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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.

Columns (8) and (9) are the initial smoothing scale in arcmin and Mpc, respectively.

48

 

 

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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)

 

 

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51

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Figure 3.6. Adaptive-Kernel Maps for the Dressler Clusters

Y (arcmin)

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Figure 3.6 (cont’d).

 

 

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Y (arcmh)

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55

 

 

 

Y (arcmin)

 

 

 

 

 

 

 

 

 

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Figure 3.6 (cont’d).

 

 

 

 

 

 

Y (11min)

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Figure 3.6 (cont’d).

 

 

 

 

 

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Figure 3.6 (cont’d).

 

 

 

 

 

 

Y (arcmin)

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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

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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.

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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. Because Dressler was interested in studying the morphology of the
galaxies in these clusters, he was forced to use a brighter limiting magnitude to ensure

accurate classification.

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69

Fig. 3.9.— Adaptive kernel maps of the HGT Sample. Map parameters are listed
in Table 3.2

Y (mm)

 

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Y (arcmin)

71

 

 

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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

 

 

 

 

 

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cores of very massive clusters.

93

Abell400
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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:

 

, : <n>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

 

 

 
   
   

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

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Abell 2022

Abell 2022

Abell 2005

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Abell 2048

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Figure 6.1. (cont’d)

191

6.3 Application to the Cluster Sample

The plots given in Figure 6.1 are constructed using the MPL method with the above
penalty function (equation 6.8). Both the projected density and the space density
are presented with V and 2 plotted on a log-log scale with the bottom 2: axis labeled
in kpc and the top labeled in arcminutes. The center of each cluster was chosen to
be the highest peak in the adaptive-kernel map of the 2 Mpc region. The solid line
is the estimate for either 2 or 11 while the dotted and dashed lines are the 95% and
97.5%, respectively, bootstrap confidence intervals. Because of the desire to avoid
spurious cores due to oversmoothing and the lack of a reliable, objective, data-based
choice for the smoothing parameter, each of the the estimates was calculated with
/\ = 1 x 10‘5. With this choice the space density profiles are deliberately under-
smoothed with further smoother left to the viewer. As a result, the estimates for u
can be seen in many cases to “fall” near the centers of the clusters. As the smoothing
parameter is increased, the density estimate in these cases will flatten out with a
value close to that of the maximum. Further smoothing of course leads profile with a
spurious, central cusp. Comparison with the numerical simulations of MT indicates
that this type of behavior is a possible indication of a density that approaches a
constant value in the core. In the caSe of a de Vaucouleur profile, which increases
all the way to the center, the undersmoothed estimates generally oscillate about the

true density.

6.3.1 Core Radius

It has been argued that a natural consequence of relaxation in a self-gravitating
system is the development of a physical core in which the mass density approaches

a constant value (King 1966). The core radius is usually defined as the radius at

192

which the projected density falls to one half the central value. Because the density
estimate may decrease toward the center, due to the estimate being undersmoothed,
the maximum value of E was used here instead of the value at the center. Unlike the
plots shown in Figure 6.1, the profiles were constructed with a range of smoothing
parameters, including those for which the density estimate was clearly undersmoothed
and those which were merely power laws. In order to avoid creating spurious cores,
the density profiles which decreased near the core and were closest to flat were used.
For clusters that showed no sign of a flattened core in any of the calculated profiles,

the smallest smoothing parameter, /\ = 1 x 10‘s, was used.

The results are listed in Table 6.1. The a: and y positions in arcminutes of the
center (chosen as the position of the maximum in the adaptive-kernel map) are listed
in columns (1) and (2). Column (3) is the estimate of the core radius, with its 95%
upper-bootstrap confidence interval listed in column (4). The median core radius for
the clusters in the HGT sample is 150 :l: 96 kpc, where the error is the one—sigma
error. This result is approximately half that of the often quoted value of 250 h'1 kpc,
or with h = 0.75, rem = 333 kpc. On the other hand it is about three times the value

obtained for some clusters with gravitational lens observations (Squires et al 1996).

193

TABLE 6.1. Core Radius of HGT clusters

 

 

Cluster :1: y 1'6 95% error a 10 error
(arcmin) (arcmin) (kpc) (kpc)

(1) (2) (3) (4) (5) (6) (7)
A 21 —2.0 -0.5 130 254 -1.74 0.086
A 76 5.4 —5.4 289 632 —1.47 0.037
A 85 -6.3 -7.0 278 378 -l.67 0.182
A 88 -0.5 -1.7 121 130 -2.67 0.226
A 104 1.2 -0.4 168 223 —1.68 0.193
A 119 1.2 —0.4 92 144 —1.76 0.135
A 121 —2.8 —0.2 155 163 —1.94 0.104
A 147 —2.7 1.2 37 209 -1.63 0.097
A 151 1.0 0.3 149 170 -1.89 0.047
A 154 -0.8 —0.1 132 155 -2.01 0.048
A 166 -0.1 —0.1 37 218 —1.68 0.212
A 168 -1.1 1.9 201 237 -2.00 0.105
A 189 —29.3 7.7 170 192 -2.05 0.101
A 193 0.4 -1.1 38 272 -1.80 0.247
A 194 —1.0 4.8 94 332 -2.06 0.095
A 225 1.7 -3.7 70 177 —l.72 0.128
A 246 2.7 —5.2 317 718 -2.37 0.076
A 274 -3.9 —0.4 354 704 -1.12 0.106
A 277 -0.5 -1.3 82 91 —1.88 0.144
A 389 0.4 -0.4 295 311 —1.82 0.082
A 399 0.7 -0.7 252 501 —1.54 0.062
A 400 0.7 —2.2 91 100 -1.82 0.147
A 401 —0.2 1.6 85 584 -l.64 0.098
A 415 —2.8 4.6 136 146 —1.77 0.037
A 496 2.7 —7.0 119 374 -1.74 0.062
A 500 3.4 —0.3 238 282 —1.58 0.190
A 514 -6.8 —0.2 66 98 —2.06 0.165
A 634 —3.2 16.1 170 205 -2.52 0.123
A 671 -0.3 -1.0 62 70 -—2.10 0.128
A 779 -6.8 -9.9 485 627 -0.83 0.186
A 787 4.7 1.1 137 291 -1.72 0.046
A 957 5.1 0.4 157 181 -2.44 0.058
A 978 -0.3 1.6 55 70 -—2.16 0.215
A 993 -0.3 1.6 357 382 -3.11 0.120
A 1020 -0.3 10.3 112 168 -2.04 0.098
A 1035 -l.1 3.7 108 146 -1.63 0.124
A 1126 -0.2 —1.0 184 244 —1.95 0.089
A 1139 -9.4 6.7 172 515 —1.93 0.172
A 1185 0.6 1.7 78 89 —2.20 0.082
A 1187 —5.4 5.0 114 155 -1.96 0.357
A 1213 —0.4 —1.1 80 91 -2.12 0.092
A 1216 8.2 —3.6 31 175 -1.41 0.220
A 1228 -17.2 -2.5 286 334 —1.60 0.066
A 1238 1.2 —0.7 94 102 —2.10 0.088

 

TABLE 6.1. (continued)

194

 

 

Cluster 2: re 95% error a 10 error
(arcmin) (arcmin) (kpc) (kpc)

(1) (2) (3) (4) (5) (6) (7)
A 1254 -1.4 1.9 41 335 --1.52 0.196
A 1257 -2.5 —0.5 42 83 —-2.55 0.085
A 1291 —0.3 1.6 80 95 -1.84 0.147
A 1318 2.1 —2.1 678 1010 -1.62 0.060
A 1364 2.1 2.7 195 220 -2.16 0.074
A 1365 -0.2 -0.7 50 116 —2.11 0.145
A 1367 —2.4 —4.0 95 127 —1.61 0.103
A 1377 -2.3 0.3 97 654 —1.53 0.062
A 1382 —2.5 -1.8 324 446 -2.14 0.088
A 1383 0.3 1.4 34 700 -1.47 0.226
A 1399 0.9 2.8 360 394 -l.63 0.046
A 1412 —2.2 -3.5 176 200 -1.50 0.174
A 1436 4.0 3.9 221 273 -—l.30 0.159
A 1468 2.2 3.9 64 551 -2.07 0.210
A 1474 -1.1 4.6 346 470 -1.68 0.032
A 1496 1.2 —0.5 66 107 -1.99 0.248
A 1541 —2.5 -l.7 187 279 -1.50 0.165
A 1644 —1.9 8.0 161 336 -1.65 0.076
A 1651 -1.5 1.0 104 164 -1.78 0.193
A 1656 2.2 —2.2 113 146 -1.78 0.081
A 1691 3.6 0.2 320 432 -1.53 0.152
A 1749 2.0 -0.3 44 81 -2.09 0.118
A 1767 -0.7 —4.7 351 722 -0.87 0.179
A 1773 —2.0 —0.7 35 243 -1.68 0.206
A 1775 0.7 0.2 186 275 -1.66 0.126
A 1793 1.0 1.0 44 115 -2.12 0.164
A 1795 2.0 -2.0 127 178 —2.09 0.146
A 1809 5.9 -3.3 67 112 —1.73 0.082
A 1831 -0.2 -1.2 213 381 -1.35 0.087
A 1837 6.9 -l.4 197 396 -1.63 0.115
A 1904 —0.2 -4.6 217 227 -1.64 0.057
A 1913 4.8 6.1 51 161 -l.57 0.180
A 1927 —1.2 —2.6 123 623 -1.47 0.137
A 1983 —4.3 -3.5 169 213 —1.83 0.112
A 1991 —0.9 -0.9 71 527 —1.57 0.108
A 1999 —5.8 4.8 101 275 —1.88 0.046
A 2005 2.3 1.2 262 419 —2.10 0.051
A 2022 0.9 4.0 63 95 -2.11 0.181
A 2028 -0.2 0.7 166 191 —1.83 0.242
A 2029 0.7 —0.2 50 321 —1.51 0.151
A 2040 —0.4 1.1 34 353 -1.74 0.172
A 2048 —0.2 —0.5 132 266 -1.88 0.078
A 2052 0.5 0.5 30 584 -1.22 0.312
A 2061 —0.7 -0.7 365 519 —1.55 0.072
A 2063 0.5 -3.6 194 267 -1.54 0.267

 

195

TABLE 6.1. (continued)

 

 

Cluster a: re 95% error a 10 error
(arcmin) (arcmin) (kpc) (kpc)

(1) (2) (3) (4) (5) (6) (7)
A 2065 3.1 —0.7 365 582 —1.55 0.072
A 2067 —0.7 4.5 90 200 -1.84 0.069
A 2089 1.6 0.7 323 350 —2.12 0.068
A 2092 0.3 0.8 314 425 -2.08 0.090
A 2107 2.0 1.2 35 313 —l.72 0.296
A 2124 0.8 1.8 102 232 —1.86 0.103
A 2142 —4.8 —5.9 204 737 —1.58 0.072
A 2147 1.4 1.4 95 153 -1.81 0.211
A 2151 —0.5 0.5 241 314 —1.67 0.113
A 2152 9.7 9.7 144 589 -1.49 0.060
A 2162 3.8 0.5 65 138 —2.19 0.170
A 2175 -0.9 —1.2 354 400 -l.68 0.061
A 2197 -21.0 --13.1 321 402 —1.70 0.099
A 2199 —l.7 4.0 81 218 -l.95 0.102
A 2255 0.6 —1.9 287 495 —l.72 0.043
A 2256 -0.3 —4.3 213 320 ,-1.33 0.164
A 2328 0.1 -0.6 284 379 —1.80 0.065
A 2347 -1.9 -2.7 61 119 -2.24 0.379
A 2382 —1.6 0.2 273 405 —1.62 0.074
A 2384 —1.6 0.2 90 177 -1.93 0.155
A 2399 1.5 —0.3 50 64 -1.91 0.292
A 2410 1.9 1.1 186 374 —l.50 0.089
A 2457 2.6 0.9 37 389 —1.55 0.140
A 2634 -5.0 1.7 160 450 —1.93 0.098
A 2657 2.1 1.2 36 237 -1.65 0.254
A 2666 —5.8 1.9 259 781 —1.85 0.123
A 2670 —1.2 —0.7 161 230 -1.44 0.141
A 2675 —2.6 —5.9 57 244 —2.07 0.166
A 2700 0.9 -1.9 166 225 —1.83 0.104

 

 

196

Examination of the data in Table 6.1 indicates the presence of two outliers: A1318
with rc = 678 kpc, and A779 at rc = 485 kpc. In both of these cases, the reason for
such a large value is poor center specification. If, instead of using the maximum in the
adaptive—kernel map, the Abell center is used, these values drop to rc = 36 kpc, and
To = 57 kpc, respectively. This variation in core radius with center specification is a
well—known problem (Beers & Tonry 1986). In an attempt to minimize this variation,
it has become customary to seek a center which can be specified independently of the
galaxy distribution, such as the position of the X-ray peak or that of the cD galaxy.
However, these alternative center specifications do not solve the problem, but merely
provide other centers to try. For instance, in this study, if the position of the cD
galaxies are used as the center, approximately half of the core radii for these clusters
can be reduced, but several become significantly larger. The explanation for this

sensitivity appears to be the presence of small-scale structure in the cores of clusters.

As previously discussed (see Chapter 3) many of the clusters in this survey show
evidence of small-scale structure in their cores. In the case of A1656, this structure is
believed to be real because of the matching peaks in the the X-ray surface brightness.
A1656 has rc = 113 kpc listed in Table 6.1, with the center at (2’ .2, —2’ .2), obtained
from the maximum density of the 2 Mpc region. Comparison with Figure 3.11 reveals
that this center misses the density peak in the smaller map by about two arcminutes.
Choosing the position of this peak as the center, the core radius is less than 50 kpc.
A similar situation applies to A400. It has been argued in this thesis that the core
structure seen in this cluster is also likely to be real. Choosing the position of the
south-western peak as the center, the core radius falls from the 91 kpc quoted in
Table 6.1 to 30 kpc. If on the other hand, the position of the dumb-bell galaxy, which
lies between the peaks in Figure 3.12, is chosen instead, rc = 140 kpc. Many more

examples could be cited from the HGT sample clusters. In these cases however, the

197

reality of the structures is less certain.

The above discussion indicates two possible conclusions. If clusters really have
constant density cores, the core substructure observed in the adaptive-kernel maps can
not be real and must be due to contamination from background /foreground objects.
Hence it is difficult to estimate the size of these cores with confidence, due to the
sensitivity of the measurement to this contamination. Previous estimates of the core
radii of clusters are unlikely to be accurate. The more likely conclusion is that the
core substructure seen in the maps is, in many cases, real. When this substructure is
taken into account, the core radii of clusters drops to values that are consistent with
those measured for clusters containing arcs. The constant density cores observed for
clusters in the past as well as those seen in the profiles of Figure 6.1 are caused by

the failure to recognize the complexity of the cluster core.

The difference between the core radius measured from galaxies and that measured
from either X-ray gas or the gravitational lens observations, has been used to support
the claim that the dark matter in clusters is distributed differently than the galaxies.
While it may indeed be the case that the two distributions differ, the core radii
measured using the galaxy projected positions, either obtained by a parametric fit or
by the nonparametric procedure above, should not be used as evidence for this. The
reason is simply that the measurement errors are too large and have been consistently
under-estimated. While the bootstrap error listed above is an improvement over the
errors obtained using a parametric approach, it needs to be realized that this error
is only an error in the density estimate, and not in the actual density. Perhaps a
better estimate of the actual error would be to include in the bootstrap procedure
variations in smoothing parameter as well as variations in the position of the center.

Such refinements would be quite expensive computationally.

198

6.3.2 Power Law Profiles and Estimation of £20

A number of studies, both numerical (Crone et al. 1994) and observational (Beers
& Tonry 1986), have found that power law fits are a good approximation for the
density profiles of galaxy clusters. Furthermore, the slope of the power law may have
cosmological implications. Simple gravitational collapse can be shown to give rise to
an r‘4 profile (Gott 1975; Gunn 1977). Observations, on the other hand, indicate
shallower profiles of 1’2 (Beers & Tonry 1986). This discrepancy can be explained
by secondary infall of surrounding material or accretion of smaller groups. Clusters
which have recently accreted material will exhibit a flatter density profile than clusters
which have not recently accreted material. As discussed in the previous chapter, in a
high-density Universe clusters continue to accrete material in the present epoch. On
the other hand, in a low-density Universe the mass should be too spread out at the

present epoch for accretion of appreciable amounts of material.

The best fit power law is calculated for each of the clusters in the sample. A line
was fit to the log-log plots of the space density profiles using a robust least-squares
algorithm. Since the estimates for V are not truly linear over the entire range, the
slope was calculated from the core radius of each cluster out to a radius of one Mpc.
Unlike the estimated values of core radius, changes in the smoothing parameter had
little effect on the slopes. The values for each cluster are listed in column (6), with
the one-sigma error in column (7 ) of Table 6.1. The median slope of -1.8 :t 0.3 was
obtained. If an average constant-density background of 30% is assumed, then Monte-
Carlo simulations indicate that the profiles will be too shallow by 0.15:1:0.05. The
background corrected slope is therefore likely to be —1.9 :1: 0.3, in good agreement

with previous results (Beers & Tonry 1986).

Comparison with numerical simulations (Efstathiou et al. 1988; Crone et al. 1994;

199

Jing et al. 1995) indicates that this is most consistent with an (2 = 1 Universe with
a power-law initial perturbation spectra P(k) o< k’z. Because of the degeneracy
between the spectral—power index and {20, a lower-density universe with a power
index in the range of —-3 to —4 cannot be ruled out. However, such a steep power
law for the initial perturbation spectra is currently inconsistent with the observations
(Henry & Arnaud 1991, Fisher et al. 1993, Peacock & Dodds 1994, Feldman et al.
1994). On the other hand, the flattened profiles seen here and in other studies using

projected galaxy positions, may be due to incomplete background subtraction.

6.4 Conclusions

Despite the common conception of clusters containing constant-density cores in the
galaxy distribution, there is no real evidence to support this. As the core of a cluster
is approached, the number of galaxies available becomes smaller and the density
estimate is poorly constrained, as is clearly demonstrated by the bootstrap confidence
curves in Figure 6.1. It is invariably this poorly conStrained region that is identified as
a constant-density core using parametric techniques. Mis-specification of the cluster
center merely leads to less galaxies near the center, a larger poorly-constrained region,
and therefore a larger core radius. Thus, the core radii listed in Table 6.1 should be
viewed as an upper-limit which is likely to be set more by the sample statistics than

by a physical size of the core.

Chapter 7

CONCLUSIONS

7.1

This thesis calls into question the concept of clusters of galaxies being modeled as
isothermal spheres, Michie-King distributions, or with any other equilibrium model.
Clearly the idea that a galaxy cluster can be described by the two parameters of
core radius and velocity dispersion is an extreme over-simplification, and probably
not even a useful approximation to actual clusters. The growing body of evidence
suggests that clusters are still in the process of formation and accretion of material, as
evidenced by the large fraction of clusters with substructure and the flattened density
profiles, continues at the present epoch. While both of these can be taken as evidence

of a high cosmic density parameter ()0, possible alternative explanations exist.

The studies in this thesis have been based on the positions of galaxies in clusters.
However, the mass in luminous galaxies is only expected to make up about 1% of the
total cluster mass. Given this, it is quite possible that the galaxy positions have an
entirely different distribution from that of the total mass. If the gravitational lens
observations find steeper slopes are the norm, we would be forced to accept a low-
density, open cosmology. Although the results are still somewhat uncertain, studies

of several clusters with arcs indicate an isothermal profile away from the core, or

200

201

2, is a good approximation (Bonnet et al. 1994, Smail et al. 1992, Tyson &

p oc r”
Fisher 1995, Luppino & Kaiser 1997, Squires et al 1996), consistent with the results

found here.

Another possibility is the poorly-known effects of small-scale structure on the
collapse times of larger objects. If small-scale structure can delay the collapse of
galaxy clusters till the present epoch, then the substructure seen in clusters, as well
as the flattened density profiles, could be due to material that was accreted long ago

and just now collapsing.

7 .2 Future Work

The most obvious extension of the work contained in this thesis is to apply this type
of cluster analysis to the entire Abell catalog. The existence of the APS project, as
well as other similar projects such as the APM survey, beg for such an extension to
be carried out. The importance of a consistent set of contour maps is hard to over
estimate. Any researcher studying Abell 787, for instance, should be aware that the
Abell center appears to be near of low-density trough with two high-density peaks to

the east and west.

With the significant groups identified in the projected surface density, it remains
for projection effects to be eliminated with the gathering of redshifts. Although a K-S
test on the distribution of magnitudes can be used to estimate whether distributions
of the groups are radically different, as in this thesis, evidence exists that there is not
a universal luminosity function for galaxy clusters (Jones & Mazure 1996). It appears
some clusters are simply fainter than other clusters. Therefore, the classification of
groups as foreground/background in this thesis may be more misleading then than

informative, as is demonstrated in the case of A1837 and A1836.

202

For the clusters with X-ray observations, detailed comparison could be made be-
tween the X-ray maps and the adaptive-kernel maps. This may help clear up some of
the questions regarding background contamination, as well as indicate clusters that
have undergone recent mergers. Furthermore, a detailed comparison needs to be done
between the X-ray-derived mass profiles and the projected galaxy number-density

profiles where both profiles are calculated using a fully nonparametric technique.

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