0.0377 A 85 00 09.1 —09 38 -72.08 4 1 I cD 0.0556 A 88 00 00.4 —26 20 —87.80 3 1 III 0.1086 A 104 00 47.1 +24 15 —38.35 4 1 II-—III F 0.0822 A 119 00 53.8 -—01 32 -64.11 3 1 II—III C 0.0446 A 121 00 55.0 -07 17 -69.83 4 1 III I 0.1048 A 147 01 05.6 +01 55 —60.42 3 0 III I 0.0441 A 151 01 06.4 —15 41 —77.62 3 1 II cD 0.0526 A 154 01 08.3 +17 24 --44.95 3 1 II B 0.0612 A 166 01 12.1 —16 33 -77.90 4 1 III F 0.1156 A 168 01 12.6 -00 02 —62.05 3 2 II—III I 0.0457 A 189 01 21.1 +01 24 -60.19 4 1 III I 0.0349 A 193 01 22.5 +08 27 —53.25 4 1 II cD 0.0478 A 194 01 23.0 —--01 46 ~63.10 1 0 II L 0.0178 A 225 01 36.2 +18 38 -42.56 4 1 II—III I 0.0692 A 246 01 42.1 +05 34 -54.62 4 1 II—III F 0.0753 A 274 01 52.2 —06 32 —64.29 4 3 III I 0.1289 A 277 01 53.3 —07 38 -65.04 3 1 III I 0.0947 A 389 02 49.1 —25 07 -—63.04 4 2 II F 0.1160 A 399 02 55.2 +12 49 -39.47 3 1 I—II cD 0.0725 A 400 02 55.0 +05 50 —44.93 1 l II—III I 0.0231 A 401 02 56.2 +13 23 —38.87 3 2 I cD 0.0752 A 415 03 04.4 —12 15 —54.89 4 1 II cD 0.0788 A 496 04 31.3 +13 22 -36.49 3 1 I cD 0.0326 A 500 04 36.8 -22 12 -38.49 4 1 III I 0.0666 A 514 04 45.5 -20 32 —36.02 3 1 II—III F 0.0697 A 634 08 00.5 +58 12 +3364 3 0 III F 0.0266 A 671 08 25.4 +30 36 +3311 3 0 II—III C 0.0497 A 779 09 16.8 +33 59 +44.41 1 0 I—II cD 0.0201 A 787 09 23.5 +74 38 +3620 4 2 II F 0.1355 A 957 10 11.4 -—00 40 +4288 4 1 I—II L 0.0437 A 978 10 18.0 —06 17 +40.35 3 1 II F 0.0527 A 993 10 19.4 —04 43 +4169 3 0 III I 0.0530 A 1020 10 25.2 +10 40 +5233 4 1 II-III I 0.0650 A 1035 10 29.2 +40 29 +5846 3 2 II-III F 0.0799 A 1126 10 51.3 +17 08 +6098 4 1 I—II B 0.0828 A 1139 10 55.5 +01 47 +52.66 3 0 III I 0.0376 A 1185 11 08.1 +28 57 +67.76 2 1 II C 0.0349 A 1187 11 08.9 +39 51 +6585 3 1 III I 0.0791 A 1213 11 13.8 +29 33 +6901 2 1 III C 0.0484 A 1216 11 15.2 —04 12 +51.14 4 1 III F 0.0524 A 1228 11 18.8 +34 37 +6944 1 1 II—III F 0.0344 A 1238 11 20.4 +01 23 +5642 4 1 II C 0.0716 A 1254 11 23.8 +71 22 +44.46 3 1 III I 0.0628 A 1257 11 23.4 +35 37 +7005 3 0 III F 0.0339 A 1291 11 29.3 +56 19 +57.77 3 1 III F 0.0586 TABLE 2.1. (continued) 16 Cluster RA (1950) DEC (1950) b D R BM RS 2 (1) (2) (3) (4) (5) (6) (7) (8) (9) A 1318 11 33.7 +55 15 +5900 3 1 II C 0.0189 A 1364 11 41.1 —01 30 +5680 4 1 III C 0.1070 A 1365 11 41.8 +31 11 +7488 4 1 III F 0.0763 A 1367 11 41.9 +20 07 +7304 1 2 II—III F 0.0205 A 1377 11 44.3 +56 01 +5911 3 1 III B 0.0509 A 1382 11 45.6 +71 43 +4482 4 1 II cD 0.1046 A 1383 11 45.5 +54 54 +6017 4 1 III I 0.0598 A 1399 11 48.6 —02 50 +5645 4 2 III I 0.0913 A 1412 11 53.1 +73 45 +4307 4 2 III C 0.0839 A 1436 11 57.9 +56 32 +5947 3 1 III I 0.0646 A 1468 12 03.1 +51 42 +6420 4 1 I C 0.0853 A 1474 12 05.4 +15 14 +7417 4 1 III I 0.0778 A 1496 12 10.9 +59 33 +5718 4 1 HI I 0.0961 A 1541 12 24.9 +09 07 +7086 4 1 I-II B 0.0892 A 1644 12 54.6 -17 06 +45.48 4 1 II CD 0.0456 A 1651 12 56.8 —03 56 +5861 4 1 I—II cD 0.0842 A 1656 12 57.4 +28 15 +87.96 1 2 II B 0.0230 A 1691 13 09.1 +39 29 +7722 3 1 II cD 0.0722 A 1749 13 27.3 +37 53 +7679 4 1 II CD 0.0562 A 1767 13 34.2 +59 29 +5699 4 1 II CD 0.0712 A 1773 13 39.6 +02 30 +6281 3 1 III F 0.0776 A 1775 13 39.6 +26 37 +7870 4 2 I B 0.0718 A 1793 13 46.1 +32 32 +7663 4 1 III I 0.0849 A 1795 13 46.7 +26 51 +77.16 4 2 I cD 0.0631 A 1809 13 50.8 +05 25 +6355 4 1 II . cD 0.0788 A 1831 13 56.9 +28 14 +7497 3 1 III F 0.0749 A 1837 13 59.1 —10 56 +4808 4 1 I—II cD 0.0376 A 1904 14 20.3 +48 48 +6229 3 2 II—III C 0.0719 A 1913 14 24.5 +16 54 +6659 4 1 III I 0.0533 A 1927 14 28.8 +25 54 +6768 4 l I—II cD 0.0740 A 1983 14 50.4 +16 57 +6011 3 1 III F 0.0458 A 1991 14 52.2 +18 51 +6051 3 1 I F 0.0589 A 1999 14 52.6 +54 32 +5477 4 l II—III I 0.1032 A 2005 14 56.6 +28 01 +6184 4 2 III B 0.1251 A 2022 15 02.2 +28 38 +6066 3 1 III F 0.0565 A 2028 15 07.1 +07 43 +5188 4 1 II—III I 0.0772 A 2029 15 08.5 +05 57 +5055 4 2 I cD 0.0777 A 2040 15 10.3 +07 37 +51.18 4 1 III C 0.0456 A 2048 15 12.8 +04 35 +4886 4 1 III C 0.0945 A 2052 15 14.3 +07 12 +5012 3 0 I—II CD 0.0351 A 2061 15 19.2 +30 50 +57.17 4 1 III L 0.0782 A 2063 15 20.6 +08 49 +4972 3 1 II cD 0.0337 A 2065 15 20.6 +27 54 +5656 3 2 III C 0.0722 A 2067 15 21.2 +31 06 +5676 4 1 III cD 0.0726 A 2079 15 26.0 +29 03 +5653 3 1 II—III cD 0.0657 A 2089 15 30.6 +28 12 +5443 4 1 i1 CD 0.0743 A 2092 15 31.3 +31 20 +5461 4 l II—III I 0.0669 17 TABLE 2 . 1. (continued) Cluster RA (1950) DEC (1950) b D R BM RS z (1) (2) (3) (4) (5) (6) (7) (8) (9) A 2107 15 37.6 +21 56 +5150 4 1 I CD 0.0421 A 2124 15 43.1 +36 14 +5231 3 1 I cD 0.0671 A 2142 15 56.2 +27 22 +4870 4 2 II B 0.0911 A 2147 16 00.0 +16 03 +4449 1 1 III F 0.0377 A 2151 16 03.0 +17 53 +4453 1 2 III F 0.0360 A 2152 16 03.1 +16 35 +4402 1 1 II F 0.0444 A 2162 16 10.5 +29 40 +4604 1 0 II—III I 0.0318 A 2175 16 18.4 +30 02 +4442 4 1 II cD 0.0978 A 2197 16 26.5 +41 01 +4381 1 1 III L 0.0303 A 2199 16 26.9 +39 38 +43.71 l 2 I cD 0.0312 A 2255 17 12.2 +64 09 +3495 3 2 II—III C 0.0747 A 2256 17 06.6 +78 47 +31.74 3 2 II—III B 0.0550 A 2328 20 45.4 ~18 00 ~33.56 4 2 I I 0.1470 A 2347 21 26.7 ~22 26 ~44.17 4 1 III I 0.1196 A 2382 21 49.3 ~15 53 ~4694 4 1 II—III L 0.0648 A 2384 21 49.5 ~19 47 ~48.40 4 1 II—III F 0.0943 A 2399 21 54.9 ~08 02 —44.57 3 1 III I 0.0587 A 2410 21 59.4 ~10 09 ~4658 4 1 III I 0.0806 A 2457 22 33.3 +01 13 ~46.60 4 1 I~II C 0.0597 A 2634 23 35.8 +26 46 ~33.06 1 1 II CD 0.0315 A 2657 23 42.3 +08 53 ~50.29 3 1 II F 0.0414 A 2666 23 48.4 +26 53 ~33.80 1 0 I cD 0.0273 A 2670 23 51.6 ~10 41 ~68.52 4 3 I~II CD 0.0774 A 2675 23 53.0 +11 10 -49.l2 4 1 II F 0.0726 A 2700 00 01.3 +01 48 —58.63 4 1 II cD 0.0978 18 2.2 Digital Sky Surveys In an effort to make the large photographic surveys (such as POSS) conducted over the last half century or so more easily accessible and therefore more useful, a number of projects have been carried out, or are currently underway, to transfer them to an electronic form. These include the APM at Cambridge, APS at Minnesota, COSMOS at Edinburgh, PDS at STScI and PPM at the US. Naval Observatory (see Lasker 1995 for a review). The existence of these databases will greatly facilitate computerized procedures for catalog construction, and the making of finding charts. It also offers the possibility of doing science such as the large-scale distribution of stars in the Milky Way Galaxy (Larson 1996) or, as in this case, the distribution of galaxies within a large number of galaxy clusters. While there are a number of such projects, the only one suitable for the present work is the APS catalog of POSS I at Minnesota. The APS survey offers positions and magnitudes as well as a classification of objects as stars or galaxies, unlike the PDS scans at STScI which only provide images. And, although only partially completed, the APS survey is available to the public, unlike the APM survey which is only available for collaborative use. 2.3 The Minnesota Automated Plate Scanner The Automated Plate Scanner (APS) project at the University of Minnesota (Pen- nington et al 1993) uses a “flying spot” laser scanner. The light from a helium-neon laser is sent through a rapidly rotating, eight-sided prism. A lens and beam-splitting prism are used to form three 12 micron spots simultaneously on the E (red) and O (blue) plates of POSS I as well as a reticle for positional reference of the other two spots. The spots are detected using silicon photodetectors. 19 Of the three available modes of operation, the POSS I plates have been scanned in the threshold densitometry mode. That is, pixel information is saved only when the density exceeds 65% above the median background value, which corresponds roughly to 23.5 B mag arcsec"? The position of the ingress (when the density threshold has been met), the position of egress (when the density threshold is no longer met) and the pixel data in-between are recorded. Positions are measured at a resolution of 0.366 microns, which corresponds to 0.0245 arcsecs on the sky, with a repeatability error of 0.6 arcsecs. The magnitudes on each plate are estimated from the image diameter size. The integrated isophotal magnitudes of galaxies are listed as being accurate to 0.5 mag- nitudes. However, this error is just the reproducibility error in the measurement on any given plate. Due to variation in the emulsion of various plates the actual error in magnitudes is higher. Comparison of galaxies in the plate overlap regions indicate the the plate to plate variation is in some cases as high as 1.0 magnitude on the blue plates. This indicates the need for better magnitude calibration if studies involving luminosity functions or comparisons between objects on different plates is to be done. This does not however, affect the current study, except that the sampling depth in each cluster is not really a constant but may vary slightly frOm cluster to cluster. In those clusters which include data from more than one plate, the magnitudes were calibrated using the mean value of all galaxies in the overlap region. 2.4 Neural Network Star / Galaxy Classification In general, a single POSS I plate will produce on the order of 250,000 detected images. In order to classify this many objects, a fast and fully-automated procedure is needed. The solution of Odewahn et al. (1992) was to employ a neural network. 20 Neural networks are a family of artificial intelligence algorithms which are capable of performing pattern recognition. Typically, a neural network is trained using a sample of pre-classified objects. To perform the star/ galaxy separation 14 parameters of the image are input into the neural network. These are: diameter, ellipticity, average transmission, central transmission, ratio of ellipse area to area from the pixel count, the logarithm of the area from pixel count, first moment of the image, rms error of ellipse fit to transit endpoints, the Y centroid error, and the five image gradients defined as: G..- = T" ‘ T‘. (21) 7‘5 — 7‘]: where T,- is the median transmission value in an elliptical annulus and semimajor axis Ti. Although the performance of a neural network can be judged by viewing the output, it is not generally clear by what criteria the classification is being made. For instance, the original training set at APS contained stars and galaxies, but did not contain double stars. As a result, any suitably-elongated image was classified as a galaxy with high probability. Although this problem has be identified and remedied with a training set of double stars and the plates are being re-processed, some of the data used in this thesis were classified before such improvements were made. 2.4.1 Contamination from Stars The catalog of galaxy positions and magnitudes used in this thesis includes objects classified as galaxies by the neural net with a probability of 0.85 or higher. Each galaxy assigned a magnitude brighter than 19.0 on the blue plate was examined using the Digitized Sky Survey (DSS) done with the PDS machines at STScI. (Although there are plans to place the APS images online, at the time of writing these are still 21 only available for a small fraction of the online catalogs.) Objects which were actually stars or binary stars were deleted from the catalogs. From this, it was noticed that several plates had far more contamination than the expected 10-20%. A2666, on plate mlp 779, had the most misclassified stars with a contamination rate of 42%. The reason for such a large contamination is probably due to the fact that A2666 is a nearby cluster and therefore the 1.5 Mpc region covers a large area of sky, which includes relatively more optical binary and bright stars. Furthermore, A2666, at a galactic latitude of ~33, is relatively close to the plane of the Galaxy. This was a major reason to limit this study to clusters with |b| 2 30. In this and other clusters that showed a contamination rate greater than 10%, every galaxy was examined. In all 16,000 galaxies were examined, or about half of the catalog used in this thesis. Above 19.0 magnitude, an overall contamination rate of 12% was observed. The breakdown with magnitude is as follows: m S 16.5, 23%; 16.5 S m < 18.0, 19%; 19.0 S m < 18.0, 13%; and m > 19.0, 8%. It is somewhat surprising that the greatest contamination level is for galaxies brighter than 16.5 magnitude. In general this appears to be due to bright, saturated stars being classified as galaxies. Also, there is a sharp drop off in the number of stars deleted at magnitudes fainter than 19.0. This is likely to be a result of the greater difliculty encountered in the determination of which objects are stars and which are galaxies as the plate limit is approached, and should not be used to estimate contamination levels at these magnitudes. However, if it is assumed that the major source of misclassified galaxies (those which are actually double stars) remains constant with magnitude and that the overall contamination level is on the order of 15%, then the contamination left in the sample after deleting the probable stars should be near the 8% level. This is considered acceptable since misclassified stars should appear randomly (with a constant density) over the cluster and as such are very unlikely to be identified as coherent substructure. 22 There are also a number of objects in the APS catalog which are classified as galaxies, usually with with very high probability, with a magnitude of 8.00. In these cases the neural network has become confused by bright, and therefore large, stars or galaxies. When such an object was encountered and determined to be a galaxy from examination of the DSS image, the Third Reference Catalog of Bright Galaxies (de Vaucouleur et al. 1991, hereafter 3RC) was searched for a nearby galaxy. The photographic magnitude listed there (mg) was transformed to the m0 magnitude of the APS using the average offset calculated from other galaxies on the same plate which also had 7723 listed in 3RC. An average 1.0 magnitude needed to be added to m B to obtain m0. In 5 cases, an entry was not found in 3RC and these galaxies were assigned a magnitude of 16.5, the limit to which 3RC attempted to be complete plus one magnitude. Lastly, one cluster, A2079, had to be deleted from the sample due to a bright star which caused a hole in the galaxy catalog. Its map is still given for completeness but it is not used in any calculations. 2.4.2 Completeness Because the goal was to make the analysis of each cluster as consistent as possible, the magnitude limit for each cluster was set to an absolute magnitude Mo = ~16.2 + 510g h, as opposed to simply a fixed apparent magnitude. The value of this limit was obtained by finding the absolute magnitude of the faintest galaxies on the plate which contained the furthest cluster. That is, the cluster A2328 at z = 0.147 has a magnitude limit of mo = 22.2. With the magnitude limit so defined, each cluster is sampled to the same depth in the luminosity function, or about three magnitudes below the knee in the Schechter luminosity function. The luminosity function for galaxies is usually fit to an analytic function due 23 to Schechter (1976). The number of galaxies with luminosity in a range of L + dL according to the Schechter function is n(L)dL = N“(L/L’)‘°‘ exp(~L/L*)d(L/L*) (2.2) where L“ is a characteristic luminosity and is often referred to as the “knee” in the Luminosity function. Likewise the integrated luminosity function is N(L) = n*F(1 — a, L/L*) (2.3) where I‘(a, :r) is the incomplete gamma function. Two effects can cause cluster-to-cluster variation. The first is that the sensitivity of the emulsion on the photographic plates may vary either across a single plate or be- tween plates and thereby cause variations in the measured magnitudes of the galaxies. As already mentioned, this could be as much as one magnitude, as measured by the difference in magnitudes of galaxies in the plate overlap regions. The second is vari- ations in sample completeness, which is a function of the apparent magnitude cutoff used. In general there will be proportionately fewer galaxies detected at a magnitude of 22 than at a magnitude of 19. There are two reasons for this eflect. First, galaxies near the plate limit may fall below the detection limit due to random fluctuations in the background or emulsion sensitivity. Second, small compact galaxies, such as dwarf ellipticals, are more likely to be misclassified as stars at fainter magnitudes. Odewahn et al. (1993) have examined the completeness of the APS data by comparing the APS output for an area centered on the north galactic pole region with that of other studies of this region. From this it was concluded the APS data is 95% complete at an m0 = 19.5, 90% complete for 19.5 < mo < 20.0, and 80% 20 < mo < 21. Thus more distant clusters with a magnitude cutoff of 21 can have the galaxy counts depleted by as much as 20% as compared to a nearby cluster. One way 24 to avoid this completeness problem would be to use a brighter absolute magnitude for the cutoff. However, if this is done nearby clusters will have their magnitude cutoff raised as well. For some sparsely populated clusters, such as A194 or A634, this would result in too few member galaxies to carry out the substructure tests with any reliability. With this cutoff the cluster with the smallest number of galaxies is A634 with 71 galaxies. The Monte Carlo tests discussed in chapter 4 indicate that, with this number of galaxies, only very wide separations between the groups are likely to returned as significant most of the time. In retrospect, clusters with redshifts greater than 2 = 0.1 should probably not have been included. 2.5 Estimation of Background Contamination Not all of the galaxies which appear in the cluster maps will be gravitationally bound to the clusters. The presence of some galaxies will be due to the projection of back- ground or foreground galaxies onto the plane of the sky. (For convenience both background and foreground galaxies will be lumped together under the term back- ground.) An estimate of an assumed constant-density background can be obtained for each cluster from the adaptive-kernel procedure discussed in Chapter 3. The back- ground density can be taken as the density at the point with the largest bandwidth factor A,- (see section 3.5). Defined as such, this density corresponds to the lowest density region (but not necessarily the lowest density) in each map. Although other definitions of the background density are possible, this one has the advantage of being based on a density measurement for each cluster. This is quite different from esti- mates that count the number of galaxies in random areas of the sky and then assume that the background rate is constant for all clusters. It is possible that this procedure overestimates the background since we would expect that even at the lowest densities in the 1.5h‘1 Mpc region, some of the density there will be due to cluster members. 25 On the other hand, there is really no reason to assume that the background density is actually constant. Maps of the clusters may contain background groups and other clusters, providing a clumpy background. Some examples include A85, which con- tains the more distant cluster A89 (as well as the cluster A87 which is at the same distance as A85 [den Hartog 1995]), and A1999, which contains A2000. In such cases, contamination could be greater. Unidentified foreground clusters are not expected to be as big of a potential problem because they will appear larger and contribute a nearly constant density across the cluster maps, which should be well approximated by this method of background determination. It is this background estimate that the significance of the subclusters is estimated against in the program DEDICA discussed in Chapter 4. The background density estimates are listed in Table 2.2 for the Dressler clusters and Table 2.3 for the HGT sample clusters. The cluster is listed in column (1). Column (2) provides the total number of galaxies in each cluster. The number of expected background galaxies is given in column (3). Column (4) is the percentage of the total number. Column (5) is the density of the estimated background in galaxies arcmin'z. For the Dressler clusters, the estimated background varies from half that estimated by Dressler to nearly five times as much for A2256. 26 TABLE 2.2. Estimated Background — Dressler sample CIUSter N tot N back %Ntot 0' back (1) (2) (3) (4) (5) A 0014 79 17 21 0.0035 A 0076 72 10 14 0.0028 A 0119 116 15 13 0.0040 A 0151 105 15 14 0.0019 A 0154 79 17 21 0.0047 A 0168 106 6 6 0.0016 A 0194 75 18 24 0.0022 A 0376 119 25 21 0.0080 A 0400 92 11 12 0.0011 A 0496 81 15 18 0.0041 A 0539 99 17 17 0.0021 A 0548 234 24 10 0.0030 A 0592 61 12 19 0.0014 A 0754 150 26 18 0.0033 A 0838 62 38 61 0.0046 A 0957 82 16 19 0.0020 A 0978 62 12 20 0.0015 A 0979 86 18 21 0.0022 A 0993 91 27 30 0.0034 A 1069 47 8 18 0.0010 A 1139 63 10 16 0.0013 A 1142 59 8 13 0.0009 A 1185 44 15 33 0.0030 A 1377 52 12 24 0.0029 A 1631 90 23 25 0.0028 A 1644 145 19 13 0.0024 A 1656 245 22 9 0.0028 A 1736 166 18 11 0.0022 A 1913 86 26 30 0.0035 A 1983 123 20 16 0.0025 A 1991 53 9 18 0.0013 A 2040 108 20 19 0.0028 A 2063 110 13 11 0.0017 A 2151 152 13 8 0.0017 A 2256 83 25 30 0.0100 A 2589 72 20 27 0.0055 A 2634 132 27 21 0.0064 A 2657 82 17 21 0.0048 DC 0003-50 79 11 14 0.0014 DC 0103-47 53 12 22 0.0014 DC 0107-46 55 10 18 0.0013 DC 0247-31 48 15 32 0.0019 DC 0317-54 65 17 26 0.0021 DC 0326-53 161 37 23 0.0045 DC 0329-52 190 12 6 0.0014 TABLE 2.2. (continued) 27 CIUSter N tot N back %Ntot aback (1) (2) (3) (4) (5) DC 0410-62 64 24 37 0.0029 DC 0428-53 131 21 16 0.0025 DC 0559-40 112 10 9 0.0013 DC 0608-33 122 6 5 0.0008 DC 0622-64 98 12 13 0.0015 DC 1842-63 55 15 27 0.0018 DC 2048-52 216 42 19 0.0052 DC 2103-39 108 12 11 0.0015 DC 2345-28 95 30 32 0.0037 DC 2349-28 68 24 35 0.0030 Centaurus 73 18 24 0.0022 28 TABLE 2.3. Estimated Background -— HGT sample 011131391. Ntot Nback %Ntot ”back (1) (2) (3) (4) (5) A 21 291 40 14 0.0306 A 76 195 31 16 0.0045 A 85 323 64 20 0.0146 A 88 72 24 33 0.0237 A 104 151 27 18 0.0153 A 119 268 22 8 0.0036 A 121 145 34 23 0.0312 A 147 155 24 15 0.0038 A 151 342 43 13 0.0102 A 154 272 19 7 0.0069 A 166 157 19 12 0.0219 A 168 235 22 9 0.0038 A 189 157 26 16 0.0024 A 193 264 55 21 0.0108 A 194 129 8 7 0.0002 A 225 202 20 10 0.0082 A 246 103 33 32 0.0137 A 274 174 18 10 0.0246 A 277 230 43 19 0.0324 A 389 173 10 6 0.0116 A 399 254 16 6 0.0067 A 400 190 31 16 0.0014 A 401 288 23 8 0.0108 A 415 243 56 23 0.0295 A 496 226 20 9 0.0017 A 500 225 30 14 0.0114 A 514 282 24 8 0.0108 A 634 71 23 32 0.0014 A 671 293 52 18 0.0108 A 779 115 14 12 0.0006 A 787 154 29 19 0.0448 A 957 288 40 14 0.0066 A 978 295 29 10 0.0069 A 993 272 56 21 0.0135 A 1020 265 49 18 0.0175 A 1035 283 31 11 0.0169 A 1126 248 57 23 0.0349 A 1139 168 38 23 0.0047 A 1185 335 48 14 0.0037 A 1187 227 28 13 0.0150 A 1213 261 13 5 0.0025 A 1216 102 24 24 0.0057 A 1228 278 37 13 0.0038 A 1238 180 23 13 0.0101 A 1254 263 26 10 0.0087 TABLE 2.3. (continued) 29 CIUSter N tot N back %Ntot aback (1) (2) (3) (4) (5) A 1257 212 47 22 0.0046 A 1291 395 29 7 0.0069 A 1318 286 20 7 0.0055 A 1364 226 37 16 0.0358 A 1365 163 19 12 0.0094 A 1367 200 39 19 0.0015 A 1377 402 88 22 0.0198 A 1382 183 43 23 0.0398 A 1383 284 30 11 0.0092 A 1399 284 33 11 0.0229 A 1412 244 32 13 0.0189 A 1436 358 46 13 0.0161 A 1468 166 29 18 0.0175 A 1474 187 34 18 0.0182 A 1496 355 59 16 0.0437 A 1541 205 7 4 0.0048 A 1644 297 36 12 0.0061 A 1651 205 14 7 0.0083 A 1656 424 24 6 0.0011 A 1691 247 46 19 0.0203 A 1749 219 44 20 0.0130 A 1767 308 35 11 0.0146 A 1773 282 16 6 0.0080 A 1775 268 52 19 0.0213 A 1793 248 44 18 0.0269 A 1795 288 44 15 0.0142 A 1809 308 49 16 0.0259 A 1831 308 45 15 0.0205 A 1837 268 32 12 0.0038 A 1904 386 26 7 0.0111 A 1913 276 33 12 0.0080 A 1927 245 24 10 0.0111 A 1983 439 75 17 0.0123 A 1991 368 74 20 0.0215 A 1999 187 18 10 0.0164 A 2005 139 19 14 0.0250 A 2022 322 55 17 0.0148 A 2028 231 33 14 0.0167 A 2029 437 62 14 0.0309 A 2040 278 69 25 0.0121 A 2048 314 37 12 0.0280 A 2052 270 37 14 0.0038 A 2061 285 47 16 0.0233 A 2063 211 12 6 0.0012 A 2065 422 43 10 0.0188 A 2067 283 34 12 0.0153 A 2079 318 67 21 0.0247 30 TABLE 2.3. (continued) CIUSter N tot N back %Ntot 0 back (1) (2) (3) (4) (5) A 2089 158 11 7 0.0050 A 2092 267 24 9 0.0089 A 2107 275 45 16 0.0067 A 2124 298 47 16 0.0169 A 2142 311 29 9 0.0197 A 2147 465 35 8 0.0037 A 2151 388 61 16 0.0071 A 2152 471 81 17 0.0095 A 2162 124 17 14 0.0015 A 2175 448 35 8 0.0284 A 2197 313 35 11 0.0027 A 2199 389 46 12 0.0035 A 2255 417 22 5 0.0117 A 2256 451 38 8 0.0115 A 2328 131 14 11 0.0258 A 2347 90 15 16 0.0176 A 2382 197 13 6 0.0044 A 2384 127 10 8 0.0075 A 2399 256 26 10 0.0075 A 2410 235 29 12 0.0157 A 2457 248 16 7 0.0049 A 2634 411 76 18 0.0062 A 2657 171 17 10 0.0025 A 2666 171 31 18 0.0018 A 2670 255 26 10 0.0121 A 2675 182 33 18 0.0149 A 2700 129 38 30 0.0308 Chapter 3 PROBABILITY DENSITY ESTIMATION 3.1 Introduction Probability density estimation has a wide field of application. As such, it has received a great deal of interest from the statistical community. The question which density estimation attempts to answer is the following: given a sample of n independent observations, X, . . .Xn, what is the probability that the next observation will be at any given position 2:. Or, what is f (x), the probability density function (PDF), such that b P(a < X < b) = / f(:r) day. (3.1) This problem can be approached parametrically or nonparametrically. In the parametric approach, the form of the PDF is assumed and various parameters mea- sured. The most commonly applied PDF is the normal or Gaussian distribution, where the average u and the standard deviation 0 of the observations are estimated from the data. In fact, it has become so widely used that many researchers continue to use p and 0 even for distributions which are not Gaussain, and for which robust estimators for the location and scale of the data are required. If, as is often the 31 32 case in astronomy, there is no apriorz' reason to assume a particular form of f (:13) a nonparametric approach that makes as few assumptions as possible about the density being estimated is desirable. 3.2 The Histogram The oldest and most widely used form of nonparametric density estimate is the his- togram, which dates back at least to the work of Graunt in 1662. The density estimate of a histogram, f(:z:), is defined as: f(a:) = N15(no. of X, in same bin as x), (3.2) where h is the width of the bins and N is the total number of observations. In addition to specifying the bin width h which controls the smoothness of the histogram, it is also necessary to specify an origin for the bins. While the choice of origin may seem to be a trivial matter, it can have quite an effect on the shape of the histogram constructed, especially with small to moderate-sized data batches. As an example, Figures 3.1 and 3.2 show histograms constructed from 88 measured redshifts of Abell 400. The data are taken from Beers et al. (1992) and have errors on the order of 50 km 5". Both histograms are constructed using the same data and the same bin width, 300 km S“. The only difference between the plots is the choice of bin origin. In Figure 3.1 the origin of the bins has been set at 5000 km 3‘1 while that of Figure 3.2 has been shifted by 200 km s‘1 and set at 5200 km 3“. Although statistically the two histograms show the same thing, one gives the impression of bimodality while the other does not. In this case, it turns out that most choices of bin origin lead to a bimodal histogram, as correctly identified by Beers et al., and the first choice of origin was simply unfortunate. 33 Abell 400 004 IIIIIIIIIIIIITIIIIIITWIITIITIIIIIIII‘I— .003 —- -— .002 —- ~4 )— -J to. - .1 )— _— 0 Ali _ I I l I l L I J l l I I I I l I I I I l l I I I ll 14 I l I l I l l I I L I 5000 5500 6000 6500 7000 7500 8000 8500 9000 velocity (km/s) Fig. 3.1.— Histogram of A400 velocity measurements from Beers et al. (1992). The bin width is 300 km s'1 and the bin origin is at 5000 km s‘l. 34 Abell400 .005,_IIIITTIIIIIIIIIITII FVIIIIIIIITIIIITVIL p _ .004 — :4 L 4 .003 P — I. 2 - 1 .002 — —_l l—_— — C _ .001 B— — I ———l—_ 1 0 I I I I l I I I I l I I I I l I I I I l I I I I l I I I I l I I I I l I I I IJ 5000 5500 6000 6500 7000 7500 8000 8500 9000 velocity (km/s) Fig. 3.2.— Histogram of A400 velocity measurements from Beers et al. (1992). The bin width is 300 km s"1 and the bin origin is at 5200 km ‘1. 35 3.3 Generalizations of the Histogram There are two simple ways to free the histogram from dependence on bin origin. Perhaps the most obvious solution is to construct m histograms, each one with the origin shifted by 6 = h/m, and average them together as discussed by Chamayou (1980). The result is referred to as an Average Shifted Histogram (ASH). Thus: fla=mzma (u) where f,- is the histogram estimator given in equation (3.2) with bin origins of 0, h/m, 2h/m, ..., (m — 1)h/m, respectively. It is possible to rewrite this in a more computationally convenient form by defining a new bin width 6 = h/m (see Scott 1992). Then, A 1 ml f(x — _nh wm(z z)[no. of X,- in same bin as x], (3.4) i=l~ ~m where the bin is now the smaller bin and wm(z') is a weight function given by: wm(z') = 1 — (3.5) Note that unlike the histogram, with its box-shaped weight function, the weight function for the ASH is an isosceles triangle. An example of the ASH is given for the Abell 400 data in Figure 3.3 with m = 32 and bin width of 300 km 3“. Here the bimodal nature of the PDF is clear, thus justifying the above statement that the choice of bin origin in Figure 3.1 was merely unfortunate. There is also evidence of the possible third peak at higher velocity causing the density estimate to flatten off. An alternative solution is referred to by Silvermann as the “naive estimator.” Instead of rigidly fixing the bins to some arbitrary origin on the coordinate axis and counting the number of observations which lie in each bin, as in the classical 36 Abell 400 IlllIIIIITTTFTITIIITTIIIIIIFITHTTFIIHII .0006 —- — .0004 -— ~— .0002 - - I— _ 0 I I I I LI 1 I; l I I I I l I I I I l I I I I l I I I I l I I I I l I I I I 5000 5500 6000 6500 7000 7500 8000 8500 9000 velocity (km/s) Fig. 3.3.— Average Shifted Histogram of A400 velocity measurements. The bin width is 300 km s’1 and m=32. 37 histogram, in the naive estimator the bin origins are allowed to float based on the position of each data point. This is accomplished by placing a box of width 2h and height 1/(2hn) centered on each data point and summing the box heights at each coordinate position 2:. Or: fps) = 271%; w (1'3 1X1), (3.6) 1 The function w(:z:) is again a weight function and in this case is: 1 if |:1:| < l _ 2 10(3) _ { 0 otherwise. (3'7) Figure 3.4 shows a density estimate constructed in this manner, again for the A400 data. Notice that this density estimate leads to very sharp peaks which can be aesthetically unpleasant at best and misleading at worst. Also, like the histogram, its derivative is zero everywhere except at those points where it is discontinuous, in this case at each X,- d: h. Nevertheless, it clearly shows the two peaks as well as the lower density plateau at high velocity; in short, all the information shown in the ASH. 3.4 The Kernel Estimator Although both the ASH and the naive estimator discussed above are independent of the the choice of origin, they still retain the discontinuous nature of the histogram. This prevents them from being useful when derivatives of the density estimate are sought, as in the peak identification procedure employed in Chapter 4. The discon- tinuities in both the naive estimate and the ASH arise from the discontinuity of the weight functions: the box shape in the naive estimator and the histogram or the triangle shape in the ASH. This problem can be overcome by generalizing the weight functions to different shapes which are themselves continuous. Thus we can gener- 38 Abell 400 l I I l I I l I I I I I7 I I I I I I I I I l I I l I I I I I I I I I I I I I .8 — - .6 —— — .4 - - 2 _— —( o . I I 14 1 I I I I l I I I I l I I I I l I 14 I LI I I I l I I I I l I I I 5000 5500 6000 6500 7000 7500 8000 8500 9000 velocity (km/s) Fig. 3.4— “Naive estimator” for the A400 velocity data. Again the bin width is set to 300 km s“. 39 alize equation (3.6) by replacing the weight function w(:r) with a continuous kernel function K (2:) where: /_°° K(x)d:c = 1. (38) Instead of placing a box over each data point and summing the boxes, a bump with shape controlled by the kernel function and width specified by the smoothing pa- rameter h is used. In the ASH, the weights must sum to m so that wm(z') could be replaced by . _ mK(i/m) ”mm“ "“1 KU/m) j=1—m i=1—m,...,m~1. (3.9) Some commonly used kernel functions are: Epanechnikov: §(1— 12:2) for :52 <1 _ 4 K05) _ { 0 otherwise ’ (3'10) Biweight: -1—(1 — 3:2)2 for lxl < l _ 16 K(:z:) _ { 0 otherwise ’ (3'11) and Normal: 1 _32/2 K (11:) = . (3.12) ——e \/ 271' A density estimate constructed by summing a series of K (X,) will inherit all of the continuous and differential prOperties of K (:17). It can be shown that the Epanechnikov kernel function (Epanechnikov 1969) min- imizes the mean integrated square error (MISE) between the density estimate and the true density, provided the optimal value of h is used. The choice of the form of 40 the kernel function is not critical since the efficiencies of the other kernels differ from that of the Epanechnikov kernel only slightly (Silvermann 1986). Thus, the choice can be made on the basis of the desired differentiability of the estimate or speed of calculation. For instance, the Epanechnikov and the biweight kernels both have discontinuous derivatives at a: = :I:1. On the other hand, the normal kernel has a continuous derivative everywhere, but suffers from infinite tails. Figure 3.5 shows the density function constructed with the A400 data employing a normal kernel. As with the ASH the two peaks are seen clearly in the kernel estimator as is the possible third, lower-density peak at higher velocity. This similarity between the ASH and the kernel density estimates is not an accident. It can be shown (Scott 1992) that in the limit as m —> 00 the ASH estimate approaches that of the kernel estimate and the two techniques are equivalent. Although the previous examples used only one-dimensional data, the same argu- ments apply in two. In fact, the problem of bin origin becomes even worse since not only is the two dimensional histogram affected by shifts in the :1: and y position of the origin, but also by rotations of the coordinate axis. Since this thesis is primarily concerned with density estimation in two dimensions, in the following discussion the kernel estimator is generalized apprOpriately. 3.5 Adaptive Smoothing Methods The kernel estimator provides a smooth density estimate which is independent of origin. However, use of a fixed value of h will yield a density estimate which is over-smoothed in high-density regions, tending to hide real structure, and under- smoothed in low-density regions, which are subject to Poisson noise. One solution to this difficulty is to vary the kernel width based on the local density. 41 Abell 400 I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I I l I I I l I I I— ._. .6 —- -- .4 — — _. ..l .2 ~ — O I I I I l I I I I l I I I I l I I I I l I I I I l I I I I l I I I I l I I I I 5000 5500 6000 6500 7000 7500 8000 8500 9000 velocity (km/s) Fig. 3.5.— Adaptive-kernel density estimate of A400 velocity data. The smoothing parameter h = 150 km s“. 42 Perhaps the simplest nonparametric adaptive smoothing method is the nearest neighbor density (hereafter NND.) The NND estimate in two dimensions is defined by: fix) — ~41“)— (3.13) — 7mmc (:13)2 ’ where “(22) is the distance to the kth nearest neighbor and n is the number of obser- vations. In the NN D, it is the value of k which controls the smoothing. Astronomers generally set k = 10, though It or iii/(“+9 gives the best estimate in d dimensions (Silvermann 1986) with the “constant” of proportionality depending on the position CE. There are, however, a number of problems with the N ND estimate. The tails of the estimate will fall off very slowly, or 1", regardless of the true distribution. Thus the NND estimate over-smooths in low density regions and generally performs worse than the fixed-kernel estimator. (This is true in one and two dimensions, but for d 2 5 the NND performs better than the fixed kernel estimator at least in the case of a normal density distribution.) Furthermore, the NND is not a smooth function since the derivative is discontinuous at the points 1/2(X(j) — X (3440) where the X (j) are the order statistics of the sample (216. X (1) g X (2) g . . . g X ( m.) An alternative to the NND, first proposed by Fix & Hodges (1951) and discussed in detail in the monograph by Silvermann (1986), is to vary the kernel width based on an estimate of the local density. Construction of the adaptive-kernel density estimate proceeds as follows: Obtain a “pilot estimate” f(a:) which satisfies f(X,-) > 0 for all 2'. Define “local bandwidth factors” A,- by: /\.- = {fig-AF (3.14) 43 where g is the geometric mean of the f(X,—): logg = n’1 ilog f(X,-) (3.15) i=1 and a is a sensitivity parameter satisfying 0 S a g 1. Define the “adaptive-kernel estimate” f(:z:) by: f(x) = n-1 2.: h““A,‘“K{h‘1A,-‘l(x — X,)}, (3.16) i=1 where d is the dimensionality of the data (in this case d = 2). Note that the adaptive-kernel estimator defined above is the same as the non- adaptive (fixed) kernel estimator, except that the window width h is now replaced by hA,, the local bandwidth indicators. The result is that the window width in the adaptive-kernel estimator is decreased in high-density regions and increased in low- density regions. The amount by which the window width is decreased or increased can be altered by changing the sensitivity parameter oz. With a = 0, for instance, all the bandwidth factors become 1 and the pilot density estimate is returned. As a approaches 1, the density estimator is similar to a nearest-neighbor method, which is prone to local noise and has heavy tails. It can be shown (Abramson 1982) that a value of a = 1 / 2 provides a smaller bias in the density estimate than that obtained using a fixed kernel width in both one and two dimensions (this is not necessarily true for other choices of (1.) Thus the value of a = 1 / 2 is adopted throughout this thesis. It needs to be pointed out that current statistical research indicates that the adaptive-kernel technique outlined above is not well behaved asymptotically (as the number of observations approaches infinity.) For data sets larger than 20,000, it can be shown that the adaptive procedure performs significantly worse than a fixed kernel estimator (Scott 1992). While this is cause for concern the adaptive-kernel 44 has, at present, the best track record for a wide variety of PDF’s with N g 200 and is therefore employed in this thesis. So far in the discussion, little attention has been paid to the subject of the smooth- ing parameter size. Since the choice of smoothing parameter has the greatest effect on the accuracy of the density estimate, a great deal of effort has be expended by statisticians searching for the best possible value of h. Unfortunately, the optimal smoothing size depends on the unknown density for which an estimate is sought and no value of h (or even a prescription for finding h) will give the best estimate for all density distributions. Because of this, many statisticians recommend choosing the smoothing parameter subjectively. That is, vary h until the desired level of smooth- ness is achieved. In fact, Scott (1992) has expressed the opinion that there is no wrong choice of h, as information is gained by all values. The drawback of this method is that different researchers will no doubt have different opinions of when an estimate is “smooth enough.” There is a great temptation to oversmooth since this leads to neater looking plots. This temptation should be avoided since further smoothing can be done by eye, but an oversmoothed estimate can not be un-smoothed, and real effects in the data can be lost. To avoid this, an automatic selection is clearly desirable. Two different methods for choosing h will be applied in this thesis. The first involves choosing the optimal smoothing parameter based on minimizing the MISE for a given kernel function with respect to a particular distribution or family of distributions. For density estimation of a bivariate-normal distribution this is: h1, = A(K){1/2(a: + 03)}1/2N“l/6, (3.17) where A(K), a constant that depends on the kernel function, is 0.96 and 2.04 for the normal and biweight kernels, respectively. This prescription is often referred to 45 as the “normal rule.” Being based on the normal distribution, density estimates constructed with hn will oversmooth multimodel densities. Based on experience, Silvermann suggests using h = 0.85h,, as a good compromise, as it works well with bimodal as well as skewed distributions. This method, which is quick and simple to calculate, will be used in constructing a consistent set of contour maps for the galaxy clusters. Because most of the clusters show evidence of multimodality, a value of h = 0.75hn is used. It should be noted that the factor of 0.75 is based on experience using the adaptive-kernel technique with clusters of galaxies and is somewhat arbitrary. Clearly, there is a trade ofl. A unimodal-normal cluster will be undersmoothed to some extent, while a multi-modal cluster will tend to be oversmoothed. Use of an adaptive-kernel technique however, partially compensates for this effect, and the final density estimate is relatively insensitive to variations in kernel width within 15% to 20% of the optimal value. The other method, least squares cross validation (LSCV), involves finding the value for h = hcv which minimizes the cross validation term in the expression for the integrated square error (ISE). The ISE is given by: ISE(f) = / f2(:r)da:+ / f2(:v)d:1:~2 / f(a:) f(:1:)da:. (3.18) Because the first term depends only on the actual density it is constant with respect to changes in h. Therefore minimizing the ISE is equivalent to minimizing the term: M001) = / f(:c)2dx~2 / f(:r)f(:c)d:c. (3.19) Unfortunately, this still depends on the unknown density f (2:) To get around this, it can be shown that: E {2/f(a:)f($)da:} = E {2N‘1:f_,(X,-)} , (3.20) 46 where E is the expectation operator and f,_1 is the density estimate obtained with the ith observation deleted from the calculation. If N Mow) = / fa) - 2N4 gm.) (3.21) then under the mild assumption that the minimizer of E {Mo(f)} is near the mini- mizer of Mo, hay will minimize the ISE. Furthermore, due to a result of Stone (1984), in the limit as N —> oo, hcv will be the best possible choice of smoothing parameter (in the sense that a density estimate constructed using it will have the minimum ISE) and could not be improved upon even if f (2:) were known exactly. Despite the superior asymptotic performance of the LSCV method, it nevertheless can run into problems when applied to real data sets. In particular, Silvermann (1986) shows that for small values of h, Mo(h) can become extremely sensitive to small scale effects (such as the rounding of real numbers) in the data. He therefore recommends searching of hcv only in the range of 0.25hn < h < l.5h,,, where h" is given by the normal rule. 3.6 Application: Galaxy Number-Density Plots To find the pilot estimate of density in the cluster contour maps, the kernel estimator is employed on a 100x100 grid with a fixed window width. The window width is set automatically based on the total number of galaxies in each cluster and scaled by their average marginal variance, For this application, the kernel function is taken to be the biweight kernel: K2(X) = —1 _ 'r 2 '1‘ {3n (1 x x) forx x<1 (3.22) 0 otherwise. The biweight kernel function is employed because it gives a smoother appearing con- tour plot than the Epanechnikov calculated on the same number of grid points. 47 The contour plots in Figure 3.6 are constructed from the positions of galaxies previously published by Dressler (1980). The bar in each plot indicates the initial smoothing scale (h = 0.75h,,), the size of which varies from 0.16 to 1.11 Mpc. The positions of the galaxies classified as D or cD by Dressler are indicated by filled circles. The crosses and plus marks indicate the median positions of the groups that are returned as significant by the KMM and DEDICA algorithms, respectively, as discussed in Chapter 4. The maps are centered on the median position of all the galaxies in each cluster. Table 3.1 presents the parameters of each map. Column (1) gives the cluster name. The number of galaxies in each cluster is listed in column (2). The RA and DEC (1950 coordinates) of the median galaxy position for each cluster are in columns (3) and (4). Column (5) lists the surface density of the highest contour in galaxies per square arcmin. Column (6) lists the surface density of the lowest contour, which is set to one galaxy per resolution element. Listed in column (7) is the contour spacing. 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Wee ee 3 «3 BE < ewe fimfi Seed eeeee wwee.e be we ee+ New 2 3 e: eeeu < end ewe eefiee nweee weefie ea em ee+ eNe 5e 2 we ewem < eee flea wwee.e wwee.e eeeee he ee 3+ esm ew 3 em 33 < ee.e eeH Named Seed eSNe we ee 3+ eem nw 3 e3 eeeH < Se 8 E 8v E Awe 5 E 3 82 5898 $888338 $5882.83 $5.55}me . .888 8 :26 25s 88: can 88: we z 8:30 £82583 .H .e mamflh 50 Fig. 3.6.— Adaptive kernel maps of the Dressler sample clusters. Map paramters are listed in Table 3.1. v (m) 0 Y (ml-I) 4L 1 LLJ 51 Y (mil) Y (Immh) V (email) 0 0 X (uranln) M151 I I l I I I I l I l I [III lllllllllllll l l l J 0 X (uanh) Figure 3.6. Adaptive-Kernel Maps for the Dressler Clusters Y (arcmin) Y (min) Y (arcmin) 52 M378 1 I I I I I I—I I I I I I [OI Ij I I I I I l— —l 10 — — r - L- -l .S : : S o _ 3 '- —4 > b -l - w .10 __ _. C 2 -2o — - lllJLll lllllllljlllll -20 -10 0 10 2O Xhmmh) M498 1 40 _ _ 2° — q l- A _ t. E o __ 3 8 D > - F .20 h— r- -l I ‘l ‘40 _" l- -. I 1 l 1 1 1 1 1 l 1 f E >. X (arcmin) Figure 3.6 (cont’d). Y (arcmin) Y (arcmh) Y (min) 53 11 L1 L14 1 1 l 40 -20 0 chmln) M838 .— l l l l I l l l l l l ~40 -20 o X (arcmh) W978 x (arcmin) Figure 3.6 (cont’d). X (arcmin) .1? g a ; 4O 3 i “(@D ). . l“ 40 M979 40 20': L .3 : é o— ; I .20:— _ r . h 'l ‘ l 4° @1 . ,1 -20 0 20 40 54 Y (arcmin) X (arcmh) Aboll 1139 rTfI X (arcmln) Figure 3.6 (cont’d). l l X (arcmln) Y (min) Y (arcmh) Y (arcmin) 55 Y (arcmin) Y (arcmin) Y (arcmin) Figure 3.6 (cont’d). Y (11min) Y (min) 40 1O X (uranh) Aboll2256 T—l—IIIIIIIIIIIT—IIITIIIIIL IIIIIIIIIIIITITIIIIIIII filllllllLlJ—lLlllllllllll 4 [4; l l l l L 1 L1 1 1 l 1 1_L l l 1 I l -20 -1O 0 10 20 X(nremln) 56 Y (am-nil) Y (11min) 20 20 Figure 3.6 (cont’d). X (arcmin) Abell 2151 ITr I I T X (arcmin) 57 Abell 2634 Abell 2657 .- I I I I l I I I I I I I I .. I T I I I I T I I I I 20 — — E S 3 >. E E ). x (aremh) 000247-31 E E >. Figure 3.6 (cont’d). Y (arcmln) Y (arcmin) Y (arcmin) 58 Y (arcmin) Y (3mm) Y (arcmin) Figure 3.6 (cont’d). Y (arcmin) Y (arcmin) Y (arcmin) X (arcmin) DC1842-63 I I I I I I r I I I I I I I I T 40 — — L- —I l- -l 20 -— _ o l— 0.16 Mpc A .20 L. H _ P- 0 -I .40 _— J 1 l l I 1 i l l l 1 I l 1 l I ~40 -20 O 20 40 X (arcmin) 002106-39 59 X (arcmin) Y (arcmin) Y (arcmin) Y (arcmin) 4O _ s _ _ 20— _ _ _ _ _4 .I o x 4 L O — r— — -20 — .— l‘ 0.31 Mpc - - I——{ w 40 — J 1 l l l L L l I i l l l l i l .40 —20 O 20 40 X (arcmin) Figure 3.6 (cont’d). Y (amnln) Y (aremln) Figure 3.6 (cont’d). CENTAURUS 61 For comparison, Figures 3.7 and 3.8 show four examples of the improvement of the adaptive-kernel maps over those of GB. The GB map is shown on the left, while the corresponding adaptive-kernel map (produced with the same data) is on the right. The GB maps were constructed using 50% overlapping boxes (in essence a two-dimensional ASH with m=2) with fixed window width of 0.24, 0.48 or 0.7%”1 Mpc (the scale bar at the upper left of the GB maps indicates the width used). The plots in Figure 3.7 were chosen to illustrate how details in high—density regions could easily be oversmoothed, hiding structure in the core of the clusters. In both Abell 496 and Abell 754 the GB maps were smoothed using their smallest window width of 0.24h"1 Mpc. Although Abell 754 is obviously elongated, the structure in the core is not resolved. This structure is clearly resolved in the adaptive-kernel maps, even though the initial smoothing window is larger than that used in the corresponding GB map. Figure 3.8 illustrates how undersmoothing of low-density regions can lead to “noise”. Most commonly, this noise arises from small numbers of galaxies located in isolated regions on the outskirts of clusters, but with separations smaller than the size of the fixed window width. While it is possible that some of these density fluctuations in the outskirts of the clusters may be due to galaxies that are gravitationally bound to the cluster and not just Poisson noise of the background, they are very unlikely to be dynamically significant to the evolution of the cluster. Essentially all of this noise is eliminated in the adaptive-kernel maps. For comparison purposes the positions of the galaxies are plotted in the adaptive-kernel maps. '09‘27’ 62 l---| A496 -13°la’ I Y (aren't!) h - 04" 31'7‘3 A 754 Y (maria) c \y. .0 O “o \? l l l I l I l [.1 I. I l I . I 40 -20 O 20 X (0min) Fig. 3.7.— Comparison of adaptive-kernel maps with those of GB. In both A496 and A754 detail in the high density regions, which is oversmoothed in the GB maps, is resolved in the adaptive-kernel map. ~25-37’ * '5'58’ 63 8 "MW. .1 J Y (truth) Y (until) “548 '40 . I I L I I I I I I J [4| I I ~40 ~20 0 20 40 Xhmh) “1903 I I I I ‘° ‘ W l- . I I. o l- -l 20 "' 0 o.— l- - ._ J o _ —-1 l— all .20 h— _— .l '40 '-' o I I I I I I I I I4 I i I I I I Iq 40 ~20 0 20 40 MM) Fig. 3.8.— Comparison of adaptive-kernel maps with those of GB. In both A548 and A1983 noise in the low-density regions of the GB maps is smoothed away in the adaptive-kernel map. 64 Adaptive-kernel maps of the HGT sample clusters are given in Figure 3.9. Again, the crosses and plus marks indicated the positions of groups found significant by either KMM or DEDICA respectively. The cross and plus marks with squares and circles around them indicate possible foreground or background groups based on the magnitude distribution (see Chapter 4). The positions of the filled circles in this case indicate the galaxies identified as cD by Struble & Rood (1982). The scale bar indicates the size of the initial smoothing window in Mpc. Table 3.2 gives the same quantities for the HGT sample as Table 3.1 did for the Dressler sample, except that column (5) now lists the apparent 0 magnitude cutoff used for each cluster. For the 25 clusters which are in common between the two samples comparison of APS data with that of Dressler shows that on average the maps made from the APS data have 33% more galaxies than in the Dressler maps. This is due to the inclusion of fainter magnitude galaxies in the maps made with the APS data. 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X (arcmin) Figure 3.9 (cont’d). M5“) II [I IIIIIIIIITIIIIIT'I 1. 1.- p 1111111 0.53 Mpc IIITIITIITT klllLLlllllLl 11111111111 llllllJlL A y— N 0 ~20 -10 0 x (math) 10 M634 I] .1 —4 —1 q 11 IJLL l l l ngl 75 M787 “3.11957 Ii 0‘ .1 . fi q -—4 q 1 J1 20 __ 0.61 Mpc I 1——-1 Y (area-1h) 0 Y (aroma) O T I l 1 .5 _ _- -I : < -2o — — : 0.59 Mpc . b . — 40 —- l—-—-l — - 1 I I I I I I I I I m I I I I I I j '- l I I I I I I I I I I_ I I I I d ~10 -5 0 5 10 -20 0 20 , x (arcmin) x (mm) AD“ 978 M 990 Iva# I 3 E E E >- >- 1 1 1k 0 Hum) X(nremh) M1020 M1035 ITIIIIIIIIIIIIIIIIIIIIIIIq 2° IIII 20— — - — ; - 10— — 10r- — +- ~ r 1 - '4 A F q A +- q S : E _ 4 E. °t '1 s. o:— >' 1' -1 >' __ I j _ '10 —- r- I- " '10? L - I— " - I 0.53m: ~ .20— ._ 1- | I / -1411111111111111111111111‘ .2053‘111M1Jm111141I—J -20 -10 0 10 20 -20 -10 0 10 20 X (3min) X (arcmin) Figure 3.9 (cont’d). 76 Aboll1126 M113 zol‘Il—I‘rIVIIIII‘IrI—I IIIIIIIIIIIIIIII T O l Y (Iranln) a Y (mm) o 3 IIIIITIIIUIIIII lllIllIIlllIlesLI O \IIJLIIIII ‘5 T .20 L111111I1111I1I1 #1111111 -20 -10 40 40 X(uremh) X(uranh) Abollnas Aboll11a7 IIIIII'IIIIIIIII‘IrI ll'IjlllllL 0.57 Mpc 1 llllILli I 8 / a 111111111111111114 g 8 lllr'fiIflIIIIIIIITW 1—1 ’5" E E 0 § ° I I .20 + . O -10 ! 40 1 1 I l l l IJ_[ L 1 I 1 I l 1 1 I l I .20 -2o 0 20 o X(Irunh) X(Ircm|n) Abell 1213 M1216 W‘I’fll Figure 3.9 (cont’d). Y (M) Y (ml-1) Abell 1228 /IIIIIIII Ff 40/ 0 0— .20— r- .401... 1 X(uremh) M11254 L_IIIIIfiI [III [II [IIII TTT. V/ '1 __ 0.58 7 ”1 I————1 10— - 0— _ r- q .10 ..— -20 —1 _J 1114I1111I1411I1111I111j -20 ‘10 0 1o 20 X(urani1) M11291 — 1 1 I 1 A 1 I 1 1 _L I l 1 _ -2o 0 20 X (uranln) Y (arcmin) Y (arcmh) Y (arcmin) M1238 I I I I I I I I I I I T .4 . I 10 '— <- 0 "- _1 r- -4 ~10 ’- L I 1 I I l L LI I 141 I I -20 -1O 0 10 20 X(urcmin) “1257 w I v 40 — _ " '1 20 r— "" 0 -—4 F 1 .20 .— P .40 I— 1.. O I 1 ‘ o _4 '20 _ 0.61 Mpc _ _. 1__1 - 1 I l I 1 I 1 I 1 I I -20 O 20 X (arcmin) Figure 3.9 (cont’d). 78 “1385 I I I I I I I I I I I IT I I I T r 2° N 1 0.60 .. _ 1_____, 10 — O —* 2 1 Q : § 0 "" —I I i 2 u— .1 -10 T , O 1 r— .4 -20 _— 'f I L I I I I I I I I I I I I I I I I '20 -10 O 10 20 X(aremh) 3 E >. II 4 4 f '1 E fi >‘ .4 +11...111.1.‘ 20 0 2O X(arcmin) Figure 3.9 (cont’d). .1 ——1 "1 1 4 .1 11111141J11111111 -10 0 10 X(urernh) Aboll1438 IIIIIIIIIIIIIIIIIII IIII h -1 1- 1 L. _ Z I 0.52Mpc .1 1-—-l A IIIIIIJILIIIIlIIIILIlII11 -20 -1o 0 1o 20 x (arcmin) Ab." 1474 IIIIIIIITIII IrL —I -1 79 Y (arcmin) Y (arcmin) Y (arcmin) M1468 20IITI IIII 10 I IIII III ILIlIJII an O I Figure 3.9 (cont’d). § Y (uranin) Y (arcmin) Y (010111?!) 10 -1O 80 “10111541 — I I IIIIlrIIIIIIIIIII IIIIr IIIIIIIIII IIILLILIIIIIIl ’5‘ E ). —1 L J I I I 1 1 I I 1 1 1 I 1 1 1 1-I -10 0 10 X(urcmln) Aboll1051 M1658 IIIWII I IIIIIIjTII I‘IIT_ Q I d _ | I _ IIIIILILIILII -10 0 1o 20 X (uremh) Ab.” 1691 -I 1111L1111I1 1 1 1d] 1 1 11/1 -1o 0 1o 20 -2o 0 20 x (urcmh) X (arcmin) Figure 3.9 (cont’d). 20 10 E E o 3 >. -10 E I -10 -2o 20 1o 75‘ § 0 ;.’ -10 81 “0111767 Aboll1773 I 1 1111 11 1 1 1 1 1111 111 1 11 TI 1 1 1 1 1 /r I 'q I Ink. ZOJ I I V —‘ . : 0 ~ I. L 10 C : : E . 1- _1 .. _ S o — r 4 3 - I- - >- .. d 1— '1 '7 -. 7 7 10- ~ 0.54Mpc - ' P — ‘ I— 0.51Mpc _ "‘ '— 'I _ I. I « -2o~ p 4 411111 111111 I1 J 1 1 114 14 1 1 1 1 1 111 1 I -20 10 20 -20 -1O 0 10 20 XIMh) X(arcmln) M1775 ”0.1793 4 I I 111 1'1W111 1 a 20 I I I 1 I 1 1 1 1 I 1 r1 1 I 1 1 1 1 1- 1 h '4 E E >. E\I111IIL1M11 -20 1111I1 1 11 I1 11 1 I 1 L1 1 ~20 -1O 0 10 20 -20 -10 0 1O 20 X(urcmh) X(arcmin) M11795 M18“ [IIITWIIIIITIITIIIIIIIIII _ zog'II'I"IWIIII r- _‘ : C i 7 C 0.55Mpc : 10— A r i _ 1 : ‘ E C I _ _ S o _ - - g _4 1— .1 > _- ..J C : -10*— |——I _ t : Z 2 ~1L1 111IL111I1111I1111I11fi -2oklllIllllI1111I1111I— -20 ~10 0 1O 20 -20 -10 0 10 20 X(aremh) X(arcmin) Figure 3.9 (cont’d). Y (um-11h) Y (alum) Y (min) AMI 1831 IIIIII‘IVII lllllll 1 llllllllllllll 1|Illllll llllllllll -2o -10 10 x (um-11h) Abdl 1904 IIIIIIIIII YIITfllIIIlIIrrI IllllIlIlJI/ llllllll will!!! lllllllllllLlLLlIllll IIIIIIIIIIIIIII -20 -10 O 10 X (arcmin) 82 M1837 IIIII’IIIIIIIII Y (Iranh) Figure 3.9 (cont’d). X (lmh) Abollm 5 .— ‘5‘ E E o - ; p I- -5 I— .10 ..- :\_/\'\l ’41 TTIIIIIIIWIIIIIII llllLJlIllllIlll 0.60 Mpc I——1 l l l l 1 l l l l -10 -5 0 X (arcmin) mm UVIITIIIIITIII@II lllllJleLlllIlllll :\ lllilllllllllll -10 0 10 X (arcmh) M O 83 Y (arcmin) Y (arcmin) Y (amin) 20 10 Figure 3.9 (cont’d). J l l l l l L I_ ' l -10 O 10 X (Imh) TIIIIIIIT—IllIlllllTrI llllllll llllJlll Ii: 0 -10 O X (arcmin) 10 20 Y (arcmin) Y (arcmh) Y (uranh) -20 84 I I l l l l l I I I I I I I l l I Figure 3.9 (cont’d). -20 0 20 ~10 0 10 Xhmmh) X(Ircmh) M2061 I I I I I I I I I I I I I I I I I I 20 _ I I I 0.53Mpc - 1———1 - 10 E ' I S o L _. s .. .1 >- _ - I- -1 -10 —— _. ., -20 1 1 1 1 I 1 1 1 1 1m] 1 1 1_1 I -20 ~10 0 10 20 Namath) M2085 I I I I I I I I I I I I I I I I I 2° 0— I—. I— .1 : : Q + _ :— 10 — 0-54 Mpc _ 1- " I—i _. g, ° r i >- L. .1 F -10 — _ ; __ -1 .20 _ i ”11..1111..111.1111..11. -20 ~10 0 10 20 X(aremin) Y (arcmh) Y (aremh) 20 ' '\ P — I- -! _ q —1 F. -1 P— — >- -1 P - I- H h -l — — - .— "' ‘ 1.. 10 1 I 1 1 1 1 I 1 1 1 1 I 1 11 -20 ~10 0 10 20 X(aremh) Abell2089 gr! 1 I I I I I 1W! 1 I I I I- - _ 0.61Mpc _ — H —1 . J 11I1111I11111IA 0 10 20 X (aremh) AD." 2107 X (arcmin) 85 Y (arcmin) Y (11min) Y (arcmin) Abell 2079 111 llllIlLlIIll X (arcmln) M2092 I IIIIIIITI I I _4 ' O b 0.60 Mpc I-—I lIIIJ—LIII -20 -10 0 10 X(arcmh) Abel2124 I I I I I I I I I I I I I I I I I I I I I I I I 20 — -- L- .4 1O -— — O t— _ -1O — — .20 _ / ' _J I 1 I 1 l I l I I l I I l l .20 -10 0 10 20 X(arcmln) Figure 3.9 (cont’d). 20 -20 -4O X(arcmb) Aboll2151 I— L. '1 lIllelllIlllII 40 -20 O 20 40 X(urcmb) Aboll2162 IIIIIIIle—I;R4 .1 (L11 . 0.83Mpc - ~ l—-——l - I O I _ 1111lm111" -20 0 20 40 X (aremb) 86 Y (arcmin) Y (womb) 20 -20 I I I I- - -1 )- d - 0.56 Mpc 4 L. I-——I - '- 1 I l I 1 i l l l I l 1 l I l l l 40 -20 0 20 X (crumb) Abell 2152 I I I I I I I I I W I I Figure 3.9 (cont’d). LJII O 10 X (arcmin) 147 Y (numb) Y (numb) Y (arcmb) [WWW 20 40 20 10 10 -5 87 M2197 “21$ III—IIIIITIIIIITIIIII—I III IIIIIIIIFIIII. 0.58 q _ I.’ I-—I ~ mi — _ -— 20 ‘— I- —I b -—I I- - E I -1 I- -I v -1 - .1 >- - _._ —— -20r _ _ 1° \ - _J _ 1111 111I111I111I11 1 111I111I111IL 4O -20 0 20 4O -4O -20 0 2O 40 X(arcmb) KIM“) Abdl2255 M256 _1 I I I I I I I I I 1 1 L. 1 I I I I I 1 I I 1 : 4 2° — - 1 )— : 0.50Mpc : )- I—I q i I E ” - .4 S 1— —4 > I‘ _ _I _ In -I -20”- r 1; - —I 1 1 1 I 1 1 11 1 1 11 I 1 1 1 1 4 ~20 -10 0 10 20 X(ammb) M2328 M2347 I 1 I Id I I I I I I I I 1‘1 I 1 u 4 ’ 'I -4 10" '—‘ 3 - ‘ S OH — 3 > h - 7 I II 11 Ill _\ bLIllJLII II I .5 O _A 0 Figure 3.9 (cont’d). X (arcmin) 88 M2382 rillIIIIWIIIIIIII 20— O .10; 0.55Mpc - 1—1 11111111 _A O Y (flunk!) o llllllLlll .20— P11111L1L1111111 1111111 -20 -1O 0 10 20 X(|rcmh) mason Y (arcmin) Y (arcmin) Y (arcmin) 20 10 Figure 3.9 (cont’d). Ab“ 2410 I I I I I T I I '- W L d ._ —J 1. .. -— —J 1 1 1 l L m 1 L 1 l 1 1 1 1 1‘ -20 -1O 0 10 20 X (arcmh) Abel! 2634 IIIIIIIIIII —. .4 .1 _J .1 .1 q _ q —1 Y (arcmin) Y (arcmin) Y (arcmin) 89 Abell2657 w$IIIIII1IIIIIIr _ '1 d .1 S E “ g -1 _ >_ ‘ — _‘ —-4 40+IIL111111111L1 1 40 ~20 0 20 40 X(aremin) M12870 M2675 I Tllllilll‘rllI L I_III II [III]IIII IITI . ‘ 20/ .... — - q ‘_ 101- _. ‘ P d J 1— J 3 3 I 1 —~ So— — S s— .4 >' - q .10.. _. r - '20—- ‘1111111/1N111111L1111 _J11 1111111111m1111 -20 40 0 10 20 -20 -1O 0 1O 20 X(arem|n) X(aromin) Abe“ 2700 ”Q C370 X (arcmin) Figure 3.9 (cont’d). 90 For the clusters which are common to both samples it is of interest to compare the maps made with the different data sets. Although this comparison is made somewhat difficult by the different sizes of the maps, it is clear that most of the features visible in the Dressler maps are still visible in the maps made with the APS data. There are a number of special cases which require further attention. The peaks to the north and south of A119 are washed out in the larger APS data set, so much so that the DEDICA program no longer finds them significant, although the KMM partition remains. This could mean that the fainter magnitude cutoff used in the second map (my z 18.6 for the APS data as opposed to my 2 16.5 in the Dressler data) has increased the background to the point where the structure is being washed out. However, comparison of the estimates for background listed in tables 2.2 and 2.3 indicate the opposite is true; the Dressler map has a higher background. This is likely a consequence of the slightly larger area plotted in the APS map, which follows A119 into a lower density region near the edge of the map. The elongated peak to the northeast of A400 in the map made from the Dressler data is beginning to become resolved into two components in the APS map. Lastly, the elongation seen in the core of A1656 in the Dressler map is completely obscured in the APS map. This is not due to the larger data set employed but to the increased area plotted in the map. In Figure 3.10 the core region of the cluster is plotted using the APS data. Here the bimodal nature of the density is evident. In fact, a large fraction of the clusters in the HGT sample show a similar effect. In the case of the Coma cluster the reality of the structure in the core can be confirmed by corresponding peaks in the X-ray surface brightness map of the cluster (Davis & Mushotzky 1993). A400, shown in Figure 3.11, is another example of a cluster with apparent core substructure. In this case the EINSTEIN X-ray map is elongated in a direction similar to the small scale structure in the core of the cluster (Beers 91 et al. 1992). Further evidence of the reality of this substructure is given by the bimodal (and possibly even trimodal) appearance of the velocity distribution plotted in the examples of density estimation given above. While a more detailed study is clearly called for, this raises the intriguing possibility that most clusters could have substructure in their cores at scales of approximately 250h‘1 kpc, or the canonical size of the core radius of clusters. By choosing to search for substructure in the 1.5h‘1 region, this study misses identification of this small-scale core structure. 92 Abel|1656 I T I I I I I T l I I I I I l_ _ 0.09 Mpc _ 10 1— — g _. E 2 O — 3 >_ _. -10 _— _l 1 I 1 1 l l I J l 1 g l 1— -1O 0 10 X(arcmin) Fig. 3.10.— Core region of A1656 showing that substructure can exist even in the cores of very massive clusters. 93 Abell400 I4. I F/r'l I I Y (arcmin) X (arcmin) Fig. 3.11.— Adaptive-kernel map of the core region of Abell 400 indicating the possible existence of core substructure. Chapter 4 TESTS FOR SUBSTRUCTURE 4.1 Introduction Although the adaptive-kernel maps discussed in Chapter 3 can give an indication of the existence of substructure within galaxy clusters, the presence of peaks in the maps alone cannot prove the statistical significance of those groups. For this a test for substructure must be employed. Proving a particular feature, such as a second density peak, exists in a particular data set can be a difficult problem. In general it is a simpler task to shOw that alterative descriptions of the data are not true. This is referred to as hypothesis testing. In this chapter two tests for substructure will be discussed and applied to the projected galaxy positions of the sample clusters, both of which are hypothesis tests. The first test, KMM (McLachlan & Basford 1988, hereafter MB), is a parametric approach which fits the data to bivariate Gaussian distributions and tests whether a single Gaussian or multiple Gaussians provide the best fit to the data. Significant substructure is claimed for those clusters where a likelihood-ratio test rejects the hypothesis that the data are drawn from a single Gaussian. Although fitting the projected distribution of galaxies in a self-gravitating system to Gaussians cannot be justified from theoretical arguments, the Gaussian neverthe- 94 95 less remains the best available choice. Given the freedom to choose any mean and standard deviation, the two-dimensional Gaussian is capable of approximating a wide variety of shapes. More importantly, from the perspective of hypothesis testing as applied here, it is the most well-studied statistical distribution. Other theoretically- derived PDFS for clusters such as the Michie—King models or the isothermal sphere are mathematically more complex and have not been well-studied by statisticians. Furthermore, the observational evidence for such distributions in galaxy clusters is still open to debate (see discussion in Chapter 6.) The second test, DEDICA, is the nonparametric technique due to Pisani (1993, 1996). In this approach no assumptions are made about the shapes which the groups might have, only that each group is identifiable by a single peak in the PDF. Each detected peak, in turn, is assumed to belong to the background and the likelihood of the fit evaluated. Significant substructure is claimed for for those groups where a likelihood-ratio test rejects the hypothesis that they belong to the background. Although in this thesis these tests will be applied only in two dimensions, the :1: and y positions of the galaxies projected onto the plane of the sky, both algorithms are capable of using redshifts to test for clustering in three dimensions. With the large redshift surveys currently planned, such as SLOAN (see Bahcall 1995) and ENACS (see den Hartog 1995), it is important that these potentially very powerful techniques be well studied and tested. 4.2 The KMM Algorithm KMM implements the Expectation Maximization (EM) algorithm of Dempster et al. (1977) as described by McLachlan & Basford (1988, hereafter MB). The program is also discussed for use in the detection of bimodality in univariate data by Ashman, 96 Bird & Zepf (1994, hereafter ABZ), and has been applied extensively in the analysis of substructure in clusters by Bird (Bird 1994a, Bird 1994b, Bird 1995, Bird, Davis & Beers 1995 ). The KMM program provides a maximum-likelihood fit of a data set to a mixture of Gaussian distributions, and can be used for hypothesis testing by evaluation of a likelihood-ratio test. The number of Gaussians to be fit, 9, as well as an initial g-group partition of the data is specified by the user. Alternatively, the user can specify a first guess at the parameters of the individual Gaussians to be fit (locations and covariance matrices) along with an estimate of the mixing proportions. For this application specification of an initial partition is our preferred choice, rather than specification of the unknown positions and sizes of the groups. Furthermore, several objective partitioning algorithms exist that can be employed to specify the initial partition (see Kaufmann & Rousseeuw 1990 and references therein). If it is assumed that the data points, x1, . . .xN (the a: and y positions of the galax- ies) are independently drawn from 9 Gaussian probability density functions (PDF), G1, . . . 09, then the PDF for the superpOpulation G can be represented as: 9 f(x; (’5) = gflifdx; 9), (4-1) where f,(x; 0) is the PDF of G, and 7r,- is the fraction of the superpopulation belonging to G,. Here 0 contains the elements of the mean vectors 11, and the covariance matrices 2,- for each group, and the vector as = (1’. W (42) is the vector transpose of all unknown model parameters. The log-likelihood of the complete data can then be defined as: g n LC(¢) = Z Z Zij [108 7T1 +108 fi(xi; 9)], (4-3) i=1j=1 97 where 23,-,- is an indicator variable: 2":{1 iijEG, '3 0 if x, 3 0,. Once an initial partition has been specified by the user, KMM calculates the log- likelihood of the fit using equation (4.3). The program then proceeds to find the value of (13, say (1’)“), which maximizes the expectation of the log-likelihood conditional on the observed data and the initial fit. Using 45(1), posterior probabilities for group membership can be estimated by: , _. (1) _ Wifi(xji9) 73094) )_Zf=17rtft(xj10)’ (4.4) for i = 1... g. Here 7,-(xj; 4)) is the probability that the object with observation x]- is a member of group G,. The expectation of the log-likelihood is then re—calculated using equation (4.3) with the 2,, replaced by the 7,-(xj, ¢(1)), the posterior probabil- ities. The program then searches for the value of 43, say (15(2), which maximizes the expectation of the log-likelihood. These two steps, E (expectation) and M (maxi- mization), are repeated iteratively until LC(¢) has converged to a local maximum, provided a maximum exists (for a discussion of the convergence properties see Wu 1983). Objects are then assigned to the group for which their posterior probability of membership is the highest. The final value of LC(¢) is used to evaluate the improvement of the g-group fit over the null hypothesis that the galaxies are drawn from a distribution of go Gaussians by calculating the log-likelihood ratio A: _ LOW) *‘LdeV Mm where LC(¢)(9°) is the log-likelihood of the go group fit. The greater the value of A, the greater the improvement in the fit. 98 In order to quantify the improvement in the fit obtained by the addition of another Gaussian, the percentile (p-value) of the log-likelihood ratio can be estimated using a bootstrap procedure, as follows. Random data samples are generated under the null hypothesis that the data are drawn from a mixture of go Gaussians with means, covariance matrices, and mixing proportions specified by the likelihood estimates from the go-group fit to the original data. For each bootstrap sample, A is calculated after fitting mixture models for both go and 9 groups. The value of A from the actual data can then be compared to the null distribution of A values calculated from the bootstrap samples to find the significance. In this paper go = g — 1, thus requiring that any g-group fit be a significant improvement over the (g — 1)-group fit. It is important to note that the EM algorithm discussed above is not the only way to maximize the likelihood equation. Various other algorithms have been proposed and applied. The most well known of these are a group of algorithms based on Newton’s method (Press et al. 1986). There are also algorithms due to Fletcher & Reeves (1964) and Nelder & Mead (1965.) Six methods are compared by Everitt (1984) for the case of a mixture of two univariate normal densities. In general, convergence was fastest using Newton’s method with exact expressions of the first and second derivatives of the likelihood equation. This advantage over the EM algorithm disappears when finite-difference approximations for the derivatives are used. Both the Fletcher-Reeves and the Nelder-Mead algorithms showed a tendency to find points from which no improvement could be made even though not at a local maximum. The main advantage of the EM algorithm is that each iteration is guaranteed to improve the fit. This is not always true for Newton’s method (McLachlan & Basford 1988). In general less than 100 iterations are required for convergence, which on a Sun Sparc 2 takes less than 30 seconds. Thus speed of convergence was not a big issue. 99 4.2.1 Monte Carlo Simulations In order to assess the strengths and weaknesses of the KMM algorithm three questions need to be addressed. First, how often is KMM likely to classify a given data set as having significant substructure when such substructure does not exist? Second, for cases where real substructure exists, under what circumstances is KMM likely to fail to recover it? The former is traditionally referred to as an error of type 1, while the later is referred to as an error of type 2. Finally, what is the effect of random background / foreground contamination? In order to answer these questions a number of Monte Carlo experiments was conducted, following ABZ. Data points were drawn randomly from two-dimensional Gaussian distributions with a covariance matrix of 0n = ayy = 1.0 and 03,, = 0.0, which remained fixed for all data sets. For each case 250 data sets were generated, with the number of points set to 50, 100, 250, and 500, with an assumed constant-density background of 0%, 10% and 20% of the total number (numbers most relevant to the data sets, see Chapter 2). KMM was run on each data set for both the homoscedastic (common covariance) and the heteroscedastic (independent covariance) situations. In the homoscedastic tests, the Gaussians are forced to share a common shape, while they are allowed to have independent shapes in the heteroscedastic case. The significance of the resulting partitions was evaluated using the bootstrap procedure (1000 resamples) described in the previous section, with the modification that the analytical means and covariance matrices of the null hypothesis were used instead of the likelihood estimates. This avoided the need to bootstrap each realization of a data set, which was impractical due to the CPU time required. The experiments can be divided into two broad categories, corresponding to the different error types. Category I contains those data sets generated under the null 100 hypothesis in order to test KMM’s prOpensity to identify substructure which is not real. Category II contains data sets generated with hypothesized substructure in order to test the ability of KMM to correctly recover the input (real) substructure. In category I, data points were generated from a single Gaussian with mean (0,0) and covariance described above, and KMM was requested to find two groups. Futher- more, random data sets were generated from two equally-populated Gaussians with the mean of the first group set to (0,0) and the mean of the second group varied between (1.50,0.0) and (4.00,0.0) in steps of 6x = 0.25. Again, the covariance matrix of each group remained the same as above. In this instance, KMM was asked to iden- tify three groups. In category 11, two equally-populated Gaussians were generated as described above, and KMM was requested to find two groups. For category I experiments KMM was started with an objective partitioning of the data supplied by the program PAM (Partitioning Around Medoids). As described by Kaufmann & Rousseeuw (1990), PAM searches for a user-specified number of repre- sentative objects (the medoids). The medoids are chosen such that the dissimilarity (or distance) between the groups is maximized while at the same time the dissimilar- ity within each group is minimized. A final partition is effected by simply assigning each object to the closest medoid. In category II the initial partition of the data was obtained by assigning each object to the closest Gaussian (note that this is the same as running PAM with the medoids forced to be the centers of the Gaussians, without the CPU time required by actually running PAM). In order to compare the results of the experiments with the fits obtained using real data, a generalization of the dimensionless parameter A11 defined by ABZ is employed: (1.,- ‘ /0,'0’j , Am,- = (4-6) 101 where dij is the distance between the averages of groups 2' and j, and a,- is the standard deviation of group 2' along the vector joining the average positions of groups 1' and j. Note that with the averages and covariance matrices of the Gaussians described above, A11 is simply equal to the average a: position of the second group. For all category I experiments using homoscedastic fits, the rate of false positives (type 1 errors) at the 90% significance level remained below 10%. When groups with less the 20% of the total number of galaxies in each cluster were rejected, as done in this thesis, the rate of false detection falls to 5%. The corresponding numbers for the 95% and 99% significance levels are 3% and 1%, respectively. These error rates changed little by the rejection of small groups or the distance between the groups. In general, the effect of adding a constant density background lowered the rate of substructure detection. The results for the heteroscedastic runs show the opposite behavior. The highest error rate at the 90% significance level reached 85%. These error rates depend less on the significance level then on the separation of the groups and the background level added, with wider separations and higher background leading to higher error rates. However, the error rate is most sensitive to the small-group cutoff level. By rejecting groups with less than 20% of the total number, the error rate, in the case of 500 galaxies with a 20% background, is cut from 84% to 30%, with these values remaining constant for the 90%, 95% and 99% significance levels. Applying a 20% cutoff for small groups, the error rate only reaches the 10% level for groups with 250 or more members. It can be concluded that in the heteroscedastic case KMM has a propensity to return highly-significant groups with few members. Therefore, in large data sets a 20% small-group cutoff needs be employed because increasing the significance level does not lower the error rate. When dealing with large data sets, N g 200, the following procedure has been found to give good results. KMM 102 is run first for the homoscedastic case. If a g-group partition is found to be a good improvement (at the 90% level or better) over the (g —1)-group partition, then KMM can be run for the heteroscedastic case using the same initial partition to see if an improvement can be made over the homoscedastic fit, as judged by the Anderson- Darling statistic (see McLachlan & Basford 1988 and section 4.2.3 of this thesis). In this way the freedom to attain a better fit by allowing the groups to have different covariance matrices is retained without losing the robustness of the homoscedastic case. Furthermore, the Hawkins test described below will often give a good indication of whether a heteroscedastic fit is required. The results of the two—group, category II, homoscedastic fits with no background are in good agreement with the univariate experiments of ABZ where the p-value was obtained by assuming that the null distribution of the LRTS was distributed as x2, as opposed to the bootstrap procedure employed here. In Figures 4.1 and 4.2 the results are plotted for the homoscedastic and heteroscedastic mixture models, respectively, using a 20% small-group cutoff, with a 10% background. With N = 50, the rate of detection at the 99% significance level does not achieve 90% until A11 = 4.00, a rather large separation. The corresponding numbers for N = 100, 250 and 500 are Ap=3.25, 2.75, and 2.50, respectively. In the heteroscedastic experiments, it can be seen that the necessary separation needs to be larger for a given rate of detection. Again, the addition of a constant-density background, at least to the 20% level, lowers the significance of the fits. This result underscores the need for deep catalogs of galaxies which sample well. into the cluster luminosity function, without over-sampling background galaxies. Although these Monte Carlo experiments can provide a useful guide to situations in which KMM is likely to succeed or fail to detect substructure, too much emphasis should not be placed on them because the cases tested are quite specific. Questions 103 Homoscedastic I 1 __ L— _ .8 — o. _- o _ W ._ 3 .6 — O _ ,> _ a. _ .5 .4 —— 4-fl .- 0 O .. L I... .2 —- o l Fig. 4.1.-—— KMM success rate vs. group separation — homoscedastic case. 104 Heteroscedastic I r I I I I I I I I I I I ._v_ I l _, I I , .. ’ ' ’ ’ 0,. I, -‘ I / _ I I’ I I .— 1— ’ I, I _ 0. ’ , I O I - I vn , _ 0 I, — 3 I’ B . ‘ > I l ,’ T O. I _ I c l '— 09 II, 4.: z _ O ,’ 2 ,I _. ‘0- I I -—1 I I ———l I I I .1 I ..... I 1 I l l I l l l l I l l l l I l A 3 3.5 4 Fig. 4.2.— KMM success rates — heteroscedastic case. 105 not addressed by these experiments include how often in KMM likely to identify a single Gaussian as three or even four Gaussians, and how good is KMM at detecting substructure which is not Gaussian. Testing the vast array of possible parameter space where KMM might be applied is beyond the scope of this study. However, limited tests indicate that if a group is peakier than Gaussian, as clusters of galaxies are expected to be (Beers & Tonry 1986), KMM will perform slightly better than the above results indicate in the sense that closer groups can be identified. On the other hand, highly skewed distributions may be identified as significant substructure, especially with a large background. This can be guarded against by employing the Hawkins test as discussed in section 4.2.3 of this thesis. 4.2.2 Application of KMM to the Dressler Sample KMM was run on each of the Dressler clusters for g = 2, g = 3, and g = 4 groups. The initial partition of the data was effected by assigning each galaxy to the closest user-chosen medoid. The medoids were generally chosen to correspond to the density peaks observed in the adaptive-kernel maps. However, multiple medoids were used for each cluster to ensure that KMM converged to a global maximum. In the cases where different solutions were found, the mixture model with the highest log-likelihood was chosen. The log-likelihood ratio was calculated with go = g — 1 and the bootstrap carried out with 1000 iterations to estimate the significance of the fit. A p-value _<_ 0.05 was considered to be significant. From the Monte Carlo simulations, it can be estimated that this choice corresponds to an error rate of approximately 8%. Although a smaller error is easily achievable by applying more strict criteria, it was decided that without redshift data such refinements would not be meaningful. The clusters for which KMM rejects the null hypothesis at the 95% level are listed in Table 4.1. Although a number of clusters have more than one acceptable partition, 106 only the one with the best Anderson-Darling statistic is listed. Furthermore, only those clusters with at least two groups containing more than 20% of the total are listed. (A151 is listed in Table 4.1, even though the second group does not meet the 20% criteria, for comparison with the DEDICA results discussed below.) Column (1) lists the cluster name. Column (2) gives the number of galaxies in each group. The percentage of the total number of galaxies for each group in given in column (3). In columns (4) and (5) the :1: and 3; positions along with their respective one-a errors of the groups are listed in arcminutes. Column (6) is the significance of the partition. It is interesting to note that several of the clusters which show multiple conden- sations in their adaptive-kernel maps are returned by KMM as not having significant substructure. For instance, there are four clusters, A978, A1991, DC1842-63 and Centaurus, which appear to have two similar-density condensations near their cen- ters. These groups are fit by KMM and have A11 values of 2.1, 1.3, 0.5, and 1.7 respectively. From the Monte Carlo experiments it can estimated that the probabil- ity (assuming these structures are real) of KMM returning significant p-values to be 0.09, 0.06, 0.02, and 0.10, respectively. These numbers would of course improve with a larger number of galaxies. Therefore, the absence of a given cluster from Table 3.1 should not be interpreted to mean that the cluster does not have substructure, but that a more detailed analysis (or deeper catalog of galaxies) might be required to detect it. Centaurus for instance, is known to have substructure in its velocity distribution (Lacey, Currie & Dickens 1986), a result that might have been predicted from the adaptive-kernel map. 107 TABLE 4.1. Mixture Model Parameters - Dressler Sample Cluster N % of total :1: :1: a, y :1: 0,, S (arcmin) (arcmin) (1) (2) (3) (4) (5) (6) A 119 65 56 0.7 i 11.1 0.0 :1: 5.3 1.000 28 24 —3.1 :1: 11.1 20.5 :1: 5.3 23 20 6.8 :1: 11.1 —l6.7 :1: 5.3 A 151 88 84 -2.9 :1: 18.3 3.0 :1: 15.1 0.972 17 16 —2.6 :1: 20.0 —32.5 :1: 4.5 A 154 37 47 2.5 d: 7.6 —2.0 :h 8.8 0.999 21 27 6.7 :1: 7.6 14.5 :1: 8.8 21 27 —14.4 :1: 7.6 —15.0 :t 8.8 A 194 45 60 -4.3 :1: 21.1 0.3 :1: 21.3 0.980 30 40 6.5 :t 16.8 -3.3 :1: 17.7 A 496 51 63 -—0.2 :t 5.0 —2.2 :t 12.8 0.999 17 21 -22.9 :t 5.0 0.6 :1: 12.8 13 16 17.2 :1: 5.0 3.7 d: 12.8 A 548 157 67 —11.9 :t 19.1 8.1 i 16.0 1.000 77 33 22.2 :1: 8.5 -l9.6 :t 7.8 A 754 84 56 —12.1 :1: 13.9 —0.2 :1: 12.6 0.999 40 27 13.1 :1: 13.9 9.7 :1: 12.6 26 17 18.6 :1: 13.9 —27.4 :h 12.6 A 838 38 61 3.7 :1: 15.7 —4.0 :h 12.8 0.999 16 26 —33.1 :1: 15.7 26.5 :1: 12.8 8 13 27.5 :1: 15.7 —31.2 :1: 12.8 A 957 46 56 -—7.9 :t 20.2 —0.5 :1: 22.9 1.000 36 44 5.4 :1: 19.2 0.1 :1: 17.9 A 979 50 58 0.8 :1: 19.3 0.5 :1: 7.8 0.998 19 22 6.0 :h 19.3 -37.6 :1: 7.8 17 20 4.5 :1: 19.3 30.2 :1: 7.8 A 993 45 49 —6.4 :1: 20.8 4.9 :t 11.8 0.982 25 27 13.2 :1: 6.6 -5.5 :t 23.0 21 23 —9.6 :1: 24.1 —38.1 :1: 7.4 A 1069 26 55 2.4 :1: 12.8 —10.5 :h 15.1 0.995 21 45 5.7 :1: 20.5 8.0 :t 21.0 A 1631 69 77 0.3 :1: 16.7 4.9 d: 8.6 1.000 21 23 0.3 :1: 16.7 —24.5 :1: 8.6 A 1736 133 80 —5.1 :1: 17.0 —5.4 :1: 15.6 0.984 33 20 21.4 :t 17.0 22.8 :1: 15.6 A 2151 74 49 -1.0 :1: 16.7 —2.0 :1: 7.9 1.000 47 31 —0.1 :1: 16.7 25.7 :1: 7.9 31 20 12.0 :1: 16.7 -28.0 :1: 7.9 108 TABLE 4.1. (continued) Cluster N % of total a: :1: a; y :1: 0,, S (arcmin) (arcmin) (1) (2) (3) (4) (5) (6) A 2634 66 50 0.9 :1: 11.4 —0.6 :1: 7.5 0.953 34 26 -8.0 :1: 11.4 11.7 :1: 7.5 32 24 8.4 :1: 11.4 —16.8 :1: 7.5 A 2657 61 74 4.4 i 8.4 —0.8 i 8.9 0.960 21 26 —19.4 :t 8.4 —2.3 :1: 8.9 DC 0103-47 25 47 —19.2 :1: 10.8 0.2 :1: 9.6 0.995 16 30 13.4 :1: 10.8 -11.9 :1: 9.6 12 23 9.9 :t 10.8 32.7 i 9.6 DC 0247—31 34 71 -3.7 :t 19.8 -4.8 :1: 24.0 1.000 14 29 —0.2 :1: 2.1 0.6 :t 1.8 DC 0326-53 97 60 5.5 :1: 20.1 -l3.l :t 13.9 1.000 64 40 —11.5 :1: 20.1 21.6 :1: 13.9 DC 0428-53 89 68 —1.2 :t 20.7 3.8 :1: 19.2 0.999 42 32 0.4 :1: 3.7 -2.6 :1: 8.5 DC 0559-40 47 42 —9.5 :1: 12.6 6.4 :h 16.6 0.999 45 40 2.4 :t 11.4 -2.3 :1: 4.0 20 18 32.9 :1: 10.2 -7.8 :t 26.1 DC 0622—64 55 67 —0.2 :1: 18.9 9.3 :1: 18.9 0.992 24 24 —5.0 :1: 18.9 —25.7 :1: 9.3 19 19 —1.8 :1: 18.9 30.0 :1: 9.3 DC 2048-52 96 44 2.1 :h 19.3 -5.1 :1: 14.4 1.000 77 36 6.5 :1: 15.2 -10.4 i: 12.8 43 20 -5.0 :1: 14.7 35.5 :1: 6.4 DC 2103-39 94 87 1.9 :1: 19.7 —1.3 :1: 16.8 14 13 --7.9 :1: 11.3 3.5 d: 23.6 DC 2345-28 41 43 1.4 :1: 5.0 1.8 :1: 13.7 0.998 31 33 —15.3 :1: 5.0 —2.8 :1: 13.7 23 24 19.3 i 5.0 —1.4 :1: 13.7 DC 2349-28 42 62 —3.4 :1: 14.6 —6.9 :1: 8.0 0.973 26 38 —1.0 d: 14.6 19.7 :1: 8.0 109 Two other clusters — A400 and Coma (A1656) - are also known to have substruc- ture from detailed kinematic and X-ray surface-brightness studies, but are absent from Table 3.1. Although the adaptive-kernel map of A1656 shows an elongated plateau to the east (suggesting unresolved substructure), the adaptive-kernel map of A400 shows no evidence of substructure in its core (Beers et al. 1992 suggest that the low-density peak to the northeast is likely to be a background group). In both of these clusters, the substructure is resolved in the adaptive-kernel maps made with a deeper galaxy survey (Kriessler, Beers & Odewahn 1995). From table 4.1 it can be seen that 26 out of 56 clusters (46%) in the Dressler sample are better described by a multi—modal Gaussian fit than by a unimodal Gaussian, at the 95% confidence level (again, A151 is not counted.) The Gaussian sub-groups have a median separation from the global cluster centers of 0.611“1 Mpc. These results are in concert with those of Geller & Beers (1982, hereafter GB), although there are disagreements for individual clusters. There are six clusters — A1142, A1983, A1991, DC 0317—54, DC 0326-53, and DC 0410-62 — for which GB claim substructure which is not confirmed by KMM. A1142 has a three-group partition that is significant at the 90% level, and therefore did not make the 95% cut. A1991 has two clear peaks in the central region of the cluster, one of which contains the D / cD galaxy. As discussed above, the groups are simply be too close together in this cluster for KMM to find a significant two-group partition. 4.2.3 Application of KMM to the HGT Sample In the same manner as above, KMM was applied to the APS data for the HGT sample of clusters. Because the limiting magnitude used for each cluster was fainter than the my = 16.5 used in the Dressler sample, the HGT sample has more galaxies available and possibly larger field contamination. In an attempt to keep the error 110 rate low, the cutoff for significant structure was raised from 95% to the 99% level for this study. Furthermore, in the previous section no attention was paid as to whether or not a Gaussian was a good fit to the individual groups. The small sizes of the groups found in the Dressler sample made rejecting the the Gaussian hypothesis for the individual groups difficult and unreliable in many cases. Futhermore, the Monte Carlo experiments suggested little need for such precautions. (In any case, all of the partitions listed in Table 4.1 for the Dressler clusters meet the criteria outlined below.) The situation is different using the larger data sets offered by the APS. With some clusters having 500 members and the possibility of 30% field contamination, the Monte Carlo simulations discussed previously suggest that the errors could in these cases be much larger than desired. A simple experiment illustrates the dangers. If KMM is run on a data set which consists of 250 points drawn randomly from a bivariate Gaussian distribution and 100 points drawn from a uniform distribution, there is a high probability of a three-group partition being a significant improvement over that of a two—group partition. In these cases the heavy tails on either side of the single-peaked Gaussian have been fit as two separate Gaussians. It should be noted that this type of error is not simply a problem which pertains specifically to the KMM algorithm, but to all hypothesis tests. If the null hypothesis is fundamentally wrong, it is possible to get positive results even if the hypothesis being tested for does not pertain. To explore the goodness of the Gaussian assumption for the groups a test due to Hawkins (1981) is employed. Although more complicated than a x—square test, the Hawkins’ test does not require binning of the data and can be used to test for homoscedasticity at the same time. This test is briefly outlined below. 111 To apply the Hawkins test it is necessary to assume that the mixture model returned by KMM is the true underlying density distribution. Then the Mahalanobis squared distance is calculated between each observation and the average of the group (calculated excluding 27,5) to which it belongs. Or, if 23,5 is the jth observation in the ith group and 5:,- is the average of the 2' group, the Mahalanobis squared distance is defined as: D(£L‘i, Iii, ; S) = (.Tij — ji)($ij — 53,-)'S’1(a:,-j — 113;). (4.7) Here, the matrix S is given by 9 S = 2011 — 1)Si/(N - 9) (4-8) i=1 with S,- the covariance matrix of the ith group 29:1(902'1 — 530036 — ii)’ n,—1 5.: (4.9) As with the mean, 5',- is calculated with 32,-1- deleted in case it contaminates the esti- mates of the mean and covariance matrix. It can be shown that the quantity: (n,- — 1)l/ ("'2'pr +10 - 1) D($ij,ji; S) (4.10) is an F distribution with d and V = n—g—d degrees of freedom (dis the dimensionality of the data, in this case d = 2.) If “(6) denotes the tail area under F4,” to the right of the calculated value of equation (4.10), then under the hypothesis that the ith group is normal the aij will be distributed approximately uniformly over the interval [0,1]. A summary of this information can be given by the Anderson-Darling statistic, defined as: "i W,- = —”i — 2(21 - 1) [108020) + 108(1 - ai(m.--j+l))] /ni, (4-11) 1:1 112 where for each 2' = 1,. . . ,g, 01(1) 3 01(2) _<_ . . . g aim). According to the Monte Carlo simulations carried out by Hawkins (1981), if W,- is greater than 2.5 the normality of the ith group can be rejected at the 95% level. However, in applying the Hawkins test MB recommend not using a hard cutoff since equation (4.11) only holds exactly as N —> 00. This is because the aij can not be exactly uniform on the interval [0,1] unless there is an infinite number of them. This advice has been followed here. Groups with an Anderson-Darling statistic much greater than 3 were rejected as not significant. However, groups in the range of 2.5 to 4 were kept if DEDICA also returned the group as significant. The intention was to avoid rejecting significant groups simply because they are not well fit by a Gaussian, yet at the same time avoiding the identification of skewness or heavy tails, which usually have W,- z 7 — 12, as additional groups. The results are given in Table 4.2. Again, only the best-fitting partition is listed. In this case, 83 clusters out of 118 (70%) have a significant multimodal Gaussian fit. Of the 25 clusters that are in both the Dressler and the HGT samples, if substructure was identified in the Dressler sample then in general it was identified in the HGT sam- ple. Again this comparison is made more difficult because of the different sizes used. An illustrative case is that of A194. Since a similar magnitude cutoff is employed for both data sets (m0=17.6 is approximately mV=16.5), the larger number of galaxies in the APS data is due almost entirely from the larger area used. In the Dressler data, KMM splits the core region of the cluster into two groups. Even though the core region is more clearly elongated in the map made with the APS data, it is not partitioned. Instead, KMM is drawn to the groups which lie to the southeast (A207) and to the northwest. This same situation applies to the clusters A496, A957, and A2634. This indicates that substructure in the core regions of the clusters is being missed by the analysis done here and that a follow-up study which includes only the galaxies within 1/2 of an Abell radius show he conducted. 113 TABLE 4.2. Mixture Model Parameters for HGT Sample Cluster Ng %N¢at %Ltot a: :1: a; y :1: 0,, mm“; 1111-", S (arcmin) (arcmin) (0) (O) (1) (2) (3) (4) (5) (6) (7) (8) (9) A 21 146 50 52 —0.8 :1: 7.7 0.4 :h 4.0 19.9 18.6 1.000 82 28 28 3.0 :1: 7.7 12.1 :1: 4.0 20.1 18.9 63 22 21 —4.0 :1: 7.7 —11.7 :1: 4.0 20.2 19.1 A 85 204 63 63 -0.6 :1: 13.7 2.5 :1: 16.5 18.8 17.5 1.000 69 21 19 —22.8 :1: 3.8 —2.9 :1: 16.6 19.0 18.1 50 15 18 —9.4 :1: 14.4 —29.8 :1: 1.7 18.9 18.2 A 88 43 60 68 —0.2 :1: 7.3 -4.7 :1: 4.9 17.8 17.0 0.996 22 31 26 -7.4 :1: 7.1 6.8 :1: 3.6 18.0 18.5 7 10 6 -1.2 :1: 10.7 0.0 :h 11.4 18.4 0.0 A 104 115 76 68 —4.3 :1: 7.2 —0.9 :1: 10.0 19.9 18.8 0.992 36 24 32 14.4 :1: 3.5 2.1 :1: 9.7 19.5 19.0 A 119 142 53 56 3.7 :1: 15.9 1.1 :1: 9.4 18.6 17.0 0.999 67 25 22 5.1 :h 15.9 —23.8 i- 9.4 18.4 18.0 59 22 22 —10.1 :1: 15.9 25.9 i 9.4 18.5 17.8 A 121 44 30 28 —9.2 :1: 4.2 8.0 :1: 4.9 19.9 19.3 1.000 42 29 38 1.9 :1: 4.2 -3.0 :1: 4.9 20.3 19.6 30 21 17 —8.2 :1: 4.2 —9.7 :1: 4.9 20.0 20.0 29 20 17 8.5 :1: 4.2 10.1 :1: 4.9 20.4 20.4 A 151 184 54 59 -0.4 :1: 6.5 -5.4 :1: 16.1 19.0 17.6 1.000 93 27 26 -21.0 :1: 6.5 —-2.2 :1: 16.1 19.0 18.1 65 19 15 20.9 :1: 6.5 -—5.1 :1: 16.1 19.2 18.4 A 154 111 41 35 —0.5 :1: 12.4 -.0.3 :t 5.8 19.5 18.4 0.996 91 33 42 -4.1 :1: 12.4 —16.6 :1: 5.8 19.3 17.9 70 26 23 1.4 :t 12.4 17.0 :t 5.8 19.3 18.6 A 166 60 38 38 3.5 :l: 6.8 -2.1 :1: 3.2 20.1 19.2 0.994 52 33 34 2.8 :1: 6.8 9.0 :t 3.2 20.4 19.8 45 29 27 —1.9 :1: 6.8 —10.7 :t 3.2 20.2 19.6 A 168 114 49 48 —1.2 :1: 18.8 —0.5 :1: 8.3 18.6 17.4 0.995 74 31 25 -0.2 :t 18.8 23.6 :h 8.3 18.8 18.1 47 20 28 6.6 :1: 18.8 —24.9 d: 8.3 19.0 18.4 A 189 66 42 54 —26.7 :1: 10.3 16.9 :t 19.1 18.2 17.5 0.996 61 39 30 3.4 :1: 17.4 —0.6 :t 28.0 18.5 17.6 30 19 17 35.8 :1: 9.6 -16.1 :1: 18.7 18.3 18.3 A 193 168 64 58 0.6 :1: 13.8 0.6 :1: 15.0 18.8 17.4 0.992 56 21 15 —21.1 :1: 13.8 —18.8 :1: 15.0 19.0 18.4 40 15 27 21.3 :1: 13.8 19.8 :1: 15.0 19.2 18.9 114 TABLE 4.2. (continued) Cluster Ng %Ntot %L¢ot a: :1: a, y :1: 0y mmed mjm S (arcmin) (arcmin) (0) (O) (1) (2) (3) (4) (5) (6) (7) (3) (9) A 194 66 51 57 -3.7 :1: 34.0 0.5 :1: 40.4 16.9 16.2 0.990 34 26 26 -47.0 d: 28.9 -53.6 :1: 23.5 17.0 16.8 29 22 17 58.8 :t 22.0 59.4 :I: 22.9 17.1 17.1 A 225 142 70 69 ~10.2 :1: 8.0 -0.5 :1: 11.9 19.6 18.4 0.997 60 30 31 9.8 :1: 8.0 —0.2 :1: 11.9 19.5 18.8 A 246 82 80 80 -1.3 :t 12.8 0.3 :1: 13.1 19.8 19.3 0.999 21 20 20 18.0 :1: 4.3 17.0 :1: 4.7 20.0 0.0 A 274 122 70 68 -6.2 :h 4.4 -0.2 :1: 7.0 19.7 18.5 0.999 52 30 32 5.4 :1: 4.4 -0.8 :1: 7.0 20.0 19.1 A 277 115 50 50 1.1 :h 3.9 -0.8 :1: 8.7 19.7 18.2 0.996 70 30 34 —10.6 :1: 3.9 -1.3 :1: 8.7 20.0 18.9 45 20 16 12.4 :t 3.9 3.3 :1: 8.7 19.5 19.0 A 415 121 50 61 -2.6 :1: 9.6 8.1 :t 6.4 20.0 18.7 1.000 122 50 39 -1.1 i 9.6 -10.0 :1: 6.4 19.9 18.9 A 496 143 63 61 5.5 :1: 23.9 -6.9 :h 16.0 18.2 17.1 0.992 83 37 39 13.3 :1: 23.9 34.5 :1: 16.0 18.1 17.1 A 514 130 46 40 —9.3 :1: 7.1 —1.0 :t 9.2 19.6 18.5 0.994 78 28 35 —1.9 :t 12.6 —18.3 :t 2.7 19.5 18.5 74 26 25 11.9 :1: 6.1 3.7 :1: 10.4 19.5 18.7 A 787 111 72 59 4.4 :1: 4.5 1.5 :1: 5.6 20.4 18.9 0.995 43 28 41 —8.3 :h 2.8 0.2 :1: 7.0 20.2 19.5 A 957 143 50 54 -0.5 :h 19.5 2.0 :l: 7.6 18.6 17.5 1.000 88 31 31 -6.2 :t 19.5 -24.6 :1: 7.6 18.9 17.7 57 20 15 -3.5 :t 19.5 30.8 :1: 7.6 18.9 18.4 A 978 114 39 39 —0.3 :t 14.3 -0.4 :1: 7.0 19.0 17.9 0.996 124 42 40 -0.1 :1: 14.3 -20.6 :1: 7.0 19.2 17.7 57 19 21 4.9 :1: 14.3 19.3 :1: 7.0 19.0 18.2 A 993 215 79 77 -2.5 :b 16.4 -1.3 :1: 17.8 19.1 17.4 1.000 57 21 23 22.8 :1: 4.9 5.1 :t 17.9 19.0 18.1 A 1139 106 63 73 —14.0 :1: 13.7 -0.3 :t 22.8 18.4 16.7 0.990 62 37 27 22.6 :1: 13.7 —3.2 :1: 22.8 18.4 17.7 A 1185 155 46 52 0.9 :1: 23.8 —0.1 :1: 7.3 17.8 16.3 1.000 105 31 31 —5.7 :1: 26.6 -32.3 :t 11.7 17.8 16.7 75 22 17 —6.1 :1: 30.6 34.1 :1: 13.5 18.1 17.2 A 1213 118 45 57 2.3 :1: 8.1 1.5 :1: 17.6 18.6 17.2 0.994 94 36 28 -22.5 :1: 8.1 1.6 d: 17.6 18.9 17.9 49 19 15 24.5 :1: 8.1 -—2.4 :1: 17.6 18.8 18.4 115 TABLE 4.2. (continued) Cluster Ng %Ntot (701401 a: :1: a; y :t 0,, mmed mjm S (arcmin) (arcmin) (0) (O) (1) (2) (3) (4) (5) (6) (7) (8) (9) A 1238 77 43 49 --0.3 :1: 4.8 —1.9 :1: 12.0 19.8 18.9 1.000 57 32 27 —16.4 :1: 4.8 7.7 :t 12.0 20.0 19.6 46 26 24 15.9 :h 4.8 -7.5 :t 12.0 19.9 19.6 A 1254 211 80 70 -5.1 :1: 12.2 0.3 :1: 13.4 19.8 17.9 0.990 52 20 30 17.1 :1: 6.4 —10.9 :1: 9.3 19.6 19.0 A 1257 89 42 44 2.9 :1: 21.8 2.9 :1: 18.4 18.2 16.6 0.997 47 22 16 18.9 :1: 18.7 -36.5 :1: 8.8 18.2 17.6 39 18 15 5.5 :1: 25.1 40.5 :1: 5.9 18.4 17.9 37 17 25 —41.8 :t 5.5 6.0 :1: 26.6 18.0 17.6 A 1318 114 40 70 2.2 :1: 14.5 —12.1 :1: 9.6 19.3 17.7 0.995 104 36 20 5.5 :t 10.5 11.1 :1: 10.0 18.9 17.6 49 17 8 -23.0 :1: 4.6 10.7 :t 13.2 18.9 18.3 19 7 2 25.5 :1: 2.8 25.1 :1: 2.3 19.3 0.0 A 1364 178 79 80 0.5 :1: 7.9 2.7 :1: 8.0 19.8 18.1 1.000 48 21 20 —11.7 :1: 2.5 -7.9 :1: 5.3 19.6 19.1 A 1365 58 36 43 —2.6 :1: 8.6 —0.4 :1: 3.3 19.3 18.5 1.000 67 41 39 1.6 :1: 11.5 —12.3 :1: 4.3 19.7 18.5 38 23 18 —0.6 :1: 13.2 14.2 :h 5.1 20.0 19.6 A 1377 176 44 56 —12.1 :t 10.5 —6.8 :t 10.3 18.9 17.3 1.000 91 23 22 16.0 :1: 10.5 15.3 :1: 10.3 19.1 18.0 68 17 12 17.5 :t 10.5 —18.5 :h 10.3 19.2 18.2 67 17 10 -18.1 :1: 10.5 19.6 :1: 10.3 18.9 18.2 A 1382 84 46 56 0.3 :1: 3.5 —0.6 :1: 8.4 20.0 19.0 0.995 55 30 23 -10.9 :1: 3.5 -0.5 :1: 8.4 20.4 19.8 44 24 20 11.3 :1: 3.5 1.2 :1: 8.4 20.3 19.9 A 1399 164 58 72 —1.2 :1: 8.8 3.5 :1: 7.2 19.7 18.2 1.000 83 29 21 3.1 :1: 8.3 —10.3 :1: 4.8 20.1 18.9 37 13 7 —10.5 :t 5.5 15.8 :1: 1.6 20.3 20.1 A 1436 197 55 57 —0.4 :1: 12.6 -1.3 :1: 5.5 19.4 17.7 1.000 95 27 31 —2.9 :1: 12.6 16.5 :t 5.5 19.2 18.2 66 18 12 2.4 :1: 12.6 —18.1 :t 5.5 19.5 19.0 A 1474 76 41 39 —3.4 :1: 5.6 6.6 :1: 7.1 19.8 18.8 1.000 50 27 29 -l3.4 :1: 5.6 -7.2 :t 7.1 20.0 19.1 32 17 14 11.5 :1: 5.6 11.4 :t 7.1 19.9 19.8 29 16 18 10.3 :1: 5.6 -10.8 :h 7.1 19.3 19.4 A 1496 233 66 71 1.5 :1: 8.3 —0.4 :1: 9.1 19.8 17.8 0.997 82 23 22 -14.2 :1: 2.4 -0.7 :1: 8.1 20.0 18.5 40 11 7 12.1 :1: 3.6 12.1 :1: 3.9 20.0 19.7 116 TABLE 4.2. (continued) Cluster Ng %N¢a¢ %Ltot a: :1: a, y :1: 0y mmed mJ-m S (arcmin) (arcmin) (0) (O) (1) (2) (3) (4) (5) (6) (7) (8) (9) A 1541 37 18 9 —2.8 :1: 3.0 -1.8 :t 1.3 19.6 19.4 0.998 123 60 81 3.2 :1: 8.6 6.4 :1: 5.9 19.8 18.2 45 22 10 -3.4 :1: 10.4 —11.2 :t 4.7 19.9 19.6 A 1651 103 50 49 —0.3 :1: 4.3 0.7 :1: 10.2 19.9 18.9 0.991 54 26 29 —14.1 :1: 4.3 2.0 :1: 10.2 19.7 19.3 48 23 23 12.7 :1: 4.3 —4.3 :t 10.2 20.1 19.3 A 1656 226 53 50 5.3 :1: 16.6 0.0 :t 34.2 17.1 15.3 1.000 108 25 20 —42.9 i 16.6 10.2 :1: 34.2 17.2 16.0 90 21 30 47.1 :1: 16.6 -7.9 :I: 34.2 17.2 15.9 A 1691 119 48 60 1.0 :t 4.9 0.5 i 12.1 19.1 17.8 0.997 72 29 24 15.5 i 4.9 2.1 :1: 12.1 19.6 18.6 56 23 17 -16.6 :1: 4.9 -5.0 :1: 12.1 19.6 19.0 A 1749 116 53 56 0.7 i 14.6 1.3 :1: 6.8 19.1 18.0 0.999 65 30 31 0.2 i 14.6 —16.8 :t 6.8 19.2 18.3 38 17 13 -9.3 :1: 14.6 19.9 :h 6.8 19.7 19.3 A 1767 186 60 62 -2.8 :t 11.5 -9.5 :1: 8.3 19.4 17.9 0.999 122 40 38 —0.9 :1: 11.5 9.2 :1: 8.3 19.3 18.3 A 1773 119 42 40 4.5 d: 9.4 —1.3 :1: 11.2 19.8 18.7 1.000 111 39 38 —13.2 :t 5.2 7.3 :1: 7.8 19.8 18.6 52 18 22 —2.2 :1: 2.3 —0.6 :1: 2.6 19.4 18.8 A 1775 202 75 76 5.1 :1: 11.4 0.0 :1: 11.5 19.6 17.8 0.998 66 25 24 —14.1 :1: 5.6 —8.4 :1: 7.4 19.3 18.6 A 1795 36 12 12 2.3 i 1.4 —1.6 :t 2.4 19.3 19.2 1.000 138 48 51 -2.6 :1: 14.1 11.7 :1: 9.3 19.2 18.1 114 40 37 —2.6 :t 13.7 —-13.2 :1: 7.1 19.5 18.3 A 1809 244 79 67 8.1 :1: 7.2 -1.8 :1: 10.4 19.8 17.8 1.000 64 21 33 -10.1 :1: 7.2 4.0 :1: 10.4 19.7 18.7 A 1831 132 43 44 1.6 :1: 11.6 6.5 :1: 9.8 19.6 18.2 0.995 122 40 43 1.0 :1: 11.5 —1.5 :t 10.9 19.4 18.1 54 18 13 -3.4 i 13.0 —12.3 :1: 6.5 19.7 19.1 A 1837 182 68 70 -3.1 :t 23.1 7.9 :1: 18.0 18.6 17.1 0.995 86 32 30 -2.0 :1: 19.7 —28.7 :1: 9.0 18.7 17.9 A 1904 135 35 29 —8.0 :t 9.1 8.5 :1: 9.8 19.7 18.4 107 28 28 —2.6 :1: 11.9 —17.2 d: 4.1 19.5 18.1 88 23 30 1.5 :1: 4.5 —2.5 :1: 4.9 19.2 18.1 56 15 13 16.7 :1: 4.6 1.1 :1: 12.1 19.6 18.8 117 TABLE 4.2. (continued) Cluster Ng %N¢o¢ %Ltot :1: :1: a; y :1: a” mmed mjm S (arcmin) (arcmin) (0) (O) (1) (2) (3) (4) (5) (5) (7) (8) (9) A 1913 193 70 67 4.4 :1: 12.9 1.1 :1: 14.8 19.2 17.6 0.994 60 22 26 -13.6 :1: 10.0 12.9 :1: 10.5 19.2 18.4 23 8 7 23.7 :1: 5.0 —18.0 :1: 8.3 19.3 19.5 A 1927 154 63 53 1.9 :1: 10.2 -1.3 :h 8.5 19.8 18.4 0.999 53 22 25 -11.0 :1: 6.3 15.4 :t 4.7 19.8 19.2 38 16 22 11.2 :1: 7.5 —16.0 :1: 3.3 19.8 19.6 A 1983 328 75 78 0.4 :1: 19.8 —7.7 i 15.9 18.7 16.6 1.000 111 25 22 2.1 :1: 19.1 26.0 :1: 7.8 18.9 17.5 A 1991 235 64 68 —-0.3 :h 13.3 —5.7 :1: 11.9 19.1 17.5 1.000 96 26 24 -1.3 :t 16.4 19.3 :1: 6.6 19.4 18.2 37 10 8 23.5 :1: 2.9 -20.7 :1: 5.6 19.1 18.9 A 1999 116 62 59 —3.7 :l: 7.6 4.5 :t 4.9 20.1 18.7 1.000 71 38 41 0.0 :h 7.6 -8.4 :1: 4.9 19.8 18.7 A 2005 95 68 73 4.8 :1: 4.3 -—0.9 :t 6.8 20.3 18.9 0.998 44 32 27 —7.0 :1: 4.3 3.0 :1: 6.8 20.4 20.0 A 2022 214 66 74 -0.2 :1: 14.2 8.7 :t 9.9 19.2 17.4 0.995 108 34 26 4.6 :1: 14.2 -—14.5 :t 9.9 19.4 18.5 A 2028 164 71 68 —2.0 :1: 11.6 —1.1 :h 10.3 19.8 18.3 0.997 67 29 32 —0.5 :1: 12.2 15.7 :t 4.0 19.7 18.9 A 2048 200 64 63 —0.2 :1: 8.3 -1.4 :1: 8.1 20.1 18.4 100 32 34 9.2 :1: 5.5 9.4 d: 5.3 20.0 19.0 14 4 3 —11.7 :1: 4.6 —16.0 :1: 1.2 20.3 0.0 A 2063 117 55 53 -12.8 :t 23.2 1.1 :1: 28.6 18.3 17.3 1.000 48 23 22 30.4 :1: 10.9 —29.6 :h 15.2 18.4 18.1 46 22 25 1.7 i 4.5 ~3.2 :1: 7.2 18.0 17.6 A 2067 181 64 59 0.2 :1: 10.7 0.3 :1: 11.8 19.9 18.8 1.000 67 24 25 18.4 :1: 3.8 -14.2 :1: 4.1 19.6 19.1 35 12 16 —11.1 :1: 5.2 15.6 :1: 4.0 19.7 19.4 A 2079 154 48 53 —14.7 :1: 7.4 8.0 :1: 9.6 19.5 18.2 0.995 109 34 33 8.5 :l: 9.3 -8.5 i 6.9 19.5 18.7 29 9 8 —6.4 :1: 10.8 —20.9 :h 2.9 19.7 19.7 26 8 6 21.3 d: 2.6 13.5 :t 9.2 19.7 19.9 A 2092 167 63 71 —2.1 :t 13.1 —6.7 :t 10.0 19.3 18.1 1.000 63 24 21 -8.6 :1: 6.7 11.9 :t 8.0 19.2 18.7 37 14 9 17.8 :1: 4.5 17.9 :1: 5.3 19.7 19.5 A 2142 198 64 66 1.1 i 8.8 3.3 :1: 9.1 20.2 18.8 0.994 86 28 28 -8.2 :1: 6.5 -7.8 :1: 6.8 20.2 19.2 27 9 6 14.6 i 2.6 11.2 :h 5.0 20.4 20.4 118 TABLE 4.2. (continued) Cluster Ng %Ntot %Lto¢ :1: :1: a, y :1: 0,, mmed m jm S (arcmin) (arcmin) (0) (O) (1) (2) (3) (4) (5) (6) (7) (8) (9) A 2147 220 47 49 —-2.8 :1: 20.4 —1.9 :1: 17.2 18.4 16.6 1.000 168 36 37 -5.7 :1: 24.1 31.7 :t 9.6 18.3 16.8 43 9 7 —0.8 d: 27.4 —43.0 :t 2.9 18.7 18.1 34 7 6 -44.5 :t 1.7 0.7 :1: 24.1 18.4 18.3 A 2151 155 40 46 —1.3 :1: 22.2 -0.3 i 9.1 18.3 16.7 1.000 123 32 30 —0.7 :1: 22.2 29.3 :h 9.1 18.3 16.9 110 28 24 7.8 :t 22.2 —29.6 :1: 9.1 18.6 17.2 A 2152 180 38 38 —8.5 :1: 13.7 —8.0 :t 25.1 18.5 17.0 0.995 152 32 37 32.9 :1: 9.6 -19.8 :t 14.4 18.5 16.8 112 24 21 17.0 :t 13.5 18.0 :1: 16.4 18.5 17.4 27 6 4 —40.4 :1: 2.8 —l.4 :t 26.4 18.8 18.8 A 2175 222 50 46 -1.6 :1: 8.3 1.3 :1: 8.3 20.2 18.9 1.000 161 36 38 —7.9 i 5.7 —6.6 i 5.3 20.2 18.9 65 15 15 9.3 :h 4.3 10.5 :1: 3.9 20.4 19.6 A 2197 202 65 62 —17.3 i 17.1 -—9.7 :1: 28.5 17.9 15.9 1.000 111 35 38 26.6 :1: 17.1 3.8 :1: 28.5 17.9 16.3 A 2199 155 40 45 —5.7 :1: 18.7 6.8 :1: 16.5 17.8 16.3 0.996 93 24 19 -—5.0 :1: 24.5 -30.7 :t 14.0 18.1 16.9 93 24 23 37.1 :1: 10.2 —2.6 :1: 27.8 17.9 16.7 48 12 12 -9.4 :1: 20.1 45.7 :1: 6.5 17.7 17.2 A 2255 276 66 71 —0.6 :1: 9.8 —6.0 :1: 7.0 19.6 17.9 1.000 141 34 29 ——1.1 :t 9.8 11.3 d: 7.0 19.8 18.5 A 2256 237 53 54 4.1 :1: 13.7 0.9 i: 13.5 19.0 17.3 0.995 121 27 21 —12.6 :1: 7.9 —16.1 :t 7.5 19.3 18.0 93 21 26 —0.5 :t 4.2 —3.7 :1: 3.8 18.8 18.0 A 2347 44 49 39 -0.5 :1: 7.0 -0.8 :1: 2.7 20.6 19.8 1.000 25 28 40 -—1.9 :1: 7.0 9.9 :h 2.7 20.9 21.0 21 23 21 —0.8 :t 7.0 -—9.7 :t 2.7 20.6 21.0 A 2399 149 58 57 8.4 :1: 9.4 0.7 :t 14.9 19.0 17.7 0.999 107 42 43 -14.3 :t 9.4 —0.3 d: 14.9 19.2 17.9 A 2410 183 78 84 1.2 :1: 10.7 —2.3 :1: 9.3 19.6 17.6 0.999 52 22 16 —10.2 :1: 5.4 13.5 :1: 5.0 19.5 18.9 A 2457 148 60 63 2.1 :t 13.5 1.2 :1: 5.9 19.1 17.8 1.000 58 23 19 2.7 :1: 13.5 -18.4 :h 5.9 19.5 18.8 42 17 17 7.9 :1: 13.5 19.9 :t 5.9 19.3 18.9 A 2634 280 68 73 —9.8 :1: 22.0 —6.1 :t 26.9 18.0 16.4 0.999 97 24 20 41.4 :1: 9.0 -1.9 :t 29.0 18.2 17.1 34 8 8 —43.2 :t 7.1 36.7 :1: 7.3 18.0 17.7 119 TABLE 4.2. (continued) Cluster N9 %Ntot %Ltot :L' :I: 0’; y :I: 0'” mmed mjm S (arcmin) (arcmin) (0) (O) (1) (2) (3) (4) (5) (6) (7) (8) (9) A 2657 77 45 46 2.0 :1: 8.5 -1.4 :1: 20.2 18.5 17.8 0.999 56 33 32 —26.5 :1: 8.5 —4.9 :1: 20.2 18.5 18.0 38 22 22 28.9 :1: 8.5 —6.6 :1: 20.2 18.5 18.2 A 2666 103 60 70 —28.7 :1: 22.2 —2.7 :1: 37.1 17.8 16.5 1.000 68 40 30 31.4 :1: 16.7 0.9 :t 35.7 18.0 17.5 A 2670 132 52 57 0.0 :1: 4.4 1.9 :1: 10.8 19.0 17.9 1.000 77 30 25 —16.2 :1: 4.4 1.8 :1: 10.8 19.4 18.6 46 18 18 13.1 :1: 4.4 —3.0 :t 10.8 19.1 18.7 A 2675 100 55 50 -2.7 :1: 10.4 —6.9 :1: 8.4 19.5 18.4 0.999 82 45 50 —1.2 :1: 11.4 12.7 :1: 5.3 19.5 18.2 A 2700 66 51 51 —0.8 :1: 8.6 —1.4 :1: 5.6 19.6 18.9 1.000 43 33 35 4.3 :1: 7.9 9.0 :1: 5.2 19.8 19.2 20 16 14 -1.5 i 8.0 -15.3 :1: 1.4 19.5 0.0 120 Likewise, A151, with a larger area in the Dressler data, clearly has a low density elongation to the east which is partly resolved in the smaller and deeper map of the APS data. It is this elongation that KMM identifies as significant and not the peak to the south as in the Dressler data. The peak to the south may still be significant except that a number of its members did not make the one Abell radius cutoff. Other clusters similar to A151 include A978, A1139, A1913, A1983, and A1991. In six cases — A168, A1185, A1377, A1656, A2063, and A2256 — KMM indicates the existence of substructure that is not present in the Dressler data. While it is possible that three, A168, A1185, and A1656, are due simply to the increased size of the APS map, the others are of similar size and the complexity in the maps is most likely due the the inclusion of fainter galaxies. Whether or not these structures are real, or simply due to the background, can only be resolved with redshift measurements. However, using the procedure discussed below, it appears likely that the substructure in A168, A1377, and A2063 is due to background contamination. 4.3 The DEDICA Algorithm Many of the disadvantages associated with the use of KMM to detect substructure in galaxy clusters can be eliminated by using a method which evaluates the significance of the peaks in the probability density distribution. The easiest way to do this is a procedure similar to that adopted by GB: count a peak as significant if its density is 30 above the background density. The main drawback to such a procedure is that the significance of the group depends critically on the assumed background density. Furthermore, as in KMM, it is advantageous to be able to assign individual galaxies to a specific group and to ascribe membership probabilities, this time without making any assumptions about the form of the underlying PDF for the groups. The details of 121 such a clustering procedure have been worked out and implemented by Pisani (1996) in the program DEDICA, which is summarized below. First, the PDF of the cluster needs to be calculated. This is accomplished using the adaptive-kernel method used in Chapter 3 for construction of the contour maps. In this case it is advantageous to use the normal kernel despite its infinite support and lower efficiency since the next step requires the calculation of the derivative of f (:1:) Furthermore, more attention needs to be given to the choice of initial smoothing parameter h, since the number of groups and their significance will be dependent on the density estimate. Pisani (1996) recommends choosing h by the method of minimizing the cross validation term in the IMSE (least squares cross validation, LSCV). To accomplish this DEDICA sets h = 4h,, (hn being the smoothing window as specified by the normal rule) and reduces it by a factor of 2 with each iteration till the value which minimizes the CV term is found. However, because LSCV can be sensity to small- scale effects in the data for small smoothing parameters (Silverman 1986), in this thesis h is given a lower bound of 100 kpc. This is about the size of a large galaxy. If LSCV returns of value of h smaller than this size, it is deemed to have failed and a value of h = 0.85hn is adOpted. Without taking this precaution, h can, at times, be reduced to scales of 20 kpc or so and even the high-density peaks are assigned less than 5% of the galaxies within an Abell radius. Thus the trend appears to be that larger values of h will generally lead to larger subcluster sizes. This is due to the fact that smaller smoothing parameters lead to density estimates with larger gradients, and therefore a smaller spatial extent for the peaks found in the next step. In most of the clusters hay g 0.85hopt, and is deemed to have failed in about 40% of the cases. This poor success rate, taken with the fact that a majority of the rest of the clusters have hcv 2: how, makes it difficult to justify the extra expense in CPU time 122 necessary to calculate hcv. Furthermore, it is unclear whether the goal of minimizing the ISE of the density estimate is appropriate when calculation of the derivative of the density estimate is sought. The next step is to identify the peaks in the density estimate, and therefore possible subclusters. This is done by an iterative scheme due to Fukunaga & Hostetler (1975.) The local maxima of f (:1:) can be found by the limit of the sequence: Wm") (4.12) rm+1 = rm + a2 where 1‘0, the position vector, is set to the position vector of each data point in turn. The factor (12 controls the rate of convergence of the sequence which is optimized with: 2 gilVfUO/“TIHT (4-13) 0.2: The iterative procedure is stopped when |rm+1 — rmI/rm g 10‘s. A cluster is then defined as the group of points with positions vectors 7‘,- for which equation (4.12) converges to the same value of r. Clusters with only a single member are considered to be isolated points. Once the set of V groups, C“, and no isolated points has been identified, f (r), the PDF of the entire sample, can be defined as: f (r) = 2 MT). (4.14) :0 where f,,(r) is the PDF for each of the l/ groups: f1 = i Z K(rj.a,-;r). (4.15) N 26014 123 and fo(r) is the PDF of the background. The statistical significance of the 11th group can be evaluated using a likelihood-ratio test: (4.16) Thus defined, the likelihood ratio is distributed as chi-square with one degree of freedom (Materne 1979). The quantity LN is the sample likelihood or: i=1 [1:0 LN = H [i 13403)] 1 (4'17) and L01) is the value that LN would have if the 11th cluster were described by f0(r) and thus actually belonged to the background. Or: II N 1 I‘ll“ [“T') -(f# Ti) +37 2 K(7”j,00;7‘i) . (4.18) 1'60» As discussed in Chapter 2, the background density adOpted here is the density at the point with the largest bandwidth factor hAo in the adaptive-kernel density estimate. Another choice of the background density fo(r) could be the density due to all points defined as isolated. In many cases though, all points are assigned to one group or another with no points listed as isolated. Finally, the probability that the ith galaxy belongs to the 11th cluster can be defined as: f__u(7'i) P(z' E p): f—m). (4.19) One added degree of freedom available with the DEDICA algorithm which needs to be mentioned is the ability to merge nearby groups into a single group. The merging is accomplished by a minimum spanning tree technique. Thus the user can specify, to some extent, on what length scales DEDICA is to search for substructure by specifying a distance den-t. Density peaks which are further apart then do,“ will be considered distinct, while those closer will be merged. With do.“ set very large, 124 all galaxies will be in one large group; with dc,“ very small most galaxies will be in their own group and isolated. This is different from the behavior of KMM, where the scale of the substructure is set by the scale of the data, z'.e., by one Abell radius in this case. In the application of DEDICA to the clusters den-t is set such that all galaxies are assigned to one large group. It is then lowered and the results analyzed as each successive group is split off from the main cluster. The process is halted before the significant groups found in previous steps are split into their generally non-significant component parts. 4.3.1 Application of DEDICA to the Dressler Sample The results of the DEDICA runs are given in Table 4.3. Only groups significant at the 99% level are included. Column (1) lists the cluster. Column (2) gives the number of galaxies assigned to each group while column (3) gives the percentage of the total number in the cluster. The median a: and y positions of the galaxies in each group is listed in columns (5) and (6) along with their one sigma errors. Column (7 ) gives the significance of each group evaluated against the background which is listed in Table 2.1. 125 TABLE 4.3. DEDICA Cluster Parameters for Dressler Sample Cluster N % of total a: :1: a, y :t 0,, S (arcmin) (arcmin) (1) (2) (3) (4) (5) (6) A 14 61 77 —4.3 d: 10.1 3.3 d: 8.7 1.000 18 23 12.2 :1: 7.7 —14.1 :1: 7.7 1.000 A 76 35 49 10.3 i 8.4 —4.4 :1: 12.0 1.000 25 35 —6.0 :1: 7.3 11.8 :1: 7.9 1.000 A 119 85 73 3.6 :1: 11.6 0.6 :1: 10.1 1.000 16 14 —13.1 :1: 7.0 21.7 :1: 5.2 1.000 15 13 0.2 :1: 6.2 —17.9 :1: 4.0 1.000 A 151 72 69 0.5 :1: 15.4 2.9 :1: 10.4 1.000 21 20 2.5 :1: 10.6 -29.4 :1: 7.2 1.000 A 154 41 52 4.4 :1: 8.3 0.3 :1: 6.7 1.000 18 23 -11.7 :t 7.4 —19.4 :1: 5.9 1.000 16 20 0.2 :h 5.1 18.3 :1: 4.8 1.000 A 194 37 49 9.1 d: 16.5 —ll.0 :1: 11.5 1.000 29 39 —11.3 :1: 13.6 10.8 :1: 13.9 1.000 A 400 51 55 10.1 i 13.9 —1.8 :1: 13.8 1.000 29 32 —15.1 :1: 10.4 13.5 :1: 13.8 1.000 A 496 60 74 —0.9 :1: 16.0 4.6 :1: 12.5 1.000 21 26 0.4 :1: 4.4 —13.7 :1: 5.9 1.000 A 548 112 48 19.3 d: 10.3 -l5.9 :1: 14.2 1.000 89 38 —24.8 :1: 11.9 6.0 :1: 12.2 1.000 33 14 —7.5 :1: 10.0 25.6 :1: 6.5 1.000 A 592 29 48 —2.0 :h 14.0 4.0 :t 10.6 1.000 14 23 -21.6 :1: 7.6 -9.7 :1: 8.6 0.999 9 15 39.6 :1: 0.8 18.1 :1: 17.0 0.994 A 754 46 31 —18.1 :t 11.3 -2.1 :1: 6.9 1.000 36 24 21.3 :1: 12.8 -19.6 :1: 24.9 1.000 33 22 2.1 :1: 4.3 5.0 :h 4.2 1.000 A 838 17 27 18.7 :1: 7.1 2.1 :t 10.4 1.000 14 23 —0.7 :t 3.1 —5.0 :1: 9.5 0.999 8 13 —40.8 :t 3.5 24.2 :1: 3.1 0.998 A978 32 52 —0.5 :1: 14.2 15.4 d: 12.6 1.000 30 48 0.3 :t 15.1 —12.8 :1: 7.7 1.000 A 979 47 55 -3.8 :1: 11.5 1.5 :I: 10.3 1.000 17 20 9.8 :1: 15.8 —38.2 i 4.9 1.000 126 TABLE 4.3. (continued) Cluster N % of total a: :1: a, y :b 01, S (arcmin) (arcmin) (1) (2) (3) (4) (5) (6) A 993 28 31 —l4.6 :1: 14.5 -0.5 :1: 7.7 1.000 21 23 7.0 :1: 7.6 16.0 :t 6.0 1.000 19 21 15.5 :1: 6.0 -20.5 :1: 15.8 1.000 A 1069 15 32 10.7 :1: 6.3 4.0 :t 11.8 1.000 15 32 -3.0 :t 6.8 -20.0 :1: 7.3 1.000 A 1139 39 62 6.2 :1: 18.7 5.2 :1: 10.7 1.000 20 32 —13.5 :1: 10.8 -11.6 :1: 11.0 0.999 A1142 19 32 -4.4 :h 3.2 20.0 :1: 14.2 1.000 17 29 7.3 :1: 8.4 -23.1 :h 6.3 1.000 12 20 13.4 :1: 11.7 2.8 :1: 9.8 0.997 11 19 —21.5 :1: 5.6 -2.4 :1: 4.8 1.000 A 1631 50 56 —2.9 :t 17.6 7.6 :1: 8.5 1.000 20 22 5.9 :h 5.5 —1.6 :1: 5.8 1.000 20 22 —3.8 :1: 14.4 -26.1 :1: 7.9 1.000 A 1644 67 46 3.3 :1: 19.7 18.1 :1: 11.0 1.000 63 43 —4.7 :t 17.4 —14.0 :1: 11.2 1.000 A 1656 132 54 —3.7 :1: 11.3 —0.5 :1: 10.8 1.000 64 26 24.3 i 9.2 —10.8 :1: 19.7 1.000 A 1736 76 46 —3.2 :1: 22.0 -15.2 :t 10.5 1.000 56 34 —2.4 :1: 13.3 8.2 :1: 9.8 1.000 A 1913 29 34 5.4 :1: 4.2 2.4 :1: 8.0 1.000 19 22 -8.4 :t 6.6 -3.4 :1: 10.8 0.997 A 1983 54 44 2.8 :1: 12.7 -3.2 :1: 10.0 1.000 35 28 0.6 :1: 11.0 20.8 :1: 8.0 1.000 A 1991 26 49 8.0 :1: 14.9 -6.9 :1: 9.7 1.000 18 34 —2.0 :1: 12.9 14.2 :1: 8.6 1.000 A 2151 73 48 11.3 :1: 12.8 -8.1 :1: 13.6 1.000 46 30 —3.8 :1: 12.6 26.2 :1: 8.8 1.000 25 16 -18.5 :1: 5.2 0.3 :h 7.4 1.000 A 2256 53 64 —2.7 :1: 9.8 -3.4 :1: 3.9 1.000 26 31 4.6 :1: 6.5 7.4 :1: 4.7 1.000 A 2589 31 43 -0.7 :1: 5.7 —8.7 :1: 7.8 1.000 30 42 6.0 i 10.0 4.0 :t 4.7 1.000 A 2634 66 50 —1.4 :t 11.8 -3.4 i 7.4 1.000 38 29 -6.6 :1: 9.6 11.9 :1: 5.3 1.000 16 12 13.5 :1: 5.8 -14.4 :1: 4.8 0.999 TABLE 4.3. (continued) 127 Cluster N % of total :1: :1: 0,, y :1: 0,, S (arcmin) (arcmin) (1) (2) (3) (4) (5) (6) A 2657 58 71 —0.5 :1: 8.0 1.4 :1: 9.4 1.000 13 16 17.8 :1: 2.6 -0.1 :1: 7.3 1.000 11 13 --20.7 :1: 4.5 —11.1 i 4.7 0.999 DC 0103-47 26 49 —18.3 :t 10.5 0.0 :t 10.9 1.000 15 28 14.0 :1: 13.7 —10.3 :1: 11.4 0.999 12 23 9.6 :1: 6.6 33.4 :1: 7.5 1.000 DC 0317-54 47 72 2.8 :1: 10.7 8.2 :1: 19.5 1.000 17 26 —21.2 i 12.8 -20.0 :h 9.3 1.000 DC 0326—53 49 30 1.0 :1: 11.4 -13.8 :1: 10.0 1.000 33 20 —25.4 :1: 8.1 26.5 :1: 7.1 1.000 DC 0329-52 146 77 —3.5 :t 11.9 2.2 :1: 14.3 1.000 39 21 24.8 :1: 7.1 —17.1 :1: 14.5 1.000 DC 0410-62 31 48 7.0 :1: 14.5 —7.2 :1: 13.2 1.000 18 28 7.6 :h 18.1 36.1 :1: 8.2 0.996 15 23 —32.4 i 7.9 —2.7 :h 21.7 0.995 DC 0428-53 55 42 —0.1 :1: 9.7 -8.1 :1: 8.6 1.000 31 24 1.6 i 5.4 13.4 :1: 9.2 1.000 DC 0559—40 53 47 9.8 :1: 12.4 -5.4 :t 9.2 1.000 34 30 —12.1 :t 11.9 3.6 :1: 11.4 1.000 DC 0622-64 71 72 3.4 :t 15.9 1.7 :t 14.2 1.000 16 16 —13.0 :1: 10.8 —26.4 :1: 5.8 1.000 11 11 —10.5 :1: 8.3 32.8 :1: 8.2 0.998 DC 1842-63 25 45 ——4.0 :t 6.8 6.5 :1: 5.9 1.000 23 42 3.5 :1: 10.7 —5.1 :1: 6.7 1.000 DC 2048-52 144 67 1.4 :t 13.1 —8.8 :1: 13.0 1.000 35 16 —11.0 :1: 8.1 34.4 :1: 7.7 1.000 28 13 25.3 :1: 9.1 21.1 :1: 16.2 1.000 DC 2103-39 39 36 13.4 :1: 10.3 —3.2 :t 12.4 1.000 36 33 —8.9 i: 9.9 4.1 :1: 8.1 1.000 17 16 0.0 :1: 10.5 32.6 :1: 7.3 1.000 16 15 —18.5 :1: 5.4 —16.5 :1: 3.8 1.000 DC 2345-28 57 60 1.8 :1: 11.2 2.8 :1: 6.4 1.000 17 18 -15.9 :1: 6.9 —17.9 :1: 7.6 1.000 11 12 18.9 :1: 5.1 -9.4 :1: 3.5 1.000 DC 2349—28 31 46 3.9 :1: 10.8 —3.7 :1: 6.7 1.000 24 35 0.8 :1: 15.8 20.6 :1: 6.6 1.000 10 15 —11.8 :h 4.0 —17.8 d: 4.7 1.000 Centaurus 32 44 18.4 :t 10.7 -3.6 :1: 14.4 1.000 30 41 —13.9 :1: 8.9 2.4 :1: 19.0 1.000 128 There are 39 clusters (70%) in the Dressler sample that have significant groups at the 99% level. Many of the peaks visible in the adaptive-kernel maps which KMM was unable to return as significant, because of the small 11 values of the groups, have been identified by DEDICA. These include A76, A400, A978, A1139, A1983, A2657, A1991, DC 0317—54, DC 0329-52, DC0410—62, DC 1842-63, and Centaurus. The partitions of A14 and A1142 are returned by KMM with a significance of 90% and therefore just missed the 95% cutoff. Note that all of the clusters identified by GB as having substructure have been confirmed by DEDICA (although four clusters — A119, A2657 DC 0622-64, and DC 2048-52 — are not counted in this study since in these cases the number of galaxies in the groups is less than 20% of the total.) This is not surprising given the similarity of the two approaches. Likewise, there are 22 clusters for which DEDICA gives positive results which did not meet the GB criteria. In these cases the difference is caused by the use of the adaptive-kernel density instead of the fixed, box-car smoothed density of GB. With a smaller kernel width used in the high-density regions, the peaks can have a higher density than if the galaxies in that peak had been spread out over the larger area of the bin width used by GB. On the other hand, there are six clusters for which substructure found by KMM is not confirmed by DEDICA. These clusters are A957, A2657, DC 0247—31, DC 0608- 33, DC 2048-52, and DC 2345-28. Where these groups are identified by DEDICA, they are listed in table 4.3. In most cases the groups simply did not have enough members to make the 20% cutoff. This applies to a number of other groups identified by KMM as well, such as the group to the south in A119 and the groups to the east and west in A496. The exception to this is DC 2048-52 which does not have any similar partition given by DEDICA. Although the Monte Carlo experiments indicate that the probability of a false positive in a cluster with only 48 members is small, the fact that there is no confirming homoscedastic fit or DEDICA partition for this case 129 leads to the suspicion that the structure in this cluster is not real. 4.3.2 Application of DEDICA to the HGT Sample In a similar fashion, DEDICA was applied to the HGT clusters with the results listed in Table 4.4. Column (1) lists the cluster name. Column (2) gives the number of galaxies in each group. The percentage of the total number of galaxies for each group in given in column (3), while the percent of the total luminosity contained in each group is given in column (4). In columns (5) and (6) the a: and 3; positions, along with their respective 10 errors of the groups are listed in arcminutes. The median apparent magnitude mmed for each group is listed in column (7). The average apparent magnitude of the 10th-to-20th ranked galaxy is given in column (8). Lastly, column (9) lists the significance of each group. In all, 96 (81%) of the clusters in the HGT sample are returned by DEDICA as having significant substructure. A comparison between the Dressler and APS samples shows that the partitions found significant by DEDICA are much more stable than KMM to variations in the chosen field size of the clusters. Despited the changes in size and magnitude limit between the two samples, 12 clusters are returned with essentially the same partition. In five cases — A957, A1185, A1377, A2040, and A2063 — substructure was detected by DEDICA in the deeper survey that was not detected in the Dressler data. However, these are all likely to be due to background contamination. Other differences can be explained as groups leaving (as in A151) or entering (as in A194 and A496) the field of view. In the case of A2256, however, it is clear that the larger background population in the APS map has washed out the structure so that it can no longer be identified. On the other hand, this larger number of galaxies has enabled KMM to identify the group, which it could not in the smaller Dressler sample. 130 TABLE 4.4. DEDICA Cluster Parameters for HGT Sample Cluster N %Ntot %Lm z :1: 03 y :1: 0,, mmed mjm S (arcmin) (arcmin) (0) (O) (1) (2) (3) (4) (5) (6) (7) (8) (9) A 21 141 48 46 -2.6 :t 5.5 -0.2 :l: 5.4 20.0 18.8 1.000 69 24 25 3.4 :h 5.8 10.2 :t 4.6 20.0 19.0 1.000 39 13 17 —10.6 :1: 4.8 -12.8 :1: 3.3 19.7 19.2 1.000 A 76 90 46 50 2.1 :1: 13.9 —10.7 :1: 16.1 18.5 17.3 1.000 47 24 22 —14.8 :t 7.1 4.5 :1: 10.2 18.4 18.0 1.000 A 85 112 35 34 0.1 :1: 10.8 -1.6 :t 8.7 18.8 17.9 1.000 59 18 24 —18.1 :1: 6.5 -27.9 :1: 4.1 18.7 17.9 1.000 51 16 15 —20.4 :1: 4.5 -2.6 :h 7.3 18.8 18.2 1.000 A 88 43 60 59 1.3 :1: 7.3 —5.6 :t 5.3 18.0 17.3 1.000 24 33 35 —9.9 :1: 5.0 4.8 :1: 3.8 17.7 18.1 1.000 A 104 112 74 72 -1.6 :1: 8.8 —3.1 :1: 6.9 19.7 18.6 1.000 21 14 23 13.4 :1: 3.5 8.2 :1: 6.3 19.1 19.8 1.000 A 119 226 84 85 4.5 :1: 15.5 -—4.3 :1: 17.2 18.5 16.9 1.000 42 16 15 —18.0 :1: 12.1 27.3 :1: 8.1 18.5 18.2 1.000 A 121 59 41 35 —9.4 :1: 4.2 3.5 :t 8.8 20.0 19.3 1.000 40 28 35 1.3 :1: 4.7 —0.2 :1: 3.0 20.0 19.5 1.000 20 14 12 10.8 :t 2.9 10.4 :1: 2.8 20.2 20.6 1.000 13 9 6 -7.2 :1: 3.1 -l4.9 :h 1.7 20.3 0.0 0.998 A 147 56 36 33 —2.3 :l: 9.5 7.9 :1: 12.6 18.7 18.1 1.000 39 25 38 9.5 :1: 5.9 -12.8 :h 12.8 18.1 17.8 1.000 A 151 161 47 53 -0.8 :1: 6.6 -8.0 :1: 12.9 19.0 17.7 1.000 130 38 30 4.3 :1: 22.3 5.7 :1: 20.3 19.0 18.0 1.000 46 13 14 ~18.8 :1: 4.7 —9.9 :h 5.4 19.0 18.6 1.000 A 154 96 35 31 3.9 :1: 8.7 —2.0 :1: 7.9 19.3 18.3 1.000 83 31 39 —12.0 :1: 8.8 —14.7 :1: 7.2 19.2 18.0 1.000 80 29 25 2.0 :1: 13.5 16.2 :1: 6.2 19.3 18.5 1.000 A 166 78 50 45 1.9 :1: 8.3 -—8.2 :l: 4.5 20.1 19.2 1.000 44 28 32 5.1 :1: 5.3 9.2 :1: 3.3 20.2 19.8 1.000 27 17 20 —0.2 :t 2.4 0.1 :1: 1.9 20.0 20.1 1.000 A 168 88 37 37 -0.4 i 7.8 —1.2 :1: 9.8 18.6 17.6 1.000 44 19 19 7.0 :1: 9.9 18.3 :1: 8.1 18.7 18.3 1.000 33 14 11 —18.6 :1: 7.4 —15.6 :t 10.1 19.0 18.9 1.000 A 189 55 35 27 4.2 :1: 18.0 -7.2 :1: 31.4 18.5 17.9 1.000 37 24 40 -29.9 :1: 10.4 4.5 :t 10.3 18.1 17.8 1.000 29 18 14 36.8 :1: 8.9 —10.6 :1: 16.9 18.3 18.3 1.000 28 18 15 -16.2 :1: 11.4 27.0 :t 6.7 18.1 18.3 1.000 131 TABLE 4.4. (continued) Cluster N %N¢o¢ %L¢o¢ a: :h 0, y :1: 0,, mmed mjm S (arcmin) (arcmin) (0) (O) (1) (2) (3) (4) (5) (6) (7) (8) (9) A 193 93 35 38. —1.1 :1: 7.9 -—0.3 :1: 7.8 18.6 17.7 1.000 77 29 20 —10.7 :1: 23.7 —23.2 :t 9.4 19.0 18.2 1.000 74 28 38 2.7 :1: 23.1 23.6 :1: 7.7 19.1 18.2 1.000 A 194 51 40 50 —10.9 i 23.6 0.4 :t 20.7 16.7 16.2 1.000 33 26 17 —-15.9 :1: 47.0 —65.3 :1: 11.9 17.2 17.1 1.000 27 21 18 59.6 :1: 20.4 59.9 :h 22.8 17.0 17.1 1.000 A 225 79 39 31 —6.3 :1: 8.8 10.7 :1: 10.9 19.8 19.0 1.000 67 33 40 7.9 :1: 8.2 -6.7 :t 7.6 19.3 18.7 1.000 56 28 29 —16.1 :t 5.0 —5.7 :1: 7.5 19.5 18.9 1.000 A 246 34 33 35 3.1 :t 8.6 —2.9 :1: 6.5 19.8 19.8 0.998 25 24 24 16.2 :1: 5.6 15.6 :t 6.2 19.8 19.9 1.000 16 16 14 —8.5 :1: 6.5 18.0 :1: 3.9 19.9 0.0 1.000 12 12 14 —18.9 :h 3.1 4.6 i 5.8 19.4 0.0 0.995 A 274 60 34 29 —4.8 d: 5.3 -7.1 :1: 3.6 20.1 19.4 1.000 56 32 36 5.5 :h 4.5 0.7 :1: 6.1 19.6 19.0 1.000 44 25 26 —6.4 i 3.8 7.2 :1: 3.6 19.7 19.2 1.000 A 277 69 30 34 4.2 :h 6.6 —5.1 :1: 4.5 19.6 18.6 1.000 50 22 14 2.2 :1: 3.6 5.9 :1: 5.5 19.9 19.4 1.000 A 389 104 60 62 —0.4 :1: 6.6 0.5 :1: 4.6 20.2 19.2 1.000 38 22 25 —4.5 i 7.4 —9.5 :1: 3.2 20.2 20.0 1.000 31 18 13 7.4 :1: 4.8 10.4 :1: 2.8 20.4 20.4 1.000 A 400 52 27 28 -29.0 :1: 17.6 —1.3 :1: 196.1 17.3 16.9 1.000 38 20 20 2.5 :t 10.4 —7.1 :t 9.2 17.3 17.1 1.000 A 415 81 33 52 -3.1 :1: 6.0 5.7 :1: 4.4 19.8 18.8 1.000 69 28 20 —3.4 :1: 4.4 —9.0 :1: 5.4 20.0 19.4 1.000 A 496 110 49 51 6.3 :1: 14.0 —0.1 :1: 13.6 18.0 17.1 1.000 69 31 33 24.8 :t 20.5 39.6 :1: 10.8 18.0 17.2 1.000 A 514 101 36 32 —11.3 :1: 5.7 —2.3 :1: 6.3 19.6 18.6 1.000 58 21 27 -10.4 :t 7.0 —18.3 :1: 2.9 19.4 18.6 1.000 27 10 9 14.2 :1: 4.4 5.1 :h 4.7 19.5 19.7 1.000 A634 44 62 65 -12.6 :1: 19.0 16.1 :1: 20.8 17.9 17.6 1.000 23 32 31 38.0 :1: 19.2 —14.4 :1: 39.7 17.7 18.0 1.000 A 779 45 39 39 —10.6 :1: 16.9 —17.6 :1: 18.8 17.5 17.1 1.000 28 24 14 29.4 :1: 16.8 18.7 :t 22.7 17.5 17.6 1.000 A 787 113 73 56 4.5 :1: 4.7 2.2 :1: 5.6 20.3 19.0 1.000 27 18 32 —9.1 i 2.4 2.5 :1: 3.9 20.0 20.4 1.000 A 957 151 52 56 6.7 :t 16.9 2.3 :t 11.8 18.6 17.5 1.000 86 30 30 -17.0 i 15.0 —22.4 :1: 10.5 18.7 17.8 1.000 51 18 14 -6.7 :1: 22.0 33.2 :t 4.3 18.9 18.5 1.000 132 TABLE 4.4. (continued) Cluster N %N¢o¢ %Lto¢ :I: :1: 0,c y :1: 0,, mmed mjm S (arcmin) (arcmin) (0) (O) (1) (2) (3) (4) (5) (6) (7) (8) (9) A 978 119 40 42 1.2 :1: 13.9 2.3 :1: 8.0 18.9 17.9 1.000 107 36 36 6.9 :t 10.0 —19.6 :t 7.1 19.0 17.8 1.000 39 13 14 1.7 :1: 11.9 22.3 :1: 5.5 18.8 18.6 1.000 A 993 117 43 48 -7.5 :1: 11.4 8.0 :t 11.3 18.9 17.7 1.000 78 29 20 9.1 :1: 16.3 —22.1 :t 6.7 19.2 18.3 1.000 40 15 18 17.9 :1: 6.0 21.5 :t 6.0 18.6 18.1 1.000 A 1020 163 62 66 4.8 :1: 9.9 9.3 d: 9.4 19.5 17.9 1.000 102 38 34 —12.9 :1: 12.5 -9.7 :t 12.1 19.7 18.7 1.000 A 1126 98 40 26 —2.8 :1: 6.5 -1.9 :1: 4.1 19.8 18.7 1.000 89 36 29 5.9 :1: 11.8 11.5 :1: 4.5 19.8 18.5 1.000 41 17 40 8.8 :1: 4.4 —6.4 :t 4.3 20.1 19.7 1.000 A 1139 84 50 58 —2.7 :1: 12.4 8.2 :h 13.1 18.0 17.0 1.000 44 26 18 28.3 :1: 11.2 -12.2 :1: 26.8 18.4 18.1 1.000 20 12 12 -24.4 :1: 7.7 —22.5 :1: 6.5 18.7 19.0 1.000 A 1185 152 45 42 -20.5 :t 23.5 -19.7 :1: 39.2 17.8 16.6 1.000 126 38 41 -0.9 :1: 11.1 1.3 :t 8.9 17.7 16.5 1.000 A 1187 78 34 36 3.2 :1: 9.0 —2.9 :1: 7.1 19.7 18.7 1.000 56 25 21 -16.6 :1: 2.9 —6.4 :1: 7.0 19.6 18.9 1.000 36 16 16 —7.1 :1: 6.8 16.2 :1: 2.6 19.6 19.3 1.000 31 14 14 -5.5 :h 3.9 5.5 i 2.8 19.4 19.4 1.000 A 1213 139 53 63 9.0 :1: 12.4 2.9 :1: 13.5 18.6 17.1 1.000 63 24 18 —18.8 :1: 7.7 13.3 :1: 10.4 18.8 18.2 1.000 59 23 18 -19.4 :t 21.8 -23.1 :1: 10.8 18.7 18.0 1.000 A 1216 47 46 45 14.0 :1: 9.4 -5.7 :1: 13.1 19.3 18.9 1.000 47 46 48 —12.3 :1: 10.5 6.0 :1: 17.0 19.1 18.7 1.000 A 1238 82 46 52 —0.5 :L- 6.5 0.3 :t 9.3 19.7 18.8 1.000 42 23 21 15.2 :1: 5.6 —13.8 :h 8.4 19.9 19.7 1.000 35 19 16 —16.8 i 2.8 17.5 :1: 3.3 20.1 20.0 1.000 A 1254 88 33 24 —3.0 :1: 5.7 4.7 i 7.7 19.8 19.0 1.000 73 28 25 —15.8 :1: 7.7 -12.5 d: 8.3 19.6 18.8 1.000 66 25 37 15.6 :t 7.2 -10.2 d: 8.7 19.6 18.7 1.000 A 1257 57 27 35 -36.8 d: 10.3 10.6 :1: 28.8 17.9 17.2 1.000 53 25 18 23.4 :1: 16.2 30.7 :1: 14.5 18.6 17.7 1.000 54 25 20 17.1 :1: 20.1 —35.4 :t 9.9 18.2 17.5 1.000 48 23 26 -3.6 :h 9.9 0.4 :1: 11.9 17.9 17.2 1.000 A 1291 103 26 31 —1.0 :1: 7.5 —5.0 :1: 8.6 18.8 17.5 1.000 94 24 20 -8.0 :1: 7.5 9.1 :1: 6.3 18.9 18.0 1.000 A 1318 116 41 63 —0.1 :1: 12.1 -9.7 :1: 9.4 19.1 17.5 1.000 74 26 20 5.0 :1: 8.5 12.9 d: 7.1 19.0 18.1 1.000 133 TABLE 4.4. (continued) Cluster N %Ntot %Ltot a: :t 0,, y :1: 0,, mmed mjm S (arcmin) (arcmin) (0) (O) (1) (2) (3) (4) (5) (6) (7) (8) (9) A 1364 98 43 49 0.6 :1: 4.6 2.9 :1: 5.4 19.7 18.5 1.000 71 31 28 —10.8 i 3.4 -6.7 i 6.0 19.8 18.9 1.000 A 1365 49 30 37 0.8 :1: 4.5 -1.5 :h 3.5 19.3 18.6 1.000 48 29 28 -2.6 :1: 8.9 -12.2 :h 3.9 19.4 18.8 1.000 25 15 16 -15.7 :1: 3.6 5.3 :h 5.6 19.4 19.7 1.000 16 10 8 18.8 :1: 2.6 —13.7 :1: 5.7 19.7 0.0 1.000 15 9 7 -0.6 :1: 2.1 15.8 :t 2.8 19.7 0.0 1.000 A 1367 130 65 72 5.3 :h 25.6 -3.6 :1: 21.6 17.0 15.8 1.000 42 21 19 —3.0 :1: 22.7 30.6 :t 11.3 17.5 17.3 1.000 A 1377 165 41 57 —5.0 :1: 9.0 -1.3 i: 10.0 18.7 17.1 1.000 134 33 23 19.5 :1: 8.7 2.9 :t 22.4 18.9 17.7 1.000 103 26 20 —23.9 :1: 7.1 4.0 :1: 21.4 19.1 18.3 1.000 A 1382 78 43 50 0.6 :1: 5.3 -3.8 :t 5.0 20.0 19.2 1.000 56 31 29 —7.0 :1: 6.9 7.2 :1: 5.4 20.4 19.7 1.000 25 14 11 11.6 :1: 2.7 7.5 :1: 4.7 20.2 20.3 0.999 A 1399 83 29 25 0.8 :1: 4.9 2.5 :1: 4.9 19.6 18.6 1.000 79 28 20 4.8 :t 6.5 -9.6 :1: 4.8 19.9 18.9 1.000 61 21 12 —8.7 i: 5.9 13.8 :L- 3.8 20.1 19.5 1.000 A 1436 170 47 45 1.8 :t 7.4 -3.0 :1: 7.0 19.2 17.9 1.000 145 41 46 ~68 :1: 15.6 11.1 :h 10.3 19.3 17.8 1.000 A 1468 95 57 49 2.1 :h 10.3 —6.9 :1: 6.8 19.8 18.7 1.000 53 32 40 —6.9 :1: 7.8 8.6 :1: 5.6 19.7 19.0 1.000 A 1474 134 72 69 3.4 :t 9.5 4.9 :t 10.9 19.6 18.3 1.000 53 28 31 —13.3 :1: 4.6 —7.1 i 8.0 19.7 19.0 1.000 A 1496 173 49 59 4.0 :1: 7.1 -2.6 :1: 7.1 19.5 17.8 1.000 125 35 30 —12.2 :1: 4.6 2.8 :t 9.9 19.9 18.3 1.000 A 1541 98 48 76 5.9 :1: 8.3 8.3 :1: 5.3 19.7 18.4 1.000 95 46 22 -4.6 :1: 5.9 —2.8 :1: 5.9 20.0 19.0 1.000 A 1644 156 53 52 2.6 :1: 19.5 —11.9 :1: 11.4 18.7 17.3 1.000 88 30 31 -3.7 :1: 10.3 9.1 :t 7.2 18.7 17.6 1.000 A1651 142 69 69 0.6 :1: 9.0 —1.7 :1: 7.6 19.8 18.6 1.000 48 23 23 --11.8 :1: 9.4 11.9 :1: 10.5 19.9 19.5 1.000 A 1656 196 46 45 0.4 :1: 20.6 -3.6 :t 20.1 17.0 15.3 1.000 165 39 43 30.5 :I: 32.4 3.0 :1: 51.5 17.1 15.4 1.000 57 13 10 —46.9 :1: 12.1 14.0 :1: 14.1 17.1 16.4 1.000 A 1691 163 66 75 4.2 :1: 9.1 -1.8 :1: 10.5 19.2 17.7 1.000 34 14 10 —17.6 :1: 4.1 —13.8 :1: 6.3 19.6 19.4 1.000 32 13 9 13.9 :1: 5.3 16.4 i 3.1 19.5 19.4 1.000 134 TABLE 4.4. (continued) Cluster N %Nto¢ %Lm a: :l: 0, y :1: 0,, mmcd mjm S (arcmin) (arcmin) (0) (O) (1) (2) (3) (4) (5) (6) (7) (8) (9) A1749 128 58 66 8.0 :1: 10.9 0.8 :1: 10.4 19.0 17.7 1.000 63 29 26 —15.3 :1: 9.8 —15.5 :1: 9.9 19.2 18.5 1.000 A 1767 277 90 92 -—0.6 :t 10.7 -0.3 :1: 12.4 19.3 17.8 1.000 31 10 8 -19.1 :1: 3.6 -16.3 :1: 5.0 19.4 19.4 1.000 A 1773 133 47 51 0.0 i 6.7 -0.8 :1: 5.8 19.6 18.4 1.000 100 35 34 ~14.1 :1: 5.3 8.6 :1: 8.1 19.7 18.7 1.000 A 1775 181 68 68 7.7 :t 10.7 1.2 :1: 12.1 19.6 17.9 1.000 87 32 32 —12.7 :1: 6.6 —8.0 :1: 7.9 19.4 18.4 1.000 A 1793 135 54 54 0.6 :1: 5.8 2.3 :1: 6.2 19.6 18.5 1.000 47 19 27 5.7 :1: 8.0 -14.0 :1: 4.1 20.1 19.8 1.000 46 19 12 —15.5 :1: 3.2 1.7 :1: 10.2 20.2 19.9 1.000 A 1795 108 38 41 —7.8 :1: 13.3 15.2 i 8.2 19.2 18.2 1.000 89 31 34 3.3 :1: 6.2 -3.5 :1: 6.8 19.2 18.3 1.000 65 23 17 —10.8 :1: 9.0 —14.0 :t 5.8 19.7 19.0 1.000 A 1809 108 35 28 7.3 :1: 6.3 —4.1 :1: 4.3 19.7 18.6 1.000 61 20 19 1.3 :1: 3.9 3.5 :1: 4.9 19.5 18.7 1.000 A 1831 105 34 33 6.0 :1: 8.4 7.1 :1: 7.3 19.4 18.3 1.000 85 28 38 0.3 :1: 7.7 —l.4 i: 6.2 19.3 18.2 1.000 72 23 21 —10.2 :1: 6.7 -12.3 :1: 5.1 19.5 18.8 1.000 A 1837 89 33 33 -19.0 :1: 17.5 18.3 :1: 12.6 18.5 17.6 1.000 73 27 29 —5.6 :1: 10.2 —26.8 :1: 9.9 18.6 17.9 1.000 53 20 18 8.2 :t 7.7 -2.8 :1: 7.0 18.7 18.3 1.000 A 1904 161 42 52 2.8 :1: 6.5 —-3.7 :1: 8.8 19.2 17.7 1.000 149 39 27 —0.4 :1: 16.6 10.3 d: 11.1 19.7 18.4 1.000 76 20 22 —10.3 :1: 6.7 —16.3 :1: 5.2 19.3 18.3 1.000 A 1913 101 37 38 7.2 :1: 7.2 5.0 :1: 8.6 19.0 17.9 1.000 66 24 17 -5.6 :1: 8.4 —11.3 :1: 7.6 19.3 18.7 1.000 55 20 21 —13.8 :1: 7.9 12.8 :t 8.0 19.2 18.7 1.000 A 1927 148 60 56 2.8 :1: 10.4 -1.6 :1: 7.9 19.7 18.4 1.000 59 24 28 —10.8 :1: 6.5 14.9 :1: 5.4 19.8 19.1 1.000 38 16 15 10.5 :t 7.5 -16.0 :1: 3.1 19.8 19.6 1.000 A 1983 211 48 47 6.7 :1: 16.7 12.4 :1: 17.8 18.7 16.9 1.000 103 23 22 -5.3 i 11.1 -2.5 :1: 5.4 18.6 17.4 1.000 68 15 19 —24.8 :1: 8.4 —22.4 :1: 9.4 18.5 17.8 1.000 A 1991 135 37 42 —2.7 :1: 7.5 -3.9 :1: 8.9 18.9 17.8 1.000 128 35 31 -0.3 :1: 18.3 17.1 :1: 8.0 19.3 18.1 1.000 85 23 23 16.9 :1: 7.8 -16.3 :1: 7.6 19.1 18.2 1.000 20 5 5 -20.1 :1: 5.4 —24.3 i 3.1 19.2 19.8 1.000 135 TABLE 4.4. (continued) Cluster N %N¢ot %Ltot a: :t 0,; y :1: 0,, mmed mjm S (arcmin) (arcmin) (0) (O) (1) (2) (3) (4) (5) (6) (7) (8) (9) A 1999 102 55 53 —5.9 :1: 7.4 5.2 :1: 5.0 20.0 18.9 1.000 85 45 47 0.5 :1: 7.6 -7.4 :1: 5.1 19.8 18.5 1.000 A 2005 106 76 82 —0.7 :t 7.4 3.2 i 5.8 20.2 18.7 1.000 33 24 18 6.1 :1: 4.1 -8.2 :1: 3.4 20.2 20.1 1.000 A 2022 137 43 60 0.9 :l: 8.8 4.3 :1: 6.7 19.0 17.4 1.000 75 23 17 9.4 :1: 9.8 —l6.0 :1: 7.4 19.5 18.8 1.000 A 2028 76 33 32 -0.1 :1: 5.0 0.6 :1: 5.5 19.9 18.8 1.000 75 32 40 —10.6 :1: 7.6 11.7 :t 6.3 19.6 18.7 1.000 A 2029 186 43 41 —2.8 i 6.9 -2.0 :h 6.2 19.6 18.4 1.000 119 27 25 15.0 :1: 6.4 -5.7 :t 13.1 19.7 18.6 1.000 67 15 13 5.2 :1: 4.2 14.3 :1: 3.9 19.7 19.1 1.000 A 2040 108 39 42 —2.4 :1: 8.6 -0.9 :h 12.5 18.7 17.8 1.000 70 25 26 17.3 :1: 14.4 23.5 :1: 10.3 18.9 18.3 1.000 59 21 18 21.2 :1: 9.0 -19.3 :1: 9.7 19.1 18.7 1.000 A 2048 127 40 44 -0.3 :1: 5.0 -3.3 i 5.5 19.8 18.6 1.000 121 39 38 8.3 :1: 6.2 8.9 d: 5.6 20.0 19.0 1.000 A 2063 134 64 67 -0.3 :1: 21.9 -3.9 :h 14.6 18.1 17.3 1.000 51 24 18 16.3 :1: 28.4 -40.8 :1: 7.9 18.4 18.1 1.000 26 12 14 -31.3 :1: 15.3 38.4 :1: 10.4 18.1 18.4 1.000 A 2065 181 43 44 3.2 :t 4.5 —0.5 :1: 6.6 19.6 18.6 1.000 136 32 30 -0.1 :1: 16.2 15.1 :1: 7.7 19.7 18.7 1.000 A 2067 136 48 45 3.1 :1: 7.5 4.3 :1: 9.8 19.8 18.9 1.000 75 27 28 17.7 :h 4.9 —14.1 :h 4.7 19.7 19.1 1.000 42 15 19 —12.3 :1: 4.9 15.1 :1: 5.2 19.6 19.2 1.000 30 11 8 -16.3 :1: 4.5 -14.3 :1: 6.3 20.0 20.0 1.000 A 2079 142 45 49 —-15.7 :t 7.7 9.9 :1: 9.5 19.5 18.2 1.000 121 38 35 5.4 :1: 9.9 -6.2 :1: 6.5 19.6 18.7 1.000 A 2089 86 54 59 0.5 :1: 8.9 4.2 :b 8.4 19.5 18.7 1.000 37 23 22 -12.1 i 6.8 -7.3 :1: 6.2 19.6 19.4 1.000 A 2092 95 36 37 —0.3 :1: 7.6 0.1 :1: 5.2 19.2 18.3 1.000 50 19 17 -11.6 :1: 6.3 16.3 :1: 5.1 19.4 18.8 1.000 A 2107 106 39 45 3.8 d: 10.9 1.9 :h 7.6 18.4 17.5 1.000 69 25 25 2.5 :t 19.5 -28.1 :h 7.5 18.8 18.1 1.000 69 25 20 —29.2 :1: 8.5 12.8 :1: 23.6 18.7 18.0 1.000 A 2124 138 46 48 0.1 :1: 8.2 2.7 :1: 6.8 19.3 18.2 1.000 75 25 26 —11.8 :h 9.5 —18.0 :1: 6.2 19.5 18.7 1.000 A 2142 134 43 42 -0.2 :1: 7.7 -4.6 :1: 5.5 20.1 19.0 1.000 68 22 18 11.9 :1: 4.3 8.7 :1: 5.6 20.3 19.7 1.000 63 20 24 -7.2 :t 6.4 12.4 :1: 5.0 20.1 19.4 1.000 136 TABLE 4.4. (continued) Cluster N %Ntot %Lto¢ a: :1: 0, y :1: 0,, mmed mjm S (arcmin) (arcmin) (0) (O) (1) (2) (3) (4) (5) (6) (7) (8) (9) A 2147 156 34 33 ~28.3 :1: 11.0 19.0 :t 18.4 18.2 17.0 1.000 111 24 24 7.6 :1: 15.6 28.7 :1: 9.7 18.3 17.2 1.000 105 23 26 4.9 :1: 9.0 —7.3 :1: 14.1 18.3 17.0 1.000 A 2151 118 30 29 4.9 :1: 24.2 27.3 :1: 10.1 18.3 17.0 1.000 116 30 24 15.5 :1: 18.5 -27.8 :1: 10.6 18.5 17.2 1.000 82 21 30 2.6 :1: 7.9 -0.8 :t 6.9 18.2 17.1 1.000 50 13 12 -24.7 :1: 8.4 0.0 :1: 9.5 18.1 17.6 1.000 A 2152 162 34 39 31.9 d: 10.4 -19.9 :1: 14.4 18.4 16.8 1.000 112 24 21 8.9 :t 15.2 17.9 :1: 10.4 18.4 17.5 1.000 84 18 18 -7.3 i 9.6 -7.7 :1: 9.2 18.3 17.3 1.000 A 2162 51 41 44 —0.3 :1: 25.0 25.6 :1: 12.7 18.0 17.6 1.000 32 26 33 —5.2 :1: 16.4 —13.1 :t 12.5 18.5 18.4 1.000 A 2175 121 27 27 -0.7 :1: 3.1 -0.6 d: 3.3 20.1 18.3 1.000 102 23 23 -7.8 :1: 2.8 —-5.4 :1: 2.8 20.3 18.4 1.000 79 18 18 8.4 :t 4.4 9.4 :1: 4.2 20.3 18.9 1.000 A 2197 152 49 51 -14.5 :1: 25.3 -28.5 :1: 16.3 17.7 16.0 1.000 91 29 32 10.9 :1: 18.2 3.5 :1: 11.5 17.6 16.6 1.000 36 12 7 27.3 :1: 16.6 42.3 :1: 9.7 18.3 18.2 1.000 A 2199 130 33 38 -9.9 :1: 15.2 9.7 :1: 12.7 17.8 16.5 1.000 93 24 23 16.2 :1: 12.9 -16.6 :t 17.2 17.8 16.8 1.000 A 2255 315 76 78 —3.8 :1: 8.5 -2.4 :1: 9.8 19.6 17.9 1.000 50 12 14 2.6 :1: 4.3 14.6 :1: 3.3 19.4 18.9 1.000 A 2256 304 67 69 -0.4 :1: 10.7 —2.8 2h 8.0 19.0 17.2 1.000 71 16 10 —11.3 :1: 7.6 —21.8 i 3.9 19.4 18.6 1.000 A 2328 82 63 63 2.3 :1: 5.6 —2.6 :1: 4.8 20.2 19.1 1.000 30 23 12 -7.0 :1: 2.5 3.9 :1: 3.7 20.3 20.3 1.000 19 15 25 0.0 d: 2.6 8.6 :1: 2.0 20.2 0.0 1.000 A 2347 41 46 58 —3.7 :1: 7.3 6.6 :1: 5.3 20.5 20.1 1.000 29 32 27 —1.0 :1: 5.1 —8.6 :1: 3.4 20.5 20.6 1.000 15 17 11 7.8 :1: 3.8 —0.3 :1: 2.5 21.0 0.0 0.997 A 2382 95 48 51 3.1 :1: 11.3 —8.6 :1: 7.7 19.3 18.5 1.000 54 27 26 6.2 :1: 6.1 7.5 :h 7.4 19.4 18.9 1.000 16 8 9 —14.8 :1: 4.8 19.3 :t 3.6 19.2 0.0 1.000 A 2384 54 43 42 -1.4 :t 4.7 1.7 :1: 4.2 19.9 19.4 1.000 32 25 30 2.3 :1: 5.8 -10.1 :t 4.1 19.9 19.9 1.000 31 24 24 2.8 :t 9.6 13.4 :1: 4.1 20.1 20.0 1.000 A 2399 57 22 23 -5.3 :1: 6.7 —7.9 :1: 8.7 19.2 18.5 1.000 55 21 24 9.6 :1: 8.4 14.8 :1: 8.0 18.9 18.1 1.000 50 20 24 5.1 :1: 6.0 -1.2 :t 4.3 18.6 18.0 1.000 137 TABLE 4.4. (continued) Cluster N %Ntot %Ltot :1: :1: 0, y :1: 0y mmed mJ-m S (arcmin) (arcmin) (0) (O) (1) (2) (3) (4) (5) (6) (7) (8) (9) A 2410 134 57 71 1.9 :1: 11.3 0.1 :1: 7.2 19.3 17.6 1.000 60 26 17 -11.3 :1: 5.2 12.9 :1: 5.7 19.5 18.9 1.000 38 16 11 4.2 :1: 7.8 -14.4 :1: 3.5 19.7 19.5 1.000 A 2457 122 49 51 0.7 :1: 8.6 2.9 :1: 7.6 19.2 17.8 1.000 91 37 37 12.8 :1: 10.4 —4.0 :t 19.3 19.2 18.0 1.000 30 12 11 -19.5 :1: 5.2 —4.2 :1: 11.2 19.1 19.1 1.000 A 2634 119 29 33 —0.4 :1: 10.8 -0.1 :1: 16.0 17.9 16.9 1.000 90 22 23 —26.8 d: 15.6 -33.7 :t 14.4 17.9 17.0 1.000 56 14 10 44.1 :1: 8.6 18.3 :t 9.2 18.3 17.9 1.000 A 2657 78 46 47 2.5 :1: 11.6 —0.3 :h 9.7 18.4 17.8 1.000 38 22 20 —25.2 :1: 9.6 -17.6 d: 13.3 18.5 18.3 1.000 A 2666 62 36 28 35.0 :1: 15.7 4.3 :1: 36.9 18.0 17.5 1.000 44 26 42 —46.4 :1: 14.1 31.7 :1: 17.7 , 17.3 16.6 1.000 43 25 17 —16.6 :1: 23.0 -45.1 i 14.9 18.0 17.8 1.000 22 13 13 —7.7 :1: 11.1 1.7 :1: 10.0 17.7 17.9 0.999 A 2670 115 45 50 0.1 :t 4.4 3.9 :t 7.7 19.0 18.1 1.000 58 23 22 -14.7 i 5.1 9.4 :1: 7.3 19.3 18.7 1.000 46 18 18 9.5 :1: 6.0 -11.9 :1: 5.9 19.2 18.8 1.000 A 2675 78 46 47 2.5 :t 11.6 -0.3 :1: 9.7 18.4 17.8 1.000 38 22 20 —25.2 :1: 9.6 —17.6 :1: 13.3 18.5 18.3 1.000 A 2700 36 28 32 3.4 :t 3.8 —3.2 :1: 3.6 19.6 19.3 1.000 19 15 15 —6.3 :1: 2.3 3.0 :t 3.6 19.6 0.0 0.995 14 11 11 3.2 :1: 3.5 8.9 :L- 1.8 19.6 0.0 0.993 138 4.4 Background / Foreground Cluster Identification With the larger groups and unbinned magnitudes of the APS data, identification of possible background / foreground groups can be carried out by applying a Kolmogorov- Smirnov (K-S) test to the magnitude distributions for the galaxies assigned to the different groups. In theory, if a group is sufficiently far away from the main cluster, along the line of sight, its galaxies will on average be fainter than that of a closer group of galaxies. Use of the K—S test in this fashion depends on the assumption that all clusters and subclusters share a common luminosity function. If galaxies in some clusters or subclusters are intrinsically fainter than others, the results of the K—S test on the magnitudes will be misleading. In general it appears this assumption holds, although there are a number of exceptions (Schechter 1976). Recently, Jones & Mazure (1996) have used the ESO Nearby Abell Cluster Survey (ENACS) to examine this assumption in detail. They conclude that galaxy clusters do not have a universal luminosity function, with significant variations occurring at both the bright and faint ends of the distribution. However, the middle of the distribution, from the 10th- ranked galaxy to the 20th-ranked galaxy appears to be the most stable region. Thus, they recommend using the average of the 10th to 20th ranked galaxies, or: 1 20 m,,,, = 1_1-gum” (4.20) where the m,- are the sorted magnitudes, to determine a redshift-independent distance estimate to clusters. They again warn that there are exceptions and that in some clusters the galaxies are simply fainter at all magnitudes. Despite these shortcomings, without a massive redshift survey the magnitude distribution is the most reliable way to determine distances to the clusters. Furthermore, subclusters of galaxies that are intrinsically brighter or fainter would, in and of themselves, be interesting in the clues they may hold for galaxy formation. 139 Thus, with the above caution, the K-S test has been applied to each of the groups identified by either KMM or DEDICA. Any groups which could be rejected as being drawn from the same distribution at greater than the 90% level were considered to be not physically associated with the main cluster. To assign the groups as either back- ground or foreground, the median magnitude and mm in each group were examined. In general these two numbers agreed. However, as examination of Tables 4.2 and 4.3 reveals, there were a number of cases where they gave opposite results. In these cases, the ratio of the number fraction and the luminosity fractions for the groups was examined. Ratios greater than 1 were considered background, while ratios less than one were classified as foreground. The groups classified as background are plotted in Figure 3.9 with an open circle and those that are foreground with an open square. A number of clusters have other Abell clusters within an Abell radius and are so labeled in Figure 3.9. In the cases where redshifts were quoted in the literature it is possible to check the results of the K—S test. A85 has two clusters which appear nearby, A87 and A89. Only A87 has a redshift quoted at 2:0.055 which is quite close the the value of A85 at z=0.0518. A89 however, is likely to be a background group even though the K-S test fails to reject it. This may result from contamination of the group with galaxies that actually belong to A85, and contamination of A85 with galaxies that actually belong to A89. The case of A1837 and A1836 should provide a warning. Although both the DEDICA and KMM partitions reject the hypothesis that the magnitudes of the two clusters are drawn from the same population at greater than the 95% level, the KMM partition has both mmed and mJ-m greater for A1836 indicating it as a background object, while the Opposite is true for for the DEDICA partition. In actual fact A1836 with z=0.0362 is at roughly the same redshift as A1837A at z = 0.03722 but is foreground to a second component to A1837 with 2:0.0718. The only other cluster with a redshift that are unambiguously identified 140 by the substructure tests is the binary cluster A2675 and A2678 which have nearly the same redshift and are likely to be gravitationally bound. The K-S test makes no distinction between the two magnitude distributions. 4.5 Comparison of Results With background and foreground groups removed, the best estimate for the fraction of clusters in the HGT sample with significant substructure is 64:1:15%. The estimate of error is the internal error between KMM and DEDICA. This error is only a lower limit because it does not include the errors associated with the K-S determination of background groups. Here a group was considered background/ foreground if either the KMM group or a nearby DEDICA group failed to pass the K-S test criteria. According to the K-S test, 20% of the clusters in the sample are contaminated with background groups within an Abell radius. This is midway between the numerical results of van Haarlem (1996), which suggest contamination at the 30% level, and the X-ray results of Briel (1993), which found contamination at the 10% level. Therefore, the results of the K-S test are not wildly off from what is expected. 4.6 Comparison to other Studies In order to compare the present results with those of other studies the characteristics of the statistic used need to be taken into account. For instance, the study of Rhee et al. (1991) found that of their six tests for substructure the test with the highest rate of detection was the Lee test (described by Fitchett 1988), with 10% of the sample clusters having substructure. With all the tests included, 26% of the sample showed some evidence of substructure. However, there are a number of important differences between that study and the work in this thesis. First, in an attempt to keep 141 background contamination low, Rhee et al. only considered the 100 brightest galaxies in each cluster. Second, substructure needed to be significant at the 99% level to be considered “real” by Rhee et al. As seen with the Monte Carlo experiments above, with only 100 galaxies in a cluster the 99% significance level is probably too restrictive for KMM, and thus may also be for a number of the tests employed by Rhee et al. More importantly, some of the tests used were only sensitive to very specific kinds of substructure. The Lee test is only sensitive to bimodal structures; multi—modal structures tend to lower the significance of the statistic (Fitchett & Webster 1987). Other tests employed, such as the percolation test and the angular separation test may not be sensitive enough to the structures they were designed to detect. Thus, the higher percentage of substructure detected here is due the increased power of the tests employed, especially the ability of KMM and DEDICA to fit clusters with more than two subclusters, as well as the more complete sampling of the luminosity function. In fact, of the 61 clusters common to both studies, 19 were identified by Rhee et al. as containing substructure. All of these 19‘ clusters have been identified by either KMM or DEDICA as containing substructure, though four are probably due to background contamination. Other recent studies have tended to find more substructure than Rhee et al., in better agreement with the present results. The study by Salvador-Sole et al. (1993) found that 50% of the 15 Dressler clusters they looked at had substructure at the 95% significance level. If Abell 1736 is removed from consideration (since it was analyzed as two separate clusters by SSG), the KMM and DEDICA results differ from those of SSG for only three clusters. Substructure is found in A1644 with a significant four-group partition from KMM (with three of the groups having less than 20% of the total number) and a two-group partition from DEDICA. Other tests which use velocity information, such as the A-statistic and the e-statistic (Bird 1993), and X-ray 142 data (Davis 1994), confirm the existence of substructure in the cluster. Although a two-group partition is found by KMM for DC 0247—31, it is likely that this structure is not real. Finally, DEDICA finds a significant second peak in the PDF of A1656. 4.7 Conclusions There are several conclusions to be drawn. First, the ability of DEDICA to separate and test the significance of close groups of galaxies is clearly superior to that of KMM. Second, DEDICA is not as sensitive as KMM to the choice of boundary for the clusters. On the other hand KMM has more power than DEDICA to detect substructure in the presence of background contamination, as seen in the case of A2256. One potential problem with the current version of DEDICA is the selection of the smoothing parameter. The derivative of the density, as well as the density, is important in the peak identification procedure. Scott (1992) shows that accurate calculation of the derivative requires a larger smoothing window and more data points than accurate calculation of the density. Furthermore, as will be discussed in the next chapter, Merritt & Trembly (1994) find that when calculating the derivative of the kernel density, the rules for choosing the smoothing parameter do not work very well. Thus it is likely that the current implementation of the LSCV technique for finding the smoothing parameter is undersmoothing and thus obtaining a steeper gradient in the density than the true gradient. In general, this leads to a higher significance of the groups, and smaller group sizes. It is important to stress that the strength of a two-dimensional analysis lies not in the ability of the statistics to establish, once and for all, whether a given cluster does or does not contain substructure. A complete analysis needs to take advantage 143 of all available data, including redshifts and X-ray surface-brightness maps. The purpose here is to take the fullest possible advantage of the readily-available galaxy position data offered by digitized sky surveys, in order to provide a guide to clusters which might or might not harbor substructure. Once this type of analysis has been carried out the researcher can more efficiently select clusters for a large redshift survey, identify which galaxies within a cluster should have redshifts measured, predict how the X-ray map for a given cluster is likely to appear, and have a framework within which to discuss possible deviations. This same type of analysis could be done on data from numerical simulations (such as those described by Pinkney et al. 1996 or van Haarlem 1996), where problems of interpretation are similar to those encountered in the study of real clusters. These algorithms have several advantages over alternative techniques for the de- tection of substructure in projected galaxy positions. They can fit any number of sub- groups, unlike the Lee test, which is only sensitive to bimodal structures. Secondly, the KMM algorithm is very robust. Although small numbers of outlying galaxies can perturb the parameters of the fit (the estimated means and covariance matrices of the groups) since all galaxies are assigned to at least one of the groups, KMM very rarely returns such groups as significant. Because KMM fits the groups to two-dimensional Gaussian distributions, a wide variety of shapes can be fit, from spherical to rather elongated structures. DEDICA has even more flexibility in this respect since it does not need the Gaussian assumption. Finally, unlike the method of SSG, these methods are very visual. The positions, shapes, and sizes of the identified groups can be seen in the adaptive-kernel maps and compared to X-ray maps for the clusters, unlike the centroid shift methods which can only give a positive or negative result. The disadvantages of this, or any two-dimensional analysis, is potential contami- nation from foreground or background galaxies. While the Monte Carlo experiments 144 indicate that a constant-density background of less then 15% lowers the significance of substructure, foreground/ background clusters pose a more serious problem. With the larger catalogs of the APS data, very distant clusters can be detected by applying a K—S test to the magnitude distributions for each group. Ultimately however, X-ray and / or redshift data will need to be considered to confirm the results. Furthermore, merger events occurring along the line of sight will not be detected. Lastly, KMM fits the groups to two-dimensional Gaussians. Departures of the actual density profiles from Gaussian will reduce the usefulness of the partitions obtained. While this can be guarded against by employing the Hawkins test when the individual groups have a large number of galaxies, for small groups the results are likely to be misleading. These results indicate that a large fraction of the clusters in the sample of galaxy clusters exhibits evidence of substructure in their projected galaxy distributions. This substructure is very often seen in the core of the clusters, even if not identified in this study. As a result of this, and with the possibility of line of sight mergers, the 64% fraction of clusters with significant substructure is likely to be a lower limit. However, a great deal of redshift information will need to be gathered in the coming years to confirm or deny these results. Chapter 5 ESTIMATION OF THE COSMIC DENSITY PARAMETER {20 5. 1 Introduction One of the reasons for studying substructure in clusters of galaxies is to place con- straints on the curvature of the universe. In this chapter the possibility of using the fraction of clusters with presently-detectable substructure to estimate {20 is explored. First the theory is described, then the cluster catalog in this thesis is used to obtain an estimate of $10. Other possible explanations for large amounts of substructure are also discussed. The argument given below is based on the work of Gunn &. Gott (1972) and Richstone, Loeb & Turner (1992, hereafter RLT). 5.2 The Theory From General Relativity, the equation of motion in a Freidmann universe with the Robertson-Walker metric is given by: dzr 47erirf’ _ A a = ___3_r 2 + ‘3‘” (5.1) where r,- and p, are the separation of two fundamental observers (z’.e., observers that are expanding with the Universe) and the density at any given time t,-. A is the 145 146 cosmological constant. The analysis of RLT has shown that A has little effect on the results for flat cosmologies, i. e. Q+A = 1. Therefore, for convenience the cosmological constant is assumed to be zero in what follows. Integrating, this equation becomes: 87er,-r,- _, ——r , 4:2E 1' + 3 (5.2) where E is a constant of integration and has units of specific energy. Equation (5.2) holds for any spherically-symmetric, homogeneous matter distribution with no pressure. It is standard to define: 7" 87rGfi where ,5 is the mean background density. In terms of the redshift z = ro/r — 1: H(z) = Ho(1+ z)(/1+ {202, (5.4) and —1—’—”°—)] 4. (5.5) 11(2) 2 [1+ 00(1+z Equations (5.4) and (5.5) hold only for a matter-dominated universe; 2'.e. after re- combination, or 2 § 103. For 2 >> 5151 and defining the small quantity 6 as: << 1, (5.6) equation (5.4) can be approximated by: 9(2) z 1 — 6(2). (5.7) It is convenient to characterize the perturbations as the fractional overdensity 6 defined as: 6=€—L (s& p 147 Then for the perturbations, equation (5.2) can be rewritten as: = \/2E + 9:19,, + a). (59) Upon substitution of equation (5.7) this becomes: __e___-,- + 6 .-.,ng 214+ __._, (5...) where u = r/r,. At this point in the calculation RLT set the constant of integration so that 11,- : Hi. In other words the perturbation is initially expanding with the Hubble flow of the background Universe. This choice has been criticized in a study by Bartelmann et al. (1993, BES). These authors argue from the Zel’dovich approximation (Zel’dovich 1970, Buchert 1989, 1992) that in fact the existence of a density perturbation at time t,- implies the existence of a potential perturbation, the gradient of which gives rise to a velocity perturbation. They therefore conclude that the time scale for collapse in the RLT analysis is too large. BES find: a = Hail/2K1 — e,- + 6,) + (e,- — c6,)u]1/2, (5.11) where the constant c is 5/3 in the analysis of BES and is 1 if the perturbation is assumed to be expanding with the Hubble flow. Inverting equation (5.11) and integrating, the time scale for collapse is given by twice the time needed to reach maximum expansion at um”, or: 1/2 umane 'u, 0 [(1 -— 61+ 6.) + (e.- — 06,-)u]1/2’ (5-12) where the initial time t,- is small compared to the present age of the Universe and has therefore been set to 0 for simplicity. The maximum expansion is found by setting 148 (5.11) equal to 0. Thus: 1—€i+6i (:6, _ 6: , (5.13) umax = from which it can be seen that perturbations with 6 _<_ (ei/c) will never collapse. Integrating (5.12) and keeping only the leading terms in e,- and 6,: ~ 71' ~ Hi(C(Si - 6;)3/2. 7' (5.14) The time T to collapse can be written in terms of the present age of the Universe to as: 71' T(C§ — 6J3”, ,. = T/to : (5.15) where T = toH,. Further progress cannot be made until adopting some distribution for the density perturbations 6,. The choice of a Gaussian distribution is again one of convenience. Although there exists the possibility of someday testing this assumption with a higher- resolution microwave satellite, the present COBE results cannot resolve perturbations on the scale of galaxy clusters. Nevertheless, the Gaussian assumption should be able to give a good first-order estimate. With a Gaussian probability distribution: .2 1 H33.) .15., (5.16) the probability of finding a perturbation 6,- 2 6’, where 6' at any arbitrary threshold, is: P(6,- Z 5!) = gerfc [Wily—)3] . (5.17) 149 Solving equation (5.15) for 6,- as a function of t’ gives the minimum perturbation size necessary to collapse before t’: 6,-(t’) — 1 [(%)2/3 + a] . (5.18) — 2 Substituting this for 6’ in equation (5.17 ) yields the probability for density perturba- tions to collapse before t’: P(t’) = gerfc {9713(3) [(%)2/3 + a] } . (5.19) And finally, the fraction of present day clusters which have collapsed within the last time interval t’ is given by: m1) = 19(1) 101:1? " ’I) (5.20) Equation (5.20) can be evaluated numerically for any {2 and time interval t’ once a value of 0, the standard deviation of the distribution of density perturbations, is known. RLT choose 0 such that P(1) gives the correct fraction of the Universe currently in virialized clusters of mass m 1 x 1015h‘1M9. That faction is given by: , : M , 5.21 p000 ( ) where (n) is the number density of rich clusters of mass M and pcflo is the mean den- sity of the universe. With a number density of 6 x 10‘6h3 Mpc‘3 from Bahcall (1988), and a mean cluster velocity dispersion of 750 km 3‘1 giving a mass of 1015h‘1MQ and a critical density pc = 1.9 x 10‘29h2 g cm‘3, RLT find f = 0.021951. This choice is criticized by BES who argue that the number density of currently- collapsed clusters is poorly known. Indeed the number density quoted from Bahcall is calculated for Abell clusters with richness class greater than 0 and z s 0.08. From the discussion of the completeness of the Abell catalog given in Chapter 1, this could 150 be in error by 10 to 30%. Furthermore, a similar measurement given by Postman et al. (1992) finds (71) = 1.2 x 10‘5h3 Mpc‘3, or about twice that of Bahcall. Because of these uncertainties, BES argue that it is better to calculate 0 from the assumed power spectrum of the primordial density fluctuations. BES find that the argument of RLT is strengthened by their analysis. If the more conservative method of RLT is adopted, it can be seen that the use of c = 5/3 merely has the effect of changing 0 to (3/5)0 and has no effect on the probability of collapse given by equation (5.20). Thus, a lack of understanding in the collapse time scales of clusters is normalized out, at least to some extent, in the final calculation. This is not true for errors in the estimate of the current number density of rich clusters of galaxies discussed above, nor for the times scales for relaxation in clusters discussed below. Along with the errors in the normalization discussed above, the major source of error in determining (2 via the percentage of clusters with substructure is the estimate of the time for such structures to be eliminated by dynamical processes. RLT adopt a value of t/to = 0.1 or about the crossing time of a rich cluster. As discussed in Chapter 1 this is only a lower limit on the relaxation time; substructure can not be eliminated on time scales shorter than the crossing time. Numerical simulations suggest that this time is likely to be much higher, in the range of 4 to 10 crossing times, depending on the density profiles adopted for the model clusters (Nakamura et al. 1995). The shortest time scales for erasure of substructure are for those clusters with small core radii and steep density profiles. Given the results obtained from clusters acting as gravitational lenses which indicate very small core radii, a value of t/to = 0.4 is adopted here. 151 5.3 Discussion The results are shown in Figure 5.1, assuming that 64% :1: 15% of the clusters have presently-detectable substructure. The solid line indicates the 64% fraction of clusters with substructure, while the dotted lines are the estimated error. The solid curve is calculated using the Bahcall (1988) normalization while the normalization used in the dashed cureve is due to Postman et al. (1990). The results indicate 90 g 0.4 — 0.6, though (2 = 2.5 is not ruled out. However, this result needs to be viewed with a great deal of caution. First, the fraction of clusters with substructure presented here is likely to be an underestimate since line of sight mergers will be missed. Second, the relaxation times for clusters is very poorly known. Although a number of numerical simulations have been performed, many of the results can be called into question because of lack of resolution or arbitrary initial conditions. Also, most of these numerical simulations have been conducted for head-on collisions only: substructure resulting from collisions with a non-zero impact parameter is likely to last longer. Furthermore, the model of gravitational collapse of a Gaussian perturbation is very specific. Although BES have shown that the argument of RLT is affected little by the generalization to the collapse of ellipsoidal perturbations, the effects of small scale substructure on the collapse times of larger objects is not yet fully understood. In a CDM dominated universe, structures are expected to grow hierarchically, from the merging of smaller objects into larger ones. A necessary by—product of this is the creation of small scale structures. As shown by Peebles (1990) gravitational collapse of larger objects in the presence of these smaller clumps can be delayed. Given the above uncertainties in the relaxation times, it may be interested to turn the question around. If a value of {20 is assumed, these results can be used to place limits on the relaxation time scales of the projected galaxy positions in clusters. 152 6F Fig. 5.1.— Fraction of clusters with substructure vs. 90. The solid line is for the normalization of Bahcall while the dashed line is for the normalization due to Postman et al. The dotted lines show the estimated error in the fraction of clusters with substructure. 153 As measured by cluster dynamics, (20 z 0.20. To reproduce the current fraction of clusters with substructure, the relaxation time of rich clusters needs to be on the order of 5.8-7.4 crossing times (depending on the normalization used) or about 6.6 x 109h‘1 years. Chapter 6 RADIAL NUMBER-DENSITY PROFILES 6.1 Introduction Estimation of the radial density profiles of clusters is important for several reasons. First, numerical simulations which seek to explore dynamical evolution and mergers of clusters generally assume a mass distribution function for the model clusters. Typ- ical forms adopted are ones which involve the density approaching a constant in the core of the cluster. Some models which have been chosen in the past include the mod- ified Hubble law, the Michie-King models, and the non-singular isothermal sphere. A number of the results, such as the survivability of substructure, depend critically on the adopted size of the core radius. In general, smaller core radii lead to shorter relaxation times because groups passing near these cores will be more efficiently dis- rupted by tidal forces. Furthermore, several clusters are now known to contain arcs of background galaxies. These arcs are formed by the foreground cluster acting as a gravitational lens. The size and curvature of these arcs depend sensitively on the form of the potential well of the lensing object. Since the light reacts to all gravitating material, whether it is due to dark or luminous matter, for the first time it is possible to compare the distribution of dark matter to the distribution of galaxies in clusters. 154 155 Furthermore, clusters which have a density cusp at their centers or very small core radii, are much more likely to act as lensing sources. Because the background, lensed objects, are often times very faint they require long exposures with large telescopes to detect. A great deal of observing time could be saved if a way existed to identify, from a large sample of clusters, which ones were likely to be detectable lensing ob- jects. Lastly, formation theories of cD galaxies depend sensitively on the form of the potential well near the cores of clusters. The existence of a density cusp in the core of a cluster makes merger events between galaxies much less likely and mass accretion by tidal stripping more important. In all of these applications it is crucial to know, not the projected profile which is measured, but the true space-density profile. Because only projected positions are available, it is necessary to make assumptions about the three-dimensional geometry. Statisticians refer to this type of problem as “ill-conditioned.” Typically, spherical symmetry is assumed and the de—projection is carried out using Abel’s equation: 1 00d}: dB 00‘) = -; r WW, (6.1) where V is the space density, r is the three dimensional radius and 2 is the projected distribution with R the projected radius. Note that this depends on the derivative of E and not directly on E. It is customary to use a parametric approach when measuring the projected den- sity profiles of clusters. The form is chosen (usually one of the above mentioned forms) and various parameters measured. However, there is a great deal of danger in such a procedure. Even fits which are statistically “good” in a x2 sense can have significant deviations from the parametric model. The statistical literature is full of examples where use of a parametric model masks information in the data which is inconsistent with the model (for example see Gasset et al. 1984). In this case the 156 problem is compounded because the Abel inversion requires the derivative be calcu- lated and small errors in E can become large errors in u (Anderssen & Jakeman 1975, Wahba 1990). These problems were addressed by Merritt & ’Ifemblay (1994). They recommend using a completely nonparametric approach based on a maximum likelihood density estimate. Use of this method allows the smoothing to be carried out directly on the estimate of 11 without the necessity of calculating 2 first. What follows is a brief description of the use of maximum likelihood in density estimation and details of the approach used by Merritt & Tremblay. 6.2 Maximum Penalized Likelihood Estimator Given the success of the maximum likelihood approach used to estimate the PDF of the Gaussian decomposition of clusters in Chapter 4, the question arises: can the maximum-likelihood technique be employed to obtain a nonparametric density estimate? The answer is yes, with a modification. Given a set of n independent observations, X,- . . . X”, the likelihood that a curve g represents the underlying density is: £0) = [1900)- . (62) Unfortunately, this likelihood has no finite maximum. A little thought reveals, for instance, that the likelihood will approach infinity as h —> 0 in any of the kernel density estimates. It is easy to see with a box-shaped kernel: - 1 157 and Hf(X.-) 2 (5,3)” (6.4) Therefore, the maximum-likelihood density estimate is a sum of Dirac delta functions placed at the positions of the observations. While this preserves all of the information in the data, it is not a useful probability density function. The solution, first applied to density estimation by Good and Gaskins (1971), is to penalize any density estimate which is not smooth to obtain the Maximum Penalized Likelihood (MPL) estimate. Thus, the maximum is sought for the function: log L10) = flognx.) — APU). (65) i=1 where P( f ) (the penalty function) is some function which quantifies the roughness of the curve f and A is a smoothing parameter. Typically P( f ) is chosen to depend on the squared derivatives of f. Good & Gaskins (1971) suggested using the penalty function: P( f) = [m “if 7}] 2 dz, (6.6) —oo which will penalize density estimates with large curvature. It is also advantageous to have P( f ) depend on the logarithm of f so that the density estimate is forced to be positive. Furthermore, the fluctuations in the estimate are penalized via their relative size and not their absolute size. Like the choice of kernel function in the kernel density estimate discussed in Chap— ter 3, any reasonable choice of the penalty function can be used to provide good es- timates of the density as long as the proper smoothing parameter is used. However, as the smoothing parameter A is increased, the shape of the density estimate will 158 tend toward the shape of the penalty function, yielding in effect a parametric density estimate. For example, the penalty function defined by: P(f) = j: [(%)3logf,]2dx (6.7) is zero if and only if f is a normal distribution. Thus as A approaches infinity, the MPL estimate will be a normal distribution with the mean and variance of the data. The behavior of the limiting estimate for large smoothing parameter differs from that of the kernel-based density estimators which approach a constant value as h is increased. In the construction of the maps of Chapter 3 or the identification of peaks in the DEDICA algorithm of Chapter 4 this eflect made little difference. It becomes important in constructing radial density profiles when the primary interest is in the behavior near the core. The kernel-based estimator will in general return an estimate which approaches a constant density near the core unless the smoothing parameter is made very small, which leads to a rather noisy estimate. This problem can be overcome by using MPL with a penalty function of the form employed by MT: P(V) = / [dzlogV/dlogr2}2d(logr), (6.8) which is zero when u = ar‘b. Thus an oversmoothed estimate will be the best-fit power law approximation. Note also that it is the space density 11 that is being penalized and therefore estimated. By calculating the maximum likelihood estimate for the space density directly from the observations, the effect of compounding the errors when calculating the derivative of the estimate of E can be avoided. An estimate of E, can then be obtained by integrating the estimate of V, which is a well-conditioned problem. Note that this does not change the nature of the problem 159 in the sense that it is still ill-conditioned. It is merely that the smoothing is now being performed directly on the derivative of 2. As with the kernel estimators, the quality of the density estimate will depend mostly on the choice of smoothing parameter. In this case there are really two den- sity estimates which are sought: the estimate of 2 and the estimate of V, or in actual fact the derivative of 23. Although they are related the smoothing parameter that provides the best estimate for the one will not necessarily provide the best estimate for the other. Scott (1992) shows that the derivative of the density requires a larger smoothing parameter and larger number of points to achieve the same MISE as com- pared to the density estimate. Furthermore, the Monte Carlo simulations performed by MT show that none of the prescription for choosing the smoothing parameter work well in this application. They recommend constructing a number of profiles and only accepting as real those features that appear over a range of smoothing parameters. 160 Fig. 6.1.— Nonparametric number-density profiles - HGT clusters IIII I I IIIIIIII r IIII I I IIIIII "IIII I I I" I «FR .05.], 4|". old" 1- 'r ,g’d" .."o b . -1 P . d I'. -4 If. i _, a s. e . w —r .5 ’ _ h ‘s I.’ 3 . f . 1- If I,‘ _ h- J." 1'~ ‘- I- k ’: \ I. ‘ ,g,?" \ -—I~° " rt". ‘ -( F "_.~ ‘. ‘ I, \ ‘1 . d I": .2“ -( lllllIJ"l 11111111 1 1 ILl‘llill 1 11111111 1 ‘1 I T ‘I '1‘ 1 O O O O O O o- v- v- v- v- '- II II I I I I I III I I I I I II II, I I I l H r t , i n -4 O P _, P .( -1 .1 fl «0 3- " -1 F _ = " q o r- .0 _ < I3 " '- l.' f 5' : r- 9, I! h g. ‘ I. Q ': 1| — O '- I.‘ ;, .1 v- '_: .' ' -1 If" _‘ I .1 I_' . | I n 2 | -I 111111 1 1 ’-l'11111 1M l1111111 1 '1 ‘1 I I O O O O o- v- v- P I I a 2" — O F A ' ~ a-i _ '1 b d __ .. P N ’ ‘ = " m r 2 L 4 fl —( O Pr .1 F P d .. ,‘f‘ ‘. \ I_- ‘. \ " 1111 1 1 1 111-1'11 11111111 1 ‘~l111111 I 1 11111111 7 T T I I T O O O O O o- P v- v- u- o- I III I I I I I q %_ —4 O -( P D -1 ,- -1 - d _ 4 F N " 7 - 3 ’ ‘ 2 . 4 n ._.__ 1‘ 9 P ‘ r . r - ,_ t.1‘ 3: _. Jill I 1 1 '5111 l l' 1 1111111 1 i 1 1' '1 7 O O O O F v- v- I— r (kpc) R (m) ' (W) R (W) 161 Abell 88 Abell 88 Abell 85 Abell 85 10’ : s; 1 4’ _.4 ._ 'r‘ _ I j __ V. 1. . I ‘. ”31“~ _, 1 [hill] 1 11111114 1 ml 1 1 1111111 I 1 111111 '1' 7 I 7 T O O O O O '- c- o- 0- II- A 10 ITTTIII IIIIIII I IIIIIII I I 43.44:» I 10’ fl 2' d :I -. h v .1 d -'I -'I .. :1 ,7 .. I'I ’ 4 1:" ~ 1—- .' >- g! —‘ O '- 1‘. d F )— '- j :- ‘-. C 111111 '1 l1111111 1 7 '1' 7 I O O O O '- o- '- .- _IIIII I I IIIIIII I I IIIIIII I I :flIIIITI I I [IIIIII I I J TIIrI I I I'll] I I I I ['IIIIII I T I .. / .i .7 ,v 1 - I a .’ -—1 O I - F I .1 I .. .1 O,_ _-, .. P_ ' "1 I- .l .'I - .‘I “ ‘.‘ I 1- l' ’ W 3 -1 I: '1‘ P '1 i' ‘2 -1 “. 51 I: i. 1- |: :I I |-. ;. —1 O I; I. . v- “. -I '1 '5 -I —( ': :1 r : ‘ '— 1111Ll l IllLdl 1‘11 1111111 1 l T 7 1 1 O O O O o- v- v- v- r (km) R (kpc) ' (km) R 0‘96) Figure 6.1. Nonparametric density profiles Abell 119 Abell 1 19 Abell 1 O4 Abell 104 IIIII I f IIIIIII I I lIIIIIl I I lllllII I I IIIIII 'f 4" h n _ — O .1 F q 0 _1— d F .1 A » § '- " v F L >- 1 L. N 1- " O I- _‘ ,— 1.’ a ’1. I" _ " I.“ ._ \\ q ‘. \ 1111111 1 111l111 1 1.111111 1 1 ‘1111111 1 1 11111111 1 1' 7 1 1 I 7 O O O O O u- I- v- v- v- '- 11 III I I [IID ; I I I b- OI I- T“ 2 .. .1 I1 0 .2: ‘ r- _‘ rs ~ 1 1- " v > K _- n h a- . — o .1 P .1 . ‘1 C .' 1 .4 111111 L1 1111111 1 1 1111111 1 1 O O O O c- v- v- '- IIIIII I I IIIIIII I I [IIIIIII I IIIIIII ITI IIII'II'I I I. I .‘ I I 3’; n O '- #1 #1111 ‘ ‘ —1° qF d _ 4 ._ \‘ .1 11111111 1. 11111111 1 11111111 1 7 7 I I I 7 O O O O O O P P v- v- '- K IIIIII I I [TrrrII _"o ..v- 2: 1 r d t 1 i -I b '1 . ~ I —° v-I—- ( | -.- r I " F 15 I ‘ >- 1; I _ 1111 1 111‘11111 11'. 11111111 1 T 1' 7 I O O O O o- v- P v- 1 r (kpc) R (be) 162 Abell 147 Abell 147 Abell 121 Abell 121 IIIIII I IIIIIIII I IIIIII I I [IIIIIIrI 7le c.1- 11111 1 1 *— I- .4 ,- In _ _ o .4 P —1 I- 3 1 I: -_ I 11.1 1 1 111 1111 1 1111111 1 1 " 11.11111 1 1 11111111 L '1 n I 1 ‘1 T 1 T O O O O O O P v- v- v- v- '- A III I I I I III I I I I I \ I I I I I IT I ‘1 F i ‘1 o n h d o - F q d t ‘ 1- .1 p- b '1 b— _ -« P I 1,0 or”, .' ~ I; O I- II‘; __“ F I,-' .1 1" 4 a l_' -4 1111111‘1 111111 1 1111111 1 1 n '1‘ 1 1 1' O O O O '- P v- '- IIIIII I I [IIIIII I I IIIIIII I I IIIIII I I I ‘_ II-KII I .\ '.\ Jr .1 .‘I .‘g \ _‘O .. I .. v- 1- 1 -. _ .1 _‘ _ \ '- 1_ \ ‘ ..\ h 'I..‘ .1 . \ P : \ . u. ‘ u. I I- .. r! _‘ I I ’f " a p’ " ,._ J _ o b ’1. q F I- <3 * ‘a. j I- ‘k. "e. '1 11111 1 1 1111111131 11111111 1 n 7 '1' 1 I I I O O O O O '- '- v- I- v- v- I IIIII I I I IIIIIII I I IIIII‘ 1‘ I I :1 ,v 1 f I H 5" —1 o 1- 51 < " 1- , I 4 . 1 at ‘ 1 , -4 >- ' . I ’- d b 4 L. .. cub—- ,’ H 1- ..” 1 2 1- ,'.' d r- 1:. >- V' .1 ‘1 -I "J l 1111 1 1 1 1111 1111 1 1 1 1 7 ‘1' 7 I O O O O P v- I- v- r (“I”) R (m) f (W) R (we) Figure 6.1. (cont’d) Abell 154 Abell 154 Abell 151 Abell 151 11111 1 111111 1 1111111 1 111111! 1111111 F l 1' , :l‘ 1- -" I a ---0 q.- .. _- fl .. ,_ .. '- A " § .- 1 v L " -1 h ~ —40 “F . ..L— . '- ‘ \ -4 1111111 1: 111 111 1 ‘J1111111 1 11111111 1 11111111 1 1’ ‘1' 1 Y T '1 O O O O O O o- v- v- F v- o- A 11TT111 1 11111111 r 11111111 {1 (v r q —-0 _" -1 9... -+ " -< .. ,_ .1 1. q A 1 § 4 v r C "' -1 .. u I, o l.’ .1:- ,.. 1.- j v-— c' " ’3 .' .1 1111111 1 111111111 1‘- 11111111 1 T '1' 7 1 O O O 0 v- P d. '- a .10 _‘v- .1 .4 .4 d .1 A d g 5 .1 I —40 q“ .1 '1 wh— t ‘\ -‘ 1111LL 11511111 11111111 L‘ 11111111 1 [1111111 1 'f 7 T 1 '1‘ O O O O O O P '- v- '- P v- 4 JIrTII 1 1 W n —O ...- . . d .1 d 3 " v C .1 - —O qP .. d v- ] — 11111 1 1 1 1' 7 Y 1 O O O 0 v- c- v- v- 163 Abell 168 Abell 168 Abell 1 66 Abell 166 11111 1 11111111 1 11111111 1 11111111 T 1:71 1:1 b ' J. H h .4 O .‘ P . q '1 O_ . FI- _ d h 1- -1 p d .. 'o *' ,e- _. v- 11" ~ 11 v. 1. \ .1 ‘- \ -4 111 1 1 11111511 1 11111 1 1 111111-1‘1 1 11111111 1 an '1' 7 1 T ‘1 . O O O O o- v- v- v- v- v- A 111 1 1 T p- a 1 ~ ° .. v- .1 1 .1 b t 4 p- d r- 1- -1 L 1. —'b 1 F .. .’ .1 .' | q 1' l " 1 111111 ‘ 1 111.1L41 1 1111111 1 7 ? 7 7 O O O O '- v- v- v- . I- 111111 1 1 11111111 1 [1111111 1 [1111111 If: 115111 1 Q OF- H F- a—A O 1- .. r- b- d L- -1 d p d 1- 4 h .1 b . " f I. I? |'. F— I: C b 1.‘ — O P I. -1 '- 1- ,’;' - t .' .- £ I’- \ .' . \ . 1111L14L '1H111L1 1111111 1 1 1111111‘11‘ I [111111 1 1 T '1' 1 '1' T O O O O O '- v- '- v- I- '- fl 1111 1 1 [111111 1 I TT11N 1 1 r J J 1| I 2’— -' 0 " —1 O .- 4 '- .- J ,_ J b ‘ d y. d in FF— . _ — O 1- t .1 '- r- 1; q 1- : " a. b "I .1 llllll l l ".111 ll 1 1 1111111 1 l v I u 1 7 T O O O O '- I- '- f ('19:) R (kpc) ' (kpc) R (We) Figure 6.1. (cont’d) Abefl193 Abefl193 Abefl189 Abefl189 111 1 T 111111 1 111 1 1 11111 1 1 11‘ 1' T 1‘ 1‘" 1' .1 1' $ .\ '.\ 'J 9 so ."I .1 '- ’ fl .1 o d .— v-_ .1 h d p - i .- f /" - ‘. ,1. r '2 ‘2 I1 N ,. ,_. _ o I: -1 '- \; -< ‘[ .‘\ "‘ I: '2‘ -1 1 111 " “ l Wit-H1111 1 111111’1 1 1111111 1 lllLfi .1 111111 1 1 '1' 7 1 T T 'o‘ O O O O O c- u- v- v- v- 4 1111111 1 [1111111 1 [111111 1 >- .1 9 .’ ! a1 _. I .1 o .-, 1 " ,-. '1 .1 o -1 _r 1 " -1 .. _ d .- h ‘ b 3- .1 -1 '- III 1 1 111111 I l l O O u- v- 111 1 1 ["1111 I I [111111 1 I [111111' 1'; [111111 1 1 I, " .- 1- n O 10 I1Il[ I l T 11_1_111 1 ~ — O 1 -‘ " I.’ -1 1- ': .1 1'. \‘1, “ 111111 1 1'1 1111111 7 1 T 7 '3 T O O O O O O O- P '- '- v- o- ‘ .JITIIrT 1 [1111111 1 yr I .‘1 r- ..., v- H —4 O .1 v- .1 .. d ‘ d p. - . . .. b 'q‘a '."‘ o" '1 .. r ' ~ '- r I: 1 -- O I; .. v- I' ' _‘ I.‘ I ’ 1.’ 1 - 13' 1 _4 111111 1 1" 1111 11 1111111_1 1 a- ? Y 7 O O O O o- P v- r- ' (kpc) R (Rae) ' (kpc) R (be) 164 Abefl225 Abefl225 Abefl194 Abefl194 1111111 1 [111111 1 1 [1111111 r [111111] I 19 fiT 1W1] .1:[11111 1 1 I '1 1f 1 q d p- I —O ...- PI— _‘ " . .'.\ 1 - I; ‘ 1111191 11111111 1 11111111 1 11111111 1 '1' ‘1' T ‘l’ T '1‘ O O O O O O '- v- o- v- I- v- A 11111 [Hm—r1 1 CI —0 -P o d H: . p 1 " -< .. .1 1. » J .. C —O - t ~ 2 . F- I- .— Q ~ _ o _.'o -4 '- ‘ v- . hf .4 Fr— _‘ 5'. -I " . q I; ., ‘ .1 - "m. f q ‘ '._ l- ._ 11111 1 1" 11111 1 1 --111t111 1 1 11111111 1 1111111 1 l 11111L1"1 111111‘L‘11_ 1111111 1 1 1111111 1 1 11111111 1 I I T I 'I‘ I 'I‘ 7 I I I 'f O O O O O O o O O o o 0 v- c- I- I- v- v- v- v- r- v- r- l a JIIIII I I II—IIIII I I IIIIIII IIII I I P h- ‘I n '- —I O -" o .. P . .- 1— ‘ S— .1 -1 h _ I- .. _ _1 P " I!) r- '1 g]... ‘ v = f ‘ K o h- h P _‘ .0 ‘ .. < h P b a " —I% —1 O " '- u o- l- -1 F— -I -I '- .1 .- « . '-. I 1111 1 1 11111141 11 11 1 1’ 1111111 1 u ' I I I 7 1 O O O O O O 0' P v- v- v- .- IIII I I IIIIII I I I [IIIII II I [IIII I III ? IIIIIII I I III I I I [IIIII I I I [IIIII I I I IIIIIIrI’f wJIlIIIII I I I r- f ."l‘ l- n n —1 o —4 o _‘ I- .. u- S_ ‘1 gr— CI p _‘ 1- P .. n 4 F .. d .4 A o h- -4 b g V I- - v = F .. 1 w 1 +- p_' n I,‘ a .1 ,'. < 1.‘ F ", r I‘. "- a on ":- — O —1 O r. d P ' d F pp— 1:". " v-I—- " .1 l- -I I- .1 " r'.' . \ ‘ . I- r' ', \ -I h i ‘. ‘ d 11111 1 1 '1111111 14 111111 1 1 [111111'1 1 111.111 1 1 11111 1".1 11 11 11 1 111111 1 1 1 111.111) 1 J 1111111 1 1 '1’ 7 I 1' I '1‘ '1 7 I I ‘1 'i O O O O O O O O O O O O '- O- '- v- '- v- v- v- I- v- '- 1 A IIIIII I I IIIIII I I I WIII I I [IIIII I I I IIIIII I I I b h a II A O --t O -4 F q '— -1 a,_ -l 91;- 1 ".. . F' -4 '- q >- 1- " '- ,. A h -I A 3 _ -I ’ 1 " " v = - 4 L a g . _ . I n -l " .I '- w--— l : fi— 1 1- ‘ " ‘ h ‘1 -1 II 11111 114‘1 1111111 1 "111111 1 1 1111511 ‘I' 7 I T 7 1' I O O O O O O O '- v- v- v- v- v- r- R (m) Abell 500 Abell 500 Abell 496 Abell 496 IIIIIII I IIIIIII1 I IIIIIIFI r IIIIIII III -‘IIIIIII I f .' L 1" ‘l N --10 JP 4 .4 - d .. i- -I p L I: '1 1— 1.. ‘ b ~ —-0 q.- Ph- - 1- -, \ - fl _, . 1urf 11111111 1 11111111 1 1111111 1 1 11111111 L k I 7 I I I 1 O O O O O O o- v- o- v- v- '- ‘ IIIIII I T IIIITIII I IIITIII I “71 , 1 .I ,. H -—O _‘F . o“ q ,— _ - .. y- «I b .1 .. d 1— 1 I T.- v—-— 4 : .- .' " 111111 1 1 111-1'!" 1 1 1111111 1 1 I '1' 7 I O O O O '- v- v- '- IIIITI I IIIIIIII I IIIIIIII I IIIIIII'I I." IIIIIIII I n I .1 I II b a ,. _o “F 1- 1-4 .1 4 -‘ .. F d P d b h C . --1O ’.' 4’ I.‘ '4 ' (y ‘. \ '1 \'-_ ‘. \ _‘ 111111 1 "41111111 11111111 1 111111‘1L1 11111111 1 I I I I '1‘ I O O O O O O O- I- o- v- v- u- R ITTI I I H h _‘o 11' . 1- d .1 .1 b ‘1 .. b P d b 1- . ' —40 ‘P .1 b 4 _l 11111 1 L 1 1 a. T O O o- v- ? (kpc) R (kpc) r (we) R (W) 167 Abell 634 Abell 634 Abell 514 Abell 514 IIIWT TIIIIII I I IIIIIII I I IIII'V I .' I n -' ‘ u. l’ 1' —‘ .- I I I \‘ _1 o 1- .‘ ‘ .1 v- : \ .1 .‘ 1 _1 .' I F o I d I I' 1 .1 eI— ~ .. .. h L- .1 of f r f 4- O I— ; . I? d O 1- 1'. -1 '- 1‘ t1 -. s I- ~~b~ -'.' ‘\ d 1111111 1 1111111Y~1 11 1 111111P1 1 11111111 1 a '1' I I I '1‘ I O O O O O O '- v- v- v- v- d IIIII I IIIIIIII I IIIIIII 3‘ r, I! ~ 1' - ll —1 O I. _1 '- d d I- '4 d ‘1 9L 1 b r F d .- .. r. _'b .. . .1 v- ! .1 ‘ - ‘ ~ P- ‘ '. ‘ 11111 1 1 1 ‘ 111 11L111 1111111 1 1 n 1 7 I I O O O O u- v- u- '- IIIIII I I IIIIIIII T IIIIIIII I I": TITIIII I .1 .- -'I .'I 17 J "o ‘. d v- o" ,’.' -I P" ". .I " I- 5 ‘Q -I .- ‘_ ‘a I- '.\ an 1. 11' .'I I. .'I . " .v .‘a .‘l " '3, .1 o i- .1 1. n — O " .1 v- C" - - e "L ,1“ : \ .. - f’. :0 \\ -‘ 1111111 1l111111.1 111111J 1 1 1111111 1 1 11111111 1 7 1' I I I I O O O O O o- F P o- P 4 IIIIII I I IIIIIII I I IIIIIII I N N — o .1 p 0 q pt: -1 "' '4 _ ‘1 - d I- .1 b 4 p n -—1 O .1 ' F 4 1- -< P- -I 111 1 1 1 1 ‘1 I O O '- v- r (kpc) R (kpc) ' (be) R (We) Figure 6.1. (cont’d) Abell 779 Abell 779 Abell 671 Abell 671 IIII I IIIIIII I IIIIIII I7 III III I IIIIIT l I 1; I, I" - K ‘1 1 I ' II —1 O .1 v- .4 ‘1 q q .11 A 1 g 5 .1 db _1 F .1 . J ‘ '-_.. ‘\ '1 1311.1 1 1 11 1111 1 11111111 1 1111111 1 1 11111111v 1‘ I I I '1‘ I I O O O O O O c- v- u- v- v- 4 IITIII I If, IIIIIII I I 1' :1 1- :1 ~ 3 ' n 1- .; —l O .' .1 v- 4 V .1 . r -< _‘ A ,. 1 1 v t f. - C |'_ 1 ‘ : ‘ . I: 1 '- I' ." I" -I " 1_1. :1 4?" I. on b ’7' 1' —1 0 1’3" 3' j.- _ I-’ :l '5 :' J '2 1' -I * 1'1 1 111111 1111111 1111L11; I I I I O O O O o- v- v- P "II—II I IIIIIIII I [IIIIIII I IIIIIIII I III IN] I 1- I." ,1 .1 _ II _1 O —1 C. d I! 21— ‘ _ '4 ' .1 A .. - § " V b s " -< h fl .. d ‘v .1 J 11"..- ."l '1 111.1"1 1111113111 1111111L1 1111111 1 1 11111111 1 '1' 7 I I I '1‘ O O O O O O P P '- F I- c- l I IIIIIIII I [IIIIIII I IIIIII P I. p —1 O _ I- .. q _I P. d 1" _‘ A I 1 i " v C .. p _ I —1 O 4 F .1 .1 '53”. '1" ,' -< ’- 1 11111'11 1' ’ 1111111 1 1 1111_1_L ‘1' I O O '- r- ‘1' o .. I 168 Abell 957 Abell 957 Abell 787 Abell 787 IIIII I I IIIIIII I I IIIIIII I I IIIIIII I I IIIIII‘ I“ I 5 " .‘I 5" ., q I- _ o d P .1 .1 G._ -4 fl- ‘ r b ’- '1 p b q r ..r .1 I: “ '- v. -4 \‘1 v. \‘ -1 v, -. \ -1 11111 1 1 1 11111 11 1 LL‘L 11111 1 l 111L1l1 1 1 111111 I l 1 v: "1' I 1 I 7 O O O O O 0 v- .- P P o- v- I II I I T T I I I I I I Q —1 O 1- ‘ r- q q .1 e— 4 _ H h d 1. '1 b _ II I- 1.-' -—1 9 I; '1 I; '1 r. -4 I I'- l 1 1 1L1 1 4; L111 l 1 1 1 111 1 1 1 1 '1' 7 T I O O O O '- v- I- '- ‘ 111111 L 11111111 1111111 1 1 111111 ‘ 11111111 1 I '1' I I I I O O O O O O P v- v- I- v- v- I IIIIII I T I q.— ~- I. F —‘o I- .10- - d 1- .. . b d F 1 p- .1 ._ -'_ .. -_ u I- V. _o I- ‘ 4" 1- .1 -‘ 1. 1 : 111111 1 1111111 11111111 1 I '1' I I O O O 0 v- '- v- v- ? (M) R ('96) ' (R96) R (be) Figure 6.1. (cont’d) IIIII I I I IIIIIII r I [IIIII I I K, {VIIII T I [IIIII I I I \ "l, a -—O _F .1 ('3 " a « ~ — _ § — v g b .1 a —10 _‘F 1 P'P" '3' I 4 111.111 1 111111 1 1 11111111 1, 11111111 1 11111111 1 7 I I T '1‘ 7 O O O O O O '- v- u- v- v- v- I r r IIIIII I I FIIIII I I {31III1'TI T 1 ’1 " 1 1 n I ‘ 4" ‘1 :l d 4 5' 1 o_ .v “ Fu- .1 m p ‘1 Q h .1 A m . § = 1.. '1 v 0 _ z 2 ~ p- 1' I: —O .1.- .1 d 1— -1 111 1 1 1 11111 1 1 1 l O O O o- v- '- I II ' _IIIIj I TIIIIrr‘I [ITITIII I lIIIIrIf’, IFII I n -—O u' .4 d ‘ Q ‘ a « 1 .. . 1 - V L d an ~40 qF '1' 7 I T I ’1‘ O O O O O 0 v- v- v- v- P v- d IIIIII I I I p " a _O .1— 2— ‘ Q 2 ‘ rs - — A 08 _ g = _ v _ C .1 1. " I —O ...- W» 4'. ".l 111 15:1 111111 1 1 1 111111 1 1 1 '1' 7 Y O O O O '- v- o- o- 169 IIIIII I 1 IIIITIII I [1111111 I III‘IIIT‘ 1.1 [111111 I I ' ’J D . _"o o_ -* " F4 -4 1. '4 a 1 1 O E ~ F - II _ a) . 2 « _"o q F FL— _‘ >- ’: ,_‘ " 1' ’1 ‘.\ -1 111111 {.1 111111 1 1 1111111 1 1 11111111 1 111111 1 1 1 T 7 V T Y '1‘ O O O O O 0 v- v- v- v- v- v- a nrrTT—FV 111111 T r III 1131 _ I |"' 1 a _1 O o_ ‘ " '— -4 : C1 to ' ‘ m ” ‘ O ' « P = F “ 3 * < * h _'o -7 ;" ’- d r- . 1',’ -1 111111 1 1 1111111 1 1-’ 1111111 1 1 7 1' '1' Y o O O 0 v- v- u- v- 1 1111111 1 nun I I 111111 I I IIIIII 1 .IIIII 1 I I l l ,1 _J‘ I ll -— O .1 '- d 521- 1 o - ‘ N F .1 O - « v- .- = " .. 0 :2 - 1 L. I —1 O .4 v- f '1— ! . 1 '- ‘ ‘. \ -1 "1111111 1'1 1111 11 1 11111111 1 ‘ 1111111 1 1 11111111 1 '1' ‘3 7 7 T '1‘ O O O O 0 v- r- P v- I- '- II III I I I r '11] II I I r .. II —1 O .. o- -4 2— 1 1' -1 O . h d 8 1 . v- 1. = L ‘ G 2 ~ « r “O r; 1 .— 4- .1 It. ' F— r_-' l -1 " ’.' 3' .1 111111 1 1111111 1 1 1111111 1 1 7 7 1 ‘f O O O O '- u- o- v- ? (W) R (we) r (W) R (be) Figure 6.1. (cont’d) "III I I IIIIII I I IIIrI I I IIIIII I "III I. I l 1' 1' 1 I g,’ ~ '1’"— ’r .I‘J’ -I h F ~ I .- ._ _ O >- -. v- - 2 G _ :2 «I? l" .. IE I -I " '- p- '0-0 E b a = P 'e’,,¢-“' ..‘ | .1 g o " '57-" | v I- I - I 2 ~ ’- \.. I. 1 P ‘h‘ ’ . \ ¢\ .4 I- ‘~ 3‘ " s~ -\ q ‘ _ ‘s 1’ ..I v- 1‘ or :I -I ‘h- fl I- ',' J ’ I r- ‘ 1 ‘ . I'11111 1 1 11111111 I 1111111 1 I 1111111‘1-1 111111 1'31 1 .. '1' 7 I T T 7 O O O O O O c- v- v- I- u- v- A IIrI I I I I IIIIT I I :1 I I I _ F: .l d .. .. F d .. O .. 1 .- u‘ q o: " ' ‘ . . I d (O '3‘ 5 I .. A v- —_ I .- . 3 4.. I ' I ‘1 E '- : I E . o = b "9,?" " I -I a 0 " $7“ 1 ' v .v' : I "' I. l.‘ : I ~— v- x I- ‘ .‘ ; I .. \ 1 ; I : P- ‘ i ' -I l '- : I q I ~ ; I v-I ' '. I ' .- o>- I : 2 I - ‘: I 1 1 I .- I ~ ; I .4 I- I Z ' 1' I ; I " ' l ' ' l 11 ll 1 l l 1 ll 1 1 l 1 1 ‘ 1 11 l l l 1 1_1 '— 7 7 I I O O O O o- v- v- '- IIIIII I I "III—I I r IIIIIII I ITII .’ IIIIII . I I I" , I I III J J I n — O O,_ 1 '- .— .. P -I " -I 8 I I P r .1 A P I. .1 § = V § 1- L .. II. I.‘ I.‘ u I —I O v-l— I '1 " *' r . -« " It; “ -I “1111’1' I 11111 1 1 1 11111111 1‘ 1111111 1 1 1111111 1 1 '1' 7 T 1’ T '1‘ O O O O O O '- v- c- P v- 4 rl I I I I I I I III I I I I I I II I I I I I I I I I i n .4 0 °__ -4 V. r-_ -I l- d " -4 CD .- N I b P -I A F I g = L- v a: .1 h “ -—4 O "F‘ a F >- a I- d 1- .I I 1 l 1 1 - n I I I I O O O O I- I- v- v- 170 I IIIIII I I IIIIII I IIIIIII I IIIIIII { IIIIII _ 1 F l 1 ' .J I v H -I O o_ d " v- -I t q ,- d 3 ~ I F F 1 1- ,_ - q — O .. .0 < ‘ .. I —I O .4 1- HI— .- r C l 7 I T Y O O O O 0 v- v- P o- '— I! I IIITIII I T IIIIII n [IIITII P. I 2 J f N , .» M o o I :7 ' " F: l _' -4 I- i 1.1 -I 1 h * z :4 ‘ m l- '3 {I -I F r- l; .1 d v- __ ‘1 1, = i; .I 0 .- I: .D _"l .. < _-. r- I I l I G ,’ . I. -‘ O ._.' I "I "' H:— I‘ ,.' ll 4 I- ‘I’ la". ," q . fl. .. .. 1 1111-11 1~ J111111 I 1 1111111 '1' 7 7 O O 0 v- v- v- 3 L-IIIIII I [IIIIIIT I IIIIIIII I ITIIIII If}! ’VIIIIII I y. n I- — 0 . v- d h d 8 I F '1 = " 0 t 2 I « )- D- H I- —I o q v- all 4 _ e 1'. I J “er1 11111111 1 11111111 1 1111111 1 1 1111111 1 1 7 I 7 Y 'E I O O O O 0 v- v- v- v- v- v- I IIIIII I I IIIIIII I I IIIIIIIIII I I- ,0 p- q I— -—4 O .4 F d _ II .. 8 . F d v- 2}. = ’ - 0 I 2 V- .1 h N 1— ... o .. v- I- _ ‘ " P .‘r ‘1 111111 1 fr 11131111 1 1 1111111 1 1 7 ? T T O O O O o- v- v- v- r (kpc) R (kpc) ' (kpc) R (be) Figure 6.1. (cont’d) Abefl1216 Abefl1216 Abefl1213 Abefl1213 IQ 11111 1 7 I T T '1‘ T O O O O O O o- v- P v- v- v- ‘ IrfirI—T f lIITrTI I r ’11 I I 1 I 3' I H F —1 0 j" d O.— " F- ‘ b b A - 3’1 ,_ " v 4" I. :1 K .‘I .‘1 " I. ,o- I ‘fif', . I I." : ' 1- ". : I I r: 3 I _‘h I} > 1 " '— I: : I '1 I; ; I -I I; ; I 4 "—11111 1 “- 11 A ' 111111 I 1 1 I- 7 7 I O O O O .- F I- v- I IIIII I I I [nun I I IIIIIII I I [IIIIII I I )1! I"! I .V >- 1- I I- O .— . S— - P t .1 A L " g I L L . .. u ... «a O .1 F . I? . .1 l.‘ 1 ’ 1111 1"- 111 111 1211111111 1 11111111 1 11111111 1 'I' 7 T T 7 '1‘ O O O O O O ’- o- v- v- v- v- I IIIIIII IIIIIII I J‘lIIII I H .. -I O .1 F d . d S— - b ,. _1 A ' 1 b "‘ v P C 1- -I b N b — ° , P r" .I'.’ : 0' " ' .. o ~ 1 I. I l.‘ J I — 1111-1 1 -11111 1 1 L L 1111 1 1 1 1 1 'c' 7 Y '1 O O O O C- P C- '- 171 Abefl1238 Abefl1238 Abefl1228 Abeu1228 "III I I I IIIIIII I I IIIIII I I I [IIIII I3.“ ["111 I I I y‘ '-,\ h -3 .1 I- .1 F .. r H I. q I- r -I A . . § 1. p- d r .."o 4- " F .— '1 . " .. \ 1" 'I " 1. "-.\ '1 ' J 11111111'31 111111LLJ 11111111 1 11111111 1 n s I 7 I 7 I 1 O O O O O O P v- v- v- o- v- d IIIIII I III’I I IIIIIII I I IIIIIII Ir-J I I N(Hw) _. .1!- v-Iu— " P -I I- -I 1111 1 1 l T I O O 0- .- IIIII I I IIIIIIII I IIIIIIII I IIIIIII I," PIIIII I I .0 1- ’ J ,- —-I~O .1!- ,, 4 1.1 -I d .4 A _ § 1- "‘ v p L r- » .1 h h _3 q' '- II 1' " ‘ - .' . \ .1 1111 1.1 11111111 1 111111 1 LIIIJII 1 11111111 1 7 I I T '1‘ I O O O O O O I- v- v- v- v- P l III!!! I IIIIIIII T IIIIII'JI I ,. " II .10 .‘F " 1 .1 d _ 1 A ~ I» " -‘ v D ,_ I I- r " r ‘.' P- I: If 1" .— 5’ u 9‘ —° .F A' - b ‘1 I,‘ .1 l.‘ I: .1 1111111 I 1_1-11111 1 I O O O o- '- Figure 6.1. (cont’d) Abell 1257 Abell 1257 Abell 1254 Abell 1254 1- HIIII I I [ITIIII I I IIIIIII I I 0“ IIIIIIIEI' T'_\ IIIIIIII I ‘. i‘.‘\ _\ r (be) - .4.- .. .. . q I: .-' ,' -1 11111411 1111111131 11111111 1 11111111 1 11111111 1 ‘I I I I '1‘ T O O O O O O u- o- v- v- v- v- A IIIIII I [IIIIIII I TFTIVfTI I 3 1' ..'o .1 P d p q d 4 21- * " - 3 1- " v ' I: .1 1- p y- _‘o .40- .1 .. .1 .4 1111 1 1 1 1 1 1 1 O O o- v- I II I IIIIIII I I [IIIIIIrT IIIIIII I If! III'FIJII I I .‘III ”I 1- ." .'l I —O .1 I- q S— ‘ F d h . .. I- .1 A . § " v D L P d L. _"o .1 V. fl '1»— h .‘ “ .' ‘1 -‘ 111111 1 11111-111 11111111 1 111111111 11111111 1 1' 7 I I I '1‘ O O O O O O '- u- v- v- v- o- l IIIII—rI I IIIIIIII 17: IIIIIIII I .:I n O P 1_LL1 11 q .1 d I -—-O 1 2" I ' q I 1 -I 11111 1 ’1 1112111 1 1 1111111 1 1 I I I O O 0 v- v- 10" R (Roe) 172 Abell 1318 Abell 1318 Abefl1291 Abell 1291 JIIIII I I IIIIITI I T p ,. p 111111 1-1" 11111111 L 111111 IIIIII I I I \ I I 1 1 11111111 ‘1 11111111 1 1441 11L 1 11111 a” I I I I '1‘ ‘I O O O O O O o- v- v- v- v- v- 11 IITII Ifi IIIIIII 3 IIIIIII I I 3 $1 1 ' 3 a . —-4° .1" . . ‘31- I : .1 h -4 .. 1- cf :1. 4 f 1 '- 'I _1 -1 I I. " F- ": E. . -1 I: '2. >— 1: :l 1' E: _'b '. :1 ‘P *- :l -I '1 :l q 1- -_ 1— Q" .' -4 "1111111 1 1111111 1'1 1111111 1 1 '1' I I I O O O O v- v- o- v- 1 1111111 I IIIIIII I IIIIIIII IIIIII I III III L l I P J } 10" I I 10" 10" ’- I jTIIIIII I 11111 1 L _. qF ‘ 1' - r .- _ r .I .. '- 1 11111.11'1 1 1 ’1’ 1111111 1 L 111111 1 1’ I O O '- v- ? (kpc) R (I196) ' (kpc) R (be) Figure 6.1. (cont’d) Abell 1365 Abell 1365 Abell 1364 IIIII I I I IIIIIII I I IIIIII I I I 173 "III I l ,7 ” IIII I I .5? .5" 111111 f (ROG) .1 1- a —< 0 .fi F v-_ -4 In t ,3 ~..1 j I: 2‘. 1111.111 11111111 1" 11111111 1 11111111 1 111l1111 L I '1' I I I I O O O O O O 0- u- o- o- '- v- C IIIIII I I 'IIIIII I I lIIIIIIfilifiv .. >— / .-'I n — O ... '- O._ -1 .- F' -1 1- -4 L— ,. -1 1- _1 g 1' _4 v '- K - —O .1.- Fr— _' -I I- . -I h- :I ‘ P111111 1 l '1111 1 1 1:11 111111 14L 1 T I 7 I O O O O '- '- o- '- f (we) ". \ \ 11111111 1 1111111 1 1“; 111111 1 1 1111111 1 1 '1111111 1 I '1' I I I O O O O O O o- v- o- v- v- v- 4 IIIIIIf I IIIIITI I I [III pI I I I l I, I 3 _4 : ,r I- h d .4 .. .1 h d g " _‘ A v- § b = " V 0 11: n r- < ‘ r :- r 5' 5’ F— {1.51, _no 1- :1 u If fl "F 1- ‘: :I d I. 2' " " '1 2' .. l‘111111 1 1 111111 12.1 1111111 1 1 I '1' I I O O O O '- v- v- .- IIIIIII r IIIIIII I IIIIIII I I'IIII I IIII ,. I I ".1 I" 1_1 .1, _- cl G —0 q” q q 0_ “ R FF -1 1- 0‘) 1- A v- ,_ § = L v 0 1. 2 ’ r- . O '- 7 I I I I I O O O O O O '- v- v- P P F ‘ TrII I I I I I H r .0 up -1 2L 4 .1 1,: I I 1') -4 A " 31 = 1 I v 3 .. d h 1- C —-O . ..— ‘ -I E d I: : cl W111111 1 '5111111- 1 4111114411 '1‘ '1' I I O O O 0 v- v- v- P 3 IIIII IIIIII I IrIIII III IT II I I I II I I I [II I MW {1' I F . _ —10 .IF q k I .4 1‘ I 8 I I A " 3 = ‘ V o d. .. .0 >- < ~ ‘ 1- 1- b C P _o .1!- q 1- ’..f “ 1-1 In ... ‘ 111111 1'-’ 1111 111 1 11111111 1 11111111 1 11111111 I I I I I I O O O O O F u- '- v- P v- d IIIII I I [IIIIIrI I IIIIIII I I ., I .', .. n __ _o q" .. P d .1 13 ~ 4 C0 .1 " .- 1 = ‘ v 8 3‘ "‘ 1- < ’ ‘ 1- h p u _ —+° .1!- .1 h _‘ In '1 ‘ 1111111 1 '11111151'1 11111111 1 '1' I I I O O O O o- '- I- .- Figure 6.1. (cont’d) Abell 1383 Abell 1383 Abell 1382 Abell 1382 I I IIIIIII I IIIIII I I IIIII I IV]. I I I I" I?) I b q — O _ P .. OI,_ " FI- _‘ F .4 L 1 .1 A ~ 1 " V " L p- 1 . I" I.‘ a l'. — O I d " If. -4 ,. C: q '- 1_1". -‘ ‘ _ "H 1 -4 1111 1 1 ”1111 1 1 11111111 1 11111L11‘J 1111111 1 L p I I I I I 0 O O O O O c- v- v- v- v- I III I T I TIT—III I I I 1' I. .— O .1 v- .4 Oh " F}- .1 " -I P A I- "‘ § 1- "‘ v '- K v- '1 r C -— 0 ‘ P .. .. FT— .. 111LL1 1 111111 1 1 1 O O O 0 v- v- v- .- ? I I I I 'f O O O O O O I- P v- v- P v- A [III] I I I IIIITII I I IIIIIIIV I j I .' y n —O .. 4’ b - 1 "' .4 " -I b .1 >- d .“ I I- / _‘1 u- I.‘ | I: :' I; 'l I' 5' I :1 ’ ~ I a "L— t r . _o ’2 .' . 4" " I: - | -I F L. '- 1 -I I- I: '. I I" 1. d H1111 1 1 1 11111'«1 1 ‘ 1111111 1 1 7 'I' 1' I O O O O '- v- v- P ' (hoe) R (kpc) 174 Abell 1412 Abell 1412 Abell 1399 Abell 1399 IIIII I I IIIIIIII r '1”!!! I I IIIIII‘_ r'.‘ IIIITIT '- '. 3| q —0 °,_ - '- V-‘b ‘ p .I 1- cl b d P .1 I- .. ,. d I- - —O F— d F I I I- I I- r'3' “.2. “\ 1er1 1 1 1111 1 11111111 1 11111111 1 1111111 1 1 C h l I I I I l O O O O O u- o- o- v- I- v- I IIIIII I I IIIIIIrfi I IIIIIII “II >- 0. I I '1 r 1 I a 0 —o I p F":_— .. I. .1 h- '1 b d p .. I . I- d h C —-O .10- _ fil- .. ’ 1 ' , l ‘ 111111 1 1411111 1 11111114 7 I 'I' I O O O O P c- v- .- ,IIIIIII I TT I IIIII I "II ',I I IIIIII I I WW I" I , I "0 SI— 1.- .. d __ .. _ . _ '1 _ 4 q I- . . I —O F— .1" b 'l-‘ 1 ,. ’l ._\ " ‘111 4 11111.51 1 11111 1111111 1 1 - 11111 1 11111111; I I I I I I O O O O O o- v- o- v- I- v- I I-IIIIIII I [IIIIIII I "o e— 1 — _ .. .1 h P '1 4 h .4 b ‘ I- L .1 ‘5 O f _“o HI— 9 .41- .. ': P- '.‘ 1 I- ‘J '1 e -I I- I 111111 1 1 111111” 1 1 T 'I‘ I O O O '- '- ? (be) R (kpc) r (km) R (hoe) Figure 6.1. (cont’d) Abell 1468 Abell 1468 Abell 1436 Abell 1436 TTII T IIIIIIII I IIIIIIII l IIIIIY; $.“~IHHIII I .'. \ -: .‘I’ -.I ." a ,' -—— j : 41'" I‘..\ J 1111111 1 111111 1 1 1111111 1 L Jun-11 1 1 1111111 1 1 '1' 7 7 7 T '1‘ O O O O O P P I- '- v- v- A 1 [1111111 1 ;,I IIIIITT 1- .1 _I q —- O S— u v- : :1 d h .2 - 1 Q . g . v- _ - :v 1 - '1 3 '1 < ‘1'. fl .— : I 1 : I ’ I ' -—“o I d .. v-r— ‘ ' -4 1- ,.- ‘ 1 1112111 11 1 1 1 111111 1 T 7 T O O 0 v- I- v- I ' 006) R (we) ' (he) R (1'96) Figure 6.1. (cont’d) 176 IIIITI [IIIIII I I IIIIIII I r 111}th I IIIIIII I I IIIII I I lllllll I I IIIIIII I T IIIIIII}, .' IIIIII II T F “ " a. b fl 0 ’- —n° -1 '- .‘ 1- _1 ’- '1 -1 —1 A d 3 g— -: “03 1' u 1- o h '1 A o -1 A '- r- a '- o = " ‘ 5 = ‘ 3 O I- L 0 2;— s n _ . .o _ _ < < I- . h .. N r —: 9 " —I.O ’ P -< 1- , :1 /‘-‘ i' \ 1 t: 1 .1 ‘. I.” . \ -1 1111.5’ 111111 1 1 11111111 \ 1111111 1 1 1111111 1 1 ”1111 1 If ”11 1 1 1 "1111111 1 1 11111111 1 1111111 1 1 7 T 7 7 '1‘ '1 '1' 7 ? '1‘ T O O O O O O O O O O O O '- v- v- t- I- P u- o- v- v- P '- 11 A 11111 T 1 1111111 1 1 lil'T‘ 1 JITIrl Tfi / h .. .. n L —1 O '- —.no -I " .. v- 1 * ‘ - m d 3 SE" ‘ In ' ‘ o I- " A m a A r 1- § P a = P- " V = 4 a, .. 11: 0 21': n: _ d 2 1' d h r- I_- ’- 1; . .. 1- . - . on a r- : ' — 9 —1 o . '1 '- : : * - : 1 '1 l '1 r . .1 1 _‘ 1111111 1 (11111"1 1 1 11111111 1 #111111 1 1141111 L .‘1 1111111 1 1 1- ': 7 7 -.- 1 7 O O O O O O O 0 v- v- P '- v- P F I 3 111117 r "[1111 I "1111 l "111 1 1V IIITTrI T "1111 I 111 I I "7111 1 I I '51 1 1111111 1 I F l I _ T" I I I I V n" 5‘ z! a _:I a ...I o .-',’ 1 o S— .1 P o_ 4‘ -I '- h 1 '~- ‘ >- 4 . - E d D - '- d P ,_ -‘ V I. 10 .. In 1- .1 A o .. A '- § '- 1- § = “ v = " V r .0"- l- O '- L .f - n q 1 x < _ I I If f . “ N II« —1 O (F -—I 0 F1— , .1 '- I-' -I '- F . FF— ' 1. -1 t -4 I.‘ "' I 1‘ t - . f, g x . I. 2“ _‘ :‘ 3. ‘ -1 11111 1 L 1 lel 1 1 111111 1 1 "‘ 11111111 1 11111111 1 1111’ 1 111L111 f 1 11111111 1 11111111 1 '1' 7 7 ? T '1‘ 7 7 Y T T '1‘ O O O O O O O O O O O O '- I- '- I- '- v- v- v- '- P '- 1 A ,JUIVII 1 1 [111111 1 I 1111111’7‘1 1111'. 1' 1 ‘117111 f 1 11111111" 1 P ’ 1 3 1' q 9 a _- O ,1 —1 O 91.— ‘ " °— -: ‘ " r- ‘1 Fr- :l d d .. ' - v- : ‘ '- ' - d " -1 Q I. In u) q A :0 ' ~ A v- 1- § 1- I. § = ,_ '1 v = - v .8 1 g P I: < 1- ” " < -I t‘ 1- I q t u A I c —- O f 0 P1P- : .1 P f ' 1 I- I; _4 F— ‘f a _‘ '.' 1' l.’ 1 ': " 1- I; I ‘ 'Z -4 1- l-' '- ' -‘ 11111 1 1 111115111 1 11111 1 1 111.1111 1 2 (1111111 L4 7 '1' ‘1 T '1' 7 7 O O O O O O O '- v- v- v- '- .- Figure 6.1. (cont’d) #1111111 IIIHIYT filIIIIIII I PIIIIII I II-III I J 1’, J7 I” 4" 1' 1.: “J a’ :' ...‘o I" I. 4" . d .1 Sr .1 g1 1 . L- N ,_ -1 F = P ‘ o p 1- ...“o q.- . d v—-— ‘ _ 'r I. ‘ . 1111101 111111111 11111111 1 11111111 1 1111111 1 1 '1 7 1 T 1 '1 O o O O O O I- o- v- '- v- v- I I IIIIIIII I IIIIIIII I . IIIIITT I q —10 ‘F 1 o_ ‘1 a: "- ‘ V I '1 h ,_ -4 P _ h d - o I- 5 - ‘ p- - —° ‘F .1 r .I, -4 v- I' _.'I q 1 111114111 1 #5111111 1 1 1 111111 1 T 7 1 O O 0 v- v- v- I IIIIIII I IIIIII I IIIIIII I IIII 'I I., IIIIIII I F I l' E “I 1- A; OI -—q° p - 1 1- a F r- .1 a, P o -1 F - d E < 1 ‘ .. 6" -‘“° F”.- .v- p —1—- ’9’ d b- ..' . " I- ‘1" 2‘ 111111 1 ‘311111111 1 11111111 111111111 1 11111111 L '1' 7 1 1 1 '7 O O o O 0 P P v- o- v- 0- fl IIIIII I I [111111 I, IIIIII I I I :1 .. ,1: " GI —40 .1!- 91- " h d h d v- . _1 m . In _ 1 P - h -1 - 0 1" F I: g l.‘ 1 A. I: >- I.‘ I: l.' . . g: —o t" "' F“_ ,f . .4 P I" ' -4 I: 1 — 1111'111 11111 1 11111111 1 '1' 7 1 7 o O O O I- v- v- v- R (be) ' (RFC) R (hoe) 177 IIIIII I I [I'll]! I I llIlrI—Il I lrIIIIrI g7 ["1111 I p r, «‘1 II — O .1 '- 91:- -I __ -1 .I b (0 _ _‘ k " -4 ,— = F d 0 >- 2 ‘ F . — O .. «J— (A I 1- ‘” “.‘\ d 1111111 1 ‘111111 1 1 11111141 1 11111111 1 1111111 1 1 '1' 7 1 T T '1‘ O O O O O O P u- v- v- v- v- R ITIII I I [IIITII I I [IIIIII I a — O H F 2— ~ '- 1 b «0 I ‘ 1‘ 1' ‘ h 1‘ .1 F - = ‘ ¢ _ fl . f r r 1' ’.' n ‘I: -—I O 'f -1 P v-II— - 1- fi' 1' '1. 111111 1 1 1111111 1 1 111111 1 1 1 O I 0 1 I I 1 O O O O P v- v- v- :1 111 1 1 1 1 111 1 ‘ 1 1 I I I III" I 1 II I I [111!!!" I7 I" II I I b- a —-A O _ F q . Ii 1- cl ‘0 .. 1‘ J v- " = *- " 0 y- ? 1 1. a: I: I" ’3 u I; —-4 0 ’If-" -1 P '1— 1’3” : y- I_o' . '- 1'. ..\ .1 11111 1 11111L1 14 11111111 1 111111131 1 111111 1 1 1 '1' 7 T T 1 7 O O O O O O v- v- v- v- u- '- ‘ IIIIII I I r IIIIIII I I " a — 0 q F O ‘ '1— - 1‘ I ‘ . 2 I . P * l = - 7 -* 0 tr ,5" l 5' I- l 3' I :I l :l O ' If 3‘1 _‘ O I; 1. fi " l‘ I. -1 "'- I: I. .1 b "o l I- l‘ ' " 111111’1 1111111 1 11111111 1 '1' 7 1 T O O O O u- u- v- v- 3 (kpc) ' (kpc) R 0‘96) Figure 6.1. (cont’d) 1"!!! I I IIIIIIT I r ["1111 I I '"llIIT I [IIIIJ II, I H — O 2— . , I- '1 *- -1 ”’ -1 O) - - a: _ h .1 A v- I» § = __ “ V 0 5 2 ~ .. .1: Ph- " '- 1- -1 1" I _ " 1' I" ‘2‘) q a ‘\ 11111 1 1 111111 1 1 [IIIHLLL 11111111 1 1111111 1 1 7 ‘1' T T T T O O O O O O '- F v- v- v- I IrrII—FI I IIIIFII I I IIIIIII ];I ht I 1 1 1 x a 0 f _. O .' .. '- v-Ir ' .1 _ -1 " c1 8 ” 1 h b _‘ A 1- » § = " V 0 ' 11: 2 - r- ~ —1 0 v-:-— " '- 1- . -4 '- . - 4 1411111 1 1'~’ 11111'1‘1 1 1 11111111 1 T '1' T T O O O O '- v- v- v- IIIII II I IIIHII I I [IIIIII I I IIIHIIT'II J41"!!! I I .. fl —1 O .1 v- .1 a: -1 p d \ ID ,. , ‘ I 1- .1 A v- ’ B, = 1 « v 0 1. 2 ’ 1 I —1 O - " .. .. r f ' ‘ ‘ 1- ’.l' '.> \ 1111 1 1 ’1' 1111111 1 1 1111111 1 1 311111511 1 1111111 1 1 T '1' T T T '1‘ O O O O c- v- o- v- r- v- A IIIIII I I [ITT—TII 1 IIITIII I I V c' .. 1 a I .‘I —- O I :1 - " I :1 .. 1 :1 .1 " ‘3 :I ‘ 1- :1 m _ I f: .1 k 1' '1 _‘ A ‘- 1. .' § = - ." ‘ v 0 c 2 ” 1 'f 1- 9' 1, 1'5 " 1- "‘ O 15 " '- w-U— '3. - 1. t " 1- I; .1 11111'51 1 1 1111111 1 1 ‘1' 7 T ? O O O 0 v- o- c- v- 178 IIIIIII I IIIIIIII I IIIIIII I I [IIIII‘ I‘q [IIIIIII I I 1‘! a I —10 .1!- .. 8 ~ G - P = d O 2 1 . -—10 fl- f _ -1 E- r, ‘z‘\ -1 ‘I 1- ." : 111111 111111111 111111111 111111111 111111111 ‘1' 7 T T T '1‘ O O O O O O '- v- v- u- I- v- 4 IIIIIrIfi IIIIIIII I jIIIIIIIIIj - 1‘1 9 E 12 : 3 8 : - Q - .1 p = E " 0 2 - h. . —10 .1.- F1— -1 : -1 - 11. - 1 1 1111111 1 111 1 1 11111111 T '1' '1’ T O O O O '- v- v- v- 3 I I I IIIII I IIIIIIII I IIIIIIII I [III I I ’7 TIII I—I I .1" _ _"o qF .4 F 1 ID 1- . Q 1- h _ F i- = . o I- 2 - 1 C —-O .1!- d . . d F»- 'f, VI‘1.“‘\‘ -1 111111‘1 111111 11111111~1 11111111 1 11111111 1 T 7 I T T '1‘ O O O O O O I- v- v- o- v- A IIIII I I IIIIIIII IIIIIII I I 3 I 0 -—-O .1.- q_ ~ .. .4 In I q 0) 1- h I- " P .. = “ o b 2 ~ ~ b N —O ‘1' 5 1 " L '. J 11111 1 1 1111511 1 11 1111111 1 1 7 T '1‘ T O O O O o- v- v- .- I’ (kpc) R (W) P (m) R (be) Figure 6.1. (cont’d) 1- _'o d F P — 1 b d 1') ‘ m a: .1 A v- _ .§ = - v " L '- .1 .. ..."o q F " -4 11‘ 1 \ .1 ,'.' '1 \ .4 11111111 1 1111 11 1 1 11111111 1 11111111 1 1111111111 7 T T T '1‘ T O O O O O O u- v- v- P v- ’- ‘ IT T I IIIII I I r, I I I .. ,1; _."o ‘ ' P - .1 N - .1 2 o- ~ “b w _ g = 1- " V 3 ’ " < h ‘ 1- .‘I --.' I: ‘ I 1- l: I I.’ : I: :: —1~O l.‘ 2. - ' '- L' I. -1 '1. d r I. ’.‘ j ‘1 111111 1 1 ’v' 1111 ~fi 1111111 1 1 '1' 7 7 ’ O O O O '- F v- v- I IIIII I I I"!!! I I I [Will I I f [IIIIII I g ["1111 I I . ' J H — O _. v- 21— - D ‘ p- -1 v- r- - (9 1- Q 1. .1 A P K = ’ 1 5’ 2. - ~ ( ‘ .. _'o q ' F ‘ I". '1 \ " I '-_\ 1111 1 1 11‘ 1111111 1 1111111 1 1 1111151 1 1 1111111 1 1 '1' 7 T T T '1‘ O O O O O O '- v- v- o- u- v- R IIIIII I I \, IIIIIII I I p ,1 '0 a a — O .1 v- .1 1- -1 .- . j I" o .. -1 A w § = ’ < v 3 - m < r . ‘ I‘ .' p ,‘ n I. , 1." 1 c - 5‘ '1 - O ,f z. . '- v-1— ‘ 31 " I- ,. I “ p [‘1' 11 _ H1111.i‘ 1 1 11111111 1 1 11111111 1 T 7 I 1 O O O 0 v- v- v- o- 179 Abefl1913 Abefl1913 Abefl1904 Abefl1904 _IIIIII I [WHIII I IIIIIIII I IIIIIIII I, 91111 I J " a —10 ..v- .. .1 S— ‘ ’ .1 b ‘1 b p 4 r d 1- 1- 111 —O .10- .1 . ._ \ v-h— ‘1. \ 1111111," 1111 1 11111111--1 11111111 1 111111111 I 1 7 1 1 1 '1‘ O O O 0 O o r- v- P I- I- v- A IIIII I I I I I .. n —O .1" .1 .1 O .-'_- " _ .. 1- -1 h b d L. .1 p- " II «0 qt- .1 - <1 ‘I I". 1" l .1 11111.? 111.111 1 111111 1 1 1' '1‘ T T O O O O '- v- v- .- IIIIIII I 11111111 I IIHIIII I IIIIIIIII I; IITTIIII I f .i 1- 0| —1° .1!- 21— ‘ r- d 1- q '- c1 h .1 .. I- I —10 .1.- . P1— 1 1 1‘ j " '. \ 111111 1 11111111 1 ‘11111 11 1 211111111 1 11111111 1 '1' ’1' T 1' '1‘ O O O O O c- P '- v- P v- a IIITII IIIII‘I I I I .. I —10 .1.- b d h d h. ,. .. b J . .1 G —-O .10- 1 Fr -4 F q 11111 1111111 1 1 O O O o- c- P f (kpc) R (RFC) ' (MC) R (Roe) Figure 6.1. (cont’d) Abefl1983 Abefl1983 Abefl1927 Abefl1927 '(Nn) IIIII I IIIIIII I IIIIIII I IIIIIII I 1 1 1 er' _ —% .1 o- q .1 S— ‘ .. I- .1 A - § 1- " V 1- L b q 1- , «E _1 w .1 d d 1111111 1 11111111 7 '1' T T T '1‘ O O O O o O o- v- v- v- u- v- I II I I I I I I I I I q .. 12 q 3— ‘ I ~ A - 31 p d v -- K _ 4 . “o .. . 1 F I' 2 .1 V.’ 2 -1 1111111 1 ‘ 11111111 1 11111111 1 7 1 1 T O O O O o- c- I- c- I IIIIII I I lllllll l I IIIIIIIW I IIIIIII'U I; IIIIITI I I f .‘I b .1. J a — O 1‘ p 9:: .1 b . I- _ - L 1 1- I I I, I T I t i _ 1 .1 .12 L x u-I -I " 1- 1.’ 3| -1 I..' :1 .1 11111111 11 1111 1 11111111 1 11111111 1 11111111 I T T T T '1‘ o O O O O O '- r- F v- v- I IIIIIfiI I IIIIIII I] IIIIIII I I r- l . a —-0 up .1 l- .1 '- a1 . d .. 1- ‘ 3 P d V 1. C _E q’ '1. ‘ h .1 h -I "1 111111 1 1 1 1' 7 T O O O P v- v- 180 IIIIIIITT _1 -1 '- q -4 .1 8 1 G .. F = -1 B < i r 3'9 b ' J 1— ” \ llllL 1111111 I 1 1111111 l"l 1.111111 1 I llllll I l I T T T T T '1‘ O O O O 0 v- o- v- v- v- v- ‘ IIIIII I I ,I I .‘I .'I gr qEI .1 " -1 ' -1 a - « G) . ~ P =: ‘ ‘ O 2 ~ 1 "h- db 1- .4.- 1' q 1' 1 P' '4 [III] 1 1 l l T T O O o- o- I IIIIII I I IIIIIII I I IIIIIII I I [IIIIII I I ‘\ H‘II I I I ‘. - II -—O .0- 21— ‘ r- ” 'l . .1 O) __ O) _ - F = ’ ‘ o L 2 - 1 1- —1.b ‘F d o d v-b— to '3. \\ .1 _llllll I 11111 111 1 11111111 1 lllllI-ll l llllllll l T '1' T T T '1‘ O O O O O O o- v- v- v- v- ‘ "IIII I IIIIIII I I P II —-O 4' d . fl .. v- __ - 8 - v- " . - '- d - 3 I < - ‘ .. II —10 ..v- d a - ‘ 1-1— ,_ ; d b- ‘ 1' 11111 1 1 111111111 1 111111] 1 1 '1‘ 7 T T O O O 0 v- '- v- ? (kpc) N(Hn) '(NK) 3(NK) Figure 6.1. (cont’d) Abell 2022 Abell 2022 Abell 2005 Abell 2005 I' _[IIIII I I [IIIIIII r 1111111 I I [IIIITIT Ir II'JIIII I I III F 'I -- O .1 .— .. .°.— . Z .. " .1 A ' § 1- '1 v p L -1 b b Q -1 O .. F Pb fl." \ " 7111111 14 1111 1' 11111111 1 111111114 11111111 1 Q T I T T T '1‘ O O O O 0 O '- P v- I- '- o- l I IIIIIII I I IIIIIII I I ,‘flIIIII "a _”o .1 P 1 .1 a— - .. 1" II '- _‘ A * § 1- ‘ V - I! ud p- p- 4b .1 '- .1 .1 M— ’.' I .1 1 11111 1 1g-‘1 1 . 111111 1 1 111111 1 " '1‘ T O O 0 v- '- '- IIIII I I PIIIII I I 'IIIIII I I IIJIII I I I I ’f O... F» '| ,. ‘ _1 O r ' ‘ " i 1' 4 + H L- -1 -1 A ‘ 3 " v 1- L flu“, d I.’ U \ \ v-I— \‘ r- ' '0 L- t. — O D- ‘1 '- b ‘ . u \ ', 1— “; -_ ‘-“ 1111L11L 11111111 1 11111111 11 1". 11111111 1 u- I 1 1 T T T T 0 O O O O '- v- v- v- v- '- A nTI r I I I I- II .. —1 O I- d w .. b .1 D d P d -1 A * 35 " v f- C .. v-‘I— h- I . — O .. '1 " c1 .. d '- -4 1111 1 1 1 1 1 T O O o- .- 181 Abell 2029 Abell 2029 Abell 2028 Abell 2028 IIIIII I T 1- IIIIIII I T [IIIIII I I "III I 1 .‘ ".1 .‘I IIIIII I I I 1 1111 .1 1- ~ -— O .1 F F— _1 b ' . -4 I- '_ '. : 2 d 1111111 1 '-111111 1 1 1111111 1 1 111111111 11111111 1 T '1‘ T T T '1‘ O O O O O O u- o- o- v- v- P l IITITT I I IIIIIIT I a — 0 d F 21— ~ b N 1- q 1- 1. 4 r- -1 " _ j 3 V v- I .1 .. 1. .50 I; 4 "' I-lv— "I q h I 4 __ c If -1 "1111111 1 111111£1 '1 '1' '1' T O O O O P v- v- '- IIIIII I I IIIIIII I I IIIIIII I I [IIIIII;I ITIIIIII F ( '1 f :1 .‘I I a l —-4 O .i' 1 " S'— 1 j '- I " .1 . .. -1 L .1 A q g 1- .1 1. h —'o .4 F F'— -1 r- .- ..“‘ -1 1- f '1"\ d I— '. ‘ 11111131 1 1111111 1'-1‘ 11111111 1 lllllllLl '1' " T T T '1‘ O O O O O O o- o- o- v- '- fl IIIIII I I ITTIIII I I IIIIIIT T“T - 2 :1 :1 51 fl :1 —1 O J 1 F e— - " -1 - fl .. p a: k d 3 1- '1 v L a .1 b V. —1 O .1 '- v-1— -1 1' —4 1- ;. ; . l_.' . 1LLL 1 1 111.11 1 1 1 1~| 11111 1 1 1 1 '1 '1' T T O O O O c- '- '- Figurc 6.1. (cont’d) Abell 2048 Abell 2048 Abell 2040 Abell 2040 IIIII I I [IIIIII I I IIIIIII I I [IIIII I‘_I IIIIIII I I IIIII I I IIIIIII I I IIIIIIII I IIIIIIJ TI I'IIIIII I I f I" e' :l ."a’ '- f ’0 n o_ — O ..."o H'- .‘ v- _‘ ,- _ _ '- d - d P ‘ b 1. " '- r. q l- " v = P. -1 L F _ 4 q - " a tub—- _‘ o —I O ,- -1 '- 4 o- r- “ ’— P '.: '. \ : -. 3 b I. I I ‘.' \ I I '.‘ I 111111 1 1 11111 1 1 1 111111 111‘ 1111111 L 1 1111111 1 111111 J Imu- 1 1111111 1 1 211111111 1 1111111 1 1 I. 7 u I T T '1‘ '1' 7 I '1 I '1‘ O O O O O O O O 0 O O O o- v- v- v- v- v- '— v- v- v- F F 4 a III I rI I TIITI I I I IIIII I II ‘.I I III! I I I I IIIII I I I I r . 1— .'I Q o o,_ —1 0 — 0 P_ 4 " _ v- _ .1 2.. .. F 4 h - __ 4 A 8 r- f 4 § N e .- > . .1 ’- .4 V = :' I ~ 3 ~ . :1 1- “ ¢~ . 4 < (‘4' :I I- (' I. I: :' I: :l 1- -1 " 5 ‘. - Pv— -‘ o :l I: -‘ o .- ‘ " .1 2. "' b- " VHF- f. :‘ 3 .. " 1'- 31 ‘ C [.1 -l -4 " r- 11 51 1111111 1 1111111 1 11111114 - 111111 . U 1 I u 1’ '1‘ I O O O O O o o- v- I- P '- III I I III II I Her III II I I III IIIII I I IIII I I IIIIIII I IIIIII‘I IIII I I r 1' 1 I" fl I W' \ f I" l 1' .~ 111.1 I >- ' :1 T ‘ -' l . a "‘ n v- .1 O — o d '- -1 "’ .1 ’- d -1 .1 d N g— d 3 CI p r- -4 A o 2__ -4 * § N t ~ ‘ v = _ " - u 0 r L. c1 3 I- d - P on a r- -—1 O —I O .1 P _1 F q . -1 0.”. - .1 r. ,',r“ ‘2 \ d ’1‘; ‘.. \ .4 ”'_A' “I .1 f IIILII I I IIII‘I'I I IIIIII II I IIIIII I I- I ‘1 IIIIII I I I IlllIl I ’I"- IIIII I I IIIIII I I I IIIIIILI I IIIIII I I I '1' 7 T I '1‘ '3 I T T ’? T O O O O O O O O O O O O '- v- '- v- '- c- v- P o- P I- l d TIrI I I 'IIIIIII I ;IIIII I I III I I IIIIIII I I \I I I .— " ~ ‘9 .' c .I “ 1' q " 6 II p. ." -—1 o 6 —I O J .. '- ..', .1 P : c1 _ ' - I, —1 u 1' C! N -1 d d - A 8 g - .1 d . g N . t .- 1- '- v = '- - i ‘ _ z o ’ .1 n * ' ’ 1. -1 < _ I 1 A c 51 r- ' ( 1| l’ j' I: ‘l a " . 2 u '- L‘ d O 0". 2: —1 0 {K d P 1.: 3 .1 v- v-" * > {~' in 4 I”. .1 4.‘ .' .4 ’.‘ a'- I. -1 l.’ -4 I" . #111111 1 .1"? I 1111111 1 1 1111111 1 er4111 -' 11111111 1 u '1' 7 I T u 7 I T O O O O O O O o- v- v- v- v- v- '- v- H 182 I’ (W) R (39¢) r (kpc) R (he) Figure 6.1. (cont’d) IIIII I Abefl2065 IIIIIII I I IIIIII I I I IIIIII I I| I .1 IIIIII I .4 l‘ I LIIIII I 10’ ‘ -1 -< d .1 IIIIILII L 1' 7 I I I '1‘ O O O O O v F P P P w R IIIIIII I IIIIIII I I IIIIIII I I. II P '_'I w ." q —40 q— 51 q 4 - - ID .- 0 . 1 3 ~ 1 = " I O 1- 2 1 —"o ‘F 4 H:- ‘ b ‘ HILII I I I 7 I O O I- v- I IIIII I I IIIIIIII I [IIIIIII I [IIIIIIE‘I I (:AIIIIIIII I I h :I .‘I J. u ' a —«O .4.- .1 - d -4 g ‘ d N 9': = 1. I Q I- .D ’ . < » . 1- I —O _‘F _ I d f '. \ I. 2 \ “ IIIIII I-’I III 114 1 IIILIJJI '1 ‘ IIIIIILI L IIIIIII I I "1 I 7 I '1‘ T O O O O O O C- P v- v- v- v- ‘ FIIIIII I I TIIIIII I I lIIII'!I I l d p b a -—10 _‘v- .1 P d 8 ‘ C d_ N 1. = ' d . o b 2 1 b b n —O ~P ‘ I 1: ‘ I" -1 IIIIII I I '11 LLLI‘ I IIIIIII I I '1' 7 1 7 O O O 0 o- o- v- v- ! 00¢) R (hoe) ' (m) R (kpc) 183 Abefl2089 Abefl2089 Abefl2067 Abefl2067 11111T 11T1 1111111 111‘ I I" 1 1 I 1 [111 ‘1 1 .‘11111111 1 .. 1 2 .4 'I , a , —4 .. o_ 3 III .. . .J h .1 1. I. '1 .. I .fi"” '1 V. I- "‘ é . l I. 1- I: I “ - ‘. ‘\ do a. “ ‘ .- PI— -' ‘. “ -1 1- | . .2 ‘\ ‘1 h I t .2. \\ 4 '1111111 1 I111L111 1 1 I1111'11 1‘4 '1' 7 Y 7 T '1‘ O O O O O O P v- v- P v- A 1111111 T [111111 1 1 .. q -—1 0 d P O,_ .. .— h u t 1 1- - _ . " -1 l‘. O. 1- ‘5 1' -1 6 h 9 .' ‘. '- . I “. —1° 1': " F v-1— 1: ‘ "" If -1 '- 2' .1 11111 1'- 1 11111111 L '1 1' 7 1' O O O 0 v- P v- '- 111—r 11111 1 111111 1 "HITT‘ 1 111111 1 1" 1 1 , .1: b- :l .‘l .I .D a — O .1 P O -1 -1— . 1- -1 .. .1 .. .. 1- u: b . . I —< 0 d .- v-‘h— '5' q i- ’ ' . \ n .’ .’ \ b I..' \ -1 311111131 1 I11u1'11 1 [1111111 1 I111111 1 1 '1‘ 7 T T 1' '1‘ O O O O O o- v- r- v- v- a 1 [111111 1 1 '111111 11 [11111 1 1- .‘I ,‘I ,‘I ‘I Q , — O .1 '- O,__ -1 Fl?- 4 P- c1 )- . h- p .1 '- d b .. .. .‘I' l N 1 —~ C) 1 -" I a F ‘3‘ ‘ F p , , I- I_-' : I '4 l_ I1111’1‘1 1 111-‘11LL1 1 1 I111111 .l' O p T o '- I’ (be) R (live) ' (59¢) R (m) Figure 6.1. (cont’d) ,111111 1 1 111111 1 1 11111 1 T 11 1 111111 1 1 1' 1"? 1' '\ h t :1 .'I a Q" —1 O 1- - u- q q 1~ ‘ d O 2»— P I .. A N _ § = ,. “ v L 1- g - h 1. on _1 O F- _‘ '- .. I, “ I._¢' _‘ 11111 1“" 1 1 1 1111111 1 1 7 I T I '1‘ 7 O O O O O O '- P I- v- v- '- ‘ 1_1) 1 1 1 1 1 1 1 1 D- a -- O 1- ‘ p d d l\ I o 2— ‘ P P -1 A N " L. = 1- " v 1. C d .. b . -—4 O H I, _‘ I I; 1 IIIII L L I l" IIILLL I I I , 1 O O u- c- r .- 111111 1 1 [1111111 1 [1111111 1 [111111; ”11111111 1 1‘ a" p '1 .1 31 1. -_\ I 1‘ n 1’ b — O a p / . 3 1- '.-',.) _‘ P “"4 3 1 1 o - w" 4 " o1 - 3 o = 1- / '4 v I L P " - 6' 11' 1- .‘1 ff .'.\‘ a '. 1“ — O ‘1‘- '.~ ‘ "' 'J ‘2‘. ‘..\‘ c1 :— ‘Q. '3‘ '4 '- \‘_ .'\‘ .1 IIIIII I I IIIIII I I I ‘IIIIIJJ I IIIIIILJ I I_IIIIl I I4 '1' '1’ I I '1‘ O O O O O O '- I- I- '- v- v- I 111111 1 11111111 11111111 I h a —- O .1 v- 4 2— « F d b g . . O " _ A N - 8 .I = I- " V C h .. 1- fl — 0 d F '0— .1 .1 LIILI I I IIIIIII I I O O O O I- v- v- v- 184 Abell 21 42 Abell 2142 IIrII I IIIIIII I I IIIIII I I I IIIIIIfI fr IIIIIII I I r .w P ,p _.7 .‘I , ' n 9.1— " ‘ ° - '- h . _ D r- .- \ d b _ \ 'I ; 1 1 l - b :I I- '1 q .. .r’ f. 1.’ _ h‘. _J o F: r.. -1 F 1 '. '1 - “.3 '. 1 .1 <- '. 1 .J t a. ‘ IIIIIII I IIIIIII "~l IlIIII l i IIIIIIII 1 1111111 I 1' '1’ I T T '1‘ O O O O O O '- o- v- v- v- '- 1 IIIIIII l I IIIII .111 10' Y 1 O O u- v- 3 111111 1 1111111 1 1111111 1 111111‘ I 111111 1 I I I t “ I"[ < s _"o .P 21— ‘ V ' . N I- v- " ~ 0] ,. = e ‘ 3 < h ‘ _'o .4.- . v—-— .1 - ’1' -1 11111141 '1' 7 I 1’ T '1‘ O O O O O O u- v- P v- v- P A 111111 1 1 ‘IIITTI fl —0 ..v- 21— ‘ " 4 V ' .. N I— v- " ~ N b = Cl 0 2 - - L 1'9. '- I’ I 'C'. -1 11111 1 1 11111111 '1 'f Y O O O o- P '- ? (kpc) R (be) r (we) R (hoe) Figure 6.1. (cont’d) Abell 2151 Abell 2151 Abell 2147 Abell 2147 "III! I IIIIII I IIIIII I I I IIIIII I _ 1' 1' 1m", 1 1' 1- 4‘0 1' " 1 .1 J 1 2~ « 1- b _ J 1- 1- -1 p 1- a —O L "' q 1.: '1 ‘ l h .. ‘ ILIII l lIlllllll l Illlllll l \IJIIJIII 1 11111111 1 1' 1 1 1 '1‘ 1 O O O O O O '- u- v- v- v- P 4 PIIITII I [IIIIII IIII'IyI l I p) 1. n O '- IIIII l I I I 1' f I a 1 —10 I -"’ 1- 9' -1 ,‘.' -1 I," -4 IIIIILI 1 [11111 I111111L; :- 1 Y I O O O O F- u- .- v- I 11111 1 1111111 1 1111111 1 111 -: 11111111 _ 1 1 1" p ,1 " a —.O “P h d .. .. .. .. '- n —O ...P 1- '1 I, d I.‘ .\ 1; us‘ -1 11111’1 I1 1111 1 11111111 1 11111111 1 I11111111 7 1 1' 1 "1‘ 1 O O O O O o- v- v- v- u- v- 4 11111 1 1 1111111 1 1 [111 1 1 1 . / -1: SI 4'- . d :1 .‘1 I I1 .1 :1 a '1 :1 S— u -4 h :l 1- ' ‘ h h 1. 4 b 1- . I 1’: —‘o I.‘ _‘p .. I: «1 I". i (o t -1 1111111 1 11 I1111111 1 1' 7 Y 7 O O O 0 v- .- v- v- f (be) R (be) r ('19:) R (be) 185 Abell 2162 Abell 2162 Abell 2152 Abell 2152 IIIIII I I lIIIIII I I [IIIIII I I 1111] 1 1 f (be) — O .. II- 7 I I I '1‘ I O O O O O P I- o- r- v P ‘ PuI_II I I I r IIIII I I I [IIII I }1 I 1- 9", 1 1- ‘3 d F q I- '1 .4 9*— ‘ 3 I ‘ v 1- J g P . ,. _1'b . .1 P I w v' : - ,. ., If 2' .1 111111 1_1‘1 I11111 1-1' I111111 1 I_ 1 7 1 I O O O O I- P v- I- I III IT IIII I IIIII I I III ( I IIIII I I _ 1'" 1 1' 1 ~ 1' IIIIII 1- 1 A ~ 3 1- " v L s P 4 - I l - C ..J'b .1 P ' .1 f." -1 ’1'} ‘2' -‘ IlIIII I I IIIMII I I IIIIIII I IIIIIIII I I IIIIJII LL '1’ I 1 I '1‘ I O O O O O O u- - o- n- v- u- I IIIrTI I I IIIIIII I I .315 .. c- .1 .1 q 3 "' V C l‘ r- :. 1'; _B I: -1 F 1- ! a I .. T. 1_- " IIIIIII I IIIIIIIC I 1' 7 I O O O '- v- v- 3 Figure 6.1. (cont’d) Abell 2197 Abell 2197 Abell 2175 Abell 2175 IIIII I I lIIIII—I II—ffi IIIIII TJ IIIIIII I _ 1' WW I I '._\ r- u 3" ~ ’- 31. —4 O ." u '- I‘ll - 'l L 3'] .1 .I .. .1 1 J “P ’." II: .4 1- ‘3. \. >- ‘1‘ ‘r 1. 1 .31: .4 o- .4 L , . s ‘ . 111 l 1 11111111 1 11111111 ’1 111-1‘ 1 1111111 1 1 7 T 7 T '1‘ T O O O O O O I- v- F v- I- u- l IIII I I ‘r I IIIII I I T I .. 1- I1 1- —1 O .1 '- .4 _ -1 .1 -1 1 SF: .4 1. h r- . T P- y." I; 1' v. O y. db 1' b d . -1 111114L'1-1 1 L 11111111 1 7 T 7 O O O o- v- v- I IWT I T IIIIIII I I IIIIII I I I j I I IIIIII I I I 'I a —1 O p .4 '- i- -1 b I P I1 r- q ’- 1 h 1- -1 u 1— -‘ O '7 4 - E d l L- P: . \ _ +- '-. \ -1 1111 1 l 111 1 1 11111111‘L 11111111 1 1111111 L '1‘ ‘1' 7 1 T '1‘ O O O O O o- v- '- v- v- v- K II I I Ifi I IIIII I I I I q i. —4 O —4 F - -1 i- d I- d 1- .- 1 b -I f I. ¢ 3 (' . . cl v-u— (t 1: - O b ' -, cl F " I." 1 '1 1- !: .1 d 1’ S .. F 111111 I 1 111111 1‘. 111111J_L 1 . 1 7 7 Y O O O O 1- P o- v- ? 09¢) R (“96) ' (m) R (kw) 186 Aball 2255 Abell 2255 Abell 2199 Abell 2199 IIIII "IIIII I IIIIIII I IIIIIII I [,T ITIIWT l l 1' , .- n —O 4" d p 1- -1 1- u 1- < b '1 p d 1- r' " I "' I '. ". 1; - \- \'-. —1° \' ‘ 4P V'H— ‘ ..J L: 1 .1 I '. 1 A y. 1‘ . 1111111 1 111K111] 1 11L] 1 3111111111 1 11111111 1 1' 7 I T T '1‘ O O O O O O '- v- '- F '- '- I IIIIII I [”11th _'o 4' .4 - d .. '- d 1- 4 " J .. .1 b d " f 1 I '. 1 .3 1 4.. Fl— " _‘ - I 1. 1. “ _ t .1 1111111 1111111111 v C 1 1 7 I O O O 0 v- I- v- .- 111 r 1111 11 I 1 I .‘l I T 1 .- 111 I I I [111111 [111111JV lflilr y- 1. ‘3 “P ‘ 1" .1 d ‘ ‘ h _ ‘1 .. ~ 1 h F _ 'b .- .1 1- . -1 . o. \ .1 1111 LJfllll 1 1 1111-11111‘11 11111111 1 111111111 ? I T T '1‘ 7 O O O O O 1- v- o- v- v- '- I [IIITI I I I 1 11111 I 1 r 3' f = 1- ; I ‘5 I 3', .. I- f 7' a f' . _ 1‘. :1 -4 f 2‘ - 11411 1 1 111111 1 -'1.' J111111 1 '1" 7 7 T O O O O '- v- v- '- ' (RFC) R (kpc) ' ('96) R (39¢) Figure 6.1. (cont’d) 187 8.253 .3 85mm “as: .. .23 x A25 . G9: a q1d4 d d d 1_1 [d1 _<4d1 d “IO“ —11 u 1 q q d1 d —a q d - .IOF ‘41}! 1 11 1T q ‘4‘ ‘14 '0’ —- - 11‘ d d d J —udd d I 1 IA I 1 I 1 1 I 1 I 1 1 I l 1 n H I L H I 1 w 1m Claw 1 1m IOP I 1. I. 1 1 ’ . «u .. 1 1 ..... 1 11.. u m ‘\ 1 1 \ l \1 n 1 Yo. \\ ..u o; .. \111 1 01°F 01°F I . 1 l 1. ‘ 3 K I 1 I 1 " nl o. o— I 7 I 1 r 70w I 1 I fl 1 I (A. 1 "W Li ...h.l ..M 1 I o. 9 7 1 ch. 1 I J I 1 I II..III..I..II.. I 1 I 1 I I. I 1 .I 1 I 1 r 1 n u I 1 - 1 I 1 I n —P bi: P P P - —P 1 I b 1%: P b b E b P r! —~ - h h b p - n —h h n h - 1 A p ~10 w + Top .10 w p .10 p NOON =on< Nam =Qn< ham =02 hva =02 Aunts L A095 1 A5528 .. A3523 3 no. .9 or no— no, 0. 2 . .. 9 F p. —q<< ‘ q 11 d 4 qd