University of Birmingham › ~ajrs › papers › ajrs_mini_thesis.pdf · 2010-07-23 · Synopsis Virialized systems, such as clusters and groups of galaxies, are the largest gravitationally

THE X-RAY SCALING PROPERTIES OF VIRIALIZED SYSTEMS

by

ALASTAIR JOHN ROY SANDERSON

A thesis submitted toThe University of Birmingham

for the degree ofDOCTOR OF PHILOSOPHY

Astrophysics and Space Research GroupSchool of Physics and AstronomyThe University of BirminghamSeptember 2002

Synopsis

Virialized systems, such as clusters and groups of galaxies, are the largest gravitationally bound objects inthe Universe. As such, they provide an ideal laboratory for investigating the formation and evolution of structureon the largest scales. We present the largest detailed studyof the X-ray properties of virialized systems to date,mapping the deprojected density and temperature distribution of the gaseous intracluster medium, thus enablingthe gravitating mass profile to be determined. For a subset ofour sample, we have also measured the distributionof the galaxies themselves, allowing us to calculate the stellar mass contribution.

We find clear evidence of a departure from the simple expectations of self-similarity. The intracluster mediumis more spatially extended and systematically less dense insmaller haloes, and there is evidence of an entropyexcess in the hot gas. Our results favour a significant role for both non-gravitational heating and radiative coolingin modifying the properties of this gas, although we report only weak evidence for an enhanced star formationefficiency in groups. We also find evidence of a systematic variation in the dark matter concentration betweenvirialized haloes of different masses, consistent with a hierarchical structure formation scenario.

Acknowledgements

I am very grateful to a number of people who have helped me in one way or another through the course of thisPhD. Firstly, I would like to thank my supervisor, Trevor Ponman, for all his enthusiasm and support. I havereally enjoyed working with him and have learnt a lot in the process. Special thanks also go to my collaborators,Alexis Finoguenov, Ed Lloyd-Davies and Maxim Markevitch, for their invaluable contributions towards the paperspresented in this thesis. On a related theme, I am very grateful to Ewan O’Sullivan, Steve Helsdon, Bruce Fairleyand Fill Humphrey for fruitful discussions on various aspects of my work, and to Jo Hartwell and Ben Maughanfor proof reading parts of this thesis. My thanks go to my school teachers, Mr Haines and Mr Findlay-Palmer, forsparking an interest in Science in general, and Physics in particular. I am also indebted to Bob Vallance, DavidGeddes and Billy Wilson, for their help maintaining the computer hardware and software that I have used in thecourse of my research.

A large number of people have helped to make the last four years in Birmingham very enjoyable for me. Inaddition to many of those already mentioned, they include current and former office mates – David Acreman,Simon Ellis and Julian Pittard, and other members of the Astrophysics group – Robin Barnard, Richard Brown,Ken Elliott, David Henley, Richard Ingley, Helen Mapson-Menard, Chris Messenger, Louisa Nolan, John Osmond,Ian Robinson and Molly Stockton-Chalk, as well as current and former housemates – Steve Corner, Jon Fenton,Emma Jones, Glyn Thomas, Nick Burton, Natalie Andrew, ChrisCutler and Edmund Green. My thanks also goto the various members of the galaxies and clusters group at Birmingham, for helpful discussions and feedback onpractice talks.

Finally I would like to thank my parents for all the encouragement and support they have given me over theyears.

Contents

Preface 1

1 An Introduction to Virialized Systems 21.1 Mass components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 3

1.1.1 Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 31.1.2 Dark matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 41.1.3 The intergalactic medium . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 5

1.2 The Formation and Development of Structure . . . . . . . . . . .. . . . . . . . . . . . . . . . . 81.2.1 Hierarchical formation . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 81.2.2 Self-similarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 91.2.3 Scaling relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 10

1.3 Feedback and Interaction . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 101.3.1 Cooling and star formation . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 101.3.2 Enrichment of the intergalactic medium . . . . . . . . . . . .. . . . . . . . . . . . . . . 121.3.3 Energy injection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 131.3.4 Gas cooling revisited . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 13

1.4 The Birmingham-CfA Cluster Scaling Project . . . . . . . . . .. . . . . . . . . . . . . . . . . . 14

2 Gas Fraction and theM−TX Relation 162.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 162.2 The Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 172.3 X-ray Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 21

2.3.1 Cluster models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 212.3.2 Cooling flow correction . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 232.3.3 Markevitch sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 232.3.4 Finoguenov sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 242.3.5 Lloyd-Davies & Sanderson samples . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 25

2.4 Consistency Between Sub-samples . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 272.5 Final Model Selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 302.6 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 30

2.6.1 Gas distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 302.6.2 TheM−TX relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.6.3 The effects of non-isothermality . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 372.6.4 Virial radius . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 392.6.5 Extrapolation bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 41

2.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 422.7.1 Galaxies vs. groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 44

2.8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 45

3 Mass Composition and Distribution 463.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 463.2 3D Galaxy Density Calculation . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . 47

3.2.1 APM data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 503.2.2 NED data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 503.2.3 Surface density fitting . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 51

3.3 Optical Luminosity Calculation . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 523.3.1 Conversion between bands . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 52

iii

iv CONTENTS

3.3.2 Determination of luminosity normalization . . . . . . . .. . . . . . . . . . . . . . . . . 533.4 Results: Spatial Mass Distribution . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 54

3.4.1 Stellar distribution . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 543.4.2 Gas distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 543.4.3 Dark matter distribution . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . 543.4.4 Total density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 563.4.5 Mass-to-light ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 56

3.5 Results: Integrated Scaling Properties . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . 573.5.1 Optical luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 573.5.2 Mass-to-light ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 583.5.3 Star formation efficiency and gas loss from haloes . . . .. . . . . . . . . . . . . . . . . 613.5.4 Baryon fraction and constraints onΩ0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 633.5.5 Central density concentration . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . 64

3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 653.6.1 Implications for heating/cooling . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 653.6.2 Halo formation epoch . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 66

3.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 66

4 Entropy and similarity in galaxy systems 684.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 684.2 Sample and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 694.3 Entropy and Temperature Distributions . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . 694.4 Scaling Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . 714.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 75

4.5.1 Cooling models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . 754.5.2 Preheating models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 764.5.3 Star formation models . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 78

4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 79

5 Mass, Velocity Dispersion and Temperature Scaling Properties 815.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 815.2 Calculating velocity dispersion . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . 825.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 83

5.3.1 σ−TX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 835.3.2 σ−M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.3.3 σ−LB,j . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 885.3.4 TheMgas−TX relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 905.3.5 Abell richness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 925.3.6 Bautz-Morgan type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 92

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 935.4.1 Velocity dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 935.4.2 Implications for heating/cooling of the IGM . . . . . . . .. . . . . . . . . . . . . . . . . 95

5.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . 96

6 Conclusions 976.1 Summary of Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 976.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 97

A Miscellaneous Paper Details 99

List of Figures

1.1 An optical image of the cluster Abell 2218, taken with theHubble Space Telescope. . . . . . . . . 41.2 An X-ray image of the Coma cluster, taken with theXMM-Newtonsatellite . . . . . . . . . . . . 61.3 A cosmological simulation showing the formation of structure in aΛCDM Universe . . . . . . . . 91.4 X-ray/optical overlay images of 6 typical systems in theCluster Scaling Project sample . . . . . . 15

2.1 A comparison of the emission-weighted X-ray temperatures from this work, with those from theoriginal Finoguenov and Markevitch analyses . . . . . . . . . . . .. . . . . . . . . . . . . . . . 28

2.2 A comparison of the gas density and temperature profiles in four clusters common to the Marke-vitch, Finoguenov and Lloyd-Davies samples . . . . . . . . . . . . .. . . . . . . . . . . . . . . 29

2.3 The gas density power law index parameter (β) as a function of system emission-weighted X-raytemperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . 31

2.4 Mean gas fraction within 0.3R200 as a function of system X-ray temperature . . . . . . . . . . . . 312.5 Mean gas fraction within R200 as a function of system X-ray temperature . . . . . . . . . . . . . . 322.6 Cumulative gas fraction as a function of scaled radius (R/R200) . . . . . . . . . . . . . . . . . . . 332.7 Total mass as a function of X-ray temperature for three different temperature prescriptions, and

measured within 0.3R200 andR200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.8 Total mass withinR2500 as a function of emission-weighted X-ray temperature . . . . .. . . . . . 382.9 Total mass as a function of emission-weighted X-ray temperature, evaluated within 0.3R200, as-

suming an isothermal intergalactic medium . . . . . . . . . . . . . .. . . . . . . . . . . . . . . 382.10 Upper panel: Gas fraction withinR200 as a function of emission-weighted X-ray temperature

(within R200). Lower panel:Gas fraction withinR200, for an isothermal IGM . . . . . . . . . . . 392.11 MeasuredR200compared with the virial radius as estimated from a scaling of the formR200∝

√kT,

calibrated using numerical simulations . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . 402.12 The ratio between our measuredR200 and the predicted virial radius (see previous caption), as a

function of the ratio inT(r) betweenr = 0 andr = 0.3R200. . . . . . . . . . . . . . . . . . . . . 412.13 Azimuthally averaged X-ray surface brightness profiles for two bright clusters – Abell 1795 and

Abell 2029 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 43

3.1 The variation of stellar matter density with scaled radius . . . . . . . . . . . . . . . . . . . . . . 553.2 The variation of gas density with scaled radius . . . . . . . .. . . . . . . . . . . . . . . . . . . . 553.3 The variation of dark matter density with scaled radius .. . . . . . . . . . . . . . . . . . . . . . 563.4 The variation of overdensity with scaled radius . . . . . . .. . . . . . . . . . . . . . . . . . . . 573.5 The variation of mass-to-light ratio with scaled radius. . . . . . . . . . . . . . . . . . . . . . . . 583.6 Total gravitating mass as a function ofLB,j luminosity . . . . . . . . . . . . . . . . . . . . . . . . 593.7 TotalLB,j luminosity withinR200 as a function of system temperature . . . . . . . . . . . . . . . . 593.8 Mass-to-light ratio withinR200 as a function of system temperature . . . . . . . . . . . . . . . . . 603.9 The fraction of baryons in the form of stars as a function of system temperature . . . . . . . . . . 623.10 The ratio of stellar to dark matter mass withinR200, as a function of system temperature . . . . . . 623.11 The ratio of gas to dark matter mass withinR200, as a function of system temperature . . . . . . . 633.12 Baryon fraction withinR200 as a function of system temperature . . . . . . . . . . . . . . . . . . 643.13 The ratio ofrc to R200 as a function of system temperature . . . . . . . . . . . . . . . . . . . . .653.14 The variation ofR200 as a function of system temperature . . . . . . . . . . . . . . . . . . . . .. 66

4.1 Gas entropy (normalized by system temperature) as a function of scaled radius. . . . . . . . . . . 704.2 Gas entropy (normalized by(1+T/T0)) as a function of scaled radius. . . . . . . . . . . . . . . . 714.3 The variation of gas temperature (normalized byT200) with scaled radius. . . . . . . . . . . . . . 724.4 Gas entropy at 0.1R200 as a function of system temperature: raw data points . . . . . . .. . . . . 72

v

vi L IST OF FIGURES

4.5 Gas entropy at 0.1R200 as a function of system temperature: grouped data points, comparing mea-suredR200 with estimatedR200 from the NFW formula . . . . . . . . . . . . . . . . . . . . . . . 73

4.6 Gas entropy atR500 as a function of system temperature . . . . . . . . . . . . . . . . . . . . .. . 74

4.7 Gas entropy atR500, normalized byM2/3500, as a function of the total mass withinR500 . . . . . . . . 75

4.8 Gas Entropy at the radius enclosing a mass of 3×1013h−150 M⊙ as a function of system temperature 76

5.1 Velocity dispersion as a function of system temperature. . . . . . . . . . . . . . . . . . . . . . . 875.2 Velocity dispersion as a function of total mass withinR200 . . . . . . . . . . . . . . . . . . . . . 895.3 Velocity dispersion as a function ofLB,j within R200 . . . . . . . . . . . . . . . . . . . . . . . . . 895.4 Total gas mass withinR500 as a function of system temperature . . . . . . . . . . . . . . . . . . . 905.5 Total gas mass withinR200 as a function of system temperature . . . . . . . . . . . . . . . . . . . 915.6 Histograms of Abell number counts and Bautz-Morgan classification . . . . . . . . . . . . . . . . 935.7 X-ray gas temperature as a function of Abell galaxy number counts . . . . . . . . . . . . . . . . . 94

List of Tables

2.1 Some key properties of the 66 objects in the cluster scaling sample . . . . . . . . . . . . . . . . . 202.2 Summary of results for the power lawM−TX fitting using different mean temperature prescriptions

and integration radii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . 352.3 Summary of results for the 1-dimensional X-ray surface brightness fitting within different radii . . 42

3.1 Some key properties of the 32 objects in the optical sample . . . . . . . . . . . . . . . . . . . . . 49

5.1 Basic information, plus velocity dispersions and gas masses for the full sample . . . . . . . . . . 865.2 Summary of results for the power lawσ−TX, σ−M andσ−LB,j fitting . . . . . . . . . . . . . . 885.3 Summary of results for the power lawMgas−TX fitting . . . . . . . . . . . . . . . . . . . . . . . 92

vii

Preface

The work presented in this thesis was completed by the authorbetween 1998 and 2002, under the supervision ofProf. T. J. Ponman at the University of Birmingham. This thesis contains 4 chapters which have been preparedas individual papers, intended for publication in refereedjournals. These papers form a series based on ‘TheBirmingham-CfA Cluster Scaling Project’, which constitutes the bulk of the author’s PhD work. This project is acollaboration between Birmingham University and the Harvard-Smithsonian Center for Astrophysics (CfA). Thepapers appear in this thesis essentially identical to the versions intended for publication, except that the referenceshave been combined for all chapters into a single section forthe whole thesis. Each paper has its own acknowledg-ments section, which is included in appendix A, together with a list of co-authors and associated affiliations.

It should be noted that a small degree of duplication of material is inevitable as a consequence of the structure ofthis thesis, since each paper incorporates its own introduction, which may overlap slightly with material presentedin the general introduction to the thesis in chapter 1.

Chapter 2 The Birmingham-CfA cluster scaling project - I: gas fraction and the M−TX relation – This paperwas written by the author, based on work undertaken as part ofthis thesis. Prof. T. J. Ponman providedextensive discussions on the interpretation of the results. The three other co-authors, Dr. Alexis Finoguenov,Dr. Edward Lloyd-Davies and Dr. Maxim Markevitch provided the majority of the basic X-ray data, andgave advice on certain aspects of the data analysis, as well as contributing towards revisions of the originalmanuscript. This paper has been submitted for publication in Monthly Notices of the Royal AstronomicalSociety(MNRAS).

Chapter 3 The Birmingham-CfA cluster scaling project - II: mass composition and distribution– This paper waswritten by the author, with extensive discussions contributed by Prof. T. J. Ponman on the interpretation ofthe results. Dr. Edward Lloyd-Davies provided some of the basic optical data. This paper will shortly besubmitted for publication.

Chapter 4 The Birmingham-CfA cluster scaling project - III: entropy and similarity in galaxy systems– Thispaper was written by Prof. T .J. Ponman, although all the dataanalysis was performed by the author,who also contributed towards the interpretation of the results. It is included in this thesis, in accordancewith University regulations, as an essential part of the Cluster Scaling Project. This paper will shortly besubmitted for publication.

Chapter 5 The Birmingham-CfA cluster scaling project - IV: mass, velocity dispersion and temperature scalingproperties– This paper was written by the author, with discussions contributed by Prof. T. J. Ponman on theinterpretation of the results. This paper will be submittedfor publication in the near future.

1

Chapter 1

An Introduction to VirializedSystems

To gain a proper understanding of any system requires an analysis of the fundamental components that characteriseit on the largest scale, and the processes that give rise to them. In the case of the Universe, the dominant force onlarge scales is gravity, which shapes the development of structure through the collapse and subsequent formationof virialized1 objects: where equilibrium is maintained by the balance between the gravitational energy associatedwith the mass of the system and the kinetic energy of its individual components. The largest such organizedstructures are clusters of galaxies, in which many hundredsor thousands of individual galaxies are gravitationallybound by the presence of a large mass concentration arrangedin an extended ‘halo’.

The first indications that galaxies existed in bound congregations derive from Messier (1850), who documenteda concentration of ‘nebulae’ in the vicinity of what is now known as the Virgo Cluster. However, it wasn’t untilmuch later that these ‘nebulae’ were identified as being external to our own galaxy (Hubble, 1925). Subsequently,these objects were subject to systematic study, culminating in the first statistically complete catalogue of richgalaxy clusters by Abell (1958), which continues to be of great value to this day. While early studies were basedon optical observations, it was realised that clusters of galaxies might radiate significant amounts of energy asX-rays (Cavaliere, Gursky, & Tucker, 1971). Shortly thereafter, the launch of the first space-based X-ray telescopedemonstrated that extended X-ray sources were associated with clusters, comparable in scale to the size of thegalaxy distribution (Kellogg et al., 1972). The study of poorer clusters and groups of galaxies followed soon after(e.g. Turner & Gott, 1976), extending the range of observations of virialized systems to much lower mass scales.More recently still, the study of large samples of individual galaxy haloes has become possible with higher qualityX-ray telescopes (e.g. Beuing et al., 1999).

The X-ray waveband is ideal for the study of virialized systems, since the hot gas which emits this radiationextends throughout the entire halo of gravitating matter (see section 1.1.3). Furthermore, groups and clusters ofgalaxies are more easily identified in X-rays, since opticalobservations of the individual galaxies are susceptibleto confusion between background and cluster members (Lucey, 1983; Struble & Rood, 1991). The mass scaleof the virialized systems studied in this thesis spans the range∼ 1012–1015M⊙ (Solar masses), corresponding tosizes from∼100 kpc for a galaxy-sized halo2, up to several Mpc for a rich cluster. The equilibrium state of thesegravitationally bound objects is governed by thevirial theorem, which states that

2K +W = 0, (1.1)

whereK is the total kinetic energy andW is the total potential energy of the system. Only when this condition issatisfied will the system exist in a stable, relaxed3 state.

The number of virialized haloes of a given mass that can form is dependent on the total density of the Universe,ρtot, which is a key cosmological parameter. It is conventional to express this quantity as a fraction of the ‘critical’density needed to provide just enough self-gravity to eventually halt the expansion of the Universe, which is givenby

ρcrit =3H2

0

8πG, (1.2)

1From ‘virial’, a certain function relating to a system of forces and their points of application – first used by Clausius inthe investigation ofproblems in molecular physics.

21 parsec (pc) = 3.086×1016m3A system is relaxed when no information remains about the initial orbits of most of its particles.

2

1.1. MASS COMPONENTS 3

whereG is the gravitational constant. Assuming a value of Hubble’sconstant ofH0 = 70 km s−1 Mpc−1 , givesρcrit = 9.21×10−27Kg m−3. The ‘density parameter’ is then simply

Ω ≡ρtot

ρcrit. (1.3)

If Ω > 1, the Universe will eventually collapse back in on itself (a‘closed universe’), whileΩ < 1 implies it willcontinue expanding forever (an ‘open universe’) (see section 1.2).

1.1 Mass components

Since gravity is the dominant force in the formation of galaxies and galaxy systems, the composition by massof these objects is of great importance in understanding their behaviour. The mass budget of virialized systemsconsists of three main components, in approximately the following proportions:

Stars: 2–5 % (h70−1)

Hot gas: 10–15% (h70−5/2)

Dark matter: 80–85% (h70−1)

Together the stars and hot gas comprise thebaryonicmatter, i.e. the ‘normal’, interacting matter in the Universe.As clusters of galaxies are the largest gravitationally bound objects that exist, they constitute a reasonably fairsample of the Universe. Thus, the above proportions are typical of the mass composition of the Universe as awhole. The properties of each of these components will now bedealt with, in turn.

1.1.1 Stars

The optical luminosity arises from individual stars, with masses in the range∼0.1–100M⊙, which are gravita-tionally bound in collections numbering typically∼108−12, to form a galaxy. Individual galaxies are the smallestcategory of virialized object dealt with in this thesis. Themajority of these galaxies are themselves arranged ingroups (Tully, 1987), numbering some 10–30 objects; more massive still are clusters of galaxies, containing hun-dreds or thousands of these objects. Although large clusters are much rarer than groups of galaxies, they are moreamenable to detailed study, as their luminosities are considerably larger.

The well-defined morphology-density relation (Dressler, 1980) provides clear evidence of the close link be-tween galaxies and the environment in which they are located; early type (elliptical) galaxies are more commonin dense cluster cores, consistent with a formation historyresulting from the merger of two, roughly equal-masslate-type (spiral) galaxies. The latter objects predominantly inhabit the outskirts of clusters (and are very commonin the field galaxy population), where the likelihood of suchencounters is much lower. Galaxy mergers are morecommon in the group environment, since the lower velocity dispersion of these systems is more conducive to grav-itational interaction. The process of orbital decay and merging in dynamically evolved groups eventually leads tothe formation of a ‘fossil group’ (Ponman et al., 1994; Vikhlinin et al., 1999b), characterised by a giant ellipticalgalaxy embedded in a group-sized X-ray halo. The vicinity ofsuch an object is bereft of all but the smallest galax-ies (more than 2 magnitudes fainter), which have been consumed by the central galaxy (Jones, Ponman, & Forbes,2000).

The motions of individual galaxies within the cluster can beused to estimate its gravitating mass, throughapplication of the virial theorem (equation 1.1). Such virial mass estimates rely on a measurement of the line-of-sight velocity dispersion of the galaxies within the cluster, σclus, given by

σ2clus =

1N−1

N

∑i=1

(vi −v)2 , (1.4)

wherev is the line-of-sight recession velocity of a galaxy – as inferred from its redshift – andv is the mean recessionvelocity of all the galaxies. In Chapter 5 we investigate thescaling properties of this parameter, and compare itwith other measures of the depth of the halo potential.

The difficulty with using this method to calculate the virialmass is its susceptibility to bias. Firstly, it isassumed that the line-of-sight velocity dispersion of the system is a fair representation of the true, 3-dimensional

4 CHAPTER 1. AN INTRODUCTION TOV IRIALIZED SYSTEMS

Figure 1.1: An optical image of the cluster Abell 2218, takenwith the Hubble Space Telescope. Note the bright dominant galaxy at the centreof the cluster (to the left of the image). A number of curved arcs are visible around the periphery of the cluster core; these are the result ofgravitational lensing of distant galaxies located behind the cluster. Image courtesy of NASA and the Space Telescope Science Institute.

dynamics of the system; if galaxy orbits are predominantly radial or tangential, rather than isotropic, the mass canbe over- or underestimated accordingly. Secondly, the presence of substructure within the cluster can strongly biasthe overallσclus (Bird, 1995). Since the mass is a strong function ofσclus (see section 1.2.3), such inaccuracies arefurther magnified.

1.1.2 Dark matter

Early estimates of the amount of material contained within clusters of galaxies produced surprising results. Mea-surements of the galaxy velocity dispersion (equation 1.4)in the nearby Coma and Virgo Clusters led to inferredmasses far in excess of the total mass contained within the galaxies themselves (Zwicky, 1933; Smith, 1936).Although subsequent detection at X-ray wavelengths of diffuse gas in between the galaxies (see section 1.1.3)ameliorated the situation somewhat, the problem of the ‘missing mass’ was not resolved. Typical mass-to-lightratios for galaxy systems are in the range 150–350h70 M⊙/L⊙ (e.g. Edge & Stewart, 1991a) (see also Chapter 5),and in fact, the vast majority (>∼85 per cent) of the gravitating mass exists in the form of non-luminous material –otherwise referred to as ‘dark matter’ – which cannot be directly detected.

Since there is no way to observe this material directly, we are only able to infer its properties from its gravi-tational influence. None the less, strong constraints can beplaced on its spatial distribution in virialized systems.Numerical simulations (Navarro, Frenk, & White, 1995; Moore et al., 1999b; Fukushige & Makino, 2001) indicatethat the dark matter is well described by a universal profile,of the form (Zhao, 1996; Jing & Suto, 2000)

ρ(r) =ρ0

xn (1+x)3−n (1.5)

whereρ0 is the central density,x= r/rs andrs is a characteristic scale radius. Forr ≪ rs the profile is characterisedby a cusp, withρ ∝ r−n. However, in the limit ofr ≫ rs, the profile scales asρ ∝ r−3, for all values ofn.Equation 1.5 is the generalised form of the so-called NFW profile (Navarro et al., 1995), which is obtained forn =1. Subsequent, higher resolution simulations have revealed a steeper power law index in the cusp of the dark matter,which is characterised byn∼ 1.5 (Moore et al., 1998, 1999b; Lewis et al., 2000; Fukushige & Makino, 2001). Thepresence ofn>

∼1 density cusps in clusters of galaxies has been demonstrated using X-ray (e.g. Markevitch et al.,1999) as well as gravitational lensing observations (Squires et al., 1996), but examples of clusters with rathershallower cusps also exist (Sand, Treu, & Ellis, 2002).

Although equation 1.5 provides a good description of the distribution of dark matter in clusters and groups


of galaxies, it breaks down on the smallest scales. Observations of the rotation curves of dwarf galaxies showclear evidence for a constant-density core in the dark matter profile (e.g. Salucci & Burkert, 2000), which is welldescribed by a so-called ‘Burkert profile’ (Burkert, 1995),of the form

ρ(r) =ρ0r3

c

(r + rc) (r2 + r2c)

, (1.6)

whererc is the core radius. The incompatibility of dark matter distributions on large and small scales has beentermed the ‘core catastrophe’ and cited as evidence of the failure of cold dark matter (CDM) cosmology (Mooreet al., 1999a) (see section 1.2). While this could be the signature of weakly-interacting dark matter particles(Spergel & Steinhardt, 2000), it has recently been shown that such a core can be produced by flattening of thecentral cusp due to dynamical friction (El-Zant, Shlosman,& Hoffman, 2001) – a process which is expected tobe more efficient in smaller systems (Bird, 1995). Inadequate modelling of dynamical friction could also give riseto the over-abundance of dwarf galaxy-sized haloes predicted by numerical simulations of dark matter in CDMuniverses (Klypin et al., 1999; Moore et al., 1999a).

It is worth noting that ‘hidden’ baryonic matter could account for at least some (although not all) of the darkmatter. For example, one possible candidate is cold molecular gas in spiral galaxies (Pfenniger, Combes, & Mar-tinet, 1994).

1.1.3 The intergalactic medium

The volume between the galaxies in a virialized halo is filledwith hot, diffuse gas, at a temperature∼ 107−8 K.This intergalactic medium4 (IGM) is optically thin, with a particle density of∼10−3 cm−3 (Sarazin, 1988), andconsequently emits X-rays via thermal bremsstrahlung emission (see section 1.1.3). The gas achieves such largetemperatures through shock heating and compression, in thecollapse and formation of the halo; a process whichenergises the IGM to a level which reflects the depth of the potential well – thevirial temperature. Although it isextremely rarefied by terrestrial standards, the size of thepotential well retaining this material is sufficiently largefor many of these systems to be amongst the brightest extended X-ray sources in the Universe (see the catalogueof Ebeling et al., 1996, for example), with X-ray luminosities of order 1043−45 erg s−1. The temperature of theIGM increases with the halo mass, and approximately spans the range 0.3<

∼kT<∼20 keV, from the coolest groups

and individual galaxies, up to the richest clusters. These energies are small compared to the rest mass energy of anelectron (511 keV), and hence the plasma particles are not significantly relativistic.

At intermediate and lower temperatures (kT<∼5 keV) the dominant component in the X-ray spectrum of the gas

is emission lines from highly ionized species (‘metals’) within the plasma. For example, the iron line complexesaround 6.8 keV– resulting mainly from the Fe+24 and Fe+25 ions – are particularly noticeable at a gas tempera-ture of∼5 keV (Raymond & Smith, 1977). At higher temperatures, however, increased ionization significantlyreduces the emission line contribution, and the bremsstrahlung continuum emission dominates. It is conventionalto parametrize element compositions with a fixed proportiondefined by metal abundances in the Sun. Thus, gasmetallicities are usually quoted as some fraction of the Solar value (Z⊙), typically∼0.3 for many galaxy systems(Fukazawa et al., 1998) (see section 1.3.2).

The sound crossing time within gas in a virialized system is less than a Hubble time (i.e. approximately the ageof the Universe), so it follows that the IGM is relaxed and thus in hydrostatic equilibrium within the potential well.Under these conditions, the density and temperature structure of the IGM can be used to infer the gravitating massprofile, which is given by

Mgrav(r) = −kT (r) rGµmp

[

dlnρdlnr

+dlnTdlnr

]

, (1.7)

(Sarazin, 1988), whereµ is the mean molecular weight of the gas,mp is the proton mass,k is Boltzmann’s constantandG is the gravitational constant. In practice, this is achieved by fitting analytical models to describe the gasdensity and temperature as a function of radius (see section2.3.1 for further details). Since the IGM smoothlytraces the gravitational potential in a relaxed halo, this method of calculating its mass is more accurate and reliablethan measurements based on the virial estimator (section 1.1.1).

4hereafter we use this term to refer to the diffuse gas betweenthe galaxieswithin a collapsed halo.


Figure 1.2: A composite X-ray image of the Coma Cluster, taken with the orbitingXMM-Newtonobservatory. A number of distinct sourcesassociated with galaxies in the cluster are identifiable, embedded in a bright, extended halo of diffuse emission from the intergalactic medium.A small group of galaxies can be seen falling into the main cluster in the bottom right hand corner. 10 arcminutes, as indicated by the scalebar, corresponds to roughly 300 kpc at the redshift of the cluster. Note: the rectangular, grid-like pattern delineatesthe boundary between thedifferent CCD chips on the telescope detector, and is not associated with the cluster. Image courtesy of ESA.

X-ray emission from hot gas

The bremsstrahlung emissivity of a plasma of temperature,T, at a frequencyν is given by

εν ∝ gZ2neni√

Te−( hν

kT ), (1.8)

wherene andni are electron and ion number densities,Z is the charge on the ion andg ≈ 1 is the Gaunt factor,which corrects for quantum mechanical effects and distant collisions, and is weakly dependent on both temperatureand frequency (Kellogg, Baldwin, & Koch, 1975).

The total X-ray luminosity per unit volume can be obtained byintegrating equation 1.8 over all frequencies, togive

Z ∞

0ενdν ∝ neni

√T (1.9)

which leads to the total luminosity, via integration over the volume of the halo

LX ∝Z

Vneni

√TdV. (1.10)

It is clear from this that the X-ray luminosity of the gas willprincipally depend on the square of the gas density (as


ne ∼ ni, for a plasma mainly composed of hydrogen), and more weakly on temperature. This strong dependence ofemissivity on density has important consequences for the rate at which the gas loses energy via cooling, which canalter the properties of the IGM significantly (see section 1.3.1). Fig. 1.2 shows an X-ray image of the large, nearbyComa Cluster, as observed by theXMM-Newtontelescope. The sharp contrast of the cluster emission against thebackground is clearly visible, as is the smooth distribution of the IGM, compared to the discretely-sampled galaxypopulation seen in Fig. 1.1.

Entropy

The entropy of the hot gas is a fundamental quantity which serves as a powerful diagnostic of cluster physics. En-tropy is a macroscopic concept, which is linked to the microscopic properties of a system – in this case the motionsof individual plasma particles in the gas – and is normally associated with statistical physics (see Mandl, 1994, fora textbook treatment). There are many different configurations of microscopic states (microstates) which can giverise to the same macroscopic state (macrostate). For a volume,V, of gas in equilibrium, withN particles and atotal energy ofE, the number of microstates comprising a macrostate specified by E,V,N is given byΩ(E,V,N),which is referred to as thestatistical weightof this particular macrostate. Each microstate is assumed to have aequal probability of occurring and therefore, the most probable macrostate will be the one which can occur for thelargest number of different microstates, i.e. the one with the greatest statistical weight.

An isolated system will correspondingly evolve towards a state of maximum entropy, defined by Boltzmann as

S(E,V,N) = k lnΩ(E,V,N), (1.11)

wherek is Boltzmann’s constant. This process amounts to a loss of information, since all different initial mi-crostates will eventually converge on the most probable macrostate. As such, entropy constitutes a quantitativemeasure of disorder, and thus a lower entropy system can be said to represent a more ordered state. For practicalpurposes, a more specific version of equation 1.11 is required. Starting from the fundamental thermodynamicrelation,

TdS= dE+PdV, (1.12)

and incorporating the specific heat capacity at constant volume, given by

CV =

(

∂E∂T

)

V, (1.13)

together withCP−CV = R, the gas constant, whereCP is the specific heat capacity at constant pressure, it can beshown that the entropy for an ideal gas reduces to

S= CV lnT

ργ−1 , (1.14)

whereγ = CPCV

= 5/3. For the purposes of astronomical analyses, the followingsimplified expression for entropy isadopted, which ignores constants and logarithms (e.g. Ponman, Cannon, & Navarro, 1999)

S′ =kT

n2/3e

, (1.15)

wherene is the electron number density of the gas andkT is the familiar adopted definition of ‘temperature’.

Entropy is conserved in the adiabatic collapse and formation of a virialized halo, whereby shock heating andcompression of the gas raises it according to the mass of the system. Consequently, it is expected to scale linearlywith the virial temperature of the system. Departures from this proportionality point to the influence of processesother than heating from gravitational collapse in alteringthe thermodynamic history of the gas. Within a cluster,the entropy of the IGM must increase monotonically with radius (or remain constant) if convective stability is tobe maintained. Thus the lowest entropy gas gravitates to thecentre of the potential which, because of its low tem-perature and high density (equation 1.15; section 1.1.3), is highly susceptible to rapid cooling (see section 1.3.1).


1.2 The Formation and Development of Structure

Since dark matter is the principal mass component in the Universe, its properties dominate the formation of largescale structures through gravitational collapse. The key factor is the intrinsic energy (velocity dispersion) of thedark matter, which sets the smallest mass scale on which material can become bound in virial equilibrium. Hotdark matter, for example light neutrinos, would have relativistic particle velocities that would lead to a top-downstructure formation scenario, in which the least massive haloes form from the fragmentation of larger systems.The alternative, cold dark matter, predicts the reverse – dark matter particle velocities are negligible in the earlyUniverse, permitting the virialization of the smallest haloes at the earliest times.

The prevailing view is that the Universe is dominated by colddark matter and that, correspondingly, the for-mation of structure proceeds in a bottom-up fashion (e.g. Blumenthal et al., 1984; Davis et al., 1985; Cole &Lacey, 1996). The seeds for the formation of collapsed objects began as microscopic fluctuations in the otherwisehomogeneous Universe at the epoch of recombination. At thispoint the Universe became sufficiently cool to al-low electrons to remain bound to atomic nuclei, which vastlyincreased the mean free path of photons, essentially‘freezing in’ all structure, by preventing further interaction between matter and radiation. The fluctuations whichlater formed structures are imprinted on the cosmic microwave background (first discovered by Penzias & Wilson,1965), and were detected with the COBE satellite a decade ago(Smoot et al., 1992). The spectrum of the initialdensity perturbations at recombination is believed to be Gaussian. As the Universe expands, these fluctuationsgrow and eventually collapse non-linearly. The smaller mass perturbations are the first to collapse and they doso inside of larger perturbations, which themselves collapse at later times (Cole, 1991). This pattern of growth isreferred to ashierarchical formation.

Subsequent refinements to the model of standard CDM cosmology have been necessary, based on measure-ments of the mass density of the Universe,Ω0, compared to the total density,Ω (equation 1.3). Since there isa wealth of observational evidence to favour a low mass density of Ω0 ∼ 0.3ρcrit (e.g. de Bernardis et al., 2000;Turner, 2002) and there are very good reasons to prefer a flat Universe (i.e.Ω = 1) (Sievers et al. 2002; see alsoPeacock 1999), it follows that some extra component is required to make up the deficit. This contribution is re-ferred to as the cosmological constant (Λ) and is essentially a ‘dark’ energy associated with the vacuum of space,which causes the expansion of the Universe to accelerate. Given the above constraints, it follows that the densitycontribution of the vacuum energy isΩΛ ∼ 0.7; a value which is confirmed by measurements of the accelerationof the Universe, using distant supernovae as ‘standard candles’ (see Leibundgut, 2001, for a review). This generalarea of cosmology is currently advancing rapidly, with tighter constraints on key parameters becoming availablefrom a number of independent sources. In addition, the forthcoming launch of theMAP andPLANCK satelliteswill revolutionise our understanding of the cosmic background radiation and its role in the development of structurein the Universe.

1.2.1 Hierarchical formation

Galaxies represent the smallest class of dark-matter dominated virialized system. Therefore, in a hierarchicalformation scenario, they are the fundamental building blocks of which larger objects are composed (see Coles &Lucchin, 1995, for example). Observations of high-redshift clusters have revealed a high level of galaxy mergingoccurring (e.g. van Dokkum et al., 1999), consistent with the expectations of hierarchical formation. On largermass scales, there is evidence of substructure within and interactions between clusters of galaxies, from X-rayobservations (e.g. Jones & Forman, 1999), indicating that asignificant fraction of clusters are undergoing mergersat the present epoch. Even clusters are themselves observedin bound congregations – superclusters of galaxies(Raychaudhury, 1989; Raychaudhury et al., 1991), althoughthese assemblies have not yet had sufficient time toreach virial equilibrium at the present epoch.

Cosmological numerical simulations have provided great insight into the process by which structures evolve inthe Universe. Fig. 1.3 shows the spatial distribution of baryonic material at the present epoch for aΛ-dominatedCDM Universe. A rich network of structure is clearly visible, arranged in a complex web of filaments and knotsof material. The positions of virialized haloes occur at theintersection of large filaments (the brightest regions inFig. 1.3). Only these clumps of material are dense enough to be easily detected – the remainder has too low anX-ray surface brightness to be seen above the cosmic background with current generation telescopes. However,despite its low density, this warm-hot intergalactic medium (WHIM), as it has been termed, probably containsmost of the baryons in the Universe (Dave et al., 1998; Bryan& Norman, 1998; Cen & Ostriker, 1999; Croftet al., 2001; Dave et al., 2001). Measurement of deuterium abundances in high-redshift hydrogen clouds, coupled

1.2. THE FORMATION AND DEVELOPMENT OFSTRUCTURE 9

Figure 1.3: A cosmological simulation of structure formation in aΛCDM Universe atz= 0. The mass density is shown, consisting of 2463

particles in a a cube with 240 Mpc on a side. Image courtesy of J.Colberg; from the Virgo consortium (Jenkins et al., 1998)

with Big Bang nucleosynthesis calculations imply a Universal baryon density ofΩb = (0.04±0.008)h−270 (Burles,

Nollett, & Turner, 2001; O’Meara et al., 2001), whereas observations of the baryon content of galaxies and clustersonly account forΩb < 0.011 (Fukugita, Hogan, & Peebles, 1998). The deficit is explained by the material ‘hiding’in filaments and low surface brightness groups.

1.2.2 Self-similarity

An important consequence of hierarchical formation is the concept ofgravitational self-similarity. This holdsthat mass distributions of smaller virialized systems (like individual or small groups of galaxies) resemble simple,scaled down versions of their larger counterparts (such as rich clusters of galaxies). In the context of an expandingUniverse, where the mean density decreases monotonically with time, this concept can be subdivided into self-similarity between objects of different masses formed at the same epoch and self-similarity of objects of the samemass formed at different epochs.

Numerical simulations of ensembles of virialized haloes collapsing at the same epoch indicate that the darkmatter scales self-similarly across a very wide range of halo masses (e.g. Navarro et al., 1995). However, the effectsof a change in the mean density of the Universe with redshift give rise to modest departures from self-similaritybetween haloes of the same mass, formed at different epochs.This has been confirmed with numerical simulations(Cole & Lacey, 1996; Navarro, Frenk, & White, 1997; Avila-Reese et al., 1999; Bullock et al., 2001; Jing & Suto,2000; Jing, 2000) and analytical models of structure formation (Salvador-Sole, Solanes, & Manrique, 1998; Łokas,2000) as well as with X-ray observations of real virialized systems (Sato et al., 2000).

The concept of self-similarity can be extended further to properties of the hot gas and stellar material, whichare expected to track the underlying mass profile. However, since such baryonic matter is susceptible to complexphysical processes which do not alter the dark matter directly, it is expected that this simple model will break downunder these circumstances. None the less, the application of this principle has been shown to be extremely effectivein predicting the scaling properties of virialized haloes (Press & Schechter, 1974). Once again, there are twodifferent manifestations of self-similarity – the extent to which baryonic matter traces dark matterwithin haloesand the scaling of baryonic mass and related propertiesbetweenhaloes of different mass. N-body simulations


predict that the gas traces the dark matter (and thus scales self-similarly), when only the effects of gravity andshock heating are incorporated (Navarro et al., 1995). Thus, observations of the spatial distribution of the IGMwith respect to the gravitating mass profile provide a directprobe of the effects of non-gravitational physics on thehot gas.

1.2.3 Scaling relations

Motivated by the principle of self-similarity, well definedrelationships between various observable quantities ofvirialized haloes can be derived. These so calledscaling relationslink integrated quantities like X-ray luminosity,temperature, halo mass and velocity dispersion, to predictthe properties of galaxy systems based only on theircharacteristic size. Since the total gravitating mass is difficult to calculate, X-ray temperature is often used as amuch more readily obtainable proxy.

Starting from the basis that haloes formed at the same epoch have the same mean density, is follows thatM/R3v

is constant, whereM is the characteristic mass andRv the characteristic radius of the halo.Rv is known as thevirial radius, and can be used to scale the density profiles of virialized systems to produce an identical, self-similarprofile. The virial radius is well approximated byR200 – the radius enclosing a fixed overdensity of 200 timesthe critical density (equation 1.2) (Navarro et al., 1995).The X-ray temperature is a good estimator of the virialtemperature, which is given byT ∝ GM/Rv. SinceM ∝ R3

v, this impliesT ∝ R2v. This leads to the following

predictionsRv ∝

√T (1.16)

andM ∝ T3/2 (1.17)

(see also Chapter 2).Furthermore, if the hot gas traces the gravitating mass profile and predominantly radiates energy via bremsstrahlung

emission (see section 1.1.3, above), then its X-ray luminosity, LX ∝ M2gasR

−3v T0.5. This can then be simplified to

LX ∝ f 2gasT

2 (e.g. Ponman et al., 1999), wherefgas is the gas fraction, i.e. the ratio of gas mass to total mass, whichis assumed to be constant in the case of self-similarity. Therelation between luminosity and temperature is thus

L ∝ T2. (1.18)

Similarly, if the galaxies within a virialized system are assumed to possess a kinetic energy of exactly half thegravitational potential energy of the total mass (i.e. the virial theorem: equation 1.1), it follows thatσ2

clus ∝ M/Rv,whereσclus is the line-of-sight velocity dispersion of the galaxies inthe cluster (section 1.1.1). This gives rise to

σclus ∝√

T (1.19)

andσclus ∝ M1/3. (1.20)

Theσ−TX andσ−M relations for virialized systems are dealt with in more detail in Chapter 5.

1.3 Feedback and Interaction

That the properties of the IGM in virialized systems do not behave according to expectations based on simplescaling laws (see Chapters 2–5), points to the influence of physical processes other than gravity and shock heatingin modifying the gas. Such processes ultimately originate from the stars that have formed out of the IGM, andtheir influence implies the need to understand feedback mechanisms in seeking to explain the scaling properties ofgalaxies, groups and clusters.

1.3.1 Cooling and star formation

Since the bremsstrahlung X-ray emissivity of the hot gas scales with the square of its density (section 1.1.3), theprocess of gas cooling is intrinsically unstable, because it acts to increase the density, which leads to increasedcooling. The characteristic time-scale over which this process will occur is referred to as the cooling time, which

1.3. FEEDBACK AND INTERACTION 11

is given bytcool ∝ ρgasTX/LX , whereρgas is the density of the gas,LX is the X-ray luminosity, andTX is itstemperature. However, since the gas luminosity is itself proportional toρ2

gas

√T, it follows that

tcool ∝√

Tρgas

. (1.21)

In numerical and semi-analytic cosmological models, the effects of this process lead to all baryonic materialcooling to form stars by the present day (Cole, 1991; White & Frenk, 1991; Blanchard, Valls-Gabaud, & Mamon,1992). On smaller scales, the difficulty persists – for example, semi-analytical models of halo formation predictthe existence of central, dominant galaxies which are substantially brighter than those observed, as a result ofovercooling of gas (Kauffmann, White, & Guiderdoni, 1993).The solution to this problem is the introductionof feedback associated with star formation, which is capable of modifying both the temperature and density ofthe gas, and thus prevent it cooling (Cole, 1991; White & Frenk, 1991; Springel & Hernquist, 2002). None theless, the implementation of this feedback in numerical models is difficult, since it is an inherently sub-resolutionphenomenon for cosmological simulations of large scale structure formation.

Within virialized systems, cooling has important implications for the inner regions of the halo, wheretcool issubstantially less than a Hubble time (Edge, Stewart, & Fabian, 1992; White, Jones, & Forman, 1997; Peres et al.,1998). If left relatively undisturbed, a centralcooling flowis likely to develop (see Fabian, 1994, for a review),where a bulk inflow of material is established to maintain equilibrium as energy is rapidly radiated away. Thecharacteristic signatures of such a feature are a decrease in the X-ray temperature, coincident with an enhancementin the surface brightness on scales of 10–100 kpc, associated with the increased emissivity of the cooling gas.Cooling flows are found in∼70 per cent of clusters (Peres et al., 1998; White et al., 1997) and are very rarelyassociated with clusters that show clear signs of disruption or merging. The presence of a cooling flow can severelybias attempts to measure the global cluster (virial) temperature or gas density distribution. However, even whenattempts are made to correct for their presence, there is some debate about the outcome: for example, Allen &Fabian (1998) claimed that modelling out the cooling component led to a self-similarL−TX relation in clusters,in contrast to Markevitch (1998), who concluded that doing so was not sufficient to flatten the relation toLX ∝ T2.

The role of gas cooling has received increased attention recently, in light of new results from theXMM-NewtonandChandratelescopes. Observations of a number of cluster cooling flows have revealed an absence of gas attemperatures significantly lower than∼1–2 keV (see Bohringer et al., 2002, and references therein). Re-heatingof gas by a central active galactic nucleus (AGN) (e.g. Valageas & Silk, 1999; Churazov et al., 2002; Reynolds,Heinz, & Begelman, 2002; Bruggen & Kaiser, 2001; Blanton, Sarazin, & Irwin, 2001) is one mechanism whichcan explain this discovery, especially given that AGN are known to be common in cooling flow clusters (Burns,1990). Although this mechanism is energetically feasible,however, the interaction between bulk outflows fromAGN and the IGM is poorly understood (Brighenti & Mathews, 2002) and such a coupling may be insufficientlyeffective to suppress the cooling at the levels required.

Another promising candidate is heat conduction, which has recently been identified as as a plausible meansof regulating runaway cooling in the cores of massive clusters (Fabian, Voigt, & Morris, 2002b). In this scenario,heat is transferred from the ambient IGM to the cooling gas soas to reduce the temperature gradient betweenthe two. The key issue for heat conduction is the presence of magnetic fields, whose strength and configurationis critical in determining the effectiveness with which energy can be transported. The maximum rate at whichconduction can occur – the Spitzer rate (Spitzer, 1962) – renders this process extremely effective, but the efficiencyis rapidly reduced if particles have to cross field lines. This process may also be the same mechanism whichacts to strongly suppress cooling in large galaxies, thus inhibiting further star formation, which is otherwise aproblem in semi-analytic models of galaxy evolution (e.g. Kauffman et al., 1999). The effectiveness of heatconduction has important consequences for sedimentation of heavy ions in the core of the IGM, which are discussedin section 1.3.2.

Once cooled, the gas must inevitably drop out of the hot plasma phase of the IGM and form much densermaterial. Ultimately, the accumulation of cold gas fuels the process of star formation, although a significant masscomponent may reside in the form of dust (Edge et al., 1999) ormolecular gas, which has recently been detected inthe central galaxies of cooling flow clusters (Edge, 2001). These non-stellar baryon reservoirs are not significantlyluminous at optical wavelengths, which may explain why starformation described by a standard initial massfunction cannot be taking place throughout the cooling flow (Johnstone, Fabian, & Nulsen, 1987; Fabian, 1994).However, at least some of the missing soft X-ray flux in cooling flows may be emitted in the ultraviolet, opticaland infrared band, through rapid cooling and mixing of the IGM with cold gas (Fabian et al., 2002a).


1.3.2 Enrichment of the intergalactic medium

X-ray spectra of the intergalactic medium show clear evidence of emission lines in addition to the bremsstrahlungcontinuum in clusters (e.g. Fukazawa et al., 1998), groups (e.g. Hwang et al., 1999) and individual galaxies (Mat-sushita, Ohashi, & Makishima, 2000). Such features are the unmistakable signature of highly-ionized heavy el-ements in the plasma, with abundances greatly in excess of those expected from primordial nucleosynthesis (seeKurki-Suonio, 2002, for a review). It is even possible to infer the temperature of the gas from direct measurementsof these emission lines (Molendi et al., 1999). Since these metals must have been synthesised via nuclear fusion instars, some feedback mechanism is required to explain theirpresence in the IGM.

That mechanism is outflow of gas in a galactic flow, resulting from supernova-driven stellar winds. Directevidence for such ‘superwinds’ exists for a number of actively star forming (‘starburst’) galaxies in the localUniverse (Lehnert & Heckman, 1996; Dahlem, Weaver, & Heckman, 1998). The gas from such an outflow is richin metals, and as it interacts with the IGM it pollutes it withthese elements. Such a process is expected to be fairlylocalised to the starbursting galaxy, but if it occurred at asufficiently early epoch, merger-induced mixing couldhave evenly distributed the metals throughout the IGM. Studies of the Lyman alpha (Lyα) forest absorption linesin distant quasar spectra have established that the IGM is enriched at the level of∼10−3 to 10−2Z⊙ by z∼ 2−3(Cowie et al., 1995; Songaila & Cowie, 1996; Rauch, Haehnelt, & Steinmetz, 1997; Dave et al., 1998), i.e. prior tothe formation of larger groups and clusters. The enrichmentof the pre-virialized IGM may be caused by galacticoutflows, which have been observed at high redshift (Pettiniet al., 2001), or could possibly be the result of an earlyphase of star formation – so called population III stars (Loewenstein, 2001).

Observations of the metallicity structure of the IGM provide evidence of its ongoing contamination with heavyelements. Spatially resolved spectroscopy has revealed significant radial gradients in the global abundance ofiron in a number of groups and clusters, from theASCA(Finoguenov & Ponman, 1999; Finoguenov, Arnaud, &David, 2001a; Dupke & White, 2000),ROSAT(Buote, 2000),BeppoSAX(De Grandi & Molendi, 2001; Irwin &Bregman, 2001),Chandra(David et al., 2001; Sanders & Fabian, 2002) andXMM-Newton(Kaastra et al., 2001)X-ray satellites. The presence of such gradients is incompatible with purely pre-virialized enrichment, unlesscaused by ion sedimentation (e.g. Fabian & Pringle, 1977; Qin & Wu, 2000), where heavier species graduallymigrate towards the centre of the gravitational potential,leading to a partitioning of elements. This process is verysensitive to the structure of magnetic fields within the IGM and is only likely to operate effectively where such fieldlines adopt a predominantly radial alignment. Such a configuration may indeed exist within the central coolingzone of clusters, where the inward bulk motion of the gas can stretch the field lines in a radial direction (e.g.Fabian et al., 2002b), which may explain the observed abundance gradients found in the inner regions of a numberof clusters. The process of ion sedimentation is also closely linked to the role of heat conduction in suppressinggas central cooling in clusters, as mentioned in section 1.3.1.

Analysis of separate element abundances is able to distinguish between products from type Ia and II supernovaeas the source of wind-blown metals (Mushotzky et al., 1996; Renzini, 1997; Finoguenov & Ponman, 1999; Dupke& White, 2000); in particular, the presence of iron and silicon are the tell-tale signs of SN1a and SNII origins,respectively. Observations indicate that type II ejecta are distributed evenly, whereas type Ia products are morecentrally concentrated in the IGM of relatively relaxed clusters and groups (Finoguenov & Ponman, 1999; Dupke& White, 2000). This can be understood in terms of an early phase of enrichment by an old population of stars,prior to the formation of these systems, spreading the material from SNII outflows uniformly. Subsequent type Iasupernovae, from a younger population of second-generation stars, then provide a continuous injection of metalsinto the interstellar medium, which are released via ram-pressure stripping of the galaxies within the cluster (Gunn& Gott, 1972; Gaetz, Salpeter, & Shaviv, 1987; Stevens, Acreman, & Ponman, 1999). Since ram-pressure isproportional toρ2

gasvgal, this process is more efficient in the inner (denser) regionsof the IGM, where galaxymotions (vgal) are greatest, leading to a radially decreasing gradient away from the centre (Finoguenov & Ponman,1999; Dupke & White, 2000). This process has been observed inthe gas dynamical simulations of Metzler &Evrard (1994, 1997), which incorporate ongoing galaxy winds.

The presence of central cooling has also been linked with metallicity gradients (e.g. Finoguenov & Ponman,1999; Allen et al., 2001a; Irwin & Bregman, 2001) hinting at the role of merging activity in erasing both ofthese types of features, through mixing of the gas. A consistent picture is emerging in which the undisturbedIGM naturally evolves towards a state where a radially decreasing global abundance profile exists in tandem withsignificant central cooling (see Allen et al., 2001a, and references therein), subject to the effects of local heatingand/or conduction on the latter.

1.3. FEEDBACK AND INTERACTION 13

1.3.3 Energy injection

It has already been seen that the impact of feedback mechanisms, such as galactic winds, can have a substantialeffect on the IGM, but what consequences does this have for the energyof the gas? Clearly the kinetic energydeposited by the bulk outflow of material is a potential meansof raising the temperature of the IGM at highredshift by non-gravitational means (‘pre-heating’), thus preventing catastrophic cooling and overproduction ofstars. The key evidence for this energy injection is found inthe entropy of the gas, which is uniquely sensitiveto the effects of preheating. X-ray observations have discovered a clear excess in the entropy of the IGM in coolgroups (termed the ‘entropy floor’), which clearly exceeds levels attainable through gravitational collapse alone(Ponman et al., 1999; Lloyd-Davies, Ponman, & Canon, 2000; Xu, Jin, & Wu, 2001) (see also Chapter 4). Theabsence of low entropy gas in groups and clusters is also required to explain the reduced contribution to the cosmicX-ray background from the IGM at high redshift (Pen, 1999; Voit & Bryan, 2001a).

The presence of an entropy excess resulting from pre-heating of the IGM was originally invoked to explain thesteepening of theL−TX relation in groups compared to clusters (Kaiser, 1991). This effect would naturally leadto a reduction in the central density of the gas, which would suppress its luminosity by a proportionately greateramount in less massive systems, as well as flatten their surface brightness profiles (Ponman et al., 1999). Evidencefor a more extended IGM in smaller haloes is also found in the logarithmic slope of the gas density (as measuredby the characteristic index,β – see section 2.6.1) at large radii; groups exhibit values ofβ ∼ 0.4−0.5 (e.g. Ponman& Bertram, 1993; Helsdon & Ponman, 2000b), compared to the canonical cluster value of∼2/3 (Jones & Forman,1984). These findings are consistent with weaker capture of the energetically-boosted IGM in cooler systems,which adopts a more extended morphology as a result. The suppression of shocks associated with an increase inthe entropy of the gas may also lead to the accretion radius extending beyond the virial radius in smaller haloes(Tozzi, Scharf, & Norman, 2000). This implies a lower gas fraction in groups compared to clusters, since theaccretion (shock) boundary marks the point where the gas fraction in virialized haloes reaches the Universal meanvalue.

Although a number of studies have concluded that the heatingfrom galaxy winds is able to explain the observedtrends (Ponman et al., 1999; Loewenstein, 2000; Brighenti &Mathews, 2001; Cen & Bryan, 2001; Scannapieco& Broadhurst, 2001), it has been pointed out that the energy available from supernovae is insufficient to providethe required level of heating (Wu, Fabian, & Nulsen, 2000; Bower et al., 2001). Correspondingly, alternativemechanisms have been proposed, primarily focused on activegalactic nuclei (AGN) (see Robson, 1996, for areview) as a viable energy source (Valageas & Silk, 1999; Wu et al., 2000; Nath & Roychowdhury, 2002), althoughheating from population III stars, for example, could also be significant (Loewenstein, 2001). Some constraint onthe amount of heating that can occur is imposed by the effect of gas mixing on the metallicity structure of theIGM. However, the simulations of Bruggen (2002) indicate that the heating scenario is consistent with measuredabundance gradients.

Further constraints are provided by the existence of the Lyα forest –T ∼ 104 K (Hui & Gnedin, 1997) – whichis liable to be destroyed if energy is injected uniformly across all baryons at high redshift, heating them to∼106 K(Theuns, Mo, & Schaye, 2001). None the less, it has recently been demonstrated that galaxy wind heating at highredshift is viable, as the wind-blown bubbles that it generates preferentially affect the low density gas, leavingintact the higher density filaments responsible for the Lyα forest (Theuns et al., 2002). In any case, less energyis required to heat the gas prior to virialization, since itsdensity is much lower. Non-gravitational heating of thepost-virialized IGM requires large amounts of energy (1–3 keV per particle) to be injected into the gas in orderto reproduce the observed entropy floor in groups (Wu et al., 2000; Borgani et al., 2001; Babul et al., 2002) (seeChapter 4). Within clusters, there is direct evidence of an interaction between the IGM and the central radio sourceassociated with an AGN. Recent data from theChandraX-ray satellite has uncovered a wealth of features such ascavities excavated by subsonically expanding bubbles originating in radio jets (McNamara et al., 2000; Finoguenov& Jones, 2001; Churazov et al., 2001). However, as yet there is much less evidence of the shock heating expectedfrom supersonic motion, which is necessary to provide effective heating of the gas.

1.3.4 Gas cooling revisited

Although heating by non-gravitational processes has been shown to be capable of explaining the trends seen inscaling relations, alternative hypotheses have been considered. The most promising of these is radiative cooling ofgas (cf. Knight & Ponman, 1997), which is able to steepen theL−TX andM−TX relations with respect to self-similar expectation, exactly as observed (Muanwong et al.,2001). Although preheating models are self-consistent


in the sense that cooling is insignificant, given the effect of injecting such large amounts of energy, it does notnecessarily follow that its effects in real clusters are negligible; indeed, it has already been seen just how effectivecooling can be (section 1.3.1). In fact, predictions from models with only gas cooling and no substantial energyinjection at all have been shown to provide a reasonable match to observations (Bryan, 2000; Wu & Xue, 2002b;Muanwong et al., 2001; Dave, Katz, & Weinberg, 2002).

Just as proponents of heating models can cite the enrichmentof the IGM as evidence of the potential energyinjecting abilities of galaxy winds, so the existence of stars and ongoing star formation in galaxies, groups andclusters is proof that gas cooling is taking place. In this role, it acts to eliminate the lowest entropy gas, whichdrops out to form stars, allowing higher entropy gas to flow infrom further out and replace it. The net effect isto lower the central gas density and flatten its distribution, as well as introduce an entropy floor. Recent work hasfocused on the effects of gas cooling, in conjunction with the supernova heating from subsequent star formation, asa very effective and internally consistent explanation of self-similarity breaking (Voit & Bryan, 2001b; Voit et al.,2002). A particularly attractive feature of this model is the self-regulation of cooling that is induced by the heatingresulting from star formation – since enhanced cooling willbe balanced by the increase in heating from supernova.

We deal with the issue of gas heating and cooling in the context of the entropy of virialized systems in Chapter 4.

1.4 The Birmingham-CfA Cluster Scaling Project

The following chapters present an analysis of the scaling properties of virialized systems, as part of ‘The Birmingham-CfA Cluster Scaling Project’ – a collaboration between Birmingham University and the Harvard-Smithsonian Cen-ter for Astrophysics (CfA). The sample assembled in this project comprises 66 objects, spanning over 3 orders ofmagnitude in mass, from rich clusters of galaxies, through poor clusters and groups and down to the level of in-dividual, early-type galaxy haloes. Fig. 1.4 shows six typical systems in the sample – 2 each of clusters, groupsand galaxies. Each panel consists of an optical image of the system, with contours showing the emission from theX-ray halo superimposed.

In Chapter 2 we present the initial X-ray analysis of the sample, allowing us to reconstruct the density andtemperature structure of the IGM and hence determine the gravitating mass distribution. We use these data toexamine the behaviour of the gas fraction and theM−TX relation. In Chapter 3 we build on this by measuring thegalaxy density profile in a subset of our sample, comprising 32 groups and clusters. This allows us to determinethe stellar mass as a function of radius which, coupled with the X-ray data, enables the dark matter distributionto be inferred. We concentrate on the spatial distribution of the various mass components, as well as the scalingproperties of integrated quantities such as mass-to-lightratio and baryon fraction. Correspondingly, we investigatethe efficiency of star formation with varying halo mass.

In Chapter 4 a detailed analysis of the entropy of the IGM is presented, shedding light on the role of non-gravitational processes in modifying the gas. Our results allow us to discriminate between different models ofradiative cooling and/or energy injection to explain the observed departures from self-similarity. In Chapter 5 weinvestigate optical to X-ray correlations, focusing particularly on the galaxy velocity dispersion, as a probe of thedynamical evolution of virialized haloes. We also study therelation between gas mass and X-ray temperature –another sensitive probe of non-gravitational physics.

1.4. THE BIRMINGHAM -CFA CLUSTER SCALING PROJECT 15

Figure 1.4: Images of 6 typical systems in the cluster scaling project sample: two galaxies (top panels), two groups (middle panels) and twoclusters (bottom panels). Each panel consists of an optical(grey-scale) image, from the Digitized Sky Survey (DSS), with X-ray contours fromtheROSATsatellite (PSPC detector) overlayed.

Chapter 2

Gas Fraction and theM−TX

Relation

Abstract We have assembled a large sample of virialized systems, comprising 66 galaxy clusters, groups and elliptical galaxieswith high quality X-ray data. To each system we have fitted analytical profiles describing the gas density and temperaturevariation with radius, corrected for the effects of centralgas cooling. We present an analysis of the scaling properties of thesesystems and focus in this paper on the gas distribution andM−TX relation. In addition to clusters and groups, our sampleincludes two early-type galaxies, carefully selected to avoid contamination from group or cluster X-ray emission. We comparethe properties of these objects with those of more massive systems and find evidence for a systematic difference betweengalaxy-sized haloes and groups of a similar temperature.

We derive a mean logarithmic slope of theM−TX relation withinR200 of 1.84±0.06, although there is some evidence ofa gradual steepening in theM−TX relation, with decreasing mass. We recover a similar slope using two additional methodsof calculating the mean temperature. Repeating the analysis with the assumption of isothermality, we find the slope changesonly slightly, to 1.89±0.04, but the normalization is increased by 30 per cent. Correspondingly, the mean gas fraction within

R200 changes from(0.13±0.01)h− 3

270 to (0.11±0.01)h

− 32

70 , for the isothermal case, with the smaller fractional change reflectingdifferent behaviour between hot and cool systems. There is astrong correlation between the gas fraction within 0.3R200 andtemperature. This reflects the strong (5.8σ) trend between the gas density slope parameter,β, and temperature, which has beenfound in previous work.

These findings are interpreted as evidence for self-similarity breaking from galaxy feedback processes, AGN heating orpossibly gas cooling. We discuss the implications of our results in the context of a hierarchical structure formation scenario.

2.1 Introduction

The formation of structure in the Universe is sensitive to physical processes which can influence the distributionof baryonic material, as well as cosmological factors whichultimately govern the behaviour of the underlyinggravitational potential. By studying the properties of groups and clusters of galaxies, it is possible to probe thephysical processes which shape the evolution and growth of virialized systems.

X-ray observations of the gaseous intergalactic medium (IGM) within a virialized system provide an idealprobe of the structure of the halo, since the gas smoothly traces the underlying gravitational potential. However,this material is also sensitive to the influence of physical processes arising from the interactions between andwithin haloes, which are commonplace in a hierarchically evolving universe (e.g. Blumenthal et al., 1984). Evenin relatively undisturbed systems, feedback from the galaxy members can bias the gas distribution with respect tothe dark matter in a way which varies systematically with halo mass. N-body simulations (e.g. Navarro, Frenk, &White, 1995) indicate that, in the absence of such feedback mechanisms, the properties of the gas and dark matterin virialized haloes should scaleself-similarly, except for a modest variation in dark matter concentrationwithmass (Navarro et al., 1997). Consequently, observations ofa departure from this simple expectation provide a keytool for investigating the effects ofnon-gravitational heating mechanisms, arising from feedbackprocesses.

There is now clear evidence that the properties of clusters and groups of galaxies do not scale self-similarly:for example, theL−TX relation in clusters shows a logarithmic slope which is steeper than expected (e.g. Edge &Stewart 1991b; Arnaud & Evrard 1999; Fairley et al. 2000). A further steepening of this slope is observed in thegroup regime (e.g. Helsdon & Ponman, 2000b), consistent with a flattening in the gas density profiles, which isevident in systems cooler than 3–4 keV (Ponman, Cannon, & Navarro, 1999). Such behaviour is attributed to theeffects of non-gravitational heating, which exert a disproportionately large influence on the smallest haloes. Anobvious candidate for the source of this heating is galaxy winds, since these are known to be responsible for theenrichment of the IGM with heavy elements (e.g. Finoguenov,Arnaud, & David, 2001a). However, active galactic

16

2.2. THE SAMPLE 17

nuclei (AGN) may also play a significant role, particularly as there is some debate over the amount of energyavailable from supernova-driven outflows (Wu, Fabian, & Nulsen, 2000). Recently, theoretical work has alsoexamined the role of gas cooling (cf. Knight & Ponman, 1997),which is also able to reproduce the observed scalingproperties of groups and clusters, by eliminating the lowest entropy gas through star formation, thus allowing hottermaterial to replace it (Muanwong et al., 2001; Voit & Bryan, 2001b).

Previous observational studies of the distribution of matter within clusters have typically been limited by eithera small sample size (e.g. David, Jones, & Forman, 1995), or have assumed an isothermal IGM (e.g. White &Fabian, 1995); it appears that significant temperature gradients are present in many (e.g. Markevitch et al., 1998),although perhaps not all (e.g. White, 2000; De Grandi & Molendi, 2002; Irwin & Bregman, 2000) clusters ofgalaxies. Another issue is the restriction imposed by the arbitrary limits of the X-ray data; halo properties must beevaluated at constant fractions of the virial radius (Rv), rather than at fixed metric radii imposed by the data limits,in order to make a fair comparison between varying mass scales. In this work, we derive analytical expressionsfor the gas density and temperature variation, which allow us to extrapolate these quantities beyond the limitsof the data. However, we are careful to consider the potential systematic bias associated with this process. Ourstudy combines the benefits of a large sample with the advantages of a 3-dimensional, deprojection analysis, inorder to investigate the scaling properties of virialized haloes, spanning a wide range of masses. In this work wehave brought together data from three large samples, comprising the majority of the suitable, radially-resolved3D temperature analyses of clusters. We include a large number of cool groups in our analysis, as the departurefrom self-similarity is most pronounced in haloes of this size: the non-gravitationally heated IGM is only weaklycaptured in the shallower potentials wells of these objects.

To further extend the mass range of our analysis, we include two galaxy-sized haloes in our sample, in theform of an elliptical and an S0 galaxy. We have selected objects which are free of X-ray emission associated witha group or cluster IGM, which would be difficult to disentangle from the individual galaxy halo contribution. Thestudy of Sato et al. (2000) incorporated three ellipticals,but any X-ray emission associated with these objects isclearly contaminated by emission from the group or cluster halo in which they are embedded. Galaxy-sized haloesare of great interest as they represent the smallest mass scale for virialized systems and constitute the buildingblocks in a hierarchically evolving universe.

Throughout this paper we adopt the following cosmological parameters;H0 = 70 km s−1 Mpc−1andq0 = 0.Unless otherwise stated, all quoted errors are 1σ on one parameter.

2.2 The Sample

In order to investigate the scaling properties of virialized systems, we have chosen a sample which includes richclusters, poorer clusters, groups and also two early-type galaxies, comprising 66 objects in total. Sample selectionwas based on two criteria: firstly, that a 3-dimensional gas temperature profile was available. In conjunction withthe corresponding gas density profile, this allows the gravitating mass distribution to be inferred. Secondly, wereject those systems with obvious evidence of substructure, where the assumption of hydrostatic equilibrium is notreasonable; it is known that the properties of such systems differ systematically from those of relaxed clusters (e.g.Ritchie & Thomas, 2002).

By combining three samples from the work of Markevitch, Finoguenov and Lloyd-Davies (described in detailin sections 2.3.3, 2.3.4 & 2.3.5, respectively) together with new analysis of an additional six targets (also describedin section 2.3.5), we have assembled a large number of virialized objects with high-quality X-ray data. From thesedata, we have derived deprojected gas density and temperature profiles for each object, thus freeing our analysisfrom the simplistic assumption of isothermality which is often used in studies of this nature. The large size of oursample ensures a good coverage of the wide range of emission-weighted gas temperatures, spanning 0.5 to 17 keV.Thus, we incorporate the full range of sizes for virialized systems, down to the scale of individual galaxy haloes.The redshift range isz= 0.0036–0.208 (0.035 median), with only four targets exceeding a redshift of 0.1. Somebasic properties of the sample are summarised in table 2.1.

Since we aim to study the properties ofvirializedsystems, we require that the intergalactic gas be in reasonablehydrostatic equilibrium within the region of interest. Consequently, we preferentially select objects with minimalevidence of substructure or morphological disturbance, which might cause significant departure from this approx-imation. This also favours the assumption of a spherically symmetric gas distribution, which is implicit in ourdeprojection analysis.

18

CH

AP

TE

R2

.G

AS

FR

AC

TIO

NA

ND

TH

EM

−T

XR

EL

AT

ION

Name RA Dec. z Ta R200 rc β αb γ RXc Sampled Datae

(J2000) (J2000) (keV) kpc (arcmin) keV arcmin−1 (R200)

NGC 1553† 64.043 -55.781 0.0036 0.50+0.21−0.13 203+90

−54 1.04+0.30−0.26 0.63+0.09

−0.07 – 1.44+0.15−0.14 0.10 S P

NGC 6482† 267.954 23.072 0.0131 0.56+0.37−0.22 361+181

−105 0.22⋆ 0.48+0.03−0.04 – 1.23+0.15

−0.15 0.27 S P

HCG 68 208.420 40.319 0.0080 0.67+0.19−0.15 497+104

−83 0.37+0.12−0.11 0.46+0.02

−0.02 – 1.07+0.07−0.06 0.19 L P

NGC 1395 54.623 -23.027 0.0057 0.84+0.24−0.18 556+120

−95 0.35+0.21−0.19 0.43+0.03

−0.02 – 1.05+0.07−0.06 0.07 S P

NGC 4325 185.825 10.622 0.0252 0.90+0.07−0.07 678+81

−68 0.21+0.07−0.07 0.54+0.02

−0.01 0.00+0.02−0.02 – 0.29 S P

HCG 97 356.845 -2.326 0.0218 1.00+0.13−0.12 620+45

−37 0.04+0.03−0.01 0.41+0.01

−0.01 0.02+0.03−0.03 – 0.64 L P

IC 4296 203.393 -33.622 0.0123 1.04+0.18−0.15 529+57

−49 2.64+0.11−0.11 0.31+0.01

−0.01 – 1.18+0.14−0.14 0.45 F P,S

NGC 5846 226.385 1.696 0.0058 1.18+0.07−0.07 683+37

−33 0.42+0.02−0.02 0.55+0.02

−0.02 – 1.06+0.01−0.01 0.16 F P,S

NGC 5044 198.850 -16.386 0.0090 1.25+0.06−0.06 798+36

−33 1.66+0.51−0.28 0.49+0.01

−0.01 – 0.97+0.02−0.02 0.32 L P

HCG 51 170.587 24.293 0.0258 1.38+0.04−0.04 610+15

−15 1.81+0.07−0.07 0.30+0.01

−0.01 −0.01+0.01−0.01 – 0.81 F H,S

NGC 507 20.921 33.261 0.0164 1.40+0.11−0.08 738+28

−23 0.10⋆ 0.43+0.01−0.01 0.02+0.01

−0.01 – 0.42 L P

HCG 62 193.284 -9.224 0.0137 1.48+0.18−0.16 559+35

−31 2.26+0.09−0.09 0.30+0.01

−0.01 – 1.46+0.08−0.08 0.75 F P,S

NGC 5129 201.150 13.928 0.0233 1.54+0.41−0.35 567+71

−54 2.46+0.10−0.10 0.60+0.02

−0.02 – 1.48+0.09−0.10 0.73 F P,S

NGC 6329 258.562 43.684 0.0276 1.60+0.52−0.43 859+232

−153 2.61+0.10−0.10 0.53+0.02

−0.02 – 1.06+0.16−0.17 0.55 F P,S

NGC 2563 125.102 21.096 0.0163 1.61+0.02−0.03 627+6

−8 2.07⋆ 0.42+0.003−0.003 – 1.36+0.01

−0.01 0.52 S P

Abell 262 28.191 36.157 0.0163 2.03+0.36−0.27 998+146

−113 1.49+0.17−0.16 0.40+0.01

−0.01 – 0.80+0.07−0.08 0.46 L P

Abell 194 21.460 -1.365 0.0180 2.07+0.43−0.43 1126+246

−199 8.64+0.35−0.35 0.60+0.02

−0.02 −0.01+0.04−0.04 – 0.34 F P,S

IV Zw 0381 16.868 32.462 0.0170 2.07+0.56−0.42 892+104

−86 2.77+0.40−0.38 0.38+0.03

−0.03 0.04+0.03−0.02 – 0.58 L P

MKW 4 180.990 1.888 0.0200 2.08+0.05−0.06 842+18

−17 5.45+0.22−0.22 0.64+0.03

−0.03 – 1.29+0.02−0.02 0.56 F P,S

MKW 4S 181.647 28.180 0.0283 2.46+0.23−0.21 978+74

−65 2.64+0.11−0.11 0.51+0.02

−0.02 – 1.14+0.06−0.06 0.55 F P,S

Virgo 187.697 12.337 0.0036 2.55+0.07−0.06 1086+29

−30 2.20+0.09−0.09 0.45+0.02

−0.02 −0.01+0.00−0.00 – 0.24 F P,S

NGC 3258 155.887 -34.758 0.0095 2.57+0.12−0.12 750+25

−25 10.80+0.43−0.43 0.32+0.01

−0.01 – 1.60+0.07−0.07 0.53 F P,S

NGC 6338 258.825 57.400 0.0282 2.64+1.92−1.55 893+121

−278 1.93+0.30−0.26 0.53+0.04

−0.03 – 1.25+0.09−0.25 0.52 S P

Abell 539 79.134 6.442 0.0288 2.87+0.22−0.21 1305+124

−104 5.21+0.21−0.21 0.69+0.03

−0.03 – 1.04+0.06−0.06 0.40 F P,S

MKW 9 233.122 4.682 0.0397 2.88+0.68−0.55 1246+284

−212 0.83+0.03−0.03 0.52+0.02

−0.02 – 0.97+0.09−0.09 0.41 F I,S

AWM 4 241.238 23.946 0.0318 2.96+0.39−0.39 1540+343

−290 1.93+0.08−0.08 0.62+0.02

−0.02 −0.07+0.07−0.07 – 0.20 F P,S

Abell 1060 159.169 -27.521 0.0124 3.31+0.11−0.1 1587+69

−40 7.49+0.06−0.01 0.72+0.01

−0.00 – 0.97+0.01−0.02 0.23 L P+G

2A0335+096 54.675 9.977 0.0349 3.34+0.3−0.27 1596+158

−139 1.40+0.06−0.06 0.65+0.03

−0.03 – 0.95+0.03−0.03 0.41 F P,S

Klemola 442 356.919 -28.138 0.0290 3.40+0.28−0.26 1513+180

−148 3.40+0.14−0.14 0.61+0.02

−0.02 – 0.94+0.06−0.05 0.18 F P,S

continued overleaf

2.2

.T

HE

SA

MP

LE

19



Abell 2634 354.615 27.022 0.0309 3.45+0.28−0.27 1189+104

−85 8.62+0.34−0.34 0.69+0.03

−0.03 – 1.29+0.09−0.09 0.50 F P,S

Abell 2052 229.176 7.002 0.0353 3.45+0.39−0.4 1507+281

−237 1.75+0.07−0.07 0.64+0.03

−0.03 −0.02+0.07−0.07 – 0.22 F P,S

Abell 779 139.962 33.771 0.0229 3.57+0.94−0.76 1075+203

−148 1.42+0.06−0.06 0.34+0.01

−0.01 – 1.02+0.13−0.14 0.28 F P,S

Abell 2199 247.165 39.550 0.0299 3.93+0.06−0.06 1223+18

−15 2.140.0010.001 0.60+0.0005

−0.0005 – 1.15+0.01−0.01 0.64 L P

Abell 2063 230.757 8.580 0.0355 4.00+0.12−0.12 1493+57

−56 3.79+0.15−0.15 0.69+0.03

−0.03 0.05+0.02−0.02 – 0.45 F P,S

HCG 94 349.319 18.720 0.0417 4.02+0.46−0.43 1151+94

−83 1.10+0.04−0.04 0.48+0.02

−0.02 – 1.17+0.05−0.05 0.68 F P,S

AWM 7 43.634 41.586 0.0172 4.02+0.75−0.62 2207+641

−420 5.28+0.23−0.07 0.59+0.00

−0.00 – 0.67+0.09−0.09 0.24 L P

MKW 3S 230.507 7.699 0.0453 4.42+0.57−0.67 1218+176

−123 4.13+1.38−1.38 0.71+0.07

−0.07 – 1.32+0.10−0.11 0.41 M P,G,S

Abell 2657 356.237 9.201 0.0400 4.53+0.61−0.45 1251+188

−108 5.68+0.31−0.31 0.76+0.02

−0.02 – 1.34+0.09−0.12 0.40 M P,G,S

Abell 780 139.528 -12.099 0.0565 4.63+0.25−0.24 2032+152

−133 1.68+0.04−0.04 0.67+0.01

−0.01 – 0.90+0.03−0.03 0.53 L P+G

Abell 3391 96.608 -53.678 0.0536 5.39+0.72−0.57 1671+306

−211 2.44+0.12−0.12 0.53+0.01

−0.01 – 0.99+0.10−0.11 0.52 M P,G,S

Abell 4059 359.250 -34.752 0.0480 5.50+0.5−0.46 1313+161

−116 2.85+0.65−0.65 0.67+0.02

−0.02 – 1.29+0.07−0.08 0.44 M P,G,S

Abell 2670 358.564 -10.408 0.0759 5.64+0.4−0.39 1647+122

−111 0.97+0.04−0.04 0.55+0.02

−0.02 – 1.04+0.04−0.04 0.61 F P,S

Abell 2597 351.319 -12.124 0.0852 6.02+0.47−0.45 1841+161

−144 1.40+0.06−0.06 0.68+0.03

−0.03 – 1.05+0.04−0.04 0.56 F P,S

Abell 119 14.054 -1.235 0.0444 6.08+0.49−0.47 1720+185

−135 6.74+0.39−0.39 0.66+0.02

−0.02 – 1.14+0.08−0.09 0.51 M P,G,S

Abell 496 68.397 -13.246 0.0331 6.11+0.35−0.43 1540+94

−69 2.85+0.14−0.12 0.62+0.01

−0.01 – 1.16+0.03−0.03 0.51 L P+G

Abell 1651 194.850 -4.189 0.0846 6.18+0.55−0.36 1777+170

−116 2.02+0.23−0.23 0.70+0.02

−0.02 – 1.10+0.04−0.05 0.45 M P,G,S

Abell 3558 201.991 -31.488 0.0477 6.28+0.37−0.3 1598+124

−87 2.46+0.36−0.36 0.55+0.03

−0.03 – 1.13+0.04−0.05 0.50 M P,G,S

Abell 3571 206.867 -32.854 0.0397 7.31+0.28−0.38 1870+101

−120 4.14+0.31−0.31 0.69+0.01

−0.01 – 1.12+0.04−0.03 0.69 M P,G,S

Abell 3112 49.485 -44.238 0.0703 7.76+1.65−3.08 1311+237

−295 1.03+0.69−0.69 0.63+0.02

−0.02 – 1.32+0.14−0.09 0.61 M P,G,S

Abell 399 44.457 13.053 0.0722 7.97+0.69−0.73 1734+149

−190 1.89+0.36−0.36 0.53+0.05

−0.05 – 1.16+0.09−0.06 0.54 M P,G,S

Abell 1650 194.674 -1.756 0.0845 8.04+1.75−1.14 1816+756

−376 2.25+0.78−0.78 0.78+0.12

−0.12 – 1.19+0.15−0.17 0.48 M I,G,S

Abell 2218 248.970 66.214 0.1710 8.28+1.82−1.33 1904+180

−149 0.90⋆ 0.59+0.01−0.01 – 1.11+0.02

−0.02 0.80 L P+G

Abell 1795 207.218 26.598 0.0622 8.54+1.66−1.05 2000+628

−290 4.01+0.20−0.21 0.83+0.02

−0.02 – 1.17+0.10−0.14 0.32 M P,G,S

Abell 665 127.739 65.854 0.1818 8.60+1.27−0.94 2273+268

−279 1.21+0.04−0.04 0.65+0.01

−0.01 – 1.02+0.08−0.06 0.66 L P+G

Abell 2256 256.010 78.632 0.0581 8.62+0.55−0.51 1814+124

−145 5.02+0.11−0.11 0.78+0.01

−0.01 – 1.27+0.07−0.05 0.56 M P,G,S

Abell 85 10.453 -9.318 0.0521 8.64+0.64−0.29 1684+160

−61 4.82+0.24−0.24 0.76+0.02

−0.02 – 1.32+0.03−0.07 0.56 M P,G,S

Abell 3266 67.856 -61.417 0.0545 9.53+0.97−0.55 1880+165

−103 5.72+0.46−0.46 0.74+0.04

−0.04 – 1.29+0.05−0.07 0.50 M P,G,S

Abell 401 44.737 13.573 0.0739 9.55+0.45−0.5 1851+113

−123 2.37+0.09−0.09 0.63+0.01

−0.01 – 1.22+0.05−0.04 0.51 M P,G,S

Abell 2029 227.729 5.720 0.0766 9.80+0.4−0.42 2266+111

−103 2.37+0.09−0.09 0.68+0.03

−0.03 0.20+0.06−0.06 – 0.91 F P,S

continued overleaf

20

CH

AP

TE

R2

.G

AS

FR

AC

TIO

NA

ND

TH

EM

−T

XR

EL

AT

ION



Abell 478 63.359 10.466 0.0882 10.95+2.15−1.82 1723+587

−332 2.34+0.23−0.23 0.75+0.01

−0.01 – 1.34+0.17−0.18 0.51 M P,G,S

Abell 2319 290.274 43.964 0.0555 10.99+0.81−1.14 1882+140

−113 2.37+0.79−0.79 0.54+0.06

−0.06 – 1.23+0.04−0.05 0.69 M P,G,S

Tri. Aus. 249.584 -64.516 0.0510 11.06+1.04−0.96 1963+266

−188 4.41+0.25−0.24 0.67+0.01

−0.01 – 1.26+0.08−0.09 0.48 M P,G,S

Abell 2142 239.592 27.233 0.0894 11.16+1.54−1.15 2216+544

−292 3.14+0.22−0.22 0.74+0.01

−0.01 – 1.18+0.10−0.13 0.46 M P,G,S

Abell 644 124.355 -7.528 0.0711 11.68+1.52−1.29 1660+299

−242 2.18+0.18−0.18 0.73+0.02

−0.02 – 1.35+0.11−0.10 0.44 M P,G,S

Abell 1689 197.873 -1.336 0.1840 12.31+1.19−0.93 2955+135

−112 0.60+0.02−0.00 0.73+0.06

−0.00 0.00⋆ – 0.42 L P+G

Abell 2163 243.956 -6.150 0.2080 16.64+3.36−1.55 2104+794

−253 1.63+0.08−0.08 0.73+0.02

−0.02 – 1.38+0.11−0.22 0.75 M P,G,S

Table 2.1: Some key properties of the 66 objects in the sample, listed in order of increasing temperature. Positions and redshifts are taken from Ebeling et al. (1996,

1998); Ponman et al. (1996) and NED. Columns 5–11 are data as determined in this work. All errors are 68% confidence.

† indicates the two galaxies.⋆ denotes no errors available, as parameter poorly constrained.1 also known as NGC 383.2 also known as Abell 4038.

a The cooling-flow corrected, emission-weighted temperature of the system within 0.3R200, as determined in this work.b Temperature gradient; positive values meanT decreaseswith radius.c Outer radius of X-ray data as a fraction ofR200d F = Finoguenov et al. , L = Lloyd-Davies et al. , M = Markevitch et al. , S = Sanderson et al. (this work)e P =ROSATPSPC, H =ROSATHRI, G = ASCAGIS, S =ASCASIS, I =EinsteinIPC; + denotes simultaneous fit

2.3. X-RAY DATA ANALYSIS 21

As a number of systems are common to two or more of the sub-samples, we are able to directly compare datafrom different analyses, allowing us to investigate any systematic differences between the techniques employed.We present the results of these consistency checks in section 2.4. The diverse nature of our sample, with respect tothe different methods used to determine the gas temperatureand density profiles, insulates our study to an extentfrom the bias caused by relying on a single approach. However, we are still able to treat the data in a homogeneousfashion, given the self-consistent manner in which the cluster models are parametrized (see section 2.3.1).

2.3 X-ray Data Analysis

The X-ray data used in this study were taken with theROSATPSPC andASCAGIS & SIS instruments. Althoughnow superseded by theChandraandXMM-Newtonobservatories, these telescopes have extensive, publiclyavail-able data archives and are generally well-calibrated. In addition, the PSPC and GIS detectors have a wide field ofview, which is essential for tracing X-ray emission out to large radii, particularly for nearby systems, whose virialradii can exceed one degree on the sky. The use of three separate detectors, on two different telescopes, enhancesthe robustness of our analysis, by reducing potential bias associated with instrument-related systematic effects.

Since this work brings together data from three separate samples, as well as analyses of six extra systems,there is considerable variation in the form in which those data were originally obtained. This necessitated asupplementary processing stage to convert the data into a unified format, in order to treat them in a homogeneousfashion. In the case of the Finoguenov sample, analytical profiles were fitted to deprojected gas density andtemperature points (see section 2.3.4 for details); for theMarkevitch sample it was necessary to calculate the gasdensity normalization for such an analytical function, from the fitted data (section 2.3.3). However, our chosenmodel parametrization – described below – was fitted directly to the raw X-ray data for the remaining systems,including the Lloyd-Davies sample (further details of the data analysis are given in section 2.3.5).

2.3.1 Cluster models

In order to evaluate the gas temperature and density in a virialized system, as well as derived quantities such asgravitating mass, at arbitrary radii, we require a 3-dimensional analytical description of these data. A core indexparametrization of the gas density,ρ(r), is used, such that

ρ(r) = ρ(0)

[

1+

(

rrc

)2]− 3

2β

, (2.1)

whererc andβ are the density core radius and index parameter, respectively. The motivation for the use of thisparametrization is essentially empirical, although simulations of cluster mergers are capable of reproducing acore in the gas density, despite the cuspy nature of the underlying dark matter distribution (e.g. Pearce, Thomas,& Couchman, 1994). However, in the absence of merging, N-body simulations offer no clear explanation forthe presence of a significant core in the IGM profile, even whenthe effects of galaxy feedback mechanisms areincorporated (Metzler & Evrard, 1997).

The density profile is combined with an equivalent expression for the temperature spatial variation, describedby one of two models; a linear ramp, which is independent of the density profile, of the form

T(r) = T(0)−αr, (2.2)

whereα is the temperature gradient. Alternatively, the temperature can be linked to the gas density, via a polytropicequation of state, which leads to

T(r) = T(0)

[

1+

(

rrc

)2]− 3

2β(γ−1)

, (2.3)

whereγ is the polytropic index andrc andβ are as defined previously.

22 CHAPTER 2. GAS FRACTION AND THE M−TX RELATION

Together,ρ(r) andT(r) can be used to determine the cluster gravitating mass profileas, in hydrostatic equilib-rium, the following condition is satisfied

Mgrav(r) = −kT (r) rGµmp

[

dlnρdlnr

+dlnTdlnr

]

, (2.4)

(Sarazin, 1988), whereµ is the mean molecular weight of the gas andmp is the proton mass. This assumesa spherically symmetric mass distribution, which has been shown to be a reasonable approximation, even formoderately elliptical systems (Fabricant, Rybicki, & Gorenstein, 1984).

Since the X-ray emissivity depends on the product of the electron and ion number densities, we parametrizethe gas density in terms of a central electron number density(i.e. atr = 0), assuming a ratio of electrons to ionsof 1.17. We base our inferred electron densities on the X-rayflux normalized to theROSATPSPC instrument,as there is a known effective area offset between this detector and theASCASIS and GIS instruments. In thosesystems where the original density normalization was defined differently, a conversion was necessary and this isdescribed below.

Once the gravitating mass profile is known (from equation 2.4), the corresponding density profile can be foundtrivially, given the spherical symmetry of the cluster models. This can then be converted to an overdensity profile,δ(r), given by

δ(r) =ρtot(r)ρcrit

, (2.5)

whereρtot(r) is the mean total density within a radius,r, andρcrit is the critical density of the Universe, given by3H2

0/8πG.

It is the overdensity profile which determines the virial radius (Rv) of the cluster; simulations indicate that areasonable approximation toRv is given by the value ofr whenδ(r) = 200 (e.g. Navarro et al., 1995) – albeit forρtot(r) calculated at the redshift of formation,zf , rather than the redshift of observation,zobs– and we adopt thisdefinition in this work. Strictly speaking, the approximationRv = R200 is cosmology-dependent but, in any case,the implicit assumptionzf = zobs is a greater source of uncertainty. In particular, there is asystematic trend forthe discrepancy between these two quantities to vary with system size, in accordance with a hierarchical structureformation scenario, in which the smallest haloes form first.The consequences of this effect are addressed insection 2.6.4. Given the local nature of our sample, the assumed cosmology has little effect on our results. Forexample, comparing the values of luminosity distance obtained forq0 = 0 andq0 = 0.5: the difference is less than5% for our most distant cluster (z= 0.208), dropping to less than 2% forz< 0.1 (i.e. for 94% of our sample).

Length scales in the cluster models are defined in a cosmology-independent form, with the core radius of thegas density expressed in arcminutes and the temperature gradient in equation 2.2 measured in keV per arcminute.It was necessary to convert to these units from actual lengthmeasurements (e.g. Mpc) for a number of systems. Ineach case, the assumed cosmology of the original analysis was allowed for in the conversion.

The contributions to the cluster X-ray flux, in the form of discrete line emission from highly ionized atomicspecies in the IGM, are handled differently between the different sub-samples. However, in all cases the gasmetallicity was measured directly in the analysis and hencethis emission has, in effect, been decoupled from thedominant bremsstrahlung component, which we rely on to measure the gas density and temperature.

The key advantage of quantifying gas density and temperature in an analytical form, is the ability to extrapolateand interpolate these and derived quantities, like gas fraction and overdensity, to arbitrary radius. Consequently,the virial radius and emission-weighted temperature can beevaluated in an entirely self-consistent fashion, andthus we are able to determine the above quantities at fixed fractions ofR200, regardless of the data limits.

Clearly, where this extrapolation is quite large (e.g. atR200) there is potential for unphysical behaviour in the gastemperature, which is not constrained to be isothermal. This is particularly true when steep gradients are involved(i.e. large values ofα in equation 2.2 or values ofγ very different from unity in equation 2.3). A linear temperatureparametrization is most susceptible to unphysical behaviour, as it can extrapolate to negative values within thevirial radius. To avoid this problem, we have identified those linearT(r) models where the temperature withinR200 becomes negative. In each case the alternative, polytropictemperature description was used in preference,where this was not already the best-fitting model.


2.3.2 Cooling flow correction

The effects of gas cooling are well known to influence the X-ray emission from clusters of galaxies (Fabian,1994). Cooling flows may be present in as many as 70 per cent of clusters (Peres et al., 1998) particularly amongstolder, relaxed systems, where merger-induced mixing of gasis not a significant effect. Consequently we expectcooling flows to be common in a sample of this nature, as we discriminate against objects with strong X-raysubstructure, which is most often associated with merger events. It is possible to infer misleading properties forthe intergalactic gas, both spatially and spectrally, if the contamination from cooling flows is not properly accountedfor. Specifically, gas density core radii – and, consequently, theβ index in equation 2.1 (see Neumann & Arnaud,1999, for example) – can be strongly biased, as can the temperature profile, particularly as central cooling regionshave the highest X-ray flux.

We allow for cooling flows using one of two approaches: eitherby excluding the data in cooling regions fromthe fit, or by modelling the contaminating emission as a separate component. Such a correction was applied to datain all of the sub-samples; further details of the different methods used are given below in the sections specific toeach of the sub-samples.

By allowing for this source of contamination, we have fully compensated for the presence of gas cooling, wherethis has been found to be a significant effect. The final cluster models therefore parametrize only the ‘corrected’gas density and temperature profiles.

2.3.3 Markevitch sample

The sub-sample of Markevitch (hereafter ‘M sample’) was compiled from several separate studies and comprisesspatial and spectral X-ray data for 27 clusters of galaxies (Markevitch et al., 1998; Markevitch, 1998; Markevitchet al., 1999; Markevitch & Vikhlinin, 1997; Markevitch, 1996). Of these datasets, 22 are included in our finalsample, the remaining systems being covered by one of the other sub-samples (the factors affecting this choice aredescribed in section 2.4).

To measure the spatial distribution of the gas, X-ray imagesof the clusters were fitted with a modified versionof equation 2.1; under the assumption of isothermality, equation 2.1 leads to an equivalent expression for theprojectedX-ray surface brightness,S, given by

S(r) = S(0)

[

1+

(

rp

rc

)2]−3β+ 1

2

, (2.6)

in terms of projected radius,rp as well as the density core radius,rc, and index,β. This is a modified King functionor isothermalβ-model (Cavaliere & Fusco-Fermiano, 1976). For all but one of the clusters, data from theROSATPSPC were used for the surface brightness fitting, as this instrument provides greatly superior spatial resolutioncompared to theASCAtelescope (for Abell 1650, no PSPC pointed data were available and anEinsteinIPC imagewas used instead).

Although strictly only appropriate for a uniform gas temperature distribution, this approach is valid since, forthe majority of the clusters in this sub-sample, the exponential cutoff in the emission lies significantly beyond theROSATbandpass (∼0.2–2.4 keV). Consequently, the X-ray emissivity in this energy range is rather insensitive tothe gas temperature, and therefore scales simply as the square of its density. These images were also used directlyas models of the surface brightness distribution, in order to determine the relative normalizations between projectedemission measures in the different regions for which spectra were fitted usingASCAdata.

Gas density data for this sub-sample were provided in the form of a King profile core radius andβ index, asderived from PSPC data, using equation 2.6. However, the density normalization was only available in the formof a central electron number density for a small number of clusters: Abell 1650 & Abell 399 (Jones & Forman,1999) and Abell 3558, Abell 3266, Abell 2319 & Abell 119 (Mohr, Mathieson, & Evrard, 1999). In the originalMarkevitch analyses, density normalization data for the remaining systems were taken from Vikhlinin, Forman, &Jones (1999a), in the form of values of the radius enclosing aknown overdensity with respect to the average baryondensity of the Universe at the observed cluster redshift. Itwas therefore necessary, for this work, to convert thesevalues into central electron densities, to provide the necessary normalization component in the cluster models.

Radii of overdensity of 2000,R′, were taken from Vikhlinin et al. (1999a) and were combined with the gas


density core radii,rc, andβ indices to determine the density normalization,ρ(0), given that

ρ(0)

Z R′

0

[

1+

(

rrc

)2]− 3

2β

4πr2dr =4πR′3

32000ρ(zobs), (2.7)

whereρ(zobs) is the mean density of the Universe at the observed redshift of the cluster. The integration wasperformed iteratively using a generalisation of Simpson’srule to a quartic fit, until successive approximationsdiffered by less than one part in 108.

The fitted gas density and temperature data for the M sample were corrected for the effects of central gascooling in the original analyses: the cluster models based on these data parametrize only the uncontaminatedcluster X-ray emission. This was achieved by excising a central region of the surface brightness data in the originalanalysis and, for the temperature data, by fitting an additional spectral component in the central regions (whererequired), to characterise the properties of the cooling gas flux. Full details of these methods can be found inVikhlinin et al. (1999a) and Markevitch et al. (1998).

Temperature data for all the clusters in this sub-sample were provided in the form of a polytropic index anda normalization evaluated at 2rc (as defined in equation 2.3). This radius was chosen as it lay within the fitteddata region (i.e. outside of any excised cooling flow emission) in all cases. These fits results are based on the2-dimensionaltemperature profile, but have been corrected for the effectsof projection. To construct cluster models,it was necessary to calculateT(0) from these normalization values, by re-arranging equation2.3 and substitutingr = 2rc to give

T(0) = T(2rc)

[

5+

[

32

β(γ−1)

]]

. (2.8)

These central normalization values were combined with the corresponding polytropic indices and density param-eters to comprise a 3-dimensional description of the gas temperature variation. Errors on all parameters weredetermined directly from the confidence regions evaluated in the original analyses.

2.3.4 Finoguenov sample

The sub-sample of Finoguenov (hereafter ‘F sample’) comprises X-ray data compiled from several sources, incor-porating a total of 36 poor clusters and groups of galaxies (Finoguenov & Ponman, 1999; Finoguenov & Jones,2000; Finoguenov, David, & Ponman, 2000; Finoguenov et al.,2001a) which were subject to similar analysis. Ofthe corresponding fitted results, 24 were used in the final sample, with the remainder taken from one of the othersub-samples (the factors affecting this choice are described in section 2.4). A combination ofROSATandASCASIS instrument data was used to determine the spatial and spectral properties of the X-ray emission respectively.

Values for the King profile core radius and index parameter were taken from surface density profile fits (usingequation 2.6) to PSPC images of the clusters, with the exception of HCG 51 and MKW 9, where no such data wereavailable and aROSATHRI andEinsteinIPC observation were used respectively. A central region ofthe surfacebrightness data was excluded for all systems, to avoid the bias torc andβ caused by emission associated withcentral gas cooling. The best-fit parameters were used to determine the 3-dimensional gas density and temperaturedistribution, via an analysis ofASCASIS annular spectra, by fitting volume and luminosity-weighted values in aseries of spherical shells, allowing for the effects of projection. In this stage of the analysis, the central coolingregion was included and an additional spectral component was fitted to the innermost bins, allowing this extra emis-sion to be modelled. A regularisation technique was used to stabilise the fit by smoothing out large discontinuitiesbetween adjacent bins. Further details of this method can befound in Finoguenov & Ponman (1999).

To generate cluster models for these objects, it was necessary to infer a central gas density normalization, aswell as an analytical form for the temperature profile. Density normalization was determined by a core indexfunction (equation 2.1) fit to the data points, using theβ index and core radius values from the PSPC surfacebrightness fits. This was achieved by numerically integrating equation 2.1 (as described in section 2.3.3) betweenthe radial bounds of the spherical shells used to determine the fit points, weighted byr2 to allow for the volume ofeach integration element. The core radii andβ index were fixed at their previously determined values andρ(0) wasleft free to vary. A best fitting normalization was then foundby adjustingρ(0) so as to minimize theχ2 statistic.Confidence regions forρ(0) were determined from those values which gave an increase inχ2 of one. Fitting wasperformed using the MIGRAD method in theMINUIT minimization library from CERN (James, 1998) and errorswere found with MINOS, from the same package. For the core radius andβ index parameters, a fixed error of


four per cent was assumed, based on an estimate of the uncertainties in the surface brightness fitting (Finoguenov,Reiprich, & Bohringer, 2001b).

Since the original density points were measured in units of proton number density, it was necessary to convertthem to electron number density for consistency between thecluster models. It was also necessary to allow for aknown effective area offset between theASCASIS andROSATPSPC instruments. This adjustment amounts to afactor of 1.2 multiplication to convert from proton number densities inferred using the former, to equivalent valuesmeasured with the latter.

An analytical form for the gas temperature profile was obtained from a mass-weighted (i.e. density multipliedby the integration element volume, using equation 2.1) fit tothe 3-dimensional data points, excluding the coolingcomponent. For 5 of the coolest groups (IC 4296, NGC 3258, NGC4325, NGC 5129 & NGC 6329), the coldcomponent was not unambiguously separated from the bulk halo contribution and so those central bins that wereaffected were excluded from the analytical fit. The best fitting temperature values were subsequently found forboth a linear and polytropic description, again based on theχ2 criterion.

The parametrization which gave the optimum (i.e. lowest)χ2 fit to the data points was used, except where thisgave rise to unphysical behaviour in the model; for three systems (Abell 1060, HCG 94 & MKW 4) the linearT(r) model led to a negative temperature withinR200, when extrapolated beyond the data region. In these cases apolytropic description was used in preference.

2.3.5 Lloyd-Davies & Sanderson samples

The sub-sample of Lloyd-Davies (hereafter ‘L sample’) comprises 19 of the 20 clusters and groups of galaxiesanalysed in the study of Lloyd-Davies et al. (2000) (Abell 400 was omitted as it is thought to be a line-of-sightsuperposition of two clusters). Of the corresponding fittedresults, 14 were used in the final sample, with theremainder taken from either the M or F samples (see section 2.4). ROSATPSPC data were analysed for all theobjects, with data from the wider passbandASCAGIS instrument included to permit the analysis of certain hotterclusters.

To extend the sample to include individual galaxies and alsoto improve the coverage at low temperatures, anadditional six objects were analysed – four groups and two early-type galaxies (this sub-sample is hereafter referredto as the ‘S sample’). The galaxy groups were drawn from the sample of Helsdon & Ponman (2000b) and werechosen as being fairly relaxed and having high-qualityROSATPSPC data available. Cooler systems, in particular,were favoured, in order to increase the number of low mass objects in the sample. The extra objects include twoearly type galaxies; an elliptical, NGC 6482 and an S0, NGC 1553. Great emphasis was placed on identifyinggalaxies free of contamination from X-ray emission associated with a group or cluster potential, in which they mayreside, since this is known to complicate analysis of their haloes (e.g. Mulchaey & Zabludoff, 1998; Helsdon &Ponman, 2000b). The most well-studied galaxies are generally the first-ranked members in groups or clusters, andit is known that such objects are atypical, as a consequence of the dense gaseous environment surrounding them:the work of Helsdon et al. (2001) has shown that brightest-group galaxies exhibit properties which correlate withthose of the group as a whole, possibly because many of them lie at the focus of a group cooling flow.

Genuinely isolated early-type galaxies are rare objects, given the propensity for mass clustering in the Uni-verse. In addition, finding a nearby example of such a system,which possesses an extended X-ray halo that hasbeen studied in sufficient detail to measureTX(r), severely limits the number of potential candidates. AlthoughNGC 1553 lies close to an elliptical galaxy of similar size (NGC 1549), there is no evidence from the PSPC data ofany extended emission not associated with either of these objects, which might otherwise point to the presence of asignificant group X-ray halo (see page 26). NGC 6482, by contrast, is a large elliptical (LB ∼ 6×1010LB⊙) whichclearly dominates the local luminosity function and which is embedded in an extensive X-ray halo (∼100 kpc). Itsproperties indicate that this is probably a ‘fossil’ group (see page 27) and as such, its properties are expected todiffer from those of an individual galaxy halo.

The data reduction and analysis for the S sample was performed in a similar way to the study of Lloyd-Davieset al. (2000), and a detailed description can be found there.The method used involves the use of a spectral ‘cube’of data – a series of identical images extracted in contiguous energy bands – which constitutes a projected view ofthe cluster emission. A three dimensional model of the type described in section 2.3.1 can be fitted directly to thesedata in a forward fitting approach (Eyles et al., 1991), in order to ‘deproject’ the emission. The gas density andtemperature are evaluated in a series of discrete, spherical shells and the X-ray emission in each shell is calculatedwith a MEKAL hot plasma code (Mewe, Lemen, & van den Oord, 1986). The emission is then redshifted andconvolved with the detector spectral response, before being projected into a cube and blurred with the instrument


point spread function (PSF). The result can be compared directly with the observed data and the goodness-of-fit is quantified with a maximum likelihood fit statistic (Cash, 1979). The model parameters are then iterativelymodified, so as to obtain a best fit to the data.

The contributions to the plasma emissivity from highly ionized species, in the form of discrete line emission,is handled by parametrizing the metallicity of the gas with alinear ramp (assuming fixed, Solar-like elementabundance ratios), normalized to the Solar value. However,the poorer spectral resolution of the PSPC requiresthat the metallicity be constrained to be uniform where onlyROSATdata were fitted (as for all six extra systems inthe S sample). For those clusters whereASCAGIS data were additionally analysed in the L sample (denotedby a‘+’ in the right-most column of table 2.1), the gradient of the metallicity ramp was left free to vary.

The use of maximum likelihood fitting avoids the need to bin upthe data to achieve a reasonable approxima-tion to Gaussian statistics: a process which would severelydegrade spatial resolution in the outer regions of theemission, where the data are most sparse. The only constraint on spatial bin size relates to blurring the clustermodel with the PSF; a process which is computationally expensive and a strongly varying function of the totalnumber of pixels in the data cube. Although the Cash statistic provides no absolute measure of goodness-of-fit,differences between values obtained from the same data set are χ2-distributed. This enables confidence regions tobe evaluated, for determining parameter errors (cf. Lloyd-Davies et al., 2000).

For the S sample, two different minimization algorithms were employed to optimise the fit to the data. A mod-ified Levenberg-Marquardt method (Bevington, 1969) was generally used to locate the minimum in the parameterspace. Although very efficient, this method is only effective in the vicinity of a minimum and is not guaranteed tolocate the global minimum. In several cases this approach was unable to optimise the cluster model parameters reli-ably and a simulated annealing minimization algorithm was used (Goffe, Ferrier, & Rogers, 1994). This techniqueenables the fit space to be searched very thoroughly in a systematic way that permits uphill moves to be made,depending on the ‘temperature’ of the algorithm, which gradually reduces as the fitting progresses. The process ofoptimisation is analogous to the minimization of the lattice energy in a quantity of material as itslowlycools andcrystallises. The energy of the material allows it move through a higher energy configuration (an ‘uphill’ move ina minimization problem), by an amount which depends on its temperature. As this temperature cools, fewer uphillmoves are accepted and the algorithm converges on the minimum. In this way, the global minimum is likely to beidentified. The algorithm is effective even with highly non-parabolic fit surfaces, which would cause problems forgradient-based algorithms such as the Levenberg-Marquardt method. However, the disadvantage of this techniqueis the computational cost associated with the very large number of fit statistic evaluations required: once the globalminimum was identified, the Levenberg-Marquardt method wasused to determine parameter errors, in an identicalfashion to Lloyd-Davies et al. (2000).

In order to determine errors on derived quantities, such as gravitating mass and gas fraction, we adopt therather conservative approach of evaluating the quantity using the extreme values permitted within the confidenceranges specified by the original fitted parameters. However,although this method tends to slightly overestimate theerrors, as can be seen from the intrinsic scatter in our derived masses in section 2.6.2, it is not liable to introduce asystematic bias into any weighted fitting of these data.

For those systems in the L and S samples where a cooling flow component was fitted, a power law parametriza-tion was used to describe the gas temperature and density variations within the cooling radius (also a fitted param-eter). To avoid unphysical behaviour atR= 0, these power laws were truncated at 10 kpc, well within the spatialresolution of the instrument (for NGC 1395 a cut-off of 0.5 kpc was used to reflect the much smaller size of itsX-ray halo).

NGC 1553

The X-ray spectra of elliptical galaxies comprise an emission component originating from a population of discretesources within the body of the galaxy, as well as a possible component associated with a diffuse halo of gas trappedin the potential well. The contributions of these differentspectral components vary according to the ratio of theX-ray to optical luminosity of the galaxy (LX /LB) (Kim, Fabbiano, & Trinchieri, 1992). Since we are interestedonly in the X-ray halo of the systems in this work, we favour those galaxies with a highLX /LB, where the emissioncan be traced beyond the optical extent of the stellar population.

A 14.5 ks PSPC observation was analysed, in which the S0 galaxy NGC 1553 appears quite far off axis,although within the ‘ring’ support structure. Some 2000 counts were accumulated in the exposure and the emissionis detectable out to a radius of 4.8 arcmin (21 kpc). Althoughits LX /LB, of 1.53×10−3, does not mark it out asa particularly bright galaxy, its X-ray halo is clearly visible and uncontaminated by group or cluster emission. In

2.4. CONSISTENCYBETWEEN SUB-SAMPLES 27

fact, this ratio is typical of non-group-dominant galaxies(cf. Helsdon et al., 2001). However, for this reason weexpect an appreciable contribution to the X-ray flux from discrete sources; Blanton et al. (2001) have recentlyfound that diffuse emission only accounted for∼84 per cent of the total X-ray luminosity in the range 0.3–1 keV,based on a 34 ks observation with the ACIS-S detector on boardtheChandratelescope.

The PSPC data show evidence of central excess emission, which is adequately described by a power lawspectrum, blurred by the instrument PSF, with a photon indexconsistent with unity. This was modelled as aseparate component, so as to decouple its emission from thatof the halo. Blanton et al. (2001) find evidence of acentral, point-like source which they fit with an intrinsically absorbed disc blackbody model. The spatial propertiesof the X-ray halo are not addressed in their analysis, but in any case the emission is only partly visible, due to thesmall detector area of the ACIS-S3 CCD chip.

NGC 6482

The elliptical galaxy NGC 6482 is a relatively isolated object, whose brightest companion galaxy within 1h−150 Mpc

is at least two magnitudes fainter. However, its X-ray luminosity is in excess of 1042h−250 erg s−1, which is very

large for a single galaxy. These properties classify this object as a ‘fossil’ group – the product of the merger of anumber of smaller galaxies, bound in a common potential well(Ponman et al., 1994; Mulchaey & Zabludoff, 1999;Vikhlinin et al., 1999b; Jones et al., 2000). Correspondingly, this system is more closely related to a group-sizedhalo – albeit a very old one (cf. Jones et al., 2000) – than to that of an individual galaxy. The X-ray over-luminousnature of this galaxy (LX /LB = 0.048) implies that the vast majority of the emission originates from its large (>∼100 kpc) halo, with a negligible contribution from discretesources.

Approximately 1500 counts were accumulated in an 8.5 ks pointing with the PSPC. During the fitting processit was found that there was a significant residual feature in the centre of the halo, which may indicate the presenceof an AGN. It was not possible to adequately model this feature with either a point-like or extended componentand it was necessary to excise a central region (radius 1.2 arcmin) of the data to obtain a reasonable fit. As a result,the core radius was rather poorly constrained and hence was frozen at its best-fit value of 0.2 arcmin for the errorcalculation stage. In addition, the hydrogen column could not be constrained and had to be frozen at the galacticvalue (7.89×1020 cm−2), as determined from the radio data of Stark et al. (1992).

2.4 Consistency Between Sub-samples

As a consequence of converting the data from the different sub-samples into a uniform,analyticalformat, we areable to adopt a coherent approach in our analysis. By extrapolating the gas density and temperature profiles, it ispossible to determine the virial radius and mean temperature (see below) self-consistently, and thus independentlyof the arbitrary data limits. Of course, this process of extrapolation can potentially introduce other biases, and thisis discussed in section 2.6.5 below. In some systems, emissivity profiles are affected by significant central coolingand we emphasize that in our analysis we have eliminated thiscontaminating component in all of our targets,in order to maintain consistency between the different sub-samples. This was achieved by using the followingmethods:

(1) Excision of a central portion of surface brightness data, to prevent bias ofrc & β (F & M samples).

(2) Fitting of an additional model to characterise the spectral (F & M samples) or combined spatial/spectral (L& S samples) properties of the central cool component flux.

In this section we present the results of an investigation into the consistency of our sample and the agreementbetween the different analyses involved.

Mean temperatures were calculated for each system, by averaging their gas temperature profiles within 0.3R200,weighted by emissivity and excluding any cooling flow component (hereafter referred to asTew; see column 5 intable 2.1). Figure 2.1 shows the temperatures determined inthis way, from the F & M samples, compared tothe corresponding values taken from the original analyses.The F sample (left panel) shows good agreement,although some discrepancy is expected, due to differences in the prescription for obtainingTew. However, twoclusters are clearly anomalous – Abell 2670 and Abell 2597. The case of A2670 is a known discrepancy, arisingfrom an unusually high background in the SIS observation. A2597 is an example of the complications of a largecooling flow, which is more readily resolved in the SIS observation than the GIS data. The values ofTew quoted


Figure 2.1: Comparison of the emission-weighted temperatures from this work, with those from the original Finoguenov (left panel) andMarkevitch (right panel) analyses.

in Finoguenov et al. (2000) for these clusters are actually based on PSPC and GIS data respectively (Hobbs &Willmore 1997 and Markevitch et al. 1998) and not on the SIS data analysed in that paper. However, to maintainconsistency we have used just the SIS data to construct our model for these clusters.

The agreement betweenTew values for the M sample (right panel) is less good, but here differences are to beexpected: the method used in this work weights the temperature profile, between 0.3R200 and zero radius, by theemissivity of the gas as determined by extrapolatingρ(r) andT(r) inwards from beyond the cooling flow region.In contrast, Markevitch et al. (1998) determine a flux weighting for their mean temperatures based on their estimateof the emission measure from thenon-cooling gas within the core region. For strong cooling flows, this gives a lowweighting to the central values ofT(r) compared to those values just outside the cooling zone. Since almost all thesystems in this sample have polytropic indices in excess of one, their gas temperaturesincreasetowards the centre,so the differences in the spatial weighting give rise to a systematic difference between values ofTew determinedwith the two methods. The overall effect of our analysis is actually to correct for the consequences of gas cooling,rather than simply to exclude the contribution from the coldcomponent to the X-ray flux. This amounts to a simplenormalization offset – the mean of the values ofTew from the M sample is 18 per cent lower than that of the valuesdetermined in this work.

To assess the consistency between the different initial analyses in our sample, we studied the models derivedfor four clusters which were common to the M, F and L samples (Abell 2199, Abell 496, Abell 780 & AWM 7),providing a direct comparison of methods. Figure 2.2 shows the temperature and density profiles for each ofthese systems – in each plot the different lines correspond to a different analysis result. It can be seen that thedensity profiles show excellent agreement in all but the verycentral regions. At the redshift of the most distantcluster (z= 0.057, for A780), 1 arcmin corresponds to roughly 60 kpc and hence these differences are confinedto the innermost parts of the data. Since these are all cooling flow clusters, any discrepancies in the core can beattributed to differences in the way the cooling emission ishandled. In any case, the effects of these discrepancieson the global cluster properties are small. The temperatureprofiles show considerably more divergence, and for theclusters A780 and AWM 7, the L sample temperature rises with radius, in contrast to the M and F sample models.In the case of A780, data from a recentChandraanalysis (David et al., 2001; McNamara et al., 2000) indicatethatT(r) does indeed show evidence of a rise with radius within the inner∼200 kpc in the ACIS-S detector data,although the ACIS-I temperature profile exhibits a drop in the outer bin, in the range 200–300 kpc.

The discrepancy between the temperature profiles of A780 andAWM 7 is exacerbated by the rise with radiusseen in the L sample models, which has the compounding effects of increasing the size ofR200, as well as steepen-ing the gravitating mass profile. However, these clusters have two of the most extreme rises inT(r) of any systemin our sample, and only 5 other systems show any significant increase in temperature with radius. While it is clear

2.4. CONSISTENCYBETWEEN SUB-SAMPLES 29

Figure 2.2: A comparison of the gas density and temperature profiles in four clusters common to the M (dotted lines), F (solid lines) and L(dashed lines) samples. The vertical lines mark the position of R200 for each of the different models.


that some clusters show evidence of a radially increasing temperature profile in their central regions, it is unlikelythat this will continue out to the virial radius. This presents a fundamental problem for a monotonic analyticalprofile, which must inevitably find a compromise: in general the fit is driven by the central regions, which havea greater flux weighting. In the case of A780, the difference in T(r) leads to a factor of 3 difference in the totalmass withinR200, between the models, although this discrepancy is reduced to 60 per cent for the mass within0.3R200. The corresponding effect on the gas fraction is also less severe, since the total gas mass increases withR200. However, for A496 – whose temperature profile is more typical of the systems in our sample – the agreementbetween the gravitating mass withinR200 for the different models is much better, varying by only 40 per cent.

2.5 Final Model Selection

In order to arrive at a single model for each system, we determined an order of preference for the sub-samples,to choose between analyses, where overlaps occurred. An initial selection was made on the basis of unphysicalbehaviour in the models; the linear temperature parametrization is prone to extrapolate to negative values withinR200, and so a number of models were rejected on these grounds. Of the remaining overlaps, we preferentiallyselect those cluster models from the L sample, as this represents the direct application of the model to the rawX-ray data and hence should be the most reliable method. Application of this criterion leaves just four remainingsystems, where an overlap occurs between the F and M samples.These were resolved on an individual basis; ineach case the analysis of the data which covered the largest angular area was chosen. Since the ability to tracehalo emission out to large radii is critical in this study, this amounts to selecting the more reliable analysis. Theparameters for each of the final models are listed in table 2.1.

2.6 Results

2.6.1 Gas distribution

Figure 2.3 shows the variation in the slope of the gas densityprofile with emission-weighted temperature for thesample. It can be seen that, for the hottest systems (> 3–4 keV),β is consistent with the canonical value of 2/3 (e.g.Jones & Forman, 1984). However, below this temperature the gas profiles become increasing flattened compared toself-similar expectation, in agreement with the work of Helsdon & Ponman (2000b). There is a strong correlationbetweenβ and temperature as measured by Kendall’s K statistic, whichgives a significance of 5.8σ.

Intriguingly, the galaxy (NGC 1553) and fossil group (NGC 6482) – the diamonds in figure 2.3 – seem todeviate from this general trend. Although there are only twopoints, these are the coolest objects in the sample andNGC 1553 in particular appears to have a value ofβ more consistent with clusters than with groups of a similartemperature. We will revisit this issue in the broader context of galaxy scaling properties in section 2.7.1.

We fitted a straight line, in log space, to the points (both including and excluding the two galaxies) using theODRPACK software package (Boggs et al., 1989, 1992), to take accountof parameter errors in both the X and Ydirections. The dotted line in figure 2.3 shows the best fitting relationβ = (0.439±0.06)T0.20±0.03, excluding thegalaxies. The index is marginally consistent with the logarithmic slope of 0.26±0.03 found by Horner, Mushotzky,& Scharf (1999) for their literature-based sample, spanning the range∼1–10 keV. The flatter slope of our datareflects the greater number of hotter clusters in our sample,where the relation tends to flatten to approximatelyβ = 2/3, indicated by the dashed line in figure 2.4.

The fit also matches the data points from the simulations of Metzler & Evrard (1997), which include the effectsof galaxy winds on the IGM, albeit with their points having a∼25 per cent higher normalization. A fit to theentire sample yields a flatter relation, given byβ = (0.482± 0.06)T0.15±0.03. Although the points seem to bereasonably well described by a simple power law, there is a considerable amount of intrinsic scatter in the data –80 per cent more than would be expected from the statistical errors alone. This may reflect a difference in the levelof energy injection between haloes, but can also be attributed to the effects of hierarchical assembly: Cavaliere,Menci, & Tozzi (1999) predict aβ-T relation with a 2σ scatter envelope, resulting from merging histories, whichis in qualitative agreement with the data in figure 2.3.

The variation inβ is also reflected in the gas fraction (fgas), evaluated within a characteristic radius of 0.3R200

(figure 2.4). There is a clear trend (significant at the 6σ level, excluding the two galaxies) for cooler systems to havea smaller mass fraction of X-ray emitting gas. However, the galaxy NGC 1553 lies well below the general clusterrelation, consistent with the coolest groups, apparently at odds with itsβ of approximately 2/3. This behaviour is

2.6. RESULTS 31

Figure 2.3: The gas density logarithmic slope parameter (β) as a function of system emission-weighted temperature. The diamonds representthe two galaxies in the sample. The dashed line indicates thecanonical value ofβ = 2/3 and the dotted line is the best fit to the points, excludingthe galaxies.

Figure 2.4: Gas fraction within 0.3R200 as a function of system temperature. The diamonds representthe two galaxies in the sample.


Figure 2.5: Gas fraction within R200 as a function of system temperature. The diamonds representthe two galaxies in the sample and the dottedline shows the unweighted mean of the whole sample.

also evident infgaswithin R200, shown in figure 2.5. By contrast, the fossil group NGC 6482 exhibits gas fractionproperties which are consistent with itsβ – i.e. slightly above groups of a similar temperature in bothcases. Forthe whole sample, the behaviour of the gas fraction withinR200 is only slightly different from that within 0.3R200;there still remains a strong (5.4σ) trend, although there is some evidence of a levelling off above∼5 keV, abovewhich the significance of a correlation drops to 3.2σ.

This can also be seen in the mean gas fraction withinR200 for those systems hotter than 4 keV, which gives

(0.163±0.01)h− 3

270 , as compared tofgas= (0.134±0.01)h

− 32

70 for the whole sample. Since the errors in the evalua-tion of this quantity are dominated by systematic uncertainties, we use anunweightedmeanfgas, which is sensitiveonly to the intrinsic scatter in the data. This behaviour suggests that virialized objects may not be ‘closed systems’,in that some of their gas might have escaped beyondR200, particularly for the coolest groups. However, it must beremembered that any effects of systematic extrapolation errors could contribute to the observed trend.

To understand the behaviour of the gas fraction across the sample, figure 2.6 shows howfgasvaries with radius,grouped into five temperature bins for clarity. Beyond∼0.2R200, the profiles lie in order of temperature such that,at a fixed radius, gas fraction decreases as temperature decreases, mirroring the trend seen in figure 2.4. This isessentially a simple normalization offset and demonstrates that the effects of energy injection are more pronouncedin less massive (i.e. cooler) systems, particularly below∼3–4 keV, as seen in figure 2.5.

The general trend is for gas fraction to rise monotonically (beyond∼0.03R200) with radius from∼0.02 in thecore to around 18 per cent atR200, for the richest clusters (kT > 8 keV). This behaviour demonstrates that thedistribution of the IGM is not similar to that of the dark matter, even in the largest haloes, but is significantly moreextended, as previously reported (e.g. David et al., 1995).This may seem surprising, given that the dark matteris dissipationless, unlike the gas; that the gas is the more extended component on all mass scales may providesome indication of the influence of feedback processes that can inject energy directly into the IGM. Such a trend isconsistent with the simulations of Metzler & Evrard (1997) who find that modelling the effects of feedback fromgalaxy winds leads to a less centrally concentrated IGM structure. However, it should be noted that simulationswhich model the effects of cluster mergers can generate a similarly extended gas distribution, without energyinjection from non-gravitational processes (Navarro & White, 1993; Pearce et al., 1994).

2.6. RESULTS 33

Figure 2.6: Spatial variation of gas fraction within a givenradius (normalized to R200), grouped by system temperature. The solid line representsthe coolest systems (including the two galaxies) (0.3–1.3 keV), increasing in temperature through dashed (1.3–2.9 keV), dotted (2.9–4.6 keV),dot-dashed (4.6–8 keV) and finally dot-dot-dot-dashed (8–17 keV).

2.6.2 TheM−TX relation

Since the emission-weighted temperature reflects the depthof the underlying potential well which retains the X-raygas, a tight relation between system mass and temperature isexpected. It can be shown that, for the case of simpleself-similar scaling,M ∝ T3/2 (see Mohr & Evrard, 1997, for example). Observations generally reveal a steeperrelation, however, consistent with a breaking of self-similarity, as found in other scaling relations (e.g.L−TX).

ASCAtemperature profiles, on which we rely in this work, have a relatively large systematic uncertainty be-cause of the wide mirror PSF. A comparison of theASCAprofiles from one of the subsamples used here (that ofMarkevitch et al., 1998) with recentChandraandBeppoSAXresults appears to confirm the temperature decline atlarge radii (e.g. David et al., 2001; Markevitch & Vikhlinin, 2001; Nevalainen et al., 2001; De Grandi & Molendi,2002). At the same time,ASCAtemperatures in the regions immediately adjacent to the central bins in the cool-ing flow clusters appear to be systematically too high, although within their uncertainties (e.g. David et al., 2001;Arnaud et al., 2001; De Grandi & Molendi, 2002). Direct comparison is limited to a few clusters at present.

It is important to correct for the effects of any central cooling flow when calculating the characteristic temper-ature of a cluster, but is is not obvious how best to achieve this. In our analysis below, we employ three differentmethods extrapolating over, or excluding the central region, and also weighting with the gas density rather thanemissivity. The justification for using these three different prescriptions forT is as follows:

(i) emission-weighted, extrapolating over CF: this attempts to fully correct for the presence of a CF and providean estimate ofT in the absence of cooling.

(ii) emission-weighted, excising cooling region: this method of calculatingT more closely matches the CF-corrected, spectroscopic measurements which have been used frequently in previous work.

(iii) mass-weighted, extrapolating over CF: this method gives values ofT which are more naturally obtainedfrom numerical simulations, and is less sensitive than emission weighting to the properties in the densecentral core.

We have applied these methods to deriveT within two different radii:

(a) 0.3R200: the majority of our systems have X-ray emission detectableto at least this radius, which is typicalof group detection radii.


(b) R200: this represents our nominal virial radius, and more closely matches the detection radii of rich clusters.

We thus have six different methods of calculatingT, including our default method of emission-weightingT(r)within 0.3R200 (i.e.Tew, described above and listed in table 2.1).

We have combined these temperature data with our gravitating mass measurements (within both 0.3R200 andR200, as appropriate) to give a total of sixM−TX relations. Strictly speaking, the masses we derive should bescaled by a factor of(1+zf)

−3/2, to allow for the change in mean density of the Universe with redshift. We havechosen to omit this adjustment, sincezf is unknown, and the assumptionzf = zobs is prone to systematically biasthe results, as mentioned previously. We note, however, that incorporating this correction actually makes very littledifference to the best-fit parameters (Finoguenov et al., 2001b).

The set ofM−TX relations is plotted in figure 2.7, together with the best-fitpower law in each case. Thefitting was performed in log space, using theODRPACK software package, using symmetrical errors in both axesderived from the half-widths of the asymmetric errors on theoriginal values. The upper section of table 2.2 lists theparameters of the fit lines, together with the correspondingscatter about the relation, normalized to that expectedfrom the statistical errors alone. A series of 1000 random realisations of the data was generated by scattering eachpoint away from the best-fit line, using the 1σ errors in both X and Y directions. The intrinsic scatter was measuredfor the real data and for each simulated dataset, by summing in quadrature the orthogonal distance of each pointfrom the best-fit line. The real scatter was then normalized to the mean scatter from all the realisations, to give thenumbers quoted in column 5 of table 2.2. In each case, the level of scatter is fully consistent with the errors, thusjustifying the use of a weighted, orthogonal distance regression to determine the best fit.

For the emission-weighted temperature and mass within 0.3R200 (method A in table 2.2), we find a best-fitrelation of log(M/M⊙) = (12.80± 0.03)+ (1.92± 0.06)× logT for the whole sample. Exclusion of the twogalaxies has a negligible effect on this result. Excision ofthe cooling region (method B) leaves the normalizationof the M−TX relation unchanged and increases the index only marginally, to 1.94± 0.06. The use of mass-weighting to evaluateT (method C) yields a best-fit which is consistent with those ofmethods A and B. It istherefore clear that, within 0.3R200, theM−TX relation shows a significantly steeper logarithmic slope than theself-similar prediction of 3/2. The agreement between the different methods for obtainingT demonstrates therobustness of this result. The behaviour of theM−TX relation withinR200 is similar but with a somewhat lesssteep slope: all three measurements ofT (methods D,E & F) are consistent in producing a best-fit powerlaw indexof ∼1.84. The two emission-weighted methods (D & E) have identical normalizations, but the effect of using amass-weighting is to increase this value by∼60 per cent. Although yielding a flatter slope compared to theM−TX

relation within 0.3R200, this is still significantly steeper than the self-similar prediction. SinceT(r) generally dropswith radius, and more of the emission arises at large radius in cooler systems, we expectT for the latter to dropmore as we move from 0.3R200 to R200, hence flatteningM−TX.

The study by Sato et al. (2000) of 83 clusters, groups and galaxies observed withASCAfound a logarithmicslope of 2.04±0.42, using total mass withinR200 together with temperatures determined by spectral fitting.Thisvalue is more consistent with the slope for our data within 0.3R200 thanR200, which may indicate that averagingT within 0.3R200 provides a closer match with spectroscopically measured temperatures, since real X-ray dataare rarely detectable out toR200. Nevalainen, Markevitch, & Forman (2000) measure a logarithmic slope of1.79± 0.14, calculating total mass within an overdensity of 1000 (M1000), also using spectroscopically-derivedtemperatures. Their normalization, of logM = 13.15, is intermediate between our values for 0.3R200 andR200, asexpected for an overdensity of 1000. The slope of 1.78±0.09 found by Finoguenov et al. (2001b) for a sample of39 clusters (usingM500) is consistent with our relation withinR200, and their normalization of 13.28±0.05, liesslightly below our own values within this radius.

While many X-ray studies appear to suggest that the slope of theM−TX relation is steeper than the self-similarprediction of 1.5, it has been suggested that this may an artefact of the analysis. Horner et al. (1999) measure aslope of 1.78±0.05 for a sample of 38 clusters, using theβ model to estimate masses. In contrast, they find a slopeof 1.48± 0.12 for a smaller sample of 11 clusters, for which they have spatially resolved temperature profiles.They attribute the discrepancy to the simplistic assumption of isothermality and confirm the apparently self-similarslope of theM−TX relation with another sample of 27 clusters with virial massestimates. However, the virialmass estimator is known to be susceptible to bias from interloper galaxies and the presence of substructure. Inaddition, the X-ray data for their 11 cluster sample are taken from the literature, and are therefore expected to becorrespondingly heterogeneous.

The differences between the temperatures obtained with thedifferent methods can be gauged by studying theright-most two columns of table 2.2. These show the mean and standard deviation of the ratios obtained by dividing

2.6. RESULTS 35

kT weighting Fit radius Index Normalization Scattera Meanb σc

(R200) (log(M⊙))

Non-isothermalA....Emission 0.3 1.92±0.06 12.80±0.03 1.02 – –B....Emission (CF excised) 0.3 1.94±0.06 12.80±0.03 0.98 0.98 0.04C....Mass 0.3 1.97±0.06 12.85±0.03 0.78 0.93 0.14D....Emission 1.0 1.84±0.06 13.37±0.03 0.96 0.92 0.12E....Emission (CF excised) 1.0 1.86±0.06 13.37±0.03 0.90 0.91 0.15F....Mass 1.0 1.83±0.06 13.58±0.03 0.52 0.78 0.41

IsothermalA′...Emission 0.3 1.97±0.05 12.86±0.03 0.89 – –B′...Emission (CF excised) 0.3 1.98±0.05 12.86±0.03 0.89 — —C′...Mass 0.3 2.02±0.06 12.85±0.03 0.83 — —D′...Emission 1.0 1.89±0.04 13.48±0.02 0.65 — —E′...Emission (CF excised) 1.0 1.90±0.04 13.48±0.02 0.64 — —F′...Mass 1.0 1.87±0.05 13.52±0.02 0.44 — —

Table 2.2: Summary of results for the power lawM−TX fitting using different mean temperature prescriptions andintegration radii (formeasuring both mass and temperature). Primed models in the lower half of the table areisothermal models, which have identical meantemperatures to models A-F, but different total masses. Notes:aMultiples of the statistical scatter expected from the errors alone.bdenotes themean ratio of kT divided by the corresponding values obtained with prescription A andc is the standard deviation of these ratios across thesample (these numbers have been omitted from the lower half since the ratios depend only on temperature, which is unchanged).

the values ofT found with each of methods B to F with the corresponding ones determined using method A. It canbe seen that the effect of excising the cooling region, as opposed to extrapolating over it, results in an average 2per cent decrease inT, consistent with the general trend forT(r) to increase towards the centre. An even largerdrop inT– of 8 per cent – is observed when comparing the mass-weightedvalues (method C) with the baselineset (A), although the spread of ratios is increased (σ = 0.14). By averaging over the whole ofR200 (D to F), themean temperature decreases compared to method A, by 8 and 9 per cent for methods D and E (emission-weighted),respectively. The similarity between these mean ratios reflects the proportionately smaller influence of the coolingregion-excision when integrating over the entire cluster volume. The mass-weightedT within R200 shows an evengreater drop, of 22 per cent (albeit withσ = 0.41), compared to method A. This is due to gas mass dropping offless sharply than luminosity, lending greater weight to theouter regions, where the gas temperature is generallylower.

The level of scatter in ourM−TX data is consistent with, or smaller than the scatter expected just from statisti-cal errors (depending upon the way in which the temperature is weighted) – i.e. values of∼0.5–1.0 in column 5 oftable 2.2 – suggesting that our error bounds are somewhat conservative, as previously described (see section 2.3.5).This conservative approach helps to allow for extra sourcesof error – for example, simulations have shown thatdeviations from hydrostatic equilibrium introduce a 15–30per cent rms uncertainty into hydrostatic mass estimates(Evrard, Metzler, & Navarro, 1996). In any case, it can be seen that a power law is not an ideal description of thedata in several of the plots in figure 2.7. This may reflect the dominance of systematic effects when extrapolatingout to large radii, or could indicate that the data follow a different functional form. Careful inspection of figure 2.7reveals some evidence for a convex shape in some cases, suggesting that the logarithmic slope steepensgraduallyfrom the cluster to the group regime. A convexM−TX relation was predicted by the simulations of Metzler &Evrard (1997), but only where the input energy provided by galaxy winds is assumed to be fully retained as thermalenergy in the IGM: their simulations do not show this behaviour in practice, as the extra energy is predominantlyexpended in doing work redistributing the gas within the potential.

More recently, Dos Santos & Dore (2002) have developed a purely analytical model, which predicts a convex

M−TX relation of the form:M = M0T3[

1+ TT0

]−3/2, with T0 = 2 keV. This leads to a curve with a self-similar

slope of 2/3 at the high mass, smoothly increasing to an asymptotic value of 3 forkT → 0. Their model includes theeffects of non-gravitational heating on the pre-virialized IGM, as well as shock heating, and is able to reproducethe observedL−TX relation with a similar, curved fit to the data points.

Comparisons with mass measurements using data from the latest X-ray missions are rather limited at present.


Figure 2.7: Total mass as a function of temperature for threedifferent temperature prescriptions (rows), and measuredwithin 0.3R200 (leftcolumn) andR200 (right column). In each case the solid line indicated the best fitting power law. See upper half of table 2.2 for further details.

2.6. RESULTS 37

However, a recentChandrastudy by Allen, Schmidt, & Fabian (2001b) has foundM−TX andL−TX relations inagreement with the predictions of self-similarity, albeitfrom a small sample of only six rich clusters. This study isbased on analysis of both X-ray data and gravitational lensing information and finds good agreement between massestimates derived from the different methods. Allen et al. (2001b) find aM−TX logarithmic slope of 1.51±0.27within a radius of overdensity of 2500, which is approximately equivalent to 0.3R200. However, their result is notdirectly comparable to ourM−TX relations A, B & C above, since the overdensity profiles in oursample are notself-similar, soR2500actually corresponds to a different fraction ofR200 for each system.

To permit a proper comparison, we have also derived masses within R2500 and have fitted these data in anidentical way to our otherM−TX relations. Figure 2.8 shows our results, together with the five clusters from Allenet al. (2001b) which they use in theirM−TX sample. It can be seen that their data points agree well with ourvalues at similar temperatures. Fitting the five Allen et al.(2001b) clusters using the same regression techniqueas we employed above, we find a best-fitting slope of 1.56±0.16 with an intercept of logM = 13.12±0.15. Ifthe sixth cluster from their sample (3C295) is included in the fit, the best-fit slope increases to 1.64± 0.15 andintercept decreases to 13.04±0.14. This cluster was omitted by Allen et al. as it was the only member of theirsample without a confirmed lensing mass estimate. Fitting the clusters from our own sample which are hotter than5.5 keV for comparison with the Allen et al. (2001b) analysis(their coolest cluster hasT = 5.56 keV), we find alogarithmic slope of 1.84±0.14, with a normalization of logM = 12.80±0.13. This is marginally consistent withthe Allen et al. result.

If the difference in slope between the two samples of hot clusters is real, it might be related to the dynamicalstate of the samples. The sample of Allen et al. includes onlythe most relaxed clusters, where the assumption ofhydrostatic equilibrium has been independently verified bylensing mass estimates. Our own sample is less wellcontrolled, although we have excluded objects which are clearly not in equilibrium. On the other hand, it is clearfrom Figure 2.8 that the shallower slope from the Allen et al.data is a poor match to the relation for cooler clusters,whilst the steeper slope of 1.84 fits rather well across the entire temperature range.

Previous studies have suggested that the high and low mass parts of the wholeM−TX relation may be char-acterised by power laws with different slopes. The cross-over temperature between the two regimes is typically∼3 keV (Finoguenov et al., 2001b). It is not obvious from our data that there is such a break in theM−TX rela-tion, as has been found for theL−TX relation (e.g. Fairley et al., 2000, and references therein). Finoguenov et al.(2001b) find a steepening of the logarithmic slope, from 1.48±0.11 above 3 keV to 1.87±0.14 below. However,this behaviour may simply be a manifestation of a smooth transition with temperature, masked by a dearth of coolsystems in their sample, where the steepening slope is most apparent. More high quality data of the type presentedby Allen et al. (2001b), but covering a wide temperature range, will be required to establish whether theM−TX

is really convex. What our results demonstrate clearly, is that either the relation steepens towards lower masssystems, or its slope is substantially steeper than 1.5.

2.6.3 The effects of non-isothermality

To investigate directly the effects of neglecting spatial variations in gas temperature, we have generated an addi-tional set ofisothermalmodels for our sample, i.e. withα = 0, for a linearT(r), or γ = 1, for a polytropic IGM.We have used the values ofT already determined for the six different methods describedabove – with associatederrors – to define the constant value. These isothermal models have then been subjected to an identical analysis tothe original set, in order to provide a fair comparison of results.

Figure 2.9 shows theM−TX relation for the isothermal sample derived using temperatures from method A(referred to as A′). It can be seen that the convex shape evident in panel A of figure 2.7 is largely absent, andthat a tighter relation about the best-fit line is observed. The parameters of this power law fit are given in thelower half of table 2.2, together with equivalent data for the other five isothermalM−TX samples. Within 0.3R200

the logarithmic slope increases marginally for the isothermal models, but within the errors, for each of the threemethods of measuringT. However, for the two emission-weighted methods, the normalization increases by∼15per cent, although it is unchanged for the mass-weightedT. Similar behaviour is observed for theM−TX dataevaluated withinR200: the logarithmic slope is slightly steepened for the isothermal case, and the normalizationis increased – for the emission-weighted methods – by∼30 per cent. However, the mass-weighted normalizationdecreasesby 15 per cent, compared to the non-isothermal models.

It is clear from this that the assumption of isothermality leads to anoverestimateof the total mass withinR200,when an emission-weighted method is used to calculateT. A similar conclusion was reached by Horner et al.(1999), for a sample of 12 clusters, who found that isothermality overestimated the mass by a factor of 1.7 – a


Figure 2.8: Total mass withinR2500 as a function of emission-weighted temperature. The solid line is the best fitting power-law to the pointsabove 5.5 keV (dotted vertical line), i.e. excluding the grey points. The diamonds are the data of Allen et al. (2001b) andthe dashed line is ourbest-fitting power-law to these data. See text for details.

Figure 2.9: Total mass as a function of emission-weighted temperature, evaluated within 0.3R200, for an isothermal IGM (method A′). Thesolid line represents the best-fit power law. See lower half of table 2.2 for further details.

2.6. RESULTS 39

Figure 2.10:Upper panel:Gas fraction withinR200 as a function of emission-weighted temperature (withinR200 – method ‘D’).Lower panel:Gas fraction withinR200 for an isothermal IGM . The dotted lines show the unweighted mean of the whole sample.

result confirmed by Neumann & Arnaud (1999). The latter authors found that the cumulative mass within a givenradius for an isothermal cluster is significantly steeper than that of a cluster with a polytropic index of 1.25 (avalue typical of the systems in our sample – see table 2.1), with the intersection of the two occurring at∼0.35R200.Consequently, the isothermal assumption over-predicts the mass for 96 per cent of the cluster volume.

Neglecting temperature gradients in the IGM appears to havelittle or no effect on the logarithmic slope of theM−TX relation and, once again, the observed slopes are in good agreement between the three different methodsof calculatingT. This is in contrast to the prediction of Horner et al. (1999), who suggested that the assumption ofisothermality leads to a steepening in theM−TX slope, which would otherwise be self-similar (i.e. 3/2). However,they base this conclusion on an analysis of a small sample (12systems), with data drawn from a number of differentsources in the literature. We also find that the rms scatter about the best fittingM−TX relations is significantlyreduced in our isothermal models, and fully consistent withthat expected from the statistical errors. We concludethat a power law seems to provide a good description of theM−TX relationfor an isothermal IGM.

The overestimation of the total mass for the isothermal caseleads to a corresponding underestimation in thetotal gas fraction withinR200, shown in figure 2.10. The unweighted mean gas fraction for the whole sample is

(0.110± 0.01)h− 3

270 , as compared to(0.134± 0.01)h

− 32

70 for the non-isothermal case. It can also be seen that thescatter about the mean is lower for the isothermal case, although the apparent drop at∼1 keV, seen in figure 2.10,is still noticeable. The most obvious outlier on this graph is the galaxy NGC 1553 (the left-most point). For thissystem, the isothermal model results in a significantly lower fgas, which greatly increases its distance from thesample mean.

2.6.4 Virial radius

The precise location of the outer boundary of a virialized halo is difficult to quantify and is very rarely directlyobservable. The virial radius is dependent on the mean density of the Universe when the halo was formed, as wellas the adopted cosmology (Lacey & Cole, 1993). Clearly it is important to be able to define this quantity reliably,since we assume that self-similar haloes will have identical properties when scaled byRv. The radius enclosing a


Figure 2.11: PredictedR200 from the NFW formula (equation 2.9) plotted against measured R200. The solid line indicates the line of equality.

mean overdensity of 200 (R200) is proportional toRv in any given cosmology – and lies withinRv for all reasonablecosmologies (Bryan & Norman, 1998) – and scales in a identical way (Navarro et al., 1995). However, previousstudies have not always been able to determineR200, and so have relied on other means to estimate this quantity.Atight relationship betweenTew andRv (and henceR200) is expected, as both these quantities reflect the depth of thegravitational potential well in a virialized halo; self-similarity predicts thatRv ∝

√Tew (cf. the size-temperature

relation, Mohr & Evrard 1997). This proportionality has been confirmed in ensembles of simulated clusters, whichprovide a value for the normalization in the relation. One such example is the work of Navarro et al. (1995), whodeduce that

R200 = 0.813

(

TkeV

)12

(1+z)−32 h−1

70 Mpc. (2.9)

However, their simulations only included adiabatic compression and shock heating, and did not allow for the effectsof energy injection.

The correspondence between our values ofR200 as determined from the overdensity profile (listed in table 2.1)and those calculated with equation 2.9 is shown in figure 2.11. It can be seen that there is significant deviationfrom the locus of equality between these quantities, markedby the solid line. The largest discrepancy is observedin the smallest haloes, indicating that the NFW equation significantly over-predictsR200 in these systems (theeffect of extrapolation bias is addressed in section 2.6.5). This is to be expected, given thatTew for these objects ismost likely to be susceptible to bias from non-gravitational heating. To explore the reasons for the disagreementbetween the two methods for calculatingR200, we have examined the role of temperature gradients as the source ofthe scatter, given their importance in calculating the gravitating mass (see equation 2.4). We have defined a simple,quantitative measure of the departure from isothermality,which, as has already been seen, can exert a significantinfluence on scaling properties (section 2.6.3). We use the ratio T(0)/T(0.3R200), as this is very sensitive to thepresence of a temperature gradient, and the two distances involved bracket the region of interest used to calculateTew.

The relationship between this quantity and the ratio of the measuredR200 divided by the NFW predicted value,is shown in figure 2.12. There is clearly a strong anti-correlation between these quantities, significant at the 6.9σlevel. Even with the two most extreme points removed (the left and right-most points on the graph), the significanceof the relation drops only slightly, to 6.4σ. It can be seen that the most isothermal systems (clustered around the

2.6. RESULTS 41

Figure 2.12: The ratio between our measuredR200 andR200 from the NFW formula, as a function of the ratio inT(r) betweenr = 0 andr = 0.3R200. Error bars have been omitted for clarity. The lines of equality on both axes are marked; the solid line represents the locus ofisothermality.

solid vertical line) scatter around the line of equality between the two measurements ofR200. This demonstratesthat the NFW formula is valid only for nearly isothermal haloes, and that it otherwise over-predictsR200 for themost common case of a radially decreasing temperature profile.

2.6.5 Extrapolation bias

Our analysis relies on the validity of extrapolating analytical profiles fitted to an inner region of the data, in orderto compensate for the emission which is undetected. However, since the extrapolation is, in general, greater forsmaller systems, there is a potential for introducing a systematic bias in our fitting. This is particularly true of theslope of the gas density profile, which is best constrained bythe emission from outer regions of the X-ray halo,and which has been found to vary significantly with temperature (see section 2.6.1). The work of Vikhlinin et al.(1999a) has shown that there is evidence of a slight steepening of the gas density logarithmic slope with radius,which could explain part of the observed correlation between β andTew. To explore the effects of extrapolationon our data, we have performed a series of fits to the surface brightness profiles of two clusters (Abell 1795 &Abell 2029), investigating the role of the outer radius of the fitted data in constraining the fit parameters.

These clusters were selected as they are rich systems (and hence relatively unaffected by energy injection), withhigh quality data (∼100,000 and 30,000 counts, respectively), that cover a fairly large angular extent. This allowsus to trace the emission to a large fraction ofRv and means we can analyse a spatial subset of the data, withoutapproaching the resolution limits of the instrument. In addition, we have chosen systems which have cooling flowemission confined to as small a region as possible, compared to the gas halo core size, so as to minimize the biasthis contamination can have on our results.

Since we are aiming in this section only to explore the behaviour of the gas density logarithmic slope as afunction of radius, we have adopted a different approach to that described in section 2.3.5. Rather than applyinga full deprojection analysis to the data, we have fitted 1-dimensional, azimuthally-averaged, surface brightnessprofiles for each cluster. This method permits a much more direct investigation of the gas density index in theouter regions of the halo and allows us use a quantitative measure of goodness-of-fit, based on theχ2 criterion.

We obtained azimuthally-averaged surface brightness profiles for both clusters, fromROSATPSPC data, in


Cluster kT Router Router/R200 Normalization β rc red.χ2/dof(keV) (arcmin) (ct s−1/degree2) (arcmin)

A2029 9.80 18 0.68 294+9−33 0.73+0.02

−0.01 2.51+0.20−0.20 1.25/23

12 0.45 356+74−53 0.70+0.02

−0.02 2.17+0.24−0.24 1.35/13

9 0.34 399+130−77 0.68+0.03

−0.03 1.99+0.31−0.32 1.72/8

A1795 8.54 17 0.60 173+21−17 0.83+0.02

−0.02 3.72+0.25−0.24 1.04/50

12 0.42 209+41−30 0.79+0.03

−0.03 3.26+0.33−0.33 1.18/30

† 9 0.32 401+364−130 0.70+0.04

−0.03 2.18+0.53−0.59 1.09/18

rc fixed⋆ 9 0.32 169+5−5 0.83+0.01

−0.01 3.72 1.49/19

Table 2.3: Summary of results for the 1-dimensional surfacebrightness fitting within different radii. Errors are 1σ. A central region of thedata was excluded to avoid contamination of by cooling flow emission; the radii of exclusion were 2.7′ & 4′ for A2029 & A1795, respectively.†This fit was noticeably biased by the excised central regionof the data and was repeated:⋆ indicates that the core radius was frozen at itspreviously-determined best-fit value to the whole profile (see text for details).

the following way. An image of the cluster was extracted in the 0.2–2.4 keV band, and point sources above 4.5σsignificance were masked out. Using the master veto rate, thecontribution to the background from particles wassubtracted, and the image was then ‘flattened’ by dividing bythe corresponding exposure map, to correct for theeffects of vignetting. An estimate of the astrophysical background was obtained, based on an annulus extractedfrom beyond the cluster emission, with the PSPC support spokes removed. Point sources were also masked outfrom the annulus, and the remaining counts were extrapolated across the field and subtracted from the sourceimage. Finally, a radial profile was extracted – centred on the peak in the X-ray emission – in a series of fixed-width annuli, with a minimum of 50 counts per bin (see figure 2.13).

A King profile function (equation 2.6) was fitted to the data, using theQDP package (Tennant, 1999), and allthree parameters were left free to vary. A small central region of the data was excised, to prevent emission fromthe cooling flow biasing the results. The fitting was repeatedfor a subset of the data, excluding emission beyonda fixed radius,Router, so as to investigate any systematic variation inβ with radius. The results of these fits aresummarised in table 2.3, and the best-fit model to the whole image is shown, together with the data points, infigure 2.13.

It can be seen from table 2.3 that there is only marginal evidence for a systematic trend inβ with radius forA2029, with a large overlap between the confidence regions. However, A1795 seems to show a significantly lowerβ (0.70) for the innermost 9′, as compared to the profile fitted out to 17′. An explanation of this apparent flatteningcan be found in the core radius, which is significantly smaller – 2.2′, as compared to 3.7′ for the whole profile. Thisbehaviour is an artefact of the excised central cooling region, which is comparable in size to the gas core in theIGM. Consequently,rc is biased when a large part of the outer profile is excluded from the fit – and this propagatesthrough to the best-fitβ value. This is confirmed by the bottom row in table 2.3, which lists the results of re-fittingthe innermost 9′ with rc frozen at the best-fit value from the full, 17′ profile: the correspondingβ (0.83) is, in fact,identical to the slope for the whole profile. This problem is particularly pronounced for A1795 due to its fairlylarge cooling flow: the excised central region amounts to∼15 per cent ofR200. The radius of the cooling flow inA2029 is only about 10 per cent ofR200, a value which is more typical of groups, and hence the potential for biasin rc is minimal.

Lewis et al. (2000) find a similar tendency forrc andβ to decrease, when fitting truncated radial profiles ofsimulated clusters. Similarly, their analysis excludes a central portion of the data, corresponding to emission fromeither the central cluster galaxy or a cooling flow. They alsoclaim to find a steepening in the logarithmic slope ofthe gas density with radius in the outer regions, as reportedby Vikhlinin et al. (1999a); however, this discrepancyis only evident in the vicinity of the virial radius – a regionrarely probed by observations of even the hottestclusters. In any case, it is clear that, even if such a steepening is a significant effect, it is not able to introduce largesystematicrelativebiases between groups and clusters in our analysis.

2.7 Discussion

It is clear from the scaling properties we have examined, that virialized systems do not exhibit self-similar be-haviour. Theβ− T relation and the gas fraction data reveal a flattening of the gas density profiles, which is

2.7. DISCUSSION 43

Figure 2.13: Azimuthally averaged surface brightness profiles for A1795 (left panel) and A2029 (right panel). The solidline indicates thebest-fitting model (see text for details).

most obvious in the group regime. These observations are consistent with energy injection into the IGM by non-gravitational means. However, three questions remain unanswered. Firstly, what caused this heating; secondly,when did it take place and, thirdly, is self-similarity broken only below a certain critical temperature, or does thetransition occur gradually?

There are three main candidates for the origin of the self-similarity breaking. Both galaxy winds (e.g. Lloyd-Davies et al., 2000) and AGN heating (e.g. Wu et al., 2000) areable to inject energy at roughly the levels requiredto raise the entropy of gas in the central regions of the IGM. However, the role of gas cooling in imposing anentropy floor (e.g. Muanwong et al., 2001) could also be significant. Although our results in this paper provideno means of discriminating between the first two options, ourdata do allow us to address the viability of cooling.The cooling hypothesis is offered some support by our gas fraction results: the variation infgaswith Tew shown infigures 2.5 & 2.6, is consistent with the loss of gas required in the cool systems if cooling is to have a significanteffect. On the other hand, there is no evidence that groups have an excessive totalLB compared to clusters, whichwould be expected if the cooling gas ultimately formed stars(Helsdon & Ponman, 2002) (see also Chapter 3). Thismay indicate that thefgas trend is due to gas being displaced to larger radius by the effects of energy injection.

The second question raises two possibilities; either the heating took place prior to halo collapse (so called ‘ex-ternal’ or ‘pre-’heating), or most energy injection occurred after virialization (internal heating). Although a rathersimplistic distinction, these two scenarios will manifestthemselves in different ways on cluster properties, giventhe effect of injecting energy into a medium in which a significant density gradient has already been established.For example, preheating will tend to weaken the shock boundary and move it outwards as compared to internalheating (Tozzi et al., 2000). Since this boundary marks the point where the gas fraction fades into the universalvalue, this amounts to an observable signature. The evidence from fgas is tentative, given the extrapolation uncer-tainties, but there is indication of a systematic variationin the total gas fraction withinR200 with Tew. Observationsof the X-ray background suggest that the heating phase took place over a time-scale of∼107 yr (Pen, 1999), al-though the epoch of energy injection is not constrained. Conversely, observations of the entropy floor (Ponmanet al., 1999; Lloyd-Davies et al., 2000) place a strict upperlimit on zpreh, the redshift at which the preheating epochcould have taken place, ofzpreh<∼10.

The question of on what mass scale the effects of self-similarity breaking occur is more readily answeredwith our large sample. Whereas previous work has pointed to asharp transition between groups and clusters,our results do not offer much support for this hypothesis. The analytical approach of Dos Santos & Dore (2002)shows that the two mass regimes can be unified with a simple model that incorporates energy injection fromnon-gravitational processes. Their predicted scaling relations show a gradual steeping ofM−TX and L−TX,with decreasing temperature, and indicate that accretion shocks cannot be completely suppressed in groups. OurmeasuredM−TX data offer some support for a convex relation, as opposed to abroken power law, but the scatteris rather large. Whilst higher quality data fromXMM-NewtonandChandrawill doubtless shed some light onthis issue, the greatest uncertainty lies in the systematicbias associated with extrapolation toR200: it is necessaryto trace the X-ray haloes of nearby groups out toRv in order to resolve the issue satisfactorily, and this callsfor


observations with a wider field-of-view.

2.7.1 Galaxies vs. groups

In a CDM Universe, the formation of structure proceeds in a bottom-up fashion, with the smallest haloes virializinginitially and subsequent merging activity leading to the hierarchical assembly of progressively larger haloes (e.g.Blumenthal et al., 1984). Consequently, the smallest objects tend to be older, having collapsed at an earlier epoch.This then leads to differences in the scaling properties, asa result of the higher density of the Universe at that time(e.g. theM−TX relation normalization). However, in the context of a preheating prescription, invoked to explainthe breaking of self-similarity in galaxy groups, the timing of this early formation epoch is critical; it is possiblethat these objects virializedprior to the preheating phase. This would give rise to behaviour more consistent withmassive clusters of galaxies, which are sufficiently large as to be insensitive to the effects of energy injection. Onepossible candidate for the origin of this preheating is population III stars, whose formation precedes even that ofgalaxies, which may also have contributed to the enrichmentof the IGM (e.g. Loewenstein, 2001).

As has already been seen, the properties of the galaxy-sizedhaloes in this sample appear to differ from thoseof groups of a similar temperature. Specifically, the gas density index (β) is rather steeper than expected froma simple extrapolation of theβ− T relation for the whole sample. In addition, the S0 galaxy NGC1553 liesnoticeably below theM−TX relation. This can be understood in terms of a large discrepancy between its redshiftof formation and observation: there is a bias towards observing nearby objects of small mass, but these are likelyto have formed earliest of all sized haloes. For a given halo mass, the virial temperature is proportional to(1+zf)and so, for NGC 1553, a value ofzf ∼ 3–4 would increaseTew, and push it to the right of theM−TX relation, asobserved. Alternatively, the discrepancy can be attributed to the effects of non-gravitational heating onTew caused,for example, by outflows from within the galaxy itself: a value of Tew ≃ 0.3 keV (compared to the actual valueof 0.5 keV) would bring the point back on the best-fitM−TX relation for the whole sample. This explanation isfurther supported by itsLX /LB, which is low enough for the stellar population in the galaxyto have significantlyinfluenced its X-ray halo (e.g. Pellegrini & Ciotti, 1998).

In fact, the behaviour of NGC 1553 is sufficiently unusual that it points to an alternative formation mechanismto that which is usually invoked for hot gas in groups and clusters. For example, the halo could have been builtfrom supernova-driven winds, originating in its own stellar population (e.g. Ciotti et al., 1991), rather than fromprimordial material . This explanation is supported by itsLX /LB, and may explain the anomalously steepβ index,which would otherwise be flattened by the effects of energy injection prior to collapse.

In contrast, the position of the elliptical galaxy NGC 6482 on theM−TX relation is fully consistent with thebest fitting line to the whole sample. This is not surprising,as it is likely that this object is a fossil group and hencewill exhibit group-type behaviour. What is certain is that this must be a very old system, given the long time-scalefor orbital decay and merging of the galaxy members. Although not as discrepant as NGC 1553, NGC 6482 showssome suggestion that itsβ index may be high for its temperature. This could indicate that it formed sufficientlyearly to have been relatively unaffected by a phase of energyinjection, that would have occurred prior to theformation of most larger objects. There is no doubt that its halo must be predominantly primordial – itsLX /LB issufficiently high that the influence of its stellar population is negligible, in terms of contributing to the total gasmass.

Given an early formation epoch, consistent with hierarchical formation, coupled with a correspondingly largermean density, it is particularly important to consider the cooling time of the X-ray gas. At a radius of 0.1R200,NGC 1553 has a gas cooling time (tcool) of ∼6 Gyr and NGC 6482 hastcool ≃ 3 Gyr. For comparison, the coolingtimes of the next two coolest systems in our sample – HCG 68 andNGC 1395 – are 18 Gyr and∼25 Gyr,respectively. In the case of NGC 1553, this implies that someform of heating mechanism must have preventedsignificant gas cooling, if its halo was formed beforez= 1 (corresponding to a light travel time of roughly 6 Gyr).This is also supported by the fact that there appears to be no evidence of strong cooling in the core, where thedensity is even higher. Energy injection from galaxy winds could provide the heating mechanism necessary toexplain this result, as suggested above. For NGC 6482, some mechanism is also needed to prevent catastrophic gascooling, which is not observed even in the core, given that this fossil group must be a very old system. Once again,its highLX /LB rules out a significant heating contribution from supernova-driven winds. However, there is someevidence of a AGN component in this galaxy (Goudfrooij et al., 1994), which may provide a suitable reheatingmechanism to prevent the establishment of a cooling flow.

2.8. CONCLUSIONS 45

2.8 Conclusions

We have studied the scaling properties of the X-ray emittinggas and gravitating mass of a large sample of clus-ters, groups and galaxy-size haloes. The 3-dimensional variations in gas density and temperature – corrected forcontamination from cooling flows – are parametrized analytically, allowing us determine all derived quantities ina self-consistent manner. We have derived virial radii and emission-weighted temperatures from these models andare able to extrapolate the properties of the data to measurethem at fixed fractions ofRv. We have also analysedan identical set of isothermal models, to investigate the effects of neglecting spatial variations in the temperatureof the IGM. We summarise our main findings below.

(1) Beta varies strongly with temperature, although there is evidence that galaxy-sized haloes do not follow thistrend. We find a best fitting power law relation of the formβ = (0.44±0.06)T0.20±0.03.

(2) There is a 6σ correlation betweenfgaswithin 0.3R200and temperature, consistent with the variation inβ. Thistrend is weakened only slightly (to 5.4σ) by extrapolating the gas fraction toR200 although, above 4 keV,the significance of this correlation drops to 3.2σ. The meanfgas within R200 for the systems hotter than

4 keV is (0.163±0.01)h− 3

270 , compared to(0.134±0.01)h

− 32

70 for the whole sample. Under the assumptionof isothermality, the scatter betweenfgas at R200 andTew is reduced, as is the normalization, giving a mean

for the whole sample of(0.110±0.01)h− 3

270 .

(3) Observations of the variation in gas fraction as a function of radius in our sample reveal a systematic trend ingas fraction with temperature in all but the central regions( <

∼ 0.3R200). This is consistent with the observedtrend in fgaswith Tew.

(4) In our study of theM−TX relation, we employ two additional methods of calculating the average systemtemperature, one of which excludes the central region, another weighting the temperature with gas densityrather than emissivity. We apply our three different methods within both 0.3R200andR200, for both mass andT, to give a total of sixM−TX relations. We find that the logarithmic slope of the relationis steeper within0.3R200 but that, even withinR200, it is inconsistent with self-similarity. There is close agreement betweenthe measured slopes found for each of the three different prescriptions forT. For the emission-weightedT,within R200, we findM = 2.34×1013×T(1.84±0.06) M⊙. We find that the effect of assuming isothermalityon the slope is negligible, but the normalization increasesby 15 and 30 per cent for 0.3R200 and R200,respectively (cf. Neumann & Arnaud, 1999; Horner et al., 1999), indicating that the total gravitating mass issignificantly overestimated in our data when temperature gradients are neglected. In addition, the scatter inthe relation is reduced (and fully consistent with the parameter errors) compared to the non-isothermal case.The corresponding best fitting relation is given byM = 3.02×1013×T(1.89±0.04) M⊙.

(5) The relation betweenR200 andTew, as deduced from simulated clusters (Navarro et al., 1995) deviates sys-tematically from the measured values ofR200, as inferred from the overdensity profile. We find a strongnegative correlation between the ratio of the NFW predictedR200 to our measured values and a quantita-tive measure of non-isothermality (T(0)/T(0.3R200)). We show that only in the absence of a temperaturegradient do the methods agree.

(6) We address the issue of systematic bias associated with the extrapolation of the X-ray data toR200, by fittingazimuthally averaged surface brightness profiles for two clusters, within different outer radii. We find noevidence for a significant variation inβ with cluster radius and conclude that the flatter gas densityprofilesof cooler systems cannot be attributed to the generally poorer quality of data available for these objects.

(7) We find that the two galaxies in the sample display unusualproperties. We have selected these objects onthe basis of a lack of associated group or cluster halo emission, which can contaminate the galaxy halo flux.The S0 galaxy, NGC 1553 has a steepβ index and falls to the right of the mainM−TX relation, indicative ofan early formation epoch (zf ∼ 3–4) – which causes haloes of a given mass to be hotter than those collapsingat later times. It is also possible thatTew for this galaxy may have been artificially raised, probably bysupernova-driven outflows from its stellar population. Theelliptical galaxy NGC 6482 also shows a rathersteep gas density profile, but otherwise exhibits group-like behaviour, consistent with a classification as a‘fossil’ group.

Chapter 3

Mass Composition andDistribution

Abstract We investigate the spatial distribution of the baryonic andnon-baryonic mass components in a sample of 66 virializedsystems. We have used X-ray measurements to determine the deprojected temperature and density structure of the intergalacticmedium and have employed these to map the underlying gravitational potential. In addition, we have measured the deprojectedspatial distribution of galaxy luminosity for a subset of this sample, spanning over 2 decades in mass. With this combinedX-ray/optical study we examine the scaling properties of the baryons and address the issue of mass-to-light (M/L) ratio ingroups and clusters of galaxies.

We measure a median mass-to-light ratio of 249h70 (M/L)⊙ in the rest frameBj band, in good agreement with othermeasurements based on X-ray determined masses. There is no trend inM/L with X-ray temperature and only a marginallysignificant trend for mass to increase faster than luminosity: M ∝ L1.08±0.12

B,j . This implied lack of significant variation in starformation efficiency suggests that gas cooling cannot be greatly enhanced in groups, unless it drops out to form baryonicdarkmatter. Correspondingly, our results favour non-gravitational heating over radiative cooling as the primary cause ofthe observed

departure from self-similarity in low mass systems. The median baryon fraction for our sample is 0.162h−3/270 , which allows us

to place an upper limit on the cosmological matter density,Ω0 ≤ 0.25h−170

We find evidence of a systematic trend towards higher centraldensity concentration in the coolest haloes, indicative ofanearly formation epoch and consistent with hierarchical formation models.

3.1 Introduction

As the material most amenable to detailed study, the properties of optically luminous matter have long been used toinfer the distribution of mass in the Universe. Clusters of galaxies provide an excellent laboratory for this purpose,since they are sufficiently large that their global properties reflect those of the Universe as a whole.

Previous work has shown that both optical light and gas, in particular, are more spatially extended than the darkmatter within virialized systems (David et al., 1995). Evidence of a monotonic rise in gas fraction with radius hasalso been found across a wide range of mass scales in a very large sample of virialized systems (Sanderson et al.,2002, hereafter Chapter 2), in broad agreement with numerical simulations (Eke, Navarro, & Frenk, 1998; Frenket al., 1999). This behaviour has been confirmed in a recent, high qualityXMM-Newtonobservation of a relaxedcluster (Pratt & Arnaud, 2002), where the X-ray halo has beentraced out almost to the virial radius, althoughChandraobservations of extremely relaxed lensing clusters have revealed a flat gas fraction profile in a smallnumber of cases (Allen, Schmidt, & Fabian, 2002). However, the latter observations only probe the innermost∼1/3 of the halo: our data, though of poorer quality, extend beyond this region in many cases, and we benefit fromaveraging over a large ensemble of virialized systems.

While the hot gas has been shown to depart systematically from self-similarity under the influence of non-gravitational physics, the same cannot be said of the dark matter, which is not directly affected by such processes.N-body simulations indicate that the dark matter of virialized haloes should follow a universal profile, across awide range of scale sizes (Navarro et al., 1995), apart from amild trend in its concentration with mass (Navarroet al., 1997; Salvador-Sole et al., 1998; Avila-Reese et al., 1999; Jing, 2000).

Although the contribution to the baryonic mass component from the individual galaxies in virialized systemsis rather less than that from the X-ray emitting intergalactic medium, the galaxy component is of great importance,none the less. As the ultimate endpoint of gas cooling, as well as the source of non-gravitational energy injectioninto the IGM, the behaviour of the stars is closely linked to the properties of the gas. In addition, stellar sourcesare responsible for the synthesis of the metals that contaminate the IGM, which are released via supernova-driven

46

3.2. 3D GALAXY DENSITY CALCULATION 47

winds (e.g. Finoguenov et al., 2001a).However, the scaling properties of the stellar content are rather less well established than those of the IGM. In

particular, there is some debate on the mass-to-light ratioin groups and clusters and how this compares with theglobal value. Recent studies have reached differing conclusions about how this quantity varies with halo mass. Forexample, Girardi et al. (2002) report a trend towards increasedM/L in clusters, a result also favoured by Marinoni& Hudson (2002). By contrast, measurements ofM/L based on X-ray mass estimates have concluded that it isroughly universal in groups and clusters (Cirimele, Nesci,& Trevese, 1997; Hradecky et al., 2000).

The approach of a combined X-ray and optical analysis of clusters and groups of galaxies is a powerful toolfor the understanding of mass distribution in these systems. The X-ray data yield information about the behaviourof the diffuse, hot intergalactic medium (IGM) and can be used to determine the underlying gravitational potentialstructure, under the assumption of hydrostatic equilibrium. By contrast, optical measurements can be used to mapthe spatial properties of the baryons resident in luminous matter, and hence deduce their contribution to the totalmass budget. Combining these two types of complementary observations then enables the dark matter distributionto be inferred.

In Chapter 2, we presented a detailed X-ray analysis of the IGM in a large sample of 66 virialized systems,allowing us to reconstruct the properties of the IGM. We haveassembled our sample from three existing X-raystudies of groups and clusters, to which we have added a smallnumber of cool groups. To each system, wehave fitted analytical profiles to parametrize both the gas density and temperature as a function of radius. Thisallows us to put all the X-ray data on a unified footing, givingus the freedom to extrapolate the gas propertiesto arbitrary radius. As with detailed X-ray analyses, previous combined X-ray/optical studies of this nature havebeen restricted to relatively small sample sizes (e.g. Cirimele et al., 1997; Hradecky et al., 2000). In this paper, webuild on our modelling of the IGM, by incorporating the spatial distribution of stellar material, which also allowsus to determine the distribution of dark matter. Despite only covering half our original sample (some 32 groupsand clusters), our optical sample is well suited to the studyof the scaling properties of these systems, since weretain good coverage across a wide range of system masses.

Throughout this paper we adopt the following cosmological parameters;H0 = 70 km s−1 Mpc−1andq0 = 0.All quoted errors are 1σ on one parameter, unless otherwise stated.

3.2 3D Galaxy Density Calculation

Since we have already deduced the deprojected gas distribution of each member of our sample in Chapter 2, weaim to similarly determine the spatial distribution of optical light, in order to infer the stellar mass distribution.However, this task is complicated by the discrete nature of the individual galaxies – unlike the evenly distributedgas, which smoothly traces the underlying gravitational potential. In addition, the contrast against the backgroundof a cluster or group of galaxies is much lower at optical wavelengths, scaling only linearly with number density,rather than in proportion toρgas

2 in the case of X-ray emission. These two difficulties are further compoundedby the large angular size subtended by a typical group or cluster, which may exceed one degree on the sky –significantly larger than can be viewed by current generation wide-field CCD cameras. Correspondingly, therehave been relatively few detailed photometric studies of rich clusters made with CCDs (e.g. Carlberg, Yee, &Ellingson, 1997).

One way to avoid some of the above problems is to quantitatively establish cluster membership with redshiftmeasurements of individual galaxies in its vicinity. Although this avoids the need to estimate the backgroundcontribution, such an advantage comes only at the price of long observing times, in order to obtain high-qualityspectra for a large fraction of the galaxy members. For this reason such studies are restricted to a small numberof clusters (e.g. Koranyi & Geller, 2002, 2000; Mohr et al., 1996; Girardi et al., 1995; Fabricant, Kent, & Kurtz,1989). Given these limitations, we have taken a different approach to the issue of measuring the spatial distributionof galaxies and then estimating the total luminosity of the whole system. We outline below our method, which isbased on widely-available digitized photographic plate data derived from all-sky surveys.

48

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Names RA Dec. z Ta R200 Rbs LB,j M/L Referencec

(J2000) (J2000) (keV) (arcmin) (arcmin) (1011LB,j,⊙) (M/LB,j)⊙

NGC 4325 185.825 10.622 0.0252 0.90 22.4 2.6 1.22+0.6−0.6 293+160

−160 G02 (APS)

NGC 5846 226.385 1.696 0.0058 1.18 95.5 39.3 1.13+0.6−0.6 325+170

−170 H02

HCG 62 193.284 -9.224 0.0137 1.48 33.5 12.7⋆ 2.59+1.3−1.3 78+40

−40 G02 (APS)

NGC 5129 201.150 13.928 0.0233 1.54 20.5 7.8 2.56+1.3−1.3 82+48

−48 G02 (APS)

NGC 2563 125.102 21.096 0.0163 1.61 31.4 12.1 1.11+0.6−0.6 256+130

−130 H02

Abell 262 28.191 36.157 0.0163 2.03 50.4 47.0⋆ 9.71+4.9−4.9 118+65

−65 G02 (APS)

Abell 194 21.460 -1.365 0.0180 2.07 52.2 20.4 5.16+2.6−2.6 320+200

−200 G02 (COSMOS)

MKW 4 180.990 1.888 0.0200 2.08 35.5 40.3 1.12+0.6−0.6 613+310

−310 G02 (APS)

MKW 4S 181.647 28.180 0.0283 2.46 28.9 11.9 3.37+1.2−1.2 320+130

−130 H00

NGC 6338 258.825 57.400 0.0282 2.64 26.4 5.9 14.0+7.0−7.0 59+60

−50 G02 (APS)

Abell 539 79.134 6.442 0.0288 2.87 38.0 5.9 6.17+3.1−3.1 415+200

−200 G00

AWM 4 241.238 23.946 0.0318 2.96 40.2 11.5 4.75+2.4−2.4 886+540

−540 G02 (APS)

Abell 1060 159.169 -27.521 0.0124 3.31 104.8 15.0⋆ 18.9+9.5−9.5 244+120

−120 G02 (COSMOS)

Abell 2634 354.615 27.022 0.0309 3.45 32.1 12.0 25.4+12.8−12.6 76+40

−40 G02 (APS)

Abell 2052 229.176 7.002 0.0353 3.45 36.6 2.3 6.70+1.7−1.7 589+240

−240 H00

Abell 2199 247.165 39.550 0.0299 3.93 34.3 31.2⋆ 9.98+3.5−2.4 211+60

−60 H00

Abell 2063 230.757 8.580 0.0355 4.00 35.7 2.3 7.08+1.7−1.6 541+140

−140 H00

AWM 7 43.634 41.586 0.0172 4.02 105.2 4.2⋆ 7.68+3.8−3.8 1614+970

−970 G00

Abell 3391 96.608 -53.678 0.0536 5.39 26.9 4.3 25.1+12.6−12.6 213+120

−120 G02 (COSMOS)

Abell 2670 358.564 -10.408 0.0759 5.64 19.6 5.9 20.7+10.3−10.4 249+130

−130 G02 (COSMOS)

Abell 119 14.054 -1.235 0.0444 6.08 33.5 5.5 26.9+13.6−13.4 218+120

−120 G02 (COSMOS)

Abell 496 68.397 -13.246 0.0331 6.11 39.2 20.0⋆ 13.1+6.6−6.5 321+170

−170 G02 (COSMOS)

Abell 3558 201.991 -31.488 0.0477 6.28 28.7 18.2 47.7+23.8−23.8 99+50

−50 G02 (COSMOS)

Abell 3571 206.867 -32.854 0.0397 7.31 39.8 17.9 46.8+23.4−23.4 161+80

−80 G02 (COSMOS)

Abell 2218 248.970 66.214 0.1710 8.28 11.3 10.4⋆ 29.9+3.7−3.8 266+60

−60 S96

Abell 1795 207.218 26.598 0.0622 8.54 28.3 6.4⋆ 27.9+13.9−14.0 330+240

−240 G02 (COSMOS)

Abell 2256 256.010 78.632 0.0581 8.62 27.3 7.3 39.2+19.6−19.6 175+90

−90 G00

Abell 85 10.453 -9.318 0.0521 8.64 28.1 11.6 23.9+12.1−12.0 229+120

−120 G02 (COSMOS)

Abell 3266 67.856 -61.417 0.0545 9.53 29.7 12.0 29.3+14.7−14.7 261+140

−140 G02 (COSMOS)

continued overleaf

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Names RA Dec. z Ta R200 Rbs LB,j M/L Referencec

(J2000) (J2000) (keV) (arcmin) (arcmin) (1011LB,j,⊙) (M/LB,j)⊙

Abell 2029 227.729 5.720 0.0766 9.80 26.4 4.6 86.3+43.0−43.1 155+80

−80 G02 (APS)

Abell 478 63.359 10.466 0.0882 10.95 17.8 12.8⋆ 28.1+7.3−7.3 210+160

−160 H00

Abell 2142 239.592 27.233 0.0894 11.16 22.7 8.7 50.0+25.0−25.0 250+170

−170 G02 (APS)

Table 3.1: Some basic properties of the 32 objects in the optical sample, listed in order of increasing temperature. Positions and redshifts are taken from Ebeling et al.

(1996, 1998); Ponman et al. (1996) and NED. Columns 7–9 are data as determined in this work, except for those values ofrs marked with a⋆, which were taken from

Lloyd-Davies (2001). Note: values are forH0 = 70 km s−1 Mpc−1 . All errors are 68% confidence.aThe cooling-flow corrected, emission-weighted temperature of the

system within 0.3R200, as determined in Chapter 2.aThe NFW scale radius of the galaxy distribution (see equation 3.1).cReference for the luminosity data, G02 = Girardi

et al. (2002); G00 = Girardi et al. (2000); H00 = Hradecky et al. (2000); H02 = Helsdon & Ponman (2002); S96 = Squires et al. (1996).

50 CHAPTER 3. MASS COMPOSITION AND DISTRIBUTION

The process can be separated into two stages. Firstly, determination of the spatial distribution of the galaxynumber density (see section 3.2.3) and, secondly, the calculation of the normalization of this density profile (seesection 3.3.2). Although the normalization can, of course,be calculated in the spatial fitting, this makes no al-lowance for the contribution to the total luminosity from galaxies too faint to be observed. To estimate this contri-bution it is necessary to determine the luminosity functionof the observed galaxies and extrapolate this down tosome limiting magnitude, in order to correct for the missinglight. Whilst it is possible to do this using photographicplate based data, the measured magnitudes require careful calibration, due to subtle variations in the sensitivity ofthe photographic emulsion both within and between plates. Therefore, we have taken integrated luminosity valuesfrom the literature and used these to infer a 3-dimensional normalization, using our fitted profile parameters.

We assume a distribution of optical light described by an NFWprofile (Navarro et al., 1995), i.e.

ρ =ρ0

x(1+x)2 , (3.1)

wherex= r/rs andrs is a characteristic scale radius. This function rises from alogarithmic slope of -3 at large radiito a central cusp (ρ ∝ r−1) with the transition between the two regimes occurring aroundrs. This parametrization isadvantageous as it has only one free parameter (since the normalization is constrained by our maximum likelihoodfitting method – see section 3.2.3), which helps stabilise the fitting, given the relatively large background level inthe case of the plate data.

We make the assumption of spherical symmetry in the deprojection of the optical light distribution, as we didfor the X-ray analysis in Chapter 2. Since our sample has beenselected on the basis of a relaxed X-ray morphology,it is reasonable to expect a corresponding degree of regularity in the galaxy distribution. This also improves thequality of the fit, which might otherwise be degraded by the presence of significant substructure or bi-modality.However, despite this, it was not possible to obtain a satisfactory surface density fit to the photographic plate datafor a small number of cool groups; their treatment is described in section 3.2.2 below.

3.2.1 APM data

The Automatic Plate Measuring (APM) machine source catalogue1 is a digitized catalogue based on the 1st gener-ation Palomar Observatory Sky Survey (POSS I) and United Kingdom Schmidt Telescope (UKST) photographicplates. The field of view covered is extremely large – 6.2 and 5.8 degrees on a side for the plates from POSS Iand UKST, respectively – and the whole sky has been observed and catalogued in this way. As such, it is idealfor studying large scale structure at optical wavelengths,although the magnitude information it provides can berather unreliable, owing to saturation and other problems with photographic emulsions. However, we only exploitthe positional data available – which are very reliable – since we determine total luminosities for our groups andclusters from literature measurements.

To extract source positions for our fitting, we have applied very conservative selection criteria to allow forproblems with source identification. Although detection algorithms have been used to distinguish between stellarand non-stellar sources, this classification is often unreliable and can cause stars to be identified as galaxies (e.g.Caretta, Maia, & Willmer, 2000) and vice versa: in an analysis of data from the Minnesota Automated PlateScanner (APS) digitized survey of POSS I plates, Rines et al.(2002) recently found that∼3.3 per cent of objectsidentified as stars were actually galaxies, in the vicinity of the cluster Abell 2199. This confusion is exacerbatedfor faint galaxies, and consequently we only neglect sources classified as stars if they are brighter thanmb = 15,otherwise all detected sources are used. This avoids any potential systematic bias in the fitting between clusters, atthe expense of a significantly increased background level. All objects classified as noise features are excluded.

3.2.2 NED data

The problems associated with optical analysis are exacerbated at the scale of groups of galaxies, which presenteven lower contrast against the background, owing to their smaller volume. However, many well-studied groups– including most of those in our sample – are sufficiently close so as to have measured recession velocities avail-able in the literature. In such cases it is therefore possible to establish unambiguous membership of the group,thus eliminating the contamination from fore- and background objects. However, the numbers of such confirmedmembers are generally rather small and tend to be predominantly limited to the inner regions of the halo. The

1http://www.ast.cam.ac.uk/˜apmcat/

3.2. 3D GALAXY DENSITY CALCULATION 51

impact this has is mitigated to some extent by our chosen parametrization of galaxy density (equation 3.1), whichhas only a single free parameter (the scale radius,rs), since sparse data obviously provide weaker constraints onspatial fitting (Girardi et al., 1995).

It was not possible to derive NFW scale radii from the APM datafor four of our groups (NGC 2563, NGC 5129,NGC 5846 & NGC 6338) and so we took galaxy positions from the NASA Extragalactic Database (NED) for thesesystems. It was also necessary to do this for Abell 539, sinceno POSS I or UKST data were available for this poorcluster. There is potential for bias when using NED data, since the completeness properties of the database areunknown – although we are only concerned with the spatial completeness, since the luminosities for these systemsare taken from other sources. In particular, it is possible that there may be a dearth of galaxies catalogued in theouter regions of these groups, which may introduce a spurious central surface density enhancement. However, wenote that any such bias is reduced by the fact that we have usedthe radius of the outermost galaxy as the cut-offradius in the fitting, rather than the X-ray determined virial radius used to select the galaxies from NED.

3.2.3 Surface density fitting

As we are interested in the deprojected properties of the galaxy optical light, we have chosen to parametrize thespace density distribution in three dimensions and then numerically project this, for comparison with the surfacedensity of galaxies. This is advantageous, since it directly yields the required parameters, without the need forinversion. For the analysis of the APM data, a constant background term was included as a free parameter, toallow for a variation in the number of fore- or background sources caused, for example, by a differences in large-scale structure along the line-of-sight. It is important tofit this separately for each cluster, since our background(incorporating a stellar component) can vary significantlywith position on the sky. The centroid of the model wasfixed at the cluster position as listed in NED, in order to stabilise the fitting. Although the optical centroid canbe displaced with respect to the X-ray centroid, we note thatGirardi et al. (2000) find a typical error inLB withinRv of just 5 per cent, when recomputing their integrated optical luminosities using an X-ray rather than opticalcentroid.

The 3-dimensional density profile was evaluated in a series of spherical shells, by numerically integratingequation 3.1 between their radial boundaries. Each spherical shell has a corresponding annulus, with identicalradial bounds. The surface density in each annulus is calculated by summing the contributions made to it by itscorresponding shell and all those lying outside it. Hence the contribution to the surface density from projectionalong the line of sight is fully accounted for.

We have chosen to perform anunweightedfit to the data (as also used by Hradecky et al. 2000 and Koranyi& Geller 2002, for example), so that each galaxy is treated asa single point at a given position. This approachis advantageous since it avoids the need to use plate-measured magnitudes, which require careful calibration asmentioned above. By treating each galaxy identically we areimplicitly assuming that there is no luminositysegregation, i.e. that the luminosity function is everywhere the same within the cluster. This is a reasonableassumption, since any segregation is generally limited to only the very brightest galaxies – which tend to belocated more centrally in clusters (Adami, Katgert, & Biviano, 1998).

The use of an unweighted fit has important consequences for those clusters that possess a central cD galaxy.On the one hand, our method limits the potential for an extremely bright, centrally located galaxy to skew the fit,by overemphasising the central luminosity density. On the other hand, in so doing, there is a tendency to slightlyoverestimate the luminosity in the outer regions. However,the effect this has on our total luminosity estimates issmall, given the sizes of the apertures used to measure the luminosity values to which we normalize our densityprofiles (see section 3.3.2). In any case, the contribution to the luminosity function from cD galaxies is fullyaccounted for by this same method for determining the total cluster luminosity.

Although some previous studies of galaxy distributions in clusters have fitted analytical profiles with cores tothe data (e.g. Girardi et al., 1995; Cirimele et al., 1997), we favour the cusped NFW profile (equation 3.1). Notonly does it have one fewer fitted parameter – which helps stabilise the fitting, as previously mentioned – it alsoprovides a more accurate description of the dark matter distribution (Navarro et al., 1995). Simulations indicate thatthe galaxies trace the dark matter more closely than the gas does, with only a modest bias (Dave et al., 2002); theymay even be more centrally concentrated, owing to the effects of dynamical friction (Metzler & Evrard, 1997).We note that Adami et al. (1998) have shown that only the surface density of faint galaxies shows a significantpreference for a distribution with a core rather than a cusp.Since our analysis is based on photographic plateobservations, rather than the higher quality CCD data used in their work, we can therefore expect to be relativelyinsensitive to this effect.


Previous studies offer support for the presence of a cusp in the luminosity distribution. For example, Oegerle,Jewison, & Hoessel (1987) measure a power law slope of approximately -1 in the centres of 3 Abell clusters; Merritt& Tremblay (1994) find evidence of a central power law dependence of galaxy space density in the core of theComa cluster, using a non-parametric algorithm. Beers & Tonry (1986) have demonstrated that imperfect centringcan ‘erase’ the presence of a real cusp and artificially introduce a constant density core. Using two methods ofdetermining the cluster centre, they find that the majority of the 48 clusters in their sample possesses a central cuspin projected galaxy density. The inner regions of these clusters are well characterised by a power law slope of -1.

We use the maximum likelihood fitting procedure described inSarazin (1980), which we now briefly outline.There are three parameters to be determined – the scale radius, rs, the surface density normalization,σ0, andbackground surface density,Σbg (since the centroid is fixed – see above). However, only two ofthese parame-ters are independent, therefore we require an additional constraint, otherwise the maximum likelihood solution isunbounded. We apply the condition that the model reproduce the observed number of galaxies within the fittingregion. In the standard method of Sarazin (1980),rs andΣ0 are left as free parameters andΣbg is calculated so thatthe ‘area’ under the model + background equals the total number of galaxies within the fit region. However, in thecase of our NED dataΣbg is knowna priori (i.e. it is zero, since all the galaxies are confirmed group members), sowe use the alternative method detailed in section IV of Sarazin (1980). Here,Σbg andrs are left as fitted parameters,while Σ0 is instead calculated so as to reproduce the observed galaxycount.

As a result of our numerical projection technique,Σ0 is not actually a direct model parameter. However it istrivially related toρ0, the NFW central density normalization from equation 3.1. We determine the appropriatevalue ofρ0 by setting it initially to unity and calculating the ‘area’ under the projected data, excluding the parts ofany annuli lying outside the fit region, i.e. summing the product of the surface density in each annulus and its area.The ratio of the observed number of galaxies in the fit region to this number then yieldsρ0.

Fitting was performed using the MIGRAD method in theMINUIT minimization library from CERN (James,1998) and errors on the parameters were found with MINOS, from the same package. Errors were determinedfrom the increase in the Cash statistic of one, since differences between values obtained from the same data set areχ2-distributed.

3.3 Optical Luminosity Calculation

Since we have chosen to avoid using magnitude information inthe calculation of the galaxy surface density distri-bution, we require an alternative means of deriving a normalization value for our deprojected luminosity profiles.To do this we have taken integrated luminosity measurementsin fixed metric apertures from the literature, whichallow us to infer a central luminosity density. The details of this procedure are described in section 3.3.2 below.

The majority of these values are taken from the sample of Girardi et al. (2002), which incorporates most of thesample of Girardi et al. (2000). This sample comprises data from the APS and COSMOS/UKST surveys – we haveselected luminosity estimates from the latter for preference, where overlaps occurred. The only exception to this isHCG 62, for which the APS and COSMOS aperture luminosities quoted in Girardi et al. (2002) differ by a factorof 2. For this group we have chosen the APS measurement, sincethe COSMOS value leads to a mass-to-light ratio(see section 3.5.2) which is in strong disagreement with thesimple estimate of (Ponman & Bertram, 1993). Datafor three other clusters (Abell 539, Abell 2256 and AWM 7) were taken from Girardi et al. (2000). These lattersystems have two different luminosity estimates based on literature magnitudes – we chose the ‘M(red)’ sample,since this provided the largest overall number of galaxies detected for all three clusters.

A further five systems are covered by the study of Hradecky et al. (2000), based on data from the secondgeneration Palomar Observatory Sky Survey (POSS II), calibrated with CCD photometry. Two groups (NGC 2563and NGC 5846) have luminosity estimates from Helsdon & Ponman (2002) within an aperture defined by the virialradius as calculated in Helsdon & Ponman (2000b). Finally, we used the luminosity quoted in Squires et al. (1996)for the rich cluster Abell 2218. The references for each system are listed in Table 3.1, together with some keyproperties.

3.3.1 Conversion between bands

We have adopted theBj photometric band as our standard reference frame, since this was used by Girardi et al.(2002) and Girardi et al. (2000), and we takeB⊙,j = 5.33 (Girardi et al., 2002). However, our other literaturesources have used different bands and so we have converted these values into theBj band, under the assumption

3.3. OPTICAL LUMINOSITY CALCULATION 53

that the majority of the light originates in early-type galaxies. We have used the following relations to perform theconversions.

To convert from theB band to theBj band, we assume

B⊙ = 5.48 (Girardi et al., 2000)

Bj = B+0.28(B−V) (Blair & Gilmore, 1982)

B−V = 0.9, for early-type galaxies (Girardi et al., 2000)

⇒ Bj = B+0.252,

which leads toLBj

LB⊙,j

= 0.691LB

LB,⊙. (3.2)

To convert from theV band to theBj band, we assume

V⊙ = 4.82 (Allen, 1973)

⇒ Bj = V +1.152, from above (for early-type galaxies)

which leads toLBj

LB⊙,j

= 0.554LV

LV,⊙. (3.3)

For comparison withRband luminosities, we assume

R⊙ = 4.28 (Allen, 1973)

V −R= 0.55 for early-type galaxies (Girardi et al., 2000)

⇒ Bj = R+1.702, from above,

which leads toLBj

LB⊙,j

= 0.547LR

LR,⊙. (3.4)

3.3.2 Determination of luminosity normalization

With the exception of the Hradecky et al. (2000) data, all ourliterature values of luminosity are simple estimateswithin a fixed aperture on the sky. As such, these measurements represent the integrated luminosity within acylinder, defined by the aperture multiplied by the diameterof the cluster. It was therefore necessary to allow forthe projected contribution to this value from luminosity atlarge radii, when inferring values for the 3-dimensionalnormalization. We accomplished this in the following way, using our numerical projection technique describedpreviously. The normalization was set to unity and the NFW profile was integrated out to a radius of 500 arcmin(i.e.≫Rv) in a series of fixed-width spherical shells. The projected emission from these shells was then computed,in the corresponding series of annuli. The total luminositywithin the aperture radius was then calculated, by simplysumming the contributions from the annuli within that radius. Where the aperture radius lay within an annulus,the contribution from this annulus was determined by linearinterpolation, since the annulus width is small. Theappropriate normalization factor was found by dividing themeasured luminosity value by that calculated for anormalization of unity.

For the five clusters analysed by Hradecky et al. (2000), a slightly different approach was taken, since theauthors corrected their aperture luminosities for the effects of projection. Working in three dimensions, we setour normalization to unity and simply integrated our density profile fromr = 0 to a radius of 1h50 Mpc (used forall their quoted luminosities). The ratio of the aperture luminosity to the integrated value then yielded the centralluminosity density that we require.

The errors on our quoted luminosities are derived from the errors on the aperture values. Following Girardiet al. (2002), we assume a 1σ error of 50 per cent on our totalLB,j values based on their data. The same fractionalerror was assumed for the values from Helsdon & Ponman (2002), based on their own error estimates. The valuesfor the Hradecky et al. (2000) data have smaller errors, as does the Squires et al. (1996) luminosity for Abell 2218,


since these are based on higher quality POSS II and CCD observations, respectively, which are more accurate thanthe majority of our photographic plate data.

Owing to our choice of an NFW profile for the fitting, coupled with the bias towards optically rather than X-rayselected clusters in previous studies, we are extremely restricted in the number of direct comparisons which canbe made between our spatial fitting results and those available in the literature. However, we note that Adamiet al. (1998) found a scale radius of 3.7±0.6 arcmin for the cluster Abell 119, which is reasonably consistent withour value of 5.5 arcmin (see Table 3.1). This corresponds to a∼4 per cent change in the total optical luminositywithin R200, based on the aperture luminosity for this cluster as measured by Girardi et al. (2002). The impact ofuncertainties in the spatial fitting parameters is limited by the fact that our aperture luminosity values are generallyquite representative of the global value – the median difference between these aperture radii and our X-ray deter-mined values ofR200 is ∼50 per cent. In the case of Abell 119, our correction to the Girardi et al. (2002) apertureluminosity amounts to only∼10 per cent. For Abell 2256, the analysis of Oegerle et al. (1987) provides supportfor our findings, measuring an inner logarithmic slope of−0.98±0.02 for the galaxy density distribution in thiscluster. Moreover, their binned radial profile is consistent with an outer slope of -3 and a break radius similar toour value of 7.3 arcmin.

3.4 Results: Spatial Mass Distribution

We convert optical light directly into stellar mass, assuming a mass-to-light ratio for early-type galaxies of(M/L)B

= 7h70 Solar (Pizzella et al., 1997). A similar value of (8.3± 0.35)h70 was found by van der Marel (1991).UsingB⊙−B⊙,j = 0.15, from section 3.3.1 gives(M/L)Bj = 6.1h70(M/LB,j)⊙. To infer the density profile of darkmatter, we subtract the gas and stellar mass from the total mass (as determined from the X-ray data). Consequently,we only have information on the dark matter and stellar properties for the optical sub-sample of 32 groups andclusters.

3.4.1 Stellar distribution

The stellar mass density as a function of scaled radius, for the optical sample, is shown in Fig. 3.1, grouped intofive temperature bins for clarity. We have chosen slightly different temperature ranges to define our grouping binsfor the optical sample (cf. fig. 6 in Chapter 2; Fig. 3.2 and Fig. 3.4, below), in order to maintain approximately thesame number of systems in each bin.

There is a clear consistency in the shape and normalization of the profiles across the whole range of temper-atures, which are approximately co-aligned. However, close inspection of Fig. 3.1 reveals a slight excess in thestellar density of the coolest two temperature bands, compared to the more massive systems. This points towardsa possible weak trend in star formation efficiency with mass;we will revisit this issue in section 3.5.3.

3.4.2 Gas distribution

Fig. 3.2 shows the gas density profiles for the whole X-ray sample, averaged according to their mean X-ray tem-perature. It is very clear that there is a great deal of variation in the five different curves. Even amongst the hottestsystems there is a variation in the normalization of the lines, in the direction of increasing gas mass with highertemperature, although the shape of the profiles is very similar.

The two coolest temperature bands exhibit very different behaviour, having the lowest gas density within0.3R200 and showing no evidence of the central core seen in the hotterclusters. The normalization of the coolgroup lines is consistent with the trend towards a lowering of gas density compared to the clusters. This behaviourmirrors the trend seen in the gas fraction in Chapter 2, and provides strong evidence of deviation from self-similarscaling of the IGM both within and between haloes of different masses.

3.4.3 Dark matter distribution

The spatial variation of the density of dark matter can be seen in Fig. 3.3, for the optical sample. The similaritybetween the hottest four temperature bands is quite close, with only the coolest systems showing a deviation fromthe general trend, and then only within∼0.1R200. The dark matter is the most self-similar of the three masscomponents, as a consequence of its insensitivity to the types of heating and/or cooling processes which can

3.4. RESULTS: SPATIAL MASS DISTRIBUTION 55

Figure 3.1: The variation of stellar matter density with scaled radius, for theoptical sample, grouped by system temperature. The solidline represents the coolest systems (0.3–2.0 keV), increasing in temperature through dashed (2.0–2.9 keV), dotted (2.9–4.6 keV), dot-dashed(4.6–8.5 keV) and finally dot-dot-dot-dashed (8.5–17 keV).

Figure 3.2: The variation of gas density with scaled radius,for the full sample, grouped by system temperature. The solid line representsthe coolest systems (including the two galaxies) (0.3–1.3 keV), increasing in temperature through dashed (1.3–2.9 keV), dotted (2.9–4.6 keV),dot-dashed (4.6–8 keV) and finally dot-dot-dot-dashed (8–17 keV).


Figure 3.3: The variation of dark matter density with scaledradius, for theoptical sample, grouped by system temperature. The solid linerepresents the coolest systems (0.3–2.0 keV), increasing in temperature through dashed (2.0–2.9 keV), dotted (2.9–4.6 keV), dot-dashed (4.6–8.5 keV) and finally dot-dot-dot-dashed (8.5–17 keV).

influence baryonic material. Despite this, it is clear that there is evidence of an enhanced central density in thecore of the average density profiles of the coolest groups. Wereturn to this unusual behaviour in more detail insection 3.5.5.

3.4.4 Total density

To provide some comparison with the density profiles of the separate mass components we have plotted the profilesof integrated overdensity in Fig. 3.4, i.e. the mean densitywithin a given radius, normalized to the critical densityof the Universe. Since the gravitating mass profile is determined from the X-ray data, Fig. 3.4 incorporates ourwhole sample, including the two early-type galaxies.

The dark matter accounts for the majority of the gravitatingmass, so it is not surprising that the trend seen inFig. 3.3 are mirrored in Fig. 3.4. The difference between thedark matter and total mass density seen in the secondcoolest bin can be attributed to the slightly different temperature ranges in each case, used for reasons explainedpreviously.

Although the total density profiles in Fig. 3.4 are essentially self-similar in the outer regions, there is still areasonable degree of scatter. We have comparedR200 with two different radii of overdensity, for each object inour X-ray sample: we have selectedR2500andR500, which are often used in the literature (e.g. Allen et al., 2001b;Finoguenov et al., 2001b), since X-ray haloes are more readily traceable out to these radii than toR200. We findthat, on average, an overdensity of 2500 and 500 correspond to 31 and 66 per cent ofR200, respectively, withcorresponding standard deviations of 5 and 4 per cent.

3.4.5 Mass-to-light ratio

The extent to which light traces masswithin virialized haloes can be gauged by studying the mass-to-light ratio asa function of radius in these systems. Fig. 3.5 shows these profiles for the optical sample, grouped by temperatureas before. It can be seen that there is evidence of decrease inthis quantity with increasing radius, outside of∼0.1R200, in four of the five bands. Interestingly, the central range (2.9–4.6 keV) shows a clearincreasein M/Lin this region. Otherwise, these results demonstrate that the distribution of optical light is more extended than thatof the gravitating mass, as also found by David et al. (1995).It is worth noting that the simulations of Metzler

3.5. RESULTS: INTEGRATED SCALING PROPERTIES 57

Figure 3.4: Spatial variation of cumulative total density –normalized to the critical density of the Universe – with scaled radius, for thefullsample, grouped by system temperature. The solid line represents the coolest systems (including the two galaxies) (0.3–1.3 keV), increasingin temperature through dashed (1.3–2.9 keV), dotted (2.9–4.6 keV), dot-dashed (4.6–8 keV) and finally dot-dot-dot-dashed (8–17 keV).

& Evrard (1997) indicate that the stellar distribution can be lessextended than that of the dark matter, due tothe effects of dynamical friction transferring energy awayfrom the galaxies. The trend towards higher centralconcentration in the coolest systems, seen in sections 3.4.3 & 3.4.4, is seen in Fig. 3.5 as a clear central increase inmass-to-light ratio.

We have chosen to truncate the radial scaling in Fig. 3.5 at 0.03R200, rather than 0.01R200, as our opticalluminosities are likely to be unreliable in the core of the halo. This is because our unweighted galaxy densityfitting is insensitive to the excess luminosity associated with a central cD galaxy. However, we would thereforeexpect tounderestimatethe luminosity in the core, which would overestimate the mass-to-light ratio and henceflatten any drop inM/L in the centre. This suggests that the observed trend towardsa decrease in the mass-to-lightratio <

∼ 0.1R200 is probably real.

3.5 Results: Integrated Scaling Properties

Relationships between integrated properties provide an important tool for investigating the similarity betweenhaloes across a range of masses. Although quantities such asluminosity are directly observable, some allowancemust be made for the effects of projection if fair comparisons are to be made between different systems. Cor-respondingly, all our quoted luminosity values (see Table 3.1) are derived from integrating thedeprojectedlightprofiles, using the appropriate normalization calculated in section 3.3.2, to provide a more sensitive probe of scal-ing properties. Where we have compared our results with those from the literature quoted in different photometricbands, we apply the correction factors given in section 3.3.1 to determine equivalent values in theBj band.

3.5.1 Optical luminosity

A useful test of the scaling properties of the stellar distribution is the relationship between optical luminosity andmass. Fig. 3.6 shows theLB,j luminosity withinR200 for the optical sample plotted against the total mass withinR200, as calculated in Chapter 2. We performed an orthogonal distance regression to fit a stright line to the datain log space, using theODRPACK software package (Boggs et al., 1989, 1992), to take accountof errors in bothX and Y directions. The best-fitting relation is logLB,j = (−1.36±1.46)+ (0.93±0.10) logM and the scatter in


Figure 3.5: The variation of mass-to-light ratio (in theBj photometric band) with scaled radius for theoptical sample, grouped by systemtemperature. The solid line represents the coolest systems(0.3–2.0 keV), increasing in temperature through dashed (2.0–2.9 keV), dotted(2.9–4.6 keV), dot-dashed (4.6–8.5 keV) and finally dot-dot-dot-dashed (8.5–17 keV).

the data is 1.21 times that expected from the statistical errors alone. The logarithmic slope of this relation is flatterthan, but just consistent with self-similarity (i.e.LB,j ∝ M), at the 1σ level – this reflects the slight excess in stellardensity in the coolest systems referred to in section 3.4.1.

For comparison, we have also plottedLB,j against the emission-weighted temperature, as calculatedin Chap-ter 2, which we show in Fig. 3.7. Using the same regression technique, the best-fitting relation is logLB,j =(11.02± 0.01)+ (1.62± 0.14) logkT, and the scatter in the data is 0.96 times that expected from the statisticalerrors alone. Furthermore, there is no evidence of any steepening in the relation or any other systematic deviationfrom a simple power law, indicating that this provides an accurate description of the data. The logarithmic slope ofthis relation issteeperthan, if just consistent with self-similarity (i.e.LB,j ∝ T1.5, assuming that light traces mass),at the 1σ level. However, this is due to the effect of theM−TX relation slope – which is itself significantly steeperthan self-similar.

The scatter about theLB,j −TX relation is rather less than that about theM−TX relation, especially consideringthat our mass errors are quite conservative (Chapter 2). This is puzzling, but may reflect the fact that the opticalluminosity is dominated by the stellar contribution from the inner regions, where the gas temperature is also moreheavily weighted. This would produce a tighter correlationbetweenLB,j andkT than betweenLB,j andM, sincethe total mass is less sensitive to the contribution from theinner regions of the halo.

3.5.2 Mass-to-light ratio

Since light is easily observed, it is often used as a tracer ofmass in the Universe, expressed as the ratio of totalmass to optical luminosity in some photometric band. Assuming that clusters of galaxies are a fair representationof mass composition on large scales, the mass-to-light ratios in these systems can be used to estimate the totalmass density of the Universe. It is therefore important to understand the scaling properties of this quantity if suchcosmological inferences are to be unbiased.

Fig. 3.8 shows the variation in mass-to-light ratio with X-ray temperature, for the optical sample (see Table 3.1for the data). It can be seen that there is little evidence of any trend in the data; Kendall’s K statistic indicates thesignificance of a negative correlation is 0.37σ. The best-fitting power law has a logarithmic slope of−0.06±0.17,which is fully consistent with no trend. The level of scatterabout the best-fit is exactly that expected from thestatistical errors. We therefore conclude that a universalmass-to-light ratio in groups and clusters provides a good


Figure 3.6: Total gravitating mass as a function ofLB,j luminosity (both withinR200), for the optical sample. The line shows the best-fittingpower-law, which has a logarithmic slope of 0.93±0.10.

Figure 3.7: TotalLB,j luminosity within R200 as a function of system temperature for the optical sample. The line shows the best-fittingpower-law, which has a logarithmic slope of 1.62±0.14.


Figure 3.8: Mass-to-light ratio withinR200, in theBj band, as a function of system temperature for the optical sample. The solid line representsthe best fitting power law, which has a logarithmic slope of−0.06±0.17. The dotted line shows the median value for the whole sample, of249h70 (M/LB,j)⊙.

description of most of our data.It can be seen that one point in particular is very high on thisrelation: the poor cluster AWM 7 has(M/LB,j) ∼

1600(M/LB,j)⊙ albeit with rather large errors. A large mass-to-light ratio was also measured for this system byKoranyi et al. (1998), who found a value of(650± 150) h100 (M/L)⊙ for the R band, corresponding to(830±190)h70 (M/LB,j)⊙, consistent with our 1σ lower bound of∼650. However, there is quite a large amount of dustextinction (0.5 magnitudes in theB band) in the direction of AWM 7 (Schlegel, Finkbeiner, & Davis, 1998), whichmay account for its unusually low luminosity, although a correction for this absorption was applied in the originalanalysis. Exclusion of this point lowers the mean of our sample to 276±32h70 (M/LB,j)⊙. We also note that themass-to-light ratio for HCG 62 is rather low – only 78±40, which is rather less than the value esimated in Ponman& Bertram (1993). As was mentioned in section 3.3, there is a factor of two difference between the apertureluminosities quoted in Girardi et al. (2002), based on APS and COSMOS data for this system. Furthermore, wenote that the optically-determined virial radius used for their aperture is twice as large as our X-ray measuredR200, which means that we have toreducethe Girardi et al. (2002) luminosity value, according to ourfitted galaxydensity profile.

To minimize the bias caused by such discrepant measurementsin ourM/L data, we have evaluated alogarith-mic mean value, of 243+33

−29h70 (M/LB,j)⊙. This is significantly smaller than the ordinary, arithmetic mean valueof 318±52, but compares very well with the median value of 249h70 (M/LB,j)⊙. This illustrates the effect thatanomalously largeM/L systems in particular can have, when determining a representative average.

In a combined X-ray/optical study of 12 Abell clusters of galaxies, Cirimele et al. (1997) measured a meanmass-to-light ratio equivalent to 346h70 (M/LB,j)⊙, which is consistent with our results. More recently, Hradeckyet al. (2000) have also determinedM/L for a sample of 8 groups and clusters of galaxies, correctingtheir opticalluminosities for projection effects, by assuming a spatialdistribution of light identical to that they measured forthe X-ray gas. Their mean value ofM/L is equivalent to 310±45 (M/L)⊙ in theBj band, which is in excellentagreement with our own arithmetic mean. Furthermore, theirmedianM/L of ∼250 is indistinguishable from ourown median value. Hradecky et al. (2000) also conclude thatM/L is roughly independent of mass, as was alsofound by David et al. (1995), albeit based on a rather small sample in both cases. We note however, that theirassumption that the optical light traces the gas mass is not supported by our data, although this is likely to haveonly a small effect on their results.


The agreement withM/L as measured using optical mass estimates is generally less good, as pointed outby Hradecky et al. (2000). For example, Girardi et al. (2000)quote a mean value of∼175h70 (M/LB,j)⊙ fortheir sample of 105 clusters, from which many of our own luminosity estimates have been taken (since many oftheir results were incorporated in Girardi et al. (2002)). However, the disagreement with our own mean value of318± 52h70 (M/LB,j)⊙ can be attributed to a difference in luminosity as well as mass: we have extended theirquoted apertureLB,j values, according to our fitted galaxy density profiles, and evaluated an integrated luminositywithin R200 as derived from our X-ray data. On lower mass scales, optically-derivedM/L measurements are alsolower than our own value. Ramella, Pisani, & Geller (1997) find a mean approximately equivalent to 240±90h70

(M/LB,j)⊙ (99% confidence) for a very large sample of groups in the Northern CfA Redshift Survey, using virialmass estimates.

The difference between X-ray and optically derived masses can also be seen in the scaling properties ofM/L.While our results support a universal mass-to-light ratio in groups and clusters – consistent with the X-ray studiesof Hradecky et al. (2000) and Cirimele et al. (1997) – Girardiet al. (2002) find that mass increases more quicklythan luminosity, such thatM ∝ L1.34±0.03

B . A rising M/L with temperature was found by Bahcall & Comerford(2002) and the weak lensing analysis of Hoekstra et al. (2001) also concluded that group mass-to-light ratios werelower than those of clusters. However, the optical study of Carlberg et al. (1996) indicates thatM/L is universal inclusters, although their mean value, equivalent to 380±70h70 (M/LB,j)⊙ is somewhat higher than our own.

Semi-analytical models (SAMs) of galaxy formation generally predict a significant increase inM/L with halomass (e.g. Kauffman et al., 1999; Benson et al., 2000; Somerville et al., 2001). As an example of the typical be-haviour observed, Benson et al. (2000) find thatM/L reaches a minimum on mass scales of∼ 1012M⊙, increasingby a factor of 3 up to halo masses of 1015M⊙. This corresponds to a logarithmic slope of 0.16, which is only justoutside the 1σ upper bound of 0.13 from our data in Fig. 3.8, although we onlyinclude groups and clusters in ouroptical sample. This tendency forM/L to increase away from a minimum on the scale of a typical galaxy is alsoconfirmed by Marinoni & Hudson (2002). Semi-analytical models also predict thatM/L in clusters is significantlylower than the global value (Kauffman et al., 1999), suggesting that even massive clusters are a biased estimatorof the meanM/L in the Universe. That our findings do not provide a closer match to theM/L predictions ofthese models may simply reflect the over-cooling of gas in SAMs, which is a well-known problem that leads to theformation of excessively bright central galaxies, for example (Kauffmann et al., 1993).

3.5.3 Star formation efficiency and gas loss from haloes

To address the issue of star formation efficiency, we have examined the star-to-baryon ratio as a measure of theeffectiveness with which gas has been converted into stellar material. Fig. 3.9 shows this quantity plotted againstsystem temperature for the optical sample. It can be seen that there is some evidence of negative correlation, whichis significant at the 3.1σ level. However, this trend is potentially misleading, since it may be due at least in partto the variation in gas mass with temperature seen above. This means that some allowance must be made for thepossibility that gas may be lost to the system beyond the virial radius, i.e. the gas mass may be reduced withoutany corresponding increase in the stellar mass.

A better discriminator of star formation efficiency is obtained if the stellar mass is normalized to the mass ofdark matter, since this component is largely immune to bias from non-gravitational processes. We have plotted thisratio against X-ray temperature in Fig. 3.10. It is clear that the trend seen in Fig. 3.9 has largely vanished, leavingonly 0.8σ evidence for a systematic variation in star formation efficiency with halo temperature.

To identify the cause of the trend in Fig. 3.9, we have also plotted the ratio of gas to dark matter mass, shownin Fig. 3.11. A positive correlation between this ratio and the X-ray temperature is significant at the 3.3σ level.This confirms that the variation in gas fraction with temperature does indeed fully account for the trend found inthe stellar-to-baryon ratio.

The apparent self-similarity of the stellar distribution with respect to the dark matter suggests that star formationis essentially equally efficient in both groups and clusters. However, the behaviour of the gas to dark matter ratiopoints to gas loss in the cooler systems, albeit subject to significant extrapolation of the data. Together, these resultsare more suggestive of non-gravitational heating as the likely mechanism responsible for the observed breaking ofself-similariy in virialized systems, since this would more naturally account for depletion of gas in cooler systemswithout a corresponding enhanced stellar mass. The alternative, radiative cooling, would only be able to reduce gasmass by associated star formation, unless it is able to form baryonic dark matter – for example, molecular clouds(e.g. Pfenniger et al., 1994).


Figure 3.9: The fraction of baryons in the form of stars, within R200, as a function of system temperature, for the optical sample.

Figure 3.10: The ratio of stellar to dark matter mass withinR200, as a function of system temperature, for the optical sample.


Figure 3.11: The ratio of gas to dark matter mass withinR200, as a function of system temperature, for the optical sample.

As an example of the behaviour expected from cooling, we notethat the galaxy formation-regulated gas evo-lution model proposed by Bryan (2000) (and extended by Wu & Xue, 2002b) predicts a variation in the stellarmass fraction of the formfstar= 0.042(T/10)−0.35. In this model, the total baryon fraction is constant for groupsand clusters, but increased (decreased) cooling in lower (higher) mass systems leads to a lower (higher)fgas andhigher (lower)fstar. The corresponding relation for our data isfstar= (−1.63±0.08)(T/10)0.04±0.18, which showsessentially no trend, albeit with a large error on the logarithmic slope, which is only inconsistent with the Bryanvalue at the 2σ level.

3.5.4 Baryon fraction and constraints onΩ0

Using our measured stellar and gas mass data, we are able to determine the fraction of mass in baryons,fb, forour optical sample, which we show plotted against X-ray temperature in Fig. 3.12. There is evidence of a modesttrend in the data (2.5σ significance). The best-fitting power law is represented by the solid line, and is givenby log fb = (0.37± 0.15) logkT − (0.99± 0.10); the median value of 0.161 is indicated by the dotted line. Ifvirialized systems constitute a fair sample of the baryon content of the Universe, we can use this median valueto constrain the total mass density,Ω0 = Ωb/ fb. Assuming a baryon density ofΩb = 0.04± 0.008h70

−2, frommeasurements of deuterium abundances in high redshift hydrogen clouds (Burles et al., 2001; O’Meara et al.,2001), we infer a value ofΩ0 = 0.25h−1

70 . This agrees well with the results of Hradecky et al. (2000).If weimprove our statistics by combining our mean stellar fraction, of (0.034±0.006), with our mean gas fraction for

the full X-ray sample (Chapter 2), of(0.134±0.01)h−3/270 , fb increases to 0.168 andΩ0 drops to 0.23. Assuming

an unbiased measurement of the stellar and gas fraction, this represents a upper limit onΩ0, since any baryoniccomponent to the dark matter would increasefb.

This limit on Ω0 is at odds with the value of 0.30+0.04−0.03 inferred by Allen et al. (2002), based on a Chandra

analysis of six massive lensing clusters. The discrepancy arises partly from a difference in gas fraction and partlyfrom Allen et al.’s choice of stellar to gas ratio. They use a value of 0.19h0.5 for the latter quantity (Fukugitaet al., 1998; White et al., 1993), which is rather less than our measured value of 0.30± 0.04. Moreover, theirmean measured gas fraction is 0.113±0.005, compared to our mean of 0.134±0.01 for the whole X-ray sample(Chapter 2). The cause of the discrepancy in both cases is thescaling behaviour of the gas fraction – we findevidence of a rise in gas fraction with radius, whereas Allenet al. report a substantially flatfgasprofile in a number


Figure 3.12: Baryon fraction withinR200 as a function of system temperature, for the optical sample.The solid line represents the best-fittingpower law, which has a logarithmic slope of 0.37±0.15. The dotted line marks the median value of 0.162.

of their clusters, albeit restricted to a radius of overdensity of R2500. By extrapolating a constant value out toR200,no allowance is made for an increase infgas, as is predicted by numerical simulations, even when the effects ofpreheating and radiative cooling are absent (Eke et al., 1998; Frenk et al., 1999).

Furthermore, we find evidence of a decrease infgas in cooler systems, which has the effect of lowering ourmean value. Given the possible impact of non-gravitationalheating on low mass systems, the gas fraction in richerclusters is likely to be a better indicator of the universal value. If we calculate an average gas fraction for ourhottest clusters (>5 keV), we obtain an even higher value, of 0.17±0.01. Combined with our mean stellar fractionfrom above, this places a more stringent upper limit on the mass density, ofΩ0 ≤ 0.20h−1

70 .

3.5.5 Central density concentration

Although the underlying gravitational potential in virialized systems is expected to be self similar, simulationsindicate that the concentration of the dark matter should vary slightly with mass (Navarro et al., 1997). It was shownin Figs. 3.3 and 3.4 that the coolest systems in our sample do indeed appear to be more centrally concentrated – aneffect which is most pronounced in the central∼10 per cent ofR200.

To understand what is driving this behaviour, we have examined the scaling properties of the constant densitycore in the gas distribution,rc, as measured from the X-ray data in Chapter 2. The left panel of Fig. 3.13 showsrc asa fraction ofR200 plotted against system temperature. For the hottest clusters the points scatter about a self-similarmean value of roughly 10 per cent. As the temperature decreases, however, the scatter increases and<

∼1 keV thereis a very sharp drop. This trend is in excellent agreement with the predictions of the galaxy formation-regulatedgas evolution model of Bryan (2000), as studied in detail by Wu & Xue (2002b): there is a remarkable similaritybetween Fig. 3.13 and the model and data points in fig. 8 from the latter paper.

To minimize the possibility of systematic bias in this result, we have identified and excluded from Fig. 3.13those systems for which the core radius may be unreliable, owing to the presence of a central cooling emissionexcess. In 17 cases,rc was measured to besmallerthan the either the size of the radius used to excise the coolingflow or the radius within which a cooling flow component was fitted. In addition, the cluster Abell 2218 wasexcluded, since it was necessary to fix its core radius at the best-fitting value in order to stabilize the fitting duringthe calculation of parameter errors (Chapter 2), leaving a total of 48 systems plotted in Fig. 3.13. To suppress thescatter in the relation, we have grouped the points togetherto a minimum of four points per bin, giving a total of

3.6. DISCUSSION 65

Figure 3.13: The ratio ofrc to R200 as a function of temperature. The left panel shows all the individual points; the right panel shows the datagrouped to a minimum of four points per bin. The axes in plots have been scaled identically. The diamonds represent those systems where noerrors are available onrc as the parameter was poorly constrained during the fitting (denoted by an asterix in column 8 of table 2.1 in Chapter 2).

12 bins. The errors on each point are determined from the scatter in the X and Y directions. This plot is shown inthe right panel of Fig. 3.13. The underlying trend in the datais much clearer – it can be seen that the coolest bin isat least 10σ lower than the flat relation established by the six hottest bins.

It is difficult to explain this behaviour in terms of a fitting bias: had these coolest systems exhibited X-raycore radii consistent with the cluster trend, i.e.>

∼3 times larger than observed, they would very easily have beendetected. One possibility is that the values ofR200 are anomalously large. To test this we have plottedR200 againsttemperature in Fig. 3.14. The slope of this relation is somewhat steeper than the self-similar prediction of 0.5,but there is clearly no evidence of a trend towards an unusually large R200 in the cooler systems, which couldaccount for the discontinuity observed inrc/R200. The most clearly discrepant point at the low-mass end is theS0galaxy NGC 1553 (the coolest system in our sample), but it appears to have a value ofR200 which is lower thanits temperature would suggest. However, this object has unusual properties, which point to an anomalously hightemperature, which may have been boosted by energy injection from stellar winds (see Chapter 2).

3.6 Discussion

3.6.1 Implications for heating/cooling

Our results confirm the systematic breaking of self-similarity in the IGM that were observed in the gas fraction inChapter 2. Clearly the thermal history of the hot gas has beenaltered by the influence of non-gravitational physics.The most promising candidates are radiative cooling (Bryan, 2000; Muanwong et al., 2001) and energy injectionby heating (e.g. Valageas & Silk, 1999). Both mechanisms areable to account for the X-ray observations: moreefficient cooling in denser (smaller) haloes leads to a depletion of gas in the inner regions; the energetically boostedIGM is only weakly captured in the shallower potential wellsof less massive virialized systems, thus reducing theirgas fraction.

The cooling hypothesis (cf. Knight & Ponman, 1997) has received increased attention lately, particularly inrelation to the observed entropy properties of virialized systems – indeed our own results (see Chapter 4) are con-sistent with some of the predictions of such models, when theeffects of feedback from associated star formationare allowed for (Voit & Bryan, 2001b). However, the natural consequence of increased gas cooling is enhancedstar formation, unless a substantial reservoir of baryons exists in the form of less luminous matter – e.g. molecularclouds (Edge, 2001). The stellar properties of our sample ofvirialized systems appear consistent with the predic-tions of self-similar scaling, and we find no evidence for a significant variation in star formation efficiency acrossa wide range of halo masses. Therefore, we conclude that heating must have had at least some role to play insystematically modifying the properties of the IGM.


Figure 3.14: The variation ofR200 as a function of temperature. The solid line depicts the bestfitting linear relation, with a slope of 0.60±0.03.

3.6.2 Halo formation epoch

From both the dark matter and total mass density profiles, it is clear that there is an enhancement in the centralconcentration in the very coolest systems in our sample. This behaviour is consistent with hierarchical structureformation (e.g. Blumenthal et al., 1984), in which the smallest haloes collapse at the earliest epochs: the higherdensity of the Universe at this time results in a higher central density. Such a systematic variation in the epoch offormation of virialized systems has previously been observed (Sato et al., 2000).

Another possibility is that the greater ages for these systems allow more time for accumulation of extra materialon to their haloes, via accretion (Salvador-Sole et al., 1998). This process would have the effect of increasing thevirial radius, but without significantly modifying the characteristic turn over radius in the gas density profile, i.e.the X-ray core radius as a fraction ofR200 would shrink, as observed in section 3.5.5. Although small haloes arestill able to form at the present epoch, there may be an intrinsic bias towards observing older, more relaxed systems,which are likely to be brighter – indeed, we selected our X-ray sample principally on the basis of a regular X-raymorphology. Thus we might expect to find fewer examples of cool systems with large a core, compared to theirvirial radius. This may account for the absence of points around the cluster trend ofrc/R200≈ 0.1 below∼1 keVin Fig. 3.13.

The formation epoch of virialized haloes may play quite a significant role in influencing their scaling properties.This is particular true for nearby systems, where the redshift of observation,zobs, is a poor and systematically biasedestimator of the redshift of formation,zf . The trend for less massive haloes to be older may account forthe observedsteepening of theM−TX relation (see Chapter 2), in contrast to the apparently self-similar slope found in moredistant, massive clusters (Allen et al., 2001b), wherezobs is a much better measure ofzf . However, the simulationsof Mathiesen (2001) indicate that this process has a negligible effect on the temperatures of those clusters whichhave assembled 75 per cent of their final mass byz= 0.6.

3.7 Conclusions

We have conducted a detailed study of the mass composition ofa large sample of virialized systems. We haveX-ray data for our whole sample of 66 objects, and have used these to derive the gas and total gravitating massprofiles as a function of radius. Optical data for a subsampleof 32 groups and clusters have allowed us to determine

3.7. CONCLUSIONS 67

the stellar mass distribution, and thus infer the dark matter density profile.We have determined the deprojected luminosity distribution from unweighted surface density fits to galaxy

positions from the APM survey, combined with aperture luminosity measurements taken from the literature. Usingthe galaxy density profile we are able to extrapolate the light to our nominal virial radius ofR200 (as determinedfrom the X-ray data) to yield thedeprojectedoptical luminosity, in theBj photometric band.

We find clear evidence of a departure from self-similarity inthe properties of the gaseous intergalactic medium,in the direction of a decrease in gas density with decreasingtemperature. Our observations of the luminositystructure of virialized haloes indicate that the stellar composition scales approximately self-similarly across thesample. Our data provide 1σ evidence for an increase in the efficiency of star formation in galaxy groups, althoughthis quantity is constant across most of the sample. We find a logarithmic meanM/L of 243+33

−29h70 (M/LB,j)⊙, ingood agreement with other measurements based on X-ray mass estimates.

The dark matter and total mass density profiles of our sample are nearly self-similar, but for a clear centralexcess in the coolest systems (<

∼1.5 keV). We attribute this enhancement in central density concentration to asharp decline in the size of the gas core radius,rc (normalized toR200) amongst the coolest∼8 systems.

We measure a mean stellar mass fraction of 0.032±0.004 and a median baryon fraction, for our optical sample,

of 0.161h−3/270 . This allows us to place an upper limit of the mass density of the Universe ofΩ0 ≤ 0.25h−1

70 , thoughwe argue that lower values,Ω0 ≤ 0.2h−1

70 , are indicated by our data. Our results favour energy injection by non-gravitational heating as a contributary mechanism for explaining the observed breaking of self-similarity in theIGM, since radiative cooling alone would lead to an significant increase in star formation efficiency in groups, incontrast to our findings.

As improved X-ray mass estimates are obtained, withXMM-Newtonand Chandra, and the technology ofwide-field optical CCD photometry continues to advance, theprospects for very accurate determination of themass-to-light ratio and baryon content in virialized systems look good.

Chapter 4

Entropy and Similarity inGalaxy Systems

Abstract We examine profiles and scaling properties of the entropy of the intergalactic gas in a sample of 66 virialized systems,ranging in mass from single elliptical galaxies to rich clusters, for which we have resolved X-ray temperature profiles.Some ofthe properties we derive appear to be inconsistent with any of the models put forward to explain the breaking of self-similarityin the baryon content of clusters. In particular, the entropy profiles, scaled to the virial radius, are broadly similar in form acrossthe sample, apart from a normalization factor which differsfrom the simple self-similar scaling with temperature. Lowmasssystems do not show the large isentropic cores predicted by preheating models, and the high entropy excesses reported atlargeradii in groups by Finoguenov et al. (2002) are confirmed, andfound to extend even to moderately rich clusters. We discusstheimplications of these results for the evolutionary historyof the hot gas in clusters.

4.1 Introduction

In the widely accepted standard model for cosmic structure formation, the Universe evolves hierarchically, asprimordial density fluctuations, amplified by gravity, collapse and merge to form progressively larger systems.This hierarchical development leads to the prediction of self-similar scalings between systems of different massesand at different epochs. These scalings are also seen in cosmological simulations involving only gravitationallydriven evolution, including compression and shock heatingof the baryonic matter. Such simulations (e.g. Navarroet al., 1995; Frenk et al., 1999) result in haloes in which thedensity profiles of both dark matter and baryonicmaterial, when radially scaled to the virial radius (which we define here to be the radiusR200, within which themean density of a system is 200 times the critical density of the Universe) are almost identical in virialized systemscovering a wide range of masses, from individual galaxies torich clusters.

Given self-similar scalings of gas temperature and density, scaled X-ray surface brightness profiles are alsoexpected to be similar. Furthermore, a simple scaling is expected between X-ray luminosity,LX , and temperature.Assuming that the emission is dominated by bremsstrahlung,LX ∝ M2

gasR−3T1/2, or LX ∝ f 2

gasT2, where fgas=

Mgas/M is the gas mass fraction. X-ray properties of clusters deviate substantially from this simple scaling, andthe observedL−TX relation (White et al., 1997; Markevitch, 1998) is considerably steeper thanT2 in the clusterregime, and steepens further (Helsdon & Ponman, 2000a) in galaxy groups. Ponman et al. (1999) showed that thelatter effect is due to the suppression ofLX in galaxy groups, arising from a reduction in gas density in the innerregions of poor systems, relative to richer clusters.

It is instructive to view this in terms of the entropy of the intergalactic medium (IGM), which in the self-similar case should increase in a very simple scaling with the mean temperature of virialized systems. In practice,it is found (Ponman et al., 1999; Lloyd-Davies et al., 2000) that an excess in the entropy, above the self-similarprediction, is apparent in the inner regions of poor clusters and groups, outside the dense central regions in whichcooling is expected to radiate away entropy within the age ofthe Universe. This effect has been referred to as the‘entropy floor’, with the implication that additional physical processes, beyond gravity and resulting compressionand shock heating, have acted to set a lower limit to the entropy which the gas in collapsed haloes can have.

A great deal of theoretical work has been devoted to explaining this phenomenon over the past few years. Aswe will discuss in some detail later, the explanations proposed fall into three main classes: the gas has been heatedeither at an early epoch, before clusters were assembled (Kaiser, 1991; Evrard & Henry, 1991; Cavaliere, Menci, &Tozzi, 1997; Balogh, Babul, & Patton, 1999; Valageas & Silk,1999; Tozzi & Norman, 2001), or it has been heatedin situby star formation and/or energy input from active galactic nuclei (AGN) (Bower, 1997; Loewenstein, 2000;

68

4.2. SAMPLE AND ANALYSIS 69

Voit & Bryan, 2001b; Nath & Roychowdhury, 2002). Alternatively, some authors have argued (Knight & Ponman,1997; Bryan, 2000; Pearce et al., 2000; Muanwong et al., 2001; Wu & Xue, 2002a; Dave et al., 2002) that coolingalone will remove low entropy gas from the centres of haloes,producing a very similar effect to non-gravitionalheating.

In the present paper we aim to confront these models with the observed properties of the hot gas in a large sam-ple of galaxy systems, spanning a wide range in total mass. Inthe fullness of time, high quality X-ray observationsof the density and temperature structure of the IGM will be available fromXMM-NewtonandChandra. However,at present, such observations are sparse, and it is essential to have a broadly representative and wide-ranging sam-ple of virialized systems in order to study scaling properties. The value of this in the context of similarity-breakingin clusters has already been shown by a number of earlier studies.

In the present, and companion papers, Sanderson et al. (2002) (Chapter 2) and Sanderson & Ponman (2002a)(Chapter 3), we examine scaling properties derived from thelargest sample of virialized systems with resolved X-ray temperature profiles yet assembled. Following a brief description of the sample and our analysis in section 2.2(details are given in Chapter 2), we present the profiles and scaling properties for the entropy and temperatureacross our sample in sections 4.3 and 4.4. These results are used in section 2.7, along with relevant results fromChapters 2 and 3, to test the various models proposed to account for the entropy floor, and finally in section 4.6 wedraw our conclusions from this study.

4.2 Sample and Analysis

Our sample comprises 66 virialized systems, from rich clusters of galaxies, through groups and down to the level ofindividual galaxy-sized haloes. In Sanderson et al. (2002,hereafter Chapter 2), we reported a detailed study of the3-dimensional X-ray properties of this sample, based on data from theROSATandASCAobservatories, which weassembled from the work of three separate investigators (Markevitch et al., 1998; Markevitch, 1998; Markevitchet al., 1999; Markevitch & Vikhlinin, 1997; Markevitch, 1996; Finoguenov & Ponman, 1999; Finoguenov & Jones,2000; Finoguenov et al., 2000, 2001a; Lloyd-Davies et al., 2000), combined with a number of cool groups analysedspecially to provide better coverage of the crucial low end of the mass range. To each system we fitted analyticalfunctions, describing the gas density and temperature variation with radius, outside any central cooling region,which was excised or fitted separately. This approach allowsus to put the X-ray data from the three earlier studieson a unified footing, and gives us the freedom to extrapolate the gas properties to arbitrary radius. We used a betamodel to parametrize the density, and specified the temperature variation with either a linear ramp or a polytropicIGM. We have used these data to determine the gravitating mass profile and thus to calculate radii of overdensityin a self-consistent manner. Similarly, we have derived mean temperatures for each system, by averaging the gastemperature within 0.3R200, weighted by its luminosity (Chapter 2).

4.3 Entropy and Temperature Distributions

For convenience, we define ‘entropy’ in the present paper as

S= T/n2/3e keV cm2, (4.1)

which relates directly to observations. This has been referred to by a number of authors as the ‘adiabat’, since (apartfrom a constant relating to mean particle mass) it is the coefficient relating pressure and density in the adiabaticrelationshipP = Kργ. HenceS is conserved in any adiabatic process. Note that the true thermodynamic entropy isrelated to our definition via a logarithm and additive constant.

In Fig. 4.1, we overlay the scaled entropy profiles for all 66 systems in our sample. Under the assumption thatall these systems form at the same redshift, their mean mass densities should be identical. Hence in the simpleself-similar case, where all have similar profiles and identical gas fractions,Swill simply scale with mean systemtemperatureT. We apply this scaling, and scale the radial coordinate toR200 for each system, derived from ourfitted models (Chapter 2). It can be seen that the entropy profiles of the cooler systems, scaled in this way, tendto be significantly higher than those of richer clusters. To see this more clearly, in the right panel of Fig. 4.1, weshow the profiles grouped into bands of mean system temperature. In this grouped plot we have excluded the twogalaxies, whose profiles may be established by stellar wind losses, rather than by the processes operating in groupsand clusters.

70 CHAPTER 4. ENTROPY AND SIMILARITY IN GALAXY SYSTEMS

Figure 4.1: Entropy profiles for each system in our sample (left panel) derived from our fitted models, each scaled by 1/T. The line style ofeach profile denotes the mean temperature of the system, as described below. In the right panel, these profiles (excludingthe two galaxies, asdiscussed in the text) have been grouped into temperature bands: 0.3–1.3 keV (solid), 1.3–2.9 keV (dashed), 2.9–4.6 keV(dotted), 4.6–8 keV(dot-dash) and 8-17 keV (dot-dot-dot-dash). The bottom line shows the slope of 1.1 expected from shock heating. Its normalization is arbitrary.

It can be seen that there is a strong tendency for the scaled entropy to be higher, at a given scaled radius, incooler systems. Simulations and analytical models of cluster formation involving only gravity and shock heating,produce entropy profiles with logarithmic slopes of approximately 1.1 (Tozzi & Norman, 2001), which agreesrather well with the slope of our profiles outside 0.1R200. The mean profile in the 2.9–4.6 keV band is significantlyaffected by two systems, visible in the left hand panel, which have strongly rising profiles. One of these is AWM7,for which the temperature profile is subject to especially large systematic uncertainties, as a result of the strongcooling flow, as discussed in Chapter 2. The temperature model adopted, from the analysis of Lloyd-Davies et al.(2000), rises strongly with radius. As a result, the slope ofS(r) in this band is almost certainly overestimated inthe figure.

At smaller radii, our fitted models exclude the effects of cooled gas, since any central cooling region is excised,or represented by a separate component in our analysis (section 4.2). Apart from the coolest systems, our entropyprofiles generally flatten inside 0.1R200. This effect is also seen in many cosmological simulations (Frenk et al.,1999), even in the absence of non-gravitational heating andcooling processes, and appears to result from theintroduction of a core into the gas density distribution, due to transfer of energy between baryonic and dark matterduring merger events (Eke et al., 1998). Preheating models generally predict large isentropic central regions in lowmass systems, which are not seen in our data. We will return tothis point in section 4.5 below.

The preheating model of Dos Santos & Dore (2002) predicts that entropy should scale according to(1+T/T0),rather thanT, whereT0 is a constant related to the degree of preheating, which theyestimate asT0 = 2 keV, toprovide a best fit to the entropy floor data of Lloyd-Davies et al. (2000). In Fig. 4.2, we show the effect ofthis scaling on our temperature-grouped entropy profiles. This scaling does indeed bring the profiles into goodagreement, apart from the fact that our coolest systems showlittle sign of any central entropy core. Note from theleft hand panel in Fig. 4.1, that this is a general feature of almost all the entropy profiles for cool systems, ratherthan a result of averaging together systems with cores and others with strongly dropping central entropies.

It is also instructive to compare temperature profiles across the range of masses in our sample. We use the virialradii and masses calculated from our fitted models to computethe virial temperature

T200=GMµmp

2R200, (4.2)

expressed in keV. This is used to normalize each temperatureprofile, and the scaled profiles are grouped intemperature bands, to reduce the large scatter and make trends more obvious. The result is shown in Fig. 4.3.As with the grouped entropy plots, we have excluded the two galaxies from these profiles. We have also omittedAWM7 from the 2.9–4.6 keV band, since its highly uncertain (see discussion above) and strongly risingT(r)profile distorts the mean profile for the whole band.

Raising the entropy of the IGM results in increased temperatures as well as lower densities, and for a given levelof entropy increase this effect will be most prominent in lowmass systems, where the ‘natural’ shock-generated

4.4. SCALING PROPERTIES 71

Figure 4.2: The variation of gas entropy (scaled by(1+T/T0)−1) with scaled radius, grouped by system temperature. The solid line represents

the coolest systems (excluding the two galaxies) (0.3–1.3 keV), increasing in temperature through dashed (1.3–2.9 keV), dotted (2.9–4.6 keV),dot-dashed (4.6–8 keV) and finally dot-dot-dot-dashed (8–17 keV). The lower solid line (with arbitrary normalization)indicates the slope of1.1 expected from shock heating.

entropy is lower. As can be seen, our observations do indeed show such an effect at large radii – atR200 the scaledtemperatures are ordered in precisely this way. However, atsmaller radii, this is not the case, and in particular,the coolest systems display very flat temperature profiles, whilst the most massive clusters show the largest risein temperature betweenR200 and the centre. This behaviour is not what is predicted by most models, as we willdiscuss later.

4.4 Scaling Properties

The claim of the discovery of an ‘entropy floor’ in galaxy systems (Ponman et al., 1999) was based on the mea-surement of gas entropy at a scaled radius of 0.1R200, in systems spanning a wide temperature range. This radiuswas chosen to lie close to the centre, where shock-generatedentropy should be a minimum, hence maximisingthe sensitivity to any additional entropy, whilst lying outside the region where the cooling time is less than theage of the Universe, and hence the entropy may be reduced. This initial study was improved by Lloyd-Davieset al. (2000), who avoided the isothermal assumption made byPonman et al., and derived an entropy floor value of

139h−1/350 keV cm2 from a sample of 20 systems, which is essentially a subset of the present work.

In Fig. 4.4, we show the corresponding plot from our much larger sample. With the benefit of this increase insample size, the trend looks rather different from its appearance in Lloyd-Davies et al. (2000), where a dearth ofsystems in the 1.5–3.5 keV band led to the interpretation of arelation which followed the self-similar line down

from hot systems, flattening towards a floor value of 139h−1/350 keV cm2, corresponding, with our value ofH0, to

124h70−1/3 keV cm2. However, it appears from our data that the behaviour is rather that of a slope inS(T) which

is significantly shallower than the self-similar relationS∝ T throughout. Using unweighted orthogonal regression,we obtain a relationS(0.1R200) ∝ T0.65±0.05, which is marked in Fig. 4.4. The Lloyd-Davies et al. floor value liesat the bottom of this trend, but it is not clear whether it setsa lower limit to the entropy in galaxy groups, sinceno groups withT <

∼0.6 keV have been bright enough for detailed study to date. The two galaxies in our sampledo clearly have entropies which lie below the floor level, however much or all of the hot gas in these systems mayhave its origin in stellar mass loss, rather than retained primordial material (O’Sullivan, Forbes, & Ponman, 2001),so it may well be fortuitous that they fall close to the trend set by the groups and clusters.


Figure 4.3: The variation of gas temperature (scaled byT200) with scaled radius, grouped by system temperature. The solid line representsthe coolest systems (excluding the two galaxies) (0.3–1.3 keV), increasing in temperature through dashed (1.3–2.9 keV), dotted (2.9–4.6 keV,excluding AWM 7), dot-dashed (4.6–8 keV) and finally dot-dot-dot-dashed (8–17 keV).

Figure 4.4: Gas entropy at 0.1R200 as a function of system temperature. The diamonds representthe two galaxies. The solid line is the best fitto the barred points. The dotted line has a self-similar slope of 1 and is normalized to the mean of the hottest 8 clusters. The dashed line is theentropy floor of 124h70

−1/3 keV cm2 from Lloyd-Davies et al. (2000).

4.4. SCALING PROPERTIES 73

Figure 4.5: Gas entropy at 0.1R200 as a function of system temperature, excluding the two galaxies and grouped to a minimum of 8 points perbin. Barred crosses are based our measured values ofR200, whilst diamonds result from calculation ofR200 from mean system temperatureusing theT1/2 scaling of Navarro et al. (1995). The dotted line shows the self-similar slope of 1, normalized to the 8 hottest clusters.

Grouping the points into temperature bins (Fig. 4.5) makes the unbroken nature of the trend very clear. Notethat these results suggest the presence of excess entropy of∼100 keV cm2 in all temperature bands, relative tothe hottest systems. This plot also compares the effect on the relation of calculatingR200 from a measurementof the mass profile, compared to assuming a scalingR200 ∝ T1/2, as was done (employing the formula derivedfrom simulations by Navarro et al. (1995)) by Ponman et al. (1999) and Lloyd-Davies et al. (2000). As discussedin Chapter 2, the Navarro et al. formula agrees reasonably well with our measurements in rich clusters, but incool systems, the upward bias in temperature, relative to simulations such as those of Navarro et al. (1995), whichinclude only gravitational physics, causes theR200∝ T1/2 scaling to overestimateR200 leading in turn to an upwardbias in the entropy (sinceSrises with radius). This may have contributed, in previous studies, to the appearance offlattening in theS(T) relation towards lowT.

Whilst measurements close to the cluster centre provide themost sensitive probe of excess entropy, detectionof additional entropy at large radii is especially interesting, since many preheating models predict that the rise inentropy is essentially restricted to those central regionswhere shock-generated entropy is less than the floor valueset by preheating. Finoguenov et al. (2002) were the first to find evidence for excess entropy at a much largerradius,R500, corresponding (Chapter 3) to∼2/3R200. They argued that the high excess entropy observed at largeradii in groups and poor clusters indicates that their IGM isdominated throughout by the effects of preheating,with shocks playing little or no role. We aim, in this paper, to check this result with our larger sample.

Our data, shown in Fig. 4.6, confirm the existence of substantial excess entropy atR500, above a self-similarextrapolation from the values seen in rich clusters (dottedline). The trend is more clearly seen in temperature-grouped data shown in the right hand panel. The effect is apparent not only in the coolest systems, but also inquite rich clusters. In fact, the largest absolute values ofexcess entropy (∼1000 keV cm2) are seen in clusters withT ∼ 3–4 keV.

For direct comparison with Finoguenov et al. (2002), we showin Fig. 4.7 the entropy atR500 scaled byM−2/3500 ,

and grouped into mass bins to suppress fluctuations. For a setof self-similar systems, this scaling should renormal-ize the entropy to a constant value, independent of system temperature (sinceS∝ T ∝ M2/3). Clearly real clustersdo not follow this self-similar law. Whilst our plot is broadly consistent in shape with fig. 3 of Finoguenov et al.(2002), our larger sample again reveals important additional features. Finoguenov et al. concluded that excessentropy was only present in systems withM500<∼1014h−1

50 M⊙ (7×1013M⊙ for our choice ofH0), whereas it is


Figure 4.6: Entropy atR500 as a function of system temperature for each of the 66 systems, with the galaxies marked as diamonds (left panel)and grouped to a minimum of 8 points per bin, excluding the galaxies (right panel). The dotted line shows the self-similarslope of 1, normalizedto the 8 hottest clusters.

clear from our data that a trend in scaled entropy is present across the full mass range of clusters and groups.

Finoguenov et al. (2002) also examined the entropy at a fixed value of enclosed gravitating mass (3×1013h−150 M⊙)

over their sample of systems. The infall velocity of gas intoan accretion shock should be similar to the free fall ve-locity at the shock radius, which depends upon the enclosed mass and mean density of the Universe at the epoch inquestion. In clusters, an enclosed mass of 3×1013h−1

50 M⊙ lies deep within the system, whilst in small groups it canlie close toR200. Since cluster cores are generally assembled at higher redshifts than the outskirts of groups, oneexpects the gas density in the shell under consideration to be higher, and the entropy generated by the thermalisedkinetic energy is therefore lower, by a factor (1+zf ), wherezf is the redshift at which the shell was accreted. Inpractice, Finoguenov et al. (2002) found the entropy of thisshell to be bimodal across their sample, with a typicalvalue of∼300 keV cm2 for systems withT <

∼3 keV, and with considerably lower values (scattered over the range

100–300 keV cm2) in hotter systems. Such a distribution cannot be accountedfor on the basis of a 1+zf scaling,and Finoguenov et al. suggested instead that the entropy in cool systems is set by preheating, taking place atz∼ 3,after many cluster cores had already collapsed.

The corresponding plot for our sample, which is almost double the size of that of Finoguenov et al., is shownin Fig. 4.8. For direct comparison, we have derived entropies at an enclosed mass of 2.14×1013h−1

70 M⊙, allowingfor our different choice of Hubble constant. Our results look significantly different from those of Finoguenov et al.(2002). The entropy appears to be non-monotonic, with a minimum value in systems withT ∼ 3–4 keV. From apurely phenomenological perspective, this non-monotonicbehaviour can be understood in terms of mass profileswhich take the NFW form (Navarro et al., 1997)

ρ(r) =ρs

(r/rs)(1+ r/rs)2 , (4.3)

combined with entropy profiles which rise asS∝ r1.1, outside some flatter central core. In groups, the mass shellunder consideration here lies well outsiders, where enclosed mass grows roughly linearly with radius,

M(< r) ∼CrT3/2200 , (4.4)

so that the radius at which an enclosed mass of 2.14×1013h−170 M⊙ is achieved scales as

r∗ ∝ T−3/2200 . (4.5)

Since the entropy at this radius scales asS(r∗) ∝ T200r

1.1∗ , (4.6)

we expectS(r∗) ∝ T−0.65

200 , (4.7)

which gives a reasonable match to the slope atT < 3 keV, as shown in the right hand panel of Fig. 4.8.

4.5. DISCUSSION 75

Figure 4.7: Gas entropy atR500, normalized byM2/3500, as a function of the total mass withinR500 (excluding the two galaxies) and grouped to a

minimum of 8 points per bin.

In more massive systems (M200 > 2×1014h−170 M⊙, T > 3 keV), the mass shell we are studying moves inside

the scale radiusrs, so that the enclosed mass grows asr2. An analysis similar to that above then results in

S(r∗) ∝ T0.175200 , (4.8)

provided thatS(r) ∝ r1.1 at these small radii. However, we saw in Fig. 4.1 that for all but the coolest systems,S(r)flattens within the cluster core, in which case the positive slope of theS(r∗)−T relation will be steeper than 0.175,as is observed.

The above arguments seem able to explain the general form of the relation in Fig.4.8, using just the shape ofthe NFW profile, coupled with entropy profiles similar to those seen in simulations incorporating only gravity andshock heating. We therefore conclude that there seems to be no need to invoke preheating to explain this particularresult, despite the initially surprising, non-monotonic trend.

4.5 Discussion

A substantial amount of theoretical and computational effort has been directed towards the problem of similarity-breaking in groups and clusters, especially over the past four years. We now compare the various models with theresults above, and with additional information from the companion papers to this (Chapter 2 and Chapter 3), to seehow they fare.

4.5.1 Cooling models

A number of authors (Bryan, 2000; Pearce et al., 2000; Muanwong et al., 2001; Wu & Xue, 2002b; Dave et al.,2002) have suggested that it may be unnecessary to invoke additional heating to explain the entropy floor, sincecooling will remove low entropy gas from clusters. The counterintuive fact that cooling can have similar effects toheating in this regard was first noted by Knight & Ponman (1997), who found, on the basis of 1D hydrodynamicalsimulations of cluster growth, that cooling acted to flattenthe gas density profiles, especially in low mass systems,and to steepen theL−TX relation, but not sufficiently to match observations. More sophisticated 3D simulations of


Figure 4.8: Left panel: entropy at the radius enclosing a mass of 3× 1013h−150 M⊙ as a function of system temperature (excluding the two

galaxies).Right panel: the same data grouped into bins containing 8 points or more,with simple limiting trends, discussed in the text, markedfor comparison.

cluster evolution by Muanwong et al. (2001) and Dave et al. (2002) have shown that larger effects can be produced,depending upon the spatial resolution of the simulations. Dave et al. show that their simulations have converged inthis respect, although their cooling function does not incorporate emission from metal lines, which becomes verysignificant atT < 2 keV.

Cooling achieves its effect of breaking the similarity between low and high mass clusters, since, at a givengas density, the cooling time is shorter at lower temperatures. Hence, a larger fraction of the hot baryons coolsin groups, compared to richer clusters (Bryan, 2000). Daveet al. (2002) show that this is able to reproduce thecluster-groupL−TX relation quite well, except that the predicted steepening in the relation falls at a slightly lowertemperature than that observed. They also compare the predicted gas entropies at 0.1R200 with the results ofPonman et al. (1999) atT < 4 keV, and find reasonable consistency. However, agreement with the results fromthe present study (Figs. 4.4 and 4.5), which is superior to that of Ponman et al. (1999) in terms of sample size andallowance for non-isothermality, is much less good. Dave et al. find thatS(0.1R200) is raised in low temperaturesystems, but converges with the self-similar trend throughhot clusters for systems withT > 3 keV. In contrast, ourresults show a relation which is flatter than the self-similar line across the full temperature range. Bryan (2000) andWu & Xue (2002b) produce results in better agreement with ourobservations, but their analytical model is basedon an assumed variation in star formation efficiency with system mass which is much stronger than that which wederive from the subsample of our systems for which we have optical luminosities (Chapter 3).

However, the most fundamental problem faced by cooling-only models, as many of their proponents haveacknowledged, is that cosmological simulations without some form of effective ‘feedback’ of energy into thebaryonic component as stars form, have a serious ‘overcooling’ problem, which has been recognised for manyyears. At high redshift, the Universe is dense, and a large fraction of the baryons cool and form stars withinsmall collapsed haloes. For example, Dave et al. (2002) findthat in the largest systems in their simulation (withT ∼ 4 keV), almost half of the baryons have cooled out of the hot phase, whilst in small groups over 80 per cent ofthe gas has cooled. Unless this cooled gas does not form stars, the very high star formation efficiencies implied arefar above those observed in clusters (cf. discussion in Chapter 3) or inferred globally from the K-band luminosityfunction of galaxies (Balogh et al., 2001).

4.5.2 Preheating models

As pointed out by Ponman et al. (1999), since less energy is required to raise the entropy of gas by a given amountwhen its density is lower, it is energetically favourable to‘preheat’ gas before it is concentrated into a clusteror group potential. A comparison of the excess entropy required, with that potentially available from supernovaexplosions associated with star formation (e.g. Valageas &Silk, 1999) shows that a high efficiency of heatingof the intergalactic gas is required even in the most favourable circumstances – i.e. only a modest fraction ofthe supernova energy must be radiated away. However, such high efficiencies have been inferred on the basis ofobservations and modelling in the case of starburst winds (Strickland & Stevens, 2000).

A large reservoir of energy at high redshift is potentially available from the formation of massive black holes

4.5. DISCUSSION 77

in galaxies (Valageas & Silk, 1999; Wu et al., 2000), although it is at present unclear how much of this energycan be coupled into heating of the IGM. Radiative heating from AGN cannot achieve the required high entropies,but quasar outflows may do so, although the physics and demographics involved are still subject to considerableuncertainties (Nath & Roychowdhury, 2002).

Finally, cosmological simulations suggest that a large fraction of the baryon content of the Universe is cur-rently in the form of what has been dubbed the ‘warm-hot intergalactic medium’, with an entropy of the orderof 300 keV cm2 (Cen & Ostriker, 1999; Dave et al., 2001). Dave et al. (2001) show that this high entropy gas isprimarily heated, not by star-formation, but by the effectsof shocks generated during the collapse of gas into fila-ments. It seems unlikely, however, that this mechanism can account for the entropy floor found in the inner regionsof groups and clusters, since the entropy of the IGM generated in this way declines steeply with redshift (Valageas,Schaeffer, & Silk, 2002), such that it should be well below the observed floor value, at the epoch when the gasin cluster cores was accreted. This is confirmed by the recentsimulations of Borgani et al. (2002), for example,which find baryon distributions in clusters which are essentially self-similar, in the absence of non-gravitationalheating processes.

Pioneering efforts to explore the effects of preheating through simulations or analytical treatments (Metzler& Evrard, 1994; Navarro et al., 1995; Cavaliere et al., 1997)have been largely superseded by more recent work,and we concentrate here on the latter. Considerable insightcan be gained by analytical models in which preheatedgas is accreted into clusters. The model of Babul, Balogh andcoworkers has reached its most advanced stage ofdevelopment in Babul et al. (2002). This model assumes Bondiaccretion (Bondi, 1952) of preheated gas into apotential well represented by a (fixed) NFW profile. This accretion is taken to be isentropic in low mass systems,with introduction of a shock heated regime, motivated by theentropy profiles seen in numerical simulations,for systems above some critical mass. The level of preheating is tuned to achieve a good match to theL−TX

relation across a wide temperature range (0.3–15 keV). Unfortunately, the entropy level required to achieve this(330 keV cm2) is well above that actually observed in the inner regions ofgroups. Moreover, the model predictslarge isentropic cores in cool systems. The IGM in groups with M200 < 8×1013M⊙ (corresponding to∼ 2 keV,from ourM−TX relation in Chapter 2) isentirely isentropic in these models, in strong conflict with our results(Fig. 4.1).

Tozzi & Norman (2001) and Dos Santos & Dore (2002) have attempted to evaluate the effects of shock heatingon the preheated accreting gas. Dos Santos & Dore concentrate on the scaling properties of the gas entropyimmediately inside the accretion shock. They find that this is expected to scale with mean system temperatureaccording to(1+T/T0), whereT0 is an adjustable parameter related to the initial adiabat onwhich the preheatedgas lies. By choosing the preheating entropy to be 120 keV cm2, corresponding toT0 = 2 keV, they obtain agood match to theS(0.1R200)−T plot of Lloyd-Davies et al. (2000), though their curve is rather too concave incomparison with the trend seen in Fig. 4.5. In order to calculate the expectedL−TX relation from their model,Dos Santos & Dore make two key additional assumptions: the IGM is assumed to be isothermal, and entropyprofiles,S(r/R200), are assumed to have the same shape in all systems. Their resulting L−TX model steepensrather too gently to match the group data (which show a very steep slope – e.g.L ∝ T4.3 from Helsdon & Ponman,2000a) entirely satisfactorily. The isothermal assumption is in conflict with our data (Fig. 4.3) and with the resultsof cosmological simulations, which generally show a decline in temperature by about a factor 3 from the innerregions to the virial radius (Frenk et al., 1999). The assumption of similar entropy profiles, with a scale factorgiven by(1+T/T0), is in remarkably good agreement with our results, as shown in Fig 4.2, although this result ofself-similarity is anassumption, rather than an output of their model.

A more comprehensive treatment is provided by Tozzi & Norman(2001), who calculate entropy profiles basedon solving the shock jump conditions over typical accretionhistories, for systems spanning a range of final mass.They find that shock heating generates entropy profiles with acharacteristic logarithmic slopeS∝ r1.1, outsidea central isentropic core, which, for a given preheating level, occupies a larger fraction ofR200 in lower masssystems. The slope of 1.1 agrees well with our results (Fig. 4.1), and with numerical simulations (e.g. Borganiet al., 2001). However, as can be seen from Fig. 4.1, we do not see the larger isentropic cores inS(r/R200) whichappear to be a feature of all preheating models.

Nath & Roychowdhury (2002) explore the energetics and evolutionary history of preheating the IGM throughoutflows from radio loud, and broad absorption line quasars.Their conclusion is that AGN could provide the energyinput required to account for the entropy floor. However, in the context of our findings, their most interesting resultis that the specific energy injected by AGN into the gas, is expected to be higher in low mass systems, for tworeasons: the incidence of powerful AGN is expected to be higher in smaller clusters, and the fraction of theirpower expended inPdV work is larger in smaller systems. This appears to be contrary to our entropy scaling


results, shown in Figs. 4.5 and 4.6, which show that at small radii the excess entropy is similar (∼100 keV cm2)across a wide range of system masses, whilst at larger radii it is actually highest in moderately rich (T ∼ 3–4 keV)clusters.

4.5.3 Star formation models

A natural development of cooling models, which can help to address the overcooling problem, is the incorpo-ration of feedback due to star formation. In numerical studies, the successful implementation of such a schemeis extremely challenging, and has been the Holy Grail of those engaged in simulations of galaxy formation forsome years. Voit & Bryan (2001b) introduced an interesting perspective on this issue, in the context of similaritybreaking in clusters, by noting that it makes rather little difference quite how effective feedback is in heating theIGM, since any gas within virialized systems which has low entropy will have a short cooling time, and so isremoved from its location near the centre of a dark halo either by dropping out of the hot phase, or being heatedby star formation in its vicinity, such that it escapes to large radii. In particular, they noted that the entropy ofgas with a cooling time equal to the age of the Universe is∼ 100 keV cm2, in striking agreement with the entropyfloor reported in groups and clusters. Since the cooling timescales as

√T/n in systems withT > 2 keV, where

bremsstrahlung dominates cooling, whilstS= T/n2/3, it follows that a given cooling time is achieved in gas withan entropy which scales as

Scool ∝ (tcoolT)2/3, (4.9)

so that the entropy floor generated in this way is expected to be higher in hotter systems (Voit & Bryan, 2001b).This feature may help to explain the flat trend in entropy withtemperature apparent in Fig. 4.5, and perhaps eventhe fact that the excess entropy at larger radii appears to behigher in mid-mass clusters than in groups (Fig. 4.6).

This approach was developed further by Voit et al. (2002), who constructed models in which the entropydistribution is either truncated below some critical entropy (corresponding to gas cooling out, or being ejected as aresult of vigorous heating), or shifted by the addition of anentropy boost throughout (corresponding to preheatingof the entire IGM). The two different prescriptions for entropy modification produce only subtle differences inobservable properties: shifted entropy models have ratherflatter gas density profiles at large radii in poor groups,and also slightly higher gas temperatures. These differences are not distinguishable with data of the quality usedin the present study, and may well be too challenging even forfuture studies withXMM-NewtonandChandra.One impressive feature of these models is that they contain essentially no adjustable parameters, since the entropythreshold is set by equating the cooling time of gas to a Hubble time of 15 Gyr. However, the predictedL−TX

relation fails to steepen sufficiently at low temperatures to pass through the bulk of the galaxy group points, andtheM−TX relation – whilst steeper than the self-similar relation (M ∝ T1.5) – is less steep than that derived fromthe present sample in Chapter 2.

One of the most recent numerical studies of cluster structure to incorporate the effects of both cooling andfeedback is the work of Borgani et al. (2001, 2002). These simulations explore the effects of setting an entropyfloor through instantaneous preheating at a redshift of 3, and also through heating scaled to the expected supernovaenergy input in overdense regions throughout the evolutionary history of the Universe. The effects of radiativecooling are also included in one of the supernova heating runs. The main conclusion from this work, is that>∼1 keV per particle of energy input is required to give a reasonable match to the observed steepening of theL−TX relation. The authors point out that such a large energy boost appears to exceed what is expected fromsupernova input alone, even if a high heating efficiency of the IGM is assumed.

Borgani et al. find that a somewhat larger suppression ofLX in groups is achieved by injecting the same amountof energy progressively, compared to global preheating of the IGM. Entropy profiles generally agree well with theTozzi & Norman (2001) slope of 1.1, but are flattened in low mass systems by the effects of energy injection, asare gas density profiles. The gas temperature profile becomesmore centrally peaked in low mass haloes subject toadditional heating. TheM−TX relation is found to be little affected by heating, either inslope or normalization.However, cooling is found to decrease the normalization andsteepen the slope of the relation, bringing it intobetter agreement with observations. In comparison to theseresults, we certainly observe flatter gas density profilesin cool systems (Chapter 2), and Fig. 4.2 provides some tentative evidence that the entropy slope may be rathershallower in lower temperature (T < 3 keV) systems. In the case of theM−TX relation, Lloyd-Davies, Bower, &Ponman (2002) and Voit et al. (2002) show that systematic variations in the concentration parameter of dark matterhaloes with mass, may play a key role in steepening its slope,as discussed in Chapter 2.

4.6. CONCLUSIONS 79

4.6 Conclusions

Drawing the above results together, it seems that cooling-only models cannot provide a viable explanation for thesimilarity-breaking observed in clusters, since significantly enhanced stellar luminosities are not observed in coolersystems, as was demonstrated in Chapter 3. Otherwise it would be necessary to admit the presence of very largequantities of baryonic dark matter, which would have to dominate the baryon budget in groups.

Preheating models appear to have the generic property of generating large isentropic cores in low mass systems,in conflict with our observations. This seems to be an inescapable feature of simple preheating models in which theIGM is raised uniformly and ‘instantaneously’ to a high adiabat, since such gas can only be shocked when fallinginto potential wells deep enough for its motion to be supersonic. More complex preheating models may enablethis problem to be circumvented, as we discuss below.

Models involving a mixture of cooling and star formation (orpossibly AGN heating) appear more natural andpromising. However, a number of features of our model appearto conflict withall models proposed to date:

(1) The very large entropy excesses seen at large radii (Figs. 4.6 and 4.7) are a surprise. Models tend to showentropy enhancement in the inner regions of low mass systems, with a normal shock-generated entropyprofile re-establishing itself at larger radii.

(2) Closely related to this, is the fact that the entropy profiles appear to be approximately self-similar apart froma normalization constant, and in particular, that larger isentropic cores arenotseen in galaxy groups. In fact,Fig. 4.1 shows that the lowest temperature systems actuallyhave entropy profiles which appear to drop allthe way into the centre, unlike hotter systems. The fact thatthe scaling suggested by Dos Santos & Dore(2002) brings our profiles into good agreement (Fig. 4.2) is apuzzle, given that this scaling is really notjustified by their model as a scale factor for entropyprofiles.

(3) The temperature profiles shown in Fig. 4.3, are also not quite what is expected from the models. Anymechanism which gives an entropy boost should produce the strongest results in low mass systems. Since arise in entropy has to be coupled with the maintenance of hydrostatic equilibrium, the natural consequence isa rise in central temperature, coupled with a decrease in density. The density drop is certainly observed, andthere are indications that the temperature has been raised in cool systems outside the core, but the temperatureprofiles in groups seem to lack the expected central cusp. This is related to the lack of an entropy core ingroups referred to in point (2), above.

(4) The behaviour of the gas core radius, discussed in Chapter 3, is similarly unexpected. As a fraction ofR200,the gas core radius is approximately constant at∼10 per cent over most of our temperature range, but fallsdramatically, by an order of magnitude, in groups. The only paper to predict this sort of behaviour is thatof Wu & Xue (2002a), who show that fitting a convex profile whichsteepens progressively, with a betamodel, can lead to smaller fitted core radii as the outer radius of detection shrinks. In Chapter 2 we reportedtests with fits to real data for two clusters, truncated at different radii, which suggested that this effect isnot dominating our fits. However, X-ray surface brightness profiles extending to larger fractions ofR200 forgroups are required to definitively settle this issue.

What might these disagreements with the models be pointing towards? Finoguenov et al. (2002) argued that thelarge excess entropy atR500 in groups indicates that the entropy profiles in cool systemsare dominatedthroughoutby the effects of preheating. Two features of our results make us uneasy about this conclusion. Firstly, theseexcesses are now seen (Fig. 4.7) to extend up to moderately rich clusters, and to involve entropies several timesthose expected in the IGM at the present epoch. Secondly, theslopes of all our entropy profiles (Fig. 4.1) seem toscatter about the value predicted from shock heating. This is an uncomfortable coincidence if shock heating playsno role in establishing them. It is certainly possible to devise preheating scenarios which could reproduce any ofour profiles, by varying the history of energy injection, andthe density gradient of the gas into which this energywas injected, in the vicinity of systems of a given mass. However, such a solution would require a large amount offine tuning, and would also throw away the impressive successof the Voit et al. (2002) approach in accounting forthe observed entropy floor without any free parameters.

If, on the contrary, we wish to retain shock heating as the basic mechanism responsible for generating therising entropy profiles seen outside the core in both clusters and groups, how can this be achieved? Since thepost-shock temperature is essentially determined by the infall velocity of gas into a halo, which can be retarded,but not increased. The only way to raise the whole entropy profile in low mass systems, in the way implicit in


Fig. 4.1, seems to be to reduce the density of the pre-shock gas accreting into lower mass systems, relative tothat falling into rich clusters. In simple spherical collapse models, the matter turning around and accreting intohaloes at a given epoch is expected to have a given overdensity. However, in reality, cosmological simulationsshow us that accretion is far from spherical, and that much ofit takes place along filaments. Perhaps this changeof geometry makes a fundamental difference to the way in which the density of accreting gas scales with systemmass, by effectively reducing the dimensionality of the surface over which accretion takes place from 3 towards 1.If this has the effect of increasing the density of gas falling into richer systems, then the shock-generated entropyprofiles in these will be depressed as a whole, relative to those systems whose growth is fed by lower density gas.This idea is speculative, but worth pursuing.

Chapter 5

Mass, Velocity Dispersion andTemperature Scaling Properties

Abstract The results obtained from a detailed X-ray study of the intergalactic medium (IGM) in 66 virialized systems arecompared with selected optical properties. In particular,we focus on the group and cluster velocity dispersion, as a probe ofthe dynamical evolution of the gravitating mass halo.

We measure a logarithmic slope of 0.64±0.04 for theσ−TX relation, steepening to 1.18±0.24 below 2.1 keV, which isinconsistent with simple self-similar scaling (σ ∝ T0.5). Similar behaviour is seen in theσ−M andσ−LB,j relations, whichexhibit logarithmic slopes for the combined groups and clusters in our sample of 0.39±0.03 and 0.42±0.06, respectively (cf.σ ∝ M1/3).

The relation between the gas mass and its mean temperature isalso investigated, and found to exhibit a steepening withrespect to self-similar expectation (Mgas∝ T1.5). We find evidence of gradual increase in the logarithmic slope of the relationfrom 1.84±0.10 for clusters hotter than 2.1 keV, to 2.85±0.35 below this temperature, forMgasmeasured withinR500. Theinterpretation of our results is based on the effects of hierarchical structure formation, and we asses the impact of ourfindingson models of non-gravitational heating and/or cooling of the IGM.

5.1 Introduction

The distribution of mass within and between virialized systems is of great cosmological importance, although bothare difficult to measure accurately. For this reason, parameters such as X-ray temperature, luminosity and galaxyvelocity dispersion – which are more amenable to observational study – are often used as proxies for halo masswhen investigating scaling properties. While such quantities reflect the depth of the gravitational potential well,they are also sensitive to physical process like energy injection, cooling and dynamical friction, which may havelittle or no effect on the dark matter that constitutes the dominant mass component.

The distribution of the line-of-sight velocities of galaxies in a bound system has often been used to estimatemasses of clusters in dynamic equilibrium (e.g. Girardi et al., 1998), and has been shown to correlate tightly with X-ray temperature (Lubin & Bahcall, 1993; Bird, Mushotzky, & Metzler, 1995; Girardi et al., 1996; Wu, Fang, & Xu,1998; Xue & Wu, 2000). However, there is some disagreement over the extent to which theσ−TX relation agreeswith the expectations of a simple isothermal and hydrostatic (i.e. self-similar) model, which predictsσ ∝ T0.5. Anumber of authors have suggested that the relation in clusters is consistent with such a model (Edge & Stewart,1991a; Lubin & Bahcall, 1993), while others report a significantly steeper trend (Bird et al., 1995; Wu et al., 1998;Xue & Wu, 2000). The situation in groups is equally controversial, with some studies reporting an apparentlyself-similar trend (Ponman et al., 1996; Mulchaey & Zabludoff, 1998), in contrast with the more recent findings ofHelsdon & Ponman (2000b), who measure a logarithmic slope of1.1±0.2.

Departures from self-similarity in theσ−TX relation have been variously attributed to energy injection heat-ing the gas (McCarthy, Babul, & Balogh, 2002; Bialek, Evrard, & Mohr, 2001; Loewenstein, 2000; Helsdon &Ponman, 2000b), dynamical friction (Bird et al., 1995; Metzler & Evrard, 1997) or merging (Fusco-Femiano &Menci, 1995; Menci & Fusco-Femiano, 1996) altering galaxy motions, as well as the effects of orbital isotropyand halo concentration on the velocity dispersion (Łokas & Mamon, 2001). The role of dark matter concentrationin influencing the dynamics of the galaxies is particularly important, since this parameter has been shown to varysystematically with halo mass in X-ray observations (Sato et al., 2000), theoretical models (Salvador-Sole et al.,1998; Łokas, 2000) and numerical simulations (Avila-Reeseet al., 1999; Bullock et al., 2001).

Such considerations also bear upon scaling relations involving X-ray parameters, since the hot gas in the IGMtraces the underlying potential and is also susceptible to bias from non-gravitational processes. TheMgas−TX

81

82 CHAPTER 5. MASS, VELOCITY DISPERSION ANDTEMPERATURESCALING PROPERTIES

relation is a case in point, since its logarithmic slope is particularly sensitive to the effects of energy injection onthe IGM, which will lower the gas density in the core, as well as raise its temperature (McCarthy et al., 2002). Thisrelation has been shown to deviate from the self-similar prediction of Mgas∝ T1.5 in clusters (Mohr et al., 1999),although this result has not been verified in the group regime, where the effects of energy injection will be mostapparent (Loewenstein, 2000).

The X-ray properties of the galaxies, groups and clusters inthis paper are based on the data presented inSanderson et al. (2002, hereafter Chapter 2) . In that paper we undertook a deprojection analysis of the X-rayemission from the IGM in 66 virialized systems. This sample was assembled from three existing X-ray studiesof groups and clusters, to which a small number of cool groupswere added. To each system, we fitted analyticalmodels to parametrize both the gas density and temperature as a function of radius. This allowed the data from thedifferent subsamples to be treated identically and permitted the properties of the gas to be extrapolated to arbitraryradii. In the present work we have used these fitted models to evaluate the total gas mass; we examine the scalingproperties of this quantity in section 5.3.4. We summarise some basic properties of the sample in Table 5.1.

In this paper, we study the correlations between the X-ray and optical properties of the sample. In particular, weare interested in the dynamical state of virialized systemsas measured by the velocity dispersion of their galaxies(see section 5.2, below). We investigate the relation between this parameter and the X-ray temperature and totalmass (from Chapter 2). We also compare the velocity dispersion with the total optical luminosity of a subset ofour sample, comprising 32 groups and clusters. These luminosities were derived in Sanderson & Ponman (2002a,hereafter Chapter 3), where we fitted analytical profiles to describe the 3-dimensional galaxy distribution. Thesewere combined with aperture luminosity measurements, taken from the literature, to allow us to determine the totalluminosity within our X-ray determined virial radius.

In addition, have compiled a list of Abell cluster richness values for the 38 Abell clusters in our sample, takenfrom the online Cluster galaxies data base1, to which we have added an equivalent measurement for AWM 7 (fromBahcall, 1981). In section 5.3.5, we examine the correlation between this quantity and X-ray temperature andgas mass. We have also compiled a list of Bautz-Morgan types for our Abell clusters, from the same source (seesection 5.3.6), excluding Abell 478, Abell 780 or Abell 2163, for which no classification was available. We presentall our findings in section 5.3.

Throughout the analysis which follows, we adopt the following cosmological parameters;H0 = 70 km s−1 Mpc−1andq0 = 0. All quoted errors are 1σ on one parameter, unless otherwise stated.

5.2 Calculating velocity dispersion

The dynamical properties of the galaxy members in a virialized system (or stars, in the case of an individual galaxy)reflect the depth of the underlying gravitational potentialwhich binds them in equilibrium. The dispersion of theline-of-sight velocities of the galaxies about the mean canbe used to infer the total gravitating mass of the system,through application of the virial theorem (e.g. Girardi et al., 1998), although such methods are susceptible to biascaused by substructure within clusters (Bird, 1995).

We have compiled a sub-sample of 60 of our targets for which wewere able to obtain velocity dispersions. Themajority of these measurements were taken from the literature, with a further 12 derived from our own calculations.The latter systems were all groups, for which recession velocities of a representative number of galaxy memberswere readily available in the literature. Of these, three (NGC 1395, NGC 3258 & IC 4296) have been identified inthe group catalogue of Garcia (1993), based on the Lyon-Meudon Extragalactic Database Archive (LEDA). Valuesof σclus were calculated directly from the Virgo infall corrected recession velocities of the group members for theselatter three systems; an additional procedure was requiredto calculateσclus for the remaining 9 systems.

For these 9 groups we determined our own group membership from the NASA Extragalactic Database (NED).The following criteria were used to select a sample of galaxies to be used to determine the velocity dispersion ofthe system.

(i) The galaxy must lie within the virial radius (R200, as calculated in Chapter 2) of the group optical centroid(taken from NED).

(ii) The recession velocity must lie within 10σclus of the group recession velocity, given an initial estimate ofσclus.

1http://www.astro.spbu.ru/CLUSTERS/

5.3. RESULTS 83

(iii) The absolute B-band magnitude must be brighter than -16.32. This ensures reasonable completeness andcorresponds to∼90 per cent of the group luminosity function (Helsdon & Ponman, 2002); for a typicalgroup in our sample (v≈ 5000 km s−1), MB corresponds to an apparent magnitude of∼17.2.

The sample was then iteratively refined, by calculating the standard deviation of the recession velocities, ex-cluding those galaxies beyond 3σ of the mean and re-calculatingσ until it converged. The whole selection andcalculation process was then repeated using this value ofσclus, to check the consistency of the result. Since thismethod is only appropriate for relatively nearby groups (i.e. <

∼12,000 km s−1), we were unable to derive veloc-ity dispersions for the six clusters in our sample for which no literature measurements were found (Abell 644,Abell 780, Abell 1650, Abell 2597, 2A0335+096 & Triangulum Australis).

We use the following equation to calculate the error on our velocity dispersion values (σclus)

δσclus = σclus

√

N−12N2 , (5.1)

whereN is the total number of galaxies. Table 5.1 lists the velocitydispersions for the sample, together withthe appropriate reference for the data. We includeσ measurements for the two galaxies in our sample (indicatedby the † in column 1 of Table 5.1) for completeness, and to permit a comparison with groups and clusters (seesection 5.3.1).

5.3 Results

5.3.1 σ−TX

Since the same potential is responsible for retaining the galaxies and the gaseous IGM, a clear correlation isexpected between the velocity dispersion of the system and its X-ray temperature. Fig. 5.1 shows this relation forthe 60 systems for which we have determinedσ, including the two galaxies – the coolest objects in the sample.A power law was fitted to the data in log space, using theODRPACK software package (Boggs et al., 1989, 1992),to take account of parameter errors in both the X and Y directions. This reduces to a straight line of the formlogσ = AlogkT +B, whereA is the power law index andB the normalization. The logarithmic slope of the best-fitting relation to all the points is 0.64± 0.04, which is significantly steeper than the self-similar prediction ofσ ∝ T0.5. As a comparison, we have plotted the locus of equality between the specific energy of the gas comparedto that of the galaxies, given byβspec= 1, where

βspec=µmpσ2

kT, (5.2)

whereµ is the mean mass per particle andmp is the proton mass.Our slope compares well with the value of 0.67±0.09 found by Wu et al. (1998), for a large sample of 94

clusters drawn from the literature, using the same orthogonal regression technique. Similarly, Girardi et al. (1998)found a logarithmic slope of 0.62±0.04 for a sample of 55 clusters, very close to that of Girardi etal. (1996). Itwas shown by Bird et al. (1995) that the fitting method can havea significant effect on the slope of the relation. Inparticular, the use of ordinary least squares regression tends to lead to somewhat smaller values of the power lawindex – a result confirmed by Wu et al. (1998) and Xue & Wu (2000). This explains the apparently self-similarlogarithmic slope of 0.46±0.12 found by Edge & Stewart (1991a), using an ordinary least squares fit to the data,which steepens to 0.75±0.08 and 0.68±0.10 when fitted with a bisector regression with and without using errors,respectively (Bird et al., 1995). Similarly, the result ofσ ∝ T0.60±0.11 found by Lubin & Bahcall (1993), for asample of 41 clusters, changes toσ ∝ T0.87±0.08 when a weighted bisector regression is applied to the data (Birdet al., 1995).

A key issue is the scaling properties of bothσclus andkT in the group regime. It is clear from other scalingrelations – e.g.L−TX (Ponman et al., 1996; Helsdon & Ponman, 2000b) – that a markedtransition occurs at∼2–3 keV, consistent with non-gravitational heating or cooling of the IGM in virialized haloes. It is thereforeinteresting to search for evidence of a systematic change inthe slope of theσ−TX relation towards lower tem-peratures. We have chosen a temperature of 2.1 keV to delineate approximately the boundary between groups andclusters, leaving 17 systems in the former category and 41 inthe latter, excluding the two galaxies. Our results aresummarised in the top third of Table 5.2.

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SName RA Dec. z Ta σ Mgas,500 Mgas,200 BM typeb Abell no. cts Referencec

(J2000) (J2000) (keV) ( km s−1) (×1012 M⊙) (×1012 M⊙)

NGC 1553† 64.043 -55.781 0.0036 0.50 168+8−8 0.91.2

0.5 1.01.50.6 – – Paturel et al. (1997)

NGC 6482† 267.954 23.072 0.0131 0.56 302+29−27 4.24.9

2.3 5.47.43.1 – – Paturel et al. (1997)

HCG 68 208.420 40.319 0.0080 0.67 185+30−30 9.53.9

2.9 14.26.54.7 – – NED

NGC 1395 54.623 -23.027 0.0057 0.84 150+131−19 13.15.5

4.0 19.89.46.5 – – LEDA

NGC 4325 185.825 10.622 0.0252 0.90 265+50−44 23.23.3

3.2 35.98.37.8 – – Zabludoff & Mulchaey (1998)

HCG 97 356.845 -2.326 0.0218 1.00 234+29−25 18.32.1

1.9 27.54.03.0 – – Mahdavi & Geller (2001)

IC 4296 203.393 -33.622 0.0123 1.04 161+28−28 11.82.0

1.8 17.04.03.4 – – LEDA

NGC 5846 226.385 1.696 0.0058 1.18 368+72−61 24.92.5


NGC 5044 198.850 -16.386 0.0090 1.25 404+64−64 35.72.7

2.4 58.55.14.7 – – NED

HCG 51 170.587 24.293 0.0258 1.38 526+123−123 16.40.8

0.7 26.21.31.3 – – NED

NGC 507 20.921 33.261 0.0164 1.40 597+60−60 31.32.5

2.0 46.23.93.0 – – NED

HCG 62 193.284 -9.224 0.0137 1.48 376+52−46 16.01.9


NGC 5129 201.150 13.928 0.0233 1.54 294+43−38 19.75.7


NGC 6329 258.562 43.684 0.0276 1.60 336+44−40 22.05.4

4.1 28.40.70.9 – – NED

NGC 2563 125.102 21.096 0.0163 1.61 365+102−104 47.918

14 72.93926 – – Zabludoff & Mulchaey (1998)

Abell 262 28.191 36.157 0.0163 2.03 525+47−33 60.212

9.7 1152923 5 40 Girardi et al. (1998)

Abell 194 21.460 -1.365 0.0180 2.07 341+57−37 88.219

19 1655859 3 37 Girardi et al. (1998)

IV Zw 0381 16.868 32.462 0.0170 2.07 468+163−121 54.414

10 81.72116 – – Mahdavi & Geller (2001)

MKW 4 180.990 1.888 0.0200 2.08 501+67−48 56.62.1

2.2 68.83.53.4 1 – Koranyi & Geller (2002)

MKW 4S 181.647 28.180 0.0283 2.46 422+110−63 76.49.6

8.7 1081816 1 – Koranyi & Geller (2002)

Virgo 187.697 12.337 0.0036 2.55 632+41−29 87.74.2

4.1 1478.07.0 5 – Girardi et al. (1998)

NGC 3258 155.887 -34.758 0.0095 2.57 189+32−32 40.02.3

2.4 48.54.14.0 – – LEDA

NGC 6338 258.825 57.400 0.0282 2.64 614+56−56 64.659

38 82.110048 – – NED

Abell 539 79.134 6.442 0.0288 2.87 629+70−52 16522

20 2564943 5 50 Girardi et al. (1998)

MKW 9 233.122 4.682 0.0397 2.88 621+190−190 13650

37.5 22310072 1 – NED

AWM 4 241.238 23.946 0.0318 2.96 400+68−46 21148

49 421140140 1 – Koranyi & Geller (2002)

Abell 1060 159.169 -27.521 0.0124 3.31 610+52−43 27518

10 4613822 5 50 Girardi et al. (1998)

2A0335+096 54.675 9.977 0.0349 3.34 – 2744238 46888

76 – – –

Klemola 442 356.919 -28.138 0.0290 3.40 341+106−80 23137

34 3998874 5 117 Wu et al. (1998)

continued overleaf

5.3

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Name RA Dec. z Ta σ Mgas,500 Mgas,200 BM typeb Abell no. cts Referencec

(J2000) (J2000) (keV) ( km s−1) (×1012 M⊙) (×1012 M⊙)

Abell 2634 354.615 27.022 0.0309 3.45 700+97−61 15018

18 1944034 3 52 Girardi et al. (1998)

Abell 2052 229.176 7.002 0.0353 3.45 561+87−73 23750

50 395130130 2 41 Wu et al. (1998)

Abell 779 139.962 33.771 0.0229 3.57 473+76−52 91.127

21 1435539 2 32 Wu et al. (1998)

Abell 2199 247.165 39.550 0.0299 3.93 801+92−61 1565.0

3.0 2116.06.0 1 88 Girardi et al. (1998)

Abell 2063 230.757 8.580 0.0355 4.00 667+55−41 27014

13 3833233 3 63 Girardi et al. (1998)

HCG 94 349.319 18.720 0.0417 4.02 773+132−132 12820

17 1753327 – – NED

AWM 7 43.634 41.586 0.0172 4.02 740+53−44 445110

93 1240470350 1 35 Koranyi & Geller (2002)

MKW 3S 230.507 7.699 0.0453 4.42 610+69−52 17849

37 2087953 1 – Girardi et al. (1998)

Abell 2657 356.237 9.201 0.0400 4.53 661+230−171 19655

34 2268952 5 51 Mahdavi & Geller (2001)

Abell 780 139.528 -12.099 0.0565 4.63 – 5215448 965120

110 – 39 –

Abell 3391 96.608 -53.678 0.0536 5.39 663+195−112 32982

63 536190130 1 40 Girardi et al. (1998)

Abell 4059 359.250 -34.752 0.0480 5.50 845+280−140 22054

39 2618357 1 66 Wu et al. (1998)

Abell 2670 358.564 -10.408 0.0759 5.64 852+48−35 33941

39 5157871 2 142 Girardi et al. (1998)

Abell 2597 351.319 -12.124 0.0852 6.02 – 4797063 718130

120 5 43 –

Abell 119 14.054 -1.235 0.0444 6.08 679+106−80 40558

48 586140100 4 69 Girardi et al. (1998)

Abell 496 68.397 -13.246 0.0331 6.11 687+89−76 31535

27 4205942 1 50 Girardi et al. (1998)

Abell 1651 194.850 -4.189 0.0846 6.18 1006+118−92 45671

51 64713092 2 70 Wu et al. (1998)

Abell 3558 201.991 -31.488 0.0477 6.28 977+39−34 33544

32 4708056 1 226 Girardi et al. (1998)

Abell 3571 206.867 -32.854 0.0397 7.31 1045+109−90 54350

61 75490110 1 126 Girardi et al. (1998)

Abell 3112 49.485 -44.238 0.0703 7.76 552+86−63 22492

110 259130120 1 116 Wu et al. (1998)

Abell 399 44.457 13.053 0.0722 7.97 1116+89−83 43666

84 599120140 2 57 Girardi et al. (1998)

Abell 1650 194.674 -1.756 0.0845 8.04 – 548410220 691710

330 2 114 –

Abell 2218 248.970 66.214 0.1710 8.28 1222+147−109 561120

90 794170130 3 214 Girardi & Mezzetti (2001)

Abell 1795 207.218 26.598 0.0622 8.54 834+85−76 716380

190 922690310 1 115 Girardi et al. (1998)

Abell 665 127.739 65.854 0.1818 8.60 821+233−130 858150

160 1350320330 5 321 Girardi & Mezzetti (2001)

Abell 2256 256.010 78.632 0.0581 8.62 1348+86−64 57573

90 688120140 4 88 Girardi et al. (1998)

Abell 85 10.453 -9.318 0.0521 8.64 969+95−61 48092

35 54914052 1 59 Girardi et al. (1998)

Abell 3266 67.856 -61.417 0.0545 9.53 1107+82−65 638100

65 764170100 2 91 Girardi et al. (1998)

Abell 401 44.737 13.573 0.0739 9.55 1152+86−70 57565

74 731110120 1 90 Girardi et al. (1998)

Abell 2029 227.729 5.720 0.0766 9.80 1164+98−78 98565

64 1340150150 1 82 Girardi et al. (1998)

continued overleaf

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SName RA Dec. z Ta σ Mgas,500 Mgas,200 BM typeb Abell no. cts Referencec

(J2000) (J2000) (keV) ( km s−1) (×1012 M⊙) (×1012 M⊙)

Abell 478 63.359 10.466 0.0882 10.95 904+261−140 533390

220 590580280 – 104 Wu et al. (1998)

Abell 2319 290.274 43.964 0.0555 10.99 1545+95−77 59584

79 767140110 4 68 Girardi et al. (1998)

Tri. Aus. 249.584 -64.516 0.0510 11.06 – 712180130 871300

200 – – –

Abell 2142 239.592 27.233 0.0894 11.16 1132+110−92 964400

230 1250730380 3 89 Girardi et al. (1998)

Abell 644 124.355 -7.528 0.0711 11.68 – 480200150 526270

200 5 42 –

Abell 1689 197.873 -1.336 0.1840 12.31 1989+245−245 1870180

140 2970280220 4 228 Wu et al. (1998)

Abell 2163 243.956 -6.150 0.2080 16.64 1698+593−439 983780

260 10701200340 – 119 Mahdavi & Geller (2001)

Table 5.1: Some key properties of the 66 objects in the sample, listed in order of increasing temperature. Positions and redshifts are taken from Ebeling et al. (1996,

1998); Ponman et al. (1996) and NED. Columns ?–? are data as determined in this work. All errors are 68% confidence.

† indicates the two galaxies, for which the velocity dispersion refers tostellar motions.1 also known as NGC 383.2 also known as Abell 4038.

a The cooling-flow corrected, emission-weighted temperature of the system within 0.3R200, as determined in Chapter 2.b Bautz-Morgan Classification (Bautz & Morgan, 1970): 1=I; 2=I-II; 3=II; 4=II-III; 5=III.c Reference for velocity dispersion.

5.3. RESULTS 87

Figure 5.1: Velocity dispersion as a function of system temperature for 58 groups and clusters (barred crosses) and two early-type galaxies(diamonds). The solid line represents the best fit to all the points, which has a logarithmic slope of 0.64±0.04. The dotted line is the best fit tothe points below 2.1 keV and the dashed line is the best fit to the data above this limit (see Table 5.2). The dot-dashed line is the locus alongwhich βspec=1 (see equation 5.2)

It can be seen that the data above 2.1 keV exhibit a logarithmic slope which is in good agreement with thatfound for the whole relation. However, the cooler systems depart noticeably from this trend, with a significantlysteeper logarithmic slope of 1.18±0.24, although with a large error. In fitting only the groups, wehave reducedthe scatter about the best fit (rightmost column of Table 5.2)to a level which is consistent with the measurementerrors onkT andσclus, which suggests that the intrinsic scatter of∼40 per cent for the whole relation is largelydue to this steepening in the group regime. An equivalent slope, of 1.1±0.2, was found by Helsdon & Ponman(2000b) for a sample of 20 loose groups of galaxies, spanningthe range 0.4–1.7 keV, also using theODRPACK

regression package. Although an apparently self-similar slope of 0.55±0.05 was found by Ponman et al. (1996)for a larger sample of 37 galaxy groups, this result was basedon an ordinary least squares fit to the data, whichis liable to artificially flatten the relation (Bird et al., 1995; Wu et al., 1998), as mentioned above. This may alsoexplain the findings of Mulchaey & Zabludoff (1998), who deduceσ ∝ T0.45±0.19 for a sample of 12 groups, usingthe same fitting technique as Lubin & Bahcall (1993).

However, a steepening of theσ−TX relation in the group regime, compared to clusters, was not found by Xue& Wu (2000), who also use weighted orthogonal distance regression to fit the data. Their logarithmic slopes are0.65± 0.03 for a sample of 109 clusters – in good agreement with our value – and 0.64± 0.08 for a sample of36 galaxy groups. While their sample is somewhat larger thanours, Xue & Wu gather X-ray temperatures from anumber of different sources in the literature. By comparison, our temperatures are more homogeneous, since wehave averaged them within a fixed fraction (0.3) ofR200 in all cases and have taken care to correct them for theeffects of central gas cooling (Chapter 2).

The two early-type galaxies in our sample – which we have excluded from the fitting – are shown as diamonds,located in the bottom left hand corner of Fig. 5.1. Of these, NGC 6482 has a much higher velocity dispersion,placing it noticeably above both the group and cluster trends; it is worth noting that this object is most likely afossil group (see Chapter 2) and thus is expected to exhibit group-like rather than galaxy-like X-ray properties(Ponman et al., 1994). However, the S0 galaxy NGC 1553 (bottom diamond in Fig. 5.1) is in good agreement withthe fit to the systems above 2.1 keV. A recent analysis by O’Sullivan (2002) has shown that theσ−TX relationfor early-type galaxies is broadly similar to that for clusters – albeit with rather large scatter – and thus somewhatflatter than the group trend. However, it must be remembered that the internal velocity dispersions of individual


Data Range Index Normalization Scattera

( keV) (logged)

σ−TX 0.6–17 0.64±0.04 2.45±0.03 1.43< 2.1 1.18±0.24 2.41±0.05 1.08≥ 2.1 0.67±0.06 2.43±0.05 1.49

σ−M 0.6–17 0.39±0.03 −2.75±0.4 1.63< 2.1 0.53±0.15 −4.67±2.0 1.57≥ 2.1 0.45±0.05 −3.73±0.7 1.54

σ−LB,j 0.6–17 0.40±0.04 −2.00±0.48 0.99< 2.1 0.41±0.26 −2.06±2.91 1.22≥ 2.1 0.42±0.06 −2.19±0.72 0.89

Table 5.2: Summary of results for the power lawσ−TX , σ−M andσ−LB,j fitting, over two different temperature ranges (approximatelycorresponding to groups and clusters of galaxies) plus the combined range. The two galaxies have been omitted from the fits; see text fordetails. Notes:aMultiples of the statistical scatter expected from the errors alone.

galaxies probe the dynamics of the stellar population and hence are less sensitive to the global properties of thehalo, which are better reflected in the temperature, at leastfor X-ray bright galaxies.

5.3.2 σ−M

Since the X-ray temperature is susceptible to bias from non-gravitational physics, it is useful also to study velocitydispersion as a function of halo mass. We have taken total mass estimates derived from the X-ray data in Chapter 2,and show them plotted againstσ in Fig. 5.2. The solid line is the best fit to the data (excluding the two galaxies),performed using the orthogonal regression method outlinedin section 5.3.1 above. We find a relation of the formM ∝ T0.39±0.03, with quite a large scatter – 1.63 times that expected from the statistical errors alone. Followingour treatment of theσ−TX relation, we have split our sample into groups and clusters and fitted each datasetseparately; the results are summarised in the middle third of Table 5.2. Given that self-similarity impliesM ∝ T3/2

and σ ∝ T1/2, it follows that σ ∝ M1/3; our logarithmic slope of 0.39± 0.03 is inconsistent with this simpleprediction, at the 2σ level. Thus the departure from self-similarity observed intheσ−TX relation cannot entirelybe due to the effect of energy injection raising the temperature of the IGM by a proportionately larger amount incooler systems.

A power law fit to the groups cooler than 2.1 keV yields a steeper logarithmic slope, of 0.53±0.15, while thesystems hotter than 2.1 keV are best-fitted by a relation of the formσ ∝ M0.45±0.05. However, it can be seen that the17 systems cooler than 2.1 keV are not exactly the same as the 17 lowest mass systems. If we fit just those systemsbelow 1014M⊙ (excluding the galaxies) we obtain an even steeper slope of 0.63±0.18. Furthermore, the level ofintrinsic scatter is almost halved (see column 5 of Table 5.2) – to 1.33 times that from the statistical errors. Nonethe less, it is still quite high, especially considering that our mass estimates are fairly conservative (see Chapter 2).We will return to this issue in the context of halo mass concentration and formation epoch in section 5.4.1.

5.3.3 σ−LB,j

To verify that the steepening seen in theσ−TX relation is not entirely due to a systematic bias in the X-raytemperature, we have plotted the velocity dispersion as a function of optical luminosity. Fig. 5.3 shows the relation,based on those groups and clusters for which we derived totalLB,j values in Chapter 3. Once again we find thatthe best-fitting power law to all the data is significantly steeper than the self-similar prediction of 1/3 (based on theassumption that light traces mass). Our logarithmic slope,of 0.40±0.04, is inconsistent with this value at∼2σ andthere is no intrinsic scatter about this fit, indicating thatit provides a good match to the data. Girardi et al. (2000)find a logarithmic slope in the range 0.4–0.5 – which is consistent with our results – using a weighted regressionfit to a sample of 89 clusters,

The fit to the clusters hotter than 2.1 keV yields an almost identical relation to that for the whole sample, witheven less scatter (see bottom third of Table 5.2). The same can be said for the systems cooler than this temperature,except that the scatter has increased somewhat. However, wenote that, as with theσ−M relation, the partition

5.3. RESULTS 89

Figure 5.2: Velocity dispersion as a function of total gravitating mass withinR200, for 58 groups and clusters and two early-type galaxies(diamonds). The solid line represents the best fit to all the points (excluding the galaxies), which has a logarithmic slope of 0.39±0.03. Thedotted line is the best fit to the groups and clusters below 2.1keV (boxes) and the dashed line is the best fit to the data abovethis limit (barredcrosses). See middle third of Table 5.2 for more details.

Figure 5.3: Velocity dispersion as a function ofLB,j within R200, for 32 groups and clusters. The Y-axis has been scaled identically to Figs. 5.1& 5.2. The solid line represents the best fit to all the points,which has a logarithmic slope of 0.40±0.04. The boxes are the groups and clustersbelow 2.1 keV and the barred crosses represent the systems hotter than this temperature. The fits to these datasets are almost indistinguishablefrom that to all the points and has not been plotted. See bottom third of Table 5.2 for more details.


Figure 5.4: Total gas mass withinR500 as a function of emission-weighted X-ray temperature. The solid line is the best-fitting power law, whichhas a slope of 2.20±0.08. The dotted line is the best fit to the points below 2.1 keV and the dashed line is the best fit to the data above thislimit (see Table 5.3)

in temperature does not split the sample cleanly – i.e. the coolest objects do not all have the lowest luminosities.It is also possible that a slight steepening in theσ−LB,j relation in groups is masked by a modest increase in starformation efficiency, as was hinted at in Chapter 3.

5.3.4 TheMgas−TX relation

The total mass contained in the IGM in virialized systems hasimportant cosmological implications, since the hotgas is the dominant contributor to the total baryon budget inthese objects. It is therefore important to understandthe scaling properties of this quantity across a wide dynamic range. Here we examine theMgas−TX relation, sincekT is a reasonable proxy for the total gravitating mass, and easy to measure for the majority of X-ray haloes whichcan be observed.

As pointed out in McCarthy et al. (2002), theMgas−TX relation is rather under-represented in analyses of thescaling properties of virialized systems, particularly inrespect of theoretical studies. Recent work has identified theMgas−TX relation as a important tool for investigating the effects of preheating (McCarthy et al., 2002), buildingon a physically-motivated analytic model of accretion of high entropy gas onto group- and cluster-sized haloes(Babul et al., 2002). Since both the total gas mass and its mean temperature are susceptible to the effects of energyinjection – in the direction of decreasingMgas and increasingkT – the slope of theMgas−TX relation is a verysensitive probe of non-gravitational heating processes.

We have derived gas masses withinR500 andR200 for our full sample – referred to asMgas,500 andMgas,200,respectively – which are listed in Table 5.1. Fig. 5.5 shows the relation betweenMgas,500 and the mean emission-weighted X-ray temperature within 0.3R200, as calculated in Chapter 2. We have fitted a power law to thesedatausing the same orthogonal distance regression method described in section 5.3.1; our results are summarised inTable 5.3. We measure a logarithmic slope of 2.20±0.08 for Mgas,500 versus temperature, which is very muchsteeper than the self-similar value of 1.5. Examination of Fig. 5.4 reveals clear evidence of a steepening in therelation for cooler systems. To check this, we have split oursample into those systems above and below 2.1 keVand fitted each dataset separately, as we did for theσ−TX andσ−M relations, in sections 5.3.1 & 5.3.2. Asexpected, there is a considerable increase in the logarithmic slope of the relation below 2.1 keV compared to thatabove this temperature – 2.85±0.35 and 1.84±0.10, respectively. In addition, the scatter about the best fitfor the

5.3. RESULTS 91

Figure 5.5: Total gas mass withinR200 as a function of emission-weighted X-ray temperature. The solid line is the best-fitting power law, whichhas a slope of 2.01±0.09. The dotted line is the best fit to the points below 2.1 keV and the dashed line is the best fit to the data above thislimit (see Table 5.3)

separate datasets is lowered to a level which is consistent with the measurement errors (see column 5 of Table 5.3).A logarithmic slope of 1.98± 0.18 was found for theMgas−TX relation withinR500 by Mohr et al. (1999),

for an X-ray flux limited sample of 45 clusters of galaxies. This value is somewhat flatter than the slope for ourwhole sample (2.20±0.08). However, this can be explained by the dearth of cooler systems in their analysis: aspointed out by Loewenstein (2000), the Mohr et al. sample comprises mainly clusters hotter than the level at whichsignificant departure from self-similarity is expected to be obvious (their coolest cluster is 2.4 keV, compared to ourpseudo ‘break’ temperature of 2.1 keV). A comparison with the slope for our hotter systems (1.84±0.10) revealsa slightly better agreement, with our clusters appearing tobe the more self-similar. Logarithmic slopes in the range∼1.7–2.7 (with an error of roughly 0.1) are predicted by the simulations of Bialek et al. (2001), with variableenergy injection – corresponding to an entropy floor (Ponmanet al., 1999) of between zero and>∼300 keV cm2.

Although we have chosen, for simplicity, to separate our sample into two temperature bands, the points in bothFigs. 5.4 & 5.5 are consistent with agradualsteepening inMgas−TX towards cooler systems. Such behaviouris seen in the analytical model of Dos Santos & Dore (2002), who predict an asymptotic slope of 1.5 in clusters,smoothly increasing toMgas∝ T3 in the cool group regime. Their model matches the data of Mohret al. (1999)well, although this sample does not extend far enough to constrain the low mass slope of the relation, as previouslymentioned. Other theoretical studies have also been shown to provide good agreement withMgas−TX observations(McCarthy et al., 2002), as have numerical simulations (Loewenstein, 2000; Bialek et al., 2001). However, wedefer discussion of their findings, in the wider context of the implications for heating and/or cooling of the IGM,to section 5.4.2.

In Fig. 5.5 we have plottedMgas as a function of temperature, this time evaluated within ournominal virialradius ofR200, rather thanR500; R500 corresponds approximately to 0.66R200 (Chapter 3). The general trends seenin Mgas,500 are reproduced, although the logarithmic slope of the relation is lower than forMgas,500 in both the cooland hot datasets, as well the whole sample (2.01±0.09; see the lower half of Table 5.3). In Chapter 2 we showedthat the logarithmic slope of theM−TX relation withinR200 was 1.89±0.04 – significantly steeper than the self-similar prediction. Therefore, it is not surprising that the Mgas−TX relation is not self-similar, since we believeit is the emission-weighted temperature and not the total gravitating mass which is responsible for the departurefrom simple expectation. However, that the slope of ourMgas−TX relation is steeper still, points to a systematicvariation in gas fraction with temperature – i.e.Mgas is not directly proportional to the total mass. This is exactly


Radius Range Index Normalization Scattera

( keV) (logM⊙)

R500 0.6–17 2.20±0.08 12.00±0.05 1.23< 2.1 2.85±0.35 11.76±0.08 0.89≥ 2.1 1.84±0.10 12.24±0.07 1.09

R200 0.6–17 2.01±0.09 12.35±0.06 1.39< 2.1 2.70±0.48 12.14±0.11 1.10≥ 2.1 1.77±0.11 12.52±0.07 1.10

Table 5.3: Summary of results for the power lawMgas−TX fitting, over two different temperature ranges and the combined range, includingthe two galaxies. Gas masses have been computed withinR500 andR200. See discussion in text. Notes:aMultiples of the statistical scatterexpected from the errors alone.

as was found in Chapter 3 and is consistent with the strong trend in the gas fraction as a function of temperatureobserved in Chapter 2. The fact that theMgas−TX relation is flatter withinR200 compared toR500 indicates thatthe gas fraction at the virial radius in the cooler systems has partially ‘caught up’ with that in the more massiveclusters. This is consistent with a weaker trend in gas fraction with temperature atR200, compared to 0.3R200

(Chapter 2).The S0 galaxy NGC 1553 (the coolest object in our sample) is anobvious outlier on theMgas−TX relation, as

it was in theM−TX relation (Chapter 2). It is possible that its whole X-ray halo has been rebuilt from stellar massloss alone (cf. O’Sullivan et al., 2001), since the mass of the stars is significant compared to the total gravitatingmass of the halo. Exclusion of this object and the other galaxy in our sample (NGC 6482) has little effect on the fitto the whole sample, although it does reduce the scatter slightly – to 1.15 times the statistical scatter. However, forthe fit below 2.1 keV, the logarithmic slope of the relation isflattened from 2.85±0.35 (2.70±0.47) to 2.73±0.37(2.50±0.49), when the galaxies are omitted from the fit withinR500 (R200).

5.3.5 Abell richness

The Abell richness of a cluster is defined as the number of galaxies in the rangem3 to m3 + 2, wherem3 is thephotometric red band magnitude of the third brightest cluster member. As such, it provides a means of quantifyingthe richness of a cluster independently of the spatial distribution or dynamics of its galaxies. This parameter wasused to select the Abell catalogue of clusters of galaxies (Abell, 1958), a subset of which form the majority of oursample of virialized systems. It serves as a useful probe of the galaxy population, as well as a proxy for clustermass, since it minimizes the dependence on physical distance to the cluster.

We present a histogram of the Abell richness of our clusters in the left panel of Fig. 5.6; the values are listedin Table 5.1. We have plotted this quantity against X-ray temperature in Fig. 5.7, and find a positive correlationof 3.4σ significance. It can be seen from the graph that the distribution of points flattens above a number countof ∼80, indicating that that this parameter is less sensitive tothe depth of the potential well in larger haloes. Abroadly similar relation between Abell richness and X-ray temperature was found by Edge & Stewart (1991a). Wenote also that the optical luminosity (derived in Chapter 3)and total mass (from Chapter 2) correlate similarly withthe number counts – both at the 3.5σ level – and a slightly stronger correlation (3.9σ) is found withMgas.

5.3.6 Bautz-Morgan type

The Bautz-Morgan (BM) classification scheme (Bautz & Morgan, 1970) was devised to catalogue clusters ofgalaxies according to the contrast between the central galaxy and the other galaxies in the cluster (see Bahcall,1977, for a review). There are three main categories, which are defined as follows:

Type I are clusters containing a centrally located cD galaxy

Type II are clusters where the brightest members are intermediate in appearance between cD galaxies and Virgo-or Coma-type normal giant ellipticals.

Type II are clusters containing no dominant galaxies.

5.4. DISCUSSION 93

Figure 5.6: Histograms of Abell number counts (left panel) and Bautz-Morgan classification (right panel). We have used the following notationfor the Bautz-Morgan type: 1=I; 2=I-II; 3=II; 4=II-III; 5=III.

In addition, there are two intermediate categories, type I-II and II-III; for simplicity, we use the notation 1=I; 2=I-II;3=II; 4=II-III; 5=III in the following analysis.

The term cD galaxy refers to those galaxies which resemble a giant elliptical with an extended, slowly decreas-ing envelope. Although much more common in rich cluster environment, cD galaxies have also been identified inpoorer clusters (Morgan, Kayser, & White, 1975; Albert, White, & Morgan, 1977). We have assigned a BM typeI to the 4 MKW and 2 AWM clusters in our sample, although we notethat some members of those samples didnot have centrally located cD galaxies (see Morgan et al., 1975; Albert et al., 1977), as is strictly required of thatclassification.

We present a histogram of the BM types for those clusters for which we have data in the right panel of Fig. 5.6.The proportions of the different types are in good agreementwith those from the sample of Edge & Stewart(1991a). It can be seen that the majority (17 out of 42) are of type I or I-II. This is not surprising, since regularclusters are more likely to have an early BM type (Bahcall, 1977), and we have selected our sample on the basis ofa relaxed X-ray morphology (see Chapter 2). However, we find no evidence of a significant correlation (i.e.< 1σ)between this parameter and cluster richness or luminosity,velocity dispersion, mass, temperature or mass-to-lightratio. This is consistent with the findings of Koranyi & Geller (2002), who note that the presence of a cD galaxyin the clusters in their sample does not seem to affect their dynamical properties significantly. They conclude thatthe formation of cD galaxies is predominantly influenced by local effects, rather than the global properties of thecluster in which it resides. The absence of a trend between BMtype and mass-to-light ratio was also reported byGirardi et al. (2000) for 86 clusters in their sample.

5.4 Discussion

5.4.1 Velocity dispersion

The velocities of the individual galaxies in groups and clusters provide useful information on the structure ofvirialized systems, in addition to that gained from studying the behaviour of the IGM. Unlike the hot gas, thegalaxies are not as susceptible to bias caused by non-gravitational heating or cooling and therefore permit animportant cross-check on the dynamical state of the haloes in these objects. However, the galaxy velocity dispersionis sensitive to other mechanisms which can explain the departures from self-similarity observed in theσ−TX andσ−M relations in sections 5.3.1 and 5.3.2 above. Here we addressthe key factors that can affect the measurementof σclus, which can be summarised as follows:

(1) Velocity substructure

(2) Metric aperture radius

(3) Velocity bias

(4) Orbital isotropy


Figure 5.7: X-ray gas temperature as a function of Abell galaxy number counts. The positive correlation between these two quantities has asignificance of 3.4σ.

The first of these is the extent to which the galaxies are in equilibrium within the halo. The presence ofsubstructure within a cluster or bulk motions of galaxies within the halo can produce misleading results, especiallyif the distribution of recession velocities is significantly non-Gaussian (Bird, 1995; Girardi et al., 1996). Given thatour sample has been selected on the basis of a fairly relaxed X-ray morphology, it is reasonable to expect that grosslarge scale substructures should not be present. We note that the fit to the clusters above 2.1 keV reduced the scatterin theσ−TX relation to a level consistent with the measurement errors.This indicates that any scatter associatedwith velocity substructure must be quite small. In addition, since such a bias is unlikely to vary systematically withhalo mass, it is unable to account for the breaking of self-similarity which we observe.

Secondly, the size of the aperture within which galaxy redshifts are measured is important, sinceσclus may varywith radius . A simulation of the formation of an X-ray cluster comparing 12 different numerical codes revealsthat in all cases the velocity dispersion of the dark matter,σdm, decreases with radius in the outer regions of thehalo (Frenk et al., 1999); a result also confirmed by the simulations of Metzler & Evrard (1997) and the analyticalmodel of Łokas & Mamon (2001). The effects of galaxy merging in the central regions of the halo can also leadto a radially decreasing velocity dispersion profile (Menci& Fusco-Femiano, 1996). Observationally, there isevidence of a dependence ofσclus on the aperture radius: Girardi et al. (1996) find an average absolute differencebetweenσclus computed at 0.5h−1

100 Mpc and the global value of∼90 km s−1, for a large sample of clusters. It isdifficult to asses the impact of this effect on our results – clearly it is likely to introduce some extra scatter intothe measurements ofσclus but, as with the problem of velocity substructure, it is not obvious that there will be anysystematic variation between groups and clusters. However, Girardi et al. (1996) find that the velocity dispersionprofile tends to flatten in the outer regions of those clusterswhich are not strongly affected by merging. Thuswide-aperture measurements ofσclus are likely to be largely insensitive to this bias.

The third issue is that of the extent to which the galaxies trace the underlying gravitational potential. This canbe quantified as the ratio ofσclus to σdm – referred to as the velocity bias. In the absence of any such bias, i.e.when the ratio is unity on all mass scales, any systematic behaviour inσclus directly reflects the properties of thedark matter distribution. However, the dark matter is largely unaffected by physical processes that can influencebaryonic matter. One mechanism which can achieve this is dynamical friction. Metzler & Evrard (1997) find thatthe galaxy distribution in their simulations is more centrally concentrated on average than that of the dark matter.They attribute this to the effect of dynamical friction transferring energy from the galaxies to the dark matter.However, even in the most extreme case, dynamical friction can alter the specific energy of the dark matter by

5.4. DISCUSSION 95

only a few per cent at most (Metzler & Evrard, 1997). Consequently, we expect to recover a relationship of theform σdm ∝ T0.5; such a trend is observed in an ensemble of simulated clusters (Eke et al., 1998), consistent withself-similarity in the dark matter.

Although the effects of dynamical friction on the dark matter are essentially negligible, this is not the case forthe galaxies, whose combined mass is very small by comparison. Metzler & Evrard (1997) find a mass dependentvelocity bias, which acts to steepen theσ−TX relation compared to the virial prediction ofσclus ∝ T0.5. Dave et al.(2002) find a weak but discernible correlation in the velocity bias, in the range 100<∼σ<

∼500 km s−1, based oncosmological simulations. Thus, there is a trend for cool groups to have their velocity dispersionunderestimatedand for clusters to have their velocity dispersionoverestimated: for σclus = 100 km s−1, the velocity dispersionis lower than the ‘true’ value by 15 per cent. Correspondingly, they find a steeper-than-self-similarσ−TX slopeof 0.88 when usingσclus. Bird et al. (1995) have suggested that dynamical friction may be proportionately moreefficient in the lower velocity dispersion environments of groups, which would lead toσclus being suppressed inthese systems. However, the velocity bias in fairly relaxedclusters cannot be very severe, since optical massestimates are broadly in agreement with those based on X-raydata (Girardi et al., 1998). Apart from dynamicalfriction, σclus can also be suppressed by the effects of galaxy merging, which can transfer orbital kinetic energyto internal degrees of freedom, thus leaving the dark matterspecific energy unchanged (Fusco-Femiano & Menci,1995).

The fourth factor affectingσclus concerns the nature of the galaxy orbits, which may not be completelyisotropic. The model of Łokas & Mamon (2001) shows that, as the orbits become more radial, the velocity disper-sion profile becomes progressively more centrally peaked. Since measurements ofσclus will tend to be weightedmore towards the inner regions of clusters and groups, radial orbits will lead to an increase inσclus. Furthermore,their model demonstrates that haloes with a higher dark matter concentration will give rise to larger values ofσclus,as the velocity dispersion profile is also more centrally peaked in these cases. By contrast, circular orbits – whichare more typical of dynamically evolved systems – show a marked decreasein the central velocity dispersionprofile.

From hierarchical formation, we expect the smallest haloesto form first and therefore be more centrally con-centrated (Salvador-Sole et al., 1998; Avila-Reese et al., 1999; Jing & Suto, 2000). Of these objects, we are lesslikely to observe very highly concentrated haloes as groups, since they would evolve quickly though merging andcooling – to form fossil groups, for example (Ponman & Bertram, 1993; Jones et al., 2000). Therefore, the effectof more circular orbits in these older systems would dominate over their increased concentration, in altering theirvelocity dispersion profiles compared to those of younger clusters with more randomised or non-isotropic pro-files. The net result would be aσ−M relation which is steeper than the self-similar prediction, with an increasedsteepening evident in the group regime, owing to the dearth of highly concentrated haloes with correspondinglylargerσclus. This behaviour is observed in our data, when theσ−M relation is split up according to mass ratherthan temperature – recall that the logarithmic slope of the groups and poor clusters below 1014M⊙ is 0.63±0.18,compared to 0.47±0.05 for the more massive systems.

In the case of theσ−TX relation, another possible explanation for the steepeningof the logarithmic slopetowards less massive haloes is the effect of preheating on the temperature of the IGM (e.g. Loewenstein, 2000).Bird et al. (1995) attribute their slope of 0.61±0.13 to a combination of protogalactic winds and dynamical friction;Helsdon & Ponman (2000b) point out that their results are consistent with the predictions of preheating models.However, since we also observe a departure from self-similarity in theσ−M relation (section 5.3.2), we concludethat the effect of energy injection systematically boosting the X-ray temperature of cooler systems is not able tofully account for the trends that we observe in theσ−TX relation.

5.4.2 Implications for heating/cooling of the IGM

The heating effect on the IGM of energy injection from galaxywinds or active galactic nuclei has also been pro-posed to explain the steepening evident in theMgas−TX relation using numerical and analytical models (Loewen-stein, 2000; Bialek et al., 2001; McCarthy et al., 2002). Theeffects of such non-gravitational heating will ad-ditionally include a significant modification of the gas density profile, leading to a reduction in the gas mass incluster cores (McCarthy et al., 2002). However the magnitude of the heating required to match the observationsis rather large. Loewenstein (2000) postulates a cluster-wide average of∼1 keV per particle in order to providea reasonable fit to the data of Mohr et al. (1999), compared to the amount of 0.3 keV per particle inferred fromX-ray observations (Lloyd-Davies et al., 2000). McCarthy et al. (2002) require an entropy floor of>∼300 keV cm2

for their model to reproduce the observedMgas−TX trend, at least double that observed (Ponman et al., 1999;


Lloyd-Davies et al., 2000). Although the entropy floor of 55–150 keV cm2, proposed by Bialek et al. (2001) ismuch more modest, this has been attributed to an underestimation of the amount of heating required, due to theinclusion of more cool systems in their simulations than were present in the comparison sample (see section 5.2 ofMcCarthy et al., 2002, for a detailed discussion).

The evidence from our own studies points to a systematic variation in the gas fraction with temperature (Chap-ter 2; Chapter 3) as also found by David et al. (1995). Although such a variation is consistent with preheatingmodels where the shock boundary extends beyondR200 in less massive haloes (Tozzi et al., 2000), a number ofpreheating models assume a universal gas fraction in groupsand clusters (Loewenstein, 2000; Bialek et al., 2001;McCarthy et al., 2002). It is possible that preheating can account for the slight steepening of theσ−TX relationcompared toσ−M andσ−LB,j. In any case, the combination of heating from energy injection and a systematicvariation in concentration with halo mass can explain at least some of the scatter in these and other scaling relations(Lloyd-Davies et al., 2002).

5.5 Conclusions

Building on a detailed X-ray analysis of the temperature anddensity structure of the intergalactic medium in asample of 66 virialized systems, we have studied the scalingproperties of galaxy velocity dispersion as a functionof X-ray temperature, total mass and optical luminosity. Inaddition, we have investigated the relation between totalgas mass and temperature, evaluating the former within two characteristic radii, enclosing a mean total density of200 and 500 times the critical density of the Universe atz= 0.

We measure a logarithmic slope for theσ−TX relation of 0.64±0.04, which is significantly steeper than thevalue of 0.5 expected for a simple isothermal and hydrostatic (i.e. self-similar) model. We use an orthogonaldistance regression technique to perform the fitting, whichallows for errors on both parameters. It has been shownthat this is much more reliable that an ordinary least squares regression, which tends to significantly underestimatethe logarithmic slope of the relation (Bird et al., 1995; Wu et al., 1998; Xue & Wu, 2000). We find good agreementwith the results of other studies based on this type of fittingtechnique. To compare the behaviour of groups andclusters, we have separated our sample according to a simpletemperature cut. We find that clusters hotter than2.1 keV have aσ−TX slope of 0.67± 0.06, compared to 1.18± 0.24 for the systems cooler than 2.1 keV. Ourgroup slope is consistent with that found by Helsdon & Ponman(2000b), who measure 1.1±0.2 for a sample of20 loose groups of galaxies.

We also examine theσ−M relation, and find a logarithmic slope of 0.39± 0.03, which is inconsistent withthe self-similar prediction of 1/3 at the 2σ level. As with theσ−TX relation, the slope above 2.1 keV is similar tothat for the full sample, whereas the systems cooler than 2.1keV exhibit a further steepening, to 0.53±0.15. Thisgeneral behaviour is also seen in theσ−LB,j relation, for which we find a slope of 0.42±0.06, with no evidenceof any intrinsic scatter. An almost identical slope is foundfor the separate fits to the data above and below 2.1 keV.Our results point to a systematic variation inσclus with halo mass (see section 5.4.1 for a full discussion), in orderto account for the slope of theσ−M relation. However, we can’t rule out the possibility that gas heating by energyinjection is also responsible for some of the steepening observed in theσ−TX relation.

Our analysis of theMgas−TX relation yields logarithmic slopes of 2.20±0.08 and 2.01±0.09 for gas massesevaluated withinR500 andR200, respectively. Using the same temperature cut of 2.1 keV from above, we find aconsiderable steepening of the relation in the group regime. We find broad agreement with the study of Mohr et al.(1999), allowing for the fact that we have many more cooler systems in our sample. We attribute the breakingof self-similarity seen inMgas−TX to the strong trend in gas fraction with temperature found inChapter 2, andnot to the effects of energy injection raising the temperature of the IGM. This is in conflict with the numericalsimulations of Loewenstein (2000) and the analytical modelof McCarthy et al. (2002), which assume a universalgas fraction. The excessive level of heating required by these models to match the observations is also inconsistentwith observations of the entropy floor in groups and poor clusters of galaxies (Ponman et al., 1999; Lloyd-Davieset al., 2000).

Finally, we investigate the correlation of Abell richness and Bautz-Morgan type with X-ray properties. We finda moderate trend between the former quantity and the X-ray temperature, consistent with the results of Edge &Stewart (1991a). However, we find no evidence of any significant trends with Bautz-Morgan type, in agreementwith the conclusions of Koranyi & Geller (2002).

Chapter 6

Conclusions

6.1 Summary of Main Results

The work presented in this thesis represents the largest detailed study to date of the X-ray properties of virializedsystems spanning a wide range of masses. We have demonstrated that these systems do not scale self-similarly, aspredicted by the simplest expectations of their formation.Specifically, the gaseous intergalactic medium (IGM) hasbeen shown to be less dense and more spatially extended in less massive haloes (Chapters 2 & 3). The entropy ofthe hot gas provides clear evidence of the influence of non-gravitational physical processes acting on this materialprior to and/or after collapse and virialization has taken place (Chapter 4). In contrast, the scaling properties of thestellar mass component are shown to be broadly consistent with self-similarity, albeit with a weak systematic trendtowards enhanced star formation in groups compared to clusters of galaxies (Chapter 3).

The variation of the velocity dispersion of the galaxies within the halo as a function of X-ray temperature, totalmass and optical luminosity has also been found to deviate from simple expectation (Chapter 5). The implicationis that the haloes of less massive systems are more dynamically evolved, as a result of having formed earlier– consistent with a hierarchical structure formation scenario. This gives rise to a systematic variation in haloconcentration with mass, as indicated by numerical and analytical models, which may also explain the steepeningof theM−TX relation with respect to the self-similar logarithmic slope of 3/2 (Chapter 2), as well as the trends intotal mass and dark matter density seen in Chapter 3.

By combining optical and X-ray data for a subset of roughly half of our main sample, we have been ableto determine the distribution of all the fundamental mass components of virialized systems: hot gas, stars anddark matter. This integrated approach provides powerful constraints on models of non-gravitational heating and/orradiative cooling, which have been invoked to explain the observed departures from self-similarity in groups andclusters. We are able to rule out purely heating-only models, in which energy is injected into the pre- or post-virialized IGM by one or both of galaxy winds and active galactic nuclei (AGN). Our results are in much betteragreement with models where significant cooling is the primary mechanism for eliminating the lowest entropy gasand modifying the density distribution of the IGM. The best match to the observations is provided by a modelwhich incorporates in a self-consistent fashion both radiative cooling and energy injection from subsequent starformation. However, there are indications that this simplescheme may provide an incomplete description of allour data.

6.2 Future Work

An important area for further exploration is the propertiesof early-type elliptical haloes, and how they comparewith those of groups and clusters. As was pointed out in Chapter 2, such objects are very rare in our Universe, sincemost X-ray bright galaxies reside in groups (and particularly in the centres of groups). Although the scaling sampleinvestigated in this thesis included only two galaxies – which were carefully selected to avoid any contaminationfrom a group or cluster X-ray halo – it has been shown that their properties are sufficiently interesting as to meritfurther study. The observed deviation from theM−TX relation of NGC 1553 may point to an early formation epoch– where the higher mean density of the Universe leads to haloes of a given mass being hotter – or could result fromsignificant non-gravitational heating from stellar winds.Such heating must have occurred at some point in orderto prevent significant mass drop-out, given the very short cooling time of the X-ray gas (∼6 Gyr at 0.1R200). Thedetailed study of isolated early-type galaxies is feasiblewith the advent ofXMM-Newton, whose field-of-view andsensitivity are ideal for the task. In addition, the unrivalled spatial resolution ofChandrawill provide invaluable

97

98 CHAPTER 6. CONCLUSIONS

information on the discrete source population in these objects and will allow us to unambiguously disentangle thisemission component from the halo contribution, as never before.

The other galaxy, NGC 6482, is even more interesting, since it is likely to be a fossil group (Chapter 2). Weare involved in a proposal to study this object with theChandrasatellite (20 ks with the ACIS-S detector), and theobservation has now been performed. The analysis of these data should prove especially enlightening, given thatthis object may well be the nearest fossil in the Universe, thus affording us an excellent opportunity to study itsproperties in great detail. In particular, it will be interesting to determine the halo dark matter concentration – as aprobe of formation epoch – as well as to investigate the possibility of a central AGN reheating mechanism, sinceno strong cooling flow exists andtcool ∼ 3 Gyr at 0.1R200.

On larger mass scales, the galaxy group regime promises to provide a vital insight into non-gravitational pro-cesses. One key area to investigate is the properties of cooling flows in low mass systems. WhileXMM-NewtonandChandraobservations have demonstrated the failure of the standardcooling flow model in clusters (see Chap-ter 1), this result awaits confirmation in groups. There are good reasons to believe things may be different here:the cooling time of the gas is likely to be lower; heating mechanisms are more effective, since the energy injectedrepresents a greater fraction of the total energy budget compared to clusters; heat conduction is less efficient atlower temperatures (Fabian et al., 2002b). It is therefore reasonable to expect that groups without AGN mighthave gas cooling at the full range of temperatures, as expected from simple models. We will be able to test thishypothesis with data from a successfulChandraproposal to look at a small number of groups with cooling flows,which do not possess strong central radio sources.

On a related theme, we have recently been awardedXMM-Newtontime to observe a poor (∼2 keV) cluster,with the aim of tracing the X-ray emission out to the virial radius. This mass regime is ideal, being small enoughto be sensitive to the effects of non-gravitational heating, but massive enough to allow the full extent of the halo tobe directly observed in a reasonable length of time. More generally, the steady accumulation of publicly availablearchival data fromXMM-NewtonandChandracontinues apace. With a small modification to the Birminghamcluster fitting software (Eyles et al., 1991), we will soon beable to apply our sophisticated model fitting techniques(see Chapter 2) to the analysis of this new generation of high-quality X-ray data on virialized systems. Withbetter data, we will be able to probe a greater volume of the halo and, correspondingly, will be less susceptibleto extrapolation bias, which is currently the major source of systematic uncertainty in the study of global clusterX-ray properties.

Appendix A

Miscellaneous Paper Details

This appendix contains the author and institution lists, keyword sections and acknowledgments from the paperspresented in Chapters 2–5.

Chapter 2

The Birmingham-CfA cluster scaling project - I: gas fraction and theM−TX

relationA. J. R. Sanderson1, T. J. Ponman1, A. Finoguenov2,3, E. J. Lloyd-Davies1,4 and M. Markevitch31School of Physics and Astronomy, University of Birmingham,Edgbaston, Birmingham B15 2TT, UK

2Max-Planck-Institut fur extraterrestrische Physik, Giessenbachstraße, 85748 Garching, Germany3Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 021384Department of Astronomy, University of Michigan, Ann Arbor, MI 48109-1090, USA

Key words: galaxies: clusters: general – galaxies: haloes – intergalactic medium – X-rays: galaxies – X-rays:galaxies: clusters

AcknowledgmentsWe thank Steve Helsdon for providing the software for measuring non-statistical scatter, and Ewan O’Sullivan foruseful discussions and input. AJRS acknowledges financial support from the University of Birmingham. Thiswork made use of the Starlink facilities at Birmingham, the LEDAS data base at Leicester and the NASA/IPACExtragalactic Database (NED).

Chapter 3

The Birmingham-CfA cluster scaling project - II: mass composition and distri-butionA. J. R. Sanderson and T. J. PonmanSchool of Physics and Astronomy, University of Birmingham,Edgbaston, Birmingham B15 2TT, UK

Key words: galaxies: clusters: general – intergalactic medium – X-rays: galaxies: clusters

AcknowledgmentsWe are grateful to Alexis Finoguenov, Ed Lloyd-Davies and Maxim Markevitch for providing the X-ray data andcontributing to the original analysis. AJRS acknowledges financial support from the University of Birmingham.This work has made use of the Starlink facilities at Birmingham, the Automatic Plate Measuring (APM) machinecatalogue at Cambridge and the NASA/IPAC Extragalactic Database (NED).

99

100 APPENDIX A. M ISCELLANEOUSPAPER DETAILS

Chapter 4

The Birmingham-CfA cluster scaling project - III: entropy and similarity ingalaxy systemsT. J. Ponman1, A. J. R. Sanderson1 and A. Finoguenov2,3

School of Physics and Astronomy, University of Birmingham,Edgbaston, Birmingham B15 2TT, UK2Max-Planck-Institut fur extraterrestrische Physik, Giessenbachstraße, 85748 Garching, Germany3Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138


AcknowledgmentsWe are grateful to Ed Lloyd-Davies and Maxim Markevitch for providing the X-ray data and contributing to theoriginal analysis. AJRS acknowledges financial support from the University of Birmingham. This work has madeuse of the Starlink facilities at Birmingham.

Chapter 5

The Birmingham-CfA cluster scaling project - IV: mass, velocity dispersion andtemperature scaling propertiesA. J. R. Sanderson and T. J. PonmanSchool of Physics and Astronomy, University of Birmingham,Edgbaston, Birmingham B15 2TT, UK


AcknowledgmentsWe are grateful to Alexis Finoguenov, Ed Lloyd-Davies and Maxim Markevitch for providing the X-ray dataand contributing to the original analysis. We also thank John Osmond for his help calculating group velocitydispersions. AJRS acknowledges financial support from the University of Birmingham. This work made use ofthe Starlink facilities at Birmingham, the NASA/IPAC Extragalactic Database (NED), the LEDA online catalogueand the Abell clusters of galaxies online data base maintained by Alexander Gubanov.

REFERENCES 101

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University of Birmingham › ~ajrs › papers › ajrs_mini_thesis.pdf · 2010-07-23 · Synopsis Virialized systems, such as clusters and groups of galaxies, are the largest gravitationally