Clusters of galaxies in large surveys: Identification and scaling laws.

(1)

Clusters of galaxies in large

Surveys: Identification and

scaling laws.

by

M.Sc. H´

ector Javier Ibarra Medel

INAOE

Thesis submitted as partial requirement for the degree

of

Ph.D in Astrophysics

at

Instituto Nacional de Astrof´ısica, ´

Optica y

Electr´

onica

February 2015

Tonantzintla, Puebla

Supervised by:

Dr. Omar L´

opez-Cruz

INAOE Researcher

Dr. Maritza Lara-L´

opez

UNAM Tenure Track

©

INAOE 2015

The author gives permission to INAOE to reproduce

and distribute copies in whole or in parts of this thesis

(2)

(3)

We characterize a sample of 91 Rich Abell Clusters using photometric and spectra measurements

from the Sloan Digital Sky Survey Data Release 7 (SDSS-DR7). We determine cluster

member-ships, cluster velocity dispersions, cluster scales, surface brightness, total masses, luminosities

and cluster luminosity functions. Also, we deal with theoretical concepts to estimate the cluster

galaxies peculiar velocities and determine the cluster caustic curves, cluster core radius, virial

radius and turn-around radius for all our cluster sample.

In addition, we identify the most massive galaxy clusters in the Galaxy and Mass Assembly

(GAMA) Survey. We develop a cluster finding technique that looks for 3D galaxy numerical

overdensities. We use the Delaunay Tessellation Field Estimator (DTFE), followed by a caustic

analysis to determine cluster masses. Additionally, the implementation of the caustic technique

allows a refinement in cluster membership and multiplicity. Therefore, we obtained an

homoge-neous sample of 93 massive clusters of galaxies from_Mcl= 3×1013−1×1015M⊙up to redshift

z _∼ 0.3. We find that the total cluster luminosity due to galaxies, as well as the luminosity

of the brightest cluster galaxy (BCG) correlate with cluster mass as: _Lcl∝ M

0.92+0−0..0832

cl ,Lbgc ∝

M0.35

+0.05

−0.25

cl , respectively. The errors in the power indexes reflect the uncertainties on the cluster

mass and luminosity estimates. Furthermore, we have generated the cluster-cluster correlation

function. Accounting for the effects of the selection function, we find a good agreement with

previous studies. Although we have covered a limited volume, we find that our optical cluster

finding technique is as effective, and covers a wider mass range, as thermal Sunyaev-Zel’dovich

Effect (tSZE) blind cluster searches.

Moreover, we explore the space defined by mass, surface brightness, size and velocity

dis-persion for virialized systems that allows a dimension reduction called the Fundamental Plane

(FP). Hence, we have generated the FP for 6,132 cluster galaxies, 93 GAMA clusters and 91

SDSS clusters as part of a comprehensive study to generate the mass versus luminosity relation

(MLR) from globular clusters to clusters of galaxies. This is the widest mass range exploration

of scaling laws that connect scales from pc. to Mpc. We have developed a suit of scripts and

programs to estimate the central galaxy stellar velocity dispersion, stellar masses, luminosity

functions and plane fits. In addition, we use the Petrosian radius as a non-parametric scale

indicator. This scale parameter overcomes the uncertainties of surface brightness profile fitting.

We find that the FP suffers from galactic non-homology effects that cause (at least for a

con-siderable part) the observed FP tilt. Nevertheless, these non-homology effects are not present

(4)

slightly tilt on the FP. Finally, we expand our scaling law analysis to low-mass galaxies, globular

clusters, galaxy groups and brighter cluster galaxies. Furthermore, we present an observational

characterization of the MLR. This relationship connects the theoretical halo mass distribution

function with the observational galaxy luminosity function. Hence, this study provides

obser-vational constrains to theoretical halo occupational models and therefore a way to explore the

(5)

Quiero agradecer a mis padres Eduardo Ibarra Martinez y Mar´ıa Ester Medel Ortega por

haberme apoyado en la realizaci´on de mis estudios, as´ı como por haberme brindado un ejemplo

a seguir. As´ı mismo quiero agradecer al Doctor Omar L´opez-Cruz por haberme guiado en el

camino de la investigaci´on cient´ıfica, y por haber sido mi asesor de forma directa o indirectamente

desde hace mas de diez a˜nos. Tambi´en quiero agradecer a mi co-asesora, la Doctora Maritza

Lara-L´opez por haberme apoyado durante mi estancia en el extranjero, y por sus valiosos

con-sejos durante mi formación como investigador. Quiero también agradecer a los doctores Raúl

Mujica, Ivanio Puerari, Roberto Alejandro Ruelas, Elsa Recillas y Miguel ´Angel Arag´on Calvo

por haber aceptado leer mi voluminosa tesis, y por sus valiosos comentarios y correcciones.

Agradezco tambi´en al Consejo Nacional de Ciencia y Tecnolog´ıa (CONACyT) por haberme

brindado el soporte econ´omico a lo largo de mis estudios de doctorado, y al Consejo de

Cien-cia y Tecnolog´ıa del Estado de Puebla (Concytep) por haberme apoyado econ´omicamente para

poder finalizar mi doctorado. As´ı mismo agradezco a mis amigos y familiares como mi hermano

Eduardo, mi hermana Diana, a Giannina que fue mi compa˜nera de cubo a lo largo de toda mi

estancia en el INAOE y la cual me tuvo que soportar, a Ricardo Ch´avez por las discusiones que

tuvimos, a Emmaly Aguilar, David S´anchez, a Chava, a Aline (a.k.a Alelin), a V´ıctor Pati˜no y

Karla, a V´ıctor Gomez, V´ıctor Mauricio, al Marco, Anaely, Vital, Milagros, Ana Torres, Jos´e

Manuel, Jos´e Miguel, y todos los del lab mm que me permitieron realizar algunas piezas en su

torno. Al departamento de formaci´on acad´emica, al departamento de Astrof´ısica y a todo el

INAOE en general por haberme brindado todos los apoyos tanto econ´omicos como acad´emicos

a lo largo de toda mi formaci´on, al Australian Astronomical Observatory por haberme apoyado

en mi estancia en Australia, a todos los que probablemente me falto mencionar, al amigo perro

(6)

(7)

Symbol

Definition

or Abbreviation

M Mass of an object

M⋆ Stellar Mass of an object

M⊙ Solar Mass

Md Dynamical Mass or Viriral Mass

MKv,d Non-homology corrected Dynamical Mass or Viriral Mass

Kv Non-homology effect factor

L Light of an object

Lx Light of an object observed at the xband

L⊙ Solar Luminosity

R Scale radius of an object

σ Velocity Dispersion

η Logarithmic slope of the MLR

µ Velocity dispersion logarithmic slope of theL −σ−Rplane

ν Scale logarithmic slope of theL −σ−R plane

a Velocity dispersion logarithmic slope of the FP

ab SB logarithmic slope of the FP

Rη Petrosian radius of an object

Rr Statistical virial radius of an object

Φ(_M) Mass Function

Φh(M) Halo Mass Function

Φsb(M|Mc) Sub-Halo Mass Function

Φ(_L) Luminosity Function

Φ(L|Mc) Conditional Luminosity Function

Φs(L|Mc) Satellite Luminosity Function

FP Fundamental Plane

CFP Cluster Fundamental Plane

SB Surface Brightness

(8)

LF Luminosity Function

MLR Mass to Light Relationship

TMLR Total Mass to Light Relationship

HMLR Halo Mass to Light Relationship

SMLR Stellar Mass to Light Relationship

SHMR Stellar to Halo Mass Relationship

CMR Colour Magnitude Relation

(9)

Chapter 1 Introduction

During the XIX century it was thought that nebula-type objects were located within the Milky

Way, and those with spiral forms apparently had an homogeneous distribution through the

sky. In the advent of the XX century the astronomer H.D Curtis made an spectroscopic study

of these spiral nebulae. He concluded that they were made by stars that cannot be resolved

because these nebulae were far away. Nevertheless, this idea was not immediately accepted

by the contemporary astronomers until Edwin Hubble measured the distance to the nebulae

in 1922. He used the period of cepheid stars within the nebulae and measured their distance

modulus. These nebulae were renamed as galaxies and have shown to have three fundamental

structures: a disc, a bulge, and a halo. The galaxy bulge has a spheroidal shape and contains

old stars that have random orbits around the galaxy center. Within the bulge, there is a low

content of gas, dust, and low star formation. The disc is formed by young stars and has a higher

content of gas (Hydrogen, Helium and Lithium), dust, and molecular clouds that feed the star

formation activity. The galaxy halo consists in gas and dark matter that extends far away to

the visible limits of the galaxy main body. Photometric studies show that the galaxy surface

brightness (SB) profiles can be modelled in two different ways, while the bulge follows a de

Vaucouleurs profile, the disk follows an exponential one.

Furthermore, Hubble not only demonstrated that galaxies are external Milky Way objects,

but he also proposed the first galaxy classification scheme based in their morphologies. The

classification system consists in three principal classes: the elliptical galaxies (E), the spiral

and barred spiral galaxies (S and SB) and the lenticular galaxies (S0 and SB0). The E types

are located in the beginning of the classification and have the name of early-types. Then, the

classification is divided in two paths: the barred spirals and non barred spirals both with the

(14)

Figure 1.1: Hubble Classification or Hubble Tuning Fork diagram. Image taken of the Lecture 11 of the

Astronomy course of the Oregon University: http://abyss.uoregon.edu/ js/ast123/lectures/lec11.html.

defines the lenticular galaxies. We show this diagram in Figure 1.1.

The early-type galaxies have a single spheroidal structure that is dominated by old stellar

populations with random orbits alike the bulges of spiral galaxies. Also, these galaxies have

an extended dark matter halo and contain a very little content of gas and dust because their

star formation was stopped at early times (Binney & Merrifield, 1998). Their sky projection

shape has an elliptical form withae andbe as its semi major and minor projected axis. These

galaxies are sub-classified as a function of their projected eccentricity asn= 10(1₋be/ae), and

determines the sequence of E0 _→E1 _→ E2 ... E7. These galaxies are common in over-dense

regions as galaxy clusters with a relative abundance of 40% with respect of late-type galaxies.

On the other hand, in low dense regions like the field their relative abundance tends to 10%

(Binney & Merrifield, 1998). Also, the most luminous elliptical galaxies (and hence the most

massive) are classified as cD galaxies that are observed in the central regions of clusters of

galaxies.

The lenticular galaxies (S0 and SB0) have a large bulge with small flat discs. Likewise

elliptical galaxies, the lenticular galaxies have a few content of gas and young stellar populations

but have a more dust contribution. Their disks can have a barred structure without a spiral

(15)

galaxies (10%). Also, the lenticular galaxies are divided in two groups: the barred and non

barred lenticular or S0 and SB0. The S0 galaxies are subdivided in three classes: S01, S02 and

S03. This classification depends on the presence of dust absorption within their disk component

and varies from S01, without dust content, to S03, with a considerable dust absorption. On the

other hand, barred lenticular galaxies are subdivided in three classes: SB01, SB02 and SB03. In

this case the classification is based in the prominence of the bar instead of the dust absorption.

The late-type galaxies have a bulge centred in a thin flat disc. The disk contain a spiral

structure with dust, gas and young stars (S). Sometimes a bar is present within the disk (SB),

in which the spiral arms starts at the edges of the bar. The late-type galaxies have a relative

abundance in the field of 80%, in contrast with their relative abundance within galaxy clusters

that is about 10%. The barred and non barred galaxies are sub-classified in 3 subtypes (a,b,c)

according to the total disk to bulge light ratio, the tight of the spiral arms and the amount of

resolved HII regions. The Sa or SBa galaxies have a predominant bulge luminosity with tight

spiral arms. In contrasts, the Sc or SBc galaxies have a small and compact bulge with loosely

tight spiral arms and with highly resolved HII regions (Binney & Merrifield, 1998).

On the other hand, we observe that galaxies are not homogeneously distributed in the

Uni-verse and are concentrated in high density regions know as galaxy clusters. For classification,

when a cluster richness is below of 50 members the cluster is considered as a galaxy group.

Galaxy groups are mostly dominated by spiral galaxies which are located within a scale of one

Mpc in radius (Mart´ınez & Saar, 2002). As an example, the Milky Way is one of the three

dominant galaxies of a galaxy group knows as the Local Group. When a group has more than

50 galaxies, it is considered as a galaxy cluster. The clusters do not have an upper richness limit

and have scales larger than 4 Mpc in radius with a variety of shapes, from spherical to irregular

forms.

There are two principal ideas try to explain the cluster formation, a monolithic collapse,

and a hierarchical merging process. The monolithic collapse explains the cluster formation as

a collapse from a primordial density perturbation. During the inflationary epoch, the matter

was almost homogeneously distributed in the Universe with small overdensiy and underdensity

regions. An underdensity region has a local density that is below of the average density of

the Universe. On the other hand, an overdensity region has a local density larger than the

average density. When overdensity regions collapse, the primordial gas begins to form the first

stars. These stars begin to clump into the first galaxies which at the same time form the galaxy

clusters. On the other hand, the hierarchical formation indicates that the primordial density

(16)

Figure 1.2: Example of the formation of the Cosmic Web. The panels from left to right and from top

to bottom illustrate a galaxy cluster formation at 0.25, 1, 4.7 and13.6 Gyr from the Big-Bang. The

panels were taken from the millennium simulation that uses10billion particles (Springel et al., 2005).

first stars begin to form. Hence, these primordial structures had to have merged following a

hierarchical assembly to form the present day galaxies. Also, these first galaxies begin to form

small galaxy groups by their gravity interaction, and therefore, the formation of galaxy clusters

consists in the assembly of these galaxy groups (Padmanabhan, 1993).

Contrary to the overdensity regions, underdensity regions cannot gravitationally collapse

since their total masses are not enough to counteract the expansion of the universe. Therefore,

these regions form expanding voids. Consequently, the initial homogeneous matter distribution

starts to have regions where galaxies begin to separate from each other like bubbles. At the same

time, this matter distribution have other regions where galaxies start to clump into clusters of

galaxies or galaxy groups. Hence, the distribution of galaxies start to form a large scale web-like

structure known as theCosmic Web, where their nodes are the galaxy clusters (van de Weygaert

& Bond, 2008). We show an example of the Cosmic Web and their evolution through the time

(17)

of these large scale structures and the galaxies that conform them.

Before the advent of CCDs, the largest cluster catalogues were those of Zwicky and Abell,

created with the aim of classifying these clusters. Both, Zwicky and Abell, selected clusters by

eyeball examination directly on photographic plates (Abell, 1958) of the Palomar Observatory

Sky Survey. Abell examined 879 pairs of photographic plates taken with the 48-inch Palomar

Schmidt Telescope. Often, this selection method has been considered inaccurate, affected by

galaxy superpositions, and hampered by severe incompleteness. However, it is remarkable that

Abell did not miss any rich cluster in the volume enclosed by 0.02_≤z_≤0.1. At redshifts higher

than z = 0.1, problems with star-galaxy separation cause problems with cluster identification

and richness estimation.

During the last decades, we have seen an increase in the use of computing in many branches

of science; astrophysics is not an exception. In the advent of large galaxy surveys, it has been

essential the use of computing for data reduction, management and visualization. Likewise, the

development of computational astrophysics has been able to simulate the universe in which we

live. These simulations use the physical theories of thousands of particles that require the use

of high performance computing. Hence, the processing and visualization of data becomes an

important task. Without these techniques, the use of the available astronomical information

would be impossible. High resolution N-body simulations (Springel et al., 2005) and large

red-shift surveys have made possible to implement more complex algorithms for cluster membership

and mass estimation (e.g., Smith et al., 2012).

These algorithms are based on the three dimensional space galaxy distribution, which allows

the treatment of substructure contamination. One of the most popular ones is the

Friends-of-Friends method (FoF, e.g., Robotham et al., 2011), which consists on linking the galaxy-galaxy

separation as a tracer of the local density. Other alternatives include the application of stochastic

geometry techniques, the most popular approaches are the Voronoi tessellation or its dual, the

Delaunay tessellation (e.g., Schaap, 2007). These tessellation methods can be applied to the

spatial galaxy distribution to trace clusters of galaxies as large overdensity peaks from the

expected background (e.g., S¨ochting et al., 2006; Gonz´alez & Theuns, 2009; Soares-Santos et

al., 2011). In addition, the Voronoi tessellations allow the study of large scale structures (LSS)

and the study of the three dimensional morphology of clusters (Arag´on-Calvo et al., 2010a,b;

Platen et al., 2011). In fact, many studies in the literature mix the advantages of the cluster

overdensity detection with a FoF, sigma clipping, or color-magnitude algorithms (Einasto et al.,

(18)

Hence, with the advent of these new techniques, we can explore the physical properties of

clusters of galaxies. One of these physical properties are the internal kinematics that govern

the evolution of clusters and their galaxy members. The internal kinematic in clusters produces

some important scaling laws (SL) governed mainly by the virial theorem (VT). The VT on

its original formulation states: the vis viva of the system equal to its virial. Which in modern

language reads: in a self gravitation system of particles, the mean kinetic energy is equal to

one-half of the mean potential energy (e.g., Collins, 1978; Cox, 1968; Chandrasekhar, 1967;

Binney & Tremaine, 1987, for derivations and extensions). A property that is hardly mentioned

is that Virialization is scale-invariant (Schroeder, 1991). This property follows by considering a

system with a homogeneous potential energy function. Therefore, the equations of motion are

unaltered by the scale (self-similarity). Then, we can consider that virialized cosmic structures

show similar properties regardless of the scale.

Overview of this thesis

The principal objective of this thesis is the analysis of scaling laws for different systems, from

stellar to galaxy clusters. With this aim, we deal with the detection and characterization of

galaxy clusters, galaxy groups, galaxies and stellar systems in general for the seek to explore

their scaling relationships. Hence, our work comprehends a variate range of topics, that goes

from pure theoretical concepts to practical observations for which we use standard and

non-standard methodologies. Hence, we implement tools to reduce, analyse, select and visualize

galaxies and galaxy clusters within the Sloan Digital Sky Survey (SDSS York et al., 2000)

and the Galaxy And Mass Assembly (GAMA Driver et al., 2009, 2011) Survey. Therefore, we

divide our principal objective into several individual objectives that connects with each other

to solve the general objective. Then, for each chapter on this thesis, we deal with an specific

problem, develop a proper methodology or methodologies to study it and present the results of

this analysis. The problems that each chapter study are the following:

• In Chapter 2 we characterize a sample of 91 Rich Abell Clusters by using data from

the SDSS Data Release 7. We deal with the definition of galaxy cluster memberships,

the estimation of the cluster velocity dispersion, clusters scales, surface brightness, total

masses, cluster luminosities and luminosity functions. Also, we deal with a theoretical an

spherical infall collapse model to determine the peculiar velocities within clusters. With

this model, we determine the cluster caustics curves in redshift space and the cluster core,

the viral radius, and the turn-around radius.

(19)

of the Delaunay Tessellation Field Estimator (DTFE). We also perform a caustic analysis

to determine cluster membership and mass. In this Chapter, we obtain 93 massive clusters

of galaxies complete up to redshiftz_∼0.3. We find that the total cluster luminosity due

to galaxies and the luminosity of the brightest cluster galaxy correlate with cluster mass.

Moreover, we have generated the cluster-cluster correlation function taking into account

the effects of the selection function.

• In Chapter 4, we explore the scaling relationships of virialized systems like the

Fundamen-tal Plane (FP). We explore these scaling relationships for 6,132 cluster galaxies, 93 GAMA

clusters and 91 SDSS clusters. Hence, we have developed a suit of scripts and programs

to estimate the galaxies stellar velocity dispersions, stellar masses, luminosity functions

and plane fits. Moreover, we expand our scaling law study to low-mass galaxies, globular

clusters, galaxy groups and brighter cluster galaxies. Hence, we present an observational

characterization of the mass versus luminosity relationship that connects the theoretical

halo mass distribution function with the observational galaxy luminosity function.

In summary, Chapter 2 contains a description of the standard methodologies and theoretical

definitions that we use in the following two chapters. Thus, the Chapter 3 and 4 are the

most important chapters of the whole thesis, and contains new methodologies and scientific

contributions. Chapter 5 summarises all the results found in this thesis. Finally, Appendixes

(20)

(21)

Chapter 2 Characterization of Rich Abell

Clusters using SDSS data.

2.1 Introduction

Clusters of galaxies are collapsed objects; therefore, their galaxies do not follow the Hubble flow.

In consequence, an observed cluster would have a disturbed estimation of its redshift-distance.

If we plot a cluster in a redshift space, we can observe the effects of the cluster collapse. Hence,

a spherical cluster acquires a prolate shape know as the finger-of-god feature; the velocity of an

infall galaxy introduces an additional component to the Hubble flow (Hamilton, 1998). Then,

we can model the galaxy infall velocity as a perturbation of the Hubble flow as:

Vob=H0R0±Vpec(r).

WhereR0 is the distance of galaxy to the cluster center andVpec is a peculiar velocity. We

model the peculiar velocity as an additional velocity component to the Hubble flow and depends

on the gravitational potential of the cluster. The galaxies that are within the gravitational

influence of the cluster falls at the center of the cluster. Then, the galaxies that lie at the back

of the cluster in the observed line-of-sight, have a negative velocity contribution with respect

of the Hubble flow. On the other hand, the galaxies that lie at the front of the cluster in the

line-of-sight have a positive contribution. This effect produces the form of the finger-of-god for

the observed clusters in redshift space.

Some authors (e.g, Gunn & Gott, 1972; Silk, 1974; Peebles, 1976) tried to modelled the

peculiar velocity as a perturbation of the Hubble flow. The basic assumption is that we can model

the perturbed Hubble flow as a function of the density field. This perturbation changes the local

(22)

be found in Plionis et al. (2008), moreover van Haarlem (1992) considered restricted numerical

simulations and the effects of triaxiality.

For the aim to study and classify clusters of galaxies, Abell (1958) created a cluster catalog

by examining 879 pairs of photographic plates from the Palomar Observatory Sky Survey taken

with the 48-inch Palomar Schmidt Telescope. Abell analysed a field of 27◦_{of the north sky and}

selects the regions where the galaxy surface number density was larger than the field. Then, he

defines four selection criteria, the Abell Richness Class (ARC), the distance, the compactness

and the galactic latitude (Saslaw, 2000). The ARC is the number of galaxies whose apparent

magnitudes arem≤m3+ 2 and is classified in 6 classes (Saslaw, 2000):

• The ARC of 0 that contains 30-49 galaxies.

• The ARC of 5 that contains more than 300 galaxies.

The distance criterion was set to covers the range of 60 to 600 Mpc due to the magnitude

limit (<20 magnitudes) of the Palomar catalog. The compactness established that all cluster

galaxy members must lie within a 1·5h−1Mpc to the cluster center. Also, Abell only selects the

numerical overdensity regions that have a galactic latitude high enough to avoid the contribution

of the Milky Way dust extinction (Mart´ınez & Saar, 2002). The Abell catalog contains 2,712

rich clusters, and only 1,682 clusters carry out the four membership criteria (Saslaw, 2000).

Now, with the advent of new technologies, the Sloan Digital Sky Survey (SDSS, York et

al., 2000) becomes the largest digital sky survey, ever done. SDSS facilities are located at the

Apache Point Observatory (APO), in New Mexico with an average seeing of 1.4′′. The main

SDSS telescope is af /7 Ritchey-Chr`etien 2.5 m telescope that provides a 3◦ field-of-view. To

date, it has released data for 357 million objects within 11,663 square degrees employing

charge-coupled devices (CCDs) as detectors. The CCD array consists in 30 SITEe/Tektronix detectors

with 2048_×2048 pixels of 24µm, arranged in six columns and five rows. Each row corresponds

to a specific band of the SDSS photometric system. The SDSS adopted a modified version of the

Gunn (1978) photometric system for its photometric catalog. Thuan & Gunn (1976) optimized

this system for extragalactic studies by picking bands that avoided the strongest night-sky lines.

(23)

sensitivity limit of the CCD. The five SDSS bands: u, g, r, i, z (final photometric calibration)

have their centres at 3,540, 4,770, 6,230, 7,630 and 9,130˚Arespectively.

Also, the SDSS team developed pipelines for data reduction and photometric analysis

(Lup-ton et al., 2002), and the point-spread function (PSF) was generated by using bright field stars.

The SDSS uses various types of flux evaluation, one of these is a 7.4′′_{arc sec}_{aperture corrected}

for PSF variations called PSF magnitude (mpsf). Also, for extended objects the SDSS provides

a model magnitude (mmodel) estimation by fitting a de Vaucouleurs or an exponential profile to

their targets. Moreover, the SDSS evaluates the fiber magnitude (mf iber) by using a 3′′ arc sec

aperture that has the same size as its spectroscopic fiber (Blanton et al., 2001). They identify

this magnitude as part of their selection criterion for spectroscopic targets. Finally, the SDSS

gives a revised Petrosian magnitude by adopting the original definition of Petrosian (1976).

On the other hand, the SDSS team optimised its spectroscopic target selection criterion to

study the large-scale structure (Strauss et al., 2002). This sample has a magnitude limit of

mr= 17·77 Petrosian magnitudes. They set this limit to achieve a complete magnitude catalog

at the dimmest magnitude available for their spectrograph to obtain a high signal-to-noise

spectrum. This magnitude completeness grants a spectroscopic galaxy survey with a redshift

deep of z _∼ 0.1 or a volume complete survey up to z _∼ 0.07. Hence, the SDSS became the

deepest optic galactic survey at the moment of its implementation (1999-2010).

Therefore, in this thesis we used the SDSS spectroscopic catalog to characterize rich clusters

of galaxies. For this purpose, we developed a suit of scripts to look for cluster membership

by taking advantage of the wealth of data from the SDSS. We implement the selection of

clus-ter members and inclus-terloper rejection by using two approaches. The first is a straightforward

procedure that uses the Colour-Magnitude Relation (CMR); whereas the second method is an

iterative process based on an earlier scheme developed by Yahil & Vidal (1977). Once we apply

the membership criteria, we calculate all the cluster masses (M), their total luminosity (L),

sizes (R) and kinematics (σ).

We divide this Chapter is divided in eight Sections. In Section 2.2 we define the cluster

sample. In Section2.3 we define the galaxy cluster membership criteria. In Section 2.4 we

define the geometrical cluster parameters. In Section 2.5 we estimate the cluster dynamical

masses. In Section 2.6 we estimate the cluster total luminosities. In Section 2.7 we analytically

model the cluster peculiar velocity and in Section 2.8 we use the cluster collapse to estimate the

cluster core,R∆and turn-around radius. Finally, in Section 2.9, we provide a summary of this

work. The cosmology that we use during this work is: Ωm = 0·27±0·03, ΩΛ = 0·73±0·021,

(24)

210 240 270 300 330 30 60 90 120 150

-60 -30 0 +30 +60

Figure 2.1: Sky distribution (in galactic coordinates) of the 125 Abell Clusters selected in this study

(marked in red). In the North Galactic hemisphere and half of the South Galactic hemisphere. The

clusters are distributed almost homogeneously over the North Galactic hemisphere.

2.2 The Sample

For this thesis, we select a sample of 125 rich Abell Clusters shown in Table 2.2. In Figure 2.1

we depict our cluster sample spatial distribution that we select under the following selection

criteria:

1. The cluster sample must lie within Abell’s (1958) statistical sample.

2. Clusters should have an Abell Richness Class (ARC) greater than 0.

3. Clusters should have a redshift between 0·02 ≤ z ≤ 0·2.

We select clusters with an ARC higher than zero in order to have a fair number of galaxies per

cluster. Also, we use the redshift range of 0·02 ≤ z ≤ 0·2 to sample deep enough inside

the cluster’s luminosity function within the survey magnitude limit. Our final sample includes

125 Abell Clusters, 28 of them are in common with The Low-Redshift Cluster Optical Survey

(LOCOS, Lopez-Cruz 2001).LOCOSwas originally intended to measure optical properties of

(25)

0.00

0.05

0.10

0.15

0.20

0

5

10

15 Cluster Redshift Distribution

Nt=125, z

0

=0.074,

σ

=0.0315

N

Cluster Redshift

Figure 2.2: Redshift distribution of the selected clusters for this thesis. The mode of the redshift

distribution isz∼0·074.

clusters have reported X-Ray counterparts. The Figure 2.2 shows the redshift distribution of

the selected clusters.

We select all galaxies within the 30′ _{of the reported cluster center. From the SDSS query}

server page1_{, we download their redshifts, model and Petrosian magnitudes, Petrosian radii,}

velocity dispersions, positions and their associated errors. For this study, we use 19,878 galaxies,

6,132 of those galaxies were selected as members of the final sample of 125 Abell clusters of

galaxies.

2.3 Cluster Membership

Early-type galaxies define a remarkably tight colour magnitude relation (CMR). One attribute

of the CMR is the progressive redness of cluster galaxies as a function of their brightness. Yoshii

(26)

Name z α δ Name z α δ

(1) (2) (3) (4) (1) (2) (3) (4)

A0085 0.0559 00:41:50.4 −09 : 21 : 00.5 A0168 0.0454 01:15:01.7 +00 : 17 : 16.0

A0257 0.0704 01:48:44.7 +14 : 01 : 46.0 A0279 0.0799 01:56:13.3 +01 : 03 : 30.0

A0634 0.0269 08:14:44.8 +58 : 06 : 47.5 A0646 0.126 08:22:48.8 +47 : 07 : 35.1

A0671 0.0500 08:28:44.0 +30 : 27 : 25.3 A0680 0.122 08:35:23.3 +36 : 47 : 43.4

A0688 0.121 08:37:01.0 +15 : 46 : 46.8 A0690 0.0806 08:39:36.4 +28 : 49 : 03.9

A0695 0.0895 08:41:10.5 +32 : 11 : 48.6 A0700 0.114 08:46:01.7 +37 : 05 : 29.5

A0779 0.0228 09:19:49.9 +33 : 45 : 30.9 A0924 0.142 10:06:44.7 +35 : 37 : 29.3

A0957 0.0985 10:13:52.9 +00 : 58 : 18.8 A0991 0.123 10:21:54.0 +18 : 58 : 28.6

A0999 0.0322 10:23:19.6 +12 : 56 : 25.0 A1020 0.109 10:27:45.8 +10 : 33 : 10.2

A1035 0.0680 10:31:46.9 +40 : 10 : 51.3 A1126 0.0841 10:54:09.0 +16 : 52 : 49.0

A1142 0.0355 11:00:59.5 +10 : 28 : 05.5 A1169 0.0589 11:08:19.2 +44 : 05 : 04.7

A1185 0.0327 11:10:45.3 +28 : 35 : 11.2 A1187 0.0758 11:12:03.4 +39 : 36 : 32.4

A1190 0.0752 11:11:42.0 +40 : 46 : 33.8 A1213 0.0470 11:16:24.1 +29 : 17 : 22.6

A1218 0.0939 11:18:06.6 +51 : 40 : 43.6 A1228 0.0394 11:22:04.7 +34 : 13 : 11.8

A1238 0.0740 11:23:03.2 +01 : 01 : 02.1 A1291a 0.0514 11:32:18.5 +56 : 06 : 44.7

A1314 0.0334 11:34:47.2 +49 : 04 : 35.9 A1318 0.0567 11:36:22.4 +55 : 02 : 27.1

A1337 0.105 11:40:13.8 +10 : 15 : 18.1 A1346 0.0983 11:41:14.5 +05 : 41 : 24.3

A1356 0.117 11:42:03.1 +10 : 25 : 49.3 A1367 0.0217 11:44:32.5 +19 : 50 : 00.1

A1377 0.0518 11:47:08.5 +55 : 42 : 53.8 A1383 0.0595 11:48:26.1 +54 : 39 : 04.5

A1385 0.0839 11:48:23.7 +11 : 25 : 48.9 A1387 0.130 11:48:53.5 +51 : 34 : 36.0

A1413 0.141 11:55:15.4 +23 : 24 : 59.5 A1424 0.0755 11:57:21.3 +05 : 04 : 27.3

A1436 0.0651 12:00:22.6 +56 : 13 : 52.9 A1468 0.0873 12:06:08.4 +51 : 28 : 38.7

A1474 0.0806 12:08:15.1 +14 : 57 : 22.6 A1496 0.0961 12:13:45.9 +59 : 14 : 12.9

A1534 0.0700 12:24:52.2 +61 : 29 : 59.5 A1538 0.135 12:26:00.6 +56 : 51 : 19.5

A1539 0.105 12:26:28.4 +62 : 37 : 30.7 A1541 0.0895 12:27:40.9 +08 : 51 : 07.2

A1542 0.119 12:27:24.2 +49 : 30 : 19.9 A1543 0.128 12:27:45.4 +30 : 22 : 54.4

A1544 0.105 12:25:45.1 +63 : 26 : 26.9 A1545 0.0951 12:28:24.7 +47 : 21 : 03.1

A1547 0.115 12:28:25.2 +26 : 49 : 36.5 A1548 0.162 12:28:34.2 +19 : 26 : 51.2

A1549 0.0623 12:30:37.0 +28 : 52 : 19.3 A1550 0.173 12:28:44.5 +47 : 43 : 22.2

A1552 0.0864 12:29:46.3 +11 : 45 : 03.7 A1569 0.0696 12:36:06.2 +16 : 33 : 33.7

A1609 0.0862 12:46:18.1 +26 : 27 : 43.0 A1630 0.0653 12:51:51.4 +04 : 33 : 27.6

A1650 0.0835 12:58:50.6 −01 : 43 : 54.6 A1656 0.0232 12:59:48.8 +27 : 58 : 20.4

A1663 0.0831 13:02:40.9 −02 : 32 : 17.2 A1689 0.0841 13:12:02.2 −01 : 13 : 15.0

A1691 0.0725 13:11:20.0 +39 : 13 : 00.3 A1750 0.0859 13:31:00.7 −01 : 50 : 35.7

A1767 0.0707 13:35:42.9 +59 : 12 : 41.1 A1773 0.0772 13:42:17.7 +02 : 15 : 18.1

A1775 0.0752 13:41:50.5 +26 : 23 : 43.7 A1793 0.0828 13:48:26.8 +32 : 14 : 45.5

A1795 0.0630 13:49:00.2 +26 : 37 : 23.4 A1809 0.0793 13:53:09.5 +05 : 12 : 02.8

A1827 0.0653 13:57:54.9 +21 : 40 : 52.8 A1831 0.0631 13:59:02.1 +28 : 02 : 36.7

A1870 0.112 14:10:42.7 +06 : 36 : 40.7 A1904 0.0718 14:22:07.0 +48 : 30 : 19.0

A1913 0.0530 14:26:59.8 +16 : 42 : 50.9 A1927 0.0951 14:31:01.1 +25 : 41 : 39.3

A1939 0.0884 14:36:57.3 +24 : 45 : 49.2 A1983 0.0451 14:52:48.3 +16 : 48 : 53.7

A1991 0.0583 14:54:32.0 +18 : 37 : 24.3 A1995 0.0918 14:53:10.1 +58 : 02 : 39.3

A2009 0.0620 15:01:28.9 +21 : 16 : 20.2 A2021 0.183 15:03:42.6 +23 : 07 : 05.1

A2022 0.0581 15:04:33.8 +28 : 26 : 57.8 A2028 0.0773 15:09:28.9 +07 : 35 : 44.2

A2029 0.0781 15:10:54.3 +05 : 47 : 32.9 A2040 0.0453 15:12:27.6 +07 : 24 : 50.5

A2048 0.0979 15:15:09.8 +04 : 22 : 50.0 A2052 0.0351 15:16:50.7 +07 : 03 : 03.4

A2056 0.118 15:19:37.7 +28 : 16 : 45.5 A2061 0.0775 15:21:23.6 +30 : 39 : 52.3

A2063 0.0344 15:23:00.1 +08 : 36 : 59.6 A2065 0.0724 15:22:43.0 +27 : 42 : 43.1

A2067 0.0739 15:23:17.4 +31 : 00 : 36.1 A2079 0.0658 15:27:40.4 +28 : 50 : 36.0

A2089 0.0736 15:32:25.9 +28 : 03 : 51.9 A2092 0.0667 15:33:33.4 +31 : 06 : 31.8

A2097 0.0653 15:35:46.8 +39 : 42 : 08.8 A2100 0.152 15:35:49.6 +37 : 31 : 53.1

A2107 0.0414 15:39:46.2 +21 : 43 : 24.9 A2110 0.0974 15:39:25.1 +30 : 44 : 19.0

A2124 0.0661 15:44:53.1 +36 : 05 : 52.7 A2142 0.0900 15:58:13.1 +27 : 15 : 14.8

A2147 0.0364 16:02:19.0 +15 : 57 : 01.9 A2149 0.0650 16:01:00.3 +53 : 57 : 58.1

A2151 0.0364 16:05:27.1 +17 : 48 : 59.1 A2152 0.0444 16:05:12.0 +16 : 26 : 36.8

A2169 0.0578 16:13:59.8 +49 : 08 : 18.8 A2175 0.0963 16:20:38.8 +29 : 53 : 08.0

Table 2.1: Redshift and positions of the initial 125 Abell Clusters sample. Column 1 is the Abell number, column 2 is the redshift of the clusters; columns 3 and 4 are the positions centers of the clusters, αis the right ascension in hours and δ is the declination in degrees in sexagecimal format, the equinox is J2000.

(27)

Name z α δ Name z α δ

(1) (2) (3) (4) (1) (2) (3) (4)

A2178 0.0968 16:20:53.7 +24 : 40 : 01.1 A2197 0.0308 16:28:39.8 +40 : 52 : 29.3

A2199 0.0305 16:28:38.9 +39 : 30 : 51.4 A2244 0.0971 17:02:51.2 +33 : 59 : 48.1

A2255 0.0802 17:13:00.1 +64 : 02 : 35.4 A2356 0.120 21:35:21.5 +00 : 13 : 51.8

A2373 0.123 21:44:54.3 +00 : 54 : 53.3 A2399 0.0580 21:57:28.9 −07 : 46 : 40.1

A2506 0.128 22:57:05.8 +13 : 18 : 05.4 A2593 0.0419 23:24:28.6 +14 : 37 : 41.0

A2670 0.0762 23:54:09.1 −10 : 24 : 03.3 A2703 0.116 00:05:22.3 +15 : 58 : 06.6

A2705 0.116 00:05:39.0 +15 : 55 : 42.6

Table 2.2: Redshift and positions of the initial 125 Abell Clusters sample. Column 1 is the Abell number, column 2 is the redshift of the clusters; columns 3 and 4 are the positions centers of the clusters, αis the right ascension in hours and δ is the declination in degrees in sexagecimal format, the equinox is J2000.

& Arimoto (1991) proposed supernova-driven winds to control the star formation rate on these

galaxies. For a massive galaxy, supernovae winds requires more time to expels its gas content;

without this gas, the star formation activity halts. As a result, the metallicity grows as a

function of the galaxy potential and therefore, by its luminosity. Consequently, cluster galaxies

prevail in a narrow color range with low dispersion (e.g. Bower et al., 1992; L´opez-Cruz, 1997;

L´opez-Cruz et al., 2004). Visvanathan & Sandage (1977) suggest, and latter L´opez-Cruz et al.

(2004) confirm the universality of the CMR for cluster galaxies atz_≤0·2. Additionally, Mei et

al. (2006) reported the CMR for clusters atz_∼1.

We have used as a first cluster membership criterion the CMR. Since we do not use spectral

measurements in this step, the CMR selection is a simple outlier rejection. To select galaxies

on the CMR, we proceed as follows:

• We perform a robust line fitting to the CMR in therversusr−z plane.

• We select the galaxies that lie within±1σstandard deviation from the fit.

• We adjust a second robust line fit to the selected galaxies and obtain the residuals.

• Finally, we select the galaxies that lie within±1σstandard deviation of the second-fit.

This method recovers galaxies within 1σfrom the CMR as we show in Figure 2.3.

The Cluster Velocity Dispersion

The selection of galaxies on the CMR is the first step to narrow down cluster members. The

second membership criterion is a revised Yahil & Vidal (1977) iterative process (hereafter as 3σ

-clipping method). For each iteration, this approach estimates the average velocity (va,i) and the

velocity standard deviation (σi) for the galaxy sample. We utilize aχ2test at 95% of confidence

(28)

12

14

16

18

20 r [mag]

0.2

0.4

0.6

0.8

1.0

1.2

1.4 r−z [mag]

−0.3−0.2−0.1−0.0 0.1 0.2 0.3 [mag]

0 10 20 30 40 50 60

Figure 2.3: Colour-Magnitude Relation (CMR) of A85. The solid line represents the estimated CMR

from our initial galaxy sample (black points). The red points represent the selected galaxies within1σ

from the CMR. The inset histogram represents the magnitude distribution of the scatter along the CMR.

test returns two upper and lower boundaries (with 95% of confidence) ofσ2_{. Next, we determine}

the initial value of σi as σi=0=

q

σ2

upper+σ2lower

2 . Then, we define two velocity limits that set a

filter window asvl=va,i±σi√αwithαas a tuning parameter. Then, we select all galaxies that

lie within the velocity limits, and we repeat the process. Therefore, once each iteration retakes,

new values ofva,i and σi provide a new definition of the velocity limits. This procedure selects

a new galaxy sub-sample in each iteration. This process finishes when the velocity limits are

the same as the immediate predecessor sub-sample. We illustrate this method in Figure 2.4.

The main parameter in this method is the tunable parameter α. In a perfect cluster, the

escape velocity is equivalent tovesc(r) =

p

2_M(r)G/r. In gravitational relaxed systems, we can

associate the velocity dispersion of the system as a function of the cluster mass and its radius

(29)

1.0 1.2 1.4 1.6 1.8 0

5 10 15 20

1.0 1.2 1.4 1.6 1.8

N

Velocity [104_km/s] _{Velocity [10}4_km/s]

Nt=154, zc=0.046, σ=591.44 km/s Nt=85, zc=0.047, σ=532.49 km/s

Figure 2.4: 3σ clipping iteration method for A1213. On the left is the original raw sample and ends

with the sample free from interlopers. The parallel vertical lines represent the velocity limits for each

iteration.

we connect the cluster escape velocity with the cluster velocity dispersion asvesc =

√

2σ. This

relation provides an αvalue of 2. However, clusters are not relaxed and, hence, αis variable.

We setαto vary between 1.75 to 6.25. Depending on theχ2_{test, the method changes the value}

ofαtoα+ 0.25 and restarts the iteration process. Whenαreaches a value larger than 6.25, the

procedure sets its value to 6.25, and a warning message is sent to a log file. This process selects

galaxies that are gravitationally bound to the cluster.

The first iteration returns a velocity dispersion that is far from the real velocity dispersion,

yet as the iterative process goes on, the contribution of interlopers rapidly diminish. When the

system converges, the background contribution must be zero. In consequence, cluster galaxies

velocity distribution from the closing iteration outlines a reliable estimation of the real velocity

dispersion of the cluster. The iterative and convergence nature of this scheme provides robustness

to our cluster velocity dispersion measures. If two clusters (or groups) lie on the same

line-of-sight, the most massive cluster prevails at the final step. Hence, this method is ideal to drop

background contamination and foreground galaxies. In contrast, this procedure is indifferent

to the presence of substructure (lumps within the cluster). In this case, we require to execute

more complex examination, e.g., the ∆₋testsubmitted by Dressler & Shectman (1988) or a 3D

(30)

Name N1 N2 N3 Name N1 N2 N3 Name N1 N2 N3

(1) (2) (3) (4) (1) (2) (3) (4) (1) (2) (3) (4)

A0085 226 183 128 A0168 183 133 84 A0257 153 114 43

A0279 88 70 41 A0634 117 85 54 A0646 105 78 38

A0671 174 137 82 A0680 62 42 37 A0688 77 57 15

A0690 148 114 69 A0695 122 96 61 A0700 83 60 31

A0779 121 89 45 A0924 93 66 13 A0957 81 65 27

A0991 97 74 48 A0999 118 90 26 A1020 182 145 60

A1035 215 169 60 A1126 137 102 42 A1142 120 89 43

A1169 182 142 69 A1185 187 143 114 A1187 138 116 62

A1190 192 155 103 A1213 154 121 85 A1218 115 92 57

A1228 171 132 79 A1238 183 140 64 A1291a 215 163 59

A1314 176 156 91 A1318 171 129 52 A1337 179 135 31

A1346 151 113 83 A1356 221 166 54 A1367 206 166 110

A1377 202 186 92 A1383 153 120 72 A1385 140 107 49

A1387 113 85 48 A1413 144 107 42 A1424 159 124 80

A1436 224 166 95 A1468 152 115 60 A1474 123 92 56

A1496 146 113 58 A1534 137 108 32 A1538 63 43 37

A1539 103 82 16 A1541 208 166 103 A1542 73 59 23

A1543 86 63 17 A1544 117 91 34 A1545 87 64 43

A1547 113 84 20 A1548 98 68 29 A1549 101 85 28

A1550 65 49 46 A1552 206 169 117 A1569 157 115 59

A1609 88 66 29 A1630 106 84 32 A1650 127 102 72

A1656 376 303 266 A1663 146 113 91 A1689 123 95 45

A1691 166 129 94 A1750 255 211 139 A1767 180 148 118

A1773 167 135 85 A1775 167 129 82 A1793 100 78 27

A1795 176 138 101 A1809 171 136 102 A1827 133 98 42

A1831 207 157 41 A1870 137 106 52 A1904 170 142 126

A1913 203 199 116 A1927 146 119 64 A1939 149 125 59

A1983 219 168 107 A1991 188 154 98 A1995 60 58 38

A2009 143 104 17 A2021 141 115 32 A2022 154 112 71

A2028 178 140 66 A2029 299 236 188 A2040 174 137 87

A2048 189 158 82 A2052 176 140 74 A2056 159 124 36

A2061 223 177 128 A2063 172 133 83 A2065 267 214 163

A2067 216 179 48 A2079 157 126 82 A2089 141 109 70

A2092 150 122 51 A2097 66 49 10 A2100 128 104 56

A2107 162 123 89 A2110 111 110 52 A2124 172 140 81

A2142 355 286 234 A2147 312 249 181 A2149 184 139 47

A2151 259 198 139 A2152 269 194 67 A2169 124 106 69

A2175 176 150 87 A2178 166 118 56 A2197 188 159 123

A2199 207 196 147 A2244 215 170 128 A2255 242 195 167

A2356 124 110 41 A2373 55 40 21 A2399 166 136 77

A2506 124 95 22 A2593 192 146 94 A2670 188 149 106

A2703 127 96 55 A2705 154 119 59

Table 2.3: List of the 125 Rich Abell clusters that we use as our initial cluster sample. In column 1 are the Abell numbers. In column 2 contains the initial number of galaxies. In column 3 are the number of galaxies that lie Color Magnitude Relation (CMR). Finally, in column 4 are the final number of cluster galaxies that returns the 3σ clipping method. Initially, 19,878 galaxies were downloaded, 15,684 lied on the CMR and 8,928 galaxies were identified as cluster members. In the case of galaxies that lie in the CMR but were rejected by the 3σ cliping method, only 4,423 (65·5% of 6,756) galaxies have high

signal-to-noise (S/N) to get its velocity dispersions. In the case of cluster galaxies, only 6,132 (68·7%

of 8,928) returns a velocity dispersion with high S/N.

Implementation of ROSTAT to test σ

In this section, we use a suite of statistical tools developed by Beers et al. (1990) to compare our

cluster’s velocity dispersions measures. This suite is known as ROSTAT2_{and offers nine robust}

(31)

Name 3σ STDEV SBIWT ADEV MAD Norm GAPPER f-Spread Psd-GAM 3S-Stdev

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

A0085 777·90 798·61 825·10 656·66 582·89 864·19 811·39 1188·1 880·76 798·61

A0168 551·35 602·58 608·42 479·47 439·05 650·92 605·51 875·69 649·14 602·58

A0257 380·77 389·59 390·53 304·71 247·79 367·38 388·78 480·29 356·03 389·59

A0279 741·78 681·75 697·53 537·55 455·39 675·16 691·36 956·85 709·30 681·75

A0634 305·64 341·06 345·82 267·62 203·25 301·33 342·30 401·10 297·33 341·06

A0646 815·39 727·85 751·88 620·33 493·19 731·20 730·92 1119·8 830·16 727·85

A0671 893·88 850·15 871·01 698·97 650·70 964·71 856·49 1168·2 865·97 850·15

A0680 - - -

-A0688 383·20 208·70 217·70 175·75 183·60 272·20 218·36 405·00 300·22 208·70

A0690 595·24 573·19 594·88 483·49 445·80 660·93 584·45 992·09 735·43 573·19

A0695 8800·9 378·77 391·01 306·76 273·60 405·63 387·43 582·29 431·65 378·77

A0700 1302·0 26·145 27·923 20·301 14·398 21·346 28·106 38·699 28·687 26·145

A0779 362·10 336·66 343·99 270·75 233·55 346·25 340·94 444·60 329·57 336·66

A0924 664·61 435·61 449·66 373·19 382·50 567·08 454·59 774·60 574·20 435·61

A0957 2536·0 1390·1 1454·6 1275·4 1317·0 1952·5 1415·3 2664·8 1975·4 1390·1

A0991 2643·7 2446·4 2581·9 2128·7 1836·3 2722·4 2486·7 4467·0 3311·3 2446·4

A0999 206·40 188·44 195·14 158·18 157·20 233·06 194·18 315·60 233·95 188·44

A1020 598·99 1287·3 1171·2 907·69 529·50 785·02 1233·4 1098·6 814·38 1216·3

A1035 621·46 1783·8 1870·7 1616·0 1420·5 2106·0 1794·0 3222·3 2388·6 1783·8

A1126 782·77 683·86 694·90 528·12 413·54 613·12 687·48 924·30 685·17 683·86

A1142 764·98 817·50 807·61 624·49 565·50 838·39 806·60 1044·8 774·57 817·50

A1169 656·08 622·43 646·11 510·04 487·49 722·75 634·78 988·79 732·98 622·43

A1185 719·07 715·63 732·44 563·42 412·50 611·56 717·14 878·70 651·37 715·63

A1187 470·69 459·91 476·58 366·13 304·79 451·89 463·63 634·50 470·34 459·91

A1190 661·86 706·96 711·70 556·67 465·60 690·28 706·90 939·60 696·51 706·96

A1213 532·49 507·87 525·57 409·84 346·50 513·71 515·75 705·59 523·05 507·87

A1218 4785·6 334·68 350·73 267·00 191·10 283·32 342·17 429·89 318·67 334·68

A1228 1264·6 249·93 249·25 196·35 166·20 246·40 250·25 315·75 234·06 228·72

A1238 539·57 569·74 558·99 435·29 346·50 513·71 564·01 661·50 490·36 533·23

A1291a 594·85 1446·2 1485·4 1246·4 1161·9 1722·6 1454·6 2482·5 1840·2 1446·2

A1314 702·73 698·19 719·10 556·52 450·30 667·60 702·28 938·25 695·51 698·19

A1318 410·08 421·82 422·99 336·75 319·35 473·46 423·04 585·59 434·09 397·71

A1337 400·00 1638·7 1777·3 1462·2 1040·4 1542·4 1646·9 3172·1 2351·5 1638·7

A1346 806·66 735·99 775·97 621·94 505·35 749·22 750·97 1118·4 829·05 735·99

A1356 719·59 2115·8 2249·4 1896·7 1521·2 2255·4 2126·5 3772·4 2796·5 2115·8

A1367 867·13 692·76 718·31 571·20 554·09 821·49 706·78 1055·7 782·57 692·76

A1377 732·09 665·01 688·67 537·18 455·70 675·61 675·53 959·25 711·08 665·01

A1383 504·84 841·89 790·61 640·39 401·24 594·88 814·46 825·59 612·00 841·89

A1385 581·51 558·58 567·50 447·84 393·29 583·09 564·60 789·00 584·87 558·58

A1387 699·77 649·09 671·68 512·59 380·39 563·97 656·91 713·55 528·94 649·09

A1413 1262·3 1224·4 1200·1 968·23 837·15 1241·1 1212·6 1474·8 1093·2 1112·7

A1424 731·42 794·08 792·81 623·48 541·50 802·81 791·62 1095·7 812·26 794·08

A1436 714·89 731·26 746·32 594·20 482·10 714·75 736·56 968·09 717·64 731·26

A1468 1180·8 1007·2 1055·1 881·09 805·35 1193·9 1026·1 1535·0 1137·9 1007·2

A1474 715·36 650·01 671·43 545·06 520·50 771·68 660·27 1104·3 818·60 650·01

A1496 424·76 387·03 399·64 305·59 239·40 354·93 388·91 481·80 357·15 387·03

A1534 384·96 328·43 341·75 272·94 273·90 406·07 339·23 582·60 431·87 328·43

A1538 - 369·66 382·49 286·79 292·05 432·98 390·93 538·95 399·52 369·66

A1539 382·04 546·77 579·97 490·26 436·04 646·47 561·20 1092·5 809·93 546·77

A1541 915·01 863·28 892·45 712·62 662·09 981·61 879·74 1289·1 955·59 863·28

A1542 523·13 485·27 487·23 376·82 371·99 551·51 491·25 626·39 464·34 485·27

A1543 481·47 366·45 375·21 287·25 298·94 443·21 376·92 597·89 443·21 366·45

A1544 587·33 644·58 637·17 486·55 418·20 620·01 641·63 821·09 608·67 644·58

A1545 - 1230·0 1180·1 902·97 628·94 932·46 1236·6 1221·8 905·78 948·79

A1547 1888·6 1295·9 1349·4 1085·0 1055·2 1564·4 1339·5 2260·3 1675·5 1295·9

A1548 1124·4 1103·4 1053·5 840·06 690·59 1023·8 1088·2 1397·1 1035·6 932·79

A1549 757·26 625·08 653·51 485·12 318·00 471·46 631·67 759·00 562·63 625·08

A1550 - 1081·3 922·17 774·72 499·50 740·54 1046·8 1127·3 835·72 787·59

Table 2.4: Velocity dispersion (σ) measurements of all ROSTAT methods for the 125 galaxy cluster sample, all in km/s units. The column 1 contains the cluster name. The column 2 contains our 3σ

clipping method measurement of σclus. The columns 3 to 8 contains the seven ROSTAT statistical methods.

(32)

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

A1552 901·99 857·08 873·08 679·92 550·80 816·60 861·00 1105·9 819·83 857·08

A1569 667·37 1501·4 1634·5 1329·1 948·90 1406·8 1503·6 2892·8 2144·4 1501·4

A1609 628·55 560·18 582·67 426·76 270·30 400·74 563·57 654·00 484·80 560·18

A1630 455·78 470·12 469·47 380·23 352·05 521·94 470·61 646·05 478·91 417·03

A1650 559·47 822·33 791·71 627·03 394·49 584·87 804·91 799·49 592·65 783·90

A1656 1043·9 1006·5 1040·9 828·47 729·00 1080·8 1020·9 1515·4 1123·3 1006·5

A1663 758·70 720·45 737·32 565·81 458·25 679·39 724·96 888·29 658·48 720·45

A1689 740·24 657·85 672·27 527·64 482·25 714·97 666·73 993·89 736·76 657·85

A1691 753·08 896·81 907·45 698·76 539·09 799·25 896·50 1146·1 849·62 896·81

A1750 897·79 872·62 911·30 740·63 664·34 984·95 891·99 1308·5 970·05 872·62

A1767 776·29 953·52 972·72 750·81 633·90 939·80 957·90 1234·0 914·78 953·52

A1773 746·35 881·16 864·56 675·16 465·59 690·28 868·02 993·44 736·43 881·16

A1775 594·85 1485·7 1572·8 1243·9 817·80 1212·4 1455·2 2714·0 2011·9 1485·7

A1793 671·75 1154·9 1020·9 830·61 477·14 707·41 1113·7 1115·7 827·05 1154·9

A1795 839·85 1170·1 995·63 852·43 611·54 906·67 1100·5 1284·0 951·81 821·81

A1809 805·52 758·72 776·85 617·06 555·89 824·16 768·00 1142·7 847·07 758·72

A1827 388·15 343·92 355·87 287·56 273·30 405·19 352·02 494·09 366·27 343·92

A1831 451·69 1986·0 2083·5 1746·1 1704·6 2527·2 2014·7 3932·6 2915·2 1986·0

A1870 895·57 907·77 920·18 718·02 649·50 962·93 912·59 1380·5 1023·4 907·77

A1904 835·91 793·01 816·01 644·09 569·70 844·62 803·35 1161·2 860·85 793·01

A1913 610·74 608·06 624·33 508·08 463·20 686·73 616·32 940·79 697·40 608·06

A1927 584·29 626·35 603·96 479·66 390·30 578·65 615·50 747·90 554·41 582·59

A1939 593·85 550·69 561·90 442·11 406·50 602·66 557·75 818·39 606·67 550·69

A1983 413·72 530·74 511·64 416·81 307·50 455·89 518·01 643·35 476·90 482·67

A1991 592·89 643·24 650·16 505·97 414·00 613·78 643·42 862·20 639·14 643·24

A1995 - 155·95 161·68 124·27 127·80 189·47 171·91 288·44 213·82 155·95

A2009 288·00 531·86 520·90 416·72 281·70 417·64 526·03 664·95 492·92 480·80

A2021 2081·5 1751·8 1744·3 1338·6 882·00 1307·6 1733·8 2385·0 1767·9 1751·8

A2022 622·76 612·48 626·32 489·64 398·70 591·10 616·80 858·00 636·02 612·48

A2028 787·91 787·40 795·82 632·16 507·60 752·55 790·54 1027·7 761·89 746·07

A2029 1124·7 1140·6 1159·3 931·40 861·29 1276·9 1149·9 1705·6 1264·3 1092·1

A2040 659·74 640·78 642·03 498·57 416·10 616·90 637·65 780·00 578·20 614·43

A2048 622·28 915·05 833·74 688·99 435·89 646·25 882·34 1005·9 745·66 875·98

A2052 639·97 660·87 677·45 523·33 452·69 671·16 667·72 952·80 706·30 660·87

A2056 502·51 489·88 482·73 383·00 255·00 378·05 484·74 604·50 448·10 489·88

A2061 813·43 856·81 891·68 705·64 560·70 831·28 860·29 1321·6 979·72 856·81

A2063 809·40 803·65 836·65 667·22 602·10 892·66 822·52 1257·0 931·80 803·65

A2065 1273·4 1253·7 1285·7 1029·2 977·99 1449·9 1271·5 1914·5 1419·2 1253·7

A2067 320·01 865·03 897·10 761·36 793·20 1175·9 884·24 1531·3 1135·1 865·03

A2079 669·03 766·81 726·14 587·99 405·00 600·44 749·61 823·79 610·67 640·17

A2089 553·25 794·44 740·03 595·41 418·05 619·79 768·55 836·10 619·79 794·44

A2092 390·35 455·83 455·80 355·64 263·09 390·06 454·36 532·94 395·06 455·83

A2097 274·43 164·46 162·81 116·94 68·849 102·07 166·61 137·69 102·07 164·46

A2100 957·14 1770·4 1550·1 1352·7 740·10 1097·2 1691·3 2012·6 1491·9 1508·7

A2107 639·41 668·99 679·42 523·07 460·20 682·28 671·30 919·50 681·61 668·99

A2110 551·87 642·08 608·07 483·32 393·29 583·09 627·36 714·30 529·50 503·11

A2124 752·51 848·38 809·33 647·88 530·70 786·80 829·50 1237·9 917·68 742·16

A2142 1058·6 1074·7 1101·0 855·43 713·10 1057·2 1081·7 1383·3 1025·4 1074·7

A2147 851·05 904·89 907·03 719·80 572·10 848·18 901·98 1157·0 857·74 904·89

A2149 255·46 396·02 355·99 286·28 209·39 310·45 379·34 423·30 313·79 355·06

A2151 801·80 784·82 810·04 653·14 613·35 909·34 798·58 1236·7 916·79 784·82

A2152 406·94 1440·5 1516·8 1234·1 750·59 1112·8 1422·0 2174·7 1612·0 1440·5

A2169 496·74 494·89 496·54 400·74 360·00 533·72 496·53 699·30 518·38 462·68

A2175 783·18 891·55 859·26 683·61 522·60 774·80 875·73 1138·6 844·06 781·21

A2178 2402·2 578·37 617·74 472·14 351·00 520·38 589·19 949·34 703·74 578·37

A2197 623·56 548·81 575·38 470·16 403·79 598·66 557·13 870·75 645·47 548·81

A2199 856·35 810·52 834·54 661·70 574·95 852·40 821·65 1169·3 866·86 810·52

A2244 1882·1 1948·0 2013·3 1578·7 1302·4 1930·9 1950·4 2882·6 2136·9 1948·0

Table 2.5: Velocity dispersion (σ) measurements of all ROSTAT methods for the 125galaxy cluster sample, all in km/s units. The column 1 contains the cluster name. The column 2 contains our3σ

(33)

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11)

A2255 1074·8 1242·7 1252·1 983·17 744·59 1103·9 1238·7 1659·0 1229·8 1242·7

A2356 567·06 640·63 667·29 546·71 476·84 706·96 649·33 990·00 733·87 640·63

A2373 5026·7 1896·5 1933·1 1355·5 722·10 1070·5 1908·8 1946·2 1442·7 1896·5

A2399 567·33 567·93 584·96 464·68 424·95 630·02 577·08 849·30 629·57 567·93

A2506 406·59 448·70 460·41 347·42 228·90 339·36 451·51 636·90 472·12 448·70

A2593 577·74 713·79 707·41 554·24 422·69 626·68 705·10 915·00 678·28 713·79

A2670 867·03 815·52 844·29 671·70 568·49 842·84 828·05 1168·2 865·97 815·52

A2703 691·47 1139·6 882·14 793·69 486·75 721·64 1056·0 997·19 739·21 627·67

A2705 630·59 584·62 599·28 465·89 336·75 499·25 587·75 738·14 547·18 584·62

Table 2.6: Velocity dispersion (σ) measurements of all ROSTAT methods for the 125 galaxy cluster sample, all in km/s units. The column 1 contains the cluster name. The column 2 contains our 3σ

statistical schemes to estimate the cluster’s velocity dispersion and its mean velocity. In this

work, we do not use the mean velocity that ROSTAT returns; instead we use only the velocity

dispersions that ROSTAT provides. The first method is the typical Gaussian standard deviation

(STDEV). The second is the median absolute deviation (MAD) that Beers et al. enunciate as

median(_|xi−m|); in this case,m is the median of the sample. The third method is a variant

of the second and employs the normal value rather than the median to achieve the normal

absolute deviation (NORM). The fourth method is the f ₋SP READ; this method recovers

the difference of the upper and lower fourths of the sample. The fifth method is the Biweight

standard deviation (SBIWT) that constitutes a part of the M-estimator’s family. This approach

uses the Φ Tukey’s Biweight function and yields the name of the method. It is a very robust

estimator designed for non-Gaussian or contaminated samples (Beers et al., 1990). The sixth is

the GAPPER method and consists to calculate the statistical parameter by gaping the sample

(see Beers et al., 1990, for more details). The seventh method (Pseudo-GAMMA) is the same

as thef ₋SP READmethod divided by 1.349 (thefspreadof a standard normal distribution).

This method is more suitable than the f₋SP READmethod for larger samples. In addition,

the procedure returns an additional dispersion estimation (ADEV) that is the standard median

absolute deviation without consider any robust statistical approaches. Also, ROSTAT returns

the 3σ(3S-Stdev) from the original definition of Yahil & Vidal (1977).

We present the ROSTAT results in Table 2.4, Table 2.5, Table 2.6 and Figure 2.5. Our

3σ clipping method agrees with the outputs of STDEV, SBIWT, NORM, GAPPER,

Pseudo-GAMMA and 3S-Stdev. In contrast, our velocity dispersion measurements present a deviation

if we compare our results with the results of ADEV, MAD andf−SP READ. We justify these

deviations because:

(34)

• The use of the median, instead of the normal value of the sample on the MAD and ADEV

methods returns an overestimation of the velocity dispersion. This overestimation is due

to possible bias caused by the foreground galaxies.

• The use of the f−SP READmethod is more efficient for low samples, in this case the

best proper method is the Pseudo-GAMMA.

Then, ROSTAT confirms (by six methods) that our procedure returns a good estimation of the

cluster’s velocity dispersion. In this case, the maximum difference between the ROSTAT results

and our 3σclipping procedure lie within_{≈ ±}150km/s.

Comparison With Other Estimates

In addition to the previous analysis, we compare our results with four previous publications,

namely: Struble & Rood (1991), Girardi et al. (1993), Zabludoff et al. (1993) and Rines &

Diaferio (2006). We present such comparisons in Table 2.8 and Figure 2.6. We find 49 matches

in the case of the catalog of Struble & Rood (1991). This catalog has the best agreement

with our cluster’s velocity dispersions with an average velocity difference (hσ3S −σS&Ri) of

−4·65km/s±148·92km/s. In the case of the Girardi et al. (1993) catalog, we find 19 matches

that have an average velocity difference of 54·56 km/s±169·96 km/s. In the same way, we

find 17 matches for the Zabludoff et al. (1993) catalog that disagree with our measurements by

92·26 km/s±254·57 km/s. This catalog returns a large contrast with our cluster’s velocity

dispersions, but also it has the lowest cluster’s matches with our sample. Also, we obtain

33 matches with the Rines & Diaferio (2006) catalog that has a constant velocity offset of

62·51 km/s±119·3 km/s. With the exception of Zabludoff et al. (1993), all the reported

cluster’s velocity dispersions lie within _±200 km/s. This dispersion is the same dispersion

that we find in the previous part. In the case of Zabludoff et al. (1993), we have not found

any one-to-one correlation between our measurements. In this case, the amount of matches is

not sufficient to recover a good statistics. In contrast, the other catalogues present a clearly

one-to-one relation. These catalogues have an adequate number of matches to justify a good

concordance with our measurements.

2.4 Geometrical Analysis and Surface Brightness Profiles

2.4.1 Total Magnitudes

To estimate absolute magnitudes and fluxes, we correct the apparent magnitudes for galactic

(35)

Name σ3σ σS&R σG σZ σR&D

(1) (2) (3) (4) (5) (6)

A0085 777·90 749·00 692·00

A0168 551·35 581·00 577·00

A0257 380·77

A0279 741·78

A0634 305·64 309·00

A0646 815·39

A0671 893·88 994·00 854·00

A0680

-A0688 383·20

A0690 595·24

A0695 8800·9

A0700 1302·0

A0779 362·10 472·00 528·00

A0924 664·61

A0957 2536·0 678·00 952·00 763·00

A0991 2643·7

A0999 206·40 261·00 417·00

A1020 598·99

A1035 621·46

A1126 782·77

A1142 764·98 417·00 610·00 579·00

A1169 656·08

A1185 719·07 783·00 709·00 207·00

A1187 470·69

A1190 661·86 636·00

A1213 532·49 598·00 568·00

A1218 4785·6

A1228 1264·6 188·00 200·00

A1238 539·57

A1291a 594·85

A1314 702·73 664·00 592·00

A1318 410·08 284·00 284·00

A1337 400·00

A1346 806·66

A1356 719·59

A1367 867·13 822·00 835·00 802·00

A1377 732·09 488·00 616·00

A1383 504·84 395·00

A1385 581·51

A1387 699·77

A1413 1262·3

A1424 731·42 619·00

A1436 714·89 501·00

A1468 1180·8

A1474 715·36

A1496 424·76

A1534 384·96

A1538

-A1539 382·04

A1541 915·01 496·00

A1542 523·13

A1543 481·47

A1544 587·33

A1545

-A1547 1888·6

A1548 1124·4

A1549 757·26

Table 2.7: Comparison between the catalogs Struble & Rood (1991), Girardi et al. (1993), Zabludoff et al. (1993) and Rines & Diaferio (2006) with the thesis 3σclipping velocity dispersions.

Clusters of galaxies in large surveys: Identification and scaling laws.