TH`ESE

Transcription

TH`ESE

THÈSE
présentée devant
L’UNIVERSITÉ TOULOUSE III - PAUL SABATIER
U.F.R. PHYSIQUE, CHIMIE, AUTOMATIQUE
en vue de l’obtention du grade de
DOCTEUR DE L’UNIVERSITÉ TOULOUSE III
(SCIENCES)
Spécialité Astrophysique
par
Oliver CZOSKE
Wide–field Observations of Clusters of Galaxies
Observations à Grand Champ d’Amas de Galaxies
Soutenue le 14 juin 2002 devant la Commission d’Examen:
M. Alain Blanchard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Président
Mme Geneviève Soucail . . . . . . . . . . . . . . . Directrice de Thèse
M. Jean-Paul Kneib . . . . . . . . . . . . . . . . Co–Directeur de Thèse
M. Peter Schneider . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rapporteur
Mme Florence Durret . . . . . . . . . . . . . . . . . . . . . . . . . Rapporteur
M. Jim Bartlett . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examinateur
M. Alain Mazure . . . . . . . . . . . . . . . . . . . . . . . . . . . . Examinateur
M. Harald Ebeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Invité
Laboratoire d’Astrophysique de Toulouse – UMR 5572
Observatoire Midi-Pyrénées
14 avenue Édouard Belin
31400 Toulouse
FRANCE
Contents
Introduction (Français)
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Introduction (English)
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I Clusters of galaxies
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Cosmological Framework
1.1 Background Cosmology . . . . . . . . . . . . . . . . . . . .
1.2 Cosmological Parameters . . . . . . . . . . . . . . . . . . .
1.2.1 High-Redshift Supernovae . . . . . . . . . . . . . .
1.2.2 Fluctuations in the Cosmic Microwave Background
1.2.3 Direct Mass Measurements . . . . . . . . . . . . . .
1.2.4 Concordance Studies . . . . . . . . . . . . . . . . . .
Résumé du chapitre en français . . . . . . . . . . . . . . . . . . .
Counting Clusters of Galaxies
2.1 Matter Power Spectrum . . . . . . . . . . . . .
2.2 Linear evolution of the density field . . . . . .
2.3 The spherical collapse model . . . . . . . . . .
2.4 Analytical estimates of the halo mass function
2.4.1 The Press–Schechter argument . . . . .
2.4.2 The excursion set approach . . . . . . .
2.5 Mass functions from numerical simulations . .
2.6 Evolution of the mass function . . . . . . . . .
Résumé du chapitre en français . . . . . . . . . . . .
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Relaxation processes
3.1 Violent relaxation . . . . . . . . . . . . . . . . . . .
3.2 The isothermal sphere . . . . . . . . . . . . . . . .
3.3 Numerical simulations: A universal mass profile?
3.4 Observational status . . . . . . . . . . . . . . . . .
Résumé du chapitre en français . . . . . . . . . . . . . .
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5
Contents
Determination of Cluster Masses
4.1 What is a cluster of galaxies? . . . . . . . . . . . . .
4.2 Gravitational Lensing . . . . . . . . . . . . . . . . . .
4.2.1 Generalities . . . . . . . . . . . . . . . . . . .
4.2.2 Strong lensing . . . . . . . . . . . . . . . . . .
4.2.3 Weak lensing . . . . . . . . . . . . . . . . . .
4.3 Distribution of the Intracluster Gas . . . . . . . . . .
4.3.1 Hydrostatic equilibrium . . . . . . . . . . . .
4.3.2 The β model . . . . . . . . . . . . . . . . . . .
4.3.3 Recent observations . . . . . . . . . . . . . .
4.4 Galaxy kinematics . . . . . . . . . . . . . . . . . . . .
4.4.1 Relaxed systems . . . . . . . . . . . . . . . .
4.4.2 Cluster substructure from galaxy kinematics
Résumé du chapitre en français . . . . . . . . . . . . . . .
Relations between observational properties
5.1 Observed relations . . . . . . . . . . . . . . . . . .
5.1.1 The LX –TX relation . . . . . . . . . . . . . .
5.1.2 The M–TX relation . . . . . . . . . . . . . .
5.1.3 The σ–TX –LX relations . . . . . . . . . . . .
5.2 The cluster temperature and luminosity functions
5.2.1 The temperature function . . . . . . . . . .
5.2.2 The luminosity function . . . . . . . . . . .
Résumé du chapitre en français . . . . . . . . . . . . . .
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II A wide-field redshift survey in the cluster Cl0024+1654 97
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Cl0024+1654: Introduction
6.1 Motivation for a wide–field spectroscopic survey
6.2 Lensing observations . . . . . . . . . . . . . . . .
6.2.1 The quintuple arc system . . . . . . . . .
6.2.2 Weak lensing observations . . . . . . . .
6.3 X-ray observations . . . . . . . . . . . . . . . . .
Résumé du chapitre en français . . . . . . . . . . . . .
A wide-field spectroscopic survey
7.1 Observations . . . . . . . . . . . . . . . .
7.1.1 Imaging . . . . . . . . . . . . . .
7.1.2 Spectroscopy . . . . . . . . . . .
7.2 Data reduction and analysis . . . . . . .
7.2.1 Reduction of spectroscopic data
7.2.2 Redshift determination . . . . .
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Contents
7.2.3 Spectroscopic measures
7.2.4 The catalogue . . . . . .
7.2.5 Completeness . . . . . .
7.3 Discussion . . . . . . . . . . . .
Résumé du chapitre en français . . .
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A high-speed collision?
8.1 The cluster environment . . . . . . .
8.1.1 Redshift distribution . . . . .
8.1.2 Distribution of spectral types
8.2 A high speed collision? . . . . . . . .
8.3 Comparison to other observations .
8.3.1 Galaxy dynamics . . . . . . .
8.3.2 X-ray observations . . . . . .
8.3.3 Gravitational lensing . . . . .
Résumé du chapitre en français . . . . . .
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Conclusions
159
Résumé du chapitre en français . . . . . . . . . . . . . . . . . . . . . . 162
III A panchromatic survey of X-ray luminous clusters at
redshift 0.2
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10 Introduction
167
10.1 Sample Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
10.2 Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Résumé du chapitre en français . . . . . . . . . . . . . . . . . . . . . . 176
11 Notes on Individual Clusters
11.1 Abell 68 . . . . . . . . . . .
11.2 Abell 209 . . . . . . . . . .
11.3 Abell 267 . . . . . . . . . .
11.4 Abell 383 . . . . . . . . . .
11.5 Abell 773 . . . . . . . . . .
11.6 Abell 963 . . . . . . . . . .
11.7 Abell 1689 . . . . . . . . .
11.8 Abell 1763 . . . . . . . . .
11.9 Abell 1835 . . . . . . . . .
11.10Abell 2218 . . . . . . . . .
11.11Abell 2219 . . . . . . . . .
11.12Abell 2390 . . . . . . . . .
Résumé du chapitre en français
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6
12 Reduction of CFH12k data
12.1 The CFH12k camera . . . . . . . . . . . . . . . . . . .
12.2 Prereduction: Bias removal, flat fielding . . . . . . . .
12.3 Removal of additive effects: Fringing and “sky ring”
12.3.1 I band fringing . . . . . . . . . . . . . . . . . .
12.3.2 Sky ring . . . . . . . . . . . . . . . . . . . . . .
12.4 Astrometry . . . . . . . . . . . . . . . . . . . . . . . . .
12.4.1 Procedure . . . . . . . . . . . . . . . . . . . . .
12.4.2 Photometric scaling . . . . . . . . . . . . . . .
12.4.3 SWarping the images . . . . . . . . . . . . . . .
12.5 Photometric Calibration . . . . . . . . . . . . . . . . .
12.6 Prospects . . . . . . . . . . . . . . . . . . . . . . . . . .
Résumé du chapitre en français . . . . . . . . . . . . . . . .
A Spectroscopic Survey on Cl0024: The Data
Contents
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List of Figures
2.1
2.2
2.3
2.4
2.5
2.6
2.7
CDM power spectrum P(k) . . . . . . .
Linear growth factor δ(z) . . . . . . . . .
Illustration of spherical collapse model .
Mass variance σ(M) . . . . . . . . . . .
Dark matter halo mass functions . . . .
Excursion set trajectories . . . . . . . . .
Evolution of halo mass function . . . . .
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4.1
4.2
Selection functions of major X-ray cluster surveys . . . . . . . .
Sketch of a gravitational lens system . . . . . . . . . . . . . . . .
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70
5.1
Evolution of the cluster temperature function . . . . . . . . . . .
93
6.1
6.2
6.3
HST/WFPC2 image of the centre of Cl0024+1654 . . . . . . . . . 103
Optical identifications of X-ray sources S6, S8 and S10 . . . . . . 108
R OSAT HRI image of Cl0024+1654 . . . . . . . . . . . . . . . . . 109
7.1
7.2
7.3
7.4
7.5
7.6
7.7
Cl0024: Example spectra and images . . . . . . . . . . .
Cl0024: Redshift comparison with Dressler et al. (1999)
Cl0024: V-I colour-magnitude diagrams . . . . . . . . .
Cl0024: Survey completeness in V magnitude . . . . . .
Cl0024 : Survey completeness map . . . . . . . . . . . .
Cl0024 : Three-dimensional distribution . . . . . . . . .
Colour images of the group of galaxies at z ' 0.49 . . . .
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
Cl0024: Redshift histogram . . . . . . . . . . . . . . . . . . . . . 136
Cl0024: Redshift vs. distance from the cluster centre . . . . . . . 137
Cl0024: Comparison of central and peripheral redshift histograms138
Cl0024: Galaxy number density maps . . . . . . . . . . . . . . . 141
Cl0024: Velocity dispersion profiles . . . . . . . . . . . . . . . . . 142
Cl0024: Scatter plots for important observables . . . . . . . . . . 143
Cl0024, simulation: Initial and final configurations . . . . . . . . 146
Cl0024, simulation: Redshift distribution . . . . . . . . . . . . . 150
Cl0024, simulation: Surface mass density profiles . . . . . . . . . 151
Cl0024: Comparison of redshift histogram to Dressler et al. (1999) 152
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8
List of Figures
10.1 Luminosity distribution of the sample and the XBACs catalogue 170
11.1 Abell 68: Central region of the CFH12k R band image . .
11.2 Spectroscopic data for Abell 68 . . . . . . . . . . . . . . .
11.3 Abell 209: Central region of the CFH12k R band image .
11.5 Spectroscopic data for Abell 267 . . . . . . . . . . . . . .
11.7 Abell 383: HST/WFPC2 image . . . . . . . . . . . . . . .
11.12 Abell 1689: Central region of the CFH12k R band image
11.21 Abell 2390 : Central region of the CFH12k R band image
12.1
12.2
12.3
12.4
12.5
12.6
12.7
12.8
12.9
12.10
12.11
12.12
12.13
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200
Layout of the CFH12k camera . . . . . . . . . . . . . . . .
Static mask showing bad columns and rows . . . . . . . .
Flowchart: CFH12k data reduction . . . . . . . . . . . . . .
I RAF script secrecor.cl . . . . . . . . . . . . . . . . . . . .
Combined R band flat field . . . . . . . . . . . . . . . . . .
Example of fringe subtraction process . . . . . . . . . . . .
Sky ring in an R band exposure of A2390 . . . . . . . . . .
Transmission curves of the CFH12k filters . . . . . . . . .
Flowchart: astrometric calibration . . . . . . . . . . . . . .
Characterisation of the astrometric transformation . . . . .
Relative photometric scales between exposures . . . . . .
Determination of photometric scale factors between chips
Determination of atmospheric extinction coefficient . . . .
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A.1 CFH12k V-band image . . . . . . . . . . .
A.2a CFH12k image subsection with redshifts .
A.2b Continued . . . . . . . . . . . . . . . . . .
A.2c Continued . . . . . . . . . . . . . . . . . .
A.2d Continued . . . . . . . . . . . . . . . . . .
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9
List of Figures
A.2e Continued
A.2f Continued
A.2g Continued
A.2h Continued
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253
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10
List of Figures
List of Tables
6.1
Identification of X-ray sources in the field of Cl0024+1654 . . . . 108
7.1
7.2
7.3
Cl0024: Observing log . . . . . . . . . . . . . . . . . . . . . . . . 117
Cl0024: Comparison of multiply observed objects . . . . . . . . 121
Cl0024: Wavelength ranges for equivalent width measurements 124
8.1
Cl0024: Distribution of spectral types . . . . . . . . . . . . . . . . 144
10.1
10.2
10.3
10.4
Physical properties of the cluster sample . . . . . . . . .
Observing logs for HST, CFH12k and XMM observations
Observing log of the CFH12k observing runs . . . . . . .
Summary of CFH12k observations . . . . . . . . . . . . .
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171
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174
175
12.1 CFH12k chip scale factors . . . . . . . . . . . . . . . . . . . . . . 226
12.2 Galactic extinction values for the cluster fields . . . . . . . . . . 231
A.1 The full spectroscopic catalogue for Cl0024+1654 . . . . . . . . . 238
11
12
List of Tables
Introduction
Les amas de galaxies sont les objets les plus massifs dans l’univers. Dans
le cadre des modèles hiérarchiques de la formation de structures dans l’univers les amas constituent aussi la classe d’objets qui se sont formés le plus
récemment. L’évolution des amas de galaxies est donc observable à petit décalage vers le rouge (redshift), z <
∼ 1, et ils sont des traceurs importants de la
distribution de la matière noire dans l’univers.
L’importance des amas de galaxies pour la recherche cosmologique s’appuie sur deux prédictions théoriques :
(1) La densité en nombre d’amas de galaxies d’une masse donnée (donc leur
fonction de masse) peut être prédite à partir de modèles linéarisés de l’évolution du champ de densité de masse avec une simple prescription pour la formation d’objets gravitationellement liés. Ceci est l’approche de Press & Schechter (1974) qui a été étendue de plusieurs façons travers les années (Bond et al.
1991; Lacey & Cole 1993). Les résultats ont été confirmés et affinés par des simulations numériques (Sheth & Tormen 1999; Jenkins et al. 2001). La fonction
de masse des amas et son évolution dépendent fortement des paramètres cosmologiques : une mesure de la fonction de masse locale contraint la normalisation du spectre de puissance, σ8 , si une estimation indépendante de la densité
moyenne ΩM dans l’univers est utilisée. L’évolution de la fonction de masse
lève la dégénérescence entre σ8 et ΩM et peut apporter des fortes contraintes
sur ΩM . Des mesures de l’évolution de la fonction de masse des amas de galaxies forment une des bases de la recherche cosmologique actuelle et future.
(2) Des simulations numériques à haute résolution de la formation des structures dans un univers composé de matière noire froide révèlent une structure
universelle de densité pour des objets gravitationellement liés sur un large
intervalle de masse, largement indépendant des paramètres utilisés dans les
simulations (ΩM , la pente du spectre de puissance, etc.). Le profil universel de
densité a un “cusp” (singularité) dans le centre (Navarro et al. 1997), même si
la pente limite exacte reste sujet de débat (Moore et al. 1998; Ghigna et al. 2000).
La prédiction d’un profil de masse universel est très robuste dans le cadre de
cosmologies dominées par la matière noire froide. La question si les galaxies et
amas de galaxies suivent réellement un tel profil concerne donc directement le
type de matière noire et des observations de profils d’amas de galaxies peuvent
13
14
Introduction
contraindre les propriétés des particules élusives dont la matière noire est composée.
Ces deux prédictions sont formulées en terme de masse. Malheureusement,
les masses des amas de galaxies ne peuvent pas être mesurées directement,
donc une investigation de ces prédictions dépend de la mesure d’autres observables qui sont liées à la masse. L’effet de lentille gravitationelle mesure des
masses directement. En revanche, la masse donnée par une analyse de lentille
gravitationelle est la somme de toute masse contenue dans un cylindre entre
la source et l’observateur. L’effet de lentille donne des masses précises pour
des amas isolés ; si plusieurs amas sont alignés le long de la ligne de visée, les
contributions des différentes composantes doivent être séparées autrement ;
sinon la masse d’un amas est surestimée si la superposition n’est pas détectée.
Des observations X de l’émission du gaz chaud dans les amas de galaxies observent une composante de matière limitée à des puits du potentiel gravitationel, donc des effets de projection ne sont pas importants. Des recherches
d’amas de galaxies en X représentent donc l’approche à construire des catalogues d’amas la moins biaisée. En revanche, la conversion des propriétés
observées en X (distribution de la brillance de surface, température, luminosité) en masse dépend de l’hypothèse essentielle que le gaz soit dans un état
d’équilibre hydrostatique. Si le gaz dévie d’un état d’équilibre, par exemple
par une collision avec un autre amas ou accrétion d’un groupe de galaxies,
l’émission en X n’est plus liée simplement au potentiel gravitationel, et les mesures en X donneront des estimations de masse biaisées. Enfin, la cinématique
des galaxies dans un amas donne aussi des estimations de masse. Si les galaxies sont dans un équilibre viriel, des mesures de la dispersion des vitesses
des galaxies peuvent être utilisées pour estimer la masse totale contenue dans
l’amas. En revanche, si la distribution des vitesses contient une composante
systématique, la dispersion des vitesses surestime la masse de l’amas.
Toutes ces méthodes d’estimation de masse sont utilisées, mais souvent
les résultats sur un même amas ne sont pas cohérents. Puisque la fonction de
masse des amas et son évolution seront mesurées sous forme de fonction de
température X, cisaillement gravitationel ou autre, la calibration des relations
entre masse et d’autres observables est un pas essentiel dans la recherche cosmologique avec les amas de galaxies.
Le manuscrit présent contribue à ce processus de deux manières. Dans
la partie II je présente une étude détaillée d’un amas de galaxies à décalage
spectral z = 0.395, Cl0024+1654. On savait que cet amas donne des masses
discrépantes dérivées à partir de son effet de lentille gravitationelle, la cinématique de ses galaxies, ou encore son émission en X. Un modèle de lentille (Tyson et al. 1998) donne un profil central de la densité dans cet amas qui contient
un cœur plat, en contradiction avec les prédictions des simulations de formation de structures avec la matière noire froide. Ce résultat a incité plusieurs
Introduction
15
auteurs à étudier des formes alternatives de matière noire (tiède, interagissant,
etc.).
J’ai réduit et analysé les données pour un survey spectroscopique à grand
champ dans Cl0024+1654. Le catalogue de redshifts et mesures spectroscopiques qui résulte de ce travail est actuellement le plus large dans un amas
à tel redshift. Il est présenté dans le chapitre 7 et reproduit dans son intégralité
dans le tableau A.1. L’analyse de la distribution des redshifts révèle une structure complexe bimodale, ce qui montre que Cl0024+1654 n’est pas un amas
simple en équilibre comme l’on avait pensé avant. La bimodalité de la distribution des redshifts nécessite la présence de (au moins) deux composantes
distinctes ; la structure détaillée de la distribution indique une forte interaction entre ces composantes. La distribution des redshifts peut être expliquée et
reproduite par un scénario de collision, où deux amas massifs entrent en collision à grande vitesse relative. Des simulations numériques (en collaboration
avec Ben Moore) ont été utilisées pour étudier le scénario et il est montré que la
distribution des redshifts est en fait reproduite dans ce scénario. En plus, la distribution projetée de masse après la collision est très semblable au profil déduit
par Tyson et al. (1998) et montre un cœur plat même si les deux amas initials
ont été construit avec un profil type matière noire froide, donc avec un cusp
central. Je discute des conséquences de ce résultat pour l’interprétation des
mesures de masse de Cl0024+1654 en particulier et pour des amas en général
dans le chapitre 9. Les résultats de cette étude ont été publiés dans deux articles, Czoske et al. (2001) et Czoske et al. (2002) ; j’ai aussi contribué à une
analyse des propriétés en X de cet amas, publiée dans Soucail et al. (2000).
Dans la partie III je présente un projet ambitieux qui a pour but d’obtenir
des observations aussi complète que possible sur un échantillon sélectionné
de façon homogène de 12 amas de galaxies lumineux en X à redshift z = 0.2.
Le travail présenté ici ne constitue qu’une partie de ce projet qui se situe dans
une collaboration internationale menée par Jean-Paul Kneib. Les bases du projet sont imagerie à haute résolution avec HST/WFPC2, imagerie multi-couleur
à grand champ avec la caméra CFH12k sur le TCFH, et des observations spectroscopiques et imageries en X avec XMM/Newton ; ces observations sont
complémentées par des observations spectroscopiques multi-objet afin d’obtenir des mesures des dispersions des vitesses et d’étudier la distribution des
redshifts dans les amas (ce qui est particulièrement important vu les résultats
obtenus dans Cl0024+1654, partie II).
Les buts principaux de ce projet sont l’étude détaillée des profils de masse
−1
des amas sur des échelles entre ∼ 10 kpc et au-delà du rayon viriel (>
∼ 1.5 h
Mpc), à partir de modélisation des arcs géants détectés dans les images HST/
WFPC2, et des analyses de cisaillement et déplétion gravitationels à partir des
images CFH12k, ainsi qu’une calibration à grande masse de la relation entre les
masses M des amas et leurs températures TX ; un aspect nouveau de ce projet
est que les masses seront mesurées de façon homogène à partir de l’effet de
16
Introduction
lentille des amas. Des mesures précédentes de la relation M–TX ont utilisé des
masses estimées à partir des propriétés X qui n’étaient donc pas entièrement
indépendantes de la mesure de température.
Dans le chapitre 10 je présenterai l’échantillon, décrierai la sélection de
l’échantillon à partir du catalogue XBACs (Ebeling et al. 1996) et je donnerai
les détails des observations faites. Ensuite je présenterai les amas individuels
et des analyses de nos mesures des redshifts dans ces amas. Les dispersions
de vitesses sont élevées dans tous les cas, ce qui montre que la sélection par
luminosité X sélectionne en effet les amas les plus massifs. En revanche, des
indications de sous-structures sont présentes dans plusieurs amas et devront
être prises en compte dans l’interprétation des autres observations. Une analyse de l’effet de lentille forte dans le centre d’un des amas (Abell 383) a été
publiée dans Smith et al. (2001).
Je me suis occupé dans le cadre de ce projet avec la prise et réduction
des images CFH12k. Puisque des méthodes de réduction de ce type d’images
mosaı̈ques n’étaient pas encore disponibles au début de ce projet, j’ai dû développer des nouvelles procédures pour réduire ces données, qui sont décrites
en détail dans le chapitre 12. Puisque les images seront utilisées pour des analyses de cisaillement gravitationel faible, la régistration astrométrique exacte
des poses individuelles dans un repère commun était la principale contrainte
à satisfaire dans la réduction des données afin de ne pas dégrader les mesures
des formes des galaxies par un alignement inexacte des poses. J’ai résolu ce
problème en utilisant des images DSS comme repères astrométriques absolus
et après minimisant les déviations de la position finale pour un large nombre
dans les images CFH12k à partir de mesures internes seulement. La déviation
rms des positions dans le repère final est de l’ordre 0.00 01, donc un vingtième
d’un pixel CFH12k. Aucun résultat scientifique des images CFH12k sera rapporté dans cette thèse, l’analyse de cisaillement faible étant toujours en cours.
La partie I fournit une compilation extensive de résultats (plutôt théoriques)
concernant les amas de galaxies, expliquant en détail la dérivation de la fonction de masse des amas, des méthodes de mesure de masse des galaxies et le
profil universel de densité, ce qui donne la motivation en particulier pour le
survey d’amas présenté dans la partie III.
Le corps scientifique de cette thèse est écrit en anglais ; je fournit un résumé
en français pour chaque chapitre.
Introduction
Clusters of galaxies are the most massive gravitationally bound objects in the
Universe. If the currently accepted hierarchical models for the formation of
structure in a cold dark matter dominated Universe are correct, they are also
the latest class of objects to have formed. The evolution of clusters of galaxies
as a class should therefore be observable at low redshift, z <
∼ 1, making them
excellent tracers for the evolution of the underlying dark matter distribution.
The importance of galaxy clusters hinges primarily on two theoretical predictions:
(1) The number density of clusters of galaxies at a given mass (i. e. the mass
function) can be predicted analytically from linearised models of the evolution of the density field with a simple prescription for the formation of gravitationally bound objects. This is the Press-Schechter approach which has been
extended in various ways over the years (Press & Schechter 1974; Bond et al.
1991; Lacey & Cole 1993). The results have been confirmed and refined with
numerical simulations (Sheth & Tormen 1999; Jenkins et al. 2001). The cluster
mass function and its evolution depend strongly on the cosmological parameter: Measurements of the local mass function constrain the normalisation of
the matter power spectrum, σ8 , if an independent estimate of the mean mass
density in the Universe, ΩM , is used. The evolution of the mass function breaks
the degeneracy between σ8 and ΩM and provides strong leverage for the determination of ΩM . Measurements of the evolution of the cluster mass function
form therefore a cornerstone of current and future research in cosmology.
(2) High-resolution numerical simulations of structure formation in a Universe
dominated by cold dark matter reveal a universal density structure for collapsed objects over a wide range of masses, largely independent of the actual parameter values used in the simulation (ΩM , slope of the power spectrum, etc.). The universal density profile is found to have a cusp in the centre
(Navarro et al. 1997), although the exact limiting slope of the density profile in
the centre is still matter of some debate (Moore et al. 1998; Ghigna et al. 2000).
The prediction of a universal profile is very robust in the context of cold dark
matter dominated cosmologies. The question whether real galaxy and cluster
halos follow this type of profile therefore relates immediately to the type of
17
18
Introduction
the dark matter and observations of cluster mass profiles put constraints on
the properties of the elusive dark matter particles.
Both these predictions are made in terms of mass. Unfortunately, cluster
masses cannot be measured directly, so that investigation of the predictions
depends on the measurement of other observables which are related to mass.
Gravitational lensing directly measures mass. However, the mass obtained
by lensing analyses is a weighted sum of all the mass contained in a cylinder between the observer and the source. Lensing provides accurate masses
for isolated clusters; however, if several clusters are aligned along the line-ofsight additional information has to be used to disentangle the separate mass
components, or, if the alignment goes undetected, cluster masses will be overestimated. X-ray observations of the emission of the hot intra-cluster gas observe a matter component which is confined to the potential wells of clusters,
so that projection effects are not important. X-ray searches for clusters of galaxies are therefore the most common and probably least biased approach to constructing cluster catalogues. However, the conversion from observed X-ray
properties (surface brightness distribution, temperature, luminosity) to mass
depends on the crucial assumption that the gas be in hydrostatic equilibrium.
If the gas has not yet settled down in the potential well or has been stirred
up by a merger with another cluster its X-ray emission will not be simply related to the gravitational potential and X-ray measurements will give biased
mass estimates. Finally, the kinematics of galaxies within a cluster provide
a simple way towards estimating cluster masses. If the galaxies are in virial
equilibrium, measurements of the galaxy velocity dispersion can be used to
estimate the total mass contained in the cluster. However, if there is a bulk
velocity component in the galaxy velocity distribution, the velocity dispersion
will overestimate the cluster mass.
All these mass estimation methods are routinely used but often yield inconsistent answers for the same cluster. Since the cluster mass function and
its evolution will be measured in the form of a X-ray temperature function,
gravitational shear function or other, the calibration of the relations between
mass and other observables is a crucial step towards cosmological research
with clusters of galaxies.
My Ph D dissertation contributes to this process in two ways. In Part II
I will present a detailed study of an individual cluster of galaxies at redshift
z = 0.395, Cl0024+1654. This cluster was known to show a discrepancy between masses derived from gravitational lensing, galaxy kinematics and X-ray
observations. A lensing model of the central mass distribution (Tyson et al.
1998) showed a flat core which is at variance with the predictions from cold
dark matter structure formation simulations, and has caused a number of authors to investigate alternative forms of dark matter (warm, self-interacting,
etc.).
Introduction
19
I reduced the data for an extensive wide-field spectroscopic survey of this
cluster and constructed the largest catalogue of redshifts and spectral properties at this redshift available to date. The catalogue is presented in Chapter 7
and listed in its entirety in Table A.1. The analysis of the redshift distribution
surprisingly revealed a complex bimodal structure, showing that Cl0024+1654
is not the simple relaxed cluster it had been assumed to be before this work.
The bimodality of the redshift distribution requires (at least) two distinct components; the detailed structure of the redshift distribution hints at a strong
interaction between these two components. The redshift distribution can be
explained and reproduced by a collision scenario, where two massive clusters
of galaxies collide at high relative speed. We investigate this scenario using numerical simulations (provided by Ben Moore) and show that the redshift distribution is indeed reproduced in this scenario. Furthermore, the projected mass
distribution seen after the collision is very similar to the profile deduced by
Tyson et al. (1998) for Cl0024+1654, in that it shows a flat core even though the
initial clusters were set up with standard cold dark matter, i. e. cuspy, density
profiles. I discuss consequences of this finding for the nature of the cold dark
matter and the interpretation of cluster mass measurements in Cl0024+1654
and in general in Chapter 9. The results of this study were published in two
papers, Czoske et al. (2001) and Czoske et al. (2002), and I also contributed to
an analysis of the X-ray properties of this cluster in Soucail et al. (2000).
Part III presents an ambitious ongoing project led by Jean-Paul Kneib which
aims at obtaining as complete a data set as possible on a homogeneously selected sample of 12 X-ray luminous clusters of galaxies at redshift z = 0.2. The
corner stones of the project are high-resolution imaging from HST/WFPC2,
wide-field multicolour imaging with the CFH12k camera on CFHT, and X-ray
spectral and imaging observations with XMM; these observations are complemented with multi-object spectroscopic observations in order to obtain velocity dispersion measurements and to investigate the redshift distribution
within the clusters (especially important in the light of the results obtained
on Cl0024+1654, Part II).
The main goals of the project are the detailed investigation of the cluster
mass profiles on scales ranging from ∼ 10 kpc to beyond the virial radius
−1
(>
∼ 1.5 h Mpc), from strong lensing analysis of the configuration of giant arcs
seen in the HST/WFPC2 images of the cluster centres, and weak lensing shear
and depletion analyses from CFH12k images; as well as a calibration of the
high-mass end of the relation between cluster mass M and the X-ray temperature TX . The essential new aspect of this project is that masses will be derived
in a homogeneous way from the lensing effects of the clusters, whereas previous measurements of the M–TX relation used masses derived from the X-ray
properties, which were therefore not independent of the temperature measurements.
20
Introduction
In Chapter 10 I present the sample, show how it was selected from the
XBACs catalogue (Ebeling et al. 1996) and give details on the observations.
I then present the individual clusters and provide analyses of the redshift
measurements. The velocity dispersions measured for these clusters are high
throughout, which confirms that our selection by X-ray luminosity indeed selects for the most massive clusters. However, hints to the presence of substructure are found in several clusters which will have to be taken into account for
the interpretation of the other observations. A strong lensing analysis of one
cluster from the sample has been published so far (Abell 383) by Smith et al.
(2001).
I was concerned in the frame of this project with obtaining and reducing
the CFH12k images. Since data reduction pipelines for this type of wide-field
imaging data were not available when the project started I had to set up my
own data reduction procedures, which are described in detail in Chapter 12.
Since the data are to be used for weak shear analyses, exact astrometric registration of the individual exposures of a field onto a common output grid was
the most important requirement in the data reduction process, so as not to
degrade the galaxy shape measurements by inaccurate alignment of the exposures. I solved this problem by using a DSS frame as the absolute astrometric reference and then minimising the deviations from the target position of a
large number of objects from the CFH12k images from internal measurements
alone. The achieved rms deviation in the target position is of order 0.00 01, i. e.
1/20 of a CFH12k pixel. No scientific results from the CFH12k images are reported in this dissertation, the weak lensing analysis of these images is still
on-going.
Part I provides an extensive compilation of (mostly theoretical) results concerning clusters of galaxies, explaining in some detail the derivation of the
cluster mass function, methods to measure cluster masses, and the universal
density profile, thus providing the motivation in particular for the cluster survey presented in Part III.
Part I
Clusters of galaxies
Chapter 1
Cosmological Framework
In this chapter I will very briefly review the isotropic and homogeneous general relativistic world models that form the background for most of the current
research in cosmology. The main aim of this chapter is to establish notation and
it is by no means exhaustive. Many pedagogical reviews of this subject exist,
for instance Peebles (1993) or Peacock (1999). My diploma thesis (Czoske 1995)
also discussed this subject and discussed methods to determine the cosmological parameters; however, recent progress in this field has turned my diploma
thesis obsolete amazingly quickly.
1.1
Background Cosmology
For the background cosmology I shall use in the following the standard Friedmann–Lemaı̂tre–Robertson–Walker (FLRW) model, which yields a set of solutions of Einstein’s equations for a spatially homogeneous and isotropic Universe. The geometry of the Universe is given by the Robertson–Walker metric:
dr2
2
2
2
2
+ r dθ + sin θ dφ
ds = dt − R (t)
1 − kr2
2
2
2
,
(1.1)
with k = +1 for spherical, k = −1 for hyperbolic, and k = 0 for flat spatial
sections respectively1 .
The kinematics of the Universe can be described through the single function R(t) (the scale factor) for which Einstein’s equations with the metric (1.1)
yield Friedmann’s equation:
2
Ṙ
k
8πG
+ 2 =
R
3
R
1 As
usual, I use c = 1 for general formulas.
23
∑ ρi (t)
i
,
(1.2)
24
Chapter 1. Cosmological Framework
where the ρi denote the energy densities in various components which differ through their equation of state pi = f (ρi ). The time evolution of ρi is then
given by the continuity equation for the energy–momentum tensor (essentially
the first law of thermodynamics applied to the adiabatic expansion of the Universe):
d ρi R3 = −pi d R3
.
(1.3)
I assume that the dynamics of the Universe are governed by two matter/energy components, on the one hand non–relativistic matter (“dust”) with
an equation of state p = 0, i. e. ρM ∝ R−3 , and comprising baryonic matter2
and non–baryonic dark matter; on the other hand a form of “dark energy”, for
example the cosmological constant Λ with an equation of state pΛ = −ρΛ , so
that ρΛ = const. The cosmological constant can be explained (in principle) as
a vacuum energy density, but more general forms of dark energy have been
considered lately, with an equation of state pX = wX ρX , where wX < −1/3 to
account for the repulsive effect necessary to cause the Universe to accelerate at
the present time (Turner & Riess 2001). As will be shown in Sect. 5.2, dark energy plays only a minor role in studies of clusters of galaxies, and I will restrict
the discussion to wX = −1, i. e. a cosmological constant.
Radiation is dynamically important only in the radiation dominated era
through until matter–radiation equivalence at redshift z ∼ 104 , but plays no
role in particular after recombination, which is the period of interest here. The
equation of state for radiation (which also includes relativistic particle species
such as light neutrinos) is prad = ρrad /3, so that ρrad ∝ R−4 .
The energy content of the Universe is usually parameterized through the
Ω–parameters, which give the energy density at the present time, t0 , relative
to the critical density ρcrit = 3H02 /8πG:
• Nonrelativistic matter:
ΩM =
8πGρM
3H02
(1.4)
• Dark energy (cosmological constant):
ΩΛ =
8πGρvac
Λ
=
2
3H0
3H02
.
(1.5)
Another Ω-parameter that is often used is related to the baryon density, ΩB .
The curvature can be then described by an analogous parameter
Ωk = −
k
R20 H02
.
(1.6)
2 The pressure of baryons is negligible at the mean density over most of the history of the
Universe. Furthermore, the amount of baryons is small compared to the amount of nonbaryonic dark matter, which is strictly pressure-less if it is constituted by weakly-interacting particles as is usually assumed.
25
1.2. Cosmological Parameters
In terms of the Ω parameters and changing the independent variable from
time t to redshift z = R0 /R(t) − 1, Friedmann’s equation can be written as
h
i1/2
Ṙ
≡ H(z) = H0 ΩM (1 + z)3 + Ωk (1 + z)2 + ΩΛ
R
,
(1.7)
where relativistic energy components have been neglected. Evaluating this
expression at the present time yields the simple relation between the Ω parameters:
ΩM + Ω Λ + Ωk = 1 .
(1.8)
The model is thus characterised by the three parameters ΩM , ΩΛ and H0 . Discussions of the behaviour of the solutions of Friedmann’s equations for different combinations of ΩM and ΩΛ can be found for instance in Bondi (1961),
Stabell & Refsdal (1966), Carroll, Press, & Turner (1992) or Czoske (1995). A
simple standard case is the Einstein–de Sitter model with ΩM = 1, ΩΛ = 0.
This model has flat spatial sections, the scale factor varies as R ∝ t2/3 and the
age of the Universe at redshift z is t ∝ (1 + z)−3/2 .
1.2
Cosmological Parameters
The initial conditions for solutions of Friedmann’s equation (1.2) are given in
terms of the scale parameter at the present time, R(t0 ) ≡ R0 3 , and the expansion rate at the present time, the Hubble constant,
Ṙ ≡ 100 h km s−1 Mpc−1 .
(1.9)
H0 = R t=t0
In recent years there has been some convergence concerning the observational
value of H0 , which is now found to be in the range 60 . . . 80; Freedman et al.
(2001) e. g. find H0 = (72 ± 8) km s−1 Mpc−1 from HST observations of Cepheid
variable stars in the Virgo cluster. The exact value of H0 is however of no importance in the following, and I shall give the appropriate scaling of physical
parameters in terms of h, defined in Eq. (1.9).
Another kinematical parameter is the “deceleration” parameter
R̈R q0 = − 2 .
(1.10)
Ṙ t=t0
In practice, cosmological observations are usually modelled in terms of the
matter/energy content of the Universe, ΩM , ΩΛ , and possibly others, which
3 This
is often normalised to R0 = 1, in which case the scale factor is usually denoted by a
(e. g. Peebles 1993).
26
then determine the kinematical properties, i. e. the time dependence R(t),
through (1.7). In the two-parameter model,
q0 =
ΩM
− ΩΛ
2
.
(1.11)
An accelerating universe has q0 < 0, a decelerating universe q0 > 0. In an
Einstein-de Sitter Universe, q0 = 1/2.
There has been a lot of progress over the past few years in the determination of the parameters ΩM and ΩΛ , especially through observations of highredshift supernovae of type Ia, fluctuations in the cosmic microwave background (CMB) and also counts of clusters of galaxies.
1.2.1
High-Redshift Supernovae
The distance modulus of an object at redshift z in the context of FLRW models
depends on the cosmological parameters H0 , ΩM and ΩΛ and can be compared
to observations if the absolute magnitude of the object is known. Usually this
is not the case, at least not to sufficient accuracy to allow determination of the
cosmological parameters. However, the absolute scale is fixed solely by H0 , so
that a plot of apparent magnitude versus redshift (the Hubble diagram) for a
class of objects with a constant, if unknown, absolute magnitude still provides
sufficient information to determine ΩM and ΩΛ .
Supernovae of type Ia (SNe Ia) are believed to be carbon/oxygen white
dwarf stars, whose mass is driven by mass accretion to the Chandrasekhar
limit, which leads to a thermonuclear explosion. Their peak luminosity is
largely determined by generic thermodynamical processes and therefore little
dependent on their individual constitution or history. SNe Ia are not perfect
standard candles since there is a correlation between peak brightness and the
rate at which their luminosity declines after the explosion. If this correlation
is corrected for, the scatter of peak brightness is reduced to 0.15 mag for a local sample. SNe Ia are intrinsically bright (MB ' −19.4), so that they can be
detected and analysed out to large redshifts. Improvements in CCD detector
technology during the 1990s have made it feasible to compile large samples of
SNe Ia out to redshift z ∼ 1 and to follow their light curves over several weeks.
Two major research groups have compiled independent samples of ' 20 to
40 SNe Ia at redshifts between 0.15 and 0.8: the Supernova Cosmology Project
(Perlmutter et al. 1995, 1999) and the High-z Supernova Search (Schmidt et al.
1998; Riess et al. 1998) (Fillipenko 2001 gives an entertaining review). The
cosmological results of both teams are consistent and favour at high levels of
confidence a Universe that is accelerating at the present epoch, has a non-zero
cosmological constant and a dynamical age of about 14.5 ± 1.5 Gyr. In particular, an Einstein–de Sitter Universe is excluded by these observations at more
than 99% confidence. The redshift range over which SNe Ia have been found
27
so far is too small to allow determination of ΩM and ΩΛ individually; instead
there is a degeneracy in the ΩM –ΩΛ plane and the best constrained combination is 0.8 ΩM − 0.6 ΩΛ = −0.2 ± 0.1 (Perlmutter et al. 1999). Assuming a flat
Universe (ΩM + ΩΛ = 1) yields the current consensus model4 with ΩM ' 0.3
and ΩΛ ' 0.7.
The main uncertainties in the determination of ΩM and ΩΛ from supernovae are however systematic, not statistical (Riess et al. 1998; Perlmutter et al.
1999). The dimming due to the larger distances in an ΩΛ dominated Universe
could be mimicked by astrophysical effects, for example intrinsic extinction in
the supernovae’s host galaxies. Since reddening is taken into account in the
analyses to provide an estimate of extinction according to the standard interstellar extinction curve, there would have to be a component of grey extinction,
i. e. independent of wavelength. Gravitational lensing by the inhomogeneous
large-scale mass distribution would on average lead to demagnification of the
supernovae by ∼ 0.1 mag. SNe Ia are not completely understood theoretically and there remains the possibility of luminosity evolution, which might
also mimic cosmological dimming. All these effects are considered too small
to challenge the basic cosmological conclusions of the supernova projects, but
need to be understood and taken into account for precise determinations of
ΩM and ΩΛ .
Searches for supernovae at larger redshifts, z >
∼ 1, hold even more promise
for cosmology since they probe the decelerating phase of the Universe before
the cosmological constant (or another form of dark energy) becomes dynamically dominant. This will allow more accurate discrimination between the
effects of cosmology and astrophysical effects such as grey extinction or luminosity evolution of supernovae (Riess et al. 2001). However, the increasing differences between the cosmological and astrophysical models at high redshifts
are counteracted by the increasing difficulty to determine accurate observational parameters for the supernovae and their light curves. For instance, one
challenge in the analysis and compilation of samples of SNe Ia at high redshifts
will be the determination of precise redshifts, as evidenced by the case of the
current record holder, SN 1997ff (Riess et al. 2001), whose redshift z = 1.7+0.10
−0.15
is essentially a photometric redshift, determined in a number of ways. While
the parameters of SN 1997ff are too uncertain to be very useful for the determination of the cosmological parameters, its brightness is sufficiently well
constrained to provide robust evidence against a grey extinction model. There
has been some discussion, however, whether SN 1997ff might be magnified
by the gravitational lens effect due a bright foreground galaxy near the line of
sight (Mörtsell et al. 2001).
4 also
referred to as the “bandwagon model”
28
The CFHT Legacy Survey5 , a wide-field imaging survey poised to start in
early 2003, is expected to find ∼ 1000 SNe Ia at redshifts between 0.3 and 0.9
over five years. A wide-field optical and near-infrared space mission, SNAP6 ,
has also been proposed to search for high-z supernovae, starting in 2010.
1.2.2
Fluctuations in the Cosmic Microwave Background
Before the recombination phase, the baryon–photon plasma present in the
early Universe underwent regular sound compression and rarefaction. The
density pattern caused by these acoustic oscillations is visible today in temperature fluctuations of the cosmic microwave background radiation (CMB),
emitted from the last-scattering surface during the epoch of decoupling between photons and baryons at z ∼ 1000 (e. g. Hu, Sugiyama, & Silk 1997,
for an overview). The standard adiabatic cold dark matter models predict a
harmonic series of peaks (“acoustic” or “Doppler” peaks) in the angular temperature fluctuation power spectrum of the CMB, the scales and amplitudes of
which depend on a whole host of cosmological parameters, such as the Hubble
constant, the density parameters ΩM and ΩΛ , the baryon density Ωb , the slope
of the initial matter power spectrum, ns , and others. Comparison between the
observed CMB power spectrum with theoretically predicted spectra can thus
be used to put constraints on the values of these parameters.
After the first detection of large-scale temperature fluctuations in the CMB
at a level of δT/T ' 10−5 by the COBE satellite (Smoot et al. 1992), a host of
ground-based experiments has measured the fluctuation spectrum on smaller
scales. In particular the experiments B OOMERANG (de Bernardis et al. 2000),
M AXIMA (Lee et al. 2001) and D ASI (Halverson et al. 2001) have measured the
spectrum to sub–degree angular scales and provided unambiguous detections
of the first three acoustic peaks.
Cosmological parameters are extracted from the measured fluctuation spectrum by comparison with theoretical predictions calculated over a grid in a
space of typically seven cosmological parameters (Stompor et al. 2001; Pryke
et al. 2001; de Bernardis et al. 2002), and confidence intervals on individual parameters are determined by marginalising over the other parameters or holding these parameters constant at their maximum likelihood values. There is a
lot of ongoing discussion in the literature over how to do this and many analysis methods have been applied to essentially the same data sets. Usually the
results are in good accord with one another.
The projected confidence regions in the ΩM –ΩΛ plane are long ellipses with
the major axis along a line ΩM + ΩΛ = const., which shows that the CMB data
effectively constrain the spatial curvature of the Universe. This is due to the
5 http://www.cfht.hawaii.edu/Science/CFHLS
6 http://snap.lbl.gov
29
fact that the separation of the acoustic peaks is set by the sound horizon at
decoupling, i. e. the distance that sound waves could have travelled before
this epoch. The angular size of the sound horizon as seen today is given by
the ratio of two distances and hence primarily depends on the geometry of the
Universe, i. e. on the curvature (Doroshkevich et al. 1978). The data are highly
consistent with a spatially flat Universe, Ωk = 0. For example, de Bernardis
et al. (2002) determine a value of ΩM + ΩΛ ≡ 1 − Ωk = 1.04 ± 0.05, using
the full B OOMERANG data set, and consistent results have been obtained from
data from M AXIMA (Stompor et al. 2001), D ASI (Pryke et al. 2001) as well as
from combined data sets from several experiments (Douspis et al. 2001).
1.2.3
Direct Mass Measurements
The matter density in the Universe can be determined fairly directly by measuring masses of individual objects or regions of space and extrapolating to
obtain a mean mass density. On galactic scales, the rotation curves of spiral galaxies and the velocity dispersion profiles of elliptical galaxies provide a
means to determine the mass profiles of galactic haloes. The masses of clusters of galaxies can be estimated in a number of independent ways, from the
kinematics of the member galaxies, the distribution and temperature of the
hot intra–cluster gas, as determined from X-ray observations, and from strong
and weak gravitational lensing (see Chapter 4 for a more complete discussion
of these methods). On both galactic and cluster scales these methods highlight
the presence of dark matter: The mass-to-light ratios determined are much
larger than typical stellar mass-to-light ratios. The mass-to-light ratio is found
to increase as one goes to larger scales and reaches values of M/L ∼ 300h on
cluster scales. Bahcall et al. (1995) argue that studies on still larger scales (see
below) show no further increase of the mass-to-light ratio, and that this value
can thus be assumed to be representative of the global value for the Universe.
On the basis of this assumption, one obtains an estimate of the total mass density in the Universe by multiplying with the mean luminosity density, e. g.
−3
8
L B = (1.93+0.8
in the B band (Efstathiou et al. 1988). The
−0.6 ) × 10 h L Mpc
result is ΩM ' 0.3, significantly short of the critical density (Bahcall et al. 1995).
A related way to estimate the mass density from clusters of galaxies is
through measurements of the fraction of baryons (intra-cluster gas and stars)
relative to the total mass of clusters of galaxies, which typically amounts to
15% (e. g. Allen et al. 2002). The baryon density in the Universe is very well
constrained from observations of the abundance of light elements (deuterium,
helium, lithium) in conjunction with models for the synthesis if these elements
in the very early Universe (big bang nucleosynthesis, BBN) and is given by
ΩB h2 = 0.106 ± 0.006 (Olive 1999). Assuming that the baryon fraction in clus-
30
ters is typical for the universal baryon fraction, one thus finds a universal mass
density corresponding to ΩM ' 0.3.
Another way to probe the mass density on small scales is to observe peculiar motions of galaxies or clusters of galaxies since these are caused by the
total matter distribution. Results from these studies scatter fairly widely, from
ΩM ' 0.3 to ΩM <
∼ 1 in at least one case.
The local abundance of rich clusters of galaxies depends on both the normalization of the matter power spectrum, σ8 , and the density of matter, ΩM .
The normalization of the power spectrum can be determined independently,
although on vastly different scales, from observations of the CMB temperature fluctuations. If one accepts the CMB normalization then the local cluster
abundance is consistent with a low matter density Universe. However, due to
the large difference between the scales on which CMB fluctuations are measured and cluster scales, this approach involves strong assumptions on the
shape of the power spectrum and can at best be considered indicative. A very
promising extension of this method is to observe the evolution of the cluster
abundance with redshift: In this case the degeneracy between σ8 and ΩM is
broken, and provides a strong handle on ΩM . This approach is discussed in
Sections 2.6 and 5.2 and provides one of the main motivations for the cluster
survey presented in Part III of this dissertation.
Finally, recent results on large galaxy redshift surveys (Sloan, 2dF) provide
measurements of the matter power spectrum on intermediate scales. This is
in a sense an equivalent measurement to the CMB fluctuations, at a different
epoch and on somewhat different scales (Peacock et al. 2001).
1.2.4 Concordance Studies
Each of the methods summarized in the previous sections provides useful
cosmological information taken by itself: Supernovae searches provide compelling evidence that the Universe is currently accelerating, the CMB fluctuations show that in all likelihood the Universe is spatially flat, and counts of
clusters of galaxies as well as dynamical analyses on various scales show that
the matter density in the Universe is significantly less than the critical density.
However, all the methods have degeneracies in the ΩM –ΩΛ plane, i. e. they
cannot determine both ΩM and ΩΛ independently. Combining results from
different methods yields much better constraints on the parameters, in particular since the degeneracies, e. g. from supernovae and the CMB fluctuations
are nearly orthogonal in the ΩM –ΩΛ plane.
Many authors have been trying to find methods to optimally extract the
values of the cosmological parameters from the variety of different data sets
(e. g. Bahcall et al. 1999; Efstathiou et al. 1999). Maybe not too surprisingly
their results differ only in detail whereas the big picture remains the same:
31
There seems to be convergence towards one point in the ΩM –ΩΛ plane, which
is around ΩM = 0.3, ΩΛ = 0.7. For example, Efstathiou et al. (1999) give ΩM =
+0.17
0.25+0.18
−0.12 and ΩΛ = 0.63−0.23 , at the 95% confidence level, from a combination
of CMB and supernova data. By contrast, Douspis et al. (2001) find ΩM ' 0.8,
ΩΛ ' 0.3, from combining CMB with cluster data, showing that there is not
yet complete consensus over the values of the cosmological parameters.
For illustrative purposes I will use the current consensus model (ΩM = 0.3,
ΩΛ = 0.7), as well as the Einstein–de Sitter model (ΩM = 1, ΩΛ = 0), which
has the great advantage that many analytical expressions are simple and can
be written in closed form.
Résumé du chapitre :
Le cadre cosmologique
Ce chapitre rappelle brièvement les bases de la cosmologie relativiste qui
forme l’arrière–plan de la recherche actuelle en cosmologie.
Le modèle standard dit de Friedmann–Lemaı̂tre–Robertson–Walker suppose que l’univers est spatialement homogène et isotrope, ce qui implique que
sa géométrie est décrite par la métrique de Robertson–Walker7 :
dr2
2
2
2
2
ds = dt − R (t)
+ r dθ + sin θ dφ
1 − kr2
2
2
2
;
les sections spatiales sont sphériques si k = 1, hyperboliques si k = −1, et
plates si k = 0.
La dynamique de l’univers est décrite par la fonction scalaire R(t) dont
l’évolution est donnée par l’équation de Friedmann
2
Ṙ
k
8πG
+ 2 =
R
3
R
∑ ρi
,
i
où les ρi sont les densités d’énergie des diverses composantes qui se distinguent
par leur équation d’état. Le système d’équations est fermé par l’équation de
continuité
3
d ρi R = −pi d R3
.
Je supposerai ici qu’il n’y a que deux composantes importantes : d’une côté
la matière non-relativiste (“poussière”), qui contient la matière baryonique et
la matière noire avec l’équation d’état P = 0, donc ρM ∝ R−3 , et de l’autre
côté l’énergie noire, sous forme d’une constante cosmologique, avec l’équation
d’état PΛ = −ρΛ et ρΛ = const. Ces deux composantes sont paramétrées par
ΩM =
7 dans
8πGρM
3H02
et
ΩΛ =
des unités oú c = 1.
32
8πGρvac
Λ
=
2
3H0
3H02
.
33
Résumé
La courbure des sections spatiales est décrite par un paramètre analogue
Ωk = 1 − ΩM − Ω Λ = −
k
R20 H02
.
L’équation de Friedmann peut donc être écrite comme
h
i1/2
Ṙ
3
2
≡ H(z) = H0 ΩM (1 + z) + Ωk (1 + z) + ΩΛ
,
R
où la variable indépendante a été transformée en décalage spectral z = R0 /R(t)−
1 (dans la suite appelé “redshift”).
Le modèle cosmologique est donc paramétré par les trois paramètres ΩM ,
ΩΛ et H0 . La constante de Hubble est définie comme
Ṙ H0 = ≡ 100 h km s−1 Mpc−1 .
R t=t0
Les observations récentes donnent des valeurs pour H0 dans l’intervalle 60 −
80 ; Freedman et al. (2001) trouvent H0 = (72 ± 8) km s−1 Mpc−1 à partir d’observations HST de cephéı̈des dans l’amas de Virgo.
La détermination des valeurs des autres paramètres ΩM et ΩΛ a fortement
progressé ces dernières années, notamment par les observations de supernovae de type Ia à grand redshift et par les observations des fluctuations de
températures du fond cosmologique.
Pour une classe d’objets avec une luminosité bien définie la forme de la
relation entre la magnitude apparente et le redshift dépend de ΩM et ΩΛ ; les
supernovae de type Ia sont de telles chandelles standards avec une magnitude absolue au maximum de MB ' −19.4, surtout après correction d’une
corrélation entre la luminosité maximale et le taux de décroissance de luminosité après le maximum. Deux collaborations ont produit des échantillons
de 20 et 40 SNe Ia avec des redshifts compris entre 0.15 et 0.8 (Perlmutter
et al. 1995, 1999; Schmidt et al. 1998; Riess et al. 1998; Fillipenko 2001). Les
résultats cosmologiques sont cohérents et indiquent avec une confiance statistique élevée que l’univers est actuellement dans une phase d’accélération avec
une constante cosmologique positive. A cause d’une dégénérescence les paramètres ΩM et ΩΛ ne peuvent pas être déterminés séparément ; la meilleure
contrainte est (0.8 ΩM − 0.6 ΩΛ ) = −0.2 ± 0.1 (Perlmutter et al. 1999). En supposant que l’univers est plat (ΩM + ΩΛ = 1) on obtient ΩM ' 0.3 et ΩΛ ' 0.7.
Les incertitudes les plus importantes dans ces études sont systématiques
et concernent des effets astrophysiques qui peuvent affecter les magnitudes
observées des SNe Ia d’une manière semblable aux effets cosmologiques, par
exemple une extinction intrinsèque dans les galaxies hôtes des supernovae ou
l’effet de lentille gravitationelle par des structures à grande échelle. Les recherches de supernovae se poursuivent à des redshifts plus élevés et vont en
particulier profiter du CFHT Legacy Survey et de la mission proposée SNAP.
34
Chapitre 1. Cosmological Framework
Le fond cosmologique (CMB) montre des fluctuations de température d’ordre δT/T ' 10−5 . Celles-ci sont dues à des fluctuations de densité et du potentiel gravitationel à l’époque de découplage entre photons et baryons à z ∼
1000. Avant le découplage le plasma couplé de baryons et photons a subi une
série d’oscillations acoustiques qui sont à la base de la forme caractéristique
du spectre de puissance des fluctuations de température, qui consiste d’une
série de pics dits “acoustiques” ou “Doppler”. Les échelles angulaires et les
amplitudes de ces pics dépendent de ΩM et ΩΛ parmi d’autres paramètres
cosmologiques.
Les expériences B OOMERANG (de Bernardis et al. 2000), M AXIMA (Lee et al.
2001) et D ASI (Halverson et al. 2001) ont mesuré les fluctuations du CMB sur
les échelles en question et ont produit des détections des pics acoustiques. Ces
mesures ont été comparées à des prédictions théoriques dans des modèles avec
typiquement 7 paramètres (Stompor et al. 2001; Pryke et al. 2001; de Bernardis
et al. 2002). Il y a encore une dégénérescence entre ΩM et ΩΛ , la meilleure
contrainte étant ΩM + ΩΛ ≡ 1 − Ωk = 1.04 ± 0.05 (de Bernardis et al. 2002), ce
qui indique que l’univers est en fait spatialement plat.
D’autres méthodes pour déterminer ΩM consistent à mesurer les masses et
les luminosités, donc les rapports masse–luminosité M/L, d’objets astrophysiques comme les galaxies ou les amas de galaxies. En supposant que cette
valeur M/L est typique de la valeur globale dans l’univers et en combinant
avec une mesure de la densité de lumière dans l’univers on arrive à une estimation de la densité de masse, donc ΩM . En fait, on trouve que M/L augmente
avec la masse des objets étudiés et atteint M/L ∼ 300h à l’échelle des amas de
galaxies, ce qui amène à ΩM ' 0.3. D’autres méthodes (fraction de baryons
dans les amas, champs de vitesse de galaxies) produisent des valeurs qui sont
généralement du même ordre mais avec des incertitudes importantes.
Vues les dégénérescences et les incertitudes dans ces méthodes individuelles
de détermination de ΩM et ΩΛ , on essaie de combiner les résultats de différentes méthodes. Ces résultats semblent converger dans le plan ΩM –ΩΛ autour
de ΩM = 0.3, ΩΛ = 0.7. Par exemple, Efstathiou et al. (1999) trouvent ΩM =
+0.17
0.25+0.18
−0.12 et ΩΛ = 0.63−0.23 ; par contre Douspis et al. (2001) trouvent ΩM ' 0.8
et ΩΛ ' 0.3, ce qui montre qu’il n’y a toujours pas un accord universel sur les
valeurs des paramètres cosmologiques.
Dans cette thèse j’utilise les modèles (ΩM = 0.3, ΩΛ = 0.7) ainsi que le
modèle Einstein–de Sitter (ΩM = 1, ΩΛ = 0) comme modèles de référence.
Chapter 2
Counting Clusters of Galaxies
Structure in the Universe is assumed to have formed through gravitational
instability. Initially, the matter in the Universe was distributed nearly homogeneously, as it still is on large scales, with tiny fluctuations that are described
by the overdensity
δ(x, t) =
ρ(x, t) − ρ(t)
ρ(t)
,
(2.1)
where ρ(t) = ρ0 (1 + z)3 = 3H(t)2 Ω(t)/(8πG) is the mean density in the Universe at epoch t. As time progresses, slightly overdense regions (δ > 0) accrete matter from their surroundings, so that they become more overdense,
hence δ increases. As long as the density fluctuations are small, |δ| 1, their
growth can be described through linear theory and δ(x, t) = δ(x, ti )D(t)/D(ti ),
with the linear growth function D(t). In regions where δ approaches unity,
nonlinear effects become important. The expansion of overdense regions will
eventually be stopped by gravity, particle orbits will turn around and reconverge. Through processes such as violent relaxation and phase mixing the
particles virialise and form a stable equilibrium configuration, which, according to the mass contained in the virialised region, we can identify as the sites
where galaxies or clusters of galaxies form.
The initial density field is assumed to have more power on small scales,
in the sense that the variance of the density field decreases when smoothed
on progressively larger scales. It will be shown that small scale perturbations
then collapse and virialise earlier than large scale fluctuations. Structure thus
forms in a hierarchical, “bottom-up” way. In the observed Universe, clusters of
galaxies are the most massive structures that are found to be in an equilibrium
state and consequently they should be the most recent objects to have formed.
For cosmology this has the advantage that the distribution of galaxy clusters
- in contrast to galaxies - can still be described by linear theory, even if their
internal structure is of course highly non-linear.
35
36
Chapter 2. Counting Clusters of Galaxies
The sites where clusters of galaxies form can be identified with peaks in the
density field. Linear theory describes how the height of peaks evolves with
cosmic time; peaks above a certain height will then be counted as clusters of
galaxies. In the following I will first briefly recall how the density field evolves
in linear theory. A simple analytically tractable model will then be developed
which describes the recollapse of homogeneous spherical overdense regions
in an expanding background Universe; this model will establish relative time
scales and provide a density threshold for regions that have formed bound
halos at a given epoch t. Then we can count clusters of galaxies in the linearly
evolved density field; I will describe the basic heuristic argument of Press &
Schechter (1974) and then the more quantitative argument of Lacey & Cole
(1993) which provides an extension to the Press-Schechter argument in terms
of an excursion set approach. This will then leave us with the mass function,
the distribution in mass of clusters of galaxies as a function of time.
The linear evolution of the density field is treated in many textbooks (e. g.
Peebles 1980, 1993; Padmanabhan 1993; Coles & Lucchin 1995; Peacock 1999).
These books also discuss the spherical collapse model and mention the Press–
Schechter argument, but do not discuss extensions and the excursion set approach.
2.1
Matter Power Spectrum
The precise origin of the density fluctuations in the very early Universe is unknown; a possibility is that they are quantum fluctuations which were magnified to classical scales during an inflationary phase.
The density fluctuations are usually assumed to be Gaussian with random
phases in Fourier space. Then the density field is uniquely specified by its
power spectrum P(k) = hδk2 i, or equivalently by the correlation function in
real space, ξ (|r − r0 |) = hδ(r)δ(r0 )i, where the usual assumptions of (statistical)
homogeneity and isotropy have been made.
In the absence of a full theory for the origin of the density fluctuations, the
shape of the primordial power spectrum can be inferred from observations of
the late-epoch power spectrum in conjunction with a model for the subsequent
evolution of the primordial power spectrum; the simplest form is the scaleinvariant Harrison–Zeldovich spectrum, Pinit (k) ∝ kns with ns = 1. Studies of
the CMB temperature fluctuations can be used to put observational constraints
on ns , for example de Bernardis et al. (2002) find ns = 0.95 ± 0.1 from the
B OOMERANG maps, consistent with the Harrison–Zeldovich form.
The primordial power spectrum is modified before the decoupling of the
baryon–photon plasma at zdec ' 1000. The exact processes are difficult and
require numerical solution of the Boltzmann equation for the mixture of collisionless dark matter and the strongly coupled baryon–photon plasma; the re-
37
2.1. Matter Power Spectrum
sult also depends on the nature (adiabatic or isocurvature, Peacock 1999) of the
initial density perturbations. In cold dark matter models, one usually assumes
adiabatic initial perturbations where matter and photons are compressed together such that δrad = 4δM /3 at all times, which implies a constant entropy
per baryon. In this case the dominant effect that modifies the power spectrum
is the rapid expansion of the Universe during the radiation-dominated phase
when the expansion time scale is texp ∼ (GρR )−1/2 , shorter than the dynamical time scale for the dark matter, tDM ∼ (GρDM )−1/2 . An overdense region
thus does not have sufficient time to decouple from the general expansion and
growth of its overdensity is suppressed. Growth of a dark matter perturbation
then resumes when matter becomes the dynamically dominant component,
that is at matter–radiation equivalence when ρR = ρDM , corresponding to a
redshift zeq = ΩM /Ωrad − 1 = 23900 ΩM h2 , for a CMB temperature T = 2.73 K
at the present time; here Ωrad includes the contribution of three relativistic
neutrino species. Growth is only suppressed on scales that are smaller than
the horizon because overdense regions that are larger than the horizon have no
knowledge about the mean expansion rate of the surrounding regions. Growth
of a small-scale fluctuation k will thus be suppressed as soon as the fluctuation
enters the horizon1 , i. e. when dH (zenter ) ∼ 1/k, and will start to grow again
at zeq ; this suppression thus concerns perturbations on scales that are smaller
than the horizon size at equivalence, deq ≡ dH (zeq ). During the radiation dominated phase, (unsuppressed) density fluctuations grow as the square of the
horizon size, so the missing growth factor for suppressed modes is roughly
∝ k2 . The resulting power spectrum is thus
P(k) ∼ Pinit (k) ∝ kns
(k deq 1)
P(k) ∼ Pinit (k) k−4 ∝ kns −4
(k deq 1)
(2.2)
.
(2.3)
Fitting formulae for numerical results for adiabatic CDM models without a
baryonic component were given by Bond & Efstathiou (1984):
1.13 −2/1.13
3/2
2
P(k) ∝ q 1 + 6.4q + (3.0q) + (1.7q)
,
(2.4)
and Bardeen et al. (1986) (Fig. 2.1):
i−1/2
ln(1 + 2.34q) 2 h
1 + 3.89q + (16.1q)2 + (5.46q)3 + (6.71q)4
,
P(k) ∝ q
2.34q
(2.5)
with q ≡ k/(ΩM h2 Mpc−1 ); the denominator in the definition of q is due to the
scaling of the horizon size deq . In the presence of baryons the denominator is
replaced by an apparent shape parameter Γh = ΩM h2 exp [−ΩB (1 + 1/ΩM )].
1 The
bang.
horizon size at time t is the proper distance that a photon has travelled since the big
38
Figure 2.1: Fitting formulae to numerical calculations of the adiabatic CDM
power spectrum due to Bardeen et al. (1986, solid line) and Bond & Efstathiou
(1984, dashed line).
This is the shape of the CDM power spectrum of the density fluctuations as
it is imprinted in the CMB temperature fluctuations. Since in linear theory the
shape of the power spectrum is unmodified at later times, this is the spectrum
that is relevant for studies of clusters of galaxies.
Eisenstein & Hu (1999) give a (complicated!) fitting formula for the power
spectrum for the full multi-component system of cold dark matter, radiation,
baryons and massive neutrinos. Both baryons and neutrinos suppress the
growth of structure on small scales, baryons in addition imprint acoustic oscillations in the power spectrum, which are most prominent in the CMB fluctuation spectrum.
For the study of clusters of galaxies it is not necessary to use the full CDM
power spectrum; since clusters cover a fairly small mass range it is sufficient to
assume that the power spectrum over the relevant range can be approximated
by a power-law, P(k) ∝ kneff , where the logarithmic slope is neff ∼ −1. In this
approximation the power spectrum is scale free and cluster properties follow
simple scaling relations depending on the slope neff (e. g. Kaiser 1986).
39
2.2. Linear evolution of the density field
2.2
Linear evolution of the density field
As long as the density fluctuations are small, |δ| 1, their evolution can be
described by linear theory and the overdensity at any time t is related to the
present overdensity δ0 via the linear growth factor, δ(r, t) = D(t) δ0 (r).
In general, the evolution of the density field ρ(r, t) is governed by an equation of motion (the Euler equation for a collisional fluid, or the Vlasov equation
for collisionless particles), the continuity equation which describes conservation of mass, the Poisson equation which relates the density field to the gravitational potential, and an equation of state which relates density and pressure P. Writing ρ(r, t) = ρb (t) (1 + δ(r, t)) and similar expressions for velocity, pressure and potential, transforming to comoving coordinates x = r/R
and linearising in the perturbations yields the linear evolution equation for
the overdensity (e. g. Peebles 1980; Peacock 1999):
Ṙ ∂δ
∇2 P
∂2 δ
+
2
=
4πGρ
δ
+
b
R ∂t
∂t2
ρb R2
.
(2.6)
After matter–radiation equivalence at zeq ' 3 × 104 , the Universe is dominated
by pressureless matter, and the pressure term on the right hand side of (2.6)
vanishes.
Eq. (2.6) can easily be solved in an Einstein–de Sitter Universe, where the
scale parameter R ∝ t2/3 , and 4πGρb = 2/(3t2 ) from the Friedmann equation
(1.2) with curvature k = 0. δ has a growing and a decaying mode which vary
as t2/3 and t−1 respectively, so the general solution is
δ(x, t) = A(x)t2/3 + B(x)t−1
.
(2.7)
In terms of redshift, the growing mode is δ ∝ R ∝ (1 + z)−1 .
For other cosmologies the solution of (2.6) is more complicated although
analytic solutions can be found for the standard cases such as flat Universes
and open Universes with no cosmological constant (Peebles 1980). In the general case, (2.6) has to be solved numerically. Heath (1977) gives integral solutions which can be written as
Z ∞
(1 + z0 ) dz0
δ1 ∝ H(z)
(2.8)
H(z0 )3
z
for the growing mode, and
δ2 ∝ H(z)
(2.9)
for the decaying mode, where H(z) is defined by the Friedmann equation (1.7).
The expression for the growing mode can be normalised to 1 at z = 0 by dividing by the integral taken from 0 to ∞, which converges for ΩM > 0.
40
Figure 2.2: Linear growth of density fluctuations as a function of redshift (1+z)
in different background cosmologies. The growth factor is normalised at the
present time, z = 0.
Fig. 2.2 shows the linear growth factor for the growing mode, normalised
at the present time, as a function of redshift for a number of different cosmological background models. In the Einstein–de Sitter Universe, δ scales as
(1 + z)−1 , as mentioned above. In flat Universes with lower matter density,
the evolution is less rapid at low redshifts and approaches (1 + z)−1 behaviour
only at z >
∼ 1 for the “standard” ΛCDM model. In the open Universe without
cosmological constant the suppression of linear growth at low redshifts is even
more pronounced.
2.3
The spherical collapse model
A simple analytically tractable model for the formation of a cluster of galaxies is that of spherical collapse of a homogeneous overdense region embedded
in a Friedmann–Lemaı̂tre–Robertson–Walker background Universe. The argument is very similar to the Newtonian derivation of the Friedmann equations
and is based on the fact that the evolution of a patch of the Universe depends
only on the local density within the patch and not on the mass distribution
outside, as long as the situation is spherically symmetric (Birkhoff’s theorem;
inhomogeneities in the outside density distribution influence the evolution of
the patch only through tidal fields, which are of higher order). The main inter-
41
2.3. The spherical collapse model
est of the model is to obtain a number of density scales that are of significance
in the cluster formation process, as well as the dependence of time scales on the
cluster mass. For simplicity I will describe the model for an Einstein–de Sitter
background model; generalisations to open and flat cosmological models are
described for example in Lacey & Cole (1993) and Eke et al. (1996). Numerical
simulations seem to indicate that the Einstein–de Sitter case is in fact sufficient
for a number of purposes (Jenkins et al. 2001).
The evolution of the size r of any spherical overdense region embedded in
an Einstein–de Sitter Universe is governed by the Friedmann equation for a
closed model:
GM
d2 r
=− 2
,
(2.10)
2
dt
r
where M is the mass of the overdense region.
The solution of equation (2.10) is the usual cycloid, given in parametric
form as
r(θ) = A (1 − cos θ) , t(θ) − t0 = B (θ − sin θ) .
(2.11)
Inserting the solution (2.11) into Eq. (2.10) yields
A3
= GM
B2
,
(2.12)
which is of course just Kepler’s Third Law in disguise.
At early times (t small, θ small), the solution (2.11) can be expanded to give
1/3 "
2/3 #
6t
9
GMt2
1−
.
(2.13)
r'
2
B
The mean density inside the region is
"
#
3M
1
3 6t 2/3
ρ=
'
1+
20 B
4πr3
6πGt2
.
(2.14)
Now, for the Einstein–de Sitter background model 6πGρb t2 = 1, so the linear
overdensity is
3 6t 2/3
δ=
,
(2.15)
20 B
which scales as t2/3 as in Eq. (2.7).
The remaining constant B, and hence A, can be expressed in terms of the
overdensity δi at some initial time ti :
B = 6ti
20δi
3
−3/2
,
A=
3ri
10δi
.
(2.16)
42
B sets the timescale for the collapse: initially more overdense regions collapse
2 ∼ k −1 in Fourier space, the
ˆ
first. For the CDM power spectrum with |δ(k)|
initial overdensity varies as hδi2 i ∝ k2 , so that small scale perturbations collapse
first; this is the basis for the hierarchical structure formation scenario.
In order to determine the true overdensity of the collapsing region as a
function of time, we compare the mean density inside the region to the mean
background density:
rb (t) 3
ρ
=
(2.17)
ρb
r(t)
Here, rb is the radius of a sphere with the same mass M as the overdense
sphere, but the mean density ρb ,
rb3 =
3M
9
= GMB2 (θ − sin θ)2
4πρb
2
,
(2.18)
where (2.11) was used for t2 . Inserting (2.11) and (2.18) in (2.17) yields the true
overdensity
9 (θ − sin θ)2
ρ
1 + δtrue (t(θ)) ≡
=
.
(2.19)
ρb
2 (1 − cos θ)3
The true overdensity (2.19) and the linearly extrapolated overdensity (2.15)
are shown in Fig. 2.3.
The overdense region reaches its maximum size at turn-around, θta = π.
At this moment, the true overdensity is δtrue (ta) = 9π 2 /16 − 1 ' 4.55 and
3
the linearly extrapolated overdensity δlin (ta) = 20
(6π)2/3 ' 1.06. Collapse
3
occurs at θc = 2π, when the linear overdensity is δc = 20
(12π)2/3 ' 1.686.
The true overdensity (2.19) formally diverges at this point; however, once the
region starts contracting non-radial motion of particles sets in and is quickly
amplified. The region stabilises due to the random motion of the particles and
virialises. The size of the collapsed or virialised region can be estimated by the
virial theorem,
W = −2K = 2Etot ,
(2.20)
where W ∼ −GM/rvir is the potential energy, K is the kinetic energy and
Etot ∼ −GM/rta is the total energy of the overdense region. The radius of the
virialised region is therefore about half the turn–around radius, rvir = rta /2.
The true overdensity of a recently virialised halo depends on the time when
this happens. A lower limit is given by the formal solution (2.11) for a free collapse. In this case virialisation occurs at θvir = 3π/2, when the true overdensity δtrue,vir ' 147. In reality the dissipative effects leading to virialisation will
delay the collapse to rvir ; if virialisation occurs at, say, tcollapse , i. e. θvir = 2π,
then δvir ' 178. This value also depends to some extent on the density parameter ΩM (Lacey & Cole 1993; Eke et al. 1996), but always turns out to lie in
2.3. The spherical collapse model
43
Figure 2.3: True and linearly extrapolated overdensities of a spherical region
embedded in an Einstein–de Sitter Universe. Time is displayed in units of
the constant B; physical times scale as the inverse of the initial overdensity
δi . The dotted line shows the evolution of the size of the overdense region on
an arbitrary linear scale. At late times (t > tta ) the evolution of the spherical region will deviate from that shown in the plot due to dissipative effects
which convert the kinetic energy of the collapse into random motion, leading
to virialisation of the sphere at a radius of half the radius at turn–around.
the range 100. . . 200. In practice, the radius r200 of a sphere enclosing a mean
overdensity of 200 is often used to estimate the virial radius rvir .
Once a patch has collapsed its physical density does not change any more,
the patch neither contracts nor expands. However, the background density
still continues to drop as ρb ∝ a−3 . The local density contrast thus grows as
(1 + δ) ∝ a3 ; e. g. in an Einstein–de Sitter universe (1 + δ) ∝ t2 . Therefore
a radius at which the mean interior overdensity has a fixed value, like r200
increases and in this sense, a cluster still continues to grow after collapse and
virialisation.
44
2.4
Analytical estimates of the halo mass function
2.4.1 The Press–Schechter argument
A simple analytical estimate of the halo mass function is given by the PressSchechter argument (Press & Schechter 1974), which is based on the spherical
collapse model. The initial density field after recombination is assumed to be
a Gaussian density field with a power spectrum P(k), for example the CDM
power spectrum described in Sect. 2.1. If top-hat filtered on a mass scale M,
related to a comoving length scale R by
4π
M=
ρ0 R3 = 1.17 × 1012
3
R
1 Mpc
3
ΩM h2 M
,
(2.21)
the filtered density field is still Gaussian with a variance σ(M). Here ρ0 =
ΩM ρcrit is the comoving mean density in the Universe. The variance σ is related to the mass through the power spectrum of the density field:
Z ∞
1
σ(M) =
k2 P(k) W 2 (k; M) dk ,
(2.22)
2π 2 0
where W(k; M) is the Fourier transform of the top hat filter with radius R given
by (2.21). Fig. 2.4 shows σ as a function of mass scale M and length scale R.
Figure 2.4: The mass variance σ in a spherical top-hat filter is plotted as a
function of the filter’s mass scale (left panel) and length scale (right panel) for
different values of the cosmological parameters. The variances are normalised
by σ8 , the variance in spheres of radius 8 Mpc (indicated by arrows), using the
cluster normalisation of Eke et al. (1996).
According to the spherical collapse model a region has collapsed and virialised when the extrapolated mean density within the region exceeds a critical
45
2.4. Analytical estimates of the halo mass function
value δc , independent of the mass contained in the region. The probability of
finding a virialised halo at a given point is thus
Z ∞
2
2
1
P(δ > δc ) = √
e−δ /2σ dδ .
(2.23)
2π σ δc
The redshift dependence of this probability can be obtained easily by replacing the constant δc by δc (z), the linearly extrapolated overdensity today, if the
overdensity was δ = δc at redshift z. For z > 0 we thus have δc (z) > δc , so that
the probability of finding a halo decreases with redshift. The number density
of halos of mass M is the product of the number density ρ0 /M of such halos if
all the mass in the Universe were present in halos of mass M, times the probability of finding a halo of mass M at a given point, dP/dM, which is related to
the probability P(> δc ) through the implicit mass dependence via σ(M). The
mass function is thus:
ρ0 dσ dP
dn
dM =
dM
dM
M dM dσ
.
(2.24)
Evaluating the derivative dP/dσ, results in the Press–Schechter mass function
r
dn
2 ρ0 d ln σ δc −δc2 /2σ2
=
e
.
(2.25)
dM
π M2 d ln M σ
The mass function can be written in a “universal” form by expressing it in
terms of ν = δc /σ(M)2 :
r
2
2 ρ0 dν dn
=
ν e−ν /2 .
(2.26)
dM
π M dM
In this form the dependence of the mass function on cosmology and redshift
(δc (z)) and the power spectrum (σ(M)) is scaled out, and the self-similarity of
the mass function becomes apparent.
The expression (2.25) contains an ad hoc factor of 2 compared to (2.24).
This fudge factor was introduced by Press & Schechter (1974) to account for
the mass that is initially in underdense regions (Since initially |δ| 1 the
amount of mass contained in underdense regions is nearly as large as that in
overdense regions. It is only during the subsequent evolution of the density
field that matter from underdense regions is accreted onto overdense regions.).
Bond et al. (1991) however argue that the fudge factor is rather related to the
clouds–in–clouds problem, which relates to the fact that particles can be included in haloes on different scales. The factor 2 is automatically accounted
for by somewhat more sophisticated approaches (Sect. 2.4.2).
The Press-Schechter function is shown in Fig. 2.52 .
2 All
the mass functions shown in the following figures were calculated using software
kindly provided by Adrian Jenkins.
46
Figure 2.5: Different theoretical estimates of the halo mass function for an
Einstein–de Sitter Universe (left) and a flat ΩΛ dominated Universe (right).
The Press-Schechter function is derived analytically, e. g. via an excursion set
approach, the Sheth-Tormen and Jenkins functions are fitting functions to results from numerical simulations.
2.4.2
The excursion set approach
A more sophisticated though still highly abstract and idealised analytical approach to estimating the mass function of collapsed dark matter halos is the
excursion set model (Bond et al. 1991; Lacey & Cole 1993; Sheth et al. 2001). In
this approach, a trajectory δ(M) is attributed to every point or particle in the
initial density field by smoothing the density with a window function of progressively larger scale centred on this point (particle).
It is easiest to understand the behaviour of these trajectories if they are expressed in terms of the variance S(M) ≡ σ2 (M) of the density field smoothed
on mass scale M (Fig. 2.6). On the largest mass scales, M → ∞, the variance
S(M) → 0 and the smoothed density field approaches the mean density, δ → 0.
Thus all the trajectories start at the origin. On smaller mass scales the fluctuations in the smoothed density field increase as do the smoothed density values
at any given point, so that the trajectory δ (S(M)) follows a random walk.
The model assumes that a particle belongs to a halo at redshift z if the
smoothed density reaches the critical value for collapse at that redshift, i. e.
if the corresponding trajectory crosses a boundary B (S(M), z) = δc (z). The
corresponding value of S(M) gives the mass of the virialised halo that the particle is in. At different redshifts the particle will in general find itself in halos
of different mass, monotonically decreasing with increasing redshift. Regions
where the trajectory is monotonically increasing can be interpreted as times of
silent accretion of matter, whereas local maxima indicate major mergers.
47
2.4. Analytical estimates of the halo mass function
Figure 2.6: Behaviour of trajectories δ (S(M)) for a density field smoothed on
progressively smaller mass scales. S(M) is the variance of the density field
when smoothed over the mass scale M. A trajectory can be read as a halo
merger history, if followed from the top right towards the origin: The halo
represented by the bold trajectory undergoes two major mergers during its
history, one at z = 0.79 and one at 0.21. At redshift z = 0 this halo has S(M) =
1.9.
The mass function of collapsed halos can be derived as follows in this
picture: The mass density of matter that is in halos of mass M is given by
Mn(M)dM, the total mean mass density in the Universe by ρ. The fraction of
mass that is contained in halos of mass M is accordingly the ratio of these two
expressions. The mass fraction at a given redshift is equal to the fraction f (S)
of trajectories that cross the boundary B (S(M), z) at S(M):
ν f (ν) =
M2
d ln M
n(M)
ρ
d ln ν
.
(2.27)
As before, the mass fraction is expressed in terms of ν = B(S(M), z)/S(M)
(Eq. 2.26). The fraction of trajectories that cross the boundary at S(M) depends on the window function that is used to create the excursion set3 . The
simplest window function would be a top-hat in real space, with radius R =
3 This
simply reflects the fact that we are dealing with a highly idealised model. Nature is
of course not trained in excursion set theory, and the practical choice of the window function
48
(3M/4πρ)1/3 . However, since the autocorrelation function of a Gaussian density field does not vanish, subsequent steps in the random walk are not independent and the determination of f (ν) is difficult. If, on the other hand, the
window function is a top-hat in Fourier space, then the random walk is truly
random, since subsequent steps just add Fourier components from spherical
shells in k-space, and these are independent. The function f (ν) can then be
found in a simple and elegant way by solving the diffusion equation with an
absorbing boundary at B (corresponding to ν = 1):
r
2
2
ν f (ν) =
ν e−ν /2
(2.28)
π
in the case of a boundary that is independent of mass (Bond et al. 1991; Lacey &
Cole 1993). This is the case for the spherical collapse model, where the critical
linearly extrapolated overdensity δc = 1.686 (for an Einstein–de Sitter Universe), independent of the mass of the virialised halo.
Combining Eqs. (2.27) and (2.28) results in the mass function
r
2 ρ d ln σ(M) δc (z) −δc2 (z)/2σ2 (M)
n(M) =
e
(2.29)
π M2 d ln M σ(M)
which is exactly the Press–Schechter function, including the fudge factor 2 (Eq.
2.25).
The interest of the excursion set or generalised Press-Schechter approach
lies in the fact that it allows the use of different kinds of filter, and can be used
to estimate merger rates, the distribution of halo formation times, etc. (Lacey
& Cole 1993).
2.5
Mass functions from numerical simulations
Analytical estimates of the halo mass function rely on very idealised and simplified models for the collapse and virialisation of bound objects. In reality
however, collapse is a very complex and highly nonlinear process which in
detail is not tractable by analytical methods. Numerical N-body simulations,
which solve the non-linear equations directly, offer a much better chance to
obtain accurate estimates of the mass function which might be compared to
reality.
The most extensive investigation to date of the mass function in numerical
simulations comes from the VIRGO consortium (Jenkins et al. 1998; Colberg
et al. 2000b). Jenkins et al. (2001) use a number of N-body simulations to derive the halo mass function over four orders of magnitude in mass. The simulations cover two cosmologies (“standard” SCDM with ΩM = 1, ΩΛ = 0, and
should be guided by the best approximation to results from numerical simulations or observations.
2.5. Mass functions from numerical simulations
49
ΛCDM with ΩM = 0.3, ΩΛ = 0.7; in both cases the initial conditions are set up
from the appropriate CDM power spectrum), and differ in the volume that is
simulated, the number of particles used (and consequently in the mass of the
particles) and thus in the mass resolution.
Jenkins et al. (2001) find that the mass function can be fitted by a universal
functional form if it is expressed in terms of σ(M), rather than M directly,
with δc = 1.686 independent of ΩM and ΩΛ ; this universality property is the
same as that of the Press–Schechter function. The dependence on the matter
power spectrum, cosmology and redshift is implicit in the rms fluctuation of
the density field, σ. Jenkins et al. (2001) find that the function
ρ d ln σ(M) dn
3.8
= 0.315 2 exp
−
−
ln
σ(M)|
(2.30)
|0.61
dM
d ln M M
fits the mass function found in the simulations to ∼ 20 . . . 30% over the range
11
−1.05 <
∼ ln σ <
∼ 1.2, corresponding to a mass range of 3 × 10 <
∼ M <
∼ 5×
15
−1
<
<
<
10 h M , a redshift range 0 ≤ z ∼ 5 and 0.3 ∼ ΩM ∼ 1.
The fitting function of Jenkins et al. (2001) is plotted in Fig. 2.5 alongside
the Press-Schechter formula. The figure highlights the deviations of the PressSchechter formula from the simulations: it predicts fewer high-mass halos
than observed and an excess of halos at the characteristic mass M∗ , corresponding roughly to σ = δc .
The analytical predictions from the excursion set approach have been improved recently to provide a better match to numerical simulations. In earlier
work, using a subset of the simulations used by Jenkins et al. (2001), Sheth &
Tormen (1999) had given a somewhat different form for the mass function,
r
2 q ! √
2
aδc −δc2 /2aσ2
σ
n(M) =
A 1+
e
,
(2.31)
2
π
σ
aδc
which is close to the form found by Jenkins et al. (2001), though valid over a
restricted mass range. Sheth et al. (2001) replaced the spherical collapse model
in the excursion set approach by a more general ellipsoidal collapse model
and found a mass function of virtually the same functional form as Eq. (2.31).
Unlike the spherical model, the ellipsoidal model results in a density threshold
δc which depends on the mass scale, i. e. a “moving” barrier B (S(M)).
The excursion set approach does not work very well on an object–by–object
basis (Sheth et al. 2001), when applied to initial density field realisations for
numerical simulations, meaning that the mass of the halo that a particle is predicted to be in is usually not the mass of the halo that the particle is actually
found in at the end of the simulation. However, the approach is found to give
accurate statistical predictions, in particular for the mass function. The reason
for the discrepancy is that a particle in general is not in the centre of its halo,
50
so that the window function centred on the particle includes matter from surrounding underdense regions, leading to an underestimate of the mass. Sheth
et al. (2001) show that the object–by–object correspondence can be improved if
only those particles are taken into account that end up at the centres of mass
of the collapsed halos.
2.6
Evolution of the mass function
The halo mass function can be used to normalise the power spectrum of the
underlying density fluctuations. In particular this is done by counting clusters
of galaxies, giving the normalisation in terms of σ8 . Eke et al. (1996) obtained
the following relations from an analysis of the X-ray temperature function of a
sample of local clusters:
−0.52+0.13ΩM
(2.32)
−0.46+0.10ΩM
(2.33)
σ8 = (0.52 ± 0.04) ΩM
for flat cosmologies, ΩM + ΩΛ = 1, and
σ8 = (0.52 ± 0.04) ΩM
for ΩΛ = 0.
Local cluster counts can only determine a combination of σ8 and ΩM , as
evidenced by Eqs. (2.32) and (2.33). The degeneracy between these two parameters can be broken by observing the evolution of the halo mass function
which is shown in Fig. 2.7 for the SCDM and ΛCDM models. The evolution
of the high-mass end of the mass function, corresponding to rich clusters of
galaxies, is strong even at low redshifts, z <
∼ 1, whereas for masses on galactic
scales the mass functions hardly evolve at all over the plotted redshift range.
This reflects the hierarchical formation of structure in cold dark matter models:
more massive halos form at progressively later times in the history of the Universe. Clusters of galaxies, which are the most massive virialised structures
in the present-day Universe, are the latest to have formed, so evolution at low
redshifts is strongest on cluster scales.
The evolution of the mass function depends strongly on the mean density
in the Universe. This is essentially due to the dependence of the age of the
Universe at a given redshift on the cosmological parameters: for lower mass
densities the Universe is older at redshift z. On the other hand, the time it
takes for an individual halo to form depends only little on the background
cosmology, through δc (z) in the spherical model, so that it forms at a higher
redshift in a Universe with lower mass density.
2.6. Evolution of the mass function
51
Figure 2.7: Evolution of the halo mass function. The halo mass function is
shown at three redshifts, z = 0, 0.5 and 1.0 for two background cosmologies,
the Einstein–de Sitter model and the Λ-dominated flat model. The mass functions are normalised using Eq. (2.32). The lower curves in each family are at
higher redshift.
Comptage des amas de galaxies
On suppose que les structures observées dans l’univers se sont formées par
instabilité gravitationnelle à partir de minimes fluctuations primordiales de
densité. Afin de décrire ces fluctuations on définit le contraste de densité
δ(x, t) =
ρ(x, t) − ρ(t)
ρ(t)
par rapport à la densité moyenne ρ(t) = ρ0 (1 + z)3 = 3H(t)2 Ω(t)/(8πG). Avec
le temps, des régions sur-denses (δ > 0) accrètent la matière de leur environnement, donc δ augmente. Quand les fluctuations de densité restent faibles,
|δ| 1, la croissance des régions sur-denses peut être décrite par la théorie
linéarisée, δ(x, t) = δ(x, ti )D(t)/D(ti ), où D(t) est la fonction de croissance
linéaire. Lorsque δ s’approche de l’unité, les effets non-linéaires deviennent
importants. L’expansion de la région sur-dense s’arrête et celle-ci s’effondre.
Par les processus de relaxation violente et de mélange de phase, les particules
se virialisent et atteignent une configuration d’équilibre stable qu’on identifie,
selon la masse de la région, comme les lieux de formation des galaxies ou des
amas de galaxies.
Dans le cas le plus simple, le champ de densité est supposé être un champ
stochastique suivant une distribution Gaussienne, donc le champ est uniquement décrit par son spectre de puissance, P(k) = hδk2 i, où l’on suppose comme
d’habitude que le champ est homogène et isotrope. En l’absence d’une théorie
complète sur l’origine des fluctuations de densité, on suppose simplement que
le spectre de puissance primordial est invariant d’échelle, P(k) ∝ kns avec ns =
1 (spectre dit de Harrison–Zeldovich). A partir des fluctuations de température
du CMB, de Bernardis et al. (2002) trouvent la contrainte ns = 0.95 ± 0.1, ce qui
est cohérent avec la forme de Harrison–Zeldovich.
Le spectre primordial est modifié avant le découplage du plasma baryons–
photons à zdec ' 1000. Le traitement du système complet de matière noire,
photons, baryons et éventuellement une composante noire chaude est complexe (Eisenstein & Hu 1999). Dans des modèles dominés par la matière froide
(et le rayonnement avant l’ère d’équivalence entre matière noire et rayonne52
53
Résumé
ment, zeq = ΩM /Ωrad − 1 = 23900 ΩM h2 ), la croissance des fluctuations est effectivement arrêtée après qu’elles entrent dans l’horizon et avant l’équivalence,
par l’expansion rapide de l’univers a cette époque. Ceci concerne donc toutes
les fluctuations d’échelle plus petite que la taille de l’horizon à zeq . L’atténuation
de la croissance pour une fluctuation d’échelle k−1 est en k2 , donc le spectre de
puissance après zeq a les formes limites
P(k) ∼ Pinit (k) ∝ kns
P(k) ∼ Pinit (k) k
−4
∝k
(k deq
ns −4
1)
(k deq 1)
.
Deux formes analytiques de P(k), ajustées à des résultats de simulations numériques, sont montrées à la figure 2.1. Ceci est la forme du spectre de puissance
CDM imprimée dans le CMB ; puisque la forme du spectre ne change plus
après découplage (sauf modifications non-linéaires à des échelles inférieures
à celle des amas), c’est elle qui est importante pour la formation des amas de
galaxies. La pente du spectre de puissance aux échelles des amas de galaxies
est d’environ −1.
L’évolution du champ de densité est donnée par l’équation de Euler pour
un fluide collisionnel (ou par l’équation de Vlasov pour un fluide non-collisionnel), l’équation de continuité qui décrit la conservation de masse, l’équation
de Poisson qui relie le champ de densité au potentiel gravitationnel, et une
équation d’état qui relie densité et pression. En exprimant la densité comme
ρ(r, t) = ρb (t) (1 + δ(r, t)) et de même pour les champs de vitesse, pression et
potentiel, on peut linéariser les équations dans les perturbations et on aboutit (après transformation en coordonnées comobiles) à l’équation linéaire de
l’évolution de la sur-densité (e. g. Peebles 1980; Peacock 1999) :
∇2 p
Ṙ ∂δ
∂2 δ
+
2
=
4πGρ
δ
+
b
R ∂t
∂t2
ρb R2
.
Le terme de pression à droite est tout à fait négligeable après l’équivalence.
Dans un univers de Einstein–de Sitter, cette équation peut être résolue facilement , donnant deux solutions indépendantes qui décrivent un mode croissant et un mode décroissant :
δ(x, t) = A(x)t2/3 + B(x)t−1
.
En terme de redshift, le mode croissant est δ ∝ (1 + z)−1 . Pour d’autres cosmologies la solution est plus compliquée et doit être effectuée numériquement.
Heath (1977) donne des solutions en termes d’intégrales (Eqs. 2.8 et 2.9).
La croissance linéaire de fluctuations est montrée à la figure 2.2. Dans les
univers à basse densité (avec ou sans constante cosmologique) la croissance
est supprimée à bas redshift par rapport au modèle de Einstein–de Sitter, comportement qu’on retrouvera dans l’évolution de la fonction de masse des amas
de galaxies.
54
Chapitre 2. Counting Clusters of Galaxies
L’évolution non-linéaire d’une région en effondrement, donc la formation
des objets gravitationnellement liés, est difficile et ne peut être suivie en détail
que par des simulations numériques. Un modèle simple de la formation des
objets qui fournit des estimations pour les échelles de temps et leur dépendance
notamment sur la masse de la région est le modèle d’effondrement sphérique.
On considère l’évolution d’une région sphérique à densité homogène plus
élevée que la densité moyenne dans l’univers. L’équation du mouvement prend
donc une forme complètement analogue à celle de l’équation de Friedmann
(sans constante cosmologique) et peut être résolue sous forme paramétrique
(équation d’une cycloı̈de). On distingue alors la solution exacte de l’équation
du mouvement de l’extrapolation linéaire (δ ∝ t2/3 ) de la sur-densité. La dernière est importante parce qu’on peut identifier des halos effondrés dans le
champ linéaire comme des régions où la densité dépasse le seuil de densité extrapolé linéairement, donné par le modèle sphérique sans connaı̂tre les détails
de l’effondrement.
Lorsque la région atteint sa taille maximale (“turn-around”, ta) la vraie surdensité est δtrue (ta) = 9π 2 /16 − 1 ' 4.55, la densité extrapolée linéairement
3
est δlin = 20
(6π)2/3 ' 1.06. Lors de l’effondrement (divergence de δtrue ), la
3
densité linéaire est δc = 20
(12π)2/3 ' 1.686. C’est ce seuil-là qui est souvent
utilisé pour définir et compter des halos dans des champs linéaires.
En réalité la région ne s’effondre pas jusqu’à la singularité mais les composantes transversales des vitesses des particules augmentent et les particules
se virialisent pour atteindre une configuration étendue stable. La conservation
d’énergie et le théorème du viriel donnent une taille de la région virialisée qui
est la moitié de la taille maximale de la région. Si la région est virialisée au
moment de l’effondrement théorique, la vraie sur-densité est δvir = 178. La
valeur exacte dépend des paramètres cosmologiques, notamment ΩM (Lacey
& Cole 1993; Eke et al. 1996), mais est toujours de l’ordre 100 . . . 200. Dans des
simulations à N corps, on utilise souvent le rayon r200 , à l’intérieur duquel la
sur-densité moyenne est de 200, comme estimateur du rayon viriel rvir .
Le temps précédant l’effondrement d’une région est dicté par la sur-densité
initiale de la région, les régions plus denses s’effondrant plus rapidement que
les autres. En combinant ce résultat avec le spectre de puissance CDM, on
aboutit au scénario hiérarchique de formation des structures : la densité ini2 ∼ k −1 , donc les structures correspondant aux faibles
ˆ
tiale varie comme |δ(k)|
échelles (grand k) s’effondrent avant celles à grande échelle.
Si le champ de densité est filtré avec un filtre dit top-hat (qui prend la valeur
1 à l’intérieur d’une sphère de rayon R, lié à une échelle de masse par M =
4π
3
3 ρ0 R , et 0 ailleurs), le champ est toujours Gaussien. La variance du champ
filtré est
1
σ(M) =
2π 2
Z
∞
0
k2 P(k) W 2 (k; M) dk
,
Résumé
55
où W(k; M) est la transformée du filtre d’échelle M. Pour estimer la distribution en masse des objets effondrés, selon Press & Schechter (1974) on compte
comme objet de masse M toute région dans le champ filtré sur une échelle M
où la densité est supérieur à la valeur critique δc = 1.689 donnée par le modèle
sphérique. Pour un champ Gaussien, la probabilité de trouver un tel objet est
simplement
Z ∞
2
2
1
P(δ > δc ) = √
e−δ /2σ dδ .
2πσ δc
La dépendance de cette probabilité du redshift peut être trouvée en substituant
la constante δc par δc (z), la densité extrapolée linéairement à l’époque actuelle
si la densité était δ = δc à redshift z. La densité de halos de masse M s’écrit
comme le produit de la densité ρ0 /M et la probabilité P(> δc ) exprimée en
terme de masse, donc
r
ρ0 dσ dP
2 ρ0 d ln σ δc −δc2 /2σ2
dn
dM =
dM =
e
.
dM
M dM dσ
π M2 d ln M σ
Ceci est la fonction de masse de Press & Schechter (1974), montrée à la figure
2.5.
L’approche simple de Press & Schechter (1974) a été étendue par Bond et al.
(1991) et Lacey & Cole (1993). Dans leur approche on considère les trajectoires
δ(S(M)), en fonction de la variance S(M) = σ2 (M). A la plus grande échelle,
M → ∞, la densité δ → 0, donc toutes les trajectoires commencent à zéro.
Lorsque l’échelle M augmente, les fluctuations de densité augmentent aussi,
et la trajectoire δ(S(M)) suit une course aléatoire.
Comme auparavant ce modèle considère qu’une particule appartient à un
halo si sa densité filtrée atteint la valeur critique δc . A des redshifts différents la
particule se trouve dans des halos de masses différentes, la masse décroissant
quand le redshift augmente. Les maxima locaux des trajectoires correspondent
à des fusions entre amas. La fonction de masse n(M) dépend de la fraction f
de trajectoires qui traversent le seuil δc à masse M, qui de son côté dépend du
type de filtre qui est utilisé. Avec un filtre de forme “top-hat” dans l’espace
de Fourier, la trajectoire suit une vraie course aléatoire et f peut être trouvé
de manière élégante comme solution de l’équation de diffusion avec un bord
absorbant indépendant de la masse. Ceci reproduit la fonction de masse de
Press & Schechter (1974). L’intérêt de l’approche généralisée est qu’elle permet
des calculs pour d’autres types de filtres ainsi que des estimations du taux de
fusions entre halos, de la distribution des époques de formation de halo, etc.
(Lacey & Cole 1993).
Des simulations numériques à N corps offrent l’approche la plus exacte à
l’étude théorique de la fonction de masse parce qu’elles résolvent les équations
du mouvement directement. Les simulations les plus importantes ont été effectuées par le consortium VIRGO (Jenkins et al. 1998; Colberg et al. 2000b)
56
Chapitre 2. Counting Clusters of Galaxies
et ont été utilisées par Jenkins et al. (2001) pour déterminer la fonction de
masse de halos CDM sur quatre ordres de grandeur dans deux cosmologies
(le modèle standard SCDM avec ΩM = 1, ΩΛ = 0, et le modèle plat avec
constante cosmologique, ΩM = 0.3, ΩΛ = 0.7, avec des conditions initiales
définies par le spectre de puissance CDM). La fonction de masse de Jenkins
et al. (2001) peut être ajustée par une fonction universelle si elle est exprimée
en terme de σ(M) avec une densité critique δc = 1.689 indépendante de ΩM
et ΩΛ (ce qui n’est pas le cas dans le modèle d’effondrement sphérique). La
formule ajustée s’écrit
dn
ρ d ln σ(M) 3.8
= 0.315 2 exp
−
−
ln
σ(M)|
.
|0.61
dM
d ln M M
Cette formule est montrée à la figure 2.5 avec la fonction de Press & Schechter
(1974). Il y a des différences entre les deux fonctions : la fonction de Press &
Schechter prédit moins de halos de grande masse que la fonction de Jenkins
et al., en revanche elle surestime le nombre de halos de masse caractéristique
M∗ , ce qui correspond à peu près à σ = δc .
Des mesures locales de la fonction de masse des amas de galaxies offrent
la possibilité de calibrer le spectre de puissance selon la variance du champ
de densité filtré sur un rayon R = 8 h−1 Mpc, ce qui correspond à une masse
typique pour les amas riches. Dans cette mesure il existe une dégénérescence
−0.5
entre σ8 et ΩM , approximativement du type σ8 ∝ ΩM
. La dégénérescence est
levée si l’on considère l’évolution de la fonction de masse, montrée à la figure
2.7. Dans le modèle ΩM = 1, ΩΛ = 0, la densité d’amas massifs évolue très
14 −1
rapidement, avec quasiment aucun amas de masse >
∼ 5 × 10 h M prédit
à z ∼ 1. L’évolution est beaucoup moins rapide dans le modèle ΩM = 0.3,
ΩΛ = 0.7, ce qui est dû surtout à l’âge de l’univers qui est plus important dans
le modèle avec ΩΛ que dans le modèle de Einstein–de Sitter.
Chapter 3
Relaxation processes
3.1
Violent relaxation
Once a dark matter halo has decoupled from the Hubble expansion it collapses
and relaxes to a virialised steady–state configuration. The dynamically dominant dark matter is (in the standard cold dark matter models) only subject
to gravitational forces and collisionless, since two-particle encounters are negligible. Its configuration is described by the distribution function f (r, v, t) in
phase space, such that f (r, v, t) d3 r d3 v gives the number of particles with position in a volume d3 r around r and velocity within d3 v around v. The evolution
of f is governed by the Boltzmann equation (also called the Vlasov equation
in the absence of collisions)
Df
∂f
∂f
=
+ v · ∇ f − ∇Φ ·
=0 .
(3.1)
Dt
∂t
∂v
Here,
R Φ is the gravitational potential which is related to the dark matter density
ρ = d3 v f (r, v, t) through the Poisson equation
∇2 Φ(r) = 4πGρ(r)
.
(3.2)
Equation (3.1) is equivalent to Liouville’s theorem: The phase space density
around any given particle (or phase element) is a constant in time, i. e. the
convective derivative D f /Dt vanishes. This is always true; the system is in a
steady-state configuration if in addition the partial derivative ∂ f /∂t vanishes.
Although two-body collisions are completely negligible in a collapsing dark
matter halo, particles do react to potential fluctuations caused by fluctuations
in the density of surrounding particles. This leads to a rapid equipartition of
specific energy and angular momentum, leading to isotropic orbits in the final
configuration. For example, the energy E = v2 /2 + Φ(r, t) (per unit mass) of
an individual particle changes according to
dE
1 dv2 dΦ
dv ∂Φ
∂Φ =
+
= v·
+
+ v · ∇Φ =
.
(3.3)
dt
2 dt
dt
dt
∂t
∂t r(t)
57
58
Chapter 3. Relaxation processes
The process is very efficient: During the first collapse phase, when the gravitational potential changes rapidly, it operates on a typical timescale of about
10% of a crossing time (Lynden-Bell 1967), so that the system settles down to a
steady-state configuration after one or two oscillations. The process has therefore been dubbed violent relaxation.
Once the system is in a steady-state configuration, ∂Φ/∂t is small, and violent relaxation effectively ceases to operate. It is interesting to note that this is
not a true equilibrium configuration; in fact, as shown in Sect. 4.7 of Binney &
Tremaine (1987) there is no true equilibrium configuration for a self-gravitating
system, because the entropy of the system can always be increased by “scattering” a few particles to large velocities and ejecting them from the system, while
the remainder of the system shrinks and becomes more centrally concentrated.
The change of specific energy (3.3) due to a rapidly changing gravitational
potential is independent of the particle mass. Violent relaxation therefore does
not result in an equipartition of energy (or temperature) as for a collisional gas,
but in an equipartition of specific energy. Collisionless gases therefore tend to
be “isothermal” in the sense that their velocity dispersion is independent of position, not their temperature. Also, there is no mass segregation in collisionless
gases. In clusters of galaxies, galaxy mass segregation is observed only in the
dense central parts, showing that two-body encounters are important here,
leading to galaxy merging and cannibalism.
3.2
The isothermal sphere
There is a wide range of steady-state solutions to the Boltzmann equation (3.1),
and which one of these is actually chosen during the collapse of an overdense
region in the Universe depends on the initial conditions and the detailed collapse history of the region. A simple example of a steady-state solution to
equations (3.1) and (3.2) is given by the isothermal sphere (Binney & Tremaine
1987). This assumes that the mass distribution is spherically symmetric and
that it is “isothermal” in the sense that the velocity dispersion is constant
throughout the halo. Taking the first velocity moment of the Boltzmann equation yields Jeans’ equation, which for spherical symmetry and an isotropic velocity distribution, hvi v j i = σr2 δij , reads
1 dΦ
d ln ρ
=− 2
dr
σr dr
.
(3.4)
Eliminating the gravitational potential Φ from this equation and the spherically
symmetric version of the Poisson equation (3.2) yields a second order linear
differential equation for the mass density:
d
4πG
2 d ln ρ
(3.5)
r
= − 2 r2 ρ .
dr
dr
σr
3.3. Numerical simulations: A universal mass profile?
59
This equation has two general solutions; one of these diverges at the centre
(r = 0) and is called the singular isothermal sphere (SIS):
ρSIS (r) =
σr2
2πGr2
.
(3.6)
The second solution of Eq. (3.5) has a finite value at the centre, but can only be
calculated numerically. There are a number of approximations to this solution,
e. g. the King approximation
"
ρKing (r) = ρ0 1 +
r
rc
2 #−3/2
,
(3.7)
which has a flat core of size rc .
3.3 Numerical simulations: A universal mass profile?
The Boltzmann equation describes probabilities and can thus be used to predict statistical properties of dynamical systems. Unfortunately, it is non-linear
and its solution is very difficult. Instead of trying to solve Boltzmann’s equation, investigations of the collapse process are usually made by self-consistent
direct numerical integration of the equations of motion for a large number
of dark matter particles moving in their own gravitational potential. This is
equivalent to solving a coupled system of linear differential equations for a
given set of initial conditions, which is straightforward in principle, but in
practice poses problems of its own due to the large number of particles necessary to achieve adequate resolution and halo statistics. N-body simulations
and the results concerning the density structure of relaxed dark matter halos
will be described next.
One can distinguish between “cosmological” simulations and simulations
of individual dark matter halos. Cosmological simulations cover large volumes and provide sufficient statistics for counting dark matter halos, but lack
the resolution for investigating the internal structure of these halos. The initial conditions are set up by randomly placing particles in the simulation box
according to an assumed power spectrum, e. g. the CDM power spectrum
(Sect. 2.1); the choice of the cosmological parameters ΩM and ΩΛ determines
the expansion rate of the simulation box. Particles then move in the gravitational field generated by all the other particles, calculated in an efficient
way so as to minimise CPU cycles. Large cosmological simulations include
those of the VIRGO consortium (Jenkins et al. 1998) with particle masses from
1.00 × 101 0 h−1 M to 2.22 × 1012 h−1 M and corresponding box sizes between ∼ 100 h−1 Mpc to 2 . . . 3 h−1 Gpc (the so-called “Hubble volume” sim-
60
ulations), covering a range of cosmological models. The results of these simulations were used by Jenkins et al. (2001) to determine the dark matter halo
mass function quoted in Sect. 2.5.
High-resolution simulations of individual dark matter halos are typically
based on lower-resolution cosmological simulations. In the final state of the
latter a dark matter halo and the particles constituting it are identified. The
corresponding region in the initial density field is identified and reinitialised
with the same statistical properties, but with particles of smaller mass. Only
this region is then re-simulated, with the effects of the surrounding mass distribution taken into account by calculating moments from the cosmological
simulation at the same time steps.
The “particles” used in high-resolution simulations of dark matter halos
have masses of at least 5 × 107 M (Ghigna et al. 2000), that is they have subgalactic masses but are many orders of magnitude more massive than any reasonable candidate particle species constituting dark matter; these are all subatomic particles. Whereas for real dark matter particles two–body encounters
are completely negligible (at least in the simplest cold dark matter scenario),
this is not true for the simulation particles. In order to prevent spurious twobody relaxation due to the lack of mass resolution in the simulations, the gravitational field of the simulation particles has to be softened, so that at distances
smaller than the softening radius the effective gravitational force is weaker
than that due to a point particle. One particular softening scheme is named
“equivalent Plummer softening”. The force between particles i and j is
Fij = − Gm2 rij
3/2
rij2 + e2
,
(3.8)
where e is the softening length. This force corresponds to the potential-density
pair of Plummer’s model (Binney & Tremaine 1987, p. 42). The softening radius sets a lower limit to the possible spatial resolution in a numerical resolution which is related to its mass resolution. Ghigna et al. (2000), for example,
use a softening length of 0.5 h−1 kpc.
Using a set of high-resolution simulations for halos covering a range of
masses in the standard CDM model, Navarro, Frenk, & White (1995) proposed
that the mass distribution of these halos follows a universal profile if distances
are properly scaled (essentially as a fraction of the virial radius, see below).
Navarro et al. (1997) extended the simulations to cover a range of cosmological models, varying the background cosmology ΩM and ΩΛ , and the shape of
the power spectrum of the initial density field. In all cases, the spherically averaged density profiles of isolated halos were well described by the universal
form proposed by Navarro et al. (1995), which has since become known as the
NFW profile. In the context of the numerical simulations (which only include
cold dark matter particles, but no baryons), the existence of a universal density
3.3. Numerical simulations: A universal mass profile?
61
profile is therefore a very robust prediction, valid over a mass range of a factor
∼ 104 in a wide variety of cosmological models.
The universal density profile proposed by Navarro, Frenk, & White (1995)
has the form
ρ(r)
δc
=
,
(3.9)
ρcrit
r/rs (1 + r/rs )2
where ρcrit is the critical density, corresponding to ΩM = 1, δc is a characteristic
density contrast, and rs is the scale radius. The profile is usually called the
NFW profile. The limiting behaviour of (3.9) is
−1
r
, r rs
ρ(r) ∼
.
(3.10)
r −3 , r rs
The most salient feature of the model is that it has a central singularity or
“cusp”.
The total mass contained within radius r is easily obtained by integrating
(3.9) by parts:
r
r/rs
3
M(< r) = 4π δc ρcrit rs ln 1 +
−
.
(3.11)
rs
1 + r/rs
The total halo mass is usually estimated by M200 , the mass contained within
a sphere of radius r200 , enclosing a mean overdensity ρ = 200ρcrit . r200 is,
in accordance with the spherical collapse model (Sect. 2.3) and results from
numerical simulations (Cole & Lacey 1996), an estimate of the virial radius.
Evaluating (3.11) at r = r200 yields a relation between the characteristic density
contrast δc and the concentration parameter, defined as c = r200 /rs :
δc =
c3
200
3 ln(1 + c) − c/(1 + c)
.
(3.12)
For a given halo mass M200 , the NFW function (3.9) thus defines a one-parameter
family of density profiles, parameterised in terms of either δc or c. Numerical
simulations show that the halo mass and the concentration parameter c (or δc )
are correlated, with low-mass halos being less concentrated than high-mass
halos (Navarro et al. 1997). For rich clusters of galaxies c ≈ 5.
Whereas most simulators seem to agree that there is indeed a universal
profile (Fukushige & Makino 1997; Huss et al. 1999; Moore et al. 1999; but see
Jing & Suto 2000 for a different view), more recent simulations with improved
mass (and consequently spatial) resolution find steeper central slopes than for
the NFW profile (Fukushige & Makino 1997; Moore et al. 1999; Ghigna et al.
2000). The profile with central slope α = −1.5 is occasionally called the Moore
profile:
" 1.5 !#−1
r 1.5
r
ρ(r) ∝
1+
.
(3.13)
rs
rs
62
3.4
Observational status
Testing the prediction of the existence and the shape of a universal mass profile
is a major observational challenge. One major obstacle is contrary to simulated
halos real halos, i. e. galaxies and clusters of galaxies, contain baryons which
constitute an additional mass component, which might well mask the dark
matter distribution in the centres of the halos. This seems to be observed in
strong lensing mass reconstructions of clusters dominated by cD galaxies, e. g.
in Abell 383 (Smith et al. 2001). Dwarf and low surface-brightness late-type
galaxies are considered to be dominated by dark matter and thus offer the
possibility to probe the dark matter distribution in a fairly clean way. Radio
observations of the rotation curves of these galaxies provide evidence for at
best a shallow cusp (Kravtsov et al. 1998), in contradiction to the hierarchical
CDM prediction. While different observations yield consistent results, it is
occasionally argued that the spatial resolution of the radio observations is not
sufficient to make strong claims about the shape of the central mass profile.
At the other end of the halo mass range a strong lensing reconstruction of the
central mass profile of the rich cluster of galaxies Cl0024+1654 by Tyson et al.
(1998) shows a flat core. In Chapter 8 I show that this core is likely due to shock
heating of the central mass distribution by a collision between two ordinary
CDM clusters which are now superposed along the line of sight; Cl0024+1654
therefore does not provide evidence against the cold dark matter picture.
The shape of the central mass distribution of galaxies and clusters of galaxies provides clues to the nature of the elusive dark matter. Cold dark matter
produces central cusps. However, no such cusp is produced if the dark matter
is self-interacting, so that there is non-zero pressure, or warm, i. e. if particle
velocities are non-negligible. Further study of the central density profiles is
therefore of great importance; gravitational lensing studies of the centres of
rich clusters of galaxies provide a particularly clean of measuring central mass
profiles and form a cornerstone of the luminous cluster survey described in
Part III.
Résumé du chapitre:
Processus de relaxation
Une fois qu’un halo de matière noire a découplé de l’expansion de l’univers, il
s’effondre et se virialise pour atteindre une configuration stable. La matière
noire, qui domine la dynamique des halos, n’est susceptible qu’aux forces
gravitationnelles. Elle est aussi non-collisionnelle, puisque les collisions entre
deux particules sont négligeables. La configuration est décrite par la fonction
de distribution f (r, v, t) dans l’espace de phase. L’évolution de cette fonction
f est donnée par l’équation de Boltzmann (aussi appelée équation de Vlasov
en l’absence de collisions)
Df
∂f
∂f
=
+ v · ∇ f − ∇Φ ·
=0
Dt
∂t
∂v
.
Ici, Φ représente le potentiel gravitationel, qui est lié à la densité de matière
noire par l’équation de Poisson
∇2 Φ(r) = 4πGρ(r)
.
L’équation de Boltzmann est équivalente au théorème de Liouville: La densité
dans l’espace de phase autour d’une particule donnée est constante, donc la
dérivée D f /Dt disparaı̂t. Ceci est toujours vrai; le système se trouve dans un
état stable si la dérivée ∂ f /∂t disparaı̂t.
Lors de l’effondrement d’un halo de matière noire, les particules échangent
énergie et moment angulaire par des processus collectifs: Même si les collisions entre deux particules sont négligeable, une particule réagit aux fluctuations spatiales et temporelles du potentiel gravitationnel dûs aux fluctuations
de densité. Par exemple l’énergie spécifique E = v2 /2 + Φ(r, t) d’une particule
varie selon
dE
1 dv2 dΦ
dv ∂Φ
∂Φ =
+
= v·
+
+ v · ∇Φ =
.
dt
2 dt
dt
dt
∂t
∂t r(t)
L’état stable final est caractérisé par une équipartition d’énergie et de moment
angulaire, donc des orbites isotropes. Le mécanisme est très efficace: pendant
63
64
la première période de l’effondrement il agit sur un temps typique de 10% du
temps dynamique du système (Lynden-Bell 1967), donc le système s’arrange
après une ou deux oscillations. Le processus a ainsi été nommé relaxation violente.
Le changement d’énergie causé par le potentiel variable est indépendant de
la masse des particules, donc le résultat est une configuration d’équipartition
d’énergie spécifique et non de température. Des gaz non-collisionnels sont
donc “isothermes” dans le sens où la dispersion des vitesses est constante. De
même, il n’y a pas de ségrégation de masse dans des gaz non-collisionnels.
Dans les amas de galaxies, une ségrégation de galaxies de différentes masses
n’est observée que dans les parties centrales où les rencontres entre deux galaxies sont en effet importantes, ce qui a pour conséquence les fusions et le “cannibalisme” des galaxies.
Le profil de masse d’une sphère isotherme est déduit depuis l’équation de
Jeans, le premier moment de vitesse de l’équation de Boltzmann, dans le cas
de symétrie sphérique et dispersion de vitesse σr isotrope et indépendante de
la distance du centre de la sphère. Avec l’équation de Poisson on trouve une
équation différentielle de deuxième ordre pour la densité ρ(r) de la sphère, qui
a donc deux solutions indépendantes. La première a une singularité au centre,
et définit le profil d’une sphère isotherme singulière (SIS)
ρSIS (r) =
σr2
2πGr2
.
La deuxième solution a une valeur finie au centre mais doit être calculée numériquement. Une approximation de cette solution est donnée par le profil de
King:
"
2 #−3/2
r
ρKing (r) = ρ0 1 +
,
rc
avec un cœur plat de taille rc .
L’équation de Boltzmann décrit des probabilités et peut être utilisée pour
prédire des propriétés statistiques d’ensembles de halos. Elle est non-linéaire
et sa solution est donc très difficile. En pratique on poursuit une autre approche, celle de la simulation numérique de l’évolution d’un gaz de particules
non-collisionnelles, par intégration directe des équations du mouvement d’un
grand nombre de particules dans leur potentiel gravitationnel commun. Ceci
équivaut résoudre un système couplé d’équations différentielles de première
ordre. Les problèmes pratiques dans cette approche sont dûs aux grand nombre de particules qui est nécessaire pour fournir une résolution suffisante et un
nombre suffisant de halos distincts pour en déterminer des propriétés statistiques.
Les particules utilisées dans les simulations ont des masses sous-galactiques
de typiquement 1010 à 1011 M et sont donc beaucoup plus massives que les
65
Résumé
particules de matière noire qui sont plutôt sous-atomiques. Si pour la vraie
matière noire les collisions sont complètement négligeable (au moins pour la
matière froide qui est normalement étudiée), ceci n’est pas le cas pour les particules dans les simulations. Pour éviter de telles collisions on coupe le champ
gravitationel des particules au-dessous d’une longueur d’assouplement. Cette
longueur limite la résolution spatiale d’une simulation et est liée à la masse
des particules, donc à la résolution en masse.
Les simulations révèlent que les halos de matière noire peuvent être décrits
par un profil de densité “universel” (dit de NFW) proposé par Navarro, Frenk,
& White (1995) sous la forme
ρ(r)
δc
=
ρcrit
r/rs (1 + r/rs )2
,
où ρcrit est la densité critique, correspondant à ΩM = 1, δc est une densité
caractéristique, et rs est le rayon d’échelle. Le profil varie comme r −1 dans le
centre et comme r −3 à l’infini. La masse totale du halo est estimée par M200 , la
masse d’une sphère de sur-densité de 200 ρcrit , comme suggéré par le modèle
sphérique. Dans les simulations le rayon r200 sépare la région virialisée d’un
halo d’une zone d’accrétion à plus grande distance (Cole & Lacey 1996). Pour
une masse M200 donnée le profil NFW forme une famille d’un paramètre, qui
peut être la densité caractéristique δc , ou le paramètre de concentration c qui
est lié à δc par l’équation (3.12). Pour les amas riches, c ≈ 5.
Navarro, Frenk, & White (1997) ont étudié la formation de halos de matière
noire à grande résolution dans un bon nombre de cosmologies avec différentes
valeurs des paramètres cosmologiques, ΩM et ΩΛ , et différents types de spectre de puissance (CDM, ou un spectre de forme kn avec différents exposants
n), et ont trouvé que le profil NFW fournit une excellente description des
profils observés dans les simulations dans tous les cas sur un grand intervalle de masse. Ceci constitue donc une très forte prédiction du scénario de
matière noire froide: si la matière noire dans l’univers est en effet de ce type,
on s’attend à ce que les profils de masse de tout objet entre galaxies naines et
amas riches suivent cette forme.
Pendant que tous les auteurs semblent être d’accord sur la forme du profil universel à grande distance du centre d’un halo, la forme exacte du profil
central reste sujet de débat. Moore et al. (1998) et Ghigna et al. (2000) trouvent
dans des simulations à plus grande résolution une pente plus raide dans le
centre, de −1.5 contre −1 pour le profil NFW.
Chapter 4
Determination of Cluster Masses
4.1
What is a cluster of galaxies?
In the preceding chapters clusters of galaxies were presented as rather abstract
entities, and dark matter was assumed to be their only component. In fact,
the term “dark matter halo” or “collapsed object” was used interchangeably
with “cluster of galaxies”. Dark matter is certainly the dynamically dominant
component in clusters; observations typically find baryon fractions in clusters
of the order of 20% (White 2000). As long as only dark matter is considered
the structure and evolution of clusters is entirely governed by gravitational
physics, making the theoretical treatment relatively simple and robust. In this
case there is a continuity between dark matter halos on galactic and on cluster
scales. However, dark matter is not the only component of clusters of galaxies, and the presence of baryons in the form of stars and galaxies and the hot
intracluster medium is essential for observations of clusters. What a cluster
of galaxies is means different things in different contexts, and the comparability between theoretical predictions, which are usually expressed in terms of
mass, and observations, which define a cluster through, for example, their Xray properties, is a fundamental problem for cosmological research based on
clusters of galaxies.
In this section I will briefly list a few definitions of “cluster of galaxies”
used in different contexts, and then discuss several methods which estimate
cluster masses from observational properties.
The Press-Schechter approach and its extensions (Sect. 2.4) assume that
clusters form at the density peaks of the initial density field smoothed on a
given length or mass scale. Counting clusters in these approaches is equivalent to counting density peaks.
Numerical simulations are based on particles and need a criterion whether
a given particle belongs to a bound halo or not. Number counts then count
distinct halos in the final particle configuration at, say, redshift 0. Jenkins et al.
66
4.1. What is a cluster of galaxies?
67
(2001) investigate two methods of assigning particles to dark matter halos:
The friends-of-friends algorithm links together all particles which have a nearest
neighbour in the set at a distance less than the linking length, which is a parameter to be specified and which is related to the mean overdensity within a
halo. The spherical overdensity method essentially smoothes the particle distribution with a spherical top-hat filter and defines halos as those regions where
the overdensity within the sphere is larger than a given threshold. The threshold is usually taken to be around 180 to 200, in accordance with the spherical
collapse model. The two methods do not in detail select the same halos; however, Jenkins et al. (2001) find that the statistical mass functions based on the
two methods agree fairly well. Another point to take into account when comparing theory and observations is the actual density threshold adopted.
Observationally, the first cluster catalogues were constructed by looking
for galaxy concentrations on the sky. Abell (1958) called a “cluster” a concentration of galaxies with at least 50 galaxies within the Abell radius rA =
1.5 h−1 Mpc with magnitudes in the range m3 to m3 + 2, where m3 is the magnitude of the third brightest galaxy within the concentration. Zwicky et al. (1961)
defined the boundary of a cluster as an isodensity contour at twice the background surface number density of galaxies and called the galaxy concentration
a “cluster” if at least 50 galaxies with magnitudes between m1 and m1 + 3 were
contained within the isodensity contour. Zwicky’s adoption of a fixed density
boundary is more in line with modern thinking (where the virial radius corresponds to a fixed overdensity) than Abell’s fixed physical radius, but Abell’s
selection criteria were stricter, making the Abell catalogue more useful for statistical purposes. Projection effects can produce galaxy overdensities on the
sky that do not correspond to bound clusters of galaxies, and both the Abell
and the Zwicky catalogues contain such cases. Modern cluster surveys based
on optical selection criteria use images in several filter bands and add colour
criteria to select clusters based on their red galaxy sequence.
Many cluster surveys identify clusters based on their X-ray emission. In
X-ray imaging surveys clusters can be fairly unambiguously identified as extended sources, although optical (spectroscopic) follow-up is always necessary
to confirm that the X-ray emission is really due to a cluster of galaxies. The exact selection criteria (most importantly a flux limit) for X-ray surveys are a
compromise between depth and width of the survey. All-sky surveys based
on the R OSAT All Sky Survey cover the largest possible sky area, but have
fairly high flux limits. Examples are the XBACs (X-ray Bright Abell Clusters,
Ebeling et al. 1996), BCS (Bright Cluster Survey, Ebeling et al. 1998, 2000) and
R EFLEX (Böhringer et al. 2001) surveys at low redshift (z <
∼ 0.3), and MACS
(MAssive Cluster Survey, Ebeling et al. 2001) at higher redshift. Deeper surveys cover smaller sky areas due to the longer exposure times needed to reach
the lower flux limits; an example is the North Ecliptic Pole survey (NEP, Voges
et al. 2001). Fig. 4.1 gives a synopsis of existing X-ray cluster surveys.
68
Chapter 4. Determination of Cluster Masses
10-11
RASS1-BS
EMSS
160 deg
ith
2
LX
>
sw
ter
10
NEP
BCS
REFLEX
BCS-E
> 0.3
z
s at
ster
clu
ino
um
yl
s
ter
MACS
at
z>
0.3
lus
ra
X-
sc
all-sky
SHARC-S
0X
-ra
lu
u
no
i
m
y
RDCS
s
10
us
s
clu
10
x
5
g
er
WARPS
10
flux limit [0.5 - 2.0 keV] (erg cm-2 s-1)
44
10-12
ny
,a
-1
Bright SHARC
z
10-13
optical detection limit
at z ~ 1.3
10-14
10
100
1000
solid angle (square degrees)
10000
Figure 4.1: Selection functions of major X-ray cluster surveys of the past
decade. Surveys are characterised by their flux limit and their survey area;
the “tails” attached to the points can be interpreted as potential flux limits
over parts of the survey area due to the varying effective exposure time in the
R OSAT All Sky Survey, which forms the basis of all the surveys (with the exception of EMSS). The dashed lines mark flux limits at which 10 or 100 X-ray
luminous clusters at redshift z > 0.3 can be expected to be found. The figure
has been kindly provided by Harald Ebeling and was published in Ebeling
et al. (2001).
For the future, cluster surveys based on the Sunyaev-Zeldovich effect or
weak gravitational lensing are envisaged. Gravitational lensing surveys will
select clusters directly by mass (Sect. 4.2); the CFHT Legacy Survey will be a
first major step towards a mass-selected cluster sample. The Sunyaev-Zeldovich
(SZ) effect is based on inverse Compton scattering of cosmic microwave background photons by the hot intracluster medium; the ICM scatters the photons
to higher energies, thus producing regions of lower (or higher, according to the
observed wavelength) radiation temperature on the sky. The great advantage
of the SZ effect is that the observed temperature decrement is independent of
redshift, allowing detection of clusters out to high redshift.
In the following sections I shall present the standard methods to derive
cluster masses from observations of the gravitational lens effect, the X-ray
emission and the galaxy kinematics.
4.2. Gravitational Lensing
4.2
69
Gravitational Lensing
The deflection of light in the vicinity of large masses is one of the central predictions of Einstein’s theory of general relativity. Close to large masses spacetime can be curved to such an extent that there are multiple null-geodesics
which connect an observer to the world-line of a distant light source; in this
case the observer sees multiple images of the source. At larger distances from
the deflecting mass, light bundles emerging from the source are (partially)
focused and sheared and the image of the source as seen by an observer is
distorted and magnified or demagnified. All these effects are readily observable in the form of multiply imaged QSOs, giant arcs and arclets in clusters of
galaxies, or a statistical distortion pattern in galaxies behind clusters of galaxies, and are subsumed under the term gravitational lensing. Because the lensing
effects depend directly on the mass distribution between the source and the
observer without regard to its dynamical state or composition (baryonic or
non-baryonic), gravitational lensing has become a very valuable tool for cosmological research.
Gravitational lensing in clusters of galaxies was first observed in the form
of the giant arc in Abell 370 by Lynds & Petrosian (1986) and Soucail et al.
(1987). Since then, many giant arcs and arclets were found in clusters and mass
models based on cluster lensing are available for many of the lensing clusters
(e. g. Kneib et al. 1993, 1996; Tyson et al. 1998; Smith et al. 2001). Coherent
weak shear patterns around clusters of galaxies were first detected by Tyson
et al. (1990) and Bonnet et al. (1994).
Useful general references for gravitational lensing include Schneider et al.
(1992) and Narayan & Bartelmann (1999). Cluster lensing is treated extensively
by Fort & Mellier (1994), Mellier (1999) and Bartelmann & Schneider (2001).
4.2.1
Generalities
If the gravitational lens is a cluster of galaxies it is safe to assume that the
deflecting mass is concentrated in a plane at the position of the cluster, where
the deflection is instantaneous, with the light rays following geodesics in the
background cosmology outside the lens plane.
Fig. 4.2 shows a sketch of the geometry of a gravitational lens system, establishing the notation. The lens equation which relates the (unlensed) source
position β to the observed image position(s) θ can be obtained from simple
geometry:
D
(4.1)
β = θ − ds α(θ).
Ds
If source and lens are at cosmological distances (as is the case when clusters
of galaxies act as gravitational lenses) all distances have to be interpreted as
angular size “distances”.
70
Figure 4.2:
Sketch of a gravitational lens system.
o. . . observer,
d. . . deflector/lens, s. . . source. The distances Dd , Dds and Ds are angular size
distances in a cosmological context, and have therefore a (temporal) orientation due to the direction of light propagation.
The deflection angle α is determined by the mass distribution of the lens:
Z
4G
θ − θ0
α(θ) = 2 Dd
d2 θ 0 Σ(θ0 )
,
(4.2)
c
|θ − θ0 |2
where Σ(θ) is the surface mass density in the object, projected along the line of
sight. Σ has dimensions of M/L2 , with physical lengths measured in the lens
plane.
For a mass distribution that is circularly symmetric around the line-ofsight, the deflection angle at an angular distance θ from the centre of the mass
distribution is given by the mass enclosed within a beam of radius θ, M(< θ):
α(θ) =
4G M(< θ)
c2 Dd θ
.
(4.3)
For a point source that is perfectly aligned with the observer and the centre of
the lens, so that β = 0, the resulting image is a ring around the lens of radius
s
4G Dd Dds
Dd θE =
M(< θE ) ,
(4.4)
ds
c2
the Einstein radius. The mean density inside the Einstein radius is called the
critical density Σcrit and provides a convenient density scale since it only depends on the relative distances between observer, lens and source:
Σcrit
M(< θE )
c2
Ds
=
=
4πG Dd Dds
πDd2 θE2
.
(4.5)
Surface mass densities are often given in terms of the convergence κ(θ) = Σ(θ)/Σcrit .
A sufficient condition for a general lens to be able to form multiple images of
background sources is that κ > 1 at some position in the lens plane.
71
Locally, the lens equation (4.1) can in general be linearised so that the lens
effect is described by the magnification matrix A:
dβ = A dθ .
(4.6)
The magnification matrix can be written as
1 − κ − γ1
−γ2
A=
−γ2
1 − κ + γ1
.
(4.7)
Here, γ1 and γ2 are the components of the shear γ, which is not a vector but
can be written as a complex number γ = γ1 + iγ2 . The eigenvalues of A are
λ1,2 = (1 − κ) ± |γ|
.
(4.8)
The effect of a gravitational lens on a small circular source (located in a region
where neither of the eigenvalues of A vanishes) can thus be pictured as being
composed of a homogeneous stretching of the radius by a factor of 1/(1 −
κ) and a stretching/squashing along perpendicular directions, the direction
and magnitude of which are determined by the shear γ. A circular source of
radius ρ is thus imaged onto an ellipse with principal axes ρ/(1 − κ ± |γ|).
Since gravitational lensing preserves the surface brightness of the source, the
change in area leads to a change (usually an amplification) of the object’s total
brightness (in terms of flux) by µ given by 1/µ = det A−1 = (1 − κ 2 ) − |γ|2 .
4.2.2
Strong lensing
Along critical lines in the lens plane, one of the eigenvalues (4.8) vanishes, and
the amplification µ diverges in the ray optical approximation. The corresponding lines in the source plane, related to the critical lines through the lens equation (4.1), are called caustics1 . The size, and to some extent the shape, of a
critical line/caustic depend on the source redshift and the background cosmology through the critical density (4.5). Sources near caustics are subject to
strong deformation and magnification, giving rise to giant arc systems in the
central parts of rich clusters of galaxies. Critical lines mark off regions of different image multiplicity: For a hypothetical source crossing a caustic from a
region of higher multiplicity into a region of lower multiplicity, two of its images approach each other as the source approaches the caustic, merge and then
disappear. In addition, images seen on opposite sides of a critical line have opposite parity, i. e. they are mirror images of each other. This fact will be used
in Sect. 6.2 to qualitatively trace the critical line corresponding to the resolved
arc system in Cl0024+1654.
1 Properly
speaking, these are anti-caustics, since in traditional ray optics caustics are defined to lie on the observer side of the deflector.
72
Two types of critical curves can be distinguished for lenses which deviate
only slightly from circular symmetry, according to which of the eigenvalues
(4.8) vanishes. Images near tangential critical curves are stretched along a direction tangential with respect to the mass distribution, whereas images near
radial critical curves are stretched in the radial direction. The Einstein circle
provides an example of a tangential critical curve. As can be seen from Eq.
(4.4), observations of tangential arcs can be used to place robust bounds on
projected cluster masses. The condition for a radial critical curve is
d M
= πDd2 Σcrit ,
(4.9)
dθ θ
showing that observations of radial arcs are useful to constrain the local slope
of the projected density profile of the lens (Williams et al. 1999). In Smith et al.
(2001), the positions of two radial arcs in Abell 383 were available to constrain
the central mass profile of this cluster (Sect. 11.4).
The strong lensing regime refers to sources that are found near caustics or
in regions where they are multiply imaged. In the lens plane, the strong lensing regime corresponds to regions in the centre of a cluster of galaxies, out to
a radius of order 100 h−1 kpc at the distance of the cluster. Modelling of observed multiple images can be used to accurately constrain the mass distribution in this region, by using as the main observational constraints the positions
of the images, their flux ratios (i. e. ratios of the local magnifications µ), and
to some extent the shapes of the images in cases where these are sufficiently
resolved. Lens modelling is typically an iterative process, where one starts out
with a fairly simple parametrised to reproduce the observational constraints
provided by one or two “obvious” image systems (giant arcs). The model
can then be used to identify less conspicuous image systems, which in turn
are used to refine the model, either by giving more robust constraints on the
parameters of the original model, or by extending the model to include additional parameters describing for example the influence of individual cluster
galaxies.
Since the position of critical lines depends on the source redshift, a lens
model can be used to predict source redshifts (Ebbels et al. 1998); on the other
hand spectroscopic redshifts for multiply imaged systems provide powerful
additional information on the structure of the lens system (in addition to confirming the reality of a multiple image system), not least for the absolute calibration of the mass profile deduced from the strong lensing analysis.
4.2.3
Weak lensing
Outside regions of multiple image formation and away from critical lines, the
lens effect of a cluster of galaxies is weak; this is the weak lensing regime. Two
73
effects can be exploited to obtain constraints on the mass profile in this regime,
the change of shape of a background galaxy, and the change of its apparent
brightness. Since neither the intrinsic shape nor the intrinsic luminosity of an
individual galaxy are known, weak lensing analyses have to be done statistically on a large number of background galaxies. I shall describe both approaches to weak lensing in turn.
Weak shear
The basic assumption here is that the intrinsic distribution of galaxy ellipticities (including orientation) has zero mean. A galaxy viewed through a cluster
of galaxies will have an ellipticity component on top of its intrinsic ellipticity
due to the shear provided by the cluster’s gravitational lens effect. A sample
of background galaxies will therefore have a mean ellipticity which deviates
systematically from zero, and which is directly related to the projected mass
distribution of the cluster. In particular, if one uses as the definition of the
complex ellipticity of a galaxy image
χ=
Q11 − Q22 + 2iQ12
Q11 + Q22
,
(4.10)
where the Qij are second moments of the image’s surface brightness, then the
expectation value of χ is related to the reduced shear g = γ/(1 − κ) through
(Schneider & Seitz 1995)
χ − 2g + g2 χ∗
= hχ(s) i = 0 ,
(4.11)
2
∗
1 + |g| − 2 Re(gχ )
where the mean intrinsic ellipticity hχ(s) i of the source galaxy population is
assumed (and in fact required) to vanish. In the linear approximation, valid
for the true weak lensing regime, Eq. (4.11) reduces to hχi ≈ g.
The convergence κ and the shear γ can both be expressed in terms of second derivatives of the projected gravitational potential and are thus related
through an integral expression:
Z
1
γ(θ) =
d2 θ D(θ − θ0 ) κ(θ0 )
(4.12)
π
with the kernel
θ22 − θ12 − 2iθ1 θ2
D(θ) =
.
(4.13)
|θ|4
Kaiser & Squires (1993) use direct inversion of (4.12) to reconstruct the density
field from the measured shear field. Other methods have been developed for
the reconstruction of the surface density, or in some cases the gravitational potential. These methods differ from the Kaiser & Squires method through their
statistical robustness with respect to observational and measurement errors.
74
Depletion
Galaxy number counts as a function of apparent magnitude have been conducted in many wave bands and now reach limiting magnitudes of ∼ 30 mag
for the deepest fields (the northern and southern Hubble Deep Fields: Williams
et al. 1996; Volonteri et al. 2000). The number counts can usually be described
as an exponential function of magnitude (a power law in flux), log n(< m) =
βm + const. over large ranges in magnitude, with slopes of β = 0.404 in the
V band and β = 0.271 in the I band (Smail et al. 1995). The most prominent
deviation from a simple linear dependence is observed in the B band where
the number counts break from β = 0.47 to β = 0.30 around B = 25 (Ellis 1997).
The presence of a gravitational lens, in particular a rich cluster of galaxies, locally modifies the observed number counts n(m) through two competing effects: On the one hand, individual galaxies are magnified, shifting the
number counts to brighter apparent magnitudes; on the other hand, the angular distances between galaxies are increased, leading to a decrease of the
galaxy number density at fixed absolute magnitude. Whether the combination
of these two effects leads to an effective increase or decrease of the observed
number counts with respect to the unlensed situation depends on the slope β
of the number counts: if for example the number counts are steep, the large
number of faint galaxies that are magnified into the magnitude range under
consideration offsets the geometrical dilution and there will be a net increase
in the number counts; this is the case for QSOs. Galaxy counts are usually
much less steep so that there is effectively a depletion of background galaxies
behind a cluster of galaxies compared to the unlensed surrounding field.
The limiting value of the number count slope β can be determined as follows (Bartelmann & Schneider 2001): Assume for simplicity that all the source
galaxies are at redshift z = ∞, which is justified by the fact that the amplification µ(zs ) for a lens at zd ∼ 0.2 varies by only about 30% for source redshifts
between 0.8 and 5; the magnification depends on the source redshift through
the angular size distance ratio (Dds /Ds )2 . Let n0 (< m) be the unlensed cumulative projected number density of galaxies brighter than m. Behind a gravitational lens with amplification µ(θ) the observed number counts will then be
given by
1
n0 (< m + 2.5 log µ(θ)) .
(4.14)
n(< m) =
µ(θ)
If the intrinsic number counts follow an exponential law, log n0 = βm,
n0 (< m + 2.5 log µ(θ)) = n0 (< m) µ2.5β
and
n(< m)
= µ2.5β−1
n0 (< m)
.
,
(4.15)
(4.16)
75
4.3. Distribution of the Intracluster Gas
For β < 0.4, flux-limited number counts behind a cluster of galaxies are therefore reduced with respect to a nearby unlensed field.
If the unlensed number counts n0 (< m) and their slope β are known, observation of n(< m) provides an estimate of the amplification field of a lensing
cluster. In the limit of weak lensing, where the convergence and in particular
the shear are small, κ 1 and γ 1, one can reconstruct the surface density
from number counts alone, κ ≈ (µ − 1)/2. Although number counts provide
a less efficient way in terms of signal-to-noise to reconstruct the cluster mass
distribution than shear analyses (Bartelmann & Schneider 2001), they are more
robust with respect to observational uncertainties, in particular seeing conditions. Ideally, one would combine shear and magnification observations to
obtain the mass distribution
κ = 1− p
1
µ(1 − |g|2 )
.
(4.17)
4.3 Distribution of the Intracluster Gas
Rich clusters of galaxies are permeated by a hot tenuous plasma with temperatures of order 107 . . . 108 K and number densities of order ng ∼ 10−3 cm−3 .
This gas emits radiation at X-ray energies through the thermal bremsstrahlung
mechanism; clusters of galaxies are amongst the brightest X–ray emitters known,
45 −2
−1
at luminosities LX <
∼ 2 × 10 h erg s . Apart from the continuum emission
the cluster gas also emits line radiation in the X-ray band, e. g. in the Fe L-shell
lines. Line radiation contributes a significant fraction to the X-ray luminosity
of poor clusters and galaxy groups; for rich clusters of galaxies, line radiation
is negligible for luminosity models, although it does provide important clues
to the metallicity and hence the history of the gas.
The emissivity of a thermalised hot plasma that emits primarily through
bremsstrahlung is (Sarazin 1988)
eff ∝ n2e T 1/2
,
(4.18)
where the electron density ne ∝ ng . Integrating this expression along the line of
sight yields the observed surface brightness. In the spherical case, the integral
is the standard Abel integral
S(R) =
Z
+∞
−∞
eff (r) dl = 2
Z
+∞
R
eff (r) r dr
√
r 2 − R2
.
(4.19)
Here, l is the coordinate along the line of sight, r is the three-dimensional distance from the cluster centre, and R = Dang θ is the projected distance at which
the line of sight passes from the cluster centre.
76
4.3.1 Hydrostatic equilibrium
To derive the distribution of the gas in the potential well of a cluster of galaxies
we assume hydrostatic equilibrium, i. e. the gas is entirely supported by its
internal pressure in the potential well Φ(r):
∇P = −ρg ∇Φ
,
(4.20)
where the gas pressure P and the gas mass density ρg = µmp ng are related by
P = ng kT for an ideal gas. Here k is the Boltzmann constant, ng is the number
density of gas particles, mp the proton mass, and µ is the mean particle mass,
typically µ = 0.63 for a completely ionised plasma of solar metallicity (Sarazin
1988). Using the equation of state for an ideal gas, (4.20) reads
kT ∇ ln ng + ∇ ln T = µmp ∇Φ .
(4.21)
The gravitational potential Φ is determined by the total mass density ρ(r) in the
cluster through the Poisson equation
∇2 Φ = 4πGρ(r)
.
(4.22)
If the total mass distribution is spherically symmetric, the potential at distance
r from the cluster centre is determined
solely by the mass contained within
Rr
the sphere of radius r, M(r) = 0 4πρr 02 dr 0 , and the equation of hydrostatic
equilibrium reads
d ln ng d ln T
Gµmp M(r)
1 dP
= kT
+
=−
.
(4.23)
ng dr
dr
dr
r2
If both the temperature profile T(r) and the gas density profile ng (r) can be
measured (from X-ray spectral and imaging observations for example), then
Eq. (4.23) can be used to derive the total mass profile M(r).
For an isothermal gas, T = const., (4.23) can be easily solved by direct
integration to yield
Z r
µmp
M(r 0 ) 0
ng = ng,0 exp −
G
dr
.
(4.24)
kT
r 02
0
The central gas density ng,0 is determined through the initial conditions and
depends on the baryon fraction in the cluster and, by extrapolation, in the
Universe.
4.3.2
The β model
A standard model for the X-ray surface brightness distribution in clusters of
galaxies is the so-called β model, introduced by Cavaliere & Fusco-Femiano
77
(1976). The main assumptions of this model are spherical symmetry and isothermality of both the collisional gas and the collisionless (dynamically dominant) dark matter. The total mass distribution then satisfies the isotropic Jeans
equation (3.4)
d ln ρ
1 dΦ
=− 2
,
(4.25)
dr
σr dr
the gas density satisfies the equation of hydrostatic equilibrium
d ln ρg
µmp dΦ
=−
dr
kTg dr
.
(4.26)
Eliminating the gravitational potential from Eqs. (4.25) and (4.26) yields the
simple relation between the gas density and the total density in a cluster
ρg (r) ∝ ρ(r) β
(4.27)
with
µmp σr2
.
(4.28)
β=
kT
More specifically the β model assumes that the total mass distribution follows the King approximation for a self-gravitating isothermal sphere (Sect.
3.2):
"
2 #−3/2
r
.
(4.29)
ρ(r) = ρ0 1 +
rc
This distribution is also suggested by the observed galaxy number density profiles in clusters.
The gas density profile is then given by
"
ρg (r) = ρg,0 1 +
r
rc
2 #−3β/2
(4.30)
and the X-ray surface brightness profile is
"
S(R) = S0 1 +
R
rc
2 #−3β+1/2
.
(4.31)
The β model usually fits observed surface brightness distributions reasonably well, with β fit ≈ 0.66, if simply considered a fitting parameter. Since
β < 1, the gas is more extended than the dark matter. Equation (4.28) provides an independent way to estimating β if the further assumption is made
that the galaxy distribution in the cluster follows the same form as the dark
78
matter distribution, equation (4.29). In this case, the dark matter velocity dispersion σr2 in (3.4) can be identified with the one-dimensional galaxy velocity
dispersion, and measurements of the latter in conjunction with a spectral measurement of the X-ray temperature yield β spect from (4.28). Typical values are
β spect ≈ 1.2, significantly larger than the values β fit found from surface brightness fitting. Possible explanations for this discrepancy invoke uncertainties in
the measurement of the galaxy velocity dispersion or the possibility that the
King approximation might not be a good description for the total mass distribution (see the discussion in Sarazin 1988).
4.3.3 Recent observations
The β model has been the standard model until quite recently. With the advent
of the latest generation of X-ray telescopes (XMM, C HANDRA) with their improved spatial and spectral resolution, it is now possible to drop some of the
assumptions that underlie the β model. Most notably temperature profiles can
now be measured, which allows in principle to derive mass estimates from
Eq. (4.23) directly. To date, there is not yet consensus as to the exact shapes
of temperature profiles in clusters of galaxies, with different authors deriving
different profiles from occasionally even the same data. Most of the studies
to date are however based on observations with the last generation of X-ray
telescopes, in particular ASCA and BeppoSAX. Markevitch (1998), e. g., found
steeply decreasing temperature profiles in a sample of 30 clusters observed
with ASCA, whereas Irwin et al. (1999) found mostly isothermal profiles in
essentially the same sample. White (2000) analysed a larger sample of 106
clusters, also observed with ASCA, and concluded that 90% of the temperature profiles were consistent with being flat. Conflicting conclusions were also
drawn from BeppoSAX data, where Irwin & Bregman (2000) claimed general
isothermality (with possibly even a slight increase at ∼ 0.3 rvir ) in a sample of
11 clusters, whereas de Grandi & Molendi (2002) found isothermal cores out
to ∼ 0.2 rvir and a rapid temperature decline at larger radii. The main reason
for these discrepancies is uncertainty about data calibration; e. g. the difference in the conclusions of Irwin & Bregman (2000) and de Grandi & Molendi
(2002) can be traced to different treatment of a support structure in the detector entrance window which obscures part of the beam. There have been few
published temperature profiles with XMM or C HANDRA so far; uncertainties
in the data calibration are still large. One example of a temperature profile
measurement with XMM/Newton (Abell 1835, Majerowicz et al. 2002) will be
discussed in Sect. 11.9.
If there is a systematic trend in the sense that the gas temperature decreases
with increasing radius as found by some authors, then the isothermal assump-
79
tion leads to an overestimation of the cluster mass of about 30% at about six
core radii (Markevitch 1998).
The major prerequisite for a meaningful mass estimate from X-ray observations is that the gas be in hydrostatic equilibrium. Unfortunately, hydrostatic
equilibrium is fairly easily disturbed, most importantly through cluster mergers which dissipate the relative kinetic energy of the collision through shock
waves driven into the ICM of the clusters. Cluster mergers have been studied with hydrodynamic simulations by Roettiger et al. (1996), Ricker (1998),
Ritchie & Thomas (2001) or Ricker & Sarazin (2001) (see also the review by
Sarazin 2001). These studies show that over ∼ 1 Gyr after the first cluster core
crossing both the X-ray luminosity and the X-ray temperature fluctuate by up
to a factor of ten. Evidence for mergers is seen in many clusters; Jones & Forman (1984) find significant X-ray substructure in at least 22% of a large sample
observed with the E INSTEIN observatory. Edge et al. (1992) suggest that clusters will typically undergo a merger every 2 to 4 Gyr, which is comparable to
the time required for the gas to return to hydrostatic equilibrium (Ritchie &
Thomas 2001). C HANDRA observations of cluster cores at high spatial resolution provide more detailed information about the structure of ongoing mergers, with evidence for shocks and cold fronts (Markevitch et al. 2000; Mazzotta
et al. 2001a,b; Markevitch & Vikhlinin 2001; Markevitch et al. 2001; Vikhlinin
et al. 2001). Cluster mergers are not always identifiable from the X-ray structure alone: in Part II I will present the case of Cl0024+1654 where evidence
for an ongoing cluster collision was only found in a wide-field redshift survey,
with the X-ray data providing no hint to any deviation from equilibrium.
In the centres of relaxed regular clusters of galaxies the gas density becomes
so high, that radiative cooling becomes important (the cooling time scale becomes shorter than the Hubble time H0−1 ). When the gas cools significantly
it contracts and the outer gas layers lose pressure support. Hence, a cooling
flow develops (see the review by Fabian 1994). In the cluster centre the gas is
expected to be in a multiphase configuration with phases at different temperatures. The increasing central gas density results in an excess X-ray surface
brightness in the cluster centre compared to a β profile fit to the outer regions
of the surface brightness profile. A temperature drop in the central regions of
cooling flow clusters has been observed, although the simple cooling flow picture has recently been challenged by the failure to detect sufficient amounts of
emission from gas below 2.7 keV in Abell 1835 (Peterson et al. 2001; Schmidt
et al. 2001).
Cooling flows can bias mass estimates for clusters of galaxies. Allen (1998)
showed that discrepancies between mass estimates from gravitational lensing
and X-ray observations for cooling flow clusters can be resolved if the effects
of the cooling flows on the X-ray properties are properly taken into account.
Discrepancies between different mass estimators persist for clusters without
80
cooling flows; for these, substructure and deviations from equilibrium have
been invoked to account for the difference. One such example where this hypothesis has been confirmed is Cl0024+1654, as discussed in Chapter 8 of this
dissertation.
4.4
Galaxy kinematics
4.4.1 Relaxed systems
The virial theorem
The standard method for estimating the mass of a self-gravitating system is
the virial theorem (e. g. Bahcall & Tremaine 1981; Heisler et al. 1985). Differentiating the quantity ∑i mi ri · ri twice with respect to time, one obtains
*
+ *
+ *
+
d2
mi ri · ri = ∑ mi ri · r̈i + ∑ mi ṙi · ṙi = 0
(4.32)
dt2 ∑
i
i
i
on average in a steady-state system.
In a self-gravitating system r̈i = −G ∑ j6=i rij /|rij |3 . Hence, the first term on
the right hand side is the total potential energy V of the system, the second
term is twice the kinetic energy T, so that the virial theorem reads
hTi + 2hVi = 0
;
(4.33)
again this is strictly true only for time averages. The potential energy in Eq.
(4.32) can be written as
−G ∑ mi ∑ m j
i
j6=i
(ri − r j ) · ri
rij3
1
r
i<j ij
= −Gm2 ∑ ∑
i
,
(4.34)
where it has been assumed that all galaxies have the same mass m, and the
equality is due to the symmetry of rij2 in the indices. Observable quantities are
the projected distance Ri of a galaxy from the cluster centre, and the line-ofsight velocity vlos,i . In the case of spherical symmetry h1/ri = (2/π)h1/Ri,
and for isotropic galaxy orbits (see below) hv2 i = 3hv2los i, hence one arrives at
an estimate for the total mass of the system
N
2
3πN ∑i=1 vlos,i
M=
2G ∑i<j 1/Rij
.
(4.35)
Heisler et al. (1985) discuss three different dynamical mass estimators, derived
from the Jeans equation. These estimators are based on the same physical
assumptions as the virial estimator, i. e. steady state, spherical symmetry and
some assumption concerning the shape of galaxy orbits, but differ to some
extent in their statistical properties.
81
4.4. Galaxy kinematics
Velocity dispersions
The velocity dispersion of a gas of self-gravitating particles with distribution
function f (r, v) is defined through
Z
2
σij = hvi v j i =
f (r, v)vi v j d3 v .
(4.36)
In the spherically symmetric case, the matrix σij2 has two independent components, the radial and tangential velocity dispersions, σr2 and σt2 , both functions of the distance from the cluster centre, r. In relaxed clusters, galaxy
orbits should be isotropic, i. e. σr ' σt , whereas in the infall region outside
the virial radius and in forming clusters radial orbits are predominant, hence
σt σr . Only the integrated line-of-sight velocity dispersion is observable; for
the spherically symmetric case with isotropic orbits, this is
R
ρ(r)σ2 (r) dz
2
R
σlos (R) =
(4.37)
ρ(r) dz
√
where R = r2 − z2 is the projected distance of the line-of-sight from the centre of the cluster.
Various estimators for the line-of-sight velocity dispersion from a sample
of galaxy redshifts zi are available (Beers et al. 1990). A galaxy with redshift
zi that belongs to a cluster with mean redshift z has a relative velocity with
respect to the cluster mean
(z − z)c
(4.38)
vi = i
1+z
in the reference frame of the cluster2 . Since the relation between velocity and
redshift is linear, one can estimate the dispersion on the redshifts and the convert to the corresponding velocity dispersion through
σv =
σz c
1+z
.
(4.39)
Estimating a cluster velocity dispersion therefore in practice means estimating the location (z) and scale (σz ) of the redshift distribution. The obvious
estimators are the usual sample mean and standard deviation; however, these
are optimum only if the underlying parent distribution is Gaussian and rather
sensitive to deviations from Gaussianity and to interlopers, i. e. foreground or
background galaxies that do not actually belong to the cluster. A common way
to reject outliers is by “3σ-clipping” of the sample: first estimates of location
and scale are computed from the full sample, then data points that deviate
2 The
total redshift (1 + zi ) is the product of the Doppler shift vi /c + 1 due to the velocity of
the galaxy within the cluster, and the cosmological redshift 1 + z.
82
from the location estimate by more than a given number of scale widths are
rejected, and new estimates are computed from the smaller sample. This is
typically done until the remaining sample is consistent with a the hypothesis
of a Gaussian parent population (Beers et al. 1990, and references therein). This
approach is criticised by Beers et al. (1990) due to the underlying normality assumption.
Beers et al. (1990) test a number of robust estimators for scale and location,
where “robustness” implies insensitivity to the assumed nature of the parent
distribution. They recommend use of biweight estimators for samples of size
upwards of N = 20. The biweight location estimator is defined as
CBI = M +
∑|ui |<1 (zi − M)(1 − u2i )2
∑|ui |<1 (1 − u2i )2
,
(4.40)
where M is the sample median and ui = (zi − M)/c MAD. MAD is an auxiliary
scale estimator, the median absolute deviation from the sample median, i. e.
the median of the distribution of |zi − M|. The value of the tuning constant c
is taken to be 6.0 (Beers et al. 1990).
The biweight scale estimator is defined as
SBI = N 1/2
h
i1/2
∑|ui |<1 (zi − M)2 (1 − u2i )4
∑|ui |<1 (1 − u2i )(1 − 5u2i )
,
(4.41)
where now c = 9.0. In Chapters 8 and 11 I use these estimators for mean
cluster redshifts and velocity dispersions.
4.4.2
Cluster substructure from galaxy kinematics
Even if the galaxies in a cluster are not in equilibrium, knowledge of their
redshift distribution is important for investigating the structure of the cluster,
providing valuable information on a third dimension of the six–dimensional
phase space. A striking example for this was found in Cl0024+1654, which is
described in Part II of this dissertation.
Many statistical methods have been invented to identify substructure in
the x–y–z distribution of cluster galaxies, where x and y refer to the projected
position on the sky, and z is the redshift. A comprehensive overview and comparative tests of a number of methods is provided by Pinkney et al. (1996).
Dressler-Shectman test
The test proposed by Dressler & Shectman (1988) aims at identifying correlations between galaxy position and velocity by comparing the local kinematical
83
4.4. Galaxy kinematics
properties (mean redshift zlocal and velocity dispersion σlocal ) with the corresponding global values. For each galaxy, zlocal and σlocal are calculated from
the redshifts of the ten nearest (in projection) neighbours plus the galaxy itself,
giving the local deviation δ as
i
11 h
2
2
δ = 2 (vlocal − v) + (σlocal − σ)
σ
2
.
(4.42)
Here, the mean redshifts have been expressed in terms of velocities3 . The
global mean velocity can be set to zero, v = 0, so that
vlocal =
c(zlocal − z)
1+z
.
(4.43)
A visual impression of the structure of the cluster can be obtained by plotting
circles of size ∝ exp δ at the galaxy positions. Groups of galaxies that deviate significantly from the global kinematic properties of the cluster then show
up as spatially clustered large circles. Examples are shown in Chapter 11 of
this dissertation where I apply the Dressler–Shectman test to the redshift data
obtained for the z ' 0.2 high–LX sample. Further examples can be found in
Dressler & Shectman (1988) or Oegerle & Hill (2001).
The δ values for the Ng individual galaxies in a cluster can be added up to
give the cumulative deviation
Ng
∆=
∑ δi
.
(4.44)
i=1
The significance of an observed ∆ value has to be determined through Monte
Carlo simulations for each cluster, by randomly shuffling the galaxy redshifts.
This erases any correlations between redshifts and positions and thus allows
an estimate of the probability whether the observed value of ∆ might have
arisen by chance, or whether it is due to true correlations between redshifts
and positions, hence to real substructure.
3 The Dressler–Shectman test was originally applied to low-redshift clusters, where redshift
is traditionally given as a “recession velocity” v = cz.
Détermination des masses des amas
Dans les chapitres précédents il a été montré que la fonction de masse des
amas de galaxies ainsi que la distribution de masse à l’intérieur des amas comprend beaucoup d’information sur la cosmologie dans laquelle les amas se
sont formés et évoluent. Malheureusement, les masses des amas ne sont pas
directement accessibles aux mesures astronomiques, donc il faut passer par
d’autres observations pour arriver à des estimations de leurs masses. Dans ce
chapitre, trois méthodes d’estimation de masse sont présentées.
Effet de lentille gravitationnelle
La théorie de la relativité générale de Einstein prédit que la lumière est
déviée par la courbure de l’espace-temps autour d’importantes concentrations
de masse. Les amas de galaxies agissent de tel façon, et provoquent plusieurs
modifications sur les images observées de galaxies plus lointaines. Dans le
centre de l’amas la densité surfacique de masse est assez élevée pour que se
puissent former plusieurs images de la même source d’arrière-plan visible par
l’observateur. Les faisceaux de lumière sont déformés par le potentiel gravitationnel, donc l’observateur voit les sources amplifiées et avec des formes
modifiées de leurs formes intrinsèques. Près du centre, ces déformations sont
larges et on observe des arcs géants. Plus loin, l’effet est faible et n’est détectable
qu’avec des méthodes statistiques. Tous ces effets sont inclus dans le terme
“lentille gravitationnelle” (gravitational lensing). Dans la partie centrale des
amas on parle de “strong lensing”, dans les parties extérieures de “weak lensing”.
La géométrie d’un système de lentille gravitationnelle est montrée à la figure 4.2. L’équation de base peut en être déduite :
β = θ−
Dds
α(θ)
Ds
,
où les distances doivent être interprétées comme des “distances” angulaires
dans un cadre cosmologique. L’angle de déflection α est directement lié à la
84
Résumé
85
distribution surfacique de masse Σ(θ) dans le déflecteur, dont l’importance
des observations de lentilles gravitationnelles dans l’astrophysique :
Z
θ − θ0
4G
d2 θ 0 Σ(θ0 )
.
α(θ) = 2 Dd
c
|θ − θ0 |2
Localement, l’effet de lentille peut être linéarisé, dβ = A dθ, avec la matrice de
magnification
1 − κ − γ1
−γ2
A=
,
−γ2
1 − κ + γ1
où la convergence κ (proportionnelle à la densité surfacique Σ) décrit une amplification isotrope de l’image, et le shear (cisaillement) γ décrit une déformation de l’image (dans l’approximation linéaire une source circulaire est observée d’avoir une forme elliptique). Sur les lignes critiques dans le plan du
déflecteur (correspondant à des caustiques dans le plan de la source) au moins
une des valeurs propres de la matrice A devient zéro. Si la source est proche
d’une caustique on observe des images fortement amplifiées et déformées ; ce
sont les arcs géants. Travers les caustiques la multiplicité d’images observées
change de deux. Plus loin du centre le shear γ est faible.
Dans le régime de strong lensing on peut modéliser la distribution centrale de masse dans un amas à partir d’observations d’images multiples et des
arcs géants. Les contraintes sur ces modèles sont les positions des images, leur
flux relatives et peut-être même leurs formes. Puisque la position des lignes
critiques dépend du redshift de la source, un modèle de strong lensing peut
être utilisé pour prédire des redshifts des sources ; contrairement, la mesure de
redshifts spectroscopiques apporte d’importantes contraintes supplémentaires
sur un modèle.
Dans le régime du weak lensing on peut distinguer deux effets différents.
Si la distribution de formes intrinsèques des galaxies d’arrière-plan a une moyenne statistique de zéro, la distribution de formes observées a une moyenne
qui est différente de zéro. Les formes sont d’habitude mesuré par les deuxièmes
moments de la lumière dans une galaxie. Avec la définition (4.10) de l’ellipticité d’une galaxie, sa valeur moyenne observée est hχi ≈ g, où g ≡ γ/(1 − κ)
est le shear réduit. A partir d’une mesure du champ de shear γ(r) on peut reconstruire la densité surfacique κ par une transformation intégrale (4.12 avec
le kernel 4.13).
L’autre effet observable est due à la magnification par la lentille gravitationelle et peut être mesuré par des comptages de galaxies d’arrière-plan en
fonction de leur magnitude. En fait, il y a deux effets en compétition : les galaxies sont individuellement amplifiées, ce qui produit une augmentation de
la densité de galaxies observées à une magnitude apparente donnée. D’autre
côté les séparations entre les galaxies sont aussi augmentées ce qui produit une
dilution de la densité à une magnitude absolue donnée. Combinant ces deux
86
Chapitre 4. Determination of Cluster Masses
effets, les comptages de galaxies observées s’écrivent comme
n(< m) =
1
n0 (< m + 2.5 log µ(θ))
µ(θ)
,
où n0 est la densité de galaxies loin d’une lentille, et µ(θ) = det A. Si les comptages intrinsèques suivent une loi de puissance, log n0 = βm,
n(< m)
= µ2.5β−1
n0 (< m)
.
Pour β < 0.4 (ce qui est le cas dans la plupart des bandes photométriques) on
observe donc une déplétion de galaxies d’arrière-plan. A partir d’une mesure
du champs de magnification µ(θ) on peut encore reconstruire la distribution
de masse.
Les deux effets de weak lensing, “weak shear” et déplétion, sont sensibles
à la même quantité physique, mais ont des différentes sensibilités vis-à-vis
les erreurs d’observation. Il serait donc fort intéressant de combiner les deux
mesures pour estimer des masses d’amas de galaxies.
Observations en X
Les amas de galaxies riches contiennent du gaz diffus à des températures
de 107 . . . 108 K et densités de 10−3 cm−3 . Ce gaz émet de la radiation dans
la domaine X par bremsstrahlung thermique, ce qui place les amas de galaxies parmi les sources les plus brillantes dans l’univers (luminosité LX <
∼ 2×
45
−2
−1
2
1/2
10 h erg s ). L’émissivité du gaz est eff ∝ ne T , donc on peut connaı̂tre
la distribution du gaz et même de la matière noire à partir d’observations X.
On suppose que le gaz est dans un état d’équilibre hydrostatique dans
le puits gravitationnelle donné par la matière noire. Dans le cas de symétrie
sphérique on aboutit à l’équation d’équilibre hydrostatique
d ln ng d ln T
Gµmp M(r)
kT
+
=−
,
dr
dr
r2
où T est la température du gaz, ng sa densité, µmp la masse moyenne des
particules du gaz, et M(r) la masse totale à l’intérieur du rayon r. Si l’on arrive
donc à mesurer les profils de densité et de température du gaz, on peut en
déduire le profil de masse totale.
Si l’on suppose ensuite que le gaz et la matière noire sont tous les deux isothermes, ils suivent des équations très similaires (l’équation d’équilibre hydrostatique avec température T pour le gaz collisionnel, l’équation de Jeans avec
dispersion de vitesse σr pour la matière non-collisionnelle), on trouve une relation très simple entre ces deux composantes, ρg ∝ ρ β , avec β = µmp σr2 /kT. Le
87
Résumé
modèle β (Cavaliere & Fusco-Femiano 1976) utilise l’approximation de King
comme modèle de la distribution isotherme de la matière noire,
"
ρ(r) = ρ0 1 +
r
rc
2 #−3/2
,
d’où on trouve pour la brillance de surface en X
"
S(R) = S0 1 +
R
rc
2 #−3β+1/2
.
Ce simple modèle donne une représentation acceptable des distributions
de brillance de surface (Jones & Forman 1984), au moins pour des observations
à relativement basse résolution spatiale. Si l’on mesure aussi la température du
gaz par des observations spectroscopiques en X et la dispersion de vitesse σr
des galaxies, on peut en déduire une valeur de β suivant l’équation (4.28). Il se
trouve que cette valeur-là est presque deux fois plus élevée que celle déduite
par un ajustement au profil de brillance (β fit ∼ 0.66) ce qui pourrait s’expliquer par le fait que l’approximation de King n’est pas un bon modèle de la
distribution de masse dans les amas. L’hypothèse d’isothermalité du gaz est
actuellement sous débat, avec des auteurs qui déterminent des différents profils de températures à partir de parfois les mêmes données. Il semble, tout de
même, que les amas sont largement isothermes jusqu’à au moins 20% du rayon
viriel.
Le fondement de toute méthode de détermination de masse à partir d’observations X est que le gaz soit dans un état d’équilibre hydrostatique. Malheureusement, les déviations de l’équilibre sont assez fréquentes. Des collisions
entre les amas perturbent la distribution du gaz et provoquent la formation
d’ondes de choc, ce qui donne lieu à des fortes fluctuations de luminosité et
température en X dans le ∼ 1 Gyr après la première collision, comme on le voit
dans des simulations hydrodynamiques des collisions (voir Sarazin 2001). Les
sous-structures, évidence de collision, sont assez fréquentes, environ 22% dans
l’échantillon de Jones & Forman (1984), observé avec le satellite E INSTEIN. Les
observations récentes avec notamment C HANDRA donnent de l’information
détaillée sur des collisions en cours dans plusieurs amas. Dans la partie II de
cette thèse je présenterai le cas de Cl0024+1654, où la collision n’a pas pu être
détectée par des observations X, mais nécessitait des mesures de redshifts sur
un grand champ autour de l’amas.
Dans les cœurs des amas riches le gaz atteint une telle densité que le temps
de refroidissement par rayonnement devient plus court que le temps de Hubble,
donc le gaz contracte et il se développe un flux de gaz des couches élevées vers
le centre, ce qu’on appelle un “cooling flow”. Le centre dense montre un excès
88
Chapitre 4. Determination of Cluster Masses
en luminosité par rapport au modèle β, ce qui introduit un biais dans les mesures de masses. Les cooling flows et des sous-structures et déviations d’état
d’équilibre ont été invoquées pour expliquer des discrépance entre des masses
déterminées avec des estimateurs différents (Allen 1998).
Cinématique des galaxies
Le mouvement des galaxies dans le puits du potentiel gravitationnel d’un
amas offre une autre (historiquement la première) méthode pour estimer la
masse de l’amas. Dans des amas en équilibre dynamique cette méthode s’appuie sur le théorème du viriel, qui dit que, moyenné sur le temps, hTi + 2hVi =
0, où T est l’énergie cinétique et V l’énergie potentielle d’une particule. A un
moment donné on remplace la moyenne temporelle par la somme sur toutes
les particules dans l’amas et on aboutit à
N
2
3πN ∑i=1 vlos,i
M=
2G ∑i<j 1/Rij
,
où vlos est la vitesse projetée sur la ligne de visée (mesuré à partir du redshift),
et R est la distance projetée du centre de l’amas. Les hypothèses essentielles
de cette approche sont que l’amas soit dans un état stable et sphériquement
symétrique, et que la distribution des orbites des galaxies soit isotrope. La dispersion des vitesses dans la formule pour la masse d’un amas est mesurée à
partir des redshifts des galaxies. Si le redshift moyen de l’amas est z, la dispersion des vitesses est liée à la dispersion des redshifts par σv = σz c/(1 + z).
Le problème pratique dans la mesure des dispersions des vitesses consiste à
éviter l’inclusion des galaxies d’avant- ou d’arrière-plan dans l’échantillon. Les
estimateurs robuste sont donc à préférer, notamment les estimateurs dits “biweight”, commandés par Beers et al. (1990).
Même si les hypothèses à la base des estimations des masses dynamiques
ne sont pas valables les relevés de redshifts dans les amas sont importants car
ils peuvent donner des informations sur la distribution des galaxies le long
de la ligne de visée, donc révéler des projections et des sous-structures. Un
exemple spectaculaire est fourni par l’amas Cl0024+1654 qui sera étudié en
détail dans la partie II de cette thèse. Il existe un nombre de tests statistiques
pour détecter des sous-structures dans la distributions spatiale et/ou en redshift des galaxies des amas. Je présente ici le test de Dressler & Shectman (1988)
qui compare les propriétés statistique des voisins d’une galaxie avec les mêmes
propriétés dans tout l’échantillon. Ce test sera appliqué dans le chapitre 11 sur
les distributions des redshifts dans un échantillon d’amas lumineux en X.
Chapter 5
Relations between observational
properties
Cluster masses are usually derived from other observational quantities, such
as X-ray temperature, the X-ray surface brightness distribution, galaxy velocity dispersion, the shear introduced in background galaxy shapes, or the
SZ decrement. The conversion between observational quantities and masses
involves theoretical models of the distribution of mass and more readily observable cluster components and fairly strong assumptions about the shape of
clusters or the dynamical state (equilibrium) of these components. It is useful
to consider statistical relations and correlations between different observable
quantities in order to test the viability of existent models and to obtain hints as
to what modifications in the models are necessary to reproduce the observed
relations.
5.1
5.1.1
Observed relations
The LX –TX relation
X-ray luminosity and temperature are the two most easily measured properties of clusters of galaxies and measurements are available for large samples of
clusters observed with X-ray satellites such as the Einstein observatory, R OSAT
or ASCA, with emerging measurements at higher accuracy forthcoming from
the XMM/N EWTON and C HANDRA observatories.
X-ray telescopes usually have low spectral resolution for imaging, so in order to convert count rates to physical fluxes one needs a model for the shape
of the spectral energy distribution of the incident radiation from the source
across the band, which depends primarily on the temperature of the gas, but
also on the gas metallicity and the hydrogen column density NH towards the
source. Temperatures can be obtained from independent high resolution spec89
90
Chapter 5. Relations between observational properties
tral measurements or can be estimated (in a somewhat round-about way) from
the LX –TX relation calibrated on a sample for which both imaging and sufficiently high-resolution spectroscopic observations exist.
The temperature and in particular the luminosity measurements can be biased by the presence of cooling flows. In cooling flow regions an additional
physical scale is introduced by the cooling time, and they are therefore not expected to follow the same relation as the outer regions of cooling flow clusters
or as clusters without cooling flows. In order to derive meaningful relations
involving X-ray properties it is therefore essential to correct for the presence of
a cooling flow, for example by simply excising the cooling flow region (Allen
1998).
Global X-ray temperature measurements are now available for sufficiently
large samples of clusters due to the high spectral resolution of the ASCA satellite. Markevitch (1998) combined ASCA temperatures with luminosities measured with R OSAT for a sample of 35 local clusters (z < 0.09) and deduced a
logarithmic slope of the LX –TX relation of 2.64 ± 0.27 (at 90% confidence) after correcting for cooling flows. Arnaud & Evrard (1999) compiled a sample
of clusters without strong cooling flows and found a slope 2.88 ± 0.15; Ettori
et al. (2001) found 2.7, again after correction for cooling flows.
Fitting models to the observed LX –TX relation (and other relations of this
type) involves linear regression on two variables, both of which carry measurement errors and intrinsic scatter. This is not a straightforward problem
and results on the fit parameters depend somewhat on what method is used
to determine the best fit (Fasano & Vio 1988; Feigelson & Babu 1992). Xue
& Wu (2000) compile a large sample of clusters and groups of galaxies from
the literature and find logarithmic slopes for the LX –TX relation of 184 clusters
between 2.54 ± 0.11 and 2.79 ± 0.08 for different estimators.
The observed relation becomes even steeper on group scales: Xue & Wu
(2000) find a slope of 5.6 for 38 groups (although this result depends strongly
on the fitting method), Helsdon & Ponman (2000) find 4.3 ± 0.5 for a combined
sample of loose and compact groups of galaxies.
The measured slopes of the LX –TX are in good agreement, but differ significantly from the simple scaling relations predicted from a theoretical model
where dark matter halos form hierarchically from an initial power spectrum
with a constant logarithmic slope n = −1 (as is appropriate for the CDM spectrum on cluster scales) and the gas is heated through dissipation of its gravitational infall energy only. This model (Kaiser 1986) predicts LX ∝ TX2 , i. e. a
significantly flatter relation than observed.
These results indicate that the self-similarity in the gas properties is broken
in groups and clusters by non-gravitational processes. Commonly invoked
processes include pre-heating of the gas by galactic winds or AGN (Kaiser
1991), or the systematic variation of the dark matter halo concentration c with
halo mass due to the variation of formation time with mass predicted by the
5.1. Observed relations
91
spherical collapse model and observed in the numerical simulations of e. g.
Navarro et al. (1997). Lloyd-Davies et al. (2002) present a simple model incorporating these effects and quite successfully reproduce the observed LX –TX
and other relations. According to this model, pre-heating is effective for low
mass systems, whereas the systematic variation of halo concentration is the
dominant effect on cluster scales.
5.1.2
The M–TX relation
The relation between cluster mass and X-ray temperature is the most essential
relation for cosmological purposes since it allows the conversion between the
theoretically predicted mass function and the more easily observable temperature function.
Eke et al. (1996) give a theoretical M–TX relation for isothermal clusters:
2/3
ΩM,0 1/3
7.75 keV
6.8
M
kTgas =
(1 + z)
.
β
5X + 3
ΩM (z)
1015 h−1 M
(5.1)
Here, β is defined as in Eq. (4.28) and X is the hydrogen mass fraction (related
to the mean particle mass µmp . The scaling M ∝ T 3/2 is as expected from selfsimilar theory (Kaiser 1986) and is generally seen in numerical simulations
(Navarro et al. 1995). The observed relation is somewhat steeper than that;
Lloyd-Davies et al. (2002) quote a slope of 1.96 ± 0.21 for 20 clusters.
The steep slope of the cluster LX –TX relation indicates that the physics of
the intracluster gas is more complicated than expected and not yet fully understood. X-ray observations of nearby clusters at high spatial resolution with, in
particular, the C HANDRA observatory reveal a multitude of substructures and
previously unseen effects in the cluster centres, such as merger shocks, cold
fronts (Markevitch et al. 2000; Mazzotta et al. 2001a,b; Markevitch & Vikhlinin
2001; Markevitch et al. 2001; Vikhlinin et al. 2001), radio bubbles (e. g. McNamara et al. 2001) etc. Even if weak substructure might not affect the mean
relations they might contribute to the scatter around these relations. It is therefore essential to calibrate relations involving cluster masses using independent
measurements of the mass, ideally from gravitational lensing, which provides
the most direct route to mass and does not depend on the dynamical state of
the cluster.
5.1.3
The σ–TX –LX relations
Measurements of the galaxy kinematics provide an independent way of estimating cluster masses. Although the use and possible accuracy of mass estimates from galaxy velocity dispersion measurements is limited, not least due
to the limited number of cluster galaxies available, it is interesting to quote
92
Chapter 5. Relations between observational properties
results on the relations between σ and X-ray observables. Xue & Wu (2000)
find LX ∝ σ5.3 for 197 clusters and σ ∝ TX0.65 for 109 clusters. Mahdavi & Geller
(2001) compile a sample of 280 clusters and find LX ∝ σ4.4 .
5.2
The cluster temperature and luminosity functions
5.2.1 The temperature function
Fig. 5.1 shows the predicted cluster temperature function from the mass function of Jenkins et al. (2001) with the theoretical M–T relation (5.1). These functions are basically just rescaled versions of the mass functions shown in Fig.
2.7 and the basic features are the same. The temperature function at the hightemperature end decreases much faster with redshift in the standard CDM case
than in the flat low-density case. The decrease in the SCDM case is very rapid
indeed, the number density of clusters with T ∼ 5 keV decreases by about a
factor of 10 out to the relative modest redshift of 0.5.
The dependence of the temperature function on the cosmological parameters was studied in detail by Eke et al. (1996) who find that the difference
between the cumulative numbers of clusters with T > 5 keV at redshifts z <
∼1
differ by more than a factor of ten, thus providing strong leverage for the determination of ΩM . The dependence on ΩΛ is by contrast rather weak, and begins
to make itself felt only at redshifts z >
∼ 1, where clusters are increasingly difficult to identify. The local normalisation of the mass and temperature functions
also depends on the cosmological parameters, however there is a degeneracy
between the dependence on ΩM and σ8 of roughly the form σ8 Ω1/2
M = const.
Local measurements of the temperature function can thus be used to normalise
the matter power spectrum by fitting σ8 with independent constraints on ΩM .
The degeneracy between these two parameters is broken, when the evolution of the temperature function can be observed. The application of this
promising method to determine ΩM hinges on the M–T relation, in particular
its evolution of redshift. In Part III of this dissertation, I will present a project
aiming to calibrate the high mass/temperature end of the M–T relation at redshift z = 0.2, which is an important step towards constraining ΩM from cluster
evolution.
5.2.2 The luminosity function
Cluster luminosity functions can be determined for any complete cluster catalogue. Larger, nearly all-sky surveys, include the Bright Cluster Survey (BCS,
Ebeling et al. 1997, 1998) for the northern hemisphere and the REFLEX survey
(Böhringer et al. 2001) for the southern hemisphere. When restricted to local
clusters the luminosity functions from both surveys agree very well. There is
5.2. The cluster temperature and luminosity functions
93
Figure 5.1: Predicted evolution with redshift of the cluster temperature function in SCDM (solid lines) and ΛCDM (dashed lines) models. This plot is
based on the mass function from Jenkins et al. (2001), shown in Fig. 2.7, and
the theoretical M–T relation, Eq. (5.1). The temperature functions are shown
at redshifts 0, 0.5 and 1.0.
still some controversy over whether evolution of the luminosity function has
been observed at higher redshift.
Relations entre observables
Les masses des amas de galaxies sont déduites à partir de mesures d’observables plus accessibles, comme la luminosité ou la température du gaz chaud,
la dispersion des vitesses des galaxies membres, ou encore du shear introduit
dans les formes des galaxies d’arrière-plan. La conversion en masse se fait
via des modèles de la structure des amas ; pour tester ces modèles et les hypothèses qui sont à leurs bases, il est utile de regarder des relations empiriques
entre différentes observables. Ceci peut aussi donner des indications comment
améliorer ces modèles.
La température TX et surtout la luminosité LX du gaz chaud sont les observables les plus facilement accessibles dans les amas, et des mesures sont disponibles pour des nombreux amas à partir d’observations avec R OSAT, ASCA,
XMM/N EWTON et C HANDRA.
Des observations donnent une relation LX –TX de la forme LX ∝ TXα avec
un exposant α d’environ 3 pour les amas riches (Markevitch 1998; Arnaud &
Evrard 1999; Ettori et al. 2001). Ceci est incompatible avec les prédictions du
simple modèle auto-similaire, dans lequel la seule source d’énergie du gaz est
d’origine gravitationnel lors de l’effondrement de l’amas. Dans ce cas, on s’attend à une relation LX ∝ TX2 (Kaiser 1986). Le fait que l’exposant observé est
différent de 2 indique qu’il y a autre sources d’énergie ; notamment le “preheating” par des vents galactiques, AGNs ou supernovae, est discuté. La relation LX –TX est encore plus raide aux échelles de groupes de galaxies, avec un
exposant d’environ 5 (Xue & Wu 2000; Helsdon & Ponman 2000).
La relation la plus importante pour des applications cosmologiques est
la relation entre masse et la température. Eke et al. (1996) donnent une relation théorique, valable pour des amas isothermes, Eq. 5.1. Le comportement
M ∝ T 3/2 est encore celui attendu dans des modèles auto-similaires. Des calibrations théoriques existent basées sur des observations en X uniquement, où
la température a été mesurée par spectroscopie, et la masse déduite à partir du
profil de brillance de surface, comme décrit dans la section 4.3. Ces mesures de
M et TX ne sont donc pas strictement indépendantes. Une meilleure approche
est une calibration où les masses sont déterminées à partir des observations
94
Résumé
95
lensing, qui sont en tout cas plus proche de la masse que des observations X.
Ceci fournit une des motivations principales pour le projet décrit dans la partie
III.
Des estimations de la fonction de masse des amas de galaxies sont fournies
par des mesures des fonctions des températures et luminosités ; la dernière est
la plus facile à mesurer, en revanche la première est moins susceptibles aux
perturbations d’équilibre dans les amas et est donc une plus robuste estimation. Le comportement général de la fonction de masse en dépendance des
paramètres ou du redshift se traduit sur la fonction de température (Eke et al.
1996), dont l’intérêt de mesurer cette fonction, qui est montrée à la figure 5.1.
Cette application dépend tout de même sur une calibration précise de la relation entre masse et température.
96
Chapitre 5. Relations between observational properties
Part II
A wide-field redshift survey in the
cluster Cl0024+1654
Chapter 6
Cl0024+1654: Introduction
6.1
Motivation for a wide–field spectroscopic survey
Clusters of galaxies are increasingly viewed not as simple isolated and relaxed
systems but as embedded in and connected to the general large-scale structure
in the Universe. This view of clusters in a larger context has consequences for
the interpretation of cluster galaxy populations, cluster dynamics and mass
estimates.
Clusters of galaxies grow by continuously accreting galaxies and groups
of galaxies from the surrounding field, mostly along filamentary structures.
In the process, galaxies are transformed from the predominantly blue, actively star-forming, spiral population characteristic of the field to the red, passive and elliptical population characteristic of the inner and denser regions
of clusters (Dressler 1980; Abraham et al. 1996; Balogh et al. 1998). Cluster
galaxy populations evolve with redshift: rich clusters at high redshift contain
a larger fraction of blue galaxies than local ones (Butcher & Oemler 1978, 1984,
Cl0024+1654 is an example of a “Butcher-Oemler” cluster). The exact nature
of the interaction of infalling galaxies with the cluster environment (hot intracluster medium, tidal gravitational field) and its influence on the morphology
of galaxies, their gas content and star-formation rates (as measured by galaxy
colours and spectral type) are as yet ill-understood; hence the interest in investigating the “infall region” beyond ∼ 1 h−1 Mpc distance from the cluster
centre, where the transition from field to cluster galaxies is taking place. The
advent of new wide-field CCD mosaic cameras available on a number of large
telescopes (e. g. CFHT, CTIO, Subaru, ESO2.2m) makes it possible to obtain
photometric and morphological information on 1 - 10 Mpc scales around the
cluster centres. However, wide-field investigation of clusters demands both
imaging and spectroscopic observations. Individual spectra of galaxies describe their spectral energy distribution and provide their redshift, which is
indispensable to produce a catalogue of cluster members with radial veloci99
100
Chapter 6. Cl0024+1654: Introduction
ties and information regarding their stellar content and star formation history.
At present there is only a limited number of clusters with more than ∼ 200
spectroscopically identified member galaxies (e. g. Abraham et al. 1996), and
especially at high redshift (z >
∼ 0.2) spectroscopically well-studied clusters become very rare, mostly due to the fact that contamination by field galaxies
increases rapidly with redshift.
The fact that clusters are not isolated systems also raises questions concerning the traditional ways of estimating masses of clusters of galaxies through
different mass estimators: gravitational lensing analyses, kinematical analyses
from redshifts of cluster member galaxies and X-ray observations.
Gravitational lensing is sensitive to the total integrated mass along the lineof-sight from the observer to the lensed sources, weighted by the appropriate
combination of angular size distances between observer, lens and source (Sect.
4.2). In the presence of massive structures other than the cluster along the line
of sight, the mass derived from gravitational lensing overestimates the mass
of the cluster proper. Large spectroscopic surveys provide additional information needed to correctly interpret the lensing analysis in this case. Metzler
et al. (2001) investigate the influence of the presence of filaments and groups
of galaxies in the vicinity of a cluster on weak lensing estimates of the cluster
mass and find that significant overestimates of M200 (up to a factor 1.5 to 2) are
possible and even likely. Similar investigations with comparable results were
conducted by Cen (1997) and Reblinsky & Bartelmann (1999).
A similar bias should be expected to affect measurements of velocity dispersions: if foreground or background groups of galaxies in the immediate
neighbourhood of the cluster are added into the redshift histogram, but are not
resolved and recognised as separate entities, the velocity dispersion of the cluster itself will be overestimated. Generally only 30 to 50 member galaxies are
used to estimate the line-of-sight velocity dispersion (and virial cluster mass;
see e. g. the large compilations of Girardi et al. 1998 and Girardi & Mezzetti
2001). Furthermore, the measured redshifts are generally concentrated within
a relatively small region within a projected radius of ∼ 500 h−1 kpc of the cluster centre. Whereas one can argue that these numbers might be sufficient for
relaxed clusters with regular spatial and velocity distributions, the various derived estimates will contain large systematic errors if unresolved substructures
are present. What can be obtained from redshift surveys is a galaxy number
density weighted line-of-sight velocity dispersion averaged along the line-ofsight. With a sufficiently large number of cluster member redshifts it is possible to measure the variation of the line-of-sight velocity dispersion with projected distance from the cluster centre (Carlberg et al. 1997), but determination of even more detailed information on the dynamical status (e. g. velocity
anisotropy profile) of a cluster requires a forbiddingly large number of redshifts (e. g. Merritt 1987).
6.1. Motivation for a wide–field spectroscopic survey
101
As more detailed observations of individual clusters of galaxies are compiled, combinations of X-ray imaging and spectroscopy, velocity dispersion
measurements and lensing mass maps frequently reveal that clusters that were
thought to be simple relaxed objects are actually more complex systems, frequently undergoing mergers or generally showing signs of substructure and
deviations from dynamical equilibrium. Evidence for recent mergers or accretion is currently accumulating through C HANDRA observations in the form of
merger shocks (e. g. Markevitch & Vikhlinin 2001; Markevitch et al. 2001) or
cold fronts (Markevitch et al. 2000; Vikhlinin et al. 2001; Mazzotta et al. 2001b).
Mergers and substructure have also been invoked to explain the discrepancy
between mass estimates from different methods that is observed in many clusters (Miralda-Escudé & Babul 1995; Wu et al. 1998).
The spectacular lensing cluster Cl0024+16541 at redshift z = 0.395 is an example of a cluster where the high mass inferred from the strong lensing mass
reconstructions is at variance with the fairly low X-ray luminosity and temperature, that indicate a total mass for the cluster which is a factor 2 to 3 smaller
than the lensing mass (Soucail et al. 2000). Both the galaxy distribution on the
sky and the X-ray morphology are regular and by themselves compatible with
a relaxed massive cluster, a notion which was further supported by the high
galaxy velocity dispersion of ≈ 1200 km s−1 found in the redshift surveys of
Dressler & Gunn (1992) and Dressler et al. (1999, hereafter D99). Cl0024+1654
was one of the first clusters in which the Butcher–Oemler effect, an excess of
blue objects in clusters at high redshift as compared to local clusters, was observed, and it was also one of the first clusters where a coherent shear field in
the background galaxy population was found (Bonnet et al. 1994). This makes
Cl0024+1654 one of the most interesting and best-studied clusters at high redshift.
In order to better understand the dynamics of Cl0024+1654, and how it is
embedded in the surrounding large-scale structure, a wide-field spectroscopic
survey was initiated in 1992 by Geneviève Soucail, Yannick Mellier and Jean–
Paul Kneib, and spectra were obtained at the Canada-France-Hawai‘i Telescope (CFHT) and the William Herschel Telescope (WHT) from 1992 to 1996.
As part of my PhD work I reduced the raw spectral data and analysed the
distribution of redshifts and spectral types in Cl0024+1654. Two papers were
published on this subject and form the basis for the following two chapters. In
the remainder of the present chapter, I will summarise previous observations
of Cl0024+1654, with particular emphasis on mass estimates and the discrepancy between these. Chapter 7 (Czoske et al. 2001) describes the observations
and data reduction and presents the catalogue which lists positions, redshifts,
1 This cluster is usually called Cl0024+1654, although the common Internet databases, such
as NED (http://nedwww.ipac.caltech.edu), list it as ZwCl0024.0+1652 as it originally appeared in Zwicky et al. (1965).
102
V and I band photometry, as well as equivalent widths for important lines,
namely [O II], [O III], Hδ, Hβ and the 4000 Å break, for 688 objects, once data
from the literature (Dressler et al. 1999) is included. The catalogue is listed in
its entirety in Appendix A, and is also available at the Centre de Données Stellaires (CDS)2 . Chapter 7 also describes the global distribution of redshifts out
to z ∼ 1 and identifies several concentrations and putative structures along the
line of sight.
The most striking result from the spectroscopic survey is the bimodality of
the redshift distribution at the cluster redshift. Cl0024+1654 is not the relaxed
isolated cluster it had been thought to be! The description of the distribution
of the cluster member redshift distribution and its interpretation in terms of
a high–speed collision of two massive clusters of galaxies is the subject of the
second paper (Czoske et al. 2002) which forms the basis for Chapter 8. This
work was done in collaboration with Ben Moore who provided the numerical
simulations used to investigate the plausibility of the scenario. Chapter 8 also
contains a discussion of the distribution of the spectral and photometric observables in the catalogue which had to be omitted from the published version
of Czoske et al. (2002) due to page constraints.
6.2
Lensing observations
6.2.1 The quintuple arc system
Cl0024+1654 acts as a gravitational lens on a background spiral galaxy, producing a spectacular quintuple arc system which makes this cluster a show-case
of strong gravitational lensing. A HST/WFPC2 image of the cluster centre is
shown in Fig. 6.1. In colour representations of this image, the tangential arcs
stand out as markedly blue compared to the yellow cluster galaxies. The radial arc is somewhat less conspicuous and one must beware not to confuse it
with another, a little brighter, blue galaxy in the cluster centre; I have spectroscopically identified the latter as a violently star–forming galaxy at the cluster
redshift (No. 379 in Table A.1, Fig. 7.1g).
The redshift of the lensed galaxy has been identified by Broadhurst et al.
(2000) and was found to be zs = 1.675. The galaxy is remarkably well resolved
and shows detailed structure suitable for providing constraints for a strong
lensing model. The asymmetry of the lensed galaxy, in particular the presence of a bright spot at one end, makes this a nice example to demonstrate the
change of image parity across a critical curve and allows a qualitative tracing
of the critical curve for the source redshift zs without the need for a quantitative lens model. The parity changes when going from image A to B, and
2 http://cdsweb.u-strasbg.fr/cats/J.A+A.htx
6.2. Lensing observations
103
Figure 6.1: HST/WFPC2 image of the centre of Cl0024+1654. The five gravitationally lensed images of a background galaxy at zs = 1.675 are marked A
through E. Galaxies 412 and 414 are cluster galaxies that perturb the lensing
potential at the location of the triple arc ABC, galaxy 379 is an extremely blue
star-forming galaxy at the cluster redshift. Here the numbers refer to the entries in the spectroscopic catalogue, Table A.1.
104
then again when going from image B to C. The critical curve thus passes between images A and B and between images B and C, with A and C lying on
the outside of the critical curve. Clearly, the lens potential in this area has a
contribution from the gravitational potential of the bright cluster galaxies 412
and 414 (Fig. 6.1), in addition to the general cluster potential. Image D has the
same image parity as image B and also lies inside the critical curve. D is also
much closer to the cluster centre than the other three tangential images. Assuming that the lens potential is axisymmetric with respect to the line of sight,
one can use the radius of the tangential critical curve, 106 h−1 kpc at the cluster
redshift, as an estimate of the radius of the Einstein ring (Eq. 4.4), resulting in
a simple but fairly robust estimate of the (projected) mass enclosed by the arc
system:
r
c2
Ds
RE ' 1.6 × 1014 h−1 M
(6.1)
M(RE ) =
4G Dd Dds
in an Einstein–de Sitter cosmology.
Several attempts have been made to reconstruct the central mass distribution in Cl0024+1654 by modelling the gravitational lens system (Kassiola et al.
1992; Wallington et al. 1995; Smail et al. 1996; Tyson et al. 1998; Broadhurst
et al. 2000). In some respects the most interesting attempt is the one by Tyson
et al. (1998) because this resulted in a rather unusual and unexpected mass
distribution. Tyson et al. (1998) took advantage of the resolved structure of
the lensed galaxy, which they represented as composed of 58 smooth disks of
light; the position and brightness of each of these disks had to be matched in all
the lensed images of the source, thus providing a large number of constraints3
on a mass model comprising not less than 512 free parameters that describe
the positions and parameters of so-called mascons (mass concentrations). Most
of the mascons in this model are associated with cluster galaxies, although
the positions of some mascons were left free to describe a global cluster mass
distribution. The resulting mass distribution is smooth, with the largest contribution in a global mass halo appertaining to the cluster itself. Several smaller
mass concentrations are apparent and are identified with individual cluster
galaxies. The unusual aspect about the mass distribution as reconstructed by
Tyson et al. (1998) is the presence of a flat core of radius rc = 35 h−1 kpc, which
is in obvious contradiction to the standard cold dark matter scenario where
dark matter halos follow a universal profile with a central cusp of logarithmic
slope −1 . . . − 1.5 (see Sect. 3.3). The mass model of Tyson et al. (1998) has
often been cited as evidence against the standard cold dark matter scenario
and has motivated investigations into scenarios with different types of dark
matter, such as warm dark matter or self–interacting dark matter (Spergel &
Steinhardt 2000; Hogan & Dalcanton 2000; Moore et al. 2000). In these models,
3 In
practice the method used a pixel-wise comparison of the modelled lens plane with the
HST/WFPC2 image of the cluster centre.
6.2. Lensing observations
105
collapse to a central cusp is prevented by pressure within the dark matter fluid,
leading to the formation of extended constant density cores. Further motivation for these studies comes from spatially resolved spectral observations of
low surface brightness and dwarf galaxies, which also show evidence for flat
central density profiles. These observations have recently been criticised as
not having sufficient spatial resolution, thus not allowing unambiguous constraints on the central slope of the mass distribution (van den Bosch et al. 2000;
van den Bosch & Swaters 2001).
Broadhurst et al. (2000) have presented a different lensing model for Cl0024
+1654 which assumes that the density profiles of individual cluster galaxies are
in fact NFW profiles. Their model incorporates mass profiles associated with
the eight brightest cluster galaxies; the sum of these mass clumps results in a
smooth mass distribution but in this case the mass distribution has no flat core
but a central cusp, consistent with an overall NFW profile. The model provides an equally good fit to the observed lens configuration as the one given
by Tyson et al. (1998). However, Shapiro & Iliev (2000) have shown that the
mass distribution inferred by Broadhurst et al. (2000) results in a velocity dispersion of more than 2200 km s−1 , which is much higher than observed in any
cluster of galaxies. The model of Tyson et al. (1998) has problems of its own,
it is in particular questionable whether the available observational constraints
justify the extremely large number of free parameters in the model. There are
certainly limits to the possible accuracy of any lens model for Cl0024+1654 because at present only one multiply lensed system is known. Broadhurst et al.
(2000) claim to have identified a candidate double image system in the cluster, but at the faint magnitudes measured for these candidates spectroscopic
confirmation of the identity of the proposed images is not feasible.
6.2.2
Weak lensing observations
Although Cl0024+1654 was one of the first clusters of galaxies where a coherent shear pattern in background galaxies was detected (Bonnet et al. 1994)
no follow-up weak lensing observations and mass reconstructions have been
published yet. Bonnet et al. (1994) measured the ellipticity of background
galaxies in the north–east quadrant of the cluster and found a coherent pattern out to a distance of ∼ 1.5 h−1 Mpc from the cluster centre. Fitting models
to the measured shear profile they obtained estimates for the projected mass
contained within that radius between 12 × 1014 h−1 M for a de Vaucouleurs
model (log Σ ∝ R1/4 ), and 20 × 1014 h−1 M for an isothermal sphere model
(Σ ∝ R−1 ).
An interesting feature shows up in the mass map constructed by Bonnet
et al. (1994), a circular perturbation of the shear field at about 60 from the cluster centre. van Waerbeke et al. (1997) reanalysed the data of Bonnet et al. (1994)
106
using a different shear recovery method and reproduced the shear perturbation at the same location as in the original analysis. There is no obvious galaxy
concentration at the location of the shear perturbation so this might be an instance of a dark cluster lens. The spectroscopic survey in Cl0024+1654 (Sect. 7)
does not show any structure in redshift space at this location either.
Several other candidates for “dark” lenses have been reported in recent
years (near Abell 1942: Erben et al. 2000) but it is still unclear whether this is a
real phenomenon or whether these features might be caused by instrumental
effects not accounted for in the data analysis. In at least one case, the shear
perturbation failed to be reproduced in a data set different from the original
one (D. Clowe, private communication).
A deep I band image of the cluster was obtained in 1995 with the UH8k
camera on CFHT (Sect. 7.1.1). Unfortunately, this image has small–scale background structure due to scattered light from a nearby bright star, making it
unusable for a weak lensing analysis. In 1999, I obtained a V band image
with the CFH12k camera which was not of sufficient depth for a measurement
of the shear pattern (the primary purpose of this observation was to provide
photometry for the spectroscopic survey presented in Chapter 7). I obtained
further CFH12k observations in 2001 in the B, V, R and I bands. These images, although taken in non-photometric conditions, have excellent seeing and
should allow an accurate measurement of the shear pattern and in particular
confirm or refute the shear perturbation reported by Bonnet et al. (1994). A
HST/WFPC2 mosaic consisting of 38 pointings (in the F814W filter, 2 orbits
per exposure) across roughly the field covered by the spectroscopic survey has
recently been obtained (principal investigator R. S. Ellis, see also Treu et al.
2001) and a weak lensing analysis of this data set is under way. While the HST
data have the advantage of superior resolution compared to the ground–based
CFH12k images they do not cover the field contiguously. The non-locality
of the shear still makes a reconstruction of the mass density field possible;
however, the accuracy of the reconstruction is obviously reduced. Combining
the high resolution of the HST data with the contiguous field coverage of the
CFH12k data is therefore particularly interesting; cross-calibration of ellipticity measurements between the two data sets being another aspect.
The collision scenario for Cl0024+1654 that we developed following the
analysis of the redshift distribution from the spectroscopic catalogue (Chapter 8) makes a definite prediction for the mass profile at large radius which
should be less steep than a typical cluster NFW profile. A measurement of the
slope of the mass profile from a weak lensing analysis therefore constitutes an
important test for this scenario.
6.3. X-ray observations
6.3
107
X-ray observations
In X-rays, Cl0024+1654 was first detected by the E INSTEIN Observatory with
a luminosity of LX = 6.8 × 1043 erg s−1 (Helfand et al. 1980). High-resolution
imaging observations were obtained between 1994 and 1996 with the R OSAT
High Resolution Imager (HRI) for a total of 116,550 seconds. The whole image
covers 280 × 280 and shows the extended cluster emission and 16 other sources,
all of which are compatible with point sources. Except for the source S1 (in the
notation of Soucail et al. 2000) all of these sources could be identified uniquely
with optical counterparts. Two of these counterparts are also in the redshift
catalogue described in Chapter 7 (Table A.1). S2 corresponds to object 218 at
z = 0.9583; this object is identified with the quasar PC0023+1653 from the catalogue by Véron-Cetty & Véron (1998). S13 corresponds to a disc galaxy at
z = 0.4081, object 350 from the catalogue. Since the publication of the paper
(Soucail et al. 2000) we have obtained a better-quality V-band image than the
one used in the paper; I provide in Table 6.1 an updated version of the published table with improved photometry. For the paper we did not dispose of
a good-quality I-band image for the source S6, since this source fell on a gap
of the UH8k image available at the time; in Fig. 6.2 I show the corresponding subimage from a recently obtained CFH12k image of Cl0024+1654, which
replaces the low-quality V-band image of this source shown in Figure 3 of Soucail et al. (2000). Also shown in Fig. 6.2 are new images of S8 and S10, which
were marred by bad columns in the UH8k image.
There are several possible optical counterparts for the source S1, the closest
to the centre of Cl0024+1654. In fact, S1 might be a marginally resolved Xray source and associated not with any individual object but with a group or
small cluster of galaxies which is falling into the main cluster along an axis
perpendicular to the line of sight. The density map of galaxies at the cluster
redshift (Fig. 8.4) shows an extension towards the north-west, which might be
connected to S1.
The central 60 × 60 of the HRI image are shown in Fig. 6.3 overlaid as contours on the CFH12k V-band image. Significant cluster emission is detected
out to a radius of about 1.0 5 from the cluster centre. The surface brightness
distribution was studied in detail by Böhringer et al. (2000) and Soucail et al.
(2000), who fitted the profile with a standard β model (see Sect. 4.3.2). Correcting for the smoothing effect of the HRI point spread function, Böhringer
et al. (2000) find a low β = 0.475+0.075
−0.050 and a surprisingly small core radius
+6.1
rc = (10.4−3.9 ) arcsec, corresponding to 33 h−1 kpc at the cluster redshift. There
is no evidence for the presence of a cooling flow in Cl0024+1654. The isophotes
have moderately elliptical shape, with ellipticity e ∼ 0.25, and a show a change
in orientation from 40◦ (north over east) outside radius ∼ 2500 to an orientation of −40◦ in the centre. Soucail et al. (2000) interpret this as an indication
108
Table 6.1: Identifications of X-ray sources detected in the R OSAT HRI image.
This is an updated version of the table published in Soucail et al. (2000). CR
is the count rate in the HRI image, converted to apparent unabsorbed flux (in
erg s−1 cm−2 ) with a conversion factor 3.35 × 10−11 erg s−1 cm−2 for a CR of 1,
assuming a power-law spectrum with spectral index of 1 for the sources, and
hydrogen absorption with NH = 4.2 × 1020 cm−2 .
δ2000
17:10:21.8
17:10:16.5
17:09:38.8
CR(s−1 )
8.7 10−4
Flux [0.5–2 keV]
2.9 10−14
S2
α2000
0:26:31.69
0:26:31.10
0:26:26.36
7.4 10−4
S3
S4
S5
S6
S7
S8
S9
S10
S11
S12
S13
S14
S15
S16
0:26:46.06
0:26:45.86
0:26:18.10
0:27:04.10
0:26:23.73
0:27:07.9
0:26:12.96
0:27:07.57
0:26:28.90
0:26:37.50
0:26:36.35
0:26:30.17
0:25:49.56
0:27:32.54
17:12:31.5
17:13:05.5
17:09:46.3
17:07:20.6
17:02:32.7
17:07:48.5
17:03:48.9
17:06:17.6
17:00:25.8
16:59:53.9
16:59:26.1
16:56:56.5
17:17:21.7
17:06:55.2
1.4 10−3
3.6 10−4
6.0 10−4
4.9 10−4
5.4 10−4
7.0 10−4
6.4 10−4
4.5 10−4
8.2 10−4
2.9 10−3
1.0 10−3
7.4 10−4
1.2 10−3
1.2 10−3
S1
2.5 10−14
V
19.38
20.40
19.90
I
17.99
18.81
19.40
V− I
1.30
1.53
0.50
4.7 10−14
1.2 10−14
2.0 10−14
1.6 10−14
1.8 10−14
2.3 10−14
2.1 10−14
1.5 10−14
2.7 10−14
9.7 10−14
3.4 10−14
2.5 10−14
4.0 10−14
4.1 10−14
20.17
21.33
—
—
21.73
—
21.37
20.30
21.25
19.69
20.80
20.36
19.82
19.07
19.34
20.63
—
—
19.78
—
20.13
19.65
19.76
19.04
19.26
19.73
19.46
18.40
0.83
0.68
—
—
1.90
—
1.22
0.63
1.35
0.62
1.52
0.63
0.38
0.66
Comment
267: z = 0.4005
282: z = 0.2132
218: z = 0.9583
(=PC0023+1653)
2E 0024.0+1643
386: z = 0.4063
2E 0023.2+1700
Figure 6.2: I-band images of the optical counterparts of the X-ray sources S6,
S8 and S10 in the field of Cl0024+1654. These images are updates of the images
shown in Figure 3 of Soucail et al. (2000).
6.3. X-ray observations
109
Figure 6.3: The central 60 × 60 of the R OSAT HRI image are shown as contours
overlayed on the CFH12k V-band image. The X-ray contours are spaced linearly, with the lowest contour at 1σ above the background level with higher
contour levels at 4σ, 7σ and so on.
110
of some physical process which causes the central gas distribution to deviate
from a regular hydrostatic distribution.
In June 1996, Cl0024+1654 was observed with the ASCA observatory, which
provided a better spectral resolution than the R OSAT HRI, necessary for the determination of a reliable X-ray temperature. The effective observation time for
the solid state imager (SIS) was 46 ksec. Soucail et al. (2000) measure a temper+4.9
ature of TX = (5.7−2.1
) keV (90% confidence level) after carefully removing the
contaminating emission of the source S1 (at a distance of 1.0 2 from the cluster
centre), using a Raymond-Smith model with a metal abundance of 0.3 solar.
Using the same model with the measured temperature TX = 5.7 keV Soucail et al. (2000) convert the HRI count rate into a measured flux of 3.5 ×
10−13 erg s−1 cm−2 and a total luminosity of LX = 6.7 × 1043 erg s−1 , which is
consistent with the value of 6.0 × 1043 erg s−1 given by Böhringer et al. (2000)
who used a lower temperature of 3.6 keV chosen so as to be consistent with the
LX –TX relation of Markevitch (1998).
Using the formalism described in Sect. 4.3.2, Soucail et al. (2000) derive
a total mass profile M(r) and give a value for the total mass inside radius
0.5 h−1 Mpc of
14 −1
M ,
(6.2)
M(< 0.5 h−1 Mpc) = 1.4+1.3
−0.6 × 10 h
and a projected mass inside a cylinder of radius 0.5 h−1 Mpc of
14 −1
M2D (< 0.5 h−1 Mpc) = 2.2+2.0
M .
−0.7 × 10 h
(6.3)
Cl0024 : Introduction
Cette partie décrit un survey spectroscopique à grand champ autour de
l’amas Cl0024+1654 à z = 0.395. Le chapitre présent introduit l’amas et donne
la motivation pour conduire ce survey spectroscopique. Ensuite je rappellerai des observations de l’effet de lentille gravitationnelle dans cet amas ainsi
que les observations en X. Les chapitres suivants décriront le survey en détail
et offrent une interprétation de la distribution de redshifts qui est le résultat
principal de cette étude. Cette partie est largement basée sur mes deux publications sur le survey spectroscopique, Czoske et al. (2001) et Czoske et al.
(2002), ainsi que l’article de Soucail et al. (2000) qui présente des observations
en X de cet amas.
Les amas de galaxies accrètent constamment des galaxies et groupes de galaxies leur environnement, transformant les galaxies bleues du champ général
en galaxies rouges, elliptiques et passives, qui sont caractéristiques de la population des centres des amas. Les populations de galaxies dans les amas
évoluent avec le redshift : des amas de galaxies à grand redshift contiennent
plus de galaxies bleues que les amas locaux. La nature exacte de l’interaction des galaxies accrétées avec l’environnement à l’intérieur d’un amas (gaz
chaud, champ de marée gravitationnelle) et son influence sur la morphologie
des galaxies, leur contenu de gaz et taux de formation d’étoiles sont à présent
mal-compris. Il y a donc un intérêt d’étudier la zone d’accrétion à l’extérieur
−1
du rayon viriel (>
∼ 1 h Mpc) où la transformation des galaxies se produit. Les
nouvelles caméras CCD mosaı̈ques qui sont désormais disponibles sur plusieurs larges télescopes (CFHT, CTIO, Subaru, ESO2.2m) permettent d’étudier
les propriétés photométriques et morphologies sur des échelles de 1 – 10 Mpc
autour des amas. Des études à grand champ des amas demandent tout de
même à la fois des observations en imagerie et en spectroscopie. Les spectres
individuels des galaxies décrivent leurs distributions spectrales d’énergie et
donnent accès à leur redshift, ce qui est indispensable afin de produire un
catalogue de membres de l’amas avec des vitesses radiales et de l’information concernant leur contenu en étoiles et histoire de formation d’étoiles. A
présent, il y a seulement un nombre limité d’amas dans lesquels plus de ∼ 200
111
112
Chapitre 6. Cl0024+1654: Introduction
membres ont été confirmés spectroscopiquement (e. g. Abraham et al. 1996),
et surtout à redshift plus élevé les amas bien étudiés spectroscopiquement deviennent rare, surtout à cause de la forte contamination de galaxies d’avantplan.
Les amas ne sont pas des systèmes isolés, ce qui a des conséquences pour
les différentes manières d’estimer des masses des amas. L’effet de lentille gravitationnelle dépend de toute masse présente le long de la ligne de visée entre
l’observateur et la source. Les surveys spectroscopiques fournissent de l’information supplémentaire qui permet d’identifier d’autres structures importantes
que l’amas. Metzler et al. (2001) ont étudié l’influence de la présence de filaments et groupes de galaxies autour des amas sur les estimations de masse
à partir de l’effet de lentille gravitationnelle faible, et trouvent que des surestimations significatives sur M200 (facteur 1.5 ou 2) sont possibles et même
probables.
Un biais similaire affecte les mesures des dispersions de vitesses si des
groupes d’avant- ou arrière-plan ne sont pas identifiés comme tels et sont inclus dans la mesure de dispersion de vitesse, qui est donc surestimée. Normalement, environ 30 à 50 redshifts sont disponibles pour estimer une dispersion
de vitesse (Girardi et al. 1998; Girardi & Mezzetti 2001). En plus, les redshifts
ne sont mesurés que pour les galaxies de la partie centrale à l’intérieur d’un
rayon projeté de ∼ 500 h−1 kpc. Ces nombres sont peut-être suffisants pour les
amas avec des distributions de galaxies et de vitesses régulières ; en revanche,
les estimations seront biaisées si des structures non-résolues sont présentes.
Un survey de redshifts donne une mesure de la dispersion pondérée par la
densité de galaxies et moyennée le long de la ligne de visée. Avec un nombre
de redshifts suffisamment large il est possible de mesurer la variation de la
dispersion de vitesse en fonction de la distance du centre de l’amas (Carlberg
et al. 1997), mais les mesures encore plus approfondies sur l’état dynamique
d’un amas nécessitent un nombre prohibitif de redshifts (Merritt 1987).
Des observations de plus en plus détaillées de l’émission en X, des dispersions de vitesse et de l’effet gravitationnel montrent souvent que des amas
apparemment réguliers et en équilibre sont en fait des systèmes complexes
qui montrent des signes de fusions avec d’autres amas ou généralement de
sous-structure. Des collisions d’amas sont surtout visibles dans les observations avec C HANDRA sous forme de chocs ou fronts froids. Plusieurs auteurs
ont fait appel à la présence de sous-structure pour expliquer les différences
entre les estimations de masse issues de différentes méthodes dans beaucoup
d’amas (Miralda-Escudé & Babul 1995; Wu et al. 1998). L’amas Cl0024+1654
est un de ces amas où l’on observe une forte discrépance d’un facteur 2 ou 3
entre des estimations de masse élevées à partir de la dispersion de vitesse et
effet de lentille gravitationnelle forte d’un côté, et les observations X de l’autre
côté.
Résumé
113
Le projet décrit ici a été initié en 1992 par Geneviève Soucail, Yannick Mellier et Jean-Paul Kneib. Les spectres ont été acquis au télescope Canada–France–
Hawai‘i et au télescope William Herschel entre 1992 et 1996. Pour cette partie
de ma thèse j’ai réduit les données spectroscopiques et analysé la distribution
de redshifts et types spectraux dans Cl0024+1654.
Observations d’effet de lentille gravitationnelle
Un système quintuple d’arcs géants est visible dans le centre de Cl0024+1654,
des images d’une galaxie spirale à z = 1.675. La figure 6.1 montre une image
HST/WFPC2 du centre de l’amas ; les arcs ainsi qu’une galaxie extrêmement
bleue de l’amas et deux galaxies elliptiques, aussi dans l’amas, qui perturbent
les arcs A, B et C, sont marqués. L’identification du rayon du système d’arcs
avec le rayon de Einstein donne une estimation simple de la masse contenue à
l’intérieur des arcs : M(RE ) ' 1.6 × 1014 h−1 M .
Les arcs sont bien résolus, donnant accès à des structures individuelles
de la galaxie source. Plusieurs modèles du système ont été publiés dans la
littérature. Le plus intéressant est celui de Tyson et al. (1998) parce qu’il trouve
une distribution de masse avec un cœur plat, ce qui est en contradiction avec
le profil universel NFW. Le modèle de Tyson et al. a souvent été cité comme
indication que la matière noire n’est pas froide comme elle est considérée dans
le scénario de CDM standard. D’autres types de matière noire ont donc été
étudiés, tiède, auto-interagissant, etc. (Spergel & Steinhardt 2000; Hogan &
Dalcanton 2000; Moore et al. 2000). Dans ces modèles la formation d’un cusp
central est inhibée par la pression du fluide de matière noire, qui forme donc
des cœurs étendus de densité constante. D’autres indications de ce type sont
proviennent des études des courbes de rotation dans des galaxies naines.
Broadhurst et al. (2000) présentent un autre modèle pour les arcs dans
Cl0024+1654 qui s’appuie sur des profils NFW associés aux galaxies les plus
lumineuses dans l’amas. Le profil de l’amas qui résulte de ce modèle a en effet
un cusp central. Par contre, Shapiro & Iliev (2000) montrent que le modèle de
Broadhurst et al. (2000) a pour conséquence une dispersion de vitesse beaucoup trop élevée.
Une analyse du champ de cisaillement gravitationnel faible a été effectuée
par Bonnet et al. (1994). Ils trouvent un profil de masse qui peut être ajusté par
un modèle de sphère isotherme ou de de Vaucouleurs, ce qui donne des masses
à 1.5 h−1 Mpc entre 12 et 20 × 1014 h−1 M . La carte de cisaillement de Bonnet
et al. (1994) montre une perturbation circulaire à environ 60 au nord-est du
centre de l’amas, qui a été confirmée par une réanalyse des mêmes données par
van Waerbeke et al. (1997). Il n’y a pas de concentration de galaxies à l’endroit
où se trouve la perturbation, il est donc possible qu’il s’agisse là d’une “lentille
114
Chapitre 6. Cl0024+1654: Introduction
noire”. J’envisage une nouvelle analyse de lentille faible dans Cl0024+1654 à
partir de nouvelles images CFH12k que j’ai obtenues récemment.
Observations en X
Une analyse de l’image R OSAT/HRI de Cl0024+1654 a été publiée dans
Soucail et al. (2000). L’image (dont la partie centrale est montrée à la figure 6.3)
couvre 280 × 280 et montre l’émission étendue de l’amas ainsi que 16 sources
ponctuelles. A l’exception de la source S1, toutes les sources pouvaient être
identifiées avec des contreparties optiques, dont deux se trouvent dans le catalogue du survey spectroscopique. Depuis la publication de l’article de Soucail
et al. (2000) j’ai obtenu une nouvelle image en V de l’amas, et je présente de
la photométrie actualisée de ces contreparties optiques dans le tableau 6.1. La
figure 6.2 remplace les figures de mauvaise qualité de l’article par des sections
d’une nouvelle image en I de Cl0024+1654 obtenue récemment.
La température du gaz dans Cl0024+1654 a été mesurée à partir d’observations ASCA et aussi publiée dans Soucail et al. (2000). Elle est de TX =
5.7 keV. Cette température est utilisée pour la conversion de taux de comptage de la HRI en flux et ensuite en luminosité, LX = 6.7 × 1043 erg s−1 . Un
modèle β donne ensuite une masse totale dans un cylindre de 0.5 h−1 Mpc de
2.2 × 1014 h−1 M .
Chapter 7
A wide-field spectroscopic survey
7.1
7.1.1
Observations
Imaging
Photometric results from two wide–field broad–band images of Cl0024+1654
in the analysis of the spectroscopic survey presented in the following chapters.
On 26 September 1995, an I-band image was obtained using the UH8k camera (Luppino et al. 1995) on CFHT. The I-band image was reduced chip by chip,
i. e. the final result is in the form of eight individual images, one for each chip
of the camera. The final images were obtained as the mean of 10 exposures
of 1200 s each, using sigma clipping (2.5σ above the mean level and 4σ below) to reject hot pixels and cosmic ray hits, and have very good seeing of 0.00 7
FWHM; however, the background is marred by stray light, presumably due
to the bright star 47 Psc (V = 5.1, spectral type M3) at ∼ 500 distance from the
centre of Cl0024+1654, although outside the field of view of the UH8k. The Iband photometric catalogue was obtained from the individually stacked chips
using the SExtractor package (Bertin & Arnouts 1996) with a threshold of 1.5 σ
and a minimum detection area of 5 pixels. The catalogue contains more than
4 × 104 objects over a field of about 28 × 28 arcmin2 , the limiting magnitude is
I ∼ 24. Unless otherwise noted, I use total magnitudes as given by SExtractor’s
MAG BEST. The internal errors on the I-band magnitudes (as given by SExtractor) are smaller than 0.05 for I < 22.7 (0.01 for I < 20.7). Note that due to the
chip-wise reduction of the I-band image, the gaps between the chips (of typical width 5-10 arcsec) were not filled in during stacking and the photometric
catalogue contains no objects from these regions.
On 15 November 1999 we obtained a 3600 s V-band image using the CFH12k
camera (Cuillandre et al. 2000) on CFHT (Figs. A.1 and A.2a-A.2h). The six
individual exposures were bias and flat-field corrected in the standard way
115
116
Chapter 7. A wide-field spectroscopic survey
using the MSCRED package under I RAF1 . The exposures were then registered
onto the Digital Sky Survey2 (DSS) image of the field and median combined.
The final image, a mosaic of all 12 chips with all the inter-chip gaps filled in,
has ∼ 0.00 7 seeing (FWHM). The V-band photometric catalogue obtained from
this image contains ∼ 3.7 × 104 objects on a field of ∼ 42 × 28 arcmin2 . The internal errors on the V-band magnitudes are smaller than 0.05 for V < 23.3 (0.01
for V < 21.1). The limiting magnitude is V ∼ 25.
In order to obtain colour information aperture magnitudes were measured
in 14 pixel (2.00 8) diameter apertures for the V- and I-band images. This diameter is sufficiently large compared to the seeing FWHM to enclose most of the
light from the object under consideration and small enough to avoid contamination by neighbouring objects in crowded regions like the cluster centre. The
objects were then matched up using a polynomial transformation of the I-band
image coordinates onto the system of the V-band image. The centres of the two
images coincide to within 40 arcsec, so that the overlapping region covers virtually the whole UH8k field. Note that the gaps from the I band image show
up in the colour-magnitude catalogue as well. The resulting colour-magnitude
catalogue contains ∼ 2.1 × 104 objects with errors on the colours of smaller than
0.05 for V < 23.3 (0.01 for V < 21.1). Star-galaxy classification over a sub-region
of the colour-magnitude plane (as relevant for the present paper) will be described in Sect. 7.2.5.
A more detailed description of the photometric catalogue (including stargalaxy separation over the whole colour-magnitude plane) is given in Mayen
et al. (2002) who use this catalogue to investigate the depletion of background
galaxies due to the gravitational lens effect of Cl0024.
7.1.2
Spectroscopy
Spectra were obtained using multi-slit spectroscopy during three observing
runs at CFHT and one at WHT. Table 7.1 shows the observing log.
Candidates for all the runs (except for run 1) were selected from a V-band
mosaic obtained at the ESO New Technology Telescope (NTT) on 17/18 October 1993, with the primary selection criterion being VNTT < 23. The seeing
on this image was rather poor, ∼ 1.00 7 FWHM. In order to reduce contamination by stars, the preparatory shallow R-band images were carefully examined
during all the CFHT runs; for runs 3 and 4, we took additional advantage of
the excellent seeing of the deep UH8k I-band image.
All the CFHT observations were done with the Multi-Object Spectrograph
(MOS, Le Févre et al. 1994) with the O300 grism. The cameras used during each
1 IRAF
is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under cooperative
agreement with the National Science Foundation.
2 http://archive.stsci.edu/dss/
117
7.1. Observations
Table 7.1: Log of the spectroscopic observing runs.
#
Date
Instrument
CCD
1
2
3
4
24–27/08/92
24–27/08/95
12–15/09/96
11–13/11/96
CFHT/MOS
CFHT/MOS
WHT/LDSS-2
CFHT/MOS
SAIC1
Loral-3
LOR1
STIS-2
Grism
Nmasks
Exp. time
(ksec)
O300
O300
med/blue
O300-1
2
6
9
3
4.5 – 7.5
6.6 – 15.6
5.4 – 7.2
6.6 – 8.1
Pixel
(µm)(arcsec)
18
15
15
21
0.377
0.314
0.357
0.440
Disp.
(Å/pix)
4.31
3.69
3.30
5.03
Std. star
Feige 110
Wolf 1346
HZ4
Hiltner 600
run and the corresponding pixel scales and dispersions are listed in Table 7.1.
Two to five exposures per mask were obtained, depending on the magnitudes
of the selected objects in each mask.
Band-limiting filters, chosen such that prominent spectral features (for instance the Ca I H/K lines blueward of the 4000 Å break) fall into the band at the
redshift of interest, allow stacking of several rows of spectra on one mask, thus
increasing the number of spectra observable in a given time. This strategy has
been successfully employed by e. g. Yee et al. (1996b, 2000). However, spectra
covering a larger range of wavelengths provide more secure redshift determinations since more absorption and emission lines can be taken into account.
This is particularly important in the presence of artifacts caused by insufficient
removal of cosmic ray hits. In the case of Cl0024+1654 at z = 0.395, the 4000 Å
break roughly coincides with the strong sky emission line [O I] λ5577, so that
insufficient subtraction of the sky line might cause a problem in the redshift
determination if only a limited wavelength band were available. Also, covering a wide wavelength range is essential in order to derive spectral types for
the galaxies and analyse in detail the spectral properties of the cluster members. For these reasons we did not use band-limiting filters and the usable
wavelength range was typically 4500–8500 Å, depending on signal-to-noise
ratio and the quality of the sky subtraction at the red end of the spectral range,
where the sky emission is dominated by molecular bands.
The observations at WHT were made with the Low Dispersion Survey
Spectrograph (LDSS-2, Allington-Smith et al. 1994), the med/blue grism and
the Loral LOR1 detector. The usable wavelength range was 4000–7500 Å,
somewhat bluer than for the MOS observations. All the LDSS-2 masks were
covered by two exposures each.
We used 100 wide slits throughout, resulting in a resolution of ∼ 13 Å, except
for run 1 where the slit width used was 1.00 5 with a correspondingly worse
resolution of ∼ 20 Å.
118
7.2
Data reduction and analysis
7.2.1 Reduction of spectroscopic data
The spectroscopic data were reduced using the semi-automatic package MUL TIRED (Le Févre et al. 1995) which in turn uses standard I RAF tasks and treats
each slit separately. The spectral images were de-biased and flat-field corrected
in the standard way. A low-order (mostly linear) polynomial fit to the sky
emission in the spatial direction was then subtracted for each individual exposure. During the fit most of the cosmic ray hits in the sky region of the
spectral images were taken care of by a sigma-rejection algorithm (pixels with
deviations of more than ±2σ from the fit were rejected before refitting); nevertheless insufficiently rejected cosmics occasionally survived the fit to produce
fake absorption features in the object spectrum.
Geometric distortions cause the true spatial/dispersion directions to deviate from the row/column directions of the CCD chip, especially at the edges
of the field. Since sky fitting was done row-wise (column-wise for the WHT
data), sky emission lines tend to be imperfectly subtracted. Isolated emission
lines (in particular [O I]λ5577, which roughly coincides with the 4000 Å break
for galaxies at the redshift of Cl0024+1654) could simply be masked for the
subsequent analysis of the spectrum, but the usable wavelength range in the
red was effectively limited by the molecular band emission from the sky.
The individual exposures were then averaged into the final two-dimensional spectrum. In those cases where more than two exposures per mask were
available, the highest pixel value was rejected, thus accounting for cosmic ray
hits. The small number of exposures per mask made cosmic ray rejection difficult and imperfect. A cosmic ray hit on the object spectrum resulted in a fake
emission feature, a hit in the sky area in a fake absorption feature. However, as
argued in Section 7.2.2, cosmics do not in general influence the redshift determination. Finally, variance-weighted one-dimensional spectra were extracted
from the combined spectral images, wavelength calibrated using lamp spectra (He/Ar at CFHT, Cu/Ar at WHT) and approximately flux-calibrated with
long-slit spectra from spectro-photometric standard stars (listed in Table 7.1).
The wavelength calibration spectra were extracted using a straight trace
(following the column/row direction of the CCD), unlike the corresponding
object spectra, where the trace followed the flexure introduced by geometric camera distortions. We verified that this causes only negligible errors by
re-calibrating several spectra with strong flexure with calibration spectra extracted using the same trace as for the object spectra.
We finally performed a check on the wavelength calibration by extracting
three spectra per mask without sky subtraction and measuring the position
of prominent sky lines using the calibration from the lamps. For the WHT
masks this revealed systematic shifts, which are due to the fact that the lamp
7.2. Data reduction and analysis
119
spectra were not taken immediately before or after the science exposures. The
positioning of the masks in the mask holder was therefore not identical during
lamp and science exposures. These systematic shifts translate to ∼ 10−3 in
terms of redshift and a correction for each mask was applied to all the redshifts
determined from this mask. No such systematic effect was found for the CFHT
masks.
Seven example spectra and the V-band images for the corresponding objects are shown in Fig. 7.1.
7.2.2
Redshift determination
Since a large fraction of the spectra in our sample have rather low signal-tonoise ratio, we decided to identify redshifts by eye, which is better at finding real absorption and emission lines amongst noise than automatic redshift
identification techniques. Also, fake emission and absorption features due
to cosmics could be identified this way by simply checking the original twodimensional spectral images. All redshifts were identified by myself3 , using a
program by Karl Glazebrook which superposes the positions of typical galaxy
emission and absorption lines for a redshift guess onto the actual spectrum.
All the identifications were checked by at least one other member of the team
(Jean-Paul Kneib and/or Geneviève Soucail). The redshifts (runs 2-4 only)
given in Table A.1 carry a flag which indicates a (somewhat subjective) level
of confidence that the redshift identification is correct. “Secure/A” redshifts
were determined from spectra where emission lines and/or several absorption lines are clearly seen, “uncertain/D” redshifts are based on a tentative
identification of a single line or possibly several weak absorption lines, and
“probable/B” and “possible/C” indicate intermediate levels of confidence.
In order to obtain a more objective estimate of the measurement (as opposed
to identification) error of our redshifts, we used the cross-correlation technique
implemented in the task XCSAO in the I RAF package RVSAO (Kurtz & Mink
1998). In all cases, the by-eye redshift identification was fed to XCSAO as the
initial redshift estimate and the allowed redshift range was restricted to about
±5% around this redshift. In this sense, XCSAO was forced to recover our redshift identification and we only used XCSAO’s error estimate.
For large values of the correlation parameter, R > 3 (Kurtz & Mink 1998),
we find that XCSAO reproduces our input redshifts very well. The mean devi
ation between by-eye redshift and XCSAO redshift is zeye −zxcsao = −0.00018
with a scatter of 0.00037, corresponding to 80 km s−1 at the redshift of Cl0024.
However, only 107 of our spectra achieve R > 3 and the redshift distribution
of these is not representative of the redshift distribution of the total sample,
low-redshift objects being more reliably identified than objects at the cluster
3 except
for those from run 1 which were identified by Geneviève Soucail
120
Figure 7.1: Example spectra and corresponding 15× 15 arcsec2 sections from
the CFH12k V-band image. The catalogue entries for these objects are listed
at the top of Table A.1. Spectra (a) and (d) are from observing run 2 (CFHT),
spectra (b), (c), (e) and (g) from run 3 (WHT), and spectrum (f) is from run 4
(CFHT). Spectra (a), (b) and (c) are examples of spectra with “secure/A” redshifts based on a large number of emission and/or absorption lines. Spectrum
(d) is an example of a “probable/B” redshift and spectrum (e) of an “uncertain/D” redshift, based on the 4000 Å break only. Spectrum (f) is for the central galaxy of the northern group at z ∼ 0.495, discussed in Sect. 7.3. Spectrum
(g), finally, is for a very blue cluster member near the centre of Cl0024+1654,
showing almost the complete Balmer series in emission. Cosmic ray hits are
marked “cr”, sky emission lines by “sky”. Note that the wavelength ranges
are different in each panel.
121
Table 7.2: Comparison of multiply observed objects. The first three lines give
numbers determined from objects observed twice during the same runs, the
last three lines compare different runs. All the redshift differences are given in
units of 10−4 . The numbers of the observing runs are those given in Table 7.1.
all qualities
|z1 − z2 | (z1 − z2 )
Runs
N
2–2
3–3
4–4
19
6
9
8.2
4.8
6.6
2–3
2 – D99
3 – D99
28
24
28
14.0
12.8
10.0
—
—
—
6±19
2±19
−6±16
“secure/A” only
N
|z1 − z2 |
19
4
7
8.2
5.8
5.4
26
22
25
10.6
9.6
13.4
redshift or beyond. XCSAO assigns an individual error estimate to each redshift, based on the width of the peak in the correlation function. For R > 3, the
median of the distribution of this error is at ∼ 0.0003 (43 km s−1 ), with the bulk
of error estimates at < 0.0005.
However, these error estimates are only useful for our “best” spectra. In
order to get a more reliable error estimate for all the spectra we can compare
redshifts determined from multiple observations either during the same run,
when the same object appears on different masks, or during different observing runs. The results of this intercomparison are shown in Table 7.2 for those
pairs of runs with useful number N of pairs: The mean absolute differences
between the redshifts are of order 1×10−3 , with no marked difference between
the values determined for all quality codes or “secure/A” redshifts only. We
note that the slits at the “southern” ends of the WHT masks were consistently
of rather poor quality, in the sense that the slit edges become quite rugged.
Redshifts from these slits are therefore less accurate than those from the central and “northern” parts of the WHT masks or from the observing runs at
CFHT. Taking this fact into account we estimate that most of our redshifts are
accurate to about ∼ 1×10−3 . The majority of our spectra were obtained during
observing runs 2 and 3; comparing the redshifts of objects observed during
both these runs we find that the systematic shift (z1 − z2 ) between the observations is consistent with zero.
Finally we compare in Fig. 7.2 the redshifts for those 54 objects that were
observed both by us and by Dressler et al. (1999). For the majority of these
objects the redshift measurements agree very well, however, there are five clear
misidentifications, with redshift deviation of more than 0.01; these cases are
discussed in detail in Sect. 7.2.4. Object 523 deviates by more than 3σ if the
five misidentifications are dropped from the sample, so we exclude this object
122
Figure 7.2: Comparison of redshift measurements for objects observed both
by us and by Dressler et al. (1999). The large panel shows the full samples, the
inset is a blow-up of the cluster region. Five clear misidentifications and one 3σ
drop-out are labelled with their numbers in the spectroscopic catalogue. The
sizes of the symbols in the inset correspond to half our redshift error estimate
of ±1 × 10−3 .
as well. The remaining 48 common objects have a mean redshift deviation of
zD99 −zour = −0.0003 with a root mean square (RMS) scatter of σ = 0.0015. The
rms scatter is consistent with the estimate for our redshift error estimate of
10−3 , assuming that the redshifts given byDressler et al. (1999) have similar
accuracy. Again, there is no evidence for a systematic shift.
7.2.3
Spectroscopic measures
We measured equivalent widths for [O II] λ3727, [O III] λλ4959, 5007, Hα (where
within the wavelength range), Hβ and Hδ, as well as the strength of the 4000Å
break. [O III], Hα and Hβ were measured semi-automatically, i. e. the continuum level was placed by visual inspection and the line integration limits were
fixed at the values given in Table 7.3. The integration ranges for [O III] and Hβ
are the same as those used by Dressler & Shectman (1987); the range for Hα
is taken from Couch et al. (2001) and does not include the neighbouring [N II]
line.
[O II] and Hδ are important indicators of ongoing (Kennicutt 1992) and recently terminated (Abraham et al. 1996) star formation within galaxies and in
particular provide essential information on the interaction of newly accreted
123
galaxies with the cluster environment. We therefore adopted a more accurate
way to determine the equivalent widths and in particular to estimate a level of
significance for the strengths of these lines. We define our equivalent widths
so that they are positive for emission lines:
Nint
Wλ =
∑
i=1
fi − fc
∆λ =
fc
Nint
∑
i=1
fi
− Nint ∆λ
fc
,
(7.1)
where f i is the flux in pixel i, Nint the number of pixels in the integration range,
and ∆λ is the wavelength dispersion in Å/pixel. The continuum level f c was
estimated as the mean value within two wavelength intervals on either side of
the line; pixels with values more than 3σ away from the mean level were iteratively rejected in order to avoid cosmic ray hits in the continuum region. The
automatic measurement of the continuum level makes it possible to estimate
an error on the corresponding equivalent widths, thus allowing an assessment
of the detection significance of the line. For this purpose we model the noise as
Poisson-distributed photon noise. This allows us to relate the variances on f i
and f c to the single-pixel signal-to-noise ratio at flux level f c , S/N = f c /σc us 2
ing the usual Poisson scaling σ2 ∝ f and σ f c = σc2 /Nc when averaged over
Nc pixels. Adding the errors due to f i and f c in quadrature we thus obtain:
2
σW
=
λ
S
N
−2 "
(Wλ + Nint ∆λ)2
(Wλ + Nint ∆λ) ∆λ +
Nc
#
.
(7.2)
Here the first term is due to the line integration and the second to the determination of the continuum level.
In the case of [O II], the signal-to-noise ratio S/N was determined between
3560 Å and 3680 Å, a region which is largely free from absorption lines. For Hδ,
the signal-to-noise ratio was measured in the range 4050 Å to 4250 Å, where
the line itself was excised between 4085 Å and 4115 Å. The latter S/N was measured for every spectrum as a global indicator of the quality of the spectrum.
The integration ranges correspond to those used by (Abraham et al. 1996). For
Hδ we used their “narrow” range.
The strength of the 4000 Å break is given as the ratio of the total flux in the
range 4050 Å< λ < 4250 Å to the total flux within 3750 Å< λ < 3950 Å. These
ranges include all the absorption lines.
All the wavelength ranges are in the rest-frame of the object. For spectra
for which no redshift could be determined we only estimated the global S/N
assuming the cluster redshift z = 0.395.
124
Table 7.3: Wavelength ranges for equivalent width measurements. All the
wavelengths are given in Å (“i/a” — interactive placement of continuum
level).
[O II]
[O III]
Hα
Hβ
Hδ
break
S/N
λcent
line
blue cont.
red cont.
3727
5007
6563
4861
4103
4000
—
3713–3741
4997–5017
6556–6570
4851–4871
4088–4116
–
4050–4250
3653–3713
i/a
i/a
i/a
4030–4082
3750–3950
–
3741–3801
i/a
i/a
i/a
4122–4170
4050–4250
–
7.2.4 The catalogue
The final catalogue is listed in its entirety in Appendix A, Table A.1. A detailed description of the catalogue entries can also be found in Appendix A.
Briefly, the catalogue lists an object number, sorted by right ascension, equatorial coordinates relative to the cluster centre, the redshift with a quality code,
V magnitude and V−I colour, equivalent widths for [O II], [O III], Hα, Hβ and
Hδ (with errors for [O II] and Hδ), the strength of the 4000Å break, a signal-tonoise ratio for the spectrum, as well as the number of the observing run during
which the object was observed plus the reference number from the catalogue
of Dressler et al. (1999) if the object was observed by these authors.
We provide new spectra for 618 objects, of which 581 have redshifts. The
global success rate is therefore 94%. Not all redshifts are equally secure though:
We qualify 435 of our redshifts (70%) as “secure” (quality code A), 28 (5%) as
“uncertain” (code D) and 86 (14%) of intermediate quality. 34 objects (5%)
turned out to be stars. The fraction of “secure” redshifts is 83% for foreground
galaxies at z < 0.37, 82% for galaxies around the cluster redshift (0.37 < z <
0.41), and drops to 53% for galaxies at higher redshift.
For completeness, we include in the catalogue the redshifts provided by
Dressler et al. (1999). The corresponding entry number from their catalogue
(prefixed by ‘D’) is listed in column 15 of our catalogue. The catalogue as
published by Dressler et al. (1999) contains 130 entries, of which 107 are cluster
members. However, we noticed several errors in their catalogue, reducing the
number of distinct objects to 1254 . 54 of these objects were observed by us as
well, usually with concordant redshifts, see the discussion in Sec. 7.2.2 and Fig.
4 Dressler et al. (1999) failed to make several identifications within their sample: thus D19
and D66 seem to be the same object, as are D21 and D65, D113 and D120, D122 and D125, D22
and D73.
125
7.2. There were however five problematic cases: Dressler et al. give the redshift
for object 376 (D7) as 0.3755, whereas our spectrum indicates a secure 0.3955.
We assume that this is a typographical error in the Dressler et al. list and adopt
our value. Our spectrum for object 416 (D33) indicates z = 0.3895 as opposed to
their 0.4035. Since Dressler et al. only give this a quality code ‘3’, we adopt our
value. Object 289 (D109) is similarly uncertain. Object 466 should be identical
to Dressler et al.’s object D37, however the spectra are completely different
giving secure redshifts of 0.3916 and 0.1840 respectively. We have to assume
therefore that D37 has wrong coordinates. A similar problem occurs for D130
(our 237).
Adding the 69 objects which were observed by Dressler et al. alone increases the size of the catalogue to 687 objects, with 650 identified redshifts.
7.2.5
Completeness
Fig. 7.3 presents colour-magnitude diagrams (CMDs) for the photometric (for
the overlap region of the CFH12k and UH8k images) and the spectroscopic catalogues. For the latter catalogue we also show separate CMDs for foreground
galaxies at z < 0.37, background galaxies at z > 0.41, galaxies at the cluster redshift (0.37 < z < 0.41) and for a newly identified group of galaxies at z = 0.495
(see Sect. 7.3).
The primary criterion for the candidate selection for the spectroscopic survey was V < 23. The completeness in V magnitude of the final catalogue is
shown in Fig. 7.4, where for each galaxy from the spectroscopic catalogue we
count the number of objects in a bin of ±0.25 mag around the galaxy magnitude in both the spectroscopic and photometric catalogues (the latter restricted
to the survey area as outlined in Fig. 7.5) and define their ratio as the total completeness at the given magnitude. Between V ' 20 and V ' 22 the completeness is roughly constant at ∼ 45 % and drops rapidly for fainter magnitudes.
Restricting the same analysis to the central area within 30 of the cluster centre (using larger bins of ±0.5 mag) shows that in this region the completeness
exceeds 80% between V ' 20 and V ' 22.
A visual impression of the variation of the completeness across the survey
area is given in Fig. 7.5 where we use an adaptive top-hat (its radius at any
given point of a grid is the geometric mean of the distances to the 10th and
11th nearest objects with spectroscopy) to compute the ratio of the numbers of
objects with spectroscopy (fixed to 10) and the number of objects in the photometric catalogue. In order to have samples with well-defined (though a posteriori) photometric selection limits we restrict the spectroscopic and photometric
catalogues for this purpose to subsamples with 20 < V < 23 and 0.6 < V−I < 2.4.
As Fig. 7.3 shows, this region of the colour-magnitude plane encloses virtually all the cluster members in the spectroscopic catalogue. Within our re-
126
stricted sample the separation between stars and galaxies within the photometric catalogue is fairly straightforward using SExtractor’s CLASS STAR parameter. Within a total of 2722 objects we find 312 stars (with an error of about
±10 due to ambiguous cases) or 11.5%. Since useful shape information was
not available for the preparation of all the spectroscopic observing runs (about
5% of the objects in the spectroscopic catalogue are stars), the whole photometric sample is used in Fig. 7.5. The completeness values are therefore actually underestimates if interpreted as completeness for spectroscopic coverage of
galaxies alone. The cluster centre is sampled at > 70% completeness in a region
of about 8000 ×15000 .
7.3
Discussion
The three-dimensional distribution of the galaxies in the redshift sample is
shown in Fig. 7.6 in the form of wedge diagrams, where the angular position
of each object on the sky has been converted to proper distance from the line
of sight, appropriate for the given redshift in a ΩM = 1, ΩΛ = 0 world model.
The cluster Cl0024+1654 shows up clearly as a sheet at z ' 0.4. The expanded
views show that Cl0024+1654 is not a simple isolated cluster but that there is
a foreground “clump” at z = 0.38, superimposed onto the main cluster and
connected to the latter via a narrow bridge. In Sect. 8 (Czoske et al. 2002)
we discuss an interpretation of this structure as being due to the foreground
cluster having passed through the main cluster.
We find a pair of compact groups of galaxies at z = 0.495 about 100 to the
north-east of the centre of Cl0024+1654 (see Fig. 7.6 at x ' 1 Mpc, y ' 2 Mpc).
The northern group includes 8 galaxies, centred at 57400 north and 20300 east of
the cluster centre, the southern group includes 6 galaxies centred at 36900 north
and 25000 (median positions); the projected distance between the groups is thus
∼ 740 h−1 kpc. The mean redshifts are zN = 0.4921 and zS = 0.4970, the formal
velocity dispersions σN = 657 km s−1 and σS = 647 km s−1 . Student’s t-test
rejects the hypothesis that the two groups have the same mean redshift at 99%
confidence, so we assume that we are really seeing two separate groups. The
velocity dispersions are presumably enhanced by tidal interaction between the
groups.
Fig. 7.7 shows colour images of the northern and southern groups created
from the I- and V-band CCD images. The galaxy at z = 0.4907 is surrounded
by three objects of similar, blue colour. It is tempting to interpret this group as
multiple images of the same background object. In this case, using the curvature radius (∼ 500 ) as an estimate for the Einstein radius and zs = 1 as a rough
guess for the redshift of the background source, we obtain 5.8 × 1012 h−1 M
for the mass within this radius.
7.3. Discussion
127
Figure 7.3: V-I colour-magnitude diagrams. The top-left diagram shows the
full photometric catalogue, the top-right diagram the full spectroscopic catalogue. The next three diagrams split the spectroscopic catalogue according
to redshift, showing foreground and background galaxies as well as galaxies
around the cluster redshift z ∼ 0.39. Note the clearly visible cluster sequence at
V− I ' 2 in the latter diagram. The bottom-right diagram shows the members
of the newly discovered group of galaxies at z ∼ 0.495 (see Section 7.3). The
parallelogram marks the subsample used in the completeness map (Fig. 7.5),
20 < V < 23, 0.6 < V− I < 2.4.
128
Figure 7.4: Completeness of the spectroscopic survey in V magnitude. For
each galaxy from the catalogue this is given as the ratio of the numbers of
galaxies in the spectroscopic and photometric catalogues in a given bin width
centred on the magnitude of the galaxy. Pluses mark galaxies taken from the
whole survey area (as outlined in Fig. 7.5), crosses galaxies within 30 of the
cluster centre. In the former case, a bin width of 0.5 mag was used, in the latter
a bin width of 1 mag.
Another overdensity in Fig. 7.6 occurs at z ∼ 0.18. These galaxies are however distributed fairly uniformly across the field with no obvious spatial concentration and are therefore just part of the general large-scale structure in the
Universe.
In the first detection of a coherent shear field around a cluster of galaxies,
Bonnet et al. (1994) found a signal to the north-east of the centre of Cl0024+1654,
indicating a concentration of mass at a point where no overdensity of galaxies
is apparent in the two-dimensional images. The direction to this dark “clump”
is indicated by a circle in Fig. A.2b and by the dashed line in Fig. 7.6. There
is no significant over-density along this line which could explain the spatial
tightness of the signal observed by Bonnet et al.
7.3. Discussion
129
Figure 7.5: Map of the completeness variation of the spectroscopic catalogue
as gray-scale with overlaid contours. The completeness at any point is determined in a circular top-hat encompassing the 10 nearest neighbours in the
spectroscopic survey; the map is smoothed with a Gaussian of width 3000 . Contour lines are spaced in 10% steps. The 50% contour is marked by a bold line,
contours at less than 50% are drawn in black, higher contours in white.
130
Figure 7.6: Three-dimensional distribution of the objects in our redshift catalogue. In the two upper panels the objects are projected onto the right ascension axis, in the lower two onto the declination axis. The upper panel of each
pair shows the large-scale distribution from z = 0 to z = 1, the lower panel
an expanded view of the environment of the cluster Cl0024 itself. The dashed
line marks the direction towards the potential perturbation detected by Bonnet
et al. (1994). Two groups at z ∼ 0.495 are marked by rectangles. The conversion from angular position on the sky to proper transverse distance was done
assuming an Einstein-de Sitter Universe with H0 = 100 km s−1 Mpc−1 .
7.3. Discussion
131
Figure 7.7: “True” colour images of the apparent centres of the northern (left)
and southern (right) groups of galaxies at z ' 0.49. The images were created
from the V and I band images, the green channel is an average of these two
images. Supposed member galaxies of the group are conspicuous by their
yellow colour, corresponding to V − I ∼ 2.4. Note the blue arc-like structure
around the galaxy at z = 0.4907.
Déscription du survey et
présentation du catalogue
Ce chapitre est basé sur la première publication (Czoske et al. 2001), en
collaboration avec Jean–Paul Kneib, Geneviève Soucail, Terry Bridges, Jean–
Charles Cuillandre et Yannick Mellier.
Dans ce travail ont été utilisées plusieurs images grand-champs de Cl0024
+1654, notamment une image en I obtenue avec la caméra UH8k au télescope
Canada–France–Hawai‘i (CFHT) en septembre 1995, et une image en V obtenue par moi-même en novembre 1999 avec la caméra CFH12k, aussi au CFHT.
Cette dernière image a été réduite et assemblée avec les méthodes décrites
dans le chapitre 12. A partir de ces images j’ai construit des catalogues photométriques contenant 4 × 104 objets en I (magnitude limite ∼ 24) et 3.7 × 104
objets en V (magnitude limite ∼ 25), ainsi qu’un catalogue combiné contenant
2.1 × 104 objets qui ont été détectés dans les deux images et pour lesquels des
couleurs V− I pouvaient être mesurées.
Les spectres ont été obtenus entre 1992 et 1996 lors de trois campagnes
d’observation au CFHT avec l’instrument MOS et d’une campagne au William
Herschel Telescope (WHT) avec l’instrument LDSS-2 (Tab. 7.1). Les observations ont été faites en mode multi-objet, avec des fentes de largeur 100 et une
résolution spectrale de ∼ 13 Å, couvrant un intervalle en longueur d’onde
d’environ 4000 Å à 8000 Å. Deux à cinq poses par masques ont été obtenues.
La figure A.1 dans l’annexe A donne la distribution des objets pour lesquels
des spectres ont été obtenues superposée sur l’image V.
Pour la réduction des données spectroscopiques j’ai utilisé la collection de
scripts MULTIRED (Le Févre et al. 1995) qui appelle des routines standard de
I RAF pour effectuer la soustraction du biais, division par un flatfield, soustraction de l’omission du ciel, combinaison des poses individuelles obtenues à travers un même masque, extraction du spectre à une dimension, calibration en
longueur d’onde, et finalement calibration en flux. Des mesures des positions
des raies du ciel ont révélé des décalages systématiques dans la calibration en
longueur d’onde des spectres obtenus avec WHT/LDSS-2 ; l’application d’un
132
Résumé
133
décalage constant est suffisante pour corriger ce défaut. La figure 7.1 montre
quelques spectres exemplaires et les images en V des objets correspondants.
La détermination du décalage vers le rouge (redshift) a tout d’abord été
faite à l’œil en comparant à une série de raies communes. Cette méthode est
toujours la plus fiable pour l’identification des redshifts à partir de spectres
de faible rapport signal-sur-bruit. L’identification de redshifts a été vérifiée
avec une méthode automatique qui fait une corrélation entre le spectre observé
et des spectres de référence (RVSAO, Kurtz & Mink 1998). Pour les spectres
pour lesquels RVSAO a donné des résultats fiables, les redshifts étaient en bon
accord avec ceux déterminés à l’œil. D’autres tests de la fiabilité des mesures
des redshifts ont été faits sur les objets observées plusieurs fois, et les objets qui
figurent dans le catalogue de Dressler et al. (1999). Les résultats de ces études
sont résumés dans le tableau 7.2 ; on trouve un bon accord entre observations
et identifications indépendantes.
Des largeurs équivalentes ont été mesurées pour les raies [O II] λ3727, [O III]
λλ4959, 5007, Hα, Hβ et Hδ, ainsi que la hauteur du break à 4000Å. Les intervalles en longueur d’onde qui ont été utilisés pour ces mesures sont spécifiés
dans le tableau 7.3. Les raies O II et Hδ sont des indicateurs importants d’un
épisode actuel ou récent de formation d’étoiles (Kennicutt 1992; Abraham et al.
1996). Nous avons donc estimé des erreurs sur les largeurs équivalentes, suivant l’équation 7.2.
Le catalogue complet ainsi qu’une description détaillée de ses contenus se
trouvent en annexe A, tableau A.1. Notre survey fournit des nouveaux spectres
pour 618 objets, dont 581 ont des redshifts mesurés. Pour donner une base
de données complète, les 125 redshifts donnés par Dressler et al. (1999) ont
été inclus dans le catalogue (marqués dans la colonne 15 du tableau A.1). 54
objets d’entre eux ont aussi été observés par nous. Lors de l’intégration des
données de Dressler et al. (1999), plusieurs erreurs ont été découvertes dans
leur catalogue ; ces erreurs ont entre-temps été confirmées et corrigées par A.
Dressler (communication privée). Ainsi notre catalogue contient des données
sur 688 objets, dont 650 ont un redshift identifié.
A cause de contraintes observationnelles la couverture du champ par le
survey spectroscopique n’est ni complète ni homogène. La figure 7.3 compare les objets contenus dans le catalogue spectroscopique avec l’ensemble
des objets contenus dans le catalogue photométrique dans le plan couleur–
magnitude. Un sous-échantillon peut être défini qui contient la majorité des
objets du catalogue spectroscopique et qui a des limites en couleur et magnitude précises : 20 < V < 23, 0.6 < V− I < 2.4. Comparant cet échantillon avec la
totalité d’objets dans les mêmes limites photométriques donne la complétude
de notre échantillon en fonction de la magnitude V (figure 7.4). Le catalogue
est globalement complet à ∼ 45% entre V ' 20 et V ' 22, et à plus de 80%
à l’intérieur de 30 du centre de l’amas. La figure 7.5 montre la variation de
la complétude travers le champs. Cette figure a été construite en déterminant
134
Chapitre 7. A wide-field spectroscopic survey
pour chaque point le rayon du cercle contenant les 10 objets les plus proches
du sous-échantillon spectroscopique et divisant ce nombre (10) par le nombre
d’objets du catalogue photométrique dans le même domaine du plan couleur–
magnitude. Le centre de l’amas a donc été couvert à plus de 70% de complétude
dans une région de 8000 × 15000 .
La distribution des objets en trois dimensions (ascension droite, déclinaison
et redshift) est montrée dans la figure 7.6. L’amas apparaı̂t nettement dans les
deux figures à grand échelle, ainsi qu’un double groupe compact de galaxies
à redshift z ∼ 0.495. La distribution d’objets autour du centre de Cl0024+1654
révèle la présence d’une structure à un redshift un peu plus bas, connectée à
l’amas principal par un “pont”, parfaitement aligné avec la ligne de visée. L’interprétation de cette configuration particulière est le sujet du chapitre suivant.
Chapter 8
A high–speed collision?
8.1
The cluster environment
8.1.1
Redshift distribution
Fig. 8.1 shows a histogram of the 300 redshifts from the spectroscopic survey
described in Chapter 7 that lie in the range 0.37 < z < 0.42, i. e. in the vicinity of
the redshift of the cluster Cl0024+1654. The redshift distribution of the cluster
galaxies is clearly bimodal, showing two peaks at z = 0.381 and z = 0.395
respectively; these peaks contain 283 galaxies.
The larger peak at z = 0.395 (0.387 < z < 0.402, hereafter referred to as component A) contains 237 galaxies, is fairly regular and resembles a Gaussian distribution as expected for a relaxed, virialized cluster of galaxies. The smaller
foreground peak at z = 0.381 (0.374 < z < 0.387, component B), by contrast,
seems too wide for the small number of 46 redshifts contained in it. This impression is confirmed by Fig. 8.2, which plots the redshift for each galaxy versus its projected distance from the centre of Cl0024+16541 . The distribution of
the galaxies in the main peak is symmetrical with respect to the central redshift
line, whereas the distribution of the foreground galaxies is roughly constant at
a rest frame velocity of ∼ −3000 km s−1 at radii larger than 30 (600 h−1 kpc), but
turns off towards smaller relative velocities to merge with the main distribution at smaller projected distances.
The 17 galaxies in the peak at z = 0.407 are more widely dispersed across the
survey field and although we cannot rule out a connection with Cl0024+1654,
it seems more likely, in the light of the scenario developed in Sect. 8.2, that they
are part of the surrounding field galaxy population.
It is remarkable that we can trace the main cluster as well as the foreground
structure out to the edge of the survey field (there remains only one object in
1
The coordinates in the catalogue are given relative to α2000 = 00h 26m 35.s 70, δ2000 =
17◦ 090 43.00 06.
135
136
Chapter 8. A high-speed collision?
Figure 8.1: Redshift histogram for 300 objects in the neighbourhood of
Cl0024+1654 (0.37 < z < 0.42). The attribution of galaxies to components A
(dark grey) and B (light grey) was done by inspection of Fig. 8.2. Whereas
the lower and upper redshift limits of components B and A respectively are
fairly secure, the exact demarcation between the two components is arbitrary
to some extent and was chosen at ∆v = −1500 km s−1 with respect to the central redshift of component A, Eq. (8.2).
the catalogue outside the range depicted in Fig. 8.2, a background galaxy). The
virial radius for Cl0024+1654 should be around 1 h−1 Mpc (Girardi & Mezzetti
2001), which means that we detect a coherent structure out to three times the
virial radius.
Redshift distribution inside 20000
From Fig. 8.2 and Fig. 8.3 (upper panel) it is obvious that the central region of
Cl0024+1654 is highly perturbed and it is impossible to separate components A
and B within ∼ 20000 (640 h−1 kpc) from the projected cluster centre. However,
the redshift distribution for the central galaxies is strongly skewed towards
negative velocities: except for two galaxies that are fairly isolated in redshift
space, the redshift distribution at positive velocities is effectively cut off at ∼
1500 km s−1 , whereas the distribution at negative velocities extends to beyond
∼ −2500 km s−1 , branching off into component B at ∼ 500 h−1 kpc from the
projected cluster centre. Quantitatively the skewness of the distribution of 161
galaxies within 20000 (0.37 < z < 0.42) is 0.54, where we use the definition given
8.1. The cluster environment
137
Figure 8.2: Redshift z plotted against angular distance R from the projected
cluster centre for the galaxies around the cluster redshift. The left axis expresses redshift as relative velocity with respect to the mean redshift of component A (Eq. 8.2), the right axis as proper line-of-sight distance at the cluster
redshift. Angular distance is converted to proper transverse distance on the
top axis. The dashed horizontal lines indicate the velocity dispersion of component A (Eq. 8.3), the dotted lines denote limits for the samples used in Sect.
8.1.1. The solid curves mark the escape velocities for masses inside radius R of
1, 3 and 5 × 1014 M . Arrows mark the position of the giant arc and the X-ray
and weak shear detection limits.
138
Figure 8.3: Redshift histograms for objects inside (upper panel) and outside
(bottom panel) a radius of 20000 (corresponding to 640 h−1 kpc) from the projected cluster centre. In the central redshift distribution there is no correspondence to the distinct peak at z = 0.38 visible in the external distribution.
by Press et al. (1992):
1
Skew(x1 . . . x N ) =
N
N
∑
j=1
xj − x
σ
3
.
(8.1)
The probability that a sample of size N = 161 drawn from a normal distribution
shows a skewness larger than that measured in Cl0024+1654 is about 0.3%.
The Shapiro-Wilk normality test (Shapiro & Wilk 1965) rejects the hypothesis
that these data points are drawn from a Gaussian parent population at 99%
confidence. The skew in the central redshift distribution is already apparent in
the histogram given by Dressler et al. (1999) (see also Section 8.3).
Redshift distribution outside 20000
In the distribution of the galaxies outside 20000 (Fig. 8.2 and Fig. 8.3, bottom
panel) the two modes are clearly separated. The mean redshift of the 71 galax-
139
ies belonging to component A at projected distances 20000 < R < 50000 is
zA = 0.3946 ± 0.0007
.
(8.2)
In Fig. 8.5 we show the two velocity dispersion profiles for galaxies in component A with either positive or negative velocities with respect to the mean redshift zA , calculated in a sliding bin containing 30 galaxies. Outside 30 (600 h−1
kpc) both profiles are flat at the same level of σ ≈ 600 km s−1 . The profile
for galaxies with negative velocities (i. e. those moving towards us, Fig. 8.5)
rises all the way to about 4500 (140 h−1 kpc), reaching a velocity dispersion of
nearly 900 km s−1 in the innermost bin. The dispersion profile of the galaxies
with positive velocities, by contrast, drops back to a value of about 600 km s−1
after having reached a maximum value of about 800 km s−1 at 20 from the centre. Although the galaxy velocities used in Fig. 8.5 have been restricted to
|v| < 1500 km s−1 , the different behaviour of the profiles for galaxies with positive and negative velocities respectively reflects the skew in the central redshift
distribution and indicates the possible presence of a bulk motion component
towards us in the central galaxy distribution.
The exact value of the velocity dispersion of the 71 galaxies within the same
distance range as used for Eq. (8.2) and velocity |v| < 1500 km s−1 relative to
the central redshift zA is
−1
σA = 561+95
−83 km s
.
(8.3)
Both zA and σA were computed using the biweight estimator (Eqs. 4.40 and
4.41 and Beers et al. 1990), the errors (95% confidence level) were estimated by
bootstrap resampling.
The skewness of the distribution of these 71 redshifts is −0.12, which should
be compared to the standard deviation of skewness for samples of this size
drawn from a normal distribution, 0.27. The Shapiro-Wilk test does not provide evidence for deviations from a Gaussian distribution.
The spatial positions of the galaxies in the foreground component B show
no preference for any direction (Fig. 8.4, middle panel). The formal velocity
dispersion for component B in the same radial distance range as used in Eq.
−1
(8.3) is σB = 554+175
based on 15 redshifts. Given that the number of
−304 km s
galaxies in component A is almost five times as large as the number in component B it is hardly conceivable that both components should have the same
velocity dispersion – their mass-to-light ratios would be extremely different.
Component B is certainly not a virialised group or cluster.
From the distribution of the external galaxies alone one might still suspect
the presence of a loose aggregate of galaxies physically disconnected from
Cl0024+1654, but well-aligned with the line of sight; however, in the centre
there are no galaxies which would correspond to the z = 0.38 peak visible in
140
the external distribution (Fig. 8.3). If the foreground component B were an independent loose system, it would show a deficiency of galaxies right in front of
the centre of Cl0024+1654, which seems unlikely in the absence of interaction
between the systems. This leads us to suspect that the two components are in
fact physically connected and that the blue tail of the skewed central distribution originates from the same physical system as the foreground component
B.
Fig. 8.2 shows the theoretical escape velocities for masses of 1, 3 and 5 ×
14
10 M enclosed in a sphere of radius R. Although these lines are merely indicative due to the assumption of spherical symmetry and the unknown tangential velocity components, it is safe to say that the galaxies in component B
outside, say, 600 h−1 kpc are not bound to the main cluster component, unlike
the negative tail of the skewed central galaxy distribution.
The galaxy density map of the main component A (Fig. 8.4) shows an
extension of the galaxy distribution towards the northwest to a distance of
10000 − 15000 . This extension is visible in images of Cl0024+1654 and we show
here that the galaxies contained in it lie at the redshift of the main cluster.
Cl0024+1654 might therefore well be comprised of three distinct subsystems:
the main cluster, component A; a foreground cluster or group, component B,
which is interacting with the main cluster; and a third group or small cluster,
which is falling onto the main cluster in the plane of the sky.
8.1.2
Distribution of spectral types
Fig. 8.6 shows scatter plots of the most important observables from the spectroscopic catalogue for the 282 galaxies with redshift 0.37 < z < 0.402. The
bimodal redshift distribution is apparent in the distributions of colour, the
equivalent widths of [O II] and Hδ and the strength of the 4000 Å break (panels 10, 4, 5, and 6 respectively), as well as in the radial distance distribution
(panel 13, which is a reprise of Fig. 8.2). In the colour distribution (panel 10),
the two modes merge at V − I ' 2, typical of the red early-type galaxy population; the red galaxies are crowded at small distances from the cluster centre
(panel 14). A similar trend is visible in the distribution of the 4000 Å break
with radial distance (panel 3). On the other hand, no such crowding is found
when plotting EW([O II]) versus V− I (panel 7), as is to be expected since central cluster populations have few star-forming regions and are dominated by
older red stars. As expected, the bluest galaxies in the cluster do show the
strongest [O II] emission. [O II] equivalent widths are distributed fairly homogeneously with clustercentric distance (panel 1), although a more precise
analysis must take into account the fact that the survey samples the field in
an inhomogeneous way, completeness being largest in the cluster centre and
generally decreasing with projected distance (see Chap. 7).
141
Figure 8.4: Galaxy number density maps for (a) component A (0.388 < z <
0.402, 237 galaxies), (b) component B (0.37 < z < 0.388, 46 galaxies) and (c)
the “field” (0.1 < z < 0.35, 141 galaxies), estimated with the generalised nearest neighbour method (Silverman 1986) with 10 neighbours for map (a) and 5
neighbours for maps (b) and (c). The maps are divided by the completeness
map (Fig. 7.5) and smoothed with a Gaussian with σ = 3000 . The grey scales
are the same for all three maps. The lowest density contour lines are at 2%
and 5%, normalised to the maximum density in the cluster component A (left
panel). The remaining contours are spaced in steps of 10%. For densities below 50% the contours are drawn in black, for higher densities in white. The
50% contour is marked by a bold white line.
142
Figure 8.5: Velocity dispersion profiles for galaxies with negative and positive
velocity with respect to the mean redshift of cluster component A, which was
held fixed at zA = 0.3946. The profiles were computed using sliding averages in bins containing 30 galaxies. The error bands give 1σ errors determined
from bootstrap resampling (10000 realisations). The curve for positive velocities was shifted by 300 km s−1 for clarity, with the corresponding velocity values marked on the right-hand axis. The bottom panel shows the ratio of the
dispersions for galaxies with negative and positive velocities.
143
Figure 8.6: Matrix of scatter plots showing the most important observables
(projected radial distance R, redshift z, colour V−I, equivalent widths for [O II]
and Hδ as well as the strength of the 4000 Å break) for the galaxies in the
vicinity of the cluster redshift.
144
Definition
2±1
3±1
23±3
72±3
Centre (183)
3±2
6±3
42±6
48±6
A (64)
0
11±5
57±8
31±7
B (35)
4±1
3±1
53±3
40±3
Field (275)
4
4
24
67
Cluster
2
10
47
41
Field
Balogh et al. (1999)
Table 8.1: Distribution of spectral types: classification by [O II] vs. Hδ (the definition of the spectral types is taken from
Balogh et al. (1999) and given in column 2). The fraction of galaxies of a given spectral type is listed as the percentage
of the total number of galaxies in the subsample (the total number is given in parentheses). See text and Fig. 8.2 for
the definitions of the samples. The numbers
are the percentages p of galaxies belonging to each class in each sample.
p
The errors are the standard deviations N p(1 − p) of the normal distribution asymptotic to the binomial distribution
with sample size N and probability p for large N.
Spectral Type
Hδ < −5 Å, [O II]< 5 Å
Hδ < −5 Å, [O II]> 5 Å
Hδ > −5 Å, [O II]> 5 Å
Hδ > −5 Å, [O II]< 5 Å
This work
K+Aa
A+emb
SF/SSBc
passived
Centre (148)
3±2
17±5
50±6
30±6
A (60)
9±5
9±5
67±8
15±6
B (33)
7±1
5±1
56±3
32±3
Field (225)
5
10
12
74
Cluster
7
11
33
45
Field
Balogh et al. (1999)
a
b
spectrum
dominated
by
K
and
A
stars;
spectrum
dominated
by A stars, with emission lines; c star-forming/short starburst; d red, passively
evolving galaxies
Definition
3±1
8±2
34±3
55±4
This work
Spectral Type
Hδ < −5 Å, br < 1.5
Hδ < −3 Å, br > 1.5
Hδ > −5 Å, br < 1.5
Hδ > −3 Å, br > 1.5
Hδ strong; b red Hδ strong; c star-forming/short starburst; d red passively evolving galaxies
bHDSa
rHDSb
SF/SSBc
passived
a blue
8.2. A high speed collision?
145
In Tables 8.1 we classify the galaxies into spectral types, defined as in Balogh
et al. (1999), using our measured values for the equivalent widths of [O II], H δ
and the 4000Å break.
We consider four samples of galaxies according to redshift and projected
spatial position; the sample boundaries for samples “Centre”, “A” and “B” are
marked by the dotted lines in Fig. 8.2. The boundary between “Centre” and
“B” is to some extent arbitrary; replacing the slanted line by a simple velocity
cut at v = −1500 km s−1 does not significantly change the numbers in Tables
8.1, however. “Field” includes all the galaxies in the sample with redshifts
0 < z < 0.55 without the cluster galaxies, 0.372 < z < 0.402 (the upper limit here
is the same as for the field sample in Balogh et al. (1999)).
Both the classifications by [O II] vs. H δ and by H δ vs. the 4000 Å break
show an excess of blue star-forming galaxies (star-forming/short starburst,
SF/SSB, and emission-line galaxies, A+em) in component “B” as compared
to the cluster centre and possibly even over the field. The distribution of spectral types in component “A” is between a typical cluster and field populations. This seems to provide support for the hypothesis that component B is a
loose galaxy overdensity with spectral characteristics typical for field galaxies.
Alternatively it is possible that the star-formation activity in the galaxies belonging to component B has recently been (re)activated and enhanced through
interaction with a denser environment.
We include in Tables 8.1 corresponding numbers from Balogh et al. (1999).
These numbers are not strictly comparable, because the selection criteria used
for the creation of our catalogue are quite different from those used by Balogh
et al. (1999)2 ; this explains the discrepancies in particular in the H δ vs. the
4000 Å break classification, although the general trends are the same. The
agreement in the [O II] vs. H δ classification on the other hand is very good,
even quantitatively.
8.2
A high speed collision?
In general, a redshift difference between well-separated clusters at cosmological distances is a combination of a cosmological redshift difference due to the
proper distance between the clusters along the line of sight and a Doppler shift
due to the relative velocities of the two clusters. In general, these two effects
cannot be disentangled uniquely, and there might be several ways to reproduce a given redshift distribution. In the following we develop a scenario that
can reproduce the observed redshift distribution in Cl0024+1654, in particular the skewed central distribution, the presence of two well-separated modes
2 Balogh et al. (1999) use data from the CNOC1 cluster redshift survey, which has limiting
Gunn r magnitudes between 20.5 and 22.0, depending on the redshift of the observed cluster
(Yee et al. 1996b).
146
Figure 8.7: The initial and final particle configurations used to simulate the
two colliding clusters. The grey scale indicates the local density of dark matter
within the box of length 12 Mpc.
away from the projected cluster centre extending to at least three times the
virial radius.
Assume that a group or cluster has undergone a near radial collision with
Cl0024+1654 and we are now viewing the merging system along the direction
of the impact. In fact, several characteristic parameters of a collision scenario
are fairly well constrained by the observed redshift distribution. The relative
velocity of the two subclusters is given by the redshift difference of components A and B and is thus of order 3000 km s−1 . Those parts of the smaller
cluster B that pass through the centre of A are more strongly decelerated in the
direction of the collision than its outer parts due to stronger dynamical friction
and they therefore show a smaller redshift difference. A tidal gravitational
shock during the crossing scatters the outer galaxies of both clusters to large
projected distance, thus increasing the extent of the cluster halos beyond their
nominal virial radius. If the transverse velocity imparted on a galaxy that is
now found at a radial distance of 3h−1 Mpc is 1000 km s−1 , then the time since
core crossing is about 3 Gyrs, and the separation between the cores of the two
clusters is roughly 5 Mpc.
The main unknown in this scenario is the mass ratio between the subclusters. There are two possible versions of this collision:
(i) A small group has passed through the core of Cl0024+1654 and has been
completely disrupted and scattered by the cluster’s tidal force. In this
case the main body of Cl0024+1654 is unperturbed and we do not ob-
8.2. A high speed collision?
147
serve a concentration of foreground galaxies belonging to the original
group because the system has been completely unbound and is scattered
to large distances perpendicular to the main cluster.
(ii) A massive cluster of approximately 50% of the mass of Cl0024+1654 collided and passed through the core of the main cluster roughly 3 Gyr ago.
This collision is insufficient to completely disrupt the impacting cluster
but its outer galaxies are scattered to large projected distances. The remaining bound core of the colliding cluster undergoes sufficient dynamical friction to turn its orbit around and in redshift space it appears to lie
at a similar distance to Cl0024+1654.
Both of these scenarios can reproduce the main features of the observed redshift distribution. However, the massive collision scenario is more interesting
because of the possibility of resolving both the mass discrepancy problem and
the conflict between the central density structure of Cl0024+1654 and predictions of hierarchical clustering models.
The mass reconstructions of Cl0024+1654 (Tyson et al. 1998; Broadhurst
et al. 2000) using the positions of multiple images of a gravitationally lensed
background galaxy show a projected mass of 1.3 × 1014 h−1 M within 106 h−1
−2
kpc and a central surface mass density
√ of 7900 h M−1 pc . For a singular isothermal halo extending to r200 /kpc ≈ 2 σ1d /(km s ) this implies a characteristic velocity dispersion larger than 1500 km s−1 in order to explain the projected
mass.
As Shapiro & Iliev (2000) point out, the situation is worse in the context of
hierarchical clustering models. Dark matter halos in cold dark matter (CDM)
type models are shallower than isothermal in their centres. In order to reproduce the observed projected mass a CDM halo with velocity dispersion of
2200 km s−1 is required. This is inconsistent with the observations presented
in this paper.
A second problem for hierarchical models is that Tyson et al. (1998) infer a
very shallow central density profile for this cluster, much flatter than the cuspy
density profiles found for clusters in CDM type models (Ghigna et al. 2000).
The central core of Cl0024+1654 has frequently been used to constrain the nature of dark matter and to argue for alternative candidates to CDM (Spergel &
Steinhardt 2000; Hogan & Dalcanton 2000; Moore et al. 2000).
A high speed encounter between two similar mass clusters can explain all
of the observations presented in Sect. 8.1 and reconcile the discrepancy between mass estimates derived for Cl0024+1654. We now explore this scenario
using high resolution numerical simulations of colliding dark matter halos to
study the evolution of the mass distribution. We construct two equilibrium
CDM halos with virial masses 9.5 × 1014 M and 5.0 × 1014 M with peak circular velocities vpeak = 1600 and 1300 km s−1 and concentrations c = 5 and
148
c = 7 respectively. Their initial separation is 3 Mpc and relative velocity is
−3000 km s−1 . The particle mass is set to m = 5 × 109 M and we use an
equivalent Plummer force softening (Sect. 3.3)
Gm2 rij
Fij = − 3/2
2
2
rij + e
(8.4)
with e = 5 kpc.
After the collision we find that the outer regions of the smaller cluster
have become unbound and are streaming radially away from the impact location. The impulse velocity perpendicular to the encounter is of the order
1000 km s−1 . Snapshots of the initial and final times are shown in Fig. 8.7.
The peak circular velocities of the bound components have fallen to 1430 and
1140 km s−1 whilst the central 1D velocity dispersions have reduced to 930 and
710 km s−1 from initial values of 1033 and 883 km s−1 respectively. At the final
time we find a total mass within a cylinder of radius 106 kpc of 8.0 × 1013 M .
The redshift distribution of 10 000 randomly selected particles at the final
time is shown in the right hand panel in Fig. 8.8. This is not a perfect match to
the observational data but the main features are present: we see a foreground
component of “galaxies” that span large projected distances from the central
region of Cl0024+1654. This component is separated by ≈ 3000 km s−1 in redshift space from the main component. The smaller cluster bound core is moving away from the main cluster at ≈ 1000 km s−1 – the reduction in speed from
the initial velocity is due to dynamical friction. A better agreement would result if the impacting cluster was slightly more massive in which case it would
suffer more friction and would be seen nearly at rest compared to the main
component. Also note that we are plotting “dark matter particles” not “galaxies”. Of course the initial configuration shown in the left panel in Fig. 8.8 also
displays a bimodal distribution of redshifts. However, this configuration is
more symmetric than either the final simulated configuration or the observed
redshift distribution; also, the radial extent, in particular of the foreground (in
redshift space) component, is much smaller than observed.
The energy from the impulsive tidal shock has been transferred into kinetic energy of the particles, heating and expanding the cold central cores of
the clusters. This leads to a flattening of the physical and projected density
profiles. In Fig. 8.9 we show the projected surface mass density profile before and after the encounter. The initial profiles are cuspy, CDM type density
profiles whereas the final profiles have nearly constant density cores in good
agreement with that inferred from the mass reconstruction (Tyson et al. 1998,
their Figure 4). The central surface mass density is 4000 M pc−2 which is very
close to the value obtained by Tyson et al. for H0 = 50 km s−1 Mpc−1 . The total
projected mass within the central 106 kpc is about 30% lower than obtained by
8.3. Comparison to other observations
149
Broadhurst et al. (2000) which could be reconciled by using a more massive
encounter and/or properly including the baryonic matter.
8.3
Comparison to other observations
The galaxy distribution of Cl0024+1654 in redshift space and on the sky, as
described in Sect. 8.1, provides strong hints that this is not a simple relaxed
system. How do the skewed central velocity dispersion and the presence of
a foreground component in redshift space affect the interpretation of other
observations and in particular mass estimates of this system?
8.3.1
Galaxy dynamics
Dynamical mass estimates rely on the assumption that the galaxies are in virial
equilibrium in the cluster’s gravitational potential well and that their velocities
are purely random.
The large velocity dispersion of σ ≈ 1200 km s−1 found by Dressler & Gunn
(1992) and Dressler et al. (1999) (hereafter D99) was based on galaxies within
50 from the projected cluster centre. As shown in Fig. 8.10, there is no direct
evidence for the bimodality of Cl0024+1654 when attention is restricted to the
central regions of the cluster. This is still true with the larger number of galaxy
spectra used here: Whereas our catalogue contains about 85% more redshifts
in this area than D99, the general shape of the histogram is the same. Both histograms are strongly skewed towards low redshifts, thus indicating the presence of substructure (Ashman et al. 1994) or a bulk velocity component. Note
that the bimodality of the redshift distribution only becomes apparent when
galaxies at larger projected distances from the cluster centre are included. In
this case it is therefore not the number of galaxy redshifts but their distribution
over a wide field that provides the clues to the complexity of this system.
The formal velocity dispersion for the 193 galaxies within 50 (Fig. 8.10) is
σcent = 1050 km s−1 . If there is indeed a bulk velocity component present in the
central velocity distribution, then mass estimates based on the formal central
velocity dispersion will overestimate the true cluster mass.
A better dynamical mass estimate might be obtained from the velocity dispersion at larger radii, where the two components A and B are clearly separated. At projected distances > 30 the velocity distribution of component A is
regular with a dispersion of ∼ 600 km s−1 . If we interpreted this value as the
velocity dispersion of a relaxed cluster, it would have only about a quarter of
the mass previously estimated for Cl0024+1654 (Schneider et al. 1986), below
mass estimates from gravitational lensing (which are roughly consistent with
a high velocity dispersion) and even below mass estimates from X-ray observations. In the context of the collision scenario presented in Section 8.2 on the
150
Figure 8.8: The “observed” redshift distribution of particles in the collision
between the simulated clusters. The left panel shows the initial conditions
whilst the right panel shows the data 3 Gyrs after the collision. This is to be
compared with Fig. 8.2.
151
Figure 8.9: The surface mass density profiles projected along the merger axis
before (solid curve) and after the collision (dashed curve). The central surface
mass density after the collision is very close to that measured by Tyson et al.
(1998).
other hand, the galaxies at large projected distance from the cluster centre are
also affected by the collision and cannot be used to derive a mass estimate
based on the assumption of dynamical equilibrium.
8.3.2
X-ray observations
The X-ray observations from R OSAT/HRI and ASCA (Soucail et al. 2000; Böhringer et al. 2000) by themselves show no indication of anything other than a
fairly small cluster of galaxies. Given the measured values for Cl0024+1654,
43 −2 erg s−1 (Soucail et al. 2000), the gas
TX = 5.7+4.9
−2.1 keV and LX = 6.8 × 10 h
does not seem too far away from the LX − TX relation of Markevitch (1998); it is
slightly too hot for its luminosity. The morphology is regular and the surface
brightness profile well fitted by a beta profile with a surprisingly small core
radius of 33 h−1 kpc. Whereas no significant substructure is required to model
the X-ray observations of Cl0024+1654, they do not provide evidence against
the collision scenario either.
The X-ray emission is difficult to predict in the context of two clusters projected along the line-of-sight several Gyr after a head-on collision. During the
collision, hydrostatic equilibrium in the gas component of the cluster(s) breaks
152
Figure 8.10: Comparison of redshift histograms of the data of D99 (corrected
for the misidentifications discussed in Sect. 7.2.4) and from Table A.1 (these
include the data of D99), constrained to the area covered by D99, clustercentric
distance < 50 . Bin widths and centres are the same as in the inset of Figure 2 of
D99.
down and the X-ray luminosity and emission-weighted temperature fluctuate
considerably (by up to a factor of 10 for LX ) and rapidly as a consequence of the
formation of shock waves and repeated expansion and compression of the core
gas (Takizawa 1999; Ricker & Sarazin 2001; Ritchie & Thomas 2001). After several Gyr however, shocks have dissipated and the gas settles down to an equilibrium configuration. The cited hydrodynamic simulations of cluster mergers consider collision speeds of about 1000 km s−1 ; in this case the dark matter
cores separate to distances of only a couple of Mpc before turning around and
eventually merging into one clump. In a high-speed collision, however, the
collisionless dark matter cores of the two clusters separate to large distance
before turning around. The gas, due to its collisional nature should experience
stronger interaction during the first crossing and thus behave quite differently
from the dark matter/galaxies and possibly also from what low-speed collision simulations predict. Ricker & Sarazin (2001) observe a slight segregation
between the gas and the dark matter cores in their simulations. This effect
should be more pronounced in a high-speed collision.
If we accept the collision scenario then a mass estimate from the X-ray observations is not possible. Detailed inclusion of a hydrodynamic treatment
of the gas component in simulations of high-speed collisions is necessary to
understand the behaviour of the X-ray emission in this case. High resolution
153
X-ray imaging with C HANDRA has recently been obtained (P. I. Hattori) but
has not been published yet.
8.3.3
Gravitational lensing
Gravitational lensing by clusters of galaxies does not rely on the matter in
the clusters being in dynamical equilibrium. However, gravitational lensing
measures the weighted integral of all the mass between the observer and the
source, and the interpretation of the measured mass value thus depends on the
detailed distribution of mass along the line-of-sight. The skew in the central
redshift distribution in Cl0024+1654 is an indication of bulk motion, possibly
due to a high-speed collision between two fairly massive clusters of a mass
ratio of roughly 2:1, or by the passage of a small group through the core of
Cl0024+1654. In the latter case the mass determined by strong lensing models
would indeed be indicative of the mass of the main cluster; however, in this
case one would expect the X-ray emitting gas to be largely unperturbed by the
collision, and one would be left with the discrepancy between the mass estimates derived from lensing and X-rays. In the former case, the lensing mass
would be the sum of the two cluster cores and therefore too large if interpreted
as representing the mass of a single cluster.
The projected mass profile for the post-merger system seen in our simulations shows two characteristics which are testable by gravitational lensing
analyses of Cl0024+1654: At small scales of several 10’s kpc, the profile shows
a flat core which is consistent with strong lensing mass models (Tyson et al.
1998), as shown in Sect. 8.2. On large scales the profile is significantly flatter
than the initial CDM profile, falling only as r −2.5 out to ∼ 3 Mpc. Previous weak
lensing analyses on Cl0024+1654 have not attempted a detailed reconstruction
of the mass profile. Bonnet et al. (1994) and van Waerbeke et al. (1997) derived
shear maps from ground-based data on the north-east quadrant of the cluster,
Smail et al. (1997) were restricted to the small field of view of HST/WFPC2.
The shear pattern found by Bonnet et al. (1994) is compatible with an isothermal sphere profile, ρ ∝ r −2 out to 3 h−1 Mpc.
A weak lensing analysis of newly obtained CFHT/CFH12k images in BVRI
is currently underway (Czoske et al. 2002, in preparation). Also, a sparsely
sampled HST/WFPC2 mosaic consisting of 38 pointings out to a distance of
2.5 h−1 Mpc from the cluster centre has recently been obtained; these data will
also be used for a weak lensing analysis (Treu et al. 2001). These data will allow
to probe the cluster’s projected radial mass profile accurately to the edge of the
survey field.
Résumé du chapitre 7 :
Une collision à grande vitesse ?
Le survey spectroscopique qui a été décrit dans le chapitre précédent révèle
que Cl0024+1654 n’est pas un amas simple (figure 7.6). Dans ce chapitre on
examine plus en détail la distribution des redshifts des galaxies et on propose
un scénario qui peut expliquer cette distribution. Il est aussi démontré que ce
scénario est capable de résoudre le désaccord entre des estimations de masse
de Cl0024+1654 à partir de différentes méthodes.
La figure 8.1 montre le histogramme des redshifts de 300 galaxies autour
de Cl0024+1654. Il est évident que cette distribution est bimodale avec un pic
principal à z = 0.395 (la composante A) qui contient 237 galaxies et qui ressemble à une distribution Gaussienne comme on l’attend pour un amas virialisé, ainsi qu’un pic d’avant-plan à z = 0.381 (composante B), qui semble trop
large vu le nombre faible de 46 galaxies qui constituent ce pic. La figure 8.2 fait
le lien entre distribution en redshift et distribution spatiale, représentée par la
distance projetée du centre de l’amas. La distribution des galaxies dans le pic
principal est symétrique par rapport à son redshift moyen ; par contre, la distribution des galaxies d’avant-plan est constante à une vitesse relative (dans le
repère de l’amas) de ∼ −3000 km s−1 à grande distance du centre seulement,
0
−1
R >
∼ 3 , (600 h kpc), mais tourne vers des vitesses relatives plus petites et
fusionne finalement avec la distribution principale vers le centre projeté de
l’amas.
Il est remarquable qu’on puisse tracer l’amas principal ainsi que la structure
d’avant-plan jusqu’au bord du champ du survey spectroscopique. Le rayon viriel de Cl0024+1654 devrait se trouver à environ 1 h−1 Mpc suivant les relations
d’échelle habituelles, donc on détecte une structure cohérente jusqu’à trois fois
le rayon viriel.
La région centrale de Cl0024+1654 est fortement perturbée et il est impossible de séparer les composantes A et B à l’intérieur de ∼ 20000 (640 h−1 kpc) du
centre projeté. La distribution des redshifts dans cette région est asymétrique
avec un “skew” de 0.54 ; si la distribution de base était une Gaussienne, un
skew d’au moins cette valeur serait réalisé avec une probabilité de 0.3%. Le
154
Résumé
155
test de Shapiro-Wilk rejette l’hypothèse que les redshifts observés soient tirés
d’une distribution Gaussienne à 99%.
Plus loin que 20000 du centre les deux modes sont nettement séparés. Le redshift moyen de la composante A est zA = 0.3946 ± 0.0007. La figure 8.5 montre
séparément les dispersions des vitesses des galaxies avec vitesse positive ou
négative par rapport à ce redshift moyen ; la cinématique de ces deux groupes
de galaxies est différente, notamment dans le centre projeté, ce qui indique la
présence d’une composante de mouvement systématique dans le centre.
Entre 20000 et 50000 du centre, la dispersion des vitesse des galaxies dans la
−1 (95% confiance) et la distribution de redcomposante A est σA = 561+95
−83 km s
shifts ne diffère pas d’une Gaussienne. Dans la même région, la dispersion de
vitesses de la composante B est de 554 km s−1 . Puisque le nombre de galaxies
contenues dans la composante B est seulement un cinquième de celui compris
dans la composante A, il est difficilement concevable que B soit un amas ou un
group de galaxies virialisé.
À partir de la figure 8.3 on argumente que la composante B est en fait liée
physiquement à la composante A et ne représente pas simplement un groupe
étendu d’avant-plan. Cette figure montre des histogrammes séparés pour le
centre et la périphérie de Cl0024+1654. Ceci confirme l’observation faite dans
la figure 8.2 qu’il y a un déficit de galaxies d’avant-plan devant le centre de
l’amas, ce qui s’explique seulement en supposant un lien physique entre les
deux composantes. La queue bleue de la distribution asymétrique centrale serait de même origine que la composante B.
Dans la carte de densité de galaxies dans la composante A (figure 8.4) apparaı̂t une extension du centre vers le nord–ouest, ce qui pourrait représenter
une troisième composante du système, un groupe de galaxies qui semble tomber vers l’amas principal dans le plan du ciel.
Dans les tableaux 8.1 les galaxies sont réparties selon leur type spectral, suivant la classification de Balogh et al. (1999) à partir des largeurs équivalentes
de [O II] et Hδ, ainsi que la hauteur du break à 4000 Å. On considère trois
échantillons de galaxies de l’amas comme indiqués dans la figure 8.2, “centre”, “A” et “B”, ainsi que les galaxies du champ. On observe dans la composante B un excès par rapport à la composante A et au centre, et même par
rapport au champ, de galaxies bleues et/ou avec des raies en émission, c’est
à dire des galaxies qui forment des étoiles actuellement ou récemment. Cette
observation est compatible avec l’interprétation de la composante B comme un
groupe séparé d’avant-plan avec des caractéristiques spectrales typiques pour
les galaxies du champ. Il est aussi possible que la formation d’étoiles dans les
galaxies de la composante B ait été réactivée et augmentée récemment par une
interaction avec un médium dense.
On propose ici qu’une collision à grande vitesse relative entre deux amas
massifs peut expliquer la distribution en redshifts observée. La vitesse des
deux amas est contrainte par la différence en redshift des deux composantes
156
Chapitre 8. A high-speed collision?
A et B et est de l’ordre 3000 km s−1 . Les parties de l’amas secondaire (B) qui
passent à travers le centre de l’amas A sont décélérées plus fortement que les
parties extérieures à cause de la friction dynamique. Un choc de marée gravitationelle disperse les galaxies extérieures des deux amas à grande distance
projetée et donc augmente la taille des amas au-delà de leur rayon viriel initial. Si la vitesse transversale d’une galaxie qui se trouve actuellement à une
distance radiale de 3 h−1 Mpc est de 1000 km s−1 , le temps passé depuis le croisement des cœurs est d’environ 3 Gyr, et la séparation actuelle entre les cœurs
est d’environ 5 Mpc.
L’inconnu principal dans ce scénario est le rapport des masses des deux
amas. Il est possible que l’amas à l’origine de la composante B ne soit qu’un
groupe de galaxies de faible masse, qui se serait désintégré complètement et
dispersé par les forces de marée de l’amas principal. Alternativement, si la
masse de l’amas B est d’environ 50% de celle de l’amas A, le cœur de B reste
intact et seulement ses galaxies extérieures sont dispersées à grande distance,
le cœur ayant subi suffisamment de friction dynamique pour renverser son
mouvement, donc il se trouve à peu près au même redshift que l’amas A.
Les deux cas peuvent également reproduire la distribution des redshift observée dans Cl0024+1654. On poursuit ici la deuxième possibilité, celle d’une
collision entre deux amas relativement massifs, parce qu’il s’avérera que ce
scénario a des conséquences intéressantes en connexion avec la modélisation
de l’effet de lentille gravitationnelle de Cl0024+1654 et ses conséquences pour
le modèle de matière noire froide. Il faut néanmoins retenir que le scénario
présenté dans la suite n’est pas unique.
Le scénario est étudié par des simulations numériques de collision entre
deux halos de matière froide (donc un profile de masse de Navarro et al. 1997)
de masses virielles 9.5 × 1014 M et 5.0 × 1014 M et des concentrations respectives c = 5 et c = 7. La séparation initiale des deux amas est de 3 Mpc,
la vitesse relative est de −3000 km s−1 . La masse des particules est fixée à
5 × 109 M , leur force gravitationnelle est modifiée à 5 kpc pour éviter des collisions deux-corps d’origine purement numérique. Après la collision on trouve
que les régions extérieures ne sont plus liées et s’éloignent radialement du lieu
de l’impact. Les situations initiales et finales sont montrées dans la figure 8.7.
La masse totale dans un cylindre de rayon 106 kpc est de 8.0 × 1013 M .
Les distributions des redshifts avant et après la collision sont montrées
dans la figure 8.8. La distribution finale montre les caractéristiques principales de la distribution observée, une composante d’avant-plan séparée de
3000 km s−1 de la composante principale, les deux s’étendant à de grandes distances projetées.
L’énergie du choc de marée est transformée en énergie cinétique des particules, ce qui chauffe et dilate les cœurs froids des amas. Ceci conduit à un aplatissement des profils de densité physique et projeté. Comme montre la figure
8.9, le profil final a un cœur de densité constante, même si les profils initials
Résumé
157
ont des cusps, caractéristiques des halos de matière noire froide. Un cœur de
densité constante est en bon accord avec le profil déduit par Tyson et al. (1998)
à partir d’une modélisation du système d’arcs géants dans Cl0024+1654.
Le scénario d’une collision a évidemment des conséquences pour l’interprétation d’autres observations de Cl0024+1654. Des surveys spectroscopiques
précédents ont trouvé une dispersion de vitesse élevée, ∼ 1200 km s−1 , donc
une masse virielle importante (Dressler et al. 1999). Pourtant ces surveys étaient
visés au centre de l’amas où, comme nous l’avons montré, les deux composantes ne sont pas séparables. Par contre la distribution de redshifts/vitesses
dans le centre est significativement asymétrique, ce qui indique la présence
d’une composante de vitesse systématique, cohérent avec le scénario de collision. Une vitesse systématique conduit à une surestimation de la dispersion
de vitesse, donc de la masse, de l’amas si celui est supposé se trouver dans un
état d’équilibre. Puisque ce n’est pas le cas, une détermination de la masse de
l’amas à partir du théorème viriel n’est pas possible.
Il est intéressant de noter que l’asymétrie de la distribution centrale des redshifts dans Cl0024+1654 apparaı̂t déjà dans l’échantillon plus petit de Dressler et al. (1999) (figure 8.10). Ce n’était donc pas le nombre élevé de redshifts
disponibles dans notre survey, mais plutôt le passage au grand champ, qui a
permis de révéler la vraie nature de cet amas.
Les observations en X de R OSAT/HRI et ASCA (Soucail et al. 2000; Böhringer et al. 2000) ne montrent pas d’indication que Cl0024+1654 pourrait être
autre chose qu’un amas régulier de masse relativement faible. Les valeurs me43 −2 erg s−1 , sont compatibles avec
surées, TX = 5.7+4.9
−2.1 keV et LX = 6.8 × 10 h
la relation LX –TX de Markevitch (1998), le gaz étant un peu trop chaud pour sa
luminosité. La morphologie est régulière et le profil de brillance de surface est
bien ajusté par un modèle β. Par contre, les observations en X ne fournissent
pas d’évidence contre le scénario de collision non plus.
Le comportement du gaz collisionnel dans une collision de deux amas est
difficile à prédire sans recours à des simulations hydrodynamiques. Après une
collision le gaz n’est plus en équilibre hydrostatique, et il y a des fluctuations
considérables de luminosité et température (Takizawa 1999; Ricker & Sarazin
2001; Ritchie & Thomas 2001). Les simulations citées considèrent des collisions
à des vitesses d’environ 1000 km s−1 , dans lesquelles les cœurs des amas ne se
séparent plus à grande distance après la première collision. Dans une collision
à grande vitesse le gaz subit une interaction plus importante que la matière
noire et on peut s’attendre à une certaine ségrégation de gaz et matière noire.
Puisque le gaz ne se trouve plus en équilibre, une détermination quantitative
de la masse de l’amas à partir d’observations X est exclue.
L’effet de lentille gravitationnelle dépend de la masse totale projetée le long
de la ligne de visée. Une mesure de masse à partir d’effet de lentille est valable
même si la matière n’est pas en équilibre dynamique ; par contre, c’est l’in-
158
Chapitre 8. A high-speed collision?
terprétation en tant que masse d’un amas qui est en question. Dans le scénario
de collision, l’effet de lentille mesure la somme des masses des deux amas.
Un succès impressionnant du scénario de collision est la création d’un cœur
de densité constante, cohérent avec le modèle présenté par Tyson et al. (1998).
À l’extérieur le scénario prédit que la densité projetée décroı̂t plus lentement
que le profil d’un halo de matière froide standard, une prédiction qui peut
être testée par des analyses de l’effet de lentille faible. J’ai obtenu récemment
des nouvelles images grand champ de Cl0024+1654 en BVRI avec la caméra
CFH12k qui permettront une telle analyse. Une autre analyse qui est en cours
utilise des observations avec le HST/WFPC2, qui consistent en 38 pointages
jusqu’à une distance de 2.5 h−1 Mpc. Une combinaison des deux types d’observation avec leurs avantages respectifs devrait amener à une détermination
précise du profil de densité de Cl0024+1654 à grande distance.
La vitesse relative entre les deux amas impliqués dans le scénario de collision est importante par rapport aux vitesses moyennes d’amas de galaxies
rapportées dans la littérature. Pourtant, dans certains cas des vitesses relatives
de l’ordre de 3000 km s−1 ont été observées, notamment dans 1E0657-56 (Markevitch et al. 2001). Cl0024+1654 est un cas extraordinaire mais pas impossible.
J’envisage une continuation de la collaboration avec B. Moore pour investiguer
des statistiques de collisions entre des halos dans des simulations numériques.
Chapter 9
Conclusions
In this Part, I presented a catalogue of spectroscopic data for 688 objects in a
field of 21×25 arcmin2 around the centre of the cluster of galaxies Cl0024+1654
at z ∼ 0.395, 295 of which lie in the vicinity of the cluster itself, in the range
0.37 < z < 0.41. The completeness of the sample exceeds 80% around the cluster
centre for V < 22, dropping to ∼ 70% for V < 23. The mean completeness over
the whole field is about 45% down to V = 22. This is therefore one of the largest
spectroscopic surveys available for a cluster at z >
∼ 0.2 and the largest at z ∼
0.4. Apart from redshifts the catalogue lists photometric data and equivalent
widths for five lines as well as the strength of the 4000 Å break, important for
determining present and recent star forming activity. The catalogue represents
a unique database for investigations into the structure of a medium-redshift
cluster of galaxies and its environment.
The main result of this study is that Cl0024+1654 is not a simple isolated
cluster, as has hitherto been assumed in interpreting kinematical, lensing and
X-ray data. Instead there is a second, less massive cluster projected onto the
centre of the main cluster. The separation of Cl0024+1654 into two components has strong consequences for the interpretation of observational data and
should in particular help to resolve the well-known discrepancy between mass
values determined using different methods for this cluster (Soucail et al. 2000).
In Chapter 8 I have analysed the galaxy distribution in Cl0024+1654 in
more detail, based on ∼ 300 galaxy redshifts and projected positions. The cluster, which was previously regarded as a prototype of a massive relaxed cluster at intermediate redshift, turns out to have a fairly complicated structure,
showing a strongly skewed redshift distribution in its central parts and two
well-separated components at larger projected distances out to ∼ 3 h−1 Mpc.
I argue that the blue tail of the central velocity distribution and the foreground component originate from the same physical system and interpret the
peculiar redshift-space distribution as the result of a high-speed head-on collision of two clusters of galaxies, the merger axis being very nearly parallel to
the line-of-sight. Using a numerical simulation from Ben Moore, we showed
159
160
Chapter 9. Conclusions
that it is possible to explain the observed redshift distribution with a highspeed collision of two rather massive clusters of galaxies with a mass ratio of
about 2:1.
Apart from reproducing the spatial/redshift distribution of the cluster galaxies, this scenario also produces a projected mass distribution which is very
close to that derived by Tyson et al. (1998) from the quintuple arc system observed in Cl0024+1654. Note that the scenario was not designed to reproduce
this mass profile. Since it is thus possible to produce a mass distribution with
a flat core from the merger of two CDM halos this eliminates one of the main
arguments against simple non-interacting cold dark matter as the dynamically
dominant component in clusters of galaxies (Spergel & Steinhardt 2000).
The distribution of spectral types in the foreground component B is more
akin to the general field population than to a cluster population. In the context of our collision scenario, these would correspond to the outer regions of
the smaller cluster which have become unbound during the impact. Even initially, these galaxies would probably not correspond to a fully transformed
cluster galaxy population. In addition, the impact might have triggered additional star formation in these galaxies. Remembering that Cl0024+1654 was
one of the most distant clusters in which the Butcher-Oemler effect was detected (Butcher & Oemler 1984; Dressler et al. 1985), this may add a new view
on this effect: The fact that ∼ 40% of the bright galaxies in the cluster are
emission line galaxies ([O II]> 5 Å) has a natural interpretation in the collision
scenario. The spectral distribution of the galaxies in component A (and centre)
are also perturbed by the collision with a possible excess of rHDS and SF/SBB
(Tables 8.1); rHDS galaxies may represent the result of a starburst induced
during the early stages of the interaction.
The relative velocity of the clusters of ∼ 3000 km s−1 implied by the redshift
distribution in Cl0024+1654 is very high. Observations and simulations find
mean peculiar velocities for clusters of order 500 km s−1 (Bahcall & Oh 1996;
Giovanelli et al. 1998; Gibbons et al. 2001; Colberg et al. 2000a). Still, colliding clusters can reach relative velocities of about 3000 km s−1 at separations of
about 1 Mpc (Sarazin 2001). Markevitch et al. (2001) have recently found a bow
shock in the galaxy cluster 1E0657-56 which implies a relative speed of 3000 to
4000 km s−1 for the collision. Cl0024+1654 is still an extraordinary but not an
impossible system.
I report the discovery of a binary group of galaxies at z ' 0.495, which
includes a gravitational arc candidate. A further overdensity in the redshift
distribution at z ∼ 0.18 can at present not be attributed to a collapsed structure,
although we cannot exclude the possibility that a centre for such a structure
exists outside the field covered by our survey. None of the structures seen in
the redshift distribution seems to be able to explain the coherent shear signal
161
at ∼ 60 to the north-east of the projected centre of Cl0024+1654, detected by
Bonnet et al. (1994).
Further work on this Cl0024+1654 is envisaged. I have obtained new deep
CFH12k images in excellent seeing conditions during the second semester 2001
(queue observations) in B, V, R and I. The images have already been reduced using the same procedure as described in Chapter 12 (except for the
pre-reduction which was done by the CFHT staff). A weak-lensing analysis of
these images will measure the slope of the mass profile to beyond the nominal virial radius, thus testing the prediction of a flat profile from the collision
scenario (Fig. 8.9). The weak shear map should also be able to pick out substructure in the field. A candidate is the extension in galaxy number density
which appears to the north-west of the cluster centre in Fig. 8.4. This region
was not included in the weak shear analysis of Bonnet et al. (1994). A shear
map based on the new data will also confirm or refute the presence of the dark
structure seen by Bonnet et al. (1994); so far their shear map and the reanalysis
of the same data set by van Waerbeke et al. (1997) remain the only indications
of the reality of this structure.
Further research on Cl0024+1654 is being conducted in the frame of an
international collaboration. This centres on the HST mosaic of 38 pointings
recently acquired. This will also be used for weak lensing analysis (a combination with the CFH12k should prove particularly interesting) but also to
measure galaxy morphologies and the dependence of the morphological mix
with distance from the cluster centre. This was initially intended to study the
density–morphology relation in a relaxed rich cluster at intermediate redshift.
The fact that Cl0024+1654 is not a simple cluster adds some difficulty and
should be taken into account in the interpretation of the data. Further spectroscopy is currently acquired by Richard Ellis and Tommaso Treu with the
Hale 5m and Keck telescopes, filling regions that were not covered by our survey and extending the database to fainter magnitudes. The aim is to study
the dependence of spectroscopic types on the clustercentric distance; as for the
galaxy morphologies, the fact that Cl0024+1654 is not a simple cluster must be
taken into account in the data interpretation but adds the chance to study in
detail the effects of a high–speed cluster collision on the galaxies’ spectroscopic
types.
The relative speed which seems to be required by the collision scenario is
rather high. We intend to investigate the statistical distribution of relative collision speeds from cosmological simulations in collaboration with Ben Moore.
This will provide the means to judge how likely a high-speed collision between
massive clusters is. Since cluster collisions might bias X-ray cluster surveys
due to the increased luminosity in the presence of merger shocks, this effect
should be studied quantitatively to ensure that the cosmological conclusions
drawn from cluster surveys are correct.
Cl0024+1654 : Conclusions
Dans cette partie j’ai présenté un catalogue de données spectroscopiques
sur 688 objet dans un champ de 21×25 arcmin2 autour de l’amas Cl0024+1654
à redshift z = 0.395, dont 295 se trouvent dans l’intervalle 0.37 < z < 0.41. La
complétude de l’échantillon dépasse 80% dans le centre de l’amas à des magnitudes V < 22, la complétude globale à ce limite en magnitude est de 45%.
Ce catalogue représente un des plus large relevés spectroscopique actuellement disponible sur un amas à z >
∼ 0.2 est le plus important à z ∼ 0.4. Le
catalogue donne aussi de données photométriques et spectroscopiques, dont
les largeurs équivalents de cinq raies importantes, ainsi que la magnitude de
la cassure à 4000 Å, ce qui est important pour la détermination de l’activité
de formation d’étoiles dans ces galaxies. Le catalogue représente une unique
base de données pour des investigations de la structure d’un amas à redshift
intermédiaire et son environnement.
Le résultat principal de cette étude est que Cl0024+1654 n’est pas un simple
amas isolé, comme l’avait été supposé auparavant dans l’interprétation des
données cinématiques, lensing et en rayon X. En revanche, la distribution des
redshifts révèle la présence d’un deuxième amas, moins massif, projeté sur le
centre de l’amas principal. La séparation de Cl0024+1654 en deux composantes
a des conséquences importantes pour l’interprétation des données observationnelles, et permet en particulier la résolution du problème des différences
de masses déterminées par des méthodes indépendantes (Soucail et al. 2000).
Dans le chapitre 8 j’ai analysé en détail la distribution des galaxies dans
Cl0024+1654, m’appuyant sur les redshifts et positions projetés de ∼ 300 galaxies. La structure tri-dimensionnelle est complexe, avec une distribution fortement asymétrique dans les parties centrales, et deux composantes bien séparées à des distances projetées plus importantes, jusqu’à ∼ 3 h−1 Mpc.
J’argue que la queue bleue de la distribution centrale des vitesses et la
composante en arrière-plan sont originaire du même système physique, et j’interprète la distribution particulière des redshifts comme le résultat d’une collision à grande vitesse de deux amas de galaxies, l’axe de la collision étant parallèle à la ligne de visée. Utilisant des simulations numériques de Ben Moore
162
Résumé
163
je montre que ce scénario peut effectivement expliquer et reproduire la distribution observée des redshifts, avec un rapport de masses des deux amas
d’environ 2 :1.
Les simulations produisent de même une distribution de masse projetée
très proche de celle qui a été déduite par Tyson et al. (1998) à partir d’une
modélisation du système quintuples d’arcs géants vu dans Cl0024+1654. Puisqu’il est possible de produire une distribution de masse avec un cœur plat par
une collision de deux halos CDM (donc sans cœur) ce résultat élimine un de
arguments majeurs contre le modèle de matière noire froide comme la composante dominante dans des amas de galaxies (Spergel & Steinhardt 2000).
La distribution des types spectraux des galaxies dans la composante d’arrière-plan est plus proche de celle de la population du champ que de celle
d’une population de galaxies d’amas. Dans le cadre du scénario de collision,
ces galaxies correspondent à des galaxies des parties extérieures de l’amas secondaire, qui ne sont plus gravitationnellement liées après l’impact. Même au
début ces galaxies ne correspondaient donc pas à une population de galaxies
d’amas complètement transformées. En plus, l’impact pourrait avoir initié une
épisode supplémentaire de formation d’étoiles dans ces galaxies. Cl0024+1654
était un des premiers amas distant dans lesquels l’effet de Butcher–Oemler fut
détecté (Butcher & Oemler 1984; Dressler et al. 1985) ; le fait que ∼ 40% des galaxies brillantes dans l’amas sont des galaxies présentant des raies d’émission
([O II]> 5Å) a une interprétation naturelle dans le scénario de collision. La distribution spectrale des galaxies dans la composante A (et dans le centre) est
aussi perturbée par la collision, avec un possible excès de rHDS et SF/SBB (tableaux 8.1) ; les galaxies rHDS peuvent être interprétées comme les résultats
d’un sursaut de formation d’étoiles initié en début de l’interaction des deux
amas.
La vitesse relative des amas de ∼ 3000 km s−1 , impliquée par la distribution
des redshifts dans Cl0024+1654, est élevée. Des observations et des simulations trouvent des vitesses particulières des amas d’ordre 500 km s−1 (Bahcall
& Oh 1996; Giovanelli et al. 1998; Gibbons et al. 2001; Colberg et al. 2000a).
Tout de même des amas en collisions peuvent atteindre des vitesses d’ordre
3000 km s−1 à des séparations d’environ 1 Mpc (Sarazin 2001). Markevitch et al.
(2001) ont récemment observé un choc dans l’amas 1E0657-56 qui implique
une vitesse relative de 3000 à 4000 km s−1 pour cette collision. Cl0024+1654 est
un système extraordinaire mais pas impossibles.
Outre l’amas même, le survey spectroscopique revèle un group binaire de
galaxies à z ' 0.495, avec un candidat d’arc gravitationnelle, ainsi qu’une surdensité de galaxies à z ∼ 0.18 qui pour l’instant ne peut pas être identifiée à
une structure effondrée. Il n’est pas exclu qu’une telle structure existe hors du
champ du relevé spectroscopique. Aucune des structures dans la distribution
des redshifts semble pouvoir expliquer le signal cohérent de cisaillement à ∼ 60
au nord-est du centre de Cl0024+1654, détecté par Bonnet et al. (1994).
164
Chapitre 9. Conclusions
D’autres études sur Cl0024+1654 sont envisagées pour l’avenir. Pendant le
deuxième semestre 2001, j’ai obtenu des nouvelles images profonde avec la
caméra CFH12k sur le CFHT, prises dans des excellentes conditions de seeing
en B, V, R et I. Les images ont été réduite avec les procédures décrites dans le
chapitre 12. Une analyse weak lensing de ces images permettra de mesurer la
pente du profil de masse au-delà du rayon viriel nominatif, ce qui fournit un
teste de la prédiction d’un profil plat du scénario de collision. Cette analyse
fournira aussi une carte de la distribution de masse dans le champ, ce qui
permettra notamment de confirmer ou refuter la réalité de la structure “noire”
vu par Bonnet et al. (1994).
De la recherche supplémentaire est conduite dans le cadre d’une collaboration internationale, qui tourne autour d’un grand mosaı̈que de 38 images
HST/WFPC2. Ceci va aussi être utilisé pour une analyse weak lensing, mais
aussi pour mesurer des morphologies des galaxies et la dépendance du mélange
morphologique de la distance du centre de l’amas. L’interprétation de ces données devra prendre en compte le fait que Cl0024+1654 n’est pas un simple
amas. Des données spectroscopiques supplémentaires sont en train d’être acquises sur les télescopes Hale 5m et Keck, remplissant des régions pas couvertes par le survey présenté ici.
La vitesse relative demandée par le scénario de collision est plutôt élevée.
J’envisage d’investiguer la distribution statistique de vitesses relatives dans
des collision vues dans des simulations cosmologiques en collaboration avec
Ben Moore. Ceci va donner une idée de la vraisemblance d’une telle collision
à grande vitesse. Puisque des collision sont en mesure de biaiser des relevés
d’amas en X à cause de l’augmentation de la luminosité en présence des chocs
de fusion, cet effet doit être étudié statistiquement pour s’assurer que le conclusions cosmologiques tirées des surveys d’amas sont correctes.
Part III
A panchromatic survey of X-ray
luminous clusters at redshift 0.2
Chapter 10
Introduction
In Part I of this dissertation I have tried to explain the significance of clusters of galaxies for current cosmological research. The mass function of rich
clusters and in particular its evolution with redshift hold important clues to
the evolution of the Universe as a whole and provide strong leverage for the
determination of cosmological parameters, in particular ΩM and σ8 . The existence of a universal mass profile for dark matter halos on scales ranging from
dwarf galaxies to rich clusters of galaxies is a strong prediction from current hierarchical structure formation models and observational confirmation or refutation of this prediction is of prime importance to evaluate the assumptions
that go into the models, e. g. concerning the nature of the dark matter.
The most robust theoretical predictions are expressed in terms of mass,
which is unfortunately difficult to observe directly1 . Predictions of observable
properties of clusters (X-ray temperatures and surface brightness distribution,
galaxy kinematics) have to include baryons and therefore involve much more
complicated physics than the relatively simple gravitational physics modelled
in numerical simulations, for example, and its theoretical treatment is usual
fraught with assumptions and simplifications, the validity of which has to be
confirmed by comparison with detailed observations.
Here I will present an ongoing ambitious project to investigate a homogeneously selected sample of a dozen massive clusters in a small redshift range
around z = 0.2 in as complete a manner as possible, assembling homogeneous data sets in a variety of wavebands and on a wide range of length
scales. The project is led by Jean-Paul Kneib, Harald Ebeling and Ian Smail
in collaboration with their respective graduate students. The corner stones are
high-resolution images from HST/WFPC2 of the cluster centres, wide-field
multicolour imaging with the CFH12k camera on CFHT and X-ray spectroscopic and imaging observations with XMM/N EWTON. These data are com1 Gravitational lensing of course measures mass directly, however this is projected mass and
attribution to collapsed objects is not always straightforward. Cl0024+1654 (Part II) provides
an example.
167
168
Chapter 10. Introduction
plemented with optical spectroscopic observations, trying to collect a large
number of redshifts for cluster members to study their dynamical state and to
obtain redshifts for giant arcs seen in the cluster centres, which is essential for
an absolute mass calibration of the strong lensing models.
The main scientific goals are measurements of the mass profiles of the clusters on scales ranging from ∼ 10 h−1 kpc (strong lensing modelling from HST
images) to beyond the virial radius (weak lensing mass reconstruction from
CFH12k images) which can then be compared to theoretically predicted profiles, most importantly the NFW profile (Chapter 3). With observations in a
variety of wavebands it will be possible to accurately calibrate the relations
between cluster mass and other observables, such as the X-ray temperature
or the galaxy velocity dispersion. An important aspect here is that our mass
estimates, being derived from gravitational lensing analyses, will be strictly
independent of the other observables; most previous studies of the M–TX relation, for example, derived masses from modelling of the X-ray emission, i. e.
essentially from the same data as the temperature measurements. It should
be noted that our sample covers only a fairly small range of masses and will
therefore not be sufficient to measure the slope of the M–TX relation; on the
other hand it will give an accurate calibration of the relation at the high mass
end, and in particular allow the investigation of the scatter around the relation
and the causes (substructure, mergers) for the scatter.
The scientific value of the survey is not limited to the clusters themselves;
for example near-infrared imaging of the cluster centre has already produced
an impressive new sample of about 60 Extremely Red Objects (EROs) at higher
redshifts, which provides important clues to galaxy formation and evolution
(Smith et al. 2002a,b).
The project is pretty much ongoing, and not many results have been published yet. Smith et al. (2001) present a strong lensing model for Abell 383, and
lensing models for the remainder of the sample are in preparation. XMM data
have been obtained for only three clusters so far and an analysis of Abell 209
is forthcoming (Marty et al., in preparation). My own task in this project was
the reduction of the CFH12k data obtained during three observing runs between February 1999 and June 2000. In Chapter 12 I shall describe in detail the
methods I have used and present a few characteristics of the resulting images.
In the remainder of this chapter I shall describe how the sample was selected and discuss in more detail the observations obtained so far. In Chapter
11 I will introduce the individual clusters in the sample and present some (preliminary) analyses of the redshift data obtained for several of them.
10.1. Sample Selection
10.1
169
Sample Selection
The aim of the project is to study a sample of the most massive clusters of
galaxies at redshift z ≈ 0.2. Since homogeneous mass measurements are (of
course!) not yet available the sample was selected by X-ray luminosity. Smail
et al. (1997) compared the tangential shear profiles of a sample of clusters with
their X-ray luminosity and found a clear (although noisy) correlation. X-ray luminosity thus provides an accurate criterion to select the most massive clusters
(temperature is in principle more directly related to cluster mass but homogeneous temperature measurements for large catalogues of clusters were not yet
available, when the project was conceived).
We choose to study massive clusters for a variety of reasons. The most massive clusters can be seen at any redshift where clusters can be detected at all
and in particular samples of high-redshift clusters will be dominated by massive systems. In order to make best use of high-redshift catalogues for cosmology the properties of massive clusters must be well-known. Massive clusters
generate a strong lensing signal, so that the mass distribution can be accurately
modelled.
We restrict ourselves to a small redshift slice in order to minimise the impact of different cluster ages within the sample. Similar samples in other redshift ranges are envisaged for the future and partially already begun, which
can be combined with the z = 0.2 sample to study the evolution of the high
mass end of the cluster population in some detail. Even if the base catalogue
is actually flux-limited, the sample will be effectively luminosity-limited if it extends over a small redshift range. The redshift slice at z = 0.2 was chosen since
a lens redshift around 0.2 maximises the lens efficiency for a source population
with mean redshift hzs i ∼ 0.8 (Natarajan & Kneib 1997).
The sample selection is based on the XBACs catalogue (X-ray brightest
Abell clusters) of Ebeling et al. (1996), an X-ray flux-limited catalogue of Abell
clusters detected in the Rosat All Sky Survey. XBACs is not, strictly speaking,
a purely X-ray selected catalogue, since it is based on the optically selected
Abell catalogue (Abell 1958) and might therefore be subject to the usual selection effects and biases inherent to the Abell catalogue. In particular, XBACs is
somewhat incomplete at high redshift due to the incompleteness of the Abell
catalogue at z >
∼ 0.25. The successor of XBACs, the Bright Cluster Survey (BCS,
Ebeling et al. 1998) is purely X-ray selected, and the fact that more than 80% of
the BCS clusters are Abell clusters indicates that this is not a major problem.
We applied in addition the following, more practical selection criteria:
• Declination −20◦ < δ < 60◦ to ensure that the cluster field can be observed at small zenith distance from CFHT on Mauna Kea (geographical
latitude 19◦ 490 3600 )
• galactic latitude |b| > 20◦ to minimise contamination of the field by stars
170
• galactic hydrogen column density NH < 10 × 1021 cm−2 in order to obtain accurate temperature measurements with XMM.
The redshift range adopted is 0.18 < z < 0.26. There are 22 XBACs cluster
in this range, seven of which are excluded by their declination, galactic latitude
or the hydrogen column density in their direction. Due to time constraints,
only 11 clusters in this range could be included in the sample. Abell 2218 is
outside the strict redshift range, however since high-quality HST/WFPC2 data
are available from the HST archive and it is an interesting and well-studied
cluster in its own right we chose to include it in the sample, bringing the sample size to 12. The location of the sample within the XBACs catalogue is shown
in Fig. 10.1 which plots X-ray luminosity versus redshift.
Figure 10.1: Luminosity distribution of the XBACs catalogue (Ebeling et al.
1996). The solid line marks the flux limit of the catalogue of 4.4 ×
10−12 erg cm−2 s−1 in the 0.1 . . . 2.4 keV band. The conversion from flux to luminosity assumes ΩM = 1, ΩΛ = 0. The redshift limits of our sample are indicated by the dashed lines. Triangles mark the members of our sample (Abell
2218 is not within the redshift limits but is observed in the same as the other
clusters). Other clusters within the redshift range were excluded for the reasons indicated by the respective symbols.
Physical properties for the clusters in our sample are listed in Table 10.1.
Redshifts, X-ray luminosities and (estimated) temperatures are taken from (Ebeling et al. 1996) or (Ebeling et al. 1998). The last column lists ASCA measurements of the cluster temperatures (from Ota 2001). One of the main goals of
our project is to replace the X-ray data with more accurate measurements with
171
10.1. Sample Selection
XMM/N EWTON. These observations should also provide us with temperature
profiles.
R OSAT/HRI maps are available for all clusters and show a wide variety of
X-ray morphologies. Abell 383 and Abell 1835 clearly contain cooling flows
visible as a sharp surface brightness peak in the centre. Other clusters, in
particular Abell 1763, Abell 2219 and Abell 209 have more extended surface
brightness distributions.
Table 10.1: Physical properties of the cluster sample. The redshifts and Xray luminosities for A 209, A 383 and A 1689 are taken from the XBACs catalogue (Ebeling et al. 1996), the remainder from the BCS catalogue (Ebeling et al.
1998). The temperatures listed in column 6 are from the same references and
are (with the exception of A 2218) estimated from the LX –TX relation. The temperatures listed in column 7 are new ASCA temperature measurements taken
from Ota (2001).
Cluster
RA
Dec
z
LX
1044
h−2
erg s−1
TX,est
TX,ASCA
keV
keV
(J2000)
(J2000)
A 68
00h 37m 06.s 85
+09◦ 090 24.00 3
0.2546
3.72
10.0
6.93+0.63
−0.59
A 209
01h 31m 52.s 55
−13◦ 360 40.00 4
0.2060
3.44
9.6
—
A 267
01h 52m 41.s 97
+01◦ 000 25.00 8
0.2300
3.43
9.7
5.51+0.44
−0.41
A 383
02h 48m 03.s 39
−03◦ 310 45.00 2
0.1871
2.01
7.5
—
A 773
09h 17m 56.s 31
+51◦ 430 20.00 8
0.2170
3.27
9.4
8.07+0.70
−0.66
A 963
10h 17m 03.s 64
+39◦ 020 49.00 8
0.2060
2.61
8.6
6.83+0.51
−0.51
A 1763
13h 35m 20.s 08
+41◦ 000 04.00 1
0.2279
3.73
10.0
8.11+0.66
−0.63
A 1835
14h 01m 02.s 08
+02◦ 520 42.00 4
0.2528
9.63
14.8
7.42+0.61
−0.43
A 1689
13h 11m 30.s 06
−01◦ 200 28.00 2
0.1840
5.18
10.8
9.31+0.45
−0.38
A 2218
16h 35m 51.s 52
+66◦ 120 15.00 2
0.1710
2.33
6.7
7.63+0.58
−0.49
A 2219
16h 40m 19.s 86
+46◦ 420 41.00 4
0.2281
5.10
11.4
9.22+0.74
−0.59
A 2390
21h 53m 36.s 86
+17◦ 410 43.00 2
0.2329
5.36
11.6
9.21+1.37
−1.04
172
10.2
Observations
The observational corner stones of the project are high-resolution imaging with
HST/WFPC2, panoramic multicolour imaging with CFH12k on CFHT and Xray spectroscopy and imaging with XMM/N EWTON (Table 10.2).
We obtained observing time with HST/WFPC2 for eight clusters from our
sample, for the remaining clusters (Abell 1689, 2218, 2219, 2390) images are
available in the HST archive. The images were taken through the F702W
filter which roughly corresponds to the Johnson R band; effective exposure
times are 7500 sec, corresponding to three orbits. The final frames have a pixel
scale of 0.00 05, an effective resolution of 0.00 15 and a 1σ detection limit within
the seeing disk of R702 ' 31 (the data reduction procedure is described in
Smith et al. 2001). The excellent resolution of the WFPC2 images permits a
detailed investigation of the structure of giant arcs and identification of multiply imaged background sources. The drawbacks of the HST observations
are the limited field of view (three chips of 8000 × 8000 each, corresponding to
170 × 170 h−2 kpc2 at z = 0.2) and the availability of images taken through
only one filter. The main use of the HST images is for the reconstruction of the
central mass distribution from strong lensing modelling (Smith et al. 2001).
The missing colour information is to some extent provided by the CFH12k
images, which also cover a much wider field of 42 × 28 arcmin2 , corresponding
to 5.3 × 3.5 h−2 Mpc2 at z = 0.2. The CFH12k camera is described in more detail
in Sect. 12.1, the data reduction procedures are the subject of Chapter 12.
The first observing run comprised one night only in February 1999. The atmospheric conditions were good, photometrically excellent, although the seeing conditions were only moderate by Mauna Kea standards, with FWHM for
star images of order 100 . This was the first semester that the CFH12k camera
was available on CFHT, and the chips were still in a preliminary configuration.
In particular some of the chips at the edges of the focal plane were of inferior
quality with very strong fringing in the I band and many cosmetic defects.
The centre of the focal plane, however, was of similar quality as in the final
configuration. Images were taken through the I and V filters, since the B filter
was not yet available.
Between the first and second observing runs the focal plane of the CFH12k
camera was reorganised, and the final chip configuration had overall excellent cosmetic properties, and significantly reduced fringing in the I band. The
second observing run of three nights in November 1999 had excellent photometric and seeing conditions, with FWHM for stellar images down to 0.00 6 in
the I band. Seeing conditions were a little worse for the three nights of the
third observing run in May/June 2000, although at 0.00 8 FWHM in the I band
still very good. In particular the second night of this run was, unfortunately,
not photometric with probably some thin cirrus at high altitude which led to
significant fluctuations in the registered signal between the individual expo-
173
10.2. Observations
Table 10.2: Observing logs for the HST, CFH12k and XMM observations. The
HST column lists the dates when the data were taken, the CFH12k column
identifies the observing run (see Table 10.3), the XMM column the scheduling
category (A1689 is observed by a different PI; GTO = guaranteed time observation) and the total exposure time accorded to each cluster. So far only three
clusters have been observed with XMM.
HST
A 68
A 209
A 267
A 383
A 773
A 963
5 Nov 1999
20 Nov 1999
12 Nov 1999
25 Jan 2000
12 Feb 2000
7 May 2000
A1763
A1835
29 Jun 2000
10 Jun 2000
A1689
A2218
A2219
A2390
archive
archive
archive
archive
CFH12k
XMM
Nov 99: BRI
Nov 99: BRI
Nov 99: BRI
Nov 99: BRI
—
Feb 99: V I
Nov 99: BRI
May 00: BRI
Feb 99: V I
May 00: R
B, 30 ksec
B, 20 ksec
B, 25 ksec
B, 30 ksec
B, 20 ksec
B, 25 ksec
May 00:
May 00:
May 00:
May 00:
BRI
BRI
BRI
BRI
B, 20 ksec
Calibration
(Jun 2000)
B’, 30 ksec
GTO, 40 ksec
GTO, 30 ksec
GTO, 20 ksec
sures. The images are still perfectly usable for weak lensing analyses, only the
photometric calibration is somewhat more uncertain than for the other runs
(Sect. 12.5).
Table 10.3 gives the observing log for the three observing runs at CFHT,
table 10.4 gives the basic characteristics of the final CFH12k images for the
sample. Unfortunately, no CFH12k could be obtained of Abell 773 which was
outside the observable right ascension ranges during the three observing runs.
At the time of writing (June 2002), XMM observations have been obtained
for 8 of the 12 cluster samples. The data reduction is in the early stages, the
method and application to Abell 209 will be described by Marty et al. (2002, in
preparation).
174
Table 10.3: CFH12k observing log. Data were taken during three observing
campaigns of seven nights in total. Conditions were excellent throughout the
Feb 1999 and Nov 1999 runs. During the May 2000 run, the image quality
was very good, however the conditions were non-photometric, in particular
during the second night. For the Nov 1999 run I also list images taken for
other projects. The V band image of Cl0024+1654 is described in Sect. 7.1.1.
Dates
Observers
Clusters
Bands
Feb 99: 23/24 Feb 1999
Ebeling/Czoske
A 963
A 1835
A 586
VI
VI
VI
Nov 99: 14–17 Nov 1999
Czoske/Smail/Smith
A 68
A 209
A 267
A 383
A 963
Cl0024+1654
A 851
Cl0818
Cl0819
BRI
BRI
BRI
BRI
BRI
V
I
BR
R
May 00: 30 May – 2 Jun 2000
Kneib/Czoske/Bridle
A1689
A1763
A1835
A2218
A2219
A2390
BRI
BRI
R
BRI
BRI
BRI
175
10.2. Observations
Table 10.4: Summary of CFH12k observations of the cluster sample. The table
lists the number of exposures, the total exposure time and the full width at half
maximum (FWHM) in the final combined image for each cluster field in each
filter. V band images were only obtained during the February 1999 observing
run. For Abell 963 we obtained I band image during both the February 1999
and November 1999 runs.
B
A 68
A 209
A 267
A 383
A 963
A 1689
A 1763
A 1835
A 2218
A 2219
A 2390
V
Nexp
ttot
FWHM
9
8
5
8
8
4
4
8100
7200
3000
7200
7200
3600
3600
—
3378
5400
2700
1.00 1
1.00 0
1.00 0
0.00 9
0.00 9
0.00 9
1.00 0
4
5
3
Nexp
5
5
1.00 1
1.00 0
1.00 1
R
Nexp
ttot
—
—
—
—
3600
—
—
3750
—
—
—
FWHM
1.00 0
0.00 8
I
ttot
FWHM
Nexp
ttot
FWHM
6
6
3
6
10
5
5
5
5
5
5
6
3600
3600
900
3600
7500
3000
3000
3000
3750
3000
3000
3600
0.00 6
0.00 7
0.00 7
0.00 7
1.00 1
0.00 7
0.00 9
0.00 8
0.00 8
0.00 8
0.00 8
0.00 9
A 68
A 209
A 267
A 383
A 963
12
12
8
10
8
7200
6600
4800
6000
4800
0.00 7
A 1689
A 1763
A 1835
A 2218
A 2219
A 2390
5
7
6
8
7
8
3000
6000
5400
6900
6300
5700
0.00 8
0.00 9
0.00 7
1.00 0
0.00 8
0.00 7
0.00 7
0.00 7
0.00 9
0.00 8
Introduction
Dans la première partie de cette thèse j’ai essaié d’expliquer l’importance
des amas de galaxies pour la recherche actuelle en cosmologie. La fonction des
amas riches et particulièrement son évolution avec le redshift tiennent d’informations importantes sur l’évolution de l’univers et pour la détermination des
paramètres cosmologiques, notamment ΩM et σ8 . L’existence d’un profil universel de masse dans des halos de matière noire froide est une forte prédiction
des modèles de formation hiérarchique des structures à grande échelle, et la
confirmation ou refutation de cette prédiction est important pour évaluer les
hypothèse à la base de ces modèles, notamment la nature de la matière noire.
Les prédictions théoriques sont exprimé en termes de masse, qui est difficilement observable directement. Des modèles théoriques du comportement des
composantes observables, galaxies, gaz chaud, sont beaucoup plus difficiles et
donc moins sûrs que ceux de la matière noire qui est soumis seulement aux interactions gravitationnellements. Il est donc essentiel de calibrer les relations
entre observables et masses des amas observationnellement.
Dans cette partie je présente un projet ambitieux qui a pour but d’étudier en
détail un échantillon de douze amas massifs dans une fine tranche en redshift
autour de z = 0.2, sélectionné de fa con homogène. Le projet, mené par JeanPaul Kneib, Harald Ebeling et Ian Smail, comprends des observations en imagerie à haute résolution avec HST/WFPC2 (sur les centres des amas), en imagerie dans plusieurs bandes sur des grands champs avec la caméra CFH12k
sur le CFHT, et imagerie et spectroscopie en rayons X avec XMM/N EWTON.
Ces données sont supplémentées par des observations spectroscopiques optiques pour amasser des nombreux redshifts de galaxies des amas, afin d’étudier leurs états dynamiques, et aussi des redshifts des arcs géants, ce qui est essentiel pour la calibration absolue des masses déduites par des modèles strong
lensing.
Les buts principaux du projets sont donc la reconstruction des profils de
masse sur des échelles de ∼ 10 h−1 kpc au-delà du rayon viriel pour comparaison avec le profil “universel” de NFW, ainsi que la calibration de la relation
M–TX , avec des masses issues des analyses lensing, donc indépendantes des
176
Résumé
177
mesures de températures en X. Outre ces buts principaux le projet a déjà produit un échantillon impressionnant d’environ 60 objets extrêmement rouges
(EROs) à redshift plus élevé, publié dans Smith et al. (2002a,b).
Le projet est travail en cours. A présent, des résultats ont été publié sur un
modèle du strong lensing par l’amas Abell 383, publication des modèles sur
les autres amas est en préparation. L’analyse des données en X sur Abell 209
est également en préparation (Marty et al. 2002), mais pas beaucoup d’observations avec XMM/N EWTON ont été acquises à présent. Mon devoir dans le
cadre de ce projet était l’acquisition et la réduction des images CFH12k, obtenues pendant trois campagnes d’observation au CFHT entre février 1999 et
juin 2000. Les procédures de réduction sont décrites en détail au chapitre 12 de
cette thèse. Une série de publications est en préparation détaillant la réduction
et analyse weak lensing de ces images.
Chapter 11
Notes on Individual Clusters
In this chapter I will present the individual clusters of our sample. For all the
clusters a subsection of the CFH12k R band image1 obtained during the course
of this project will be presented and the central morphology described. Where
available, a (preliminary) analysis of redshifts obtained for the cluster will be
presented, including biweight estimates of the central redshift and velocity
dispersion (Beers et al. 1990); the 1σ errors were estimated by bootstrap resampling. For many clusters these redshift samples are the first ever obtained.
The Dressler-Shectman test (Sect. 4.4.2) as well as plots of redshift versus distance from the projected centre (as in Fig. 8.2 for Cl0024+1654) will be used to
look for the presence of substructure. I will also cite a few interesting observations on these clusters, although this is by no means intended to be a complete
overview of the literature.
11.1
Abell 68
Abell 68 has not been studied in any detail so far. The XBACs catalogue
(Ebeling et al. 1996) only listed an estimated redshift (from the magnitude
of the 10th brightest galaxy) of z = 0.18. The subsequent BCS catalogue
(Ebeling et al. 1998) listed the first measured redshift of z = 0.2546. For this
project we obtained redshifts of 11 cluster members, giving a mean redshift of
z = 0.2513 ± 0.0010. The number of available redshifts is too small to allow
an accurate estimate of the velocity dispersion (formally σ = 808 km s−1 for
eleven galaxies). Also, application of the Dressler-Shectman test is not meaningful, so that Fig. 11.2 gives the positions of the galaxies with measured redshifts more than anything else.
Abell 68 contains a large cD galaxy elongated in the NW–SE direction (Fig.
11.1). About one arcmin to the north-west of the cD lies a compact group
1 The
characteristics of these images as well as the data processing techniques will be described in the following chapter.
178
11.1. Abell 68
179
of about five bright galaxies, so that the cluster gives a bimodal appearance.
Several blue arclets can be seen around the cluster centre, most notably a triple
arc just east of the centre of the cD galaxy. This arc actually consists of three
parallel curved arcs of which the middle one is significantly redder than the
outer ones. The source for these arcs might be a spiral galaxy with the red arc
corresponding to its bulge and the blue arcs to star-forming regions in its disk.
Figure 11.1: Central 5 × 5 arcmin2 of the CFH12k R-band image of Abell
68. This corresponds to 750 × 750 pixels of the total 12000 × 8000 (roughly)
CFH12k mosaic.
180
Chapter 11. Notes on Individual Clusters
Figure 11.2: Spectroscopic data for Abell 68
11.2
Abell 209
Abell 209 is dominated by a cD galaxy that is elongated in the NW–SE direction. There are no obvious giant arc systems. The redshift of the cluster,
z = 0.2060 is based on only two galaxy redshifts (Struble & Rood 1999); we
have not yet obtained any additional redshifts. Abell 209 was the first cluster
for which we obtained XMM data, the X-ray data analysis will be published
by Marty et al. (in preparation).
11.3
Abell 267
Abell 267 is very similar in appearance to Abell 209; again, the cluster is dominated by a giant cD galaxy, elongated in the NNE–SSW direction. There are
no giant arcs and no obviously lensed background galaxies. The previously
listed cluster redshift of z = 0.2300 was based on one galaxy redshift (Struble
& Rood 1999). We have obtained spectra for 151 objects out to ∼ 30000 from the
cD galaxy, of which 74 belong to the cluster (cluster membership is defined by
the – somewhat arbitrary – redshift cuts indicated in the left panel of Fig. 11.5).
The mean redshift of these 74 galaxies is z = 0.2269 ± 0.0006, the velocity dispersion is σ = (1125 ± 95) km s−1 . The Dressler–Shectman test (Fig. 11.5, right
panel) shows no indication for substructure.
11.3. Abell 267
181
Figure 11.3: Central 5 × 5 arcmin2 of the CFH12k R-band image of Abell 209.
182
183
11.4. Abell 383
11.4
Abell 383
Abell 383 (Fig. 11.6) is dominated by a nearly circular cD galaxy and shows a
rich and complex system of giant arcs and arclets which were used by Smith
et al. (2001) to reconstruct the central mass distribution of the cluster (Fig. 11.7).
Five multiple-image systems were identified in the cluster centre, three more
candidate systems are tentative due to their faintness. Two radial arcs are seen
in the halo of the cD galaxy, these are interpreted as counter-images to the
giant arcs B0 (which has a spectroscopic redshift of 1.01) and B1 (redshift 1.1,
estimated from the lens model). The image systems B2 and B3 are found at
larger distance from the cD galaxy and therefore lie at higher redshift (∼ 3).
Both these systems are highly perturbed by individual cluster galaxies.
The ratio of the angular positions of radial and tangential images of the
same source constrains the slope of the lensing mass profile at the position of
the radial arcs (Williams et al. 1999). In Abell 383, the (deprojected) slopes of
the density profile were found to be α ∼ −1.9 at the inner, and α ∼ −1.3 at
the position of the outer radial arc, consistent with the slope estimates from
the lens model. Surprisingly, the density profile steepens in the cluster centre!
The excess mass in the centre is attributed to the cD galaxy; when radiative
cooling becomes important (Abell 383 contains a cooling flow with a mass
−1
deposition rate Ṁ >
∼ 200 M yr ), baryons drop to the centre and contribute
significantly to the total central mass density. Mass profile predictions from
184
numerical simulations (Navarro et al. 1997; Moore et al. 1998) only take the
dark matter component into account and therefore underestimate the central
mass density if baryons are present.
The lens model comprises four major components. A cluster-scale component is predicted to have a velocity dispersion of σ0 = 920 km s−1 , a core
radius of 24 h−1 kpc and an axis ratio of 1.13. A component associated with the
cD galaxy has to be included to account for the positions of the radial arcs; this
component has a velocity dispersion of σ = 250 km s−1 and a profile consistent
with an isothermal sphere. The slope of the total mass distribution inside the
core radius of the cluster scale component, but outside the region where the cD
component dominates, is −1.3 which is between the central slopes predicted
by Navarro et al. (1997) and Moore et al. (1998) respectively. The projected
mass within a radius of 32 h−1 kpc is (1.75 ± 0.05) × 1013 h−1 M from the lens
model, in good agreement with the mass within the same radius determined
from a β model of the surface brightness distribution.
The paper (Smith et al. 2001) is included in the appendix of this dissertation.
We have obtained 41 redshifts for cluster members in Abell 383 (Fig. 11.8,
left panel). The mean redshift is z = 0.1896 ± 0.0007, the velocity dispersion
−1
is σ = 1167+130
−166 km s . Previously, the cluster redshift was based on a single
redshift measurement. Neither the z–R plot not the Dressler-Shectman test
(Fig. 11.8, right panel) show signs of substructure in Abell 383.
11.5
Abell 773
Unfortunately, no CFH12k image exists for Abell 773 because it was only marginally visible during our observing runs. The HST/WFPC2 image of this
cluster reveals a bimodal structure with many straight arcs and arclets across
the bridge between the two cluster centres. The cluster has a high fraction of
late-type galaxies. The field-of-view of the WFPC2 is rather too small for this
interesting and extended cluster, and wide-field images should be acquired in
the future.
We have obtained redshifts for 122 cluster galaxies (Fig. 11.9, left panel).
The mean redshift is z = 0.2190 ± 0.0007 (previously, the cluster redshift was
based on two galaxy redshift measurements), the velocity dispersion is found
−1
to be high, σ = 1594+117
−62 km s . However, the Dressler-Shectman plot reveals
a group of galaxies to the east of the cluster centre that seems to be dynamically
distinct from the other galaxies. Another distinct group can be seen to the west.
I apply the Dressler-Shectman test only to galaxies that belong to the cluster
(defined by the redshift cuts indicated in the left panel of Fig. 11.9), so all these
galaxies are included in the velocity dispersion estimate. Also, the bimodality
of the cluster centre seen in the WFPC2 image indicates that the cluster is not
11.5. Abell 773
185
The brighter circular structure in the lower half of the image is due to reflected
light from a bright star in the field-of-view.
186
Figure 11.7: HST/WFPC2 image of the centre of Abell 383, showing the a multitude of giant arcs, arclets and multiply imaged background sources. The
image systems B0, B1, B2 and B3 were used to model the mass distribution
(shown as contours overlaid on the image) in the cluster centre; the model includes a global cluster component as well as contributions from the numbered
cluster galaxies. Two radial arcs, B0b and B1d are seen behind the cD galaxy.
The figure is taken from Smith et al. (2001).
187
11.6. Abell 963
relaxed and that the velocity dispersion might be biased high to an ongoing
merger.
Abell 773 is known to contain a diffuse radio halo and relic sources (Kempner & Sarazin 2001; Govoni et al. 2001). A high-resolution Sunyaev-Zeldovich
map of the cluster has been obtained by Carlstrom et al. (1996).
11.6
Abell 963
The centre of Abell 963 is dominated by a cD galaxy, and contains two giant
arcs to the north and the south of the cD respectively (Lavery & Henry 1988).
The northern arc is redder than the southern arc, so that these probably belong
to two distinct source galaxies. Ellis et al. (1991) measured a redshift for the
northern arc of z = 0.771.
Struble & Rood (1999) list a mean redshift of z = 0.2048 and a velocity dispersion 1350 km s−1 based on 36 redshifts. We have obtained redshifts for 70
cluster members (Fig. 11.11), and our mean redshift z = 0.2041 ± 0.0008 and
−1 are in good accord with the values
velocity dispersion σ = 1412+100
−80 km s
listed in Struble & Rood (1999). The Dressler-Shectman test provides marginal
evidence for substructure in this cluster: In 78 out of 1000 Monte Carlo reshufflings of the data a value of the global Dressler-Shectman statistic ∆ larger than
the observed value was found.
188
11.7
Abell 1689
Abell 1689 has been studied extensively in the past. It is a very rich and extremely luminous cluster, yet it does – like Cl0024+1654 – not contain a central
cD galaxy, but has a compact group of normal elliptical galaxies at its centre.
About 1 arcmin to the north-east of the central galaxy group a second, looser
group can be seen. The cluster centre contains a host of arcs and arclets (too
faint to be seen in Fig. 11.12).
Struble & Rood (1999) list a mean redshift z = 0.1832 and an extremely high
velocity dispersion σ = 1989 km s−1 based on 66 galaxy redshifts. We have obtained 211 redshifts of cluster members, the mean redshift is z = 0.1853+0.0004
−0.0008 ,
+38
−1
the velocity dispersion is σ = 1976−56 km s . The velocity–radial distance
plot (Fig. 11.13, left panel) shows a very wide, seemingly homogeneous distribution of redshifts ranging over nearly 104 km s−1 . The Dressler-Shectman test
(Fig. 11.13, right panel) shows a clearly distinct group to the north-east of the
cluster centre, which seems to be associated with the group of galaxies seen in
Fig. 11.12. Substructure in Abell 1689 was already identified by Girardi et al.
(1997), so the measured velocity dispersion probably contains a systematic velocity component due to an ongoing cluster merger.
Gravitational lensing has been studied extensively in Abell 1689. Clowe &
Schneider (2001) and King et al. (2002) present weak shear measurements in
the cluster which are well fitted by an isothermal sphere with velocity disper-
11.7. Abell 1689
189
190
sion ' 1030 km s−1 . Miralda-Escudé & Babul (1995) model the arc configuration with two isothermal sphere components of 1400 km s−1 and 700 km s−1 respectively. Gravitational magnification effects (depletion) have been observed
by Taylor et al. (1998) and Dye et al. (2001), which seem to require a larger
velocity dispersion, ∼ 2200 km s−1 .
11.8
Abell 1763
Abell 1763 has a central cD or giant elliptical galaxy but otherwise the cluster
centre is comparatively ill-defined, with “chains” of bright galaxies heading
off in at least three directions. There are no obvious gravitational arc systems
in the cluster centre.
We have obtained redshifts for 122 cluster members, with a mean redshift
−1
of z = 0.2307 ± 0.0007 and a velocity dispersion σ = 1528+98
−58 km s . Previously, the mean cluster redshift was based on two galaxy redshifts. The
Dressler-Shectman test shows groups of galaxies that seem to be dynamically
distinct west and east of the cluster centre. The global Dressler-Shectman
statistic indicates substructure at a high level of confidence: In 1000 Monte
Carlo reshufflings of the data, the DS statistic ∆ had a value higher than the
one observed in only 5 cases.
11.8. Abell 1763
191
192
11.9
Abell 1835
Abell 1835 is dominated by a giant elliptical galaxy slightly elongated in the
north–south direction. The galaxy distribution is regular; in the centre a number of very thin long gravitational arcs are seen in the HST/WFPC2 image.
We have obtained 153 redshifts for cluster members in Abell 1835 with a
+125
−1
mean redshift of z = 0.2505+0.0002
−0.0006 and a velocity dispersion σ = 1549−55 km s .
The cluster redshift listed by Struble & Rood (1999) was based on a single
redshift measurement. The redshift distribution shows evidence for a highredshift tail and the Dressler-Shectman test hints at the presence of a distinct
group of galaxies to the north-west of the cluster centre. In all other respects,
Abell 1835 gives the impression of being a perfectly regular and relaxed cluster
of galaxies; the redshift distribution therefore warrants a more detailed investigation.
Abell 1835 is the most luminous cluster in our sample with LX = 9.63 ×
44
10 h−2 erg s−1 . From the LX –TX relation it should also be the hottest cluster,
however a recent ASCA temperature measurement resulted in a modest temperature of 7.42 keV (Ota 2001). Abell 1835 was observed by XMM/N EWTON
during the Performance Verification Phase in June 2000 (and is therefore not
within our XMM sample). An analysis of the M OS 2 and PN data was presented
by Majerowicz et al. (2002). The surface brightness morphology of Abell 1835
is very regular with nearly circular and concentric isophotes in the 0.3—3 keV
11.9. Abell 1835
193
194
band out to a radius of ∼ 0.8 h−1 Mpc, corresponding to 75% of the virial radius (r200 ). The temperature profile is flat at radii outside ∼ 250 h−1 kpc at
TX = 7.6 keV, but drops in the cooling flow region in the cluster centre to
about 4 keV. The global mean temperature outside the cooling flow region is
given by Majerowicz et al. (2002) as TX = (7.6 ± 0.4) keV, consistent with the
ASCA temperature measurement of Ota (2001). C HANDRA observations of
Abell 1835 are reported by Schmidt et al. (2001) who find a significantly higher
X-ray temperature of about 12 keV in the outer parts of the cluster. Markevitch2002 (2002) identifies a faint background flare in the Chandra data and an
insufficient correction for the effects of the point spread function in the XMM
data as the main reasons for this discrepancy.
11.10
Abell 2218
Abell 2218 is a very famous lensing cluster, with an extraordinary number
of arcs and arclets in its centre. A lens model for this cluster was presented
by Kneib et al. (1996) and required a bimodal central mass distribution, with
one mass component centred on the cD galaxy and a second one centred on a
bright galaxy about 1.0 5 to the south-east of the cD. Girardi et al. (1997) analysed
the distribution of 50 galaxy redshifts in Abell 2218 (from Le Borgne et al. 1992)
and found evidence for two distinct groups of galaxies superimposed along
11.10. Abell 2218
195
196
the line of sight, which they identify with the mass clumps modelled by Kneib
et al. (1996). The velocity dispersion of Abell 2218 is given by Le Borgne et al.
(1992) as σ = 1370 km s−1 .
11.11
Abell 2219
The optical morphology of Abell 2219 is remarkably similar to the one of Abell
2219, with a dominant cD galaxy and a second bright elliptical galaxy at ∼
1 arcmin to the south-east. A number of gravitational arcs can be seen, most
notably a straight arc between the two brightest cluster galaxies.
We have obtained redshifts for 90 cluster members, the mean redshift is
−1
z = 0.2244 ± 0.0006, the velocity dispersion σ = 1398+108
−64 km s . The DresslerShectman test hints at the presence of substructure at the 3% confidence level
(in 30 out of 1000 Monte Carlo reshufflings of the data, the global statistic ∆
was found to be larger than the observed value). The Dressler-Shectman map
shows a group of galaxies with distinct kinematical properties to the northwest of the cD galaxy, on the opposite side to the second bright elliptical galaxy.
A gravitational depletion signal (Sect. 4.2.3) at near-infrared wavelengths
was detected by Gray et al. (2000), consistent with a singular isothermal mass
distribution with σ ∼ 800 km s−1 .
11.11. Abell 2219
197
198
11.12. Abell 2390
199
11.12
Abell 2390
Abell 2390 is a cD dominated cluster with several arcs in its centre. A chain of
fairly bright galaxies extends to the north-west from the cD galaxy; the arcs on
this side of the cD are straight which points to an extension of the underlying
mass distribution in this direction. About 3 arcmin to the east of the cluster
centre lies an extended, not concentrated group of galaxies.
A large number of redshifts in Abell 2390 were obtained by Le Borgne et al.
(1991) and Yee et al. (1996a). The velocity dispersion σ = 1686 km s−1 listed by
Struble & Rood (1999) is based on 225 galaxy redshifts.
200
Les amas individuels
Ce chapitre présente en plus de détail les amas qui font parti de l’échantillon
du projet. Pour chaque amas, la partie centrale de l’image R prise avec la
caméra CFH12k dans le cadre du projet est montrée et une description de la
structure du centre est donnée. Dans les cas où un nombre suffisant de redshifts de galaxie est disponible, les statistiques principales, redshift moyen et
dispersion des vitesses, sont calculées. Le test de Dressler & Shectman (1987),
qui cherche des sous-structures dans la distribution spatiale/redshift, est exécuté pour ces amas, ce qui représente une première étape préliminaire dans
une étude plus approfondie des distributions de redshifts dans les amas.
Abell 68 n’a pas été étudié en détail auparavant. Dans le cadre de ce projet,
11 redshifts de galaxies ont été obtenus, le redshift moyen est z = 0.2513 ±
0.0010, la dispersion des vitesses est σ = 808 km s−1 (formel ; le nombre de
redshifts est trop faible pour permettre de calculer une dispersion des vitesses
fiable). Abell 68 est dominé par une large galaxie cD, avec un groupe secondaire de cinq galaxies brillantes au nord-ouest. Plusieurs arclets bleues sont
visibles, notamment un triple arc à l’est de la cD.
Abell 209 est dominé par une galaxie cD ; il n’y a pas de prominents systèmes d’arcs géants. Le redshift central z = 0.2060 est basé sur deux redshifts
seulement (Struble & Rood 1999). Une analyse des données XMM, obtenues
dans le cadre de ce projet, est en préparation (Marty et al.).
Abell 267 est dominé par une galaxie cD ; il n’y a pas d’arcs géants et pas
de galaxies multiples. Nous avons obtenu des spectres pour 151 objets à une
distance de ∼ 30000 de la cD, dont 74 appartiennent à l’amas. Le redshift moyen
est z = 0.2269 ± 0.0006, la dispersion des vitesses est σ = (1125 ± 95) km s−1 .
Le test de Dressler-Shectman ne donne aucune indication de sous-structure.
Abell 383 est dominé par une galaxie cD presque circulaire et montre un
système complexe d’arcs géants et d’arclets, qui ont été utilisés dans Smith
et al. (2001) afin de reconstruire la distribution centrale de masse dans cet amas.
Cinq systèmes d’images multiples ont été identifiés, dont deux arcs radials.
Le rapport des positions angulaires de images radiales et tangentielles de la
même source pose une forte contrainte sur la pente du profil de masse au rayon
201
202
Chapitre 11. Notes on Individual Clusters
de l’arc radial. Dans Abell 383, la pente au rayon de l’arc intérieur est de −1.9,
au rayon de l’arc extérieur −1.3, c’est à dire le profil devient plus raid au centre
de l’amas ; cet excès de masse est attribué à la présence des baryons dans la
galaxie centrale.
Nous avons obtenu 41 redshifts de galaxies appartenant à Abell 383 ; le
redshift moyen est z = 0.1896 ± 0.0007, la dispersion des vitesses est σ =
−1
1167+130
−166 km s . La distribution des redshifts ne donne pas d’indication de
sous-structure dans cet amas.
Abell 773 n’a pas pu être observé avec la CFH12k. L’image HST/WFPC2
montre une structure bimodale avec beaucoup d’arcs “étroits” entre les deux
centres. L’amas contient une large fraction de galaxies spirales. Nous avons
obtenu 122 redshifts de galaxies appartenant à Abell 773, le redshift moyen
−1
est z = 0.2190 ± 0.0007, la dispersion des vitesses est σ = 1594+117
−62 km s . Le
test de Dressler-Shectman révèle un groupe de galaxies à l’est du centre qui
semble distinct dynamiquement des autres galaxies. Joint à la bimodalité de la
distribution des galaxies, ceci montre que Abell 773 n’est pas un amas relaxé,
donc la dispersion des vitesses surestime probablement la masse de cet amas.
Abell 963 est dominé par une galaxie cD est deux arcs géants au nord et
sud de la cD, connus auparavant. Nous avons obtenu des redshifts pour 70
galaxies dans cet amas, avec un redshift moyen de z = 0.2041 ± 0.0008 et une
+100
dispersion des vitesses σ = 1412−80
km s−1 . Le test de Dressler-Shectman indique la présence de sous-structure avec une confiance marginale.
Abell 1689 est un amas qui est étudié extensivement à présent et dans le
passé. Il s’agit d’un amas très riche et extrêmement lumineux, qui par contre
ne contient pas de galaxie cD. A 1 arcmin au nord-est du groupe de galaxies
central, un deuxième groupe de galaxies brillantes existe. Le centre de Abell
1689 contient une quantité d’arcs et images multiples de sources d’arrière-plan.
211 redshifts sont disponibles dans Abell 1689, dont la plupart a été obtenu
dans le cadre de ce projet. Le redshift moyen est z = 0.1853+0.0004
−0.0008 , la disper+38
−1
sion des vitesses est σ = 1976−56 km s . La distribution des galaxies en vitesses vs. distances radiales est large et apparemment homogène. Le test de
Dressler-Shectman montre un groupe nettement distinct au nord-est du centre
de l’amas. Abell 1689 est un amas très complexe et dans toute probabilité nonrelaxé.
Abell 1763 contient une galaxie cD dans son centre ainsi que des chaı̂nes
de galaxies qui partent dans au moins trois directions. Il n’y a pas de systèmes
évidents d’arcs gravitationnels. Nous avons obtenu des redshifts pour 122
galaxies appartenant à l’amas, le redshift moyen est z = 0.2307 ± 0.0007, la
−1
dispersion des vitesses est σ = 1528+98
−58 km s . Le test de Dressler-Shectman
montre des groupes de galaxies distincts à l’est et à l’ouest du centre de l’amas ;
le test globale indique la présence de sous-structure à un niveau de confiance
élevé.
Résumé
203
Abell 1835 est dominé par une galaxie elliptique géante. La distribution des
galaxies est régulière, l’image HST/WFPC2 révèle un nombre d’arcs gravitationnels longs et minces. Nous avons obtenu des redshifts pour 153 galaxies
dans l’amas ; le redshift moyen est z = 0.2505+0.0002
−0.0006 , la dispersion des vitesses
+125
−1
est σ = 1549−55 km s . La distribution des redshifts montre une queue vers
des grands redshifts ; le test de Dressler-Shectman indique la présence d’un
groupe de galaxies distinct au nord-ouest du centre de l’amas. Abell 1835 est
l’amas le plus lumineux dans l’échantillon.
Abell 2218 est un amas bien connu, notamment par son système extraordinaire d’arcs et arclets dans son centre. Un modèle de la distribution centrale
de masse dans Abell 2218 est nettement bimodale, les deux centres de masses
étant séparé de 1.0 5. 50 redshifts sont disponibles dans cet amas (Le Borgne
et al. 1992), la dispersion des vitesses est σ = 1370 km s−1 .
Abell 2219 contient une galaxie cD ainsi qu’une deuxième galaxie brillante
à une distance de 1 arcmin. Un nombre d’arcs gravitationnels sont visibles, notamment un arc étroit entre les deux galaxies dominantes. Nous avons obtenu
des redshifts pour 90 galaxies dans l’amas ; le redshift moyen est z = 0.2244 ±
−1
0.0006 et la dispersion des vitesses σ = 1398+108
−64 km s . Le test de DresslerShectman indique la présence de sous-structure, notamment un groupe distinct de galaxies au nord-ouest de la galaxie cD.
Abell 2390 est dominé par une galaxie cD et montre plusieurs arcs dans
son centre. 225 redshifts sont disponibles dans la littérature, la dispersion des
vitesses est σ = 1686 km s−1 .
Chapter 12
Reduction of CFH12k data
12.1
The CFH12k camera
The CFH12k camera is described in detail in Cuillandre et al. (2000)1 . It is a
CCD mosaic camera consisting of twelve 2k × 4k backside-illuminated thinned
CCD chips from Lincoln Laboratories at the Massachusetts Institute for Technology, arranged in two rows as shown in Fig. 12.1. The whole field thus
covers roughly 12k × 8k pixels. The camera, which includes a focal reducer
and wide-field corrector, is mounted at the prime focus of the 3.6m Canada–
France–Hawai‘i Telescope (CFHT) which, at a focal ratio of f /4, has a pixel
scale of 0.00 206 in the centre of the focal plane for a physical pixel size of 15 µm.
The total field of view of the CFH12k is thus 42 × 28 arcmin2 .
The CFH12k contains two groups of chips as shown in Fig. 12.1. Nine chips
are made from standard epitaxial silicon (EPI), three (chips 03, 04 and 05) are
made from high-resistivity bulk silicon (HiRho). The two groups of chips have
somewhat different spectral response curves; in particular, the HiRho chips
are more sensitive (quantum efficiency is 20% higher) in the red than the EPI
chips. All the chips differ to some extent in their overall sensitivity; this will
be corrected for during the data reduction by determining photometric scaling
factors for each chip in each filter band. The varying spectral response cannot
be corrected for in this way. For high precision photometry all the chips would
have to be treated independently with individual photometric zero points and
colour terms for each chip; however, for out purposes precision photometry is
not our primary goal and we decided to build a single image from combining
all the chips disregarding the (small) colour characteristics.
The CFH12k chips have a small number of cosmetic defects, in the form
of bad columns and and rows. The static mask (provided by Jean-Charles
Cuillandre) for the May/June 2000 observing run is shown in Fig. 12.2.
1 See
also http://www.cfht.hawaii.edu/Instruments/Imaging/CFH12K/
204
205
12.2. Prereduction: Bias removal, flat fielding
06
07
08
09
10
11
03
04
05
E
N
00
01
02
Figure 12.1: Layout of the CFH12k camera. The arrows mark the output connector, defining the location of pixel (1,1) for each chip. The output for Chip03
was switched in 2000; the left arrow marks the output for our November 1999
observing run, the right arrow for our May/June 2000 run. Chips 03, 04 and 05
are made of high resistivity bulk silicon (HiRho), all the others from standard
epitaxial silicon (EPI); these two groups of chips have somewhat different sensitivity and spectral response. The layout as shown here has North at the bottom and East on the left; this is the way an MEF file is displayed by mscdisplay
and is determined by the fits keywords DETSEC which define a pixel coordinate
system for the whole chip array.
12.2
Prereduction: Bias removal, flat fielding
The pre-reduction of the CFH12k images follows standard procedures with a
number of peculiarities related to the mosaic form of the data. The process is
outlined in the first part of Fig. 12.3.
The initial data are available in the form of multi-extension fits (MEF) files,
which consist of a primary header unit which contains general information on
the exposure, and twelve extension units which contain information pertinent
to the chip (e. g. the world coordinate system which translates the physical,
pixel-based coordinate system of the chip to equatorial sky coordinates) as
well as the actual data. At the telescope the data are written in 16 bit integer
format with a size of 206 Mbyte per file. During the data reduction the data are
converted to 32 bit floating point format and the file size doubles. Thus disk
size is of primary importance for this work.
206
Chapter 12. Reduction of CFH12k data
Figure 12.2: Static mask for the May/June 2000 observing run. The masked
bad columns and rows were artificially widened by 5 pixels for display purposes only.
For the pre-reduction I use the I RAF package mscred (Valdes 1998), which
is based on ccdred, adapted for working on MEF files. mscred is adapted
to work with images from the NOAO CCD Mosaic Imager and relies on the
appropriate fits header information defining the positions of the chips within
the mosaic. For CFH12k images I found that certain header keywords had to
be modified or added. The keyword DATASEC defines the section of the chip
which contains “useful” science data (as opposed to the overscan strip or bad
boundary pixels); the CFH12k control system writes these sections to reflect
the position of the readout connection (Fig. 12.1) and thus counts pixels in
decreasing order for some chips. mscred chokes on this sections, and I had to
modify DATASEC according to
hedit *.fits[chip00] datasec [6:2049,5:4100] add- update+ verifyand equally for the other chips for mscred to work. In fact, mscred expects the
section given by DATASEC to be given by a keyword CCDSEC, so the latter had to
be added with the same value as DATASEC:
msccmd "hedit $input ccdsec ’(datasec)’ add+ verify- update+" *.fits
Finally, the fits headers contain the keyword DETSEC which specifies the position of the chip within the mosaic. For some chips (00, 01 and 07) the CFH12k
207
CFH12k images
bias, flat, science, standards
bias
flat, science, standard
Bias.fits
bias subtraction
flat
science, standard
Flat.fits
flat−field correction
Fringe.fits
fringe subtraction
DSS.fits
astrometric calibration
I band only
standard
see Figure 12.9
science
relative intensity scaling
SWarp
MEF
SEF
imcombine
Standard field
catalogues
photometric calibration
output image
Figure 12.3: Flowchart outlining the main steps in the CFH12k image reduction. The astrometric calibration is shown in more detail in Fig. 12.9. Up to the
image transformation in using SWarp, the images are in multi-extension fits
format (MEF), afterwards there is one single-extension fits file (SEF) for each
field.
208
control system wrote sections the sizes of which were incompatible with the
DATASEC defined above, so these had to be modified according to
hedit *.fits[chip00] detsec [1:2044,1:4096] add- verify- update+
These commands (contained in an I RAF script sectioncor.cl) were executed
on the raw image files as the very first step in the reduction process.
The main purpose of DETSEC is for image display purposes with the task
mscdisplay which shows the whole of the mosaic at once, although mscred
relies on the consistency and integrity of the header information to work properly. In order to display the images properly the keywords DETSEC and CCDSEC
had to be reset, this was done using the I RAF script secrecor.cl which is listed
in Fig. 12.4.
The bias is an electronic base level that is present even at zero seconds exposure time. Ideally this is just a constant across the field, however the CFH12k
shows some bias structure, especially in chip 08 in the form of a number of
“brighter” columns. I therefore chose to use a bias correction image, median
combined from at least fifteen zero exposure time images taken for each night.
The overscan strip and a few edge columns and rows were discarded by trimming the images to the section specified in DATASEC.
hedit
hedit
hedit
hedit
hedit
hedit
hedit
hedit
*.fits[chip02]
*.fits[chip03]
*.fits[chip06]
*.fits[chip07]
*.fits[chip08]
*.fits[chip09]
*.fits[chip10]
*.fits[chip11]
detsec
detsec
detsec
detsec
detsec
detsec
detsec
detsec
hedit
hedit
hedit
hedit
hedit
hedit
hedit
hedit
*.fits[chip02]
*.fits[chip03]
*.fits[chip06]
*.fits[chip07]
*.fits[chip08]
*.fits[chip09]
*.fits[chip10]
*.fits[chip11]
ccdsec
ccdsec
ccdsec
ccdsec
ccdsec
ccdsec
ccdsec
ccdsec
[6199:4156,1:4096] add- verify- update+
[2049:6,5:4100]
[2049:6,5:4100]
[6:2049,4100:5]
[6:2049,4100:5]
[6:2049,4100:5]
[6:2049,4100:5]
[6:2049,4100:5]
[6:2049,4100:5]
addaddaddaddaddaddaddadd-
verifyverifyverifyverifyverifyverifyverifyverify-
update+
update+
update+
update+
update+
update+
update+
update+
Figure 12.4: The I RAF Script secrecor.cl which resets the fits header keywords DETSEC and CCDSEC to define the correct orientation and positions of the
chips in the mosaic.
209
Figure 12.5: Combined R band flat field from the May/June 2000 observing
run.
As any CCD the CFH12k chips produce some dark current; however, the
dark current of the CFH12k is very low and we decided not to subtract dark
images which would have introduced more noise into the images.
A flat-field correction image was constructed for each filter and for each
night from images of the evening or morning twilight sky. Unfortunately it
was found that morning and evening twilight exposures did not work together, i. e. the result of dividing a morning twilight image by an evening
twilight image was not flat but showed more or less large-scale structure. On
the other hand, morning (evening) twilight flats worked very well amongst
themselves. A choice was therefore necessary to select the appropriate twilight images to correct the science images. For the November 1999 observing
run the choice was easy since the moon was waxing and therefore affected the
evening twilight flats (taken at a distance of 36◦ from the moon for the second night of this run). The May/June 2000 data on the other hand were taken
around new moon and neither the evening nor the morning twilight flats were
affected by the moon. For this run, all the twilight images for a given night
were combined to construct the final flatfield correction images. During the
first hours of the second night of this run, a cable connecting the temperature
monitor of the camera was disconnected, leading to an effective temperature
which was lower than the nominal temperature. Since the temperature af-
210
fects the pixel sensitivity, images taken before reconnection of the cable were
corrected with evening twilight flats, those taken later with morning twilight
flats.
The selected twilight images were median-combined after multiplicative
scaling by the mode of the pixel value distribution using the I RAF task flatcombine. Using the median for the combination provided the best rejection
of stars that were present in the individual twilight images. The resulting flat
field was scaled to a mean of unity across all the twelve chips. An example of
an R band flat field correction image is shown in Fig. 12.5.
The prereduction of the data was done using the MSCRED task ccdproc,
with bias subtraction, trimming and flatfielding. The flat-field correction did
unfortunately not yield perfect results, with in particular a remaining brick
pattern in chip 01. This was most likely due to the different colour of the
twilight and the night sky.
12.3
Removal of additive effects: Fringing and “sky
ring”
The main additive component in the the CFH12k images is the continuous
background which includes contributions from atmospheric and ecliptic emission as well as reflections inside the instrument. Although it is not strictly necessary to remove the background, the background level varies with time so
that when combining several exposures the sky has to be brought to a common level. Over a wide field of view, such as that of the CFH12k camera,
the background is never really constant but shows large scale gradients which
are also prone to variation over time and between exposures. Co-adding several exposures thus likely produces an “ugly patchwork” in the background
(Bertin 2001). The slowly varying background is therefore removed before Coadding the images; this is done during the transformation of the images and
the procedure is described in Sect. 12.4.
12.3.1 I band fringing
In the I band, the CFH12k images show a fairly strong fringe pattern. Fringing is produced by night sky emission line radiation, part of which is multiply reflected in the thin CCD chips and produces interference patterns akin to
the patterns seen on thin oil films, or Newton rings on glass-mounted photographic slides. Fringing due to night sky emission lines is an additive effect
and can be corrected by subtracting a suitable correction image. The fringe
pattern in the CFH12k images was found to be stable over the course of one
night that it can be isolated by median-combining images from different fields
12.3. Removal of additive effects: Fringing and “sky ring”
211
after proper removal of the continuous background which varies over a much
larger scale than the fringing. Problems in the creation of the fringe correction
images arose due to the presence of extended objects, in particular during the
November 1999 run, where all the images were centred on the cluster centres.
In the extreme case, there are pixels are covered by object emission in all the
exposures, even from different fields, so that an uncontaminated sky signal
was not available. In cases where no piece of blank sky is readily accessible, it
is best not to subtract anything, which means that objects have to be masked
in the images that go into the fringe correction image.
The procedure is as follows: I typically create three correction images per
night, each made up of about eight exposures. The number of exposures going
into a correction image has to be as large as possible in order to obtain a high
signal-to-noise ratio (where “signal” refers to the fringe pattern). On the other
hand, this number is limited to about 8 so as to avoid having to use more than
two exposures from any given field. The gain from using more images would
be small due to the large number of overlapping objects in these exposures.
There should be no overlap in the sets of exposures going into the different
correction frames.
The first step is to subtract the continuous background from the selected exposures and to mask the objects. I use the background subtraction and object
detection capabilities of SExtractor for this purpose by producing two “check
images”, “−BACKGROUND”, which is the original image with the large-scale
background model subtracted, and “SEGMENTATION”, which codes each
pixel with an integer number according to which object it is attributed to. Pixels that are not attributed to any object have the value 0 in the segmentation
image. All the pixels in the segmentation image that are attributed to an object
as well as all the pixels up to a distance of 10 pixels away (to account for faint
wings of the objects) are then set to a large negative value using the I RAF task
imrepl:
imrepl segment.fits -31000 lower=0.8 radius=10
The replaced segmentation image is then added onto the background subtracted image, which then has a mean background level of zero, modulated by
the fringe pattern, and all the pixels belonging to an object set to a large negative value (Fig. 12.6, top left). A number of exposures corrected in this way are
then median combined, ignoring pixels with values below a given level, i. e.
pixels that have been attributed to an object. Pixels which are masked out in
all the exposures are given the value 0 in the resulting fringe correction frame.
Ideally, the exposures going into the correction frame should be scaled since
the amplitude of the fringe pattern in general varies with time; however, the
variation was found to be sufficiently small so that scaling was not necessary
at this stage.
212
From each science image a correction image was then subtracted with the
proper scaling, determined by visual inspection using the task rmfringe in the
mscred package of I RAF.
Fig. 12.6 shows an example of this process.
Figure 12.6: Example of the fringe subtraction process. The image to be corrected is shown in the bottom left frame (A963, chip 09). The top left frame
shows the same (in pixel coordinates) section of another image of the same
field, with all the objects masked as described in the text. The small telescope
offset between the two exposures of the field is clearly visible. The image
shown top left is median–combined with seven others to form the fringe correction image, shown top right; note the increase in the “signal”-to-noise ratio.
The correction image is multiplied by an appropriate scale factor (0.9 in this
case) and subtracted from the original science image. The corrected image is
shown bottom right.
12.3. Removal of additive effects: Fringing and “sky ring”
213
Figure 12.7: R band exposure of Abell 2390. This figure shows the 12 chips of
the exposure after bias subtraction and flat-fielding. The sky ring due to the
variable effective width of the R filter passband is clearly visible. The layout
of the chips is as in Figure 12.1 and has north at the bottom and east to the left.
The cluster centre is visible in the lower part of chip 09. Also visible are black
strips (especially on chips 01 and 05) which mark bad columns and are masked
during the image transformation using SWarp at a later stage. In the left part
of the image a satellite trail can be seen. The vertical comet-like column on the
bottom of chip 04 is due to reflected light from a star outside the field-of-view.
Most of these defects are removed by combining several dithered exposures of
the same field.
214
Figure 12.8: Transmission curves of the CFH12k filters. The bottom panel
shows a night-sky emission spectrum based on data from Osterbrock et al.
(1996).
12.3.2 Sky ring
A problem which is in some way related to the fringing in the I-band occurs in
the R-band images. The transmission curve of the R-band filter of the CFH12k
camera has very steep flanks as shown in Fig. 12.8. The blue flank of the passband straggles the position of the Na D doublet, λ ' 5892 Å, which are fairly
strong night sky emission lines; also the red end of the passband straggles several fainter sky emission lines. The actual passband of an interference filter
depends on the angle of incidence of light on the filter, resulting in a variation
of the transmissivity at the wavelength of Na D over the field of view. This in
turn leads to an axisymmetric pattern on the scale of the CFH12k field, brighter
on the edges of the field (Fig. 12.7). Although the “sky ring” is strongest in the
R band, it is also present in V and I; the B band images are free from this effect.
The effect is essentially additive, although it also causes a small variation of
sensitivity to object emission; however, the object flux contained in the wavelength ranges corresponding to the flanks of the pass band constitutes only a
small fraction of the flux received over the whole pass band and therefore this
multiplicative effect is negligible. It is therefore only necessary to remove the
contribution from the sky emission, which is additive.
The “sky ring” might be removed in the same way as the fringe pattern
in the I-band images, by subtracting a combined correction image. However,
12.4. Astrometry
215
since it is a large-scale effect, covering the whole of the CFH12k field it can simply be considered as part of the smooth background and will thus be removed
with the background as described in Sect. 12.4. Removing a model instead of a
correction image has the advantage that no additional noise is introduced into
the image.
12.4
Astrometry
In order to obtain as precise galaxy shape measurements as possible from the
images we have to take care that these shapes are minimally degraded during the reduction process. The most precise measurements could in principle
be obtained from the individual raw exposures. However, for weak lensing
analyses we need shapes of faint background galaxies at the detection limit
of individual exposures. It is therefore necessary to combine the exposures to
increase the signal-to-noise level for the objects.
There are two possibilities to do the weak lensing measurements: The first
is to assemble all the exposures into one seamless mosaic image, covering the
whole field of view of the CFH12k camera and to measure galaxy shapes from
this image. This method has the advantage that due to the averaging over
several exposures, the fractional noise level in the sky background
is decreased
√
by roughly the square root of the number of exposures, Ne . In addition, since
the telescope was moved by up to 75 pixels (1500 ) between exposures of a given
field, the gaps between the chips of the CFH12k camera can be filled, giving
continuous coverage of the whole field.
On the other hand, in order to combine different exposures, they have to be
mapped onto a common reference grid. This mapping has to be very precise:
If the centroids of the same object, mapped from different exposures onto the
reference grid, do not coincide exactly, then the shape of the resulting image
will differ from that in the individual exposures. An elliptical galaxy will appear more (less) elliptical if the displacement vector is along the major (minor)
axis of the galaxy image. This effect will necessarily introduce random noise,
and can in no way be removed by looking at the combined image alone.
Alternatively, one can use the combined image for the detection of objects only, taking advantage of the increased signal-to-noise for faint galaxies,
and then go back to the individual exposures to actually measure the galaxy
shapes. The final galaxy shape measurement will then be the average over all
the exposures. This method has not yet been implemented due to the huge demand on CPU time for the galaxy shape measurements which will be a factor
Ne larger than in the first method.
The aim is thus to map all the exposures onto a common reference frame
such that the rms scatter of the mapped image centroids in the reference frame
is minimum. In principle, the registration of the exposures could be done using
216
only internal data, due to the partial overlap of the chips in different exposures.
For instance if an object appears near the edge of chip i in one exposure, it
might appear on the adjacent chip j in a different exposure due to the dithering.
Such chip-jumping objects can be used to determine the relative shifts and
rotations of the chips. However, the number of these objects is small, so the
derived relative rotation angles are rather uncertain.
In addition, the optical system of the telescope and camera introduces distortion into the image, the dominant component of which is a barrel distortion.
Simply shifting exposures and rotating the individual chips is not enough to
obtain an accurate alignment of objects over the whole field, in particular if
the shifts between exposures exceed the usual dither shift, as happened in a
few cases on the telescope. The solution for mapping the exposures onto a
reference frame should therefore be non-linear and remove at least the relative distortion between exposures as much as possible. This implies that we
need to determine a large number of parameters for an accurate mapping of
the individual exposure. Unfortunately, the chip layout is not stable between
exposures due to the varying gravitational force and stresses when the telescope moves. This is the “jello detector” model (Wilson et al. 2001), which
prevents one from using a large number of observations to derive a fixed set
of transformation parameters for the camera.
The absolute distortions introduced by the optical system are not important for the galaxy shape measurements. On small scales the distortion can be
linearised and described by its (spatially varying) Jacobian matrix. The deviations of the Jacobian from a shape-preserving rotation matrix are usually small.
Furthermore, distortions affect images of stars in the same way as galaxies.
Since the ideal shape of star images is known, the polarisation due to the distortion can be identified and, if the density of stars in the image is sufficiently
high, galaxy shape measurements corrected for this effect by interpolation.
I therefore distinguish between absolute astrometry and relative astrometry. Absolute astrometry refers to the deviations of the object positions in the
image as described by the World Coordinate System (WCS) from their “true”
positions as defined by a set of astrometric reference stars, such as the USNO
A2.0 catalogue. Relative astrometry refers to the alignment of an object’s images from a number of exposures when transformed to the reference coordinate system. Relative astrometry is what is most relevant for our purposes.
The procedure that I describe in the following for aligning the different
exposures onto a common reference frame is in part inspired by the work of
Nick Kaiser’s group and described in Kaiser et al. (1999). However, there are a
number of differences between the methods.
217
12.4. Astrometry
Exposure Sequence
field i , filter j
DSS image
field i
SExtractor catalogues
compact objects
SWarp to 0.205"
Shifts between
exposures
SExtractor catalogue
first
exposure
reg_to_image.pl
linear transformation
12k
DSS
all
exposures
mkmatchfiles.pl
matched catalogues
xDSS yDSS x12k y12k
comp_trafo.pl
4th order transformation
12k
DSS
2x
mkmastercat.pl
getbettercoords.pl
matched catalogues
xDSS yDSS x12k y12k
wcsctran
matched catalogues
α2000 δ2000 x12k y12k
ccmap
accurate WCS
Figure 12.9: Flowchart of the determination of an accurate world coordinate
system for the CFH12k images by registering onto a DSS image of the field.
All the catalogues and transformation solutions apply for individual chips.
218
12.4.1 Procedure
Since an internal determination of the astrometric parameters is not feasible
due to the small number of objects that can be used, it is necessary to introduce
an external reference frame in the registration process. The largest catalogue
of astrometric standard stars currently available for this purpose is the USNO
A2.0 catalogue (Monet et al. 1998) available from the United States Naval Observatory2 . This catalogue contains more than 5 × 108 stars across the whole
sky down to ∼ 20 mag in red and blue photographic bands. Typically, there are
about 50 to 80 USNO stars on each chip of the CFH12k camera, however many
of these are saturated due to their brightness and the long exposure times of
600 − 900 s that we used in our observations. If we restrict ourselves to the
USNO A2.0 catalogue the number of stars that can be used to determine is
rather low. I therefore chose the Digital Sky Survey (DSS3 ) images as the reference frames. The DSS is based on the photographic plates of the Palomar
Observatory Sky Survey (POSS-I from 1950 to 1958, POSS-II from 1987-1998),
that were digitally scanned at 1.00 7 for the first generation, and at 1.00 0 for the
second generation (the latter is also called XDSS). Where possible I used second generation DSS images as the reference frames but since the XDSS has not
been completed yet a first generation DSS image had to be used in one case
(Abell 383). The USNO A2.0 is based on the POSS plates, and nonlinear world
coordinate systems (WCS) are available for the DSS images.
In a first step the DSS image as obtained from the web (I used images from
the CADC web site4 ) are resampled to the same pixel scale as the CFH12k images, 0.00 206 per pixel. At the same time, the image is de-distorted. For this purpose I use the program SWarp (Version 1.21, Emmanuel Bertin), which reads
the WCS of the image and transforms the image according to a chosen projection. I describe SWarp in Section 12.4.3. The resulting image has the same pixel
scale as the CFH12k images and has no distortions. The latter is important for
the stability of the astrometric transformations of the CFH12k images onto the
DSS reference systems which in a first step uses pixel coordinates in the DSS
image.
Object catalogues for both the CFH12k images and the DSS reference image are constructed using SExtractor. For the determination of the astrometric
transformations I only use compact objects with FWHM between 2 and 15 pixels, magnitudes between roughly 20 and 23 magnitudes and SExtractor flags
lower than 3.
Since the WCS in the CFH12k images as written at the telescope is usually
not sufficiently accurate to permit an automatic identification of objects with
the DSS, I start the astrometric registration from scratch by first visually iden2 http://www.nofs.navy.mil/projects/ppm/USNOSA2doc.html
3 http://www-gsss.stsci.edu/dss/dss
4 http://cadcwww.dao.nrc.ca/dss/
home.htm
219
12.4. Astrometry
tifying three objects between the first exposure of a field with the DSS image.
This is easily (though somewhat tediously) done using the blinking facility of
the image display program ds9. From the coordinates of these three objects the
script reg to image.pl computes a linear transformation, consisting of a shift
(2 parameters) and a general matrix (4 parameters), from the image system to
the reference image:
xDSS = a x + B11 x12k + B12 y12k
yDSS = ay + B21 x12k + B22 y12k
.
(12.1)
(12.2)
Using this first transformation the script identifies all objects detected in both
the CFH12k image and the DSS image and recomputes the linear transformation based on all images. For the moment I restrict myself to a robust linear
transformation since this will later be used to match objects in the other exposures. In order to do this the shifts between the exposures have to be determined by identifying a star on chip 045 in all the exposures of the field and
converting the object coordinates to shifts relative to the position in the first
exposure. Normally the exposures were taken as a sequence according to a
pre-defined dithering pattern, but occasionally it was necessary to break a sequence in order to refocus the telescope or due to some other problem, so that
it is best to remeasure the shifts from the images.
The script mkmatchfiles.pl uses the linear transformation determined by
reg to image.pl on the first exposure and applies the relevant shift to create
matched catalogues for each chip in each exposure containing coordinates in
the CFH12k chip in columns 1 and 2 and the corresponding coordinates in
the DSS image in columns 3 and 4. These matched lists can now be used to
calculate non-linear transformations from the images to the DSS. It was found
that a polynomial of order 4 of the form
xDSS =
∑
aij xi y j
(12.3)
∑
bij xi y j
(12.4)
i+j≤4
yDSS =
i+j≤4
is sufficient to remove systematic residuals between the transformed and the
actual coordinates.
The residuals are dominated by the positional errors of the DSS image. The
original DSS image was scanned off the POSS plates at a pixel scale of 100 /pixel,
so the positional errors are at best of order 0.100 , which is consistent with the
rms of the residuals in the nonlinear fit.
5 Of course any other chip could be used; however, since the orientations of the pixel coordi-
nate systems vary between the chips the signs of the resulting shifts become a little confusing.
220
We can improve on this by replacing the DSS coordinates with the transformed coordinates for each object averaged over all the exposures. The standard deviation of the transformed coordinates is of order 0.05 to 0.1 pixels,
better than the positional errors of the DSS image.
There may be residual local systematic deviations from the DSS in the new
reference coordinates, however, as stated before, we are interested in good
relative alignment between the exposures and not so much concerned with
absolute astrometry. Replacing the DSS coordinates in this way also gives us
the opportunity to substantially increase the number of anchor points for the
final transformation by adding in objects that are too faint to appear in the
DSS image but are readily detected in the CFH12k exposures, each of which is
individually deeper than the DSS.
We thus need to “thread” objects through all the exposures available for
the field. The results of this identification (which is based on the transformed
coordinates) are contained in a master table for the field which lists for each
object the chip and corresponding catalogue number for each exposure. The
advantage of having such a master catalogue (or “hash table”) for matching
a number of catalogues is that it can be used to compare any entry from the
catalogues without having to redo the time-consuming identification.
The script getbettercoords.pl reads the master table to do two things:
First it computes the mean of the transformed coordinates and creates for each
chip and each exposure a matched list between the image coordinates and
the mean reference coordinate for the object. All the objects detected in the
original SExtractor catalogues can be included in these lists, not just those
also detected in the DSS image. Typically this increases the number of objects
by a factor of 3. In general an object is included if it is detected in at least
three exposures (less if the number of available exposures is low). However, if
due to the dithering an object appears in different chips in different exposures
it is automatically included since this provides an important link between the
transformations for neighbouring chips which would otherwise be completely
independent.
In addition getbettercoords.pl determines photometric scaling factors
between the exposures as described in Sect. 12.4.2 and compares the seeing
widths of compact objects to allow rejection of exceptionally bad exposures.
The new matched coordinate lists occasionally still contain spurious or
misidentified objects. They can be cleaned by reiterating the process of calculating a nonlinear transformation, threading objects and computing new reference coordinates.
So far we have worked with pixel coordinates in both the chip and the
reference systems. For the final transformation we need to have the reference
coordinates in the equatorial system. Since the reference coordinates are pixel
coordinates in the DSS image we can use the WCS of the DSS to convert them
to equatorial coordinates, right ascension and declination; this is done using
221
12.4. Astrometry
the I RAF task wcsctran6 . Finally we use the I RAF task ccmap to compute the
final transformation for each chip in each exposure. This is a polynomial fit of
order 5 in I RAF parlance, which corresponds to a 4th order polynomial as in
Equation (12.4).
Figure 12.10 illustrates the transformation from the CFH12k images onto
the reference system by considering the effect of the transformation on a small
circle at the image position. Let J = ∂(xDSS , yDSS )/∂(x12k , y12k ) be the Jacobian
matrix of the transformation (12.4), then the unit circle (cos θ, sin θ), θ ∈ [0, 2π),
is transformed onto (J11 cos θ + J12 sin θ, J21 cos θ + J22 sin θ), or
r2 = A cos2 θ + 2B sin θ cos θ + C sin2 θ
,
(12.5)
2 + J 2 , B = J J + J J and C = J 2 + J 2 . This is the equation
with A = J11
11 12
21 22
22
21
12
of an ellipse with major and minor axis directions given by the two solutions
on, say, [0, π) of
2B
.
(12.6)
tan 2θ =
A−C
Figure 12.10 plots sticks with the orientation of the major axis and length proportional to the ratio if major to minor axis minus 1 (so that an isotropic transformation, which results in a circle, is represented by a dot). The physical, optical transformation is of course the inverse of (12.4). If circles in the 12k system
are transformed onto tangentially aligned ellipses in the DSS system, then the
camera induces a radial shear on object shapes. This corresponds to a decrease
of the (radial) pixel scale on the CFH12k images with increasing distance from
the optical axis. The shear induced by the camera distortion is small, less than
2% across the whole field, and only becomes important in the outer regions.
Physically transforming the images onto the DSS system removes the CFH12k
distortion; the important result from Fig. 12.10 is the fact that the transformation (12.4 does not introduce spurious distortion due to some instability of the
solution.
In grey scale, Fig. 12.10 shows the Jacobian determinant of the transformation (12.4): the transformation demagnifies objects in the outskirts, i. e. the
transformed object will cover a smaller number of pixels in the DSS system
than in the 12k system. Again, this reflects the changing pixel scale of the
CFH12k camera across the field.
12.4.2
Photometric scaling
The aim of the reduction procedure for the CFH12k data is to create a single big image out of the twelve chips and the nexp exposures of a given field.
Ideally, one and the same object would cause the same signal on all the chips
6 The
default value for the minimum precision min sig of 7 digits is not sufficient and
should be increased to e. g. 12 digits.
222
Figure 12.10: Characterisation of the transformation from the CFH12k system
onto the DSS system (Eq. 12.4). A circle on the 12k system is mapped onto an
ellipse in the DSS system; the sticks in the figure show the orientation of the
major axis, the length is given by the ratio of major to minor axis minus one.
The reference stick in the top right corner shows an axis ratio of 1.02. In grey
scale the determinant is shown; the range is small, from 1.009 in the corners to
1.012 in the centre. The example shown is for the fourth B-band exposure of
Abell 2219.
and on all the exposures. Unfortunately this is not the case, because in general the chips have somewhat different sensitivity, and the registered signal
depends on exposure time, airmass and atmospheric extinction; these effects
can to some extent be accounted for and corrected.
The registered signal from an object in a CCD exposure is proportional (in
the linear regime) to the incident flux integrated over the exposure time. During the light’s passage through the Earth’s atmosphere the flux is attenuated
according to Bouguer’s law
f inc,λ = f 0,λ e−Eλ a(z)
,
(12.7)
where Eλ is the (wavelength dependent) extinction coefficient, z is the angular distance from the zenith (a function of time), and the airmass a(z) can be
223
12.4. Astrometry
approximated by
a(z) ' sec z ≡
1
cos z
,
z<
∼ 60 deg
(12.8)
(Kitchin 1991).
The actual signal registered on the CCD exposure is proportional to the
time integral over the incident flux over the exposure time:
S = Ai
Z
∆t
f 0,λ e−Eλ sec z dt
.
(12.9)
The fluxes f 0,λ from the objects which are of interest to us are constant over
the observing time scales. Close to the zenith, the airmass is a slowly varying
function of time and we can replace it by an effective airmass for the exposure,
(sec z)eff =
1
(sec zs + 4 sec zm + sec ze )
6
,
(12.10)
where the zenith distances on the right hand side correspond to the start, middle and end of the exposure (Stetson 19887 ). The extinction coefficient Eλ may
vary with time. If it is constant with time or varies by only a small amount
over long time scales of, say, hours, we speak of photometric conditions, and
Eλ can be assumed constant over a series of exposures. In this case a series of
science exposures covering a range of air masses can be used to determine Eλ
(12.5).
In general therefore, the images have to be scaled multiplicatively when
combining a series of exposures of the same field, so that a given object results in the same signal in all the exposures. The script getbettercoords.pl
uses the master table to determine the relevant scale factors by comparing the
instrumental fluxes of each object (FLUX BEST) between the exposures. This
results in a multiplicative scale factor for each exposure relative to the first exposure in the sequence which accounts for the differences in exposure time,
the varying air mass between exposures as well as fluctuations in sky transparency. The determination of the scale factor for each exposure is based on
several thousand objects; the distribution of the scale factors for the individual
objects is sharply peaked with a full width at half maximum corresponding to
roughly 0.03 mag. This is indicative of a photometric error (rms) in the final
co-added image of better than 0.01 mag. In principle the mode of this distribution should be used as the scale factor; in practice I found that the median
provides an excellent and more easily determinable estimator of the mode. An
example is shown in Fig. 12.11 for the five B band exposures of Abell 2219,
based on 3373 objects.
7 cited
in the online help page of the task setairmass in I RAF
224
In photometric conditions, the deviations of the scaling factors from unity
are small (once differences in exposure time are accounted for); in non-photometric conditions the scaling factors can be substantial due to strong fluctuations in sky transparency (up to 50%), and it is essential that these factors
are applied to the individual exposures before combining them. Even for data
taken in photometric conditions, the correlation between the photometric scaling factors and air mass is significant and can be used to determine the atmospheric extinction coefficient, at least in cases where the exposure sequence
covers a sufficiently large range in air mass.
Figure 12.11: Relative photometric scales of exposures 2 through 5 of Abell
2219 in the B band. The plots show histograms of the ratio of flux measured
in the exposure relative to the flux measured in the first exposure of the series.
The flux ratios are of order 2 because the exposure time for the first exposure
was only half that of the other four exposures (600 s as compared to 1200 s).
The dotted line shows the location of the median, calculated for a total of 3373
objects, of which roughly 2750 are in the range shown here; the median is an
excellent estimator of the mode of the distribution. The scale factor increases
as the air mass decreases over the exposure sequence (see Fig. 12.13).
12.4. Astrometry
225
In the same way, getbettercoords.pl finally compares the FWHM of the
objects across the exposures. Ideally this would only take stars into account,
but since this comparison is merely used as indicative here, we used all the
objects (this is not a serious matter since the catalogues were restricted to fairly
compact objects in the first place). Only in one field did the FWHM in one
exposure deviate from the remainder of the sequence (A209, R, 12 exposures
in total) to such an extent that the exposure had to be discarded.
The different chips in the CFH12k mosaic do not have exactly the same
sensitivity, given by Ai,λ in (12.9), so that under identical conditions one and
the same object would result in different signal (in ADU) on different chips.
In principle, the wavelength dependence of Ai,λ also varies for different chips;
for the CFH12k this is in particular true for the chips 03, 04 and 05, which have
different characteristics than the others (Sect. 12.1). For high precision photometry, the chips should therefore be treated independently. The photometric
requirements for our projects are not particularly high, and we therefore neglect the different wavelength dependence, and only determine global chip
scaling factors Ai (i = 0 . . . 11) for each filter used. Ideally, one would like to
use object fluxes as before; however, the number of objects that are observed
on different chips during a series of exposures is too small to derive statistically robust chip scale factors, in particular since these also depend to some
extent on the colours of the objects used. I therefore choose to determine the
chip scale factors from the sky levels in adjacent parts of the chips. All the
scales are referred to Chip 04, which has the highest quantum efficiency (Kalirai et al. 2001). First the chips adjacent to Chip 04 are scaled to the reference
chip by using the sky levels in adjacent regions on the edges of the chip, then
the scales of the chips further away are related to these (Fig. 31). The final chip
scale factors are averaged over all the exposures in a given filter taken during
an observing run and are listed in Table 12.1. The chip sensitivity typically
varies by only a few percent. Note the significantly higher sensitivity in the I
band of the HiRho chips 03, 04 and 05 compared to the other chips (Sect. 12.1).
12.4.3
SWarping the images
Now all the ingredients are in place to allow the actual physical registering of
the images onto a common output grid: the exposures have the correct world
coordinate system, the translation from pixel coordinates to equatorial coordinates, and the relative photometric scale factors between chips and between
exposures are known. For the actual transformation of the images I use the
program SWarp (Version 1.21, Bertin 2001).
In a first step the dimensions of the output frame have to be determined, so
that it contains all of the input images. This is done by running SWarp on all
the exposures to be combined in the final image. In a first step, SWarp reads
226
06
07
08
09
10
11
00
01
02
03
04
05
Figure 12.12: Scale factors between neighbouring chips are determined from
the mean sky levels measured in adjacent rectangular regions; from these a
scale factor for each chip is determined, relative to Chip 04. Multiplying the
four relative scale factors measured at the vertices of four chips in the direction
indicated by the arrows should result in unity and provides a check.
Table 12.1: The chip scale factors determined for our CFH12k observing runs.
No V band exposures were taken during the May/June 2000 observing run.
Chip
B
00
01
02
03
04
05
06
07
08
09
10
11
1.050
0.998
0.999
1.003
1.000
1.026
1.012
0.999
1.002
1.003
1.017
1.082
November 1999
V
R
1.016
0.993
0.995
1.003
1.000
1.010
0.992
0.992
0.993
1.000
0.995
1.023
1.022
0.988
0.995
1.000
1.000
1.018
1.005
0.990
0.998
1.007
0.999
1.052
I
0.943
0.944
0.934
0.977
1.000
1.008
0.961
0.938
0.908
0.916
0.948
0.943
May/June 2000
B
R
I
1.025
1.003
1.002
1.000
1.000
1.009
1.011
1.005
1.002
1.005
1.011
1.042
1.041
0.998
1.005
1.002
1.000
1.024
1.032
1.004
1.007
1.010
0.996
1.064
0.943
0.943
0.931
0.974
1.000
1.009
0.969
0.944
0.903
0.920
0.952
0.950
12.4. Astrometry
227
the world coordinate systems of the input images and computes from these
the central equatorial coordinates of the output frame, a pixel size and the size
of the output image in pixels. I chose the pixel scale as the median of the input
pixel scales; since the latter vary across the field (Fig. 12.10) the resulting pixel
scale is a little smaller than the nominal CFH12k pixel scale (0.00 205 vs. 0.00 206).
A choice has to be made regarding the projection from the spherical equatorial coordinates to the plane output image. I choose a zenithal equal area projection which has the advantage that the resulting pixel scale is constant across
the field; over the field of the CFH12k the choice is however not critical8 . In
the corners of the CFH12k field the difference between a ZEA projection and a
tangential projection is less than a tenth of a pixel.
I abort this first run of SWarp after the computation of the output frame dimensions and rerun SWarp on the individual exposures (in the form of multiextension fits files, MEF), with the results from the first run specified manually
in the configuration file. It is possible to let SWarp do the transformations of
all the images at once as well as the final combination. However in this way
SWarp produces an enormous number of temporary images, a full-sized output image for each chip in each exposure. Available disk space is easily filled
that way. For data from mosaic cameras this is however not at all necessary:
since there is no overlap between the chips, they can be combined into one
output image straightaway, each output pixel containing the original information from one and only one input pixel. This limits the number of temporary
files to twelve during the transformation of an individual exposure, plus the
number of already finished individual transformed images.
The image transformation involves necessarily a resampling of the input
images, since the shifts between the exposures are non-integer in pixels, and
in general the pixel scale changes during the transformation. Resampling
degrades the useful information contained in an image since it changes the
Fourier spectrum. An obvious consequence is noise between adjacent pixels
in the output image is correlated, whereas the noise in the input images is
strictly uncorrelated. Ideally therefore, one would like to avoid resampling
altogether. In the context of weak lensing analyses one could measure galaxy
shapes from the individual exposures, and then average over these measurements; camera distortion could be corrected for by simply subtracting a corresponding effective shear term. In principle, this method should provide the
same information as measuring galaxy shapes from a combined registered image without the degradation due to the resampling. A disadvantage would be
the presence of hot pixels and cosmic ray hits in the individual exposures, in
particular in deep images taken at high altitude as in our case; these have the
potential of heavily biassing the shape measurements of individual galaxies.
comparison with terrestrial cartography, the field of the CFH12k of 400 × 300 corresponds to an area of 74 × 56 km2
8 For
228
To my knowledge, no systematic investigation into the effects of resampling
on galaxy measurements has been conducted yet; this might be a point to investigate in the future.
For the moment we stick to the traditional way of combining the exposures
before measuring galaxy shapes, and try to minimize the image degradation
through resampling by the choice of an “optimum” interpolation kernel
f˜(x) = k(x) · f
(12.11)
where f is a column matrix containing the values of adjacent pixels and the
interpolation kernel k is a row matrix with components k i (x) = h(x − xi ). We
choose the LANCZOS3 kernel, i. e.
h=
∏
d=1,2
sinc πxd sinc
π
x
3 d
(12.12)
where the product is over pixels (x1 , x2 ) less than 3 distant from the central
pixel x. Bertin (2001) argues that this kernel preserves the signal more accurately than simpler kernels like the bilinear kernel, while creating only modest
artifacts around image discontinuities (a problem with sharply band-limited
kernels).
Running SWarp on each input MEF leaves us with an output image sized
mosaic for each exposure, plus an equal-sized mask image (Fig. 12.2), also
for each exposure. I use static masks (provided by Jean-Charles Cuillandre)
which identify bad columns and rows in the CFH12k chips. The chip cosmetics
change slightly over time, so that I used different masks for the data from
the November 1999 and May/June 2000 runs. For each exposure the mask
is transformed in exactly the same way as the actual image, so that the mask
follows the transformation of the bad columns and rows in the image. Even
more importantly, the transformed mask defines the gaps between the chips
in the exposure mosaic.
The masks are then applied to the transformed images by setting masked
pixels to large negative values. Finally, the exposures are combined into the final image using the I RAF task imcombine. This task offers a variety of methods
to reject deviant pixels, i. e. hot or cold pixels or pixels affected by cosmic ray
hits or even satellite trails. The efficiency of these methods depends primarily
on the number of exposures that are available for statistics. Unless the number
of pixels is very high (at least ten), I found that median combination provides
the most efficient cosmic ray rejection. Masked pixels are ignored by setting
the lower threshold to an appropriate value.
12.5. Photometric Calibration
12.5
229
Photometric Calibration
In Sect. 12.4.2 the relative photometric scaling between the exposures in a sequence was discussed. The absolute photometric of the CFH12k images concerns the transformation from pixel values (analogue data units, ADU) to photometric magnitudes in a standard photometric system. This involves the determination of the photometric zero-point as well as the atmospheric extinction coefficient Eλ .
The extinction coefficient is determined from a sequence of exposures of the
same field, covering a sufficiently large range in airmass. The relative scales si
are determined by the script getbettercoords.pl as described in Sect. 12.4.2.
The effective airmass for a given exposure is approximated by Eq. (12.10) with
Eq. (12.8). The zenith distance, which determines the airmass at a given moment, is given by the right ascension α and declination δ of the field, the geographical latitude φ of the telescope and the local sidereal time (LST) according
to
1
= sin δ sin φ + cos δ cos h cos φ
(12.13)
sec z
(Astronomical Almanac 1992), where h = LST − α is the local hour angle of the
field. After correction for possibly different exposure times between the exposures the photometric scales si are converted to magnitudes, ∆mi = −2.5 log si
and a linear fit of ∆mi versus (sec z)eff,i is used to determine the slope Eλ . Fig.
12.13 shows an example.
I use observations of photometric standard fields from Landolt (1992) to
determine the photometric zero points. Stetson (2000) added new magnitude measurements to the original Landolt catalogues, bringing the number
of available stars up to several hundred for some fields. Still, the distribution
of standard stars in the fields is such, that some chips of the CFH12k do not
contain any standard stars if only a single exposure is taken. In order to save
valuable observing time we did not attempt to cover all the chips with standard measurements by taking multiple exposures. This is justified since the
relative chip scales are close to unity (Table 12.1).
A world coordinate system is determined for the standard field observations in the same way as for the science exposures (Sect. 12.4). Here, the requirements are much less strict because the only use of the WCS is to identify
stars in the CFH12k images with the stars from the catalogues of Landolt (1992)
and Stetson (2000). After scaling the standard field images to an effective exposure time of 1 second the instrumental magnitudes of the standard stars are
determined using SExtractor with an initial arbitrary photometric zero point,
and then corrected for atmospheric extinction using the extinction coefficient
found before. Comparison of the corrected instrumental magnitudes with the
listed standard magnitudes yields a constant offset which is added onto the
initial arbitrary zero point to yield the correct zero point for a 1 second expo-
230
Figure 12.13: Determination of the atmospheric extinction coefficient from
comparison of object fluxes in sequences of science images (cf. Fig. 12.11). The
images are B band images from the November 1999 observing run, the resulting extinction coefficient is EB = 0.17.
sure. For each science image an individual effective zero point is determined
by scaling to the effective exposure time and correcting for atmospheric extinction at the effective airmass of the image. This zero point, as well as additional information (effective exposure time, total exposure time, number of exposures, effective airmass, extinction coefficient), was written to the reduced,
registered and combined image.
For the November 1999 observing run, the values of the photometric zero
points are in good accord with values given on the CFH12k web site. During the May/June 2000 run, however, the conditions were at least partly nonphotometric, possibly due to cirrus. The atmospheric extinction plot (Fig.
12.13) for these data, in particular the I band data taken during the second
night, does not show a simple linear behaviour but fluctuates significantly,
so that any determination of an atmospheric extinction coefficient would be
meaningless. Since the non-photometric coefficients affect the standard field
observations as well, no photometric zero point could be determined for this
run. Fortunately, the seeing quality, which is our prime concern, was not affected. The images can still be calibrated internally using the red cluster galaxy
sequence, albeit with larger uncertainty. In order to do this it is necessary to
correct for galactic extinction which further reddens the objects’ colours. Table
12.2 lists the galactic extinction values for our field; in some cases these values
are significant reaching 0.4 mag.
231
12.6. Prospects
The photometric zero point for a one second exposure were found to be:
B0 = 25.83, V0 = 25.95, R0 = 26.00, I0 = 25.78. The appropriate effective
zero points including effective exposure time and the atmospheric coefficient
appropriate for the observation were entered into the header of the final fits
file.
Table 12.2: Galactic extinction values for the cluster fields. The data are
from Schlegel et al. (1998), accessed through the NASA/IPAC Extragalactic Database (NED, http://nedwww.ipac.caltech.edu/). For completeness,
Cl0024+1654 is also listed, as well as three other clusters for which CFH12k
snapshots were obtained during the observing runs.
Abell 68
Abell 209
Abell 267
Abell 383
Abell 773
Abell 963
Abell 1689
Abell 1763
Abell 1835
Abell 2218
Abell 2219
Abell 2390
Cl0024+1654
Abell 851
Cl0818+5654
Cl0819+7054
12.6
AB
AV
AR
AI
0.400
0.083
0.108
0.143
0.063
0.064
0.120
0.039
0.128
0.105
0.105
0.478
0.246
0.070
0.390
0.131
0.307
0.064
0.083
0.110
0.048
0.049
0.092
0.030
0.099
0.081
0.081
0.367
0.189
0.054
0.299
0.100
0.248
0.051
0.067
0.089
0.039
0.039
0.074
0.024
0.080
0.065
0.065
0.296
0.153
0.044
0.241
0.081
0.180
0.037
0.048
0.064
0.028
0.029
0.054
0.017
0.058
0.047
0.047
0.215
0.111
0.032
0.175
0.059
Prospects
In June 2002 all the CFH12k were reduced and are now available for scientific
analysis. The main use of these images is for reconstructing the large-scale
mass distribution in the clusters from shear measurements on background
galaxy shapes. This is work done mainly by Sébastien Bardeau and uses software developed by Sarah Bridle (im2shape) to measure galaxy shapes and to
correct for PSF distortion, and by Phil Marshall to reconstruct the mass dis-
232
tribution from the shear measurements, using a maximum entropy method
previously developed by Sarah Bridle (Bridle et al. 1998).
Images are available in at least three passbands for all of our clusters. This
makes it possible to obtain independent mass reconstructions for the three filters, allowing an assessment of the importance of noise fluctuations and hence
of the reality of reconstructed mass peaks. However, the images taken in the
three filters are of different quality with the R band images offering the best
combination of image depth and quality (seeing FWHM), so that the final mass
reconstruction should be based on these images. The colour information provided by the images taken in the other filters will have to be used to robustly
distinguish between foreground, cluster and background galaxies. Inclusion
of foreground and cluster galaxies in the sample on which the shear measurements are made will necessarily dilute the shear signal and thus has to be
minimised.
The images can and will be used for a variety of other purposes: Galaxygalaxy lensing is based on the same measurements as the weak lensing mass
reconstruction and aims at reconstructing the mean mass distribution in cluster galaxies acting as individual lens components on background galaxies; this
is work done by Marceau Limousin.
Apart from deforming background galaxy shapes through the shear effect,
clusters also change the observed number density of background galaxies seen
through the cluster (Sect. 4.2.3). The CFH12k images provide sufficient depth
to investigate the cluster mass distributions based on this depletion effect; of
particular interest in this context are the B band images, since the galaxy number counts are steepest in this filter, thus producing a strong depletion signal.
Depletion analysis of the CFH12k images should be based on the methods developed by Christophe Mayen in the course of his PhD thesis.
The wide field of the CFH12k images offers the opportunity to look for
transient objects. Even at the telescope we noticed objects which moved a significant distance across the field during a series of exposures of the same field,
i. e. over the course of an hour or two. These objects are probably asteroids.
Obviously the length of the path observed in the images is too short to reconstruct the orbit and hence identify the object. The images of Cl0024+1654
obtained in service mode in autumn 2001 offer more advantageous time lines
to look for stationary transient objects, e. g. optical afterglows of gamma-ray
bursts. The B band images in particular were taken in four batches, of which
the first two are separated by one week, and the other two by a month respectively. In collaboration with Jean-Luc Atteia we will analyse these images and
develop methods to securely identify stationary transient objects. This work
should be seen as a feasibility study for routine analysis of images from the
CFHT Legacy Survey.
Réduction des données CFH12k
La caméra CFH12k, qui est montée au foyer primaire ( f /4) du télescope
Canada–France–Hawai‘i, consiste de douze chips CCD, dont chaque est composé de 2k×4k pixels. Avec une échelle des pixels de 0.00 206, le champ couvert
sur le ciel est 42 × 28 arcmin2 . Les chips sont de deux types différents (HiRho
et EPI).
A l’époque des observations aucun système complet pour la réduction de
ce type de données existait, donc j’ai du développer mes propres outils pour
certains étapes, en particulier pour la calibration astrométrique. Depuis, plusieurs systèmes ont été développés (par exemple Terapix, ELIXIR), qui donnent
des résultats comparable aux procédures décrites dans la suite. Les étapes dans
la réduction des données CFH12k sont montrés dans la figure 12.3.
Les premières étapes, soustraction du biais et division par un “flat–field”,
se font de fa con standard, avec le package mscred sous I RAF. Des petites modifications au niveau des informations données dans les fichier fits, contenant
les images, étaient nécessaires pour que MSCRED puisse traiter ces images.
Les images observées dans le filtre I montrent des franges d’interférence
dues aux raies d’émission provenant du ciel. Ces franges ont été enlevées
par soustraction d’une image de correction construite à partir d’un nombre
d’images scientifiques dans lesquels les objets ont été masqués avec l’aide du
logiciel SE XTRACTOR.
Un autre effet additif concerne surtout les images prises dans le filtre R : Le
filtre étant un filtre interférométrique, sa largeur dépend de l’angle d’incidence
des raies de lumière. Il est donc effectivement plus large aux bords du champ
que dans le centre. L’effet est négligeable en ce qui concerne la photométrie
des objets dans l’image ; en revanche, il ajoute une structure additive au fond
puisque les flancs du filtre R (et dans une moindre mesure du filtre V) sont
coı̈ncidents avec des raies en émission du ciel, notamment la raie Na D. Cet
“anneau du ciel” pourrait être traité de la même fa con que les franges dans
le filtre I. Puisque cette structure est de grande échelle, il est préférable de la
soustraire par le modèle global du fond du ciel dans une étape postérieure,
afin d’éviter l’addition du bruit.
233
234
Chapitre 12. Reduction of CFH12k data
L’étape la plus importante dans la réduction des images CFH12k est l’astrométrie relative des différentes poses d’un même champ. L’utilisation principale de ces images est pour une analyse weak lensing, ce qui nécessite la
mesure précises des formes de galaxies faibles dans le champs. Les formes
peuvent être dégradé de deux manières principales : (i) Le système optique
(télescope, caméra) introduit une distorsion globale, essentiellement une variation de l’échelle des pixels travers le champ (figure 12.10), ce qui ajoute un
shear aux images des objets. Cet effet est faible, il atteint environ 2% aux bords
du champ de la CFH12k. (ii) Entre les différentes poses le télescope est bougé
d’entre 600 et 1500 , afin d’éviter qu’un objet tombe sur les mêmes pixels entre
les poses. Cette démarche permet de remplir les espaces entre les chips qui ne
sont pas couvertes par une seule pose, ainsi que les parties inutilisables des
chips, dues au défaut cosmétiques. La combinaison des différentes poses requiert donc le recadrage des poses sur un repère commun ; ce recadrage doit
être fait avec la meilleure précision possible, sinon les formes des galaxies sont
dégradées fortement et irrécupérablement.
La procédure de calibration astrométrique est montré à la figure 12.9. Dans
un premier temps, on utilise un repère extérieure, donné par une image du
champ du Digital Sky Survey (DSS). Le DSS représente les plaques photographiques du Palomar Optical Sky Survey (et projets suivants couvrant le ciel
entier), digitalisées sur une échelle de pixels de 100 (deuxième génération ; la
première génération était digitalisée sur 1.00 7). Les images du DSS comportent
leurs propres distorsions astrométriques ; elles contiennent en revanche une
calibration astrométrique précises dans le “header” du fichier fits. La distorsion de l’image DSS peut donc être enlevée en transformant l’image avec le
logiciel SWarp, décrit plus tard ; en même temps, l’image DSS est magnifiée sur
la même échelle de pixel que les images CFH12k (0.00 205).
Des catalogues d’objets compacts et non-saturés sont alors construit pour
les images CFH12k et l’image de référence DSS avec SExtractor ; les translations entre les poses CFH12k d’une série de poses d’un même champ sont mesurées avec I RAF. La correspondance entre trois objets pour chaque chip de la
première pose et l’image DSS est établie à l’œil avec le script reg to image.pl,
ce qui permet de calculer une transformation linéaire entre les systèmes de coordonnées des chips de la CFH12k et l’image de référence. Le script mkmatchfiles.pl utilise ces transformations ainsi que les translations entre les poses
afin de créer automatiquement des catalogues de correspondances pour tous
les objets communs entre les images CFH12k et DSS. Le script comp trafo.pl
calcule alors une transformation de forme polynomiale de quatrième ordre
pour chaque chip de chaque pose basée sur les catalogue de correspondance,
qui minimise les déviations des positions transformées des objets par rapport
à leur positions dans l’image DSS.
Ces transformations constituent déjà une calibration astrométrique qui établit surtout le positionnement des chips individuelles entre eux et sur le ciel.
Résumé
235
Par contre, elles ne sont pas encore assez précises pour permettre une combinaison des différentes poses sans trop dégrader les formes des objets. La
précision est notamment limitée par la précision de la mesure des positions
des objets dans l’image DSS, digitalisée sur une échelle de 100 . Avec la qualité
inférieure des images photographiques à la base du DSS, les positions des objets ne sont mesurable qu’avec des erreurs d’environ 0.00 2, donc environ un pixel
CFH12k, ce qui donne aussi la précision des transformations astrométrique
dans cette étape.
Il est possible d’améliorer la précision des transformations en s’appuyant
sur des données internes aux images CFH12k uniquement, si on minimise non
les déviations des positions transformées des objets par rapport à leurs positions dans l’image DSS, mais les déviations des positions transformées entre
elles. Pour cela il est d’abord nécessaire d’identifier le même objet à travers
les poses individuelles, ce qui est fait par le script mkmastercat.pl qui crée
un catalogue maı̂tre identifiant pour chaque objet sa position dans les catalogues individuelles. Un avantage supplémentaire est que désormais tous les
objets dans les images CFH12k peuvent être utilisés, et non seulement ceux qui
sont aussi détectés dans l’image DSS, ce qui double voir triple le nombre d’objets sur lesquels s’appuie le calcul des transformation astrométrique. Dans les
meilleurs cas, ce nombre atteint environ 300 pour un chip. Le script mkmastercat.pl identifie aussi des objets qui se trouvent sur différents chips dans différentes poses, ce qui permet de faire un lien entre les transformations pour
ces chips qui seraient sinon indépendante. Le script getbettercoords.pl applique la première transformation sur les coordonnées de tous objets dans les
catalogues initiales, et en utilisant le catalogue maı̂tre calcule pour chaque
objet la position transformée moyenne, et crée donc des catalogues de correspondance pour chaque chip dans chaque pose qui opposent la position
transformée moyenne á la position dans l’image CFH12k. A partir de ces catalogues, le script comp trafo.pl calcule de nouveau une transformation polynomiale de quatrième ordre qui est donc précise à entre un dixième et un
vingtième d’un pixel. La solution peut être stabilisé en itérant la dernière étape,
ce qui vire les quelques mauvaises identifications qui peuvent être présent
dans les catalogues de correspondance.
Pour des raisons techniques, les coordonnées de référence sont transformées
en coordonnées équatoriales, α et δ, avec wcstran sous I RAF, et les transformations finales sont calculées avec CCMAP pour chaque chip de chaque poses. La
précision finale est d’environ 0.00 01.
La transformation “physique” des images est alors effectuée avec le logiciel
SWarp de E. Bertin, qui utilise les transformations calculées auparavant pour
créer pour chaque pose une image mosaı̈que d’environ 12000 × 8000 pixels,
regroupant les douze chips de la CFH12k. SWarp soustrait le fond du ciel dans
cette étape. Les mauvaises colonnes et les espaces entre les chips sont alors
masquées en remplaçant par une valeur magique, et les images sont multi-
236
Chapitre 12. Reduction of CFH12k data
pliées par des facteurs d’échelle qui ont été déterminées auparavant par le
script getbettercoords.pl en comparant le signal registré pour les mêmes
objets à travers les poses. Finalement les poses sont combinées avec imcombine
sous I RAF, pour aboutir à l’image mosaı̈que finale.
La dernière étape dans la réduction des images CFH12k concerne la calibration photométrique, ce qui implique la détermination de deux paramètres,
l’extinction atmosphérique et le point zéro photométrique. Le premier paramètre est déterminée à partir des facteurs d’échelle d’intensité en fonction de
l’airmass effective sec zeff à laquelle la pose était observée. Le point zéro est
déterminé à partir des observations de champs d’étoiles standards traitées de
la même manière que les images scientifiques.
Appendix A
Spectroscopic Survey on Cl0024: The
Data
The full catalogue listing all the obects observed for the spectroscopic survey
of Cl0024+1654, described in Chapter 7, is shown in Table A.1. The table also
contains objects from the catalogue of Dressler et al. (1999); for these I have
checked (and in some cases corrected) the redshift listed in those authors’ catalogue, remeasured equivalent widths for consistency and provided photometry and astrometry from the images described in Sect. 7.1.1. This catalogue
is available in electronic form at the Centre de Données Stellaire (CDS)1 . All
entries from the catalogue are marked with their redshifts in Figs. A.2a-A.2h
in Chapter 7.
In detail the contents of Table A.1 are as follows:
Column 1: Object number. The catalogue is sorted by relative right ascension
(Column 2).
Column 2/3: Right ascension and declination relative to α = 00h 26m 35.s 70, δ =
17◦ 090 43.00 06 (J2000), given in arcsec2 .
Column 4: Redshift.
Column 5: Redshift reliability code: A = “secure”, B = “probable”, C = “possible”, D = “uncertain”, S = “star”. For objects taken from Dressler et al.
(1999) and not observed by us, we give their quality code (ranging from
1 to 4).
Column 6: V magnitude (SExtractor MAG BEST), from CFH12k image.
1 http://cdsweb.u-strasbg.fr/cats/J.A+A.htx
2 Originally,
this was the position of galaxy 373, but a more accurate astrometric analysis
shifted the reference point by 0.00 79 to the south-east of this galaxy.
237
238
Appendix A. Spectroscopic Survey on Cl0024: The Data
Column 7: V − I colour, measured in 14 pixel (2.00 8) diameter apertures from
the CFH12k (V) and UH8k (I) images. Occasionally I magnitudes (and
hence V − I colour) are not available due to the object falling on a gap
between two chips of the UH8k (see Sect. 7.1.1).
Column 8: [O II]λ3727 equivalent width (in Å). The error was estimated using Eq. (7.2). We use the convention that equivalent widths are positive
for emission and negative for absorption lines. For objects taken from
Dressler et al. (1999) and not observed by us, we remeasured the equivalent widths ourselves from their spectra, so as to provide homogeneous
equivalent width measurements.
Column 9: [O III]λ5007 equivalent width (in Å).
Column 10: Hα equivalent width (in Å).
Column 11: Hβ equivalent width (in Å).
Column 12: Hδ equivalent width (in Å). The error was estimated using Eq.
(7.2).
Column 13: Strength of the 4000 Å break.
Column 14: Signal-to-noise ratio measured between 4050 and 4250 Å.
Column 15: Number(s) of the observing run(s) during which the object was
observed (cf. Table 7.1). Objects observed by Dressler et al. (1999) are
marked by “D” and the number from their catalogue.
Table A.1: The full spectroscopic catalogue for Cl0024+1654
num
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
αrel
δrel
z
Q
V
V− I
[O II]
[O III]
Hα
Hβ
Hδ
−498.0
−485.0
−465.1
−462.6
−462.2
−460.4
−458.6
−457.6
−457.1
−453.9
−452.0
−450.5
−450.2
−450.2
−449.2
−445.3
−444.4
−444.0
−442.7
−439.7
−438.4
−435.4
−434.8
461.2
631.1
399.8
559.2
489.9
644.2
652.0
431.1
47.4
318.2
−204.5
195.2
−627.5
543.7
−309.6
101.5
654.9
−469.4
−530.6
73.4
247.1
−367.4
681.7
0.6236
0.1756
0.3189
0.3947
0.7235
1.3939
?
0.3182
?
0.3910
0.3940
?
0.3817
0.3945
0.3590
0.3181
0.3818
0.6439
0.3585
0.2129
0.8165
0.3961
0.1567
A
A
A
A
A
A
A
A
A
D
A
A
A
A
B
C
A
D
C
A
22.38
21.27
22.16
20.97
22.63
22.43
21.90
21.81
22.12
22.14
20.61
21.71
21.67
22.11
22.13
20.16
20.07
23.60
20.94
21.04
22.32
22.51
20.75
1.73
0.75
0.95
1.97
2.02
0.87
0.82
1.59
1.07
1.98
1.89
1.53
1.24
1.47
1.97
1.81
1.32
1.09
1.46
0.74
43±3
41±2
22±4
6±2
8±3
23±4
12±2
22±1
10±5
8±1
59±3
12±3
34±10
34
13
5
15
19
20
36
10
5
13
13
4
8
10
−2±1
−10±1
−3±1
−3±1
−4±1
-
4000 Å
S/N
id.
1.0
1.2
1.1
2.0
1.2
0.0
1.5
0.0
1.4
2.2
0.0
1.5
1.5
1.5
1.0
2.0
1.0
1.5
1.1
5.0
17.7
5.6
19.8
3.5
5.4
8.1
5.9
6.1
8.8
4.1
2.2
9.7
9.8
11.9
15.8
0.9
5.5
15.4
1.2
6.8
24.3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
. . . continued
239
. . . continued
num
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
αrel
δrel
z
Q
V
V− I
[O II]
[O III]
Hα
Hβ
Hδ
−431.0
−430.8
−430.4
−430.4
−428.1
−427.1
−427.1
−426.8
−426.5
−426.2
−424.8
−424.5
−424.5
−423.9
−421.2
−420.8
−420.4
−418.4
−417.9
−415.6
−408.5
−406.2
−404.6
−403.9
−403.9
−403.4
−400.6
−400.0
−398.4
−398.3
−397.1
−394.4
−393.6
−389.0
−386.2
−386.0
−383.5
−383.1
−381.8
−381.7
−381.5
−380.9
−378.8
−377.9
−377.6
−375.6
−374.2
−372.6
−372.6
−372.6
−370.6
−368.9
−368.6
−367.2
−366.9
−366.8
−366.5
−363.0
−361.9
−361.3
−359.6
−359.4
−358.7
−358.7
−358.2
−356.7
−350.7
−349.9
−342.2
−337.2
−336.5
−336.1
−335.7
−334.9
−334.8
350.5
−136.0
−667.2
−641.8
576.2
−121.8
−304.4
453.2
447.7
167.5
−476.2
217.6
−330.1
−602.8
−190.9
−522.4
−279.7
209.7
−275.7
−317.2
−206.7
−255.2
441.7
−240.7
−354.2
167.4
35.5
−490.3
527.3
−457.6
121.0
−255.4
−506.3
316.4
−590.2
−171.3
−665.6
328.1
−410.1
−627.2
−686.6
−42.3
−364.8
−400.8
464.0
−412.3
209.7
111.7
−263.1
480.8
283.3
629.6
133.4
−381.1
−157.8
−550.6
−117.5
−232.8
613.8
−192.9
−176.0
671.5
−423.7
−453.4
−179.6
−503.3
−40.2
−349.4
−679.6
−608.8
397.9
−380.8
453.1
−601.9
480.1
0.0000
0.3186
0.3588
0.5939
0.3930
0.0000
0.3804
0.9368
0.8320
?
0.0994
0.3256
0.3592
0.1588
0.5834
0.5363
0.6805
0.3924
0.2932
0.3587
0.5487
0.3792
0.1841
0.0000
0.3593
?
?
0.2742
0.1872
0.3334
0.3003
0.5387
0.7246
0.3957
0.1673
0.0623
0.3932
0.3954
0.5830
0.3765
0.0000
0.2460
0.5239
0.3112
0.3896
0.2644
0.1069
0.3599
0.4869
0.1778
0.2280
0.7440
0.3584
0.2664
0.4445
0.0000
0.3944
0.4066
0.4086
0.3167
0.7129
0.5685
0.2656
0.2667
0.4448
0.4662
0.3944
0.3844
0.4658
0.3174
0.1780
0.2665
0.1174
0.3437
0.3581
S
A
A
C
A
S
A
B
A
A
D
A
A
A
B
D
C
A
A
D
A
A
S
A
B
A
A
A
B
A
A
A
A
A
A
A
D
S
A
A
A
A
A
A
C
A
A
A
A
A
A
A
S
A
A
A
A
B
D
A
A
A
A
A
C
B
A
A
A
A
C
A
22.05
22.35
21.66
22.50
21.18
14.49
21.13
22.39
23.26
21.79
17.87
21.94
20.59
21.13
22.01
22.48
23.21
21.88
21.83
21.37
22.69
22.25
22.36
21.91
21.35
22.21
22.50
23.06
21.40
21.48
21.61
22.35
22.18
22.55
20.11
20.23
21.01
21.63
22.08
22.97
21.72
20.33
22.64
19.92
21.21
20.72
20.33
22.10
21.17
22.06
21.32
21.35
21.12
21.75
21.65
22.24
23.46
22.27
21.98
22.91
23.71
22.68
21.36
20.16
22.49
21.42
21.58
22.09
22.26
21.61
19.94
19.75
20.68
22.42
21.01
3.49
1.60
1.12
1.77
1.10
1.04
1.37
1.06
1.54
1.14
1.82
1.75
0.76
2.03
0.87
1.64
1.81
1.33
0.77
2.80
1.49
1.74
2.59
1.09
0.93
1.38
0.96
2.16
1.55
1.71
0.89
0.74
1.90
1.85
2.25
2.21
2.52
1.25
2.21
1.64
1.90
1.41
0.87
1.36
1.61
0.95
0.84
1.55
2.00
1.53
1.79
2.94
0.62
1.60
1.21
1.07
1.12
1.50
1.28
1.72
1.17
2.25
1.36
1.52
2.00
1.00
1.24
1.27
0.73
1.16
1.50
20±4
36±4
33±6
2±2
23±2
8±2
104±7
10±1
8±2
44±2
15±4
3±2
31±2
60±8
22±2
42±11
31±2
11±1
23±3
20±4
39±4
28±4
20±2
9±2
8±1
4±1
3±2
8±2
16±1
57±6
65±2
3±1
9±2
77±15
17±3
23±2
37±6
5±2
34±3
9±2
10±3
3±3
10±3
22±4
5±1
6±2
54±8
3±1
10
13
51
5
68
9
10
14
8
4
1
20
29
29
2
109
7
-
2
22
20
21
27
6
19
41
27
18
-
6
10
5
13
9
24
4
3
10
4
4
2
1
11
15
14
2
23
6
4
4
4
-
−3±1
−11±2
−6±1
−10±2
−3±1
−13±2
−4±1
−6±1
−4±2
−4±1
−3±0
−4±1
−5±1
−4±1
−3±0
−9±2
−8±3
−1±0
−7±3
−2±0
−4±0
-
4000 Å
S/N
id.
0.0
1.5
1.4
1.4
2.1
0.0
1.2
0.0
1.3
1.4
2.4
1.6
1.1
1.7
1.4
1.1
0.0
1.2
0.0
0.0
1.1
1.2
1.4
1.1
1.4
1.6
1.4
1.1
1.9
1.9
1.4
1.8
0.0
1.3
1.6
1.5
2.0
1.2
1.6
1.5
1.2
1.0
1.4
2.3
1.8
1.7
0.0
1.1
1.2
1.2
1.2
1.4
1.1
1.1
1.7
1.5
1.6
1.5
1.6
1.4
1.5
1.6
1.5
1.2
1.5
5.2
11.7
4.5
16.1
12.0
1.2
0.8
3.2
53.0
2.7
30.8
4.8
14.8
4.5
5.7
5.4
9.2
9.8
10.3
9.3
4.6
14.4
7.1
4.5
3.8
10.1
13.3
7.4
7.3
4.7
10.1
11.3
9.0
7.5
7.6
11.8
3.9
28.6
6.5
30.6
12.6
13.8
20.1
6.7
16.7
7.0
26.6
3.4
15.8
7.9
14.2
2.3
4.9
7.6
5.8
3.5
2.7
13.6
19.4
6.0
7.7
15.2
7.9
6.0
4.7
24.6
20.4
12.3
3.2
20.4
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3, 3
3
3
3
3
3
3
. . . continued
240
. . . continued
num
αrel
δrel
z
Q
V
V− I
[O II]
[O III]
Hα
Hβ
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
−333.2
−331.6
−329.9
−326.8
−323.9
−321.6
−316.4
−315.9
−313.4
−312.3
−312.3
−309.4
−308.4
−307.4
−306.9
−306.1
−305.4
−304.8
−304.3
−303.8
−302.0
−300.3
−298.4
−297.8
−297.4
−296.2
−294.5
−292.9
−289.8
−289.8
−289.8
−288.5
−285.8
−280.2
−276.8
−275.6
−274.8
−272.4
−269.2
−264.9
−260.3
−259.3
−256.5
−252.7
−247.5
−247.2
−241.9
−241.5
−240.8
−234.9
−233.6
−229.5
−228.4
−228.4
−225.9
−225.7
−225.2
−224.3
−219.7
−213.8
−213.7
−213.1
−210.4
−207.8
−205.8
−203.5
−202.1
−200.8
−200.4
−199.8
−199.7
−198.9
−198.1
−195.8
−194.8
99.1
−106.1
−400.1
621.7
158.1
144.6
−279.6
671.6
405.7
232.0
604.9
636.3
550.7
303.0
54.7
695.5
217.4
−690.7
564.7
−227.7
349.9
−142.9
421.0
−148.3
−465.0
592.3
512.8
180.5
199.9
490.7
507.1
243.5
−706.6
−326.4
−98.2
−62.5
−510.0
180.3
−74.9
604.7
303.5
375.0
433.6
279.7
577.5
274.9
472.2
−569.3
−564.0
−213.4
−160.2
131.4
−84.8
−695.8
−315.8
−189.8
−335.3
34.7
−335.8
−260.8
−355.3
−253.6
−291.8
339.5
72.7
−108.2
−160.7
462.8
583.7
97.9
316.5
437.2
663.4
22.7
66.0
0.2134
0.3794
0.3929
0.2122
0.3963
0.3921
0.1639
0.2298
0.4445
0.3934
0.2121
0.3812
0.3051
0.7492
0.8170
0.3817
0.3935
0.2921
0.3589
0.3273
0.8922
0.3589
0.3440
0.3593
0.0000
0.3586
0.3893
0.5244
0.2674
0.2718
0.4943
0.3932
?
0.2341
?
0.3993
0.2927
0.6935
0.3925
0.3818
0.3944
0.9257
0.2128
0.1768
0.4934
0.3936
0.1724
0.4432
?
0.2480
0.3967
0.3949
0.3910
0.1388
0.2655
0.2485
0.3791
0.3938
2.5704
0.3271
0.2663
0.3265
0.4065
0.3938
0.9926
0.3775
0.5268
0.2920
0.3816
0.2456
0.7255
0.3933
0.1463
0.3953
0.3927
A
B
A
A
D
A
A
A
A
A
A
A
A
A
D
A
A
B
A
C
D
A
A
A
S
A
A
C
A
A
A
A
C
A
A
A
C
A
A
D
A
A
A
A
A
A
A
A
B
A
A
A
A
A
2
A
A
A
A
A
A
D
A
C
A
A
A
A
A
A
1
2
20.63
22.81
20.94
21.00
21.88
20.86
21.72
21.31
22.51
22.46
20.67
21.24
21.78
22.76
21.88
22.28
21.35
21.01
21.87
22.47
23.00
21.32
21.12
20.95
21.68
20.75
21.77
22.41
21.61
21.01
22.08
22.41
21.09
22.26
23.75
21.87
21.07
21.73
23.61
22.71
22.80
22.41
20.45
18.47
20.75
20.90
21.05
21.87
21.86
21.61
21.35
22.56
20.75
20.45
20.79
21.19
21.17
22.58
21.45
22.69
21.13
22.93
21.67
21.13
22.22
22.43
21.71
20.79
19.86
20.23
22.29
22.04
21.58
20.88
22.32
0.83
1.50
0.84
1.69
2.14
0.71
0.78
1.37
1.91
0.81
0.96
1.21
1.77
1.09
1.27
1.61
1.84
1.19
1.24
0.88
1.00
1.63
1.00
2.86
1.75
1.74
2.19
1.22
0.83
1.24
1.44
1.11
1.33
1.87
1.67
1.67
2.17
1.38
0.76
1.66
0.84
1.08
1.63
1.71
1.98
0.92
2.10
1.04
1.46
1.95
1.55
2.08
0.90
1.32
0.86
1.36
2.08
0.41
1.80
1.10
1.10
1.58
2.00
0.94
1.24
1.35
1.48
1.26
1.50
1.44
1.65
0.81
1.35
1.45
32±2
47±7
5±1
52±2
42±10
36±3
29±2
16±3
56±2
39±1
12±3
14±2
20±3
3±1
25±3
31±11
49±2
6±2
24±1
3±2
7±2
14±2
26±2
38±2
42±2
12±2
62±19
9±2
4±1
31±3
3±3
22±4
12±1
3±2
22±1
17±5
31±2
16±4
2±2
53±−
80±3
12±1
32±9
15±3
30±9
20±4
5±3
8±0
9±2
38±2
19±3
28±11
7±1
17±3
15
27
22
20
9
81
12
27
10
5
15
2
7
47
14
160
4
2
17
2
4
23
45
14
28
86
5
165
11
13
18
15
-
3
13
8
9
4
18
10
7
13
5
4
7
64
18
8
4
4
2
59
6
4
5
4
3
3
Hδ
−4±0
−4±1
−4±1
−4±1
−7±2
−27±12
−3±1
−3±1
−4±1
−4±0
−4±1
−4±1
−5±1
−3±0
−1±0
−4±1
−3±0
−6±2
−4±1
−4±1
6±2
−5±1
−6±1
−8±2
−7±1
−4±0
−3±0
−4±1
−4±1
-
4000 Å
S/N
id.
1.2
1.2
1.5
1.0
1.7
2.0
1.1
1.4
1.5
1.8
1.2
1.3
1.5
1.5
1.4
1.4
1.8
1.5
1.3
1.6
1.3
0.0
1.9
1.6
1.4
1.2
1.4
1.3
1.5
0.0
2.2
0.0
1.4
1.4
1.1
1.8
1.3
2.0
1.4
1.8
1.6
2.2
1.2
1.9
0.0
1.0
1.9
1.6
2.0
1.9
1.2
0.8
1.2
2.1
1.4
1.2
1.4
1.4
1.7
1.2
1.3
1.4
1.3
1.1
1.7
1.2
1.5
22.4
3.1
11.3
13.1
3.6
14.9
8.4
18.4
21.8
6.6
20.9
18.3
9.6
7.5
0.9
7.2
14.9
12.2
10.4
3.1
0.8
15.8
11.5
27.5
21.5
15.7
8.6
19.2
24.2
10.2
11.3
2.0
2.8
2.1
11.2
12.3
8.0
1.6
4.4
5.7
1.7
21.8
31.6
17.2
21.3
23.2
12.1
3.2
6.9
14.3
3.8
17.5
14.6
14.6
5.5
11.4
6.5
5.2
8.9
5.2
8.0
8.3
0.9
6.6
1.5
9.7
22.2
19.9
8.1
9.9
5.7
24.7
9.2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3, 3
3
3
3
3
3
3
3
3, 3, 3
3
3
3
3
3
3
3
D28
3
3
3
3
3
3
3
3, 3
3
3
3
2, 3
3
3
3
D54
D41
. . . continued
241
. . . continued
num
αrel
δrel
z
Q
V
V− I
[O II]
[O III]
Hα
Hβ
Hδ
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
−194.8
−193.8
−191.1
−185.3
−184.9
−184.4
−180.4
−177.9
−177.6
−177.3
−173.6
−173.3
−167.5
−165.0
−163.2
−163.1
−162.7
−158.7
−155.5
−153.5
−150.2
−149.8
−148.1
−147.2
−146.8
−143.5
−141.0
−139.3
−138.4
−137.6
−137.4
−137.4
−137.3
−135.7
−134.6
−133.6
−132.1
−131.6
−131.0
−128.7
−123.4
−123.1
−123.0
−121.7
−120.2
−119.7
−119.5
−117.1
−117.1
−116.2
−114.5
−111.5
−110.9
−109.5
−108.5
−107.1
−106.9
−105.0
−104.9
−104.8
−104.5
−102.9
−102.8
−99.0
−98.9
−95.3
−95.3
−94.6
−90.7
−89.4
−89.0
−88.7
−88.6
−87.6
−87.3
−138.5
−323.5
654.1
655.7
106.1
90.1
341.0
223.3
72.9
−60.9
257.6
255.1
−160.2
−57.0
110.9
−426.7
−16.1
230.5
4.3
−588.5
124.6
−363.9
333.5
219.8
−43.9
142.6
48.3
488.8
93.6
135.0
516.4
424.1
30.8
49.1
382.3
596.3
−46.6
−328.3
386.6
79.8
−472.5
163.1
−79.4
29.4
2.1
−61.6
−24.2
150.3
195.8
−390.1
246.8
35.6
114.9
−356.0
−35.5
204.9
463.5
572.5
183.7
353.1
69.5
684.6
29.7
−190.0
67.1
−428.1
670.5
41.7
−348.7
−72.4
49.6
−203.8
−364.5
306.2
92.4
0.0000
0.2317
0.3453
0.3450
1.4033
0.3975
0.6242
0.2717
0.3995
?
0.7451
0.3973
0.4121
0.7154
0.3943
?
0.3937
0.2461
0.3939
0.6904
0.3936
0.5329
0.3600
0.3979
?
0.3935
0.4333
0.3983
0.3920
0.3970
0.0583
0.3885
0.2279
0.0000
0.1750
0.4892
0.4137
0.1617
0.3926
0.3928
0.3448
0.3981
0.3808
0.1110
0.9583
0.3942
0.3920
0.3938
0.3970
0.3901
0.3945
0.3285
0.3957
0.3832
0.3925
0.3813
0.3600
0.1962
0.3975
0.3916
0.3958
0.2152
0.3880
0.3880
0.3913
?
0.0000
0.3960
0.2667
0.4172
0.3977
0.3965
?
0.3947
0.3950
S
A
A
A
C
A
A
A
3
A
A
A
A
A
A
A
2
A
C
A
A
A
A
C
B
A
2
A
D
2
S
A
A
A
A
A
A
A
A
2
B
A
A
A
A
A
D
A
B
A
A
A
A
A
A
A
A
A
B
A
S
A
A
C
A
A
A
3
20.87
21.65
20.95
20.40
20.78
21.50
22.21
20.80
20.40
22.86
22.88
22.24
21.96
22.58
21.63
20.95
21.17
20.63
22.36
22.41
22.66
21.75
21.65
22.06
21.55
20.89
23.10
21.59
21.22
22.47
20.56
23.29
22.99
22.90
19.61
22.48
22.41
21.25
21.33
22.17
20.16
21.24
21.99
21.39
21.62
21.93
22.53
20.46
22.41
21.48
21.99
22.44
22.03
21.35
21.83
22.09
22.05
19.29
21.90
22.21
21.59
19.83
22.39
20.70
22.05
22.14
22.25
21.53
20.33
22.34
22.13
21.68
22.30
20.91
22.79
1.53
1.04
1.22
1.12
0.92
1.17
1.68
1.26
1.49
2.56
1.49
1.12
1.74
1.95
1.31
1.44
1.47
1.63
2.62
1.84
1.57
1.70
2.02
1.43
1.21
1.97
1.91
2.11
2.11
0.84
1.08
0.83
3.14
1.36
1.20
2.08
0.85
1.35
1.34
1.53
2.12
1.77
0.98
0.66
1.99
1.85
1.73
1.66
1.05
1.21
1.70
2.01
1.58
2.10
0.88
1.13
1.14
1.99
0.84
1.95
0.86
1.20
1.64
0.98
1.27
0.17
2.04
1.41
1.38
1.89
2.11
1.65
1.05
1.51
24±3
43±2
26±1
11±2
12±2
10±1
33±2
37±2
19±4
21±1
5±2
13±2
5±2
20±1
10±3
45±9
11±2
33±2
19±1
11±2
4±1
25±5
9±1
13±11
9±2
35±4
17±2
7±3
50±3
21±1
6±0
43±4
32±1
17±3
34±2
10±2
28±1
5±3
9
43
8
6
3
3
17
31
6
4
10
49
4
2
20
12
10
5
-
18
15
0
29
0
16
38
-
7
21
19
11
5
7
2
14
10
8
4
5
6
6
7
12
7
3
9
9
9
8
-
−5±1
−6±1
−3±0
−3±0
−4±1
−7±2
−5±2
−4±1
−4±2
−5±1
−3±1
−3±1
−4±2
−8±1
−2±0
−6±2
−5±1
−2±1
−4±1
−3±1
-
4000 Å
S/N
id.
0.0
1.1
1.2
1.2
1.3
1.1
1.2
1.3
0.0
1.4
2.0
1.2
2.3
0.0
1.3
1.4
1.8
1.7
1.7
1.9
0.0
1.5
1.6
1.5
1.6
1.9
1.3
1.2
0.0
1.4
1.4
1.6
1.3
1.4
1.7
1.9
1.8
0.9
1.6
1.8
1.5
1.4
1.4
1.5
2.0
1.5
2.1
1.2
1.3
1.3
2.0
1.4
1.8
1.1
1.3
1.3
1.1
0.0
0.0
1.8
1.6
1.1
1.9
2.8
0.0
1.4
1.3
10.7
16.5
15.8
5.2
12.7
17.1
6.6
1.9
3.7
8.5
5.5
2.3
7.2
5.4
25.3
8.3
11.3
4.7
10.7
8.9
9.8
7.6
13.3
3.5
7.3
9.4
7.9
9.9
2.2
5.9
17.4
11.7
6.1
6.8
17.3
4.2
11.6
10.1
18.1
5.3
6.1
16.9
5.2
7.0
7.6
6.8
6.8
7.5
8.1
6.9
13.9
58.8
5.9
6.5
11.0
28.0
6.8
7.0
9.3
2.1
14.2
26.3
3.3
9.1
7.3
1.8
10.8
3.9
2
3
3
3
2
3
3
3
D43
2
3
3
2
3
3
3
2, 2, 3
3, 3
D60
3
3
3
3
3, D101
2
2, 3, 3
2
3
2, 3
D96
3
3
D115
2
3
3
2
3
3
3
3
2, 3
1
D122, D125
2, PC0023+1653
1
2
3, D97
3, D100
3
3, D102
2
3, D95
3
2
3
3
3
3, D99
3
2, D44
3
3, D63
3, D130
2, 3, D29
3
3
2
3
2
2, 3
3, D79
3
3
D93
. . . continued
242
. . . continued
num
αrel
δrel
z
Q
V
V− I
[O II]
[O III]
Hα
Hβ
Hδ
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
−85.8
−83.4
−83.1
−82.4
−80.7
−79.0
−77.2
−77.2
−77.0
−73.1
−72.1
−70.4
−70.1
−70.1
−68.8
−67.7
−67.1
−66.8
−66.1
−64.6
−64.5
−64.5
−63.6
−63.5
−63.4
−62.8
−62.6
−62.2
−61.8
−61.8
−60.5
−57.8
−57.8
−57.6
−57.2
−56.0
−55.9
−54.6
−54.2
−53.0
−52.7
−52.0
−51.2
−50.9
−49.7
−49.3
−48.7
−48.6
−47.1
−46.1
−46.0
−43.6
−43.6
−43.0
−42.7
−42.6
−41.4
−40.9
−39.7
−39.3
−38.5
−36.4
−36.1
−35.7
−35.5
−35.3
−34.2
−34.1
−33.1
−31.2
−30.8
−30.2
−29.9
−29.9
−29.8
485.4
−75.3
−451.3
450.2
166.1
288.6
57.1
−162.8
212.6
−419.9
328.8
−390.7
370.9
−40.1
505.5
−28.4
−141.3
−329.3
33.5
103.3
239.5
371.5
169.2
−252.8
28.4
−20.2
37.7
596.6
437.4
72.7
480.8
−26.5
444.4
38.8
161.2
136.2
134.1
86.3
−646.2
−1.3
20.1
3.0
185.0
517.7
457.6
−568.0
−415.7
−1.9
174.2
197.5
43.0
189.7
109.5
−106.6
−45.2
42.0
357.5
27.8
40.8
346.2
−58.2
−480.1
−225.8
−6.1
322.2
43.6
559.5
50.4
−401.2
−18.7
−28.1
−22.4
118.3
−25.5
20.5
0.3978
0.4000
?
0.0000
0.3921
0.3956
0.2132
0.3970
0.3902
0.6658
0.6575
0.6451
0.3994
0.3935
0.3953
0.4001
0.3815
0.3950
0.4005
0.3973
0.6231
0.3138
0.3884
0.2130
?
0.3871
?
0.4005
0.3968
0.3960
0.3847
0.3998
0.3992
0.2132
0.3906
0.3929
0.3931
0.2712
?
0.3880
0.4000
0.5806
0.3933
0.5823
0.2997
0.3798
0.3521
0.3982
0.3958
0.0000
0.3973
0.4151
0.3936
0.3929
0.3960
0.3965
0.3963
0.2149
0.3862
0.3814
0.2478
0.3915
0.4081
0.3946
0.3440
0.3940
0.4928
0.3983
0.1494
0.3902
0.2472
0.3956
0.3890
0.2471
0.3906
A
A
S
3
A
A
A
4
B
C
A
A
1
A
1
A
A
A
A
3
A
2
B
A
A
A
A
C
A
A
A
A
3
2
A
1
A
B
3
A
A
A
C
4
A
S
A
D
A
C
2
A
A
3
A
2
A
A
3
B
A
A
4
A
1
A
2
2
3
B
22.52
22.45
21.78
24.09
22.39
20.80
21.67
22.36
23.64
21.90
22.63
22.49
22.01
22.24
21.91
22.85
21.46
22.64
20.41
22.86
23.86
21.83
22.16
20.57
22.08
22.23
22.29
22.49
20.84
20.12
22.50
21.14
22.00
19.39
20.74
22.61
22.38
20.94
21.71
20.82
20.71
21.47
22.94
22.38
19.99
22.12
22.09
22.72
21.97
22.31
21.31
22.71
20.97
22.15
22.00
22.06
21.54
22.93
22.79
21.77
22.38
22.13
22.39
22.38
22.45
21.88
22.28
22.72
20.79
21.06
22.24
21.98
22.69
22.64
23.84
1.05
1.92
0.99
3.10
1.97
2.04
1.37
2.01
1.25
1.91
0.96
1.58
1.44
0.52
0.79
1.08
1.25
1.02
1.53
1.64
2.43
1.55
1.24
1.42
1.58
1.28
1.75
1.59
2.02
2.05
0.83
1.48
2.04
1.30
1.95
1.41
2.02
1.71
1.97
1.01
1.17
1.87
2.30
1.69
1.09
1.83
1.44
2.58
2.11
1.54
1.80
1.85
1.91
1.92
1.89
0.73
2.37
1.53
1.46
0.93
1.36
1.00
2.15
1.68
2.31
0.99
1.46
0.93
1.88
1.21
1.85
32±3
25±5
34±6
29±5
42±3
12±3
69±10
12±4
9±4
17±1
9±1
19±5
10±3
23±1
4±1
45±6
19±3
9±2
27±9
6±2
29±13
8±2
35±7
60±5
23±1
3±2
8±1
32±12
4±1
36±8
36±2
28±6
-
13
26
29
3
11
4
50
10
67
10
21
9
13
6
10
-
12
9
36
0
11
37
16
18
16
-
7
5
3
6
3
7
3
19
71
4
12
11
-
−4±1
−6±1
−10±2
−9±2
−10±2
−8±3
−4±1
−7±2
−2±0
−3±0
−3±1
−4±1
−3±1
−3±1
−12±4
−4±1
−3±1
−7±2
−3±1
−3±1
−4±1
-
4000 Å
S/N
id.
1.3
1.9
0.0
0.0
1.9
2.4
1.6
1.9
1.6
1.3
0.9
1.1
1.5
1.1
1.5
1.4
1.1
1.4
1.2
1.5
1.6
1.7
0.0
1.2
0.0
1.5
2.0
1.6
1.3
1.7
1.8
1.5
1.6
2.3
1.6
0.0
1.8
1.1
1.9
1.6
1.6
1.0
1.1
1.6
1.5
0.0
1.8
1.4
1.8
1.5
1.9
1.6
1.4
1.4
1.4
1.3
1.3
1.3
1.0
2.0
1.5
1.7
1.2
1.4
1.5
2.1
11.7
4.1
2.6
11.9
10.4
5.7
9.4
10.0
6.9
10.7
2.7
6.3
3.6
10.3
11.6
2.1
3.5
6.2
9.9
16.0
25.4
6.1
9.2
1.7
4.9
11.8
25.4
2.6
7.7
6.6
23.6
8.4
8.8
15.1
4.4
13.5
9.5
9.2
5.9
17.5
12.2
5.2
3.5
3.9
8.1
9.0
3.0
11.4
2.7
5.5
6.8
2.8
3.7
8.7
8.2
7.4
8.1
14.2
3.9
6.0
23.2
7.4
18.7
12.4
9.1
17.9
3.2
3.3
3
2, 1
3
3
D98
3
2, 3, D110
2, 3, D103
D83
3
3
3
3
D19, D66
3
D67
1, 3, 3
3
3, D25
3
D128
3
D81
3, D123
2
3, D64
2
3
2, 3
2, 3
3
3
3
2, D126
2
D77
D76
2, D127
3
D21, D65
3, D23
2, D109, D111
D82
3
2, 3
3
3
D45
3
2
2, 3, D46
3
2, D26
3, D87
1, D89
D47
2, 3
2
D48
3
D124
3
3, D71
D30
3
2
3
D92
3
D17
3
D15
D75
D129
3, D70
. . . continued
243
. . . continued
num
αrel
δrel
z
Q
V
V− I
[O II]
[O III]
Hα
Hβ
Hδ
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
−29.5
−29.4
−28.7
−28.1
−27.9
−27.7
−27.4
−27.4
−26.7
−25.6
−21.4
−20.9
−20.9
−20.4
−19.4
−18.9
−17.9
−16.9
−16.8
−16.8
−16.5
−16.3
−15.5
−15.5
−15.2
−15.2
−14.8
−13.0
−12.8
−12.2
−11.6
−11.2
−10.6
−10.2
−9.8
−9.8
−9.6
−7.7
−7.3
−7.0
−6.2
−4.4
−4.3
−4.0
−3.3
−2.6
−2.0
−2.0
−1.7
−1.4
−0.3
−0.1
0.6
0.7
2.9
3.9
4.3
4.9
5.6
6.3
7.7
8.9
9.3
9.5
11.6
12.3
12.8
12.8
14.5
15.1
16.6
17.9
18.2
19.6
20.5
−11.6
156.8
649.0
497.8
287.8
−180.8
424.9
153.8
−46.4
44.5
26.9
−80.7
−120.2
−218.6
39.5
−136.8
−57.6
6.3
−14.1
−147.4
−10.1
33.6
−240.6
−93.0
426.1
72.8
−516.6
−601.3
−56.4
−23.7
−12.3
−147.7
38.5
−571.6
549.9
149.1
302.3
85.9
−4.7
6.2
−75.8
199.2
−70.6
−66.3
272.1
144.5
−10.3
552.2
543.8
17.6
0.7
−28.7
−62.3
−43.4
−15.2
−5.6
3.4
53.3
103.7
−75.5
17.9
−286.2
−616.8
−554.3
−16.2
416.3
85.0
63.2
−120.0
−13.8
17.3
−489.5
29.4
−382.3
−506.7
0.3919
0.3955
0.3553
0.2250
0.8194
0.3926
2.7895
0.3955
0.3992
0.3957
0.3880
0.4040
0.3815
0.3822
0.3850
0.3949
0.3955
0.3900
0.3880
0.3827
0.3830
0.3830
0.3934
0.0000
0.2926
0.3924
0.4081
0.3416
0.3972
0.3960
0.4052
0.3988
0.3882
0.4759
0.3958
0.3960
0.3926
0.3860
0.3968
0.3897
0.3996
0.0000
0.3960
0.3960
0.3963
0.3960
0.3914
0.2153
0.2152
0.5558
0.3871
0.3968
0.3960
0.3955
0.3908
0.3973
0.3936
0.3860
0.3971
0.4014
0.3891
0.3440
0.4063
0.4061
0.3860
0.4440
0.3807
0.3964
0.0000
0.3843
0.3920
0.2668
0.3977
0.3965
0.9322
A
2
A
A
D
2
D
A
A
A
1
A
A
A
1
2
A
D
3
D
2
C
A
A
B
C
3
3
A
3
2
C
B
A
3
3
1
S
A
B
A
A
A
A
2
A
3
C
A
A
A
A
4
A
A
1
A
B
A
3
A
A
A
2
A
2
A
C
21.86
21.64
20.17
22.44
22.89
22.76
20.92
21.92
21.35
22.35
21.68
22.19
21.50
21.43
20.76
22.62
22.75
22.88
23.08
22.42
23.07
21.29
22.48
21.93
20.06
21.59
22.24
20.63
22.35
20.10
21.50
22.52
21.83
22.64
22.71
24.84
22.17
21.08
21.55
22.99
20.69
22.09
23.38
21.39
22.79
21.56
20.70
20.01
22.24
19.55
22.71
22.29
21.41
22.09
22.58
20.14
21.62
21.89
21.09
20.40
20.79
21.31
22.01
20.49
20.46
22.29
17.21
22.15
21.85
20.57
22.12
21.78
22.88
1.97
1.30
1.88
1.03
1.27
0.94
1.97
2.23
1.17
1.13
2.08
2.07
2.27
2.29
1.83
2.28
1.77
0.74
2.47
0.91
1.10
2.10
2.18
1.92
2.00
1.83
2.17
1.28
2.21
1.31
1.29
1.99
1.97
2.41
1.94
1.75
1.84
2.07
0.29
0.51
1.85
2.11
2.02
1.23
1.94
1.95
2.06
1.88
2.04
2.06
1.23
1.52
1.37
1.96
1.12
1.50
1.76
1.88
1.40
1.54
1.58
1.88
1.72
21±6
2±2
39±3
23±1
16±3
27±1
29±2
17±7
49±2
18±4
4±3
7±3
58±11
13±2
6±2
44±4
32±2
30±2
78±5
12±1
29±4
42±4
6±2
6±1
35±6
10
15
6
9
19
12
16
11
48
4
13
6
-
19
19
21
9
-
6
10
6
11
9
8
32
4
13
3
-
−3±1
−3±1
−3±1
−10±3
−5±2
−3±1
−2±1
−3±1
−5±1
−3±1
−3±1
−1±0
−4±1
−5±2
−4±2
−3±0
−3±1
−4±1
−1±1
−3±1
-
4000 Å
S/N
id.
1.9
1.6
1.8
1.3
2.3
1.9
1.1
1.3
1.9
2.2
1.2
1.2
1.9
2.0
1.7
2.5
1.8
1.3
0.0
1.1
1.8
1.4
1.8
1.6
1.7
1.7
1.4
1.6
1.1
2.2
1.8
1.3
2.0
2.0
0.0
1.8
1.6
1.9
1.2
1.1
1.2
1.6
1.8
1.7
1.1
1.7
2.0
2.1
1.8
1.3
1.1
1.3
1.6
1.2
1.6
0.0
1.6
1.1
1.6
1.6
1.7
-
13.3
6.2
16.1
6.3
1.1
29.2
6.7
8.8
6.9
16.0
9.6
5.6
17.0
18.5
6.3
5.4
5.4
13.3
5.4
17.1
19.1
8.3
5.6
8.0
8.0
6.2
8.9
6.7
1.9
5.9
7.9
14.4
11.3
13.1
10.3
1.3
9.7
24.2
10.5
13.7
19.3
4.5
3.2
5.7
13.3
5.9
7.3
8.1
3.9
30.7
18.4
4.5
5.3
10.5
10.2
21.4
10.3
8.7
14.3
8.5
14.3
0.7
2, D90
D80
3
3
3
D84
2
2, D78
2, 3, D11
2, 3, D72
D49
1, 3, D88
1, 3, D105
3, D94
D20
D86
1
2
1
D85
1
D18
3
1
2, 3
2, D22, D73
3
3
D107
D12
2
D104
D91
3
3
1
2, 3
D24
D68
D69
1
3
3, D106
3
2, 2, 3
1
2
3
3
D116
3, D14
D31
3
2, D7
2
3
3, D13
D27
3, D74
2, 1
D50
3, 3
3
3
D9
2
2, 1, 3
1
1
2, D8
D51
3
D32
3
3
. . . continued
244
. . . continued
num
αrel
δrel
z
Q
V
V− I
[O II]
[O III]
Hα
Hβ
Hδ
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
20.6
22.2
22.2
22.6
22.8
22.9
23.2
23.4
23.4
24.6
25.1
25.5
25.9
27.1
29.4
31.2
31.4
32.5
33.8
37.4
38.5
39.6
41.7
41.7
42.3
43.4
43.7
44.4
45.1
45.9
47.7
49.7
50.0
50.6
51.2
59.5
59.5
61.3
62.9
63.4
63.4
65.1
67.9
68.7
68.8
69.2
69.7
71.7
72.8
74.7
78.1
78.2
78.2
78.5
83.3
84.4
85.3
87.4
87.7
88.4
89.0
89.0
89.4
95.7
95.7
96.7
99.5
99.5
100.3
101.3
103.3
106.1
106.5
107.3
108.1
−439.5
17.8
−460.2
325.8
−253.9
53.8
−112.4
−87.1
−524.7
−657.4
−72.3
167.3
−33.8
−14.5
−65.2
−20.7
−4.9
48.1
−581.7
−51.6
16.2
−131.5
331.3
8.0
−241.1
641.2
31.1
17.0
−42.0
39.0
19.3
−30.0
−608.7
−463.1
154.9
−14.3
252.4
−0.7
−18.9
−47.6
−644.9
646.1
259.8
−266.4
−63.0
−14.0
634.7
−31.0
369.2
5.5
19.6
−24.4
−146.0
−9.5
79.0
−195.8
−188.3
106.4
777.0
62.5
10.2
182.4
267.1
159.3
−40.8
−10.8
−209.6
8.1
38.0
374.8
−398.5
715.3
−53.3
87.4
443.6
?
0.3937
0.2685
?
0.3925
0.3964
0.3910
0.3940
0.3970
0.3815
0.3921
0.6579
0.3900
0.3900
?
0.3987
0.3898
0.3895
0.4741
0.3951
0.3977
0.3911
0.3930
0.3910
0.3950
?
0.3937
0.3919
0.0000
0.3950
0.3953
0.3855
0.8466
0.1763
0.2250
0.3833
0.3440
0.3950
0.3989
0.3900
0.4076
0.0609
0.3960
0.4750
0.3910
0.1393
0.1674
0.4010
0.1171
0.3962
0.3955
0.3992
0.1666
0.3941
0.3924
0.2999
0.2983
0.3944
0.3447
0.3892
0.3935
0.3765
?
0.2860
0.3926
0.3990
0.5384
0.3931
0.1840
0.3982
0.5316
0.1121
0.3954
0.3920
0.1847
2
A
A
A
A
A
D
A
A
4
A
A
2
A
A
2
A
A
S
1
3
A
D
A
B
A
4
A
C
A
A
A
4
3
A
A
A
A
A
A
A
A
A
A
A
A
2
A
A
B
B
A
A
C
22.89
22.49
21.13
20.19
22.54
21.53
20.02
22.13
20.68
21.90
21.06
23.00
21.94
20.45
21.57
20.52
21.30
22.90
20.61
22.11
22.45
20.99
21.33
21.45
22.11
21.44
21.65
21.54
21.70
20.85
21.88
22.66
21.94
21.33
22.01
21.60
21.81
22.89
21.88
21.98
21.72
18.38
22.04
21.89
22.24
19.33
19.37
20.98
20.05
20.80
20.72
21.60
20.22
21.97
20.36
21.22
22.10
22.48
19.75
22.80
20.11
21.28
22.93
23.29
22.89
21.38
21.33
20.89
21.60
22.69
22.35
19.77
20.73
21.84
21.44
1.85
2.09
1.30
1.04
1.09
1.38
2.02
1.81
1.37
2.20
2.05
2.65
2.08
2.05
0.93
2.05
1.81
1.62
1.20
1.79
1.96
1.65
1.69
1.19
1.65
1.47
1.98
1.96
2.53
2.04
1.83
2.02
1.50
0.70
0.97
1.40
1.66
1.97
1.45
1.82
2.13
0.95
2.02
2.25
1.83
1.45
1.03
1.58
0.98
1.77
2.06
0.84
1.32
1.77
1.46
1.18
0.95
1.87
1.79
1.71
1.52
1.57
1.84
1.36
1.22
0.81
1.66
1.02
1.69
1.40
0.90
0.97
2.03
1.45
0.94
22±5
14±2
18±3
9±3
7±3
76±6
−2±3
13±3
13±2
4±2
4±2
30±9
39±3
23±2
87±4
7±2
18±1
47±10
32±4
6±2
27±1
46±5
49±14
5±2
-
2
4
15
7
66
6
7
14
125
4
21
8
16
-
11
64
13
24
4
17
11
38
31
-
4
5
16
4
22
4
5
5
5
10
43
3
2
8
10
9
8
-
−5±1
−8±2
−3±1
−2±1
−3±1
−3±1
−5±2
−2±1
−3±1
−4±1
5±1
−3±1
−4±1
−4±1
-
4000 Å
S/N
id.
0.0
1.6
1.3
0.0
1.2
1.2
1.4
1.3
1.9
2.0
1.8
0.0
1.4
1.9
1.9
1.4
1.7
1.5
1.5
1.1
1.3
0.0
1.8
0.0
1.8
1.4
1.8
1.1
1.7
1.3
1.7
1.3
2.1
2.0
1.3
1.2
1.5
1.3
0.9
1.7
1.8
1.8
1.4
1.4
0.0
1.3
1.0
1.2
1.1
1.4
1.0
1.9
1.7
-
2.5
7.2
13.7
21.4
5.4
13.7
8.8
10.4
2.9
8.2
6.9
7.1
9.5
24.7
6.7
7.8
13.7
8.0
13.3
9.0
16.2
4.4
3.2
11.9
10.3
8.7
11.0
0.4
3.4
8.8
0.0
7.9
11.4
2.6
10.7
6.8
9.4
10.7
16.8
3.4
5.3
11.9
9.2
12.0
7.8
18.1
9.7
9.7
11.9
5.2
2.5
5.4
9.0
20.6
3.6
4.9
11.2
16.7
2.3
3
D52
3
2
3
2, D16
1
2
3
3, 3
2, 1
2
1
D4
2
1
1, D6
1, D33
3
2
D53
2
2, 2
D5
3
4
1
2, D34
2
D10
D55
2
3
3
1
2, D3
2, 2
D35
2, 3
1
3
4
2
3
D1
D108
3
1
2
1
1
1, D36
2
2, 2, 3, D56
1
3
3
2, 1
4
2, 1
2, D57
2, 1
2
1
3
D2
3
2, D58
1, D37
2
3
4
2, 2, 3
1
2
. . . continued
245
. . . continued
num
αrel
δrel
z
Q
V
V− I
[O II]
[O III]
Hα
Hβ
Hδ
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
108.8
110.9
113.2
115.2
115.4
116.5
117.0
119.4
123.5
126.3
127.4
128.6
129.6
131.3
132.4
132.6
132.7
134.7
136.0
137.4
137.9
138.7
139.9
140.3
143.5
144.6
146.5
147.6
150.5
153.2
154.6
155.1
155.7
158.5
162.1
162.4
162.8
166.0
167.5
169.4
169.4
169.6
172.1
173.3
174.3
175.0
175.1
176.1
176.3
178.4
179.3
181.0
181.9
183.3
183.6
184.6
185.7
187.0
189.8
190.6
192.2
195.1
195.8
199.5
199.7
204.1
205.8
206.2
208.7
209.8
210.2
211.3
216.6
216.7
217.4
−30.4
12.7
−519.2
−17.6
−84.8
0.2
188.2
216.4
671.5
−304.1
126.2
259.5
101.0
394.0
16.1
703.2
−92.6
−619.9
−115.7
360.3
65.7
9.8
−120.9
292.7
−169.5
−484.2
45.1
−67.5
327.2
−65.6
86.8
−506.8
357.2
9.7
−123.1
−440.9
−452.0
67.5
90.9
−587.6
393.7
−385.3
305.4
0.1
−321.9
255.1
558.2
400.5
46.7
−19.3
63.4
−56.3
−156.1
−428.3
−70.8
374.3
79.5
320.4
414.9
−473.4
−496.2
341.1
30.4
30.8
599.4
−235.4
−70.3
588.7
4.4
261.6
184.5
243.4
352.0
589.7
434.3
0.3932
0.3935
0.1183
0.4010
0.3978
0.3915
0.0000
1.1887
0.1139
0.3489
0.3958
0.1769
0.2500
0.2238
0.7128
0.5331
0.3940
0.3964
0.3955
0.0000
0.2281
0.3836
0.3968
0.2568
?
0.3781
0.6557
0.3946
0.3135
0.3794
0.3930
0.3781
0.6551
0.3920
0.4948
0.3970
0.3443
0.4758
0.4460
?
0.5016
0.4426
0.1644
0.3986
0.6242
0.0000
0.4912
0.3962
0.8089
0.3933
0.3910
0.0000
1.0745
0.3784
0.1694
0.4969
0.3895
0.6573
0.3943
0.2266
0.2273
0.3968
?
0.6555
0.4951
0.4685
0.1175
0.4907
0.0000
0.2981
0.2914
0.1755
0.2569
0.4877
0.7183
C
A
A
A
2
S
D
C
A
A
A
D
D
2
A
A
C
A
S
D
A
A
A
A
A
A
2
A
A
A
2
A
C
D
A
C
A
C
A
C
S
A
B
4
A
B
S
C
A
A
A
A
A
A
A
A
A
A
A
A
C
A
S
A
A
B
A
C
A
21.46
21.77
19.85
22.44
20.05
22.06
21.45
21.19
18.85
22.35
22.45
21.02
22.08
21.33
23.22
20.59
21.00
21.87
20.34
20.70
21.57
22.83
20.84
20.44
22.25
21.04
22.80
21.25
20.62
23.60
22.43
20.47
22.09
22.50
21.66
22.13
22.10
22.88
22.14
22.57
22.98
21.64
22.92
20.62
22.70
21.78
21.32
22.82
21.95
22.47
21.79
21.62
21.56
22.10
22.24
20.94
22.21
21.70
20.11
20.93
21.78
23.47
21.30
21.37
21.75
22.00
21.03
21.87
22.34
20.78
19.98
21.29
21.81
21.96
2.18
2.19
1.24
2.01
1.93
2.09
2.33
1.76
1.17
1.16
1.06
0.75
2.11
0.92
2.66
0.85
1.95
1.26
1.96
0.83
0.89
1.98
1.02
1.23
1.11
1.99
2.15
2.00
0.95
2.63
1.93
1.59
2.64
1.33
1.64
1.28
1.09
1.25
1.26
1.93
1.87
2.16
1.21
1.48
1.83
3.07
2.35
1.64
3.25
1.59
1.70
1.91
1.24
0.98
1.07
2.12
2.09
1.73
1.18
1.12
0.95
1.00
2.60
2.23
2.39
1.47
0.84
2.48
1.94
0.97
1.68
1.39
0.98
2.33
1.60
24±7
25±4
33±6
38±16
12±11
32±8
43±4
10±3
41±8
22±2
22±2
11±2
38±3
21±4
2±2
8±1
21±5
10±2
11±3
17±5
5±2
40±6
1±2
4±3
21±2
23±3
12±2
18±4
42±7
9±4
12±2
31±11
54±14
12±1
20
38
19
6
10
4
14
14
6
15
9
18
-
16
39
7
17
11
9
10
11
12
24
-
3
12
11
12
2
8
1
3
7
5
7
4
9
8
-
−3±1
−6±2
−7±1
−7±2
−2±1
−5±1
−3±1
−7±1
−4±1
−6±1
−4±1
−6±2
−5±1
−1±1
−4±1
−3±1
−4±2
−8±2
-
4000 Å
1.5
2.0
1.7
1.8
0.0
1.9
1.3
1.2
1.2
1.2
1.5
1.7
1.6
0.0
1.5
1.1
1.3
0.0
2.4
1.6
1.6
1.1
1.4
1.8
1.7
1.8
1.3
1.3
1.1
1.2
1.0
0.0
1.2
1.7
1.2
1.2
0.0
1.8
1.3
1.3
1.6
0.0
1.3
1.4
1.9
1.3
1.1
1.2
1.1
1.3
0.0
1.5
1.3
1.7
0.0
1.1
1.6
1.3
1.4
1.0
S/N
2.3
11.6
12.6
17.2
15.2
5.5
7.2
8.2
4.5
4.5
21.9
7.1
13.4
3.8
25.8
4.8
7.9
12.4
3.0
16.8
11.5
9.3
12.8
8.8
6.8
9.1
11.9
13.3
16.2
3.3
7.4
7.2
1.8
6.2
8.1
1.8
12.9
5.4
8.1
4.9
3.4
7.8
9.0
1.0
6.1
2.4
9.5
16.9
8.1
11.7
12.4
6.3
7.5
−0.5
12.5
5.5
5.4
1.4
10.4
9.4
13.2
4.7
6.8
6.3
7.0
id.
3
1
3
2, 2
2, 3
D38
2, 1
2
4
3
2
2, 2
1
2
D117
4
2, 3
3
2, 3
2
1
2
2, 3
2, 2
3
3
2
2, 3
2, 2
D39
2, 1
3
2
D59
2
3
3
2, D112
1
3
2
3
2
2
3
2
4
2
D118
2
2, D40
2
3
3
2
2
2
2
2
3
3
2, 2
2
2, D119
4
3
2
4
2
2
2
2
2
4
2
. . . continued
246
. . . continued
num
αrel
δrel
z
Q
V
V− I
[O II]
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
218.1
219.3
220.0
222.3
226.9
227.0
228.3
229.0
230.6
231.3
232.2
234.9
238.2
239.6
240.2
240.6
242.8
244.7
246.2
251.0
251.4
253.0
254.2
254.8
258.8
260.1
260.6
261.1
262.0
267.0
267.6
268.7
271.4
277.8
278.5
280.2
281.3
281.5
282.1
285.8
286.1
286.1
286.8
287.4
291.5
292.2
297.0
299.4
301.8
302.7
308.9
310.1
310.3
310.7
314.2
317.5
317.6
317.7
319.9
320.9
322.8
328.9
332.6
332.9
338.4
342.0
344.1
350.3
351.9
356.1
361.9
362.3
365.5
368.2
373.4
−64.9
−101.7
−339.1
−416.0
245.0
44.7
−92.7
335.3
271.1
−62.9
552.1
58.7
−188.6
58.9
531.6
256.7
−13.0
33.9
279.9
545.7
424.4
345.6
11.1
112.7
−358.9
−53.9
48.3
56.2
260.8
524.9
−75.8
353.6
363.7
499.7
−56.7
−42.7
418.3
425.5
−184.9
−462.4
436.9
−493.1
312.4
−41.7
428.2
24.9
−94.2
703.2
498.2
3.8
−465.4
414.4
29.9
146.7
468.1
468.3
388.0
418.5
−421.8
−54.0
611.7
461.5
387.9
−471.1
256.7
364.2
695.5
480.9
−514.0
346.0
704.3
−518.9
−533.2
657.7
−504.3
0.1851
0.2296
0.3988
0.1848
0.3935
0.6555
0.1846
0.0000
0.4955
0.3917
0.0000
0.1823
0.3777
0.0000
0.4917
0.3797
?
0.6946
0.4000
0.4980
0.3439
0.5674
0.3633
0.1828
0.1221
0.3956
0.3947
0.3942
0.3911
0.2987
0.3921
0.5000
0.4938
0.3035
0.3926
0.2252
0.1794
0.4438
0.4954
0.2129
0.3442
0.2137
0.0000
0.2254
0.1784
0.3959
0.1837
0.1443
0.0000
0.3441
0.1802
0.1781
0.3935
0.3441
0.3974
0.0703
?
0.3910
0.2124
0.0572
0.0000
0.3758
0.4939
0.1810
?
?
0.5273
0.0000
0.3904
0.3543
0.2143
0.3962
0.3140
0.3797
0.1825
A
A
D
A
A
4
A
S
A
B
S
A
A
S
C
A
2
A
A
A
A
A
A
C
A
3
A
A
A
A
B
A
C
C
3
A
A
B
A
A
A
S
A
A
4
A
A
S
A
A
A
A
A
B
A
A
A
A
S
A
B
A
B
S
D
A
C
C
A
A
A
22.43
19.94
22.67
19.53
23.02
23.53
20.52
21.15
23.47
22.20
19.52
22.62
21.51
24.75
20.23
20.56
22.89
22.14
21.49
20.30
20.47
23.20
21.98
19.75
22.12
21.34
22.50
22.45
21.86
20.08
21.10
23.57
21.94
18.93
21.82
23.05
20.65
21.29
21.69
18.29
21.61
20.02
21.64
21.40
21.25
18.24
20.86
17.76
22.34
20.66
19.45
20.32
21.92
21.41
22.71
20.88
21.96
21.77
20.46
21.69
21.03
21.51
22.45
21.09
21.63
22.83
23.06
21.89
21.81
20.45
20.75
21.83
21.00
20.43
18.10
1.01
1.50
1.45
1.36
1.09
2.75
1.00
2.46
2.35
2.05
2.05
0.99
1.95
3.89
1.24
1.61
1.65
1.53
1.63
1.58
0.96
1.42
1.07
1.11
0.81
1.29
1.96
1.77
1.34
2.02
2.04
2.28
1.30
1.48
1.65
0.68
1.46
1.31
1.56
1.72
1.25
1.77
0.98
0.79
1.21
1.15
1.41
3.04
0.95
1.15
0.87
0.87
1.33
2.30
0.63
1.47
1.94
0.79
2.04
1.46
2.16
0.85
1.47
1.45
2.34
1.73
1.12
1.54
1.46
1.28
1.69
1.15
1.27
19±5
5±2
53±8
−3±2
8±1
33±2
10±2
51±2
62±5
24±4
9±3
30±3
13±3
14±4
6±2
22±4
44±9
10±1
16±2
6±2
12±8
44±4
3±1
34±7
40±6
15±6
30±4
26±4
20±3
20±3
24±4
−1±2
32±2
4±5
[O III]
Hα
Hβ
Hδ
19
39
20
21
11
24
14
28
1
9
12
14
6
17
13
16
3
12
2
4
11
47
10
17
12
0
4
24
5
10
13
40
4
3
-
7
18
3
14
3
5
12
7
7
10
2
3
18
7
9
5
4
15
-
−11±4
−4±0
−5±2
−2±1
−4±2
−3±1
−3±1
−5±1
−4±0
−3±1
−3±1
−6±1
−2±0
−5±1
−5±1
−4±1
−5±1
−3±1
-
4000 Å
S/N
id.
1.7
1.5
1.1
1.6
0.0
1.7
0.0
1.9
0.0
1.2
1.4
0.0
1.1
1.4
1.1
1.2
1.2
1.2
1.4
1.4
1.6
1.6
1.5
1.9
1.5
1.5
1.6
1.2
1.3
1.3
1.1
1.1
2.4
1.4
1.6
0.0
1.2
1.4
2.6
0.0
1.2
1.6
1.2
1.1
1.3
1.1
0.0
1.8
1.2
0.0
1.2
1.4
1.7
0.0
0.0
1.1
0.0
1.3
1.9
1.4
1.1
2.0
1.1
2.4
3.5
10.8
3.1
21.6
4.8
5.7
5.0
4.5
2.1
11.7
6.4
21.7
4.1
10.6
10.2
8.7
17.1
7.9
12.8
5.5
3.0
12.2
12.2
8.0
12.4
12.1
15.2
5.9
9.3
17.8
6.8
15.4
11.8
15.1
8.0
17.4
11.4
14.5
6.6
3.3
10.0
4.5
18.7
9.4
34.3
10.5
11.8
6.6
4.9
2.1
3.2
11.3
10.2
2.4
14.1
5.1
10.2
2.5
6.7
6.5
3.5
7.3
10.6
2.8
13.4
11.3
9.8
2, 2
2, 3
3
3
2
D113, D120
2, 2
2
2
2
4
2
2
2
4
2, 2
2
D121
2
4
2
2
2
2
3
2, 2
D61
2
2
4
2, 2, D42
2
2, 2
4
2
D114
2
2
2
4
2
4
2
2
2
D62
2
4
2
2
4
2
2
2, 2
2
2
2
2
4
2
4
2
2
4, 4
2
2
4
2
4
2, 2
4
4
4
4
4
. . . continued
247
. . . continued
num
αrel
δrel
z
Q
V
V− I
[O II]
[O III]
Hα
Hβ
Hδ
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
375.2
375.4
376.6
382.5
389.0
392.3
396.7
397.6
398.9
400.6
401.3
404.4
413.2
416.4
416.9
419.8
430.4
434.8
435.1
435.6
436.6
440.9
448.3
457.9
459.2
468.8
477.4
483.3
489.9
490.0
492.9
496.3
502.1
510.6
515.8
522.0
523.3
539.2
541.8
554.6
559.1
567.5
577.6
584.8
590.8
590.9
602.8
607.8
611.7
617.1
624.6
633.2
639.6
645.5
648.8
650.1
658.3
661.6
665.4
680.8
691.8
699.8
713.6
726.4
736.1
383.7
−476.0
305.8
−581.8
451.2
−545.7
−585.3
356.8
−564.2
105.2
589.7
443.9
−429.7
271.5
138.6
465.4
−514.2
−589.0
332.7
−437.8
691.2
482.4
−479.6
717.4
400.7
−427.6
198.4
−553.0
431.6
−467.6
−599.0
240.8
−413.0
439.1
−557.7
395.2
−553.9
377.0
−384.5
476.1
−683.7
333.0
−452.8
326.0
−635.9
−424.3
280.5
−498.1
363.0
−528.3
369.4
155.6
−421.6
−461.1
227.2
−592.8
267.3
−563.9
235.0
−489.3
−490.6
−482.7
−551.6
−505.7
−479.1
0.1835
0.4070
0.3900
0.3356
0.5830
0.3122
?
0.0000
0.0960
0.3957
0.1842
0.9689
0.1835
0.3436
0.3778
0.1991
0.0000
0.2118
?
0.3636
?
0.3926
?
0.3955
0.1794
0.1838
0.8090
0.1826
0.1951
0.4075
0.1683
0.1695
0.3639
0.1942
?
0.3443
0.1795
0.0000
0.4035
0.2965
0.1681
0.1829
0.3964
0.0000
?
0.1694
0.0000
0.0958
0.0000
0.3958
0.2662
0.0000
0.1843
0.1978
0.2131
0.3966
0.3086
0.2312
?
0.4934
0.6585
0.2322
0.6601
0.5122
0.2968
A
A
B
A
A
A
S
A
A
A
D
A
A
B
C
S
A
A
C
A
B
A
A
A
A
A
A
A
A
D
A
A
S
A
A
A
A
A
S
A
S
A
S
A
B
S
A
A
A
A
A
A
A
A
A
A
C
C
22.32
21.48
21.41
20.67
21.68
21.90
22.11
20.01
19.98
20.18
18.52
22.55
17.17
20.01
22.01
21.33
21.21
19.74
22.63
22.22
21.09
23.41
21.67
20.65
19.86
20.77
22.01
18.91
20.47
22.56
19.20
19.96
22.16
21.52
22.12
20.90
20.53
23.34
21.29
21.71
18.41
19.91
21.89
20.73
22.01
19.98
20.72
20.14
21.56
21.90
20.64
20.88
19.86
19.97
21.17
20.28
20.15
20.26
20.29
22.12
22.27
21.40
22.13
22.34
21.46
1.30
2.15
1.47
2.04
1.43
2.35
1.79
0.96
1.32
1.45
1.27
1.39
1.58
1.02
2.68
1.48
1.40
1.77
1.25
1.78
0.89
2.09
1.01
1.44
1.87
1.28
0.90
2.11
1.46
1.10
1.28
1.27
1.70
1.18
1.35
2.23
1.77
0.91
1.45
1.36
1.29
1.29
1.08
1.14
2.16
0.90
1.82
0.84
0.95
1.45
1.31
1.33
1.16
1.66
1.07
0.96
1.23
2.09
2.11
0.86
1.78
1.80
0.97
11±4
9±2
9±3
19±9
6±4
19±2
41±16
14±4
12±2
44±6
4±3
25±2
30±13
11±4
2±1
72±23
10±4
14±2
16±4
10±3
30±2
12±2
18±3
32±5
25±2
10±1
11±2
22±6
13±2
2±8
12
3
2
4
6
23
7
7
4
42
13
14
6
2
10
4
8
31
-
18
7
6
14
8
12
8
8
8
9
6
24
24
28
17
13
-
8
2
6
4
10
4
3
11
5
10
2
4
5
9
5
8
12
5
-
−5±1
−4±1
−3±1
−3±1
−6±2
−2±1
−5±1
6±2
9±3
−4±1
−4±1
−7±1
−5±2
−4±0
−3±1
-
4000 Å
S/N
id.
1.9
1.4
1.6
1.5
1.6
0.0
0.0
2.0
1.4
2.3
1.5
1.7
1.1
1.2
0.0
2.1
0.0
1.8
0.0
1.3
0.0
1.8
1.1
1.6
1.3
1.9
1.2
0.0
1.4
1.5
0.0
1.4
1.2
1.7
1.1
0.0
0.0
1.8
0.0
1.9
0.0
1.2
0.0
2.0
1.6
1.8
1.2
1.1
1.4
0.0
1.2
1.3
1.3
1.1
1.3
1.2
2.8
14.2
11.0
9.7
14.5
9.4
1.6
7.6
11.8
9.4
13.7
15.5
6.0
8.1
9.2
7.5
6.7
4.5
4.1
5.8
7.5
8.9
3.9
10.9
21.1
7.0
6.4
4.4
11.0
4.0
1.6
4.7
6.2
10.0
6.7
15.5
8.0
11.6
2.0
16.4
11.0
10.7
9.2
16.6
12.5
4.6
7.2
9.4
22.5
4.6
13.3
7.4
8.6
3.7
5.4
2.9
2
4
2
4
2, 2, 2
4
4
2
4
2
4
2
4
2
2
2
4
4
2
4
4
2, 2
4, 4
4
2
4
2
4
2
4
4
2
4
2
4
2
4, 4
2
4
2
4
2
4, 4
2
4
4
2
4, 4
2
4, 4
2
2
4
4
2
4
2
4
2
4, 4
4, 4
4, 4
4, 4
4, 4
4
248
Figure A.1: 220 × 250 section of the CFH12k V-band image showing the distribution of the objects in our spectroscopic sample. Expanded views of
the marked regions are shown in Figs. A.2a–A.2h as indicated in the image. The coordinates given are right ascension and declination relative to
α2000 = 00h 26m 35.s 70, δ2000 = 17◦ 090 43.00 06.
249
Figure A.2a: Subsections of the CFH12k V-band image with redshifts for members of the spectroscopic sample. The positions of the subsections on the whole
field are marked in Fig. A.1. For redshifts z < 1 only the decimals are given; for
galaxies marked ‘?’ spectra are available but no redshift could be determined.
The catalogue (Table A.1) is sorted by relative right ascension, so the entry for
any object can be easily identified by noting its coordinates.
250
Figure A.2b: Continued. The squares mark the positions of Figs. 7.7, the circle
the area around the weak shear signal detected by Bonnet et al. (1994), discussed in Sect. 7.3.
251
Figure A.2c: Continued
252
Figure A.2d: Continued
253
Figure A.2e: Continued
254
Figure A.2f: Continued
255
Figure A.2g: Continued
256
Figure A.2h: Continued
Bibliography
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TH`ESE

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