The Fundamental Plane of Early Type Galaxies in

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The Fundamental Plane of Early Type Galaxies in

Rich Abell Clusters

by

Phys. H´ ector Javier Ibarra Medel

INAOE

Thesis submitted as partial requirement for the degree of

MASTER IN SCIENCES WITH SPECIALTY OF ASTROPHYSICS

at

Instituto Nacional de Astrof´ısica, ´ Optica y Electr´ onica

April 2010 Tonantzintla, Puebla

Supervised by:

Dr. Omar L´ opez-Cruz INAOE Researcher

©INAOE 2010

The author gives permission to INAOE to reproduce

and distribute copies in whole or in parts of this thesis

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Abstract

The space defined by mass, radius and velocity dispersion for self-gravitating systems allows a dimension reduction called the Fundamental Plane (FP). These quantities are related through the Virial Theorem. Observationally, the slope of the FP is determined by the relationship between mass and luminosity, more precise the mass-to-light ratio (M/L). In this thesis, we have generated the FP for 6, 132 cluster galaxies as part of a comprehensive program to generate the FP for stellar systems, galaxies, groups, and galaxy clusters. We have also generated a control sample of 4, 423 field galaxies searching for environmental effects that could affect the FP for early-type galaxies. In this study, we have used photometric and spectra measurements from the Sloan Digital Sky Survey Data Release 7 (SDSS-DR7). We have developed an suit of scripts and programs to deal with data selection, cluster membership and galaxy velocity dispersion. We have measured the velocity dispersions of galaxies using two methods: the cross- correlation method and the direct-line-profile fitting method. We have used the Petrosian radius as a non-parametric scale indicator. The Petrosian radius overcomes the uncertainties of surface brightness profile fitting. Considering the problems that are introduced by a three-variable fit, we use a direct fitting method to determine the FP parameters (mathematically the FP has the expression log₁₀Rη = a[log₁₀σ + b hµiRη] + C). We found that the total luminosity and the mass of early-types galaxies are given by L^r ∝ M^η (η = 0·66). With the value of η, we can parametrize the FP’s coefficients a = ²²

η−1 = 0·996, b = _5η¹ = 0·3, C = −7^·68. Finally, this study finds that the fitted FP (a = 1·08, b = 0·292, C = −8^·12) agrees with the parametrized FP. This implies that the mass-to-light ratio of early-type galaxies follows M/Lr∝ L^θr(θ = ¹_η−1 = 0^·51).

The parametrization of θ as a function of η avoids the problems of the ^X_Y ∝ Y fit.

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Acknowledgements

• Agradezco primero a mis padres Eduardo y Mar´ıa por darme su apoyo incondicional en el desarrollo de este estudio de posgrado.

• Agradezco a mi director de tesis Dr. Omar López-Cruz por su apoyo en el desarrollo de este proyecto as´ı como por las incontables horas de revisión gramatical que el me dedicó.

• Agradezco a mis lectores: Dra. Elsa Recillas Pishmish, Dr. Ivanio Puerari y al Dr.

Divakara Mayya Yalia por todos sus comentarios y sugerencias.

• Agradezco a Luz y a mi hermano Eduardo por sus comentarios.

• Agradezco a mis amigos Giannina, David y Emmaly por sus comentarios y sugerencias.

• Agradezco a mis compa˜neros de posgrado por haberme escuchado en mi exposici´on (CyA) de 2 horas.

• Agradezco al CONACyT.

• Agradezco finalmente a todos los que me faltaron.

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List of Symbols Used in this Thesis

θ Exponent of the power law M/L ∝ L^θ and equals to ¹_η − 1 β Zero-Point of the power law M/L ∝ L^θ

η Exponent of the power law L ∝ M^η ξ Zero-Point of the power law L ∝ M^η

Rη Petrosian radius

µ Surface Brightness in mag/arcsec²

hµiη Average surface brightness at the Petrosian radius

σ Velocity dispersion

ηr Petrosian ratio

M Mass

L Luminosity

M^⊙ Solar Mass: 1·989 × 10³⁰kg L^⊙ Solar Luminosity: 3·84 × 10²⁶W Υ⊙ Mass-to-Light ratio in solar units Mr Absolute magnitude in r band mr Apparent magnitude in r band

b Broad function of the Direct Line Profile Fitting b_m Model broad function of the Direct Line Profile Fitting

g Galaxy spectra

t Template spectra

β Broad of the CCF of the Template

µ Broad of the CCF of the Galaxy

τ Stellar and Instrumental Broad

ι Systematic error of the Direct Line Profile Fitting method κ Systematic error of the Cross Correlation method

µ Exponent of the kinematic component of the relation L ∝ σr ν Exponent of the scale component of the relation L ∝ σr

a Slope of the Fundamental Plane

b Axis collapse coefficient of the Fundamental Plane α Virialization degree of the systems (galaxies)

δ Incompleteness luminosity ratio of the systems (galaxies)

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Chapter 1 Introduction

1.1 Fundamental Plane Historical Overview

Scaling Laws pervade Nature, they manifest themselves in many branches of Science, including Physics, Biology, Economy and, even, Sociology. In Astronomy, we have knowledge of many Scaling Relationships (SR). For example, the scaling in the spatial distribution of galaxies, ξ(r) = (ro/r)^γ (γ ≈ 1^·8, r0 ≈ 5h⁻¹M pc, r ≤ 5h⁻¹M pc), ξ(r) is the correlation function.

ξ(r) > 0 implies clustering.

Other SR are the direct result of the application of the Virial Theorem (VT). It is important note that the VT is scale invariant. If a system has an homogeneous potential energy function of degree k, then has the propriety U (α~r) = α^kU (~r). This implies that its Lagrangian for the scale α~r is the same as the Lagrangian at the scale ~r multiplied by α^k times (Schroeder, 1991). The equations of motion are unaltered by the scale (self-similarity). Considering this statements, the kinetic energy is a homogeneous quadratic function of the velocities and the potential energy (the Keplerian potential) is an homogeneous function of 1/r; therefore, the VT is scale invariant (Schroeder, 1991).

Zwicky (1937) was the first scientist who applied the VT to solve astrophysical problems.

He examined the velocity distribution of galaxies within the Coma cluster. By accounting for the mean velocity of the cluster galaxies and their light, he inferred the existence of an unseen type of matter known today as Dark Matter. Poveda (1958) extended the application of the VT to spherical galaxies assuming that their Surface Brightness (SB¹) distribution obeyed the r^1/4 law². He found that the mass-to-light ratio (M/L) had a dependency with the luminosity (L),

1The SB is expressed as I = F/A where A is the object projected area and F is the flux of the object. Other expression of the SB is µ = −2·5 log₁₀(f /A_θ) where A_θ is in angular units and f is the apparent flux of the object.

2Elliptical galaxies are extended objects; therefore, they have a SB profile. De Vaucouleurs found that the

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this relationship could be understood using the VT and the assumption that light traces mass.

By dynamics the velocity dispersion has to be correlated with the mass and in consequence with the light. Faber & Jackson (1976) also predicted that: M/L ∝ L^1/2. One year later, Tully &

Fisher (1977) found a similar relation for spiral galaxies: L ∝ Vmax³^·⁸, where Vmaxis the maximum rotational velocity of the disk. In the same year Kormendy (1977) found a relationship between the luminosity (denoted by its absolute magnitude M ) and the radius (R) for elliptical galaxies:

log R ∼ M (1.3)

The scatter of the Faber Jackson relation cannot be explained by considering observational errors alone. Terlevich et al. (1981) suggested that this scatter was caused by the contribution of a second parameter. They thought that this extra contribution was the metallicity of galaxies.

But the introduction of this extra parameter did not solve the scattered problem. The final solution for the scatter problem was reached six years later. Brosche & Lentes (1984) had applied multidimensional analysis in a study of globular clusters. They studied the relation of the galactometric distance R, the tidal radius, the concentration index, the total absolute magnitude, the color B-V, and the metallicity. Independently, Djorgovski & Davis (1987) and Dressler et al. (1987) found that the space parameter of R³, I and σ accepted a dimensional reduction (Kormendy & Djorgovski, 1989). They applied a Principal Components Analysis (PCA) and found a linear relation as one of its projections (See Fig 1.1 and Fig 1.2). Djorgovski

& Davis (1987) named this projection the Fundamental Plane (FP). It can be expressed as:

log₁₀re= 1·39 [log₁₀σ + 0·26 hµie] − 9^·71 (1.4)

radial SB profile (in magnitudes) follows the relation µ(R) = x+yr^1/4(where x and y are constants) and is named as de Vaucouleurs law or r^1/4law (de Vaucouleurs, 1948). Considering that the SB in flux is I(R) ∝ 10⁻⁰^·^4µ, then the last relation is I(R) = Ie10⁻³^·^33[(r/r^e⁾¹^/4^−1]. re is the effective radius. Assuming circular symmetry, the value of re is the half-light radius (the radius that encloses the half of light on a given galaxy) but some authors used the half-light diameter Ae= 2re. Ie is the SB at re.

3The scale parameter is the effective radius in a de Vaucouleurs profile.

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1.1. FUNDAMENTAL PLANE HISTORICAL OVERVIEW

Figure 1.1: The FP found by Djorgovski & Davis (1987) using 106 galaxies. They used the effective radius of de Vaucouleurs profile as scale parameter.

The general form of the FP is log₁₀R ∝ a [log10σ + b hµiR] + C, where the parameters a and b are independent of the cluster environment (Djorgovski et al., 1995). Which is equivalent to (using the Kormendy relation):

M (re) = −2.5 log10L = −8^·62 [log σ + 0·1 hµie] + 16·14 (1.5) On the other hand, Dressler et al. (1987) derived, in an independent way, for the FP using a least-square fit to the plane r −µ−σ for 43 galaxies in Coma and Virgo. They used the half-light diameter Aeand measured the hµieinside it with a de Vaucouleurs fitting. In many cases their apertures could not reach Ae without having a considerable sky contribution. Then, Ae was calculated by extrapolating the curve of growth. In order to prevent the pitfalls of extrapolating, they introduced a new scale indicator, which they named Dn. The Dnis the aperture where the integrated magnitude equals 20·75 magnitudes in the B band (this is not the isophote 20·75).

Dressler et al. (1987) found that the relation between these two apertures was a power law:

Dn/Ae∝ hIi^4/5e and the Dn is proportional with the velocity dispersion: Dn ∝ σ^4/3. Finally, they expressed their results as:

log₁₀Ae∝ 1^·33 [σ + 0·25 hµie] (1.6) Dressler et al. (1987) found that a dependence for the mass-to-light ratio that was expressed

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Figure 1.2: The FP by Dressler et al. (1987) using 43 galaxies of Coma and Virgo. They introduced the aperture Dn and used the relation Dn∝ σ^4/3to find the FP relation.

as:

M

L ∝ L^1/4hIi^1/20e . (1.7)

In the HJK study of 105 Coma cluster galaxies, Recillas-Cruz et al. (1990) found that L ∝ σ^αI_e^−β, where α grows with the wavelength, but β remains almost constant. The relation L − σ also changes. Finally they discussed that the dependence of the M/L with L and the SB also change, but in this case decreases. The dependence with L tends to vanish when λ ≈ 2^·2µm.

Oegerle & Hoessel (1991) generated the FP for 43 Brightest Cluster Galaxies (BCG). Ap- plying a more general fitting process, found a relation for r − σ − µ. Their FP is slightly bent at the high luminosity-end, which may be due to gravitational effects in the cluster potential-wells.

The relation among r, σ and µ is:

log₁₀re= 0·7 [log₁₀σ + 0·457 hµie] − 7.47, (1.8) rewas derived by fitting a de Vaucouleurs profile. This result may suggest that the BCG have formed in a different fashion that early-type galaxies. Oegerle & Hoessel (1991) also recovered the SR: M/L ∝ L^θ (θ = 0·24 ± 0^·002). The value of θ agrees with Eq 1.7.

In 1995 Jørgensen et al. (1996) generated the FP for cluster galaxies in a sample of 11 galaxy clusters. They looked for correlations of the cluster environmental on the FP. Their FP is:

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1.2. GROUPS AND CLUSTERS OF GALAXIES

log₁₀re= 1·24 [log₁₀σ + 0·262 hµie] + C (1.9) They derived M/L ∝ L^0.35. Jørgensen et al. (1996) also generated the FP in 3 bands (the Johnson’s B and U and the Gunn’s g); they found that the scatter of the FP was wavelength independent.

With the advent of digital surveys, a great amount of spectroscopic and photometric data are available. Bernardi et al. (2003) calculated the FP for 9000 early-type galaxies using the Sloan Digital Sky Survey (SDSS). The galaxies lie in the range of 0·01 ≤ z ≤ 0^·3 and were observed using the preliminary SDSS bands (g^∗,r^∗,i^∗,z^∗). To estimate the FP they used two fitting methods: Direct Fitting and Orthogonal Fitting. They found that the FP is the same for the all bands but depends (a dependence of the parameter a and b) on the fitting method.

In the case of the FP relation for the r^∗ band, they found:

For an Orthogonal Fitting:

log₁₀re= 1·49 [log₁₀σ + 0·20 hµie] − 8.778 (1.10) For a Direct Fitting:

log₁₀re= 1·17 [log₁₀σ + 0·256 hµie] − 8.022 (1.11) It was pointed out by Bernardi et al. (2003) that if M/L ∝ L^θ; then a = 2/(10b − 1). They also observed an evolution in the FP that reflects in a shift of the zero-point. They attributed this shift to an evolution of the SB: galaxies are higher redshifts are brighter. This evolution was proportional to ∆µ ≈ −2z.

1.2 Groups and Clusters of Galaxies

Galaxies are not homogeneously distributed through the Universe. They are clustered by the action of gravity and form large structures. Those structures can be classified according with its scale, mass or richness (the average number of galaxies that are gravitationally bounded).

If the number of galaxies is below 30, then the structures are called as groups. Groups are mostly dominated by spirals (in mass) except in compact regions. As a local example, the Milky Way (a barred spiral galaxy) is one of the three large spiral galaxies of the Local Group, which is conformed by about 30 galaxies. Groups of galaxies have sizes of about one Mpc in radius (Mart´ınez & Saar, 2002). If the number of galaxies is more than 30, then it is considered a cluster. The number of cluster galaxies is unbound and could have a radius grater than 4 Mpc.

Their shapes, have a variety of forms like spherical or irregular forms. The largest clusters also

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In Table 1.1 lists the principal parameters of groups and clusters (Bahcall, 1996).

Elliptical galaxies are common in overdense regions such as clusters of galaxies, were they could reach up to 40% of the galaxy population. In the other hand, in low dense regions their abundance drops about 10% (Binney & Merrifield, 1998). When elliptical galaxies are very luminous (and in consequence very massive), they are named cD (supergigant D galaxies) and usually are observed in the central regions of clusters of galaxies (or in the peaks of substruc- tures). However only 5% of all the galaxies in the local Universe are inside cluster of galaxies (Dressler, 1980).

1.2.1 The Abell Catalog

Abell (1958) generated a catalog of galaxy cluster whose its membership criteria minimize the projection effects. He examined 879 pairs of photographic plates of the Palomar Observatory Sky Survey that were taken with the 48-inch Palomar Schmidt Telescope. Abell analyzed about 4.26 steradians, (Bahcall, 1988) of the north sky and selected the regions where the surface number density was greater than the field. He defined four selection criteria to qualify his catalog. These criteria are listed below:

1. The richness. Is defined as the number of galaxies whose apparent magnitudes are m ≤ m3+ 2 (m3 is the magnitude of the third most bright galaxy). This richness is called the Abell Richness Class (ARC) and have six classes. ARC = 0 contains 30-49 galaxies, ARC = 1 contains 50-79 galaxies, ARC = 2 contains 80-129 galaxies, ARC = 3 contains 130-199 galaxies, ARC = 4 contains 200-299 galaxies and ARC = 5 contains 300 galaxies or more (Saslaw, 2000).

2. The compactness. It selects all the cluster members that must lie within 1·5 h⁻¹ Mpc to the cluster center.

3. The distance. It is set at 60-600 Mpc due to the m = 20 photographic magnitude limit of the Palomar catalog.

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1.3. THESIS STRUCTURE 4. The high galactic latitude. It avoids the problem of the dust extinction of the Milky Way

(Mart´ınez & Saar, 2002).

The Abell catalog contains 2,712 rich clusters where only 1,682 clusters carry out the four membership criteria.

1.3 Thesis Structure

In this thesis, we have generated the FP for 6, 132 Abell cluster galaxies. We have also generated a control sample of 4, 423 field galaxies. We are searching for environmental effects that could affect the FP for early-type galaxies. We have development tools to determine membership, velocity dispersions and non-parametric scale measurements. This study will be extended to globular clusters, dwarf galaxies, groups and cluster of galaxies. Cluster of galaxies offers a host of properties that make their ideal laboratories for evolutionary studies, see reviews in Plionis, L´opez-Cruz & Hughes (2008). This thesis is divided in 5 Chapters. Chapter 1 is a historical overview as an introduction. Chapter 2 describes the SDSS project and the adopted selection criteria for the sample of galaxy clusters. Chapter 3 describes the data reduction methods, using to generate the FP, and describes the methodology to determine the velocity dispersion of individual galaxies. Chapter 4 discusses the FP fitting process and results of the thesis. Finally, Chapter 5 presents this study conclusions.

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Chapter 2 Observations

2.1 Introduction

Large Digital Sky Surveys (DSS) have advanced study of the general properties of galaxies and their spatial distribution. One of the first large DSS was done with the Automatic Plate Measuring (APM) machine in the 1980’s; 269 photographic plates were scanned to conform the APM survey. The plates were taken with the UK Schmidt Telescope (UKST). The APM survey provided positions, magnitudes, sizes and shapes for 3 million galaxies (Turnshek et al., 1984). Following this first attempt, the Space Telescope Science Institute (STScI) designed an all-sky DSS using the photographic plates of the Palomar Observatory Sky Survey (POSS-I).

The first generation DSS consisted of 600 GB of data that eventually reached 3 Terabytes of data. In contrast, there were disadvantages in those early DSS, as an example the magnitude of the survey was limited by the poor efficiency of the photographic plates. In addition, errors on the object positions and geometrical distortions were introduced by the size and the mode of scanning. All of these problems have been overcome in the new generation of DSS.

2.2 The Sloan Digital Sky Survey

The Sloan Digital Sky Survey (SDSS) (York et al , 2000) is the largest fully digital sky survey, ever undertaken. To date, it has released measurements for 357 million objects within 11,663 squared degrees using charge-coupled devices (CCDs) as detectors. On this material, objects have been selected to get photometric and spectral information. The aim was to construct a tridimensional mapping of the local Universe. The SDSS facilities are located at Apache Point Observatory (APO) in New Mexico, whose coordinates are latitude 32^◦46^′49^′′N and longitude 105^◦49^′13^′′W at 2, 788 m of altitude. The average seeing for SDSS was about 1.4^′′. (this image

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Figure 2.1: SDSS facilities are located at Apache Point Observatory (APO, left), in the bottom left of this image is located the main telescope, in the central part of the image is visible the secondary telescope used for photometric calibrations. Details for each telescope are displayed at the right images where the top image is the secondary telescope and the bottom image is the primary telescope.

quality is poor compared with other observatories which routinely get sub-arc image quality;

but it is fine for studies of nearly galaxies)

The main SDSS telescope is an f /7 Ritchey-Chr`etien telescope with a 2·5 m primary and a 1·17 m secondary. This configuration provides a 3^◦ field-of-view (FOV). An auxiliary 0·5 m telescope was used for photometric calibrations (see Fig 2.1). The CCD array consist on 30 2048 × 2048 photometric SITEe/Tektronix CCDs with 24 µm pixels, arranged in six columns and five rows (effective area of 720 cm²). Each row corresponds to a specific band of the SDSS photometric system. The whole array sits on the Cassegrain focus of the SDSS telescope. Other 24 secondary 400 × 2048 CCDs (same pixel size) were used for astrometric calibrations. To increase the observational efficiency the observations were taken as Time-Delay-and-Integrate (TDI) drift scans. Because the images are sampled by the pixels of each column, the images can be corrected by one-dimensional flat fields (Howell, 2006). The TDI gives the opportunity to scan a wide sky field in a small amount of time. The optics of the telescope were designed to minimize the distortions introduced by the TDI drift scans (Gunn et al., 1998). Below we provide a full description of the SDSS.

The SDSS Coverage

The effective exposure time on a given object is 54·1s. Accounting for a delay of 77s between the bands; then, a given object remains on the array for 342s (Gunn et al., 1998). The scanning

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2.2. THE SLOAN DIGITAL SKY SURVEY

Release Release Photometric Photometric Spectroscopic Spectroscopic

Date Coverage Number Coverage Number

Objects Objects

deg² deg²

EDR 5 Jun 2001 462 14 × 10⁶ 386 54,008

DR1 4 Abr 2003 2,099 53 × 10⁶ 1,360 186,250

DR2 15 Mar 2004 3,324 88 × 10⁶ 2,627 367,360

DR3 27 Sep 2004 5,282 141 × 10⁶ 3,732 528,640

DR4 29 Jun 2005 6,670 180 × 10⁶ 4,783 849,920

DR5 28 Jun 2006 8,000 215 × 10⁶ 5,740 1,048,960

DR6 28 Jul 2008 9,583 287 × 10⁶ 7,425 1,271,680

DR7 27 May 2009 11,663 357 × 10⁶ 9,380 1,640,960

Table 2.1: The SDSS project consists in 7 Data Releases (DR) and an Early Data Release (EDR). The table lists the name of the release, the release date and the coverage of the photometric and spectroscopic survey of the DR.

generates (by each band) a great circular stripe of 2·5^◦wide by 90^◦long. Generating six narrow scan lines of 13^′_·5 wide with a separation of 12^′.8 and a total of 230 deg² per stripe (Yasuda et al., 2001). The total area covered by the SDSS is about 10,000 square degrees on the Northern Galactic Cap. In other words, the SDSS covers the whole sky over the Galactic latitude greater than 30^◦. A minimum galactic-extinction region was selected using the maps of Schlegel et al.

(1998). This region is centered at α = 12^h20^m, δ = +32^◦·5 with an elliptical shape of 130^◦×110^◦. The SDSS reached the Southern Galactic Cap along the celestial equator, generating a equatorial stripe at α = 20^h.7 − 4^h, δ = −5^◦to + 13^◦ (see Fig 2.3).

The Photometric System

The photometric system adopted by the SDSS is based on the Gunn (1978) system. The Gunn system was optimized for extragalactic studies by selecting bands that excluded the strongest night-sky lines (Thuan & Gunn, 1976). The SDSS photometric system covers the full range from the blue atmospheric ultraviolet cutoff to the red sensitivity limit of the CCD. The five SDSS bands: u, g, r, i, z (final photometric calibration) are centered at 3540, 4770, 6230, 7630 and 9130 ˚A (see Fig 2.2). During the preliminary phases, the SDSS bands where denominated as u^∗, g^∗, r^∗, i^∗, z^∗ due to the preliminary nature of the photometric calibrations (Fukugita et al., 1996). The overall detection limit was about 23 mag. It varies over the bands and the mean flux density in each band is hfνi = 3631 × 10⁻⁰^·^4mag Jy. The 24 astrometric CCDs have a pass-band filter identical to the r^′ band and covers a magnitude range of 8·5 to 16·9 mag (Gunn et al., 1998; York et al , 2000).

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Figure 2.2: Left: CCD configuration of the image camera of the SDSS, 30 CCDs for image and 12 for calibration. Right: CCD response curves of the five SDSS filters.

The SDSS Data Releases

The SDSS data have been released during two stages: the SDSS-I and the SDSS-II the first one was the SDSS-I and the second is the SDSS-II. Each stage consisted of several Data Releases (DRs). The SDSS-I included the Early Data Release or EDR (the prototype DR of the project), the First Data Release (DR1), the DR2, the DR3, the DR4 and the DR5. The SDSS-II consist at the moment of the DR6 and the DR7. Each DR covers a new region of the sky and includes data from the previous data release. In the future, the SDSS is planing SDSS-III, whose objective is to scan the Southern Galactic hemisphere.

2.2.1 The SDSS Photometric Catalog

The SDSS team implemented a photometric analysis software (Lupton et al., 2002) to reduce the digital images. The software corrects the images by the use of flat fields, cosmic rays, and bad pixels. In addition, the software finds prospective objects and deblending them if overlapping occurs. Finally, the software performs the photometric analysis. The point-spread function (PSF) is estimated using bright field stars. The SDSS uses an aperture of 7^′′·4 to estimate the magnitude of the objects using the PSF profile. The model magnitudes are generated by fitting a de Vaucouleurs profile or an exponential one. The SDSS denotes such magnitudes as PSF magnitudes (mpsf) or model magnitudes (mmodel) respectively. In addition the SDSS calculates 3^′′ aperture magnitudes and, these are named fiber magnitude (mf iber). Modified Petrosian magnitudes are also provided. These are generated using the original definition by Petrosian (1976) (see Section 3.3). Petrosian magnitudes were introduced to overcome SB cosmological

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2.2. THE SLOAN DIGITAL SKY SURVEY

Figure 2.3: SDSS Data Release 7. The left image is an Aittof plot of the Survey Photometric coverage and the right image is an Aittof plot of the Survey Spectroscopic coverage.

dimming (Blanton et al., 2001).

SDSS uses the difference between mpsf and mmodel to do star-galaxy separation (Scranton et al., 2002). The algorithm is based on the difference between stellar point-like objects (stars) and extended objects (galaxies). In consequence the difference ∆m = mpsf− m^model tends to be zero if the object is a star or have a value grater than a specific threshold if the object is a galaxy. The threshold is set at 0·3 mag, but in the initial releases it was set at 0·16 mag (this latter value turned out to be overoptimistic).

2.2.2 The Spectroscopic Survey

Once the galaxy identification process is completed, the SDSS implemented an algorithm to select the spectroscopic sample in an uniform and objective manner (Strauss et al., 2002). The SDSS named the spectroscopic galactic survey: the main galaxy sample. The selection criteria were optimized to study the large-scale structure and characterize the local galaxy population.

First, the algorithm rejects all galaxies brighter than mr= 14 integrated magnitude. In addition, the algorithm also rejects all objects that have a neighbor closer than 55^′′because of the physical limit to place fibers.

Then the algorithm selects all the galaxies brighter than mr= 17·77 Petrosian magnitudes.

SB and magnitude selection criteria are established to ensure a good signal-to-noise ratio on the resulting spectra. If the half-light SB µ50(the SB at the half Petrosian Flux measured in r band) is brighter than 23·0 mag/sec², then the galaxy is accepted. If the galaxy have a µ50 between 23·0 and 24·5 mag/sec², then is accepted only if the difference of the global sky (determined over the entire image frame) and the galaxy local sky (determined by median-smoothing image at a scale of 100^′′) is less than 0·05 mag or the fiber magnitude is less or equal than mr= 19·0 mag. If the galaxy have a µ50grater than 24·5, then is accepted only if the galaxy has a fiber magnitude

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Figure 2.4: Left: Optical design of the SDSS spectrograph’s, showing the slit head, the Schmidt Col- limator,the dichroic beamsplitter, the grisms and the red and blue cameras. Right: The CCD response curves of the blue and red parts.

less or equal than mr= 19·0 mag. If the fiber magnitude is less than mr= 15 mag, then the galaxy is rejected because the light of the fiber could contaminate the adjacent fibers or saturate the CCD spectrograph detector.

The SDSS Spectrographs

The SDSS has two spectrographs and each have two SITe/Tektronix 2, 048 × 2, 048 CCDs. Each spectrograph covers the spectral region of 3, 800˚A−9, 200˚A with two CCDs (see Figure2.5).

Because the spectrograph resolution is in the range of R = 1, 850 − 2, 200, then a 2, 048 pixel CCD covers a region of 1, 700 or 2, 300˚A wide. To have the desired spectrograph range, the beam was divided in a blue and red region, making possible the covering the desired range with two CCDs. The first region covers the blue part of the spectrum at 3, 800 − 6, 150˚A and the second region covers the red part of the spectrum at 5, 800 − 9, 200˚A. The resolution of R ≈ 2, 000 generates a spectral resolution of 150 km/s with a pixel size of 69 km/s.

The fibers are located in circular plates, each of them contains 640 fibers (320 per spectrograph). Each fiber has of 180µm in diameter or 3^′′ projected on the sky. The flux calibration was made with the use of standard stars within the observed field and the wavelength calibration was made using the He, Cd and Ne arc lamps with an error of 0·7 pixels (10 km/s).

They use quartz lamps for CCD flat-fielding. The total exposure time was chosen to generate a S/N = 4 per pixel. The total exposure for the fibers on the sky was 45 minutes (3 exposures of 15 minutes each). The 640 fibers of an individual plate were plugged into two cartridges that contained 320 fibers each. Each cartridge was mounted on the spectrograph. The light of the fibers passes through a Schmidt collimator and then the blue and red parts were divided by a

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2.3. GALAXY CLUSTERS SELECTED FOR THIS THESIS

Figure 2.5: One example of SDSS galaxy spectra: the galaxy SDSS J111736.67+285531.7, located in Abell 1213 at z = 0·046. Have a total luminosity of 1·27 × 10¹⁰L⊙, a mass of 1·28 × 10¹¹M⊙, a Petrosian radius of 7·18 Kpc, a velocity dispersion of 128 km/s and a SB of 21·1 mag/seg².

dichroic beamsplitter (the red light is transmitted and the blue light is reflected). The light of each region was dispersed by a grism and then was focused by the spectrograph’s camera into the CCD (see Figure 2.4). The grisms have a ruling density of 640 lines/mm for the blue part and 440 lines/mm for the red part. The two parts of the spectra were merged in the reduction process and the final SDSS spectra was obtained.

2.3 Galaxy Clusters Selected for this Thesis

For this thesis, a sample of 125 Abell Clusters were chosen from the SDSS DR7 (See Tab 2.2) using the following selection criteria:

1. The selected clusters should be part of Abell’s (1958) statistical sample.

2. The selected clusters should be isolated with Abell Richness Class (ARC) greater than 0.

3. The selected clusters should be in the redshift range between 0·02 ≤ z ≤ 0^·2.

4. The selected clusters should be at high galactic latitude (|b| ≥ 30^◦).

Clusters whose ARC is greater than zero allow us to achive efficient sampling. Richer cluster may contain over a thousand galaxies. The redshift range was chosen to sample deep enough inside the cluster’s luminosity function. In other words, the high redshift clusters may poorly sampled because the cut-off magnitude of the spectroscopic sample at 17·77 mag. Finally, the

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Figure 2.6: Sky distribution (in galactic coordinates) of the 125 Abell Clusters selected in this thesis.

In the North Galactic hemisphere and in the half of the South Galactic hemisphere. The the clusters are distributed homogeneously over the North Galactic hemisphere.

last criterion delimits the region where the galactic extinction has a minimum effect to the observed galaxies fluxes.

Our final sample contains 125 Abell Clusters, 28 clusters are in common with The Low- Redshift Cluster Optical Survey (LOCOS, Lopez-Cruz 1997). The LOCOS was originally designed to measure optical properties of low-z X-Ray luminous cluster selected from the Ein- stein sample (Jones & Forman, 1999). The clusters are homogeneously distributed through out the observed SDSS region (Figure 2.6). Only 81 clusters have X-Ray observations. All galaxies within the 30^′ of the reported cluster center were selected on the initial pass. Figure 2.7 shows the redshift distribution of the selected cluster.

2.4 Data Acquisition

Data acquisition was done by the use of a query program. The program is written in python. It is designed to download redshifts (s.z), model and Petrosian magnitudes (p.r, p.petroMag_r), Petrosian radii (p.petroRad_r), velocity dispersions (s.velDisp), positions (str(p.ra,13,8) as ra, str(p.[dec],13,8) as dec) and the associated error to each measurements. In addition, the plate (s.plate), fiber (s.fiberID) and exposition numbers (s.mjd) were also downloaded to

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2.4. DATA ACQUISITION

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

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Figure 2.7: The redshift distribution of the selected clusters for this Thesis. The mode of the redshift distribution is z = 0·08.

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 125 Abell Clusters sample. The column 1 is the Abell number, column 2 is the redshift of the clusters and columns 3 and 4 are the positions centers of the clusters, α is the right ascension in hours and δ is the declination in degrees in sexagecimal format, the equinox is J2000.

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2.4. DATA ACQUISITION identify each galaxy spectrum. The query is given below:

SELECT s.fiberID,s.mjd,s.plate,s.velDisp,s.velDispErr,s.z,s.zErr,str(p.ra,13,8) as ra, str(p.[dec],13,8) as dec,p.u,p.g,p.r,p.i,p.z,p.modelMagErr_u,p.modelMagErr_g,

p.modelMagErr_r,p.modelMagErr_i,p.modelMagErr_z,p.decErr,p.petroMag_r,p.petroMagErr_r, p.petroRad_r,p.petroRadErr_r,p.raErr, ’ugriz’ as filter

FROM BESTDR7..SpecObj as s

JOIN BESTDR7..PhotoObj AS p ON s.bestObjID = p.objID JOIN dbo.fGetNearbyObjEq(RA_input, DEC_input,Search_Rad) AS b ON b.objID = p.objID

WHERE (s.specClass = dbo.fSpecClass(’GALAXY’)) AND ( p.type = 3 OR p.type = 6)

The query program was compiled with the procedure sqlcl.py. This procedure is available in the SDSS web page and runs direct in a LINUX console. The output file was downloaded in a Coma-Separated Values (CSV ) format. This is easy to be read by the Interactive Data Language (IDL) program. The calling sequence is expressed by:

>python sqlcl.py Dir/query_input.qrt > Dir/name_output.csv

2.4.1 Spectra Downloading

The study requires the determination of the galaxies dispersion velocities. This requires an analysis of all cluster galaxies spectra. In the CSV file the fiber, the plate and the exposition date numbers were obtained. With this data, multiple lists of the required spectra were constructed.

These lists contain the web directions for each spectra:

http://das.sdss.org/spectro/1d_26/PLATE/1d/spSpec-MJD-PLATE-FIBER.fit http://das.sdss.org/spectro/1d_26/PLATE/gif/spPlot-MJD-PLATE-FIBER.gif http://das.sdss.org/spectro/1d_26/PLATE/gif/spPlot-MJD-PLATE-FIBER.ps.gz

In this case, the list contain the directions of the same spectra in the Flexible Image Transport System (F IT S or F IT ), Graphics Interchange Format (GIF ), and Post-Script (P S) format.

Data downloading is done using the UNIX command wget, applied as follows:

wget -c -i Dir/sdss-wget.lis"

All files in the list were automatically download from the DR7. The spectra F IT format consist of a one dimensional F IT file. This presentation contains the calibrated spectrum, the continuum-subtracted spectrum, and the measured parameters like the redshift and line fits.

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and algorithms for velocity dispersion determination. We have selected 125 Abell clusters from SDSS DR7, 28 clusters are in common with LOCOS Lopez-Cruz (1997), 81 clusters have X- Ray observations (e.g Einstein, ROSTAT, Chandra, XMM). We have downloaded photometric measurements for 19, 878 galaxies and 6, 132 spectra; respectively, for galaxies in a 125 cluster fields.

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Chapter 3 Cluster Membership, Velocity Dispersions, Luminosities, Masses

and Data Analysis

3.1 Introduction

We want to generate the Fundamental Plane (FP) for cluster galaxies. For this aim, in this thesis we have developed a scheme to determine cluster membership, taking advantage of the wealth of information from the SDSS. The selection of cluster members and interloper rejection were implemented using two approaches. The first one is a simple method based on the Color- Magnitude Relation (CMR); while the second method is an iterative method based on an earlier scheme developed by Yahil & Vidal (1977). Once the membership criteria were applied, we determined galaxies masses (M), luminosities (L), radii (R) and its kinematics (σ). Galaxy’s masses were determined by a straight application of the Virial Theorem.

3.2 The Color Magnitude Relation

The Color Magnitude Relation (CMR) is a very tight relation that is followed by cluster early- type galaxies. Cluster galaxies are progressively redder as a function of brightness: the brighter galaxy the redder. This effect is produced by metallicity. Yoshii & Arimoto (1991) proposed supernova-driven winds as a regulator of the metallicity in early-type galaxies. If the galaxy potential is not enough to retain its gas, then its star formation activity is ceased. If not, the galaxy could retain some part of its gas, and its star formation activity can be continue.

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color histogram was generated. The center of the histogram indicates the more populated color range. This region gives an approximation to the average color of the red sequence. If the standard deviation of the histogram approximates the CMR dispersion , then galaxies within a 1σ region of the histogram are selected as cluster members. This method fails to reject at least 10% of the interlopers, but it makes a first step in the cluster galaxy selection process.

3.2.1 The 3-Sigma-Method

The selection of galaxies on the CMR provides the first step for a finner selection of cluster members. We have applied a modified Yahil & Vidal (1977) iterative method (known as the 3σ or 3S method) as the second pass in our membership process.

On the first pass, this method estimates (at the original 30^′ aperture) the average velocity (va,i) of the sample using a standard mean and estimates the velocity standard deviation for the sample. The velocity dispersion (σi) is estimated using the standard deviation using a χ² test with 95% of confidence (assuming a Gaussian velocity distribution). This test returns two upper and lower 95% of confidence limits of σ². The initial velocity dispersion estimation is given by:

σ = s

σ²_upper+ σ_lower²

2 ; (3.1)

then, the method defines two velocity limits. These limits define a filter window as:

vl= va,i± σⁱ√

α, (3.2)

where α is a scale parameter. On the first iteration, the mean and the standard deviation changes. As a consequence, the velocity limits are redefined by new subsample. The new subsample is filtered again, and process loops through, and generates a new subsample. This process finishes when the velocity limits are the same as the immediate predecessor subsample (see Fig 3.1 and Fig 3.2).

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3.2. THE COLOR MAGNITUDE RELATION

Figure 3.1: An example of the 3-σ iteration method (In this case for A1213). On the left is the original raw sample (note the background contamination) and ends with the sample free from interlopers. The parallel vertical lines represent the velocity limits for each iteration.

The main parameter of this method is the scale parameter α. In an ideal cluster the escape velocity is equal to vesc(r) = p2M(r)G/r. The virial theorem is used to obtain the velocity dispersion as a function of the mass and radius: σ(r) = pM(r)G/r. Using this two results we find the escape velocity as a function of the velocity dispersion: vesc = √

2σ. In this case α = 2. Because the clusters are not ideal, the value of α is variable. The value of α varies between 1.75 and 6.25 and depends on the success of the χ²test. If the test fail, then the value of α is changed by α + .25 and the window selection change. In consequence, the χ² test is again applied. If α reaches 6.25, then the value of α is fixed, and a warning message is sent.

This process was designed to achieve a window that represents the region where no galaxy could escape the potential of the cluster.

The iterative nature of this method eliminates in each step galaxies which are not gravitationally bound to the cluster. The first estimations of the velocity dispersion are far from the real dispersion, but as the iterative process goes on, the contribution due to interlopers decreases rapidly (see Figure 3.1, Figure 3.2 and Table 3.1). When the process converges, the background contribution is zero and the velocity dispersion of the galaxy cluster is the best estimation of the real dispersion. The iterative and convergence nature of this method identify it as a robust method. If two cluster (or groups) lie on the same line-of-sight, the more massive cluster remains. Then, this method is ideal to eliminate contamination from background and foreground galaxies; but, it is insensitive to the presence of substructure (lumps within the cluster). In this case, a more detailed analysis is required, e.g the ∆ − test proposed by Dressler & Shectman (1988). In this study, clusters that presented strong incidence of substructure were excluded.

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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 3.1: List of the 125 Abell clusters. 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 3 − σ 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 3 − σ 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.

The Fundamental Plane of Early Type Galaxies in