simulations of binary galaxy cluster mergers: modeling real...

Simulations of Binary Galaxy Cluster Mergers: Modeling

Real Clusters and Exploring Parameter Spaces

by

John Arthur ZuHone

A dissertation submitted to the University of Chicago in conformity with the

requirements for the degree of Doctor of Philosophy.

Chicago, Illinois

March, 2009

c© John Arthur ZuHone 2009

All rights reserved

Abstract

We present an investigation of controlled N -body/hydrodynamics high-resolution

simulations of binary galaxy cluster mergers, performed using the FLASH code. In ad-

dition to analyzing the quantities directly from the simulation, we produce simulated

X-ray observations of the cluster ICM and perform standard analyses of the surface

brightness distribution and spectra of the X-ray photons emitted from the hot cluster

gas. Several lines of evidence have suggested that the galaxy cluster Cl 0024+17, an

apparently relaxed system, is actually a collision of two clusters, the interaction oc-

curring along our line of sight. We present a high-resolution N -body/hydrodynamics

simulation of such a collision. We analyze mock X-ray observations of our simu-

lated clusters to generate radial profiles of the surface brightness and temperature

to show that at later times the simulated surface brightness profiles are better fit

by a superposition of two β-model profiles than a single profile, in agreement with

the observations of Cl 0024+17. We determine from our fitted profiles that if the

system is modeled as a single cluster, the hydrostatic mass estimate is a factor ∼2-3

less than the actual mass, but if the system is modeled as two galaxy clusters in su-

perposition, a hydrostatic mass estimation can be made which is accurate to within

∼10%. Additionally, recent lensing observations of Cl 0024+17 suggest the presence

of a ring-like dark matter structure, which has been interpreted as the result of such

a collision. To determine the conditions under which such a feature would form, we

vary the initial velocity anisotropy of the dark matter particles. Our simulations show

a ring feature does not occur even when the initial particle velocity distribution is

highly tangentially anisotropic. Only when the initial particle velocity distribution is

ii

circular do our simulations show such a feature, which is consistent with the halo ve-

locity distributions seen in cosmological simulations. Lastly, we present a fiducial set

of galaxy cluster merger simulations, where the initial mass ratio and the impact pa-

rameter have been varied. By projecting the simulated quantities along the axes of the

computational domain, we produce maps of X-ray surface brightness, temperature,

projected mass density, and simulated X-ray observations. From these observations

we compute the observed X-ray luminosity and fitted spectral temperature, and fit

β-model profiles to compute estimated hydrostatic masses. From this information we

determine the effect of mergers viewed along different projections on these observed

quantities. We also construct simulated maps of galaxies, and test the power of a

commonly employed substructure statistic to probe for the existence of substructure

along the different projections during the merger. Finally, we comment on other as-

pects of our simulations, such as comparisons to existing merging clusters; and the

mixing of the intracluster medium due to merging, and resulting cluster entropy and

cooling time profiles.

iii

For Dad and Mom

iv

Acknowledgements

I would first like to thank my advisor, Don Lamb, for shepherding me over the past

few years through this process, especially through some very difficult stages. I will

always appreciate his determination to make sure that I not only succeed but excel.

I have learned some of the most valuable and hard-won lessons under his guidance.

He has worked very hard for me in more ways than I can describe.

I also owe a great debt of gratitude to Paul Ricker, professor at the University of

Illinois at Urbana-Champaign. Paul has shaped my knowledge and understanding of

computational astrophysics and the study of clusters of galaxies. Most importantly,

he has always been encouraging, and has provided me with unique opportunities to

advance my career by introducing me to the right people and always offering helpful

advice to steer me in the right direction.

I thank my thesis committee for their efforts to guide me in the development of

this project, their advice, and patience with the process.

I am also especially grateful to Hsiang-Yi Yang at the University of Illinois for

initially developing the mock X-ray observation pipeline used in this work and for

helping me to extend its capabilities.

I am grateful to Andrey Kravtsov, Douglas Rudd, Carlo Graziani, John Davis,

Joe Mohr, and Maxim Markevitch for useful discussions and advice. I also want to

acknowledge James Jee for information about his lensing study of Cl 0024+17.

I want to thank and acknowledge the efforts of the entire Flash Center team at

the University of Chicago, especially the Code Group, who have made running these

simulations possible. In this regard I will specifically single out Anshu Dubey, Paul

Rich, Klaus Weide, Lynn Reid, and Chris Daley for special thanks. I will miss going

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out to lunch with you all. I would also like to thank Katherine Riley at Argonne

National Laboratory for her assistance in getting the FLASH code running on the

BlueGene/P computer at ANL.

I want to thank my parents, Dan and Shelley ZuHone, my grandparents, and

my greater extended family for their years of love, encouragement, and prayers, and

always pushing me to put my best efforts forward.

Calculations were performed using the computational resources of Argonne Na-

tional Laboratory, Lawrence Livermore National Laboratory, and Los Alamos Na-

tional Laboratory.

This work is supported at the University of Chicago by the U.S. Department of

Energy (DOE) under Contract B523820 to the ASC Alliances Center for Astrophys-

ical Nuclear Flashes. JAZ was supported by the Department of Energy Computa-

tional Science Graduate Fellowship, which is provided under grant number DE-FG02-

97ER25308.

vi

Contents

Abstract ii

Acknowledgements v

List of Figures viii

List of Tables ix

1 Introduction 1

1.1 Clusters of Galaxies . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Cosmological Structure Formation and

Cluster Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Observations of Galaxy Cluster Mergers . . . . . . . . . . . . . . . . 41.4 Idealized Simulations of Galaxy Cluster

Mergers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.5 Description of Presented Simulations . . . . . . . . . . . . . . . . . . 5

2 Simulations 8

2.1 Cluster Merger Simulation Method . . . . . . . . . . . . . . . . . . . 82.1.1 Treatment of Hot Gas . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Treatment of Collisionless Dark Matter . . . . . . . . . . . . . 92.1.3 Treatment of Gravity . . . . . . . . . . . . . . . . . . . . . . . 102.1.4 Treatment of the Grid . . . . . . . . . . . . . . . . . . . . . . 112.1.5 Cluster Initial Conditions . . . . . . . . . . . . . . . . . . . . 11

2.2 Simulated X-ray Observations . . . . . . . . . . . . . . . . . . . . . . 12

3 A Line-Of-Sight Galaxy Cluster Collision 16

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163.2 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3.1 Qualitative Description . . . . . . . . . . . . . . . . . . . . . . 203.3.2 Mock X-ray Observations . . . . . . . . . . . . . . . . . . . . 21

vii

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.4.1 The Surface Brightness and Temperature Profiles . . . . . . . 263.4.2 The Reliability of Hydrostatic Mass Estimates . . . . . . . . . 313.4.3 The Effect of A Slightly Off-Axis Collision . . . . . . . . . . . 313.4.4 Implications for X-ray Surveys . . . . . . . . . . . . . . . . . . 32

4 Dark Matter Rings In Colliding Clusters 36

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.4.1 Relevance to Observations of Cl 0024+17 . . . . . . . . . . . . 434.4.2 Comparison with “Ring Galaxies” . . . . . . . . . . . . . . . . 43

5 A Parameter Space Exploration: Variations in Mass Ratio and Im-

pact Parameter 46

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 465.2 Methods and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 48

5.2.1 Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . 485.2.2 Mock X-ray Observations . . . . . . . . . . . . . . . . . . . . 525.2.3 Caveats . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545.3.1 Qualitative Description . . . . . . . . . . . . . . . . . . . . . 545.3.2 Time Dependence of Global Cluster Observables and Derived

Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 605.3.3 A Cluster Substructure Test: The κ-statistic . . . . . . . . . . 735.3.4 ICM Mixing . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 825.4.1 Comparisons of Simulated Mergers to Real Clusters . . . . . 825.4.2 Evolution of Luminosity, Temperature, and Mass Estimates . 895.4.3 Substructure Tests and Merger Geometry . . . . . . . . . . . 925.4.4 ICM Mixing, Cluster Entropy Profiles, and Cooling Time . . 96

6 Summary and Conclusions 100

A Mock Observation Generation and Verification Study 112

B Projected Temperature and Mass Density Maps 116

viii

List of Figures

3.1 Slices of gas density. . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Slices of gas temperature. . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Mock 40 ksec raw counts image. . . . . . . . . . . . . . . . . . . . . . 233.4 Surface brightness vs. projected radius. . . . . . . . . . . . . . . . . . 243.5 Single and double β-model fits. . . . . . . . . . . . . . . . . . . . . . 253.6 Temperature profile vs. projected radius. . . . . . . . . . . . . . . . . 273.7 Cl 0024+17 X-ray surface brightness profile. . . . . . . . . . . . . . . 273.8 Slice through the z = 0 coordinate plane of the simulation of temper-

ature at t = 3.0 Gyr. . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.9 Radial profile of mass-weighted temperature. . . . . . . . . . . . . . . 293.10 Cl 0024+17 Temperature profile. . . . . . . . . . . . . . . . . . . . . 303.11 Number of redshifts per bin vs. redshift, σz = 0.005 and σz = 0.01. . 343.12 Number of redshifts per bin vs. redshift, σz = 0.02 and σz = 0.08. . . 34

4.1 Slices through the z = 0 coordinate plane of dark matter density . . . 404.2 t = 1.5 Gyr snapshot of the β = 1/2 simulation. . . . . . . . . . . . . 414.3 t = 1.5 Gyr snapshot of the β = 0 simulation. . . . . . . . . . . . . . 414.4 t = 1.5 Gyr snapshot of the β = -3/2 simulation. . . . . . . . . . . . 414.5 t = 1.5 Gyr snapshot of the β = -5/2 simulation. . . . . . . . . . . . 424.6 t = 1.5 Gyr snapshot of the β = -17/2 simulation. . . . . . . . . . . . 424.7 t = 1.5 Gyr snapshot of the β = −∞ simulation. . . . . . . . . . . . . 424.8 Angle-averaged DM density profiles. . . . . . . . . . . . . . . . . . . . 44

5.1 Schematic representation of initial merger geometry. . . . . . . . . . . 515.2 Examples of simulated Chandra observations. . . . . . . . . . . . . . 535.3 X-ray spectral temperature vs. time for the 1:1 mass ratio mergers. . 635.4 X-ray spectral temperature vs. time for the 1:3 mass ratio mergers. . 635.5 X-ray luminosity vs. time for the 1:1 mass ratio mergers. . . . . . . . 645.6 X-ray luminosity vs. time for the 1:3 mass ratio mergers. . . . . . . . 665.7 Estimated β-model mass within r500 vs. time for the 1:1 mass ratio

mergers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

ix

5.8 Estimated β-model mass within r500 vs. time for the 1:3 mass ratiomergers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.9 Estimated and actual masses within r500 vs. time for simulation S1. . 715.10 Estimated and actual masses within r500 vs. time for simulation S2. . 715.11 Estimated and actual masses within r500 vs. time for simulation S3. . 715.12 Estimated and actual masses within r500 vs. time for simulation S4. . 725.13 Estimated and actual masses within r500 vs. time for simulation S5. . 725.14 Estimated and actual masses within r500 vs. time for simulation S6. . 725.15 Computed κ-statistic vs. time for the 1:1 mass ratio mergers. . . . . . 755.16 Computed κ-statistic vs. time for the 1:3 mass ratio mergers. . . . . . 755.17 Computed significance of the κ-statistic vs. time for the 1:1 mass ratio

mergers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.18 Computed significance of the κ-statistic vs. time for the 1:3 mass ratio

mergers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.19 The degree of gas mixing in each simulation, at a time t = 10 Gyr. . 815.20 MACS J0025.4-1222: Optical, X-ray, and lensing. . . . . . . . . . . . 835.21 Projected mass density and X-ray emission for the equal-mass, head-on

merger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.22 Comparison of real and simulated temperature profiles for MACS J0025.4-

1222. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.23 Possible merger scenario for Abell 2163. . . . . . . . . . . . . . . . . . 875.24 Simulated bubble-plot for possible merger scenario for Abell 2163. . . 875.25 Galaxy positions and projected mass density contours in the head-on

mergers shortly after initial core passage. . . . . . . . . . . . . . . . . 885.26 Projected mass density in the head-on mergers shortly after initial core

passage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 895.27 β-model mass profiles for varying values of the core radius rc. . . . . 915.28 Galaxy bubble plots for the 1:1 mass ratio merger with zero impact

parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.29 Galaxy bubble plots for the 1:1 mass ratio merger with impact param-

eter b = 464 kpc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.30 Galaxy bubble plots for the 1:3 mass ratio merger with zero impact

parameter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 955.31 Galaxy bubble plots for the 1:3 mass ratio merger with impact param-

eter b = 932 kpc. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.32 Entropy profiles for at t = 10 Gyr. . . . . . . . . . . . . . . . . . . . 985.33 Cooling time profiles for at t = 10 Gyr. . . . . . . . . . . . . . . . . . 99

A.1 Isothermal cluster temperature profile. . . . . . . . . . . . . . . . . . 115

B.1 “Spectroscopic-like” temperature maps in the z-projection for simula-tion S1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

x

B.2 Projected mass density maps in the z-projection for simulation S1. . . 117B.3 “Spectroscopic-like” temperature maps in the x-projection for simula-

tion S1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118B.4 Projected mass density maps in the x-projection for simulation S1. . 118B.5 “Spectroscopic-like” temperature maps in the z-projection for simula-

tion S2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119B.6 Projected mass density maps in the z-projection for simulation S2. . . 119B.7 “Spectroscopic-like” temperature maps in the y-projection for simula-

tion S2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120B.8 Projected mass density maps in the y-projection for simulation S2. . . 120B.9 “Spectroscopic-like” temperature maps in the x-projection for simula-





tion S4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125B.18 Projected mass density maps in the z-projection for simulation S4. . . 125B.19 “Spectroscopic-like” temperature maps in the x-projection for simula-

tion S4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126B.20 Projected mass density maps in the x-projection for simulatoin S4. . 126B.21 “Spectroscopic-like” temperature maps in the z-projection for simula-





tion S6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

xi

B.30 Projected mass density maps in the y-projection for simulation S6. . . 131B.31 “Spectroscopic-like” temperature maps in the x-projection for simula-

tion S6. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132B.32 Projected mass density maps in the x-projection for simulation S6. . 132

xii

List of Tables

3.1 Initial Cluster Parameters . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Fitted Cluster Parameters for a Single β-model . . . . . . . . . . . . 253.3 Fitted Cluster Parameters for a Double β-model . . . . . . . . . . . . 263.4 Fitted Average Spectral Temperatures . . . . . . . . . . . . . . . . . 263.5 Estimated and Exact Masses at R = 35” . . . . . . . . . . . . . . . . 323.6 Fitted parameters for Redshift Histograms with Varying σz . . . . . . 343.7 ∆χ2 for Different Exposure Times and Redshifts . . . . . . . . . . . . 35

4.1 Velocity Anisotropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.2 Initial Cluster Parameters . . . . . . . . . . . . . . . . . . . . . . . . 39

5.1 Initial Cluster Parameters . . . . . . . . . . . . . . . . . . . . . . . . 505.2 Initial Cluster Parameters . . . . . . . . . . . . . . . . . . . . . . . . 505.3 κ-statistic Tests on Simulated Single Clusters . . . . . . . . . . . . . 79

A.1 Fit to Single Cluster Test, texp = 240 ks . . . . . . . . . . . . . . . . 113A.2 Fit to Double Cluster Test, texp = 240 ks . . . . . . . . . . . . . . . . 114A.3 Systematic Trends in Radial Fits with Increasing Exposure Time . . . 114A.4 Spectral Fit for a Single Cluster, texp = 4 Ms . . . . . . . . . . . . . . 114

xiii

Chapter 1

Introduction

1.1 Clusters of Galaxies

Galaxy groups and clusters are the largest objects in the universe that are gravita-

tionally bound. Groups and clusters may contain anywhere from tens to thousands of

galaxies as seen in visible light. However, nearly a century of investigation of galaxy

clusters has demonstrated that there is literally “more than meets the eye.” Observa-

tions of clusters in the X-ray band have shown that most of the baryonic component is

in the form of plasma (e.g. Kellogg et al. 1972; Forman et al. 1972; Mitchell et al. 1976;

Serlemitsos et al. 1977; Sarazin 1988). This gas is diffuse (n ∼ 10−2 − 10−4 cm−3),

hot (T ∼ 107 − 108 K), and magnetized (B ∼ 0.1 − 3µG) (Carilli & Taylor 2002).

The presence of the hot gas is confirmed by observations in the microwave band, as

cosmic microwave background (CMB) photons are inverse-Compton scattered by the

electrons in the plasma and cause a corresponding decrement in the CMB intensity

(Sunyaev & Zeldovich 1972; Birkinshaw et al. 1984). In addition, measurements of

galaxy velocity dispersions and distortion of background galaxies due to gravitational

lensing by clusters indicate that most of the matter in galaxy clusters is in the form

of non-baryonic “cold dark matter” (CDM) (for early indications of this, see Zwicky

1937; Bahcall & Sarazin 1977; Mathews 1978).

Due to their size and composition, clusters of galaxies provide a good represen-

tation of the material properties of the universe as a whole. Thus, they are useful

1

2

for resolving important questions of cosmology and fundamental physics, as well as

being interesting in their own right. First, clusters are useful tracers of cosmic evolu-

tion. Since galaxy clusters comprise the largest objects whose masses can be reliably

measured, they can be used to determine the amount of structure in the universe on

scales of 1014 − 1015M⊙. Comparisons of the cluster mass distribution in the present

day to the mass distribution at earlier times yields the time dependence of structure

formation and can be used to constrain cosmological parameters. Combined with

other observational techniques such as analysis of the CMB and distance measure-

ments of Type Ia supernovae, powerful constraints can be derived on the amount of

dark matter and dark energy in the universe as well as properties such as the dark

energy equation of state (Voit 2005).

Second, due to their deep gravitational potential wells, clusters are essentially

“closed boxes” that retain most of their gaseous and dark material. Thus, they are

excellent cosmic laboratories for studying various important questions in astrophysics.

X-ray observations of clusters can be used to study the turbulence, magnetic fields,

and viscous properties of the ICM (Markevitch & Vikhlinin 2007). Gravitational lens-

ing and observations of hard X-rays and gamma rays provide observational constraints

on the nature of dark matter (e.g. self-interaction and/or annihilation cross-section)

(Markevitch et al. 2004; Randall et al. 2008).

1.2 Cosmological Structure Formation and

Cluster Mergers

Within the CDM paradigm of cosmological structure formation, structures in the

universe form in a “bottom-up” fashion. The smallest structures collapse first and

build up over billions of years into larger and larger structures. Today, the largest

structures formed in this fashion are clusters and groups of galaxies. Because this

process is still ongoing, many clusters of galaxies show current or recent signs of

merging with other clusters. Two prominent examples of recent note are 1E 0657-56

(the “Bullet Cluster”) (Markevitch et al. 2002; Clowe et al. 2006), and Abell 520

3

(Markevitch et al. 2005; Mahdavi et al. 2007).

Cluster mergers have important implications for astrophysics. On the one hand,

merging complicates the determination of cluster masses. Masses are most easily de-

termined for a large sample of clusters by determining “scaling relations” between

the mass of the clusters and “mass proxies.” These quantities, such as X-ray temper-

ature TX , X-ray luminosity LX , and the recently proposed YX (Kravtsov et al. 2006)

are expected to have simple power-law scalings under the assumption of hydrostatic

equilibrium (Kaiser 1986). These mass relations are strictly accurate only under this

assumption.

As clusters merge, the observable quantities used as mass proxies are altered from

the expected scalings, which rely on the assumption of hydrostatic equilibrium. The

emissivity of the gas is ǫX ∝ ρ2gΛ(T ), where for the assumption of pure bremsstrahlung

emission Λ(T ) ∝ T 1/2. Variations in both ρg and T will alter the observed X-ray

luminosity. Mergers between clusters create shocks which compress and heat the gas,

increasing both ρg and T . Therefore, mergers boost the observed X-ray luminosity

LX and X-ray spectral temperature TX , most significantly at the core passage time

of the merger (Ricker & Sarazin 2001). At late times, re-accreting material ejected

by the merger will gradually increase the X-ray luminosity and temperature (Poole

et al. 2007).

Many relaxed galaxy clusters have bright, cool cores due to the energy losses from

bremsstrahlung. The gas cools, sinks deeper in the potential of the cluster, and the

central luminosity increases as the density increases. It has been shown that the

position of a cluster’s observable quantities relative to the mean scaling relation is

correlated with the morphology of the gas core (McCarthy et al. 2004). Mergers

could disrupt these cool cores, moving them away from or toward the scaling relation

average. In addition, clusters with complex density and temperature distributions

due to merging may have this complexity hidden due to projection effects. Since

the X-ray luminosity is strongly dependent on density, temperatures obtained from

fitting models to the X-ray spectrum will be biased towards the denser gas seen in

projection. A particular but rare case of this situation is when merging clusters are

co-aligned along the line of sight. Understanding the effect of mergers on the X-ray

4

scaling relations is vital in accounting for the effects of the temperature and luminosity

boosts, direction of projection on the sky, and the state of the cluster’s core.

On the other hand, cluster mergers make it possible for observers to probe the

nature of dark matter and the material properties of the ICM. For example, X-ray

observations of the ICM combined with weak lensing analysis that demonstrate a

clear separation between the bulk of the cluster mass and the bulk of the baryonic

component (such as the “Bullet Cluster” (1E 0657-56), Clowe et al. 2006) can be used

to place constraints on the self-interaction cross section of the dark matter particles

(Springel & Farrar 2007; Markevitch et al. 2004; Randall et al. 2008). Mergers also

often result in sharp features such as shock fronts and “cold fronts” in the ICM,

which can be used to place constraints on the material properties of the gas such as

the magnetic field strength and its viscosity and conductivity (Markevitch & Vikhlinin

2007).

1.3 Observations of Galaxy Cluster Mergers

Many of the recent advances in the study of the ICM have been accomplished with

the Chandra X-ray Observatory. In the nearly ten years since launch in 1999 Chandra

has made groundbreaking discoveries in X-ray astronomy, including observations of

supernova remnants, black holes, and clusters of galaxies (Weisskopf et al. 2002). The

high angular resolution of Chandra (∼ 0.5 seconds of arc per pixel, corresponding

to a few kpc for nearby clusters) has allowed astrophysicists to probe the ICM in

exceptional detail. This development has opened up a host of new scientific questions

concerning the ICM for simulations to address, since this is a regime where the spatial

resolution of the observations is comparable to or sometimes even better than the

resolution of the simulation. This also has motivated the forging of more direct links

between simulation and observation. One technique being increasingly adopted is

the use of simulated observations of X-ray photons constructed from hydrodynamic

simulations (see, e.g. Gardini et al. 2004; Nagai et al. 2007a). This allows essentially

direct comparisons between the simulations and already existing observations and

serves as a guide for future observations.

5

Recently, many studies of individual clusters have made use of datasets from mul-

tiple observing techniques to determine the cluster properties (see, e.g. Govoni et al.

2004; Jee et al. 2007; Mahdavi et al. 2007; Bradac et al. 2008; Owers et al. 2008a,b,

for some examples). These studies combined information from optical, X-ray, and

radio wavelengths to provide a fuller picture of the dynamics of clusters, especially

merging ones. This includes analysis of substructure in the galaxy distribution, weak

and strong-lensing reconstructions of the cluster mass distribution, density and tem-

perature profiles and maps from X-ray observations, and constraints on magnetic

field strength from observations of radio halos. Simulation-based studies of clusters

which are able to reproduce mock observations of these various kinds are invaluable

for making direct connections between the simulated models and the real clusters.

1.4 Idealized Simulations of Galaxy Cluster

Mergers

Though cosmological simulations are common, and offer one avenue for studying

the merging process, for specific cluster mergers or a restricted parameter space of

mergers they are not ideal, as it is difficult to tease apart the effects of the many

individual merging events that occur. A second type of simulation that is more

appropriate is an idealized simulation where two or more model clusters are set up

in an isolated box along a merging trajectory. This kind of simulation is useful for

simulating specific cluster merger scenarios as well as exploring parameter spaces over

fundamental merger characteristics (such as mass ratio, impact parameter, merger

energy) in a controlled fashion (e.g, Roettiger et al. 1997; Ricker & Sarazin 2001;

Ritchie & Thomas 2002; Poole et al. 2006, 2007, 2008). Such simulations have been

used to place constraints on quantities such as the dark matter self-interaction cross

section (Randall et al. 2008) and investigate possible mechanisms for generating cold

fronts in relaxed clusters (Ascasibar & Markevitch 2006). It is this kind of simulation

that we employ in this study.

6

1.5 Description of Presented Simulations

In this study we present a set of idealized merger simulations with the FLASH

code that fulfill these twin objectives of simulating specific merger scenarios as well

as exploring a more general merger parameter space. The specific cluster merger we

choose to simulate is Cl 0024+17, recently observed by Chandra (Ota et al. 2004)

and subjected to a number of weak lensing studies (Tyson et al. 1998; Broadhurst et

al. 2000; Comerford et al. 2006; Jee et al. 2007). This apparently relaxed cluster has

strong observational indications that it is actually two clusters that have undergone

a merger with the merging axis along the line of sight. Czoske et al. (2001, 2002)

demonstrated that the redshift distribution of the cluster galaxies in Cl 0024+17

is bimodal, and Ota et al. (2004) demonstrated that the X-ray surface brightness

distribution of the cluster is better represented by a sum of two cluster models for

the surface brightness rather than just one. In Chapter 3 we detail the results of a

simulation of such a galaxy cluster merger involving the dynamics of both the dark

matter and the gas, and determine whether or not one would be able to discern the

existence of multiple cluster components using mock X-ray observations. We also

use these observations to determine the accuracy of a mass estimate based on this

assumption of two clusters projected along the line of sight on the sky.

Additionally, a weak lensing analysis recently presented by Jee et al. (2007) re-

vealed a ringlike structure in the projected matter distribution. They proposed that

dark matter from the cores of the clusters had been disrupted and ejected from the

systems by the collision, and they demonstrated that such features could be repro-

duced in a simulation of a collision of two pure dark matter halos. From this evidence

they suggested the current state of the system is ∼1-2 Gyr past the pericentric passage

of the cluster cores. In Chapter 4 we explore a parameter space over the initial veloc-

ity distribution of the dark matter particles to determine under what conditions such

a feature might appear in the dark matter distribution and whether these conditions

would be expected to exist in the real universe.

Finally, in Chapter 5 we extend our simulations of cluster mergers into a more gen-

eral parameter space where we perform a set of “fiducial” cluster merger simulations

7

similar to Roettiger et al. (1997), Ricker & Sarazin (2001), Ritchie & Thomas (2002),

and Poole et al. (2006, 2007, 2008). We explore a parameter space in initial mass

ratio of the merging clusters and initial impact parameter. Our aim is to investigate

the effects of these merger events on boosts in temperature and luminosity, as well

as the effect on estimated hydrostatic masses. We extend and build upon the results

of these earlier studies by constructing mock observations of our simulated clusters

along three spatial projections, as opposed to only the projection perpendicular to

the merger plane. In doing so we can address scenarios similar to the one proposed

for Cl 0024+17 as well as others. In addition, we create mock galaxy catalogs from

our simulations and use a commonly employed test of substructure, the κ-statistic,

to test the ability of this technique to detect substructure in galaxy clusters due to

mergers. In this case we also view the simulations along the different directions of the

simulation box to determine the effects of projection on the value of the κ-statistic

and its significance. We comment on some aspects of our merger simulations and

how they may shed light on a few observed merging clusters. Lastly, we also briefly

address other aspects of our simulations, such as the mixing of the ICM and the

generation of entropy due to mergers. In Chapter 6 we make our conclusions and

summarize this work.

Chapter 2

Simulations

2.1 Cluster Merger Simulation Method

We performed our simulations using FLASH, a modular, parallel hydrodynamics/N -

body astrophysical simulation code developed at the Center for Astrophysical Ther-

monuclear Flashes at the University of Chicago (Fryxell et al. 2000). FLASH uses

the Message-Passing Interface (MPI) library for inter-processor communication and

parallel I/O using the HDF5 and PnetCDF libraries, and is portable and scalable on

a variety of platforms. In the following paragraphs we detail the different modules of

FLASH that were employed in our simulations.

2.1.1 Treatment of Hot Gas

In our simulations the ICM is modeled as a compressible, inviscid fluid, and is

described by the Euler equations of hydrodynamics (written here in conservative

form):

∂ρ

∂t+ ∇ · (ρv) = 0 (2.1)

∂ρv

∂t+ ∇ · (ρvv) + ∇P = ρg (2.2)

∂ρE

∂t+ ∇ · [(ρE + P )v] = ρv · g (2.3)

8

9

where ρ is the fluid density, v is the fluid velocity, P is the pressure, E is the sum of

the internal energy ǫ and kinetic energy per unit mass,

E = ǫ+1

2|v|2 (2.4)

g is the acceleration due to gravity, and t is the time coordinate. We assume an ideal

gas equation of state, where the pressure is given by

P = (γ − 1)ρǫ (2.5)

and where we have assumed γ = 5/3. For the gas composition we assume a fully

ionized primordial composition of hydrogen and helium for which the mean atomic

mass µ = 0.592.

FLASH solves the Euler equations using the Piecewise-Parabolic Method (PPM)

of Colella & Woodward (1984), which is a higher-order version of the method de-

veloped by Godunov (1959). Godunov’s method uses a finite-volume spatial dis-

cretization of the Euler equations together with an explicit forward time difference.

Time-advanced fluxes at cell boundaries are computed using the numerical solution

to Riemann’s shock tube problem at each boundary. Initial conditions for each Rie-

mann problem are determined by assuming the non-advanced solution to be piecewise-

constant in each cell. Using the Riemann solution has the effect of introducing ex-

plicit nonlinearity into the difference equations and permits the calculation of sharp

shock fronts and contact discontinuities without introducing significant nonphysical

oscillations into the flow. Since the value of each variable in each cell is assumed

to be constant, Godunov’s method is limited to first-order accuracy in both space

and time. PPM improves on Godunov’s method by representing the flow variables

with piecewise-parabolic functions. The FLASH implementation of PPM is formally

accurate to second order in both space and time.

2.1.2 Treatment of Collisionless Dark Matter

Most of the mass in clusters of galaxies is believed to be in the form of “dark

matter”, which is believed to be essentially collisionless. To model the dark matter

10

component in our simulations, we set up a system of massive particles. FLASH

includes an N -body module which uses the particle-mesh method to solve for the

forces on gravitating particles (an introduction to this technique is given in Hockney

& Eastwood (1988)). The governing equations for the ith particle are

dxi

dt= vi (2.6)

midvi

dt= Fi (2.7)

where xi is the particle position, vi is the particle velocity, mi is the particle mass,

and Fi is the force on the particle. To solve these differential equations we employ a

variable-timestep leapfrog method. With time-centered velocities and stored acceler-

ations from the previous timestep this method is second-order in time.

In order to provide the coupling between the particles and the mesh-based quan-

tities (including the gravitational acceleration that is the sole force acting on the

particles), a mapping scheme between the particles and the mesh is needed. For

this purpose we use the “Cloud-in-Cell” method, which is a simple linear weighting

from nearby grid points. The weights are defined by considering only the region of

one “cell” size around each particle location; the proportional volume of the parti-

cle “cloud” corresponds to the amount allocated to/from the mesh. Two kinds of

mapping are used in these simulations: the particle masses are mapped to the dark

matter mass density grid variable, and the gravitational acceleration grid variable is

mapped to the particle accelerations.

2.1.3 Treatment of Gravity

The gravitational coupling between the dark matter and the gas is the major

driving force behind the dynamics of the simulation. The gravitational acceleration is

given by g = −∇φ, where φ is the gravitational potential. To obtain the gravitational

potential for a time-dependent, generalized mass distribution, Poisson’s equation must

be solved at every timestep:

∇2φ = 4πG(ρg + ρDM) (2.8)

11

where G is Newton’s gravitational constant, ρg is the fluid density and ρDM is the den-

sity of dark matter. In our simulations Poisson’s equation is solved using a multigrid

solver included with FLASH (Ricker 2008), which is appropriate for general source

distributions such as those found in galaxy cluster mergers. Multigrid solvers use

a hierarchy of discretizations to accelerate convergence of relaxation methods. The

method employed in the FLASH code is based on the algorithm presented in Huang

& Greengard (2000). The Poisson equation is solved in our simulations assuming

isolated boundary conditions, which assumes φ→ 0 as r → ∞.

2.1.4 Treatment of the Grid

FLASH uses adaptive mesh refinement (AMR), a technique that places higher

resolution elements of the grid only where they are needed. In our case we are

interested in capturing sharp ICM features like shocks and “cold fronts” accurately,

as well as resolving the inner cores of the cluster dark matter halos. As such it is

particularly important to be able to resolve the grid adequately in these regions. AMR

allows us to do so without needing to have the whole grid at the same resolution.

In all our simulations, we refine our adaptive mesh on two criteria. Where gas

is included, we adopt a second-derivative criterion for refining on discontinuities in

density, temperature, and pressure to capture shocks and contact discontinuities such

as ”cold fronts”. In order to avoid refining heavily on regions that are of low density

and therefore contribute negligibly to the X-ray emission, we suppress this refinement

at number densities lower than n ∼ 10−5 cm−3. Additionally, we refine and derefine

blocks based on the number of dark matter particles they contain. Blocks that acquire

a number of particles above a certain threshold are refined and those that lose particles

below a certain threshold are de-refined.

2.1.5 Cluster Initial Conditions

For each simulation, our clusters are initially modelled as spherically symmet-

ric configurations of gas and dark matter in hydrostatic equilibrium. For the dark

matter, we assume a radial profile motivated from cosmological simulations and ob-

12

servations of clusters (typically the NFW profile (Navarro, Frenk, & White 1997),

however see Section 4.2). Particle positions are generated by random sampling from

this radial profile. The particle velocities are sampled directly from the density pro-

file’s corresponding velocity distribution function, which for a spherically-symmetric

configuration is a function of the particle’s specific energy and angular momentum.

In addition, for simulations that also include gas, we assume observationally-

motivated radial profiles for the gas, using a physical quantity such as density, tem-

perature, or entropy as the basis for an analytical radial profile, and then integrating

the equation of hydrostatic equilibrium and the equation of state to derive the profiles

for the other relevant physical quantities. Details of these procedures for the different

simulations are given in their respective sections.

After these profiles are derived, a few steps are taken to ensure the validity of the

configuration. First, we check that the hydrostatic equilibrium condition is satisfied

by the gas. Secondly, we integrate over the particle distribution function F(x,v),

which gives

ρDM(r) =∫

F(x,v)d3v (2.9)

Recovering the original density profile for the dark matter ensures the distribution

function has been calculated correctly. Finally, a single cluster is ran in isolation to

check for the stability of the profiles over the length of time of the merger run. For

these tests we note that the profiles are generally stable, with minor variations in

the inner two zones due to force smoothing and minor steepening at the outer cutoff

radius of the profile.

2.2 Simulated X-ray Observations

To make meaningful comparisons between our simulation results and X-ray ob-

servations of real clusters, we construct mock X-ray observations of the simulation

output in the cases where our simulation contains gas. To do this we use MARX, a

suite of programs created by the MIT/CXC group that simulates the on-orbit per-

formance of the Chandra X-ray Observatory. MARX provides detailed ray-tracing

13

simulations of astrophysical sources as observed by Chandra.

Our mock X-ray observation generation procedure consists of two steps: we first

use the FLASH input of density and temperature to create a flux map in sky coordi-

nates for a range of energies. In doing this we follow closely the procedure outlined

in Gardini et al. (2004). Secondly, we use this flux map as a “user source” in MARX

to generate the photon energies, positions, and times for our simulated observation.

The FLASH AMR grid is regridded to a uniform grid at the highest resolution in

the simulation. We then choose a direction along which the cluster is observed and

the physical quantities are projected. For each cell in our dataset the emissivity of

the plasma is given by

ǫ = nenHΛ(T, Z) (2.10)

where ne and nH are the electron and hydrogen densities, respectively, and Λ(T, Z)

is the power coefficient which depends on temperature and metallicity, which are

assumed constant over one FLASH cell size. This coefficient is calculated using the

MEKAL model (Mewe, Kaastra, & Liedahl 1995). Using this relation, the photon

luminosity (photons s−1 cm−2 keV−1) at a given energy hν is given by

Lγν =

∫

VǫγνdV

′ = ΛγνEM (2.11)

where the quantity EM ≡∫

VnenHdV

′ is the emission measure. The measured flux

of photons at a specific energy at a redshift z is given by

F γν =

(1 + z)2Lγν(1+z)

4πD2L

(2.12)

where DL is the luminosity distance for the given redshift and cosmology. The flux

maps are generated for photons in 198 separate energy bins from 0.1-10.0 keV, giving

an energy resolution of ∆E = 50 eV, compared to the ∆E ≈ 100 eV resolution of

Chandra.

This flux map is then used to generate photons in MARX using the “user source”

implementation. The map is taken as an input and used as a distribution function

for the photons to determine their position in sky coordinates, energy, and time of

detection. In the cases considered here, the FLASH zone size is larger than the

14

size corresponding to a Chandra pixel. Within this zone size, photon positions are

uniformly distributed. When performing spatial analyses, the photon image is re-

binned by this factor.

For our simulated Chandra observations we use the simulated ACIS-S detector and

HRMA mirror system. For simplicity, effects such as foreground galactic absorption

and point source contamination are not included. However, we do include a diffuse

X-ray background component. The background photons are uniformly distributed in

space from a MEKAL plasma model corresponding to a constant flux of C = 8.5×10−9

counts s−1 cm−2 at a temperature of T = 40 keV, a redshift z = 0, and a metallicity

Z = 0.0. The output of photon events is given as a FITS file that can be read and

analyzed in the same way as real Chandra observations. To analyze our simulated

data, we use CIAO 3.4 and XSPEC 12.3.

For each mock cluster observation that we generate, we make measurements of

the relevant observed quantities that enable us to make comparisons between our

simulated clusters and the observed scaling relations. We perform two kinds of data

reductions: a spatial fit to the cluster surface brightness profile that provides a mea-

surement of the gas density and a fit to the cluster spectrum that yields the temper-

ature of the cluster gas and an estimate of the X-ray flux.

Specifically, we make an estimate of the cluster mass based on a isothermal β-

model for the cluster. We fit the surface brightness profile to a β-model, and take TX

as the temperature for the entire cluster. The density profile for a β-model is

ρg(r) =ρc

[

1 +(

rrc

)2]3β/2

(2.13)

which under the isothermal assumption corresponds to a surface brightness profile of

SX(r) =Sc

[

1 +(

rrc

)2]3β− 1

2

(2.14)

This is an oversimplified model, and previous investigations have demonstrated that it

tends to underestimate the masses of relaxed clusters (e.g. Bartelmann & Steinmetz

1996; Kay et al. 2004; Hallman et al. 2006; Rasia et al. 2006; Poole et al. 2007;

15

Piffaretti & Valdarnini 2008, and others). Its relevance consists in the fact that for

low counts observations fitting to more sophisticated models might not be possible,

and our desire to compare our results to previous investigations.

If the cluster is assumed to be in hydrostatic equilibrium and spherically symmet-

ric, then the estimated mass is

MX(≤ r) = −kT (r)r

Gµmp

[

d log ρg(r)

d log r+

d log T (r)

d log r

]

(2.15)

Where G is Newton’s gravitational constant, k is Boltzmann’s constant, mp is the

proton mass and µ is the mean molecular weight of the plasma. Under the further

assumption of an isothermal β-model, the resulting mass profile is

Mβ(≤ r) =3βkTr

Gµmp

[

(r/rc)2

1 + (r/rc)2

]

(2.16)

Details of a verification study to verify our mock observation generation and fitting

procedures are given in Appendix A.

Chapter 3

A Line-Of-Sight Galaxy Cluster

Collision

3.1 Introduction

One cluster of galaxies that has recently attracted attention is Cl 0024+17, a clus-

ter at a redshift z = 0.395 exhibiting both weak and strong lensing. Early attempts

at reconstructing the mass profile of this system using lensing (Tyson et al. 1998;

Broadhurst et al. 2000; Comerford et al. 2006) suggested a conflict with the predic-

tions of the standard CDM model due to the flattening of the density profile in the

inner regions of the cluster. Cosmological simulations assuming CDM indicate that

galaxy cluster mass profiles should exhibit a nonzero logarithmic slope in the inner

regions (e.g. Navarro, Frenk, & White 1997). This apparent discrepancy (and oth-

ers) led to suggestions that the CDM paradigm would need to be modified (Spergel &

Steinhardt 2000; Hogan & Dalcanton 2000; Moore et al. 2000), for example to include

self-interaction of the dark matter.

Czoske et al. (2001, 2002) demonstrated that the redshift distribution of the clus-

ter galaxies in Cl 0024+17 is bimodal and suggested that the system is composed of

two clusters undergoing a collision along the line of sight. They also performed a sim-

ulation demonstrating that such a scenario reproduces not only the bimodal redshift

distribution but also the flattening of the central density profile. X-ray observations

16

17

of the cluster using Chandra (Ota et al. 2004) and XMM-Newton (Zhang et al. 2005)

revealed that the surface brightness profile is better fit by a superposition of two ICM

models rather than one. They suggested on the basis of the isothermal temperature

profile that the individual clusters of the system had returned to equilibrium after

the collision and that consequently the encounter must have occurred several Gyr

ago. Recently, Jee et al. (2007) also demonstrated that if the system is modeled

as two ICM profiles in superposition, the mass determination based on hydrostatic

equilibrium agrees with the result from their lensing analysis.

If the merger scenario for Cl 0024+17 is correct, it raises several important ques-

tions: how long after the collision would the clusters appear relaxed, and correspond-

ingly when would a mass estimate based on hydrostatic equilibrium yield an accurate

measurement? What effect does viewing such a collision along the line of sight, with

both cluster components in superposition, have on the observed density and tem-

perature structure? We seek to provide answers to these questions by simulating

a similar high-speed collision between two clusters of galaxies. Most importantly,

we include the dynamics of the cluster gas, which were not included in the previ-

ous simulations. In this investigation, we focus on the gas dynamics of the collision,

particularly the morphology of the merger as viewed along the line of sight and the

density and temperature structure as seen in X-rays. From this information we be

able to directly address the question of the reliability of hydrostatic mass estimates of

Cl 0024+17 under the assumption of a single cluster component and the assumption

of two co-aligned clusters.

In Section 3.2 we describe the initial conditions for this particular galaxy cluster

merger simulation. In Section 3.3 we describe the evolution of the collision over the

course of 3 Gyr, and fit surface brightness and temperature profiles to the cluster gas.

In Section 3.4 use these profiles to estimate masses from hydrostatic equilibrium and

compare them to the actual mass. In addition, we discuss the implications of this

result for X-ray surveys. For this simulation we assume a flat CDM cosmology with

h = 0.5, Ωm = 1.0, as in Ota et al. (2004).

18

3.2 Initial Conditions

The cluster dark matter halos are initialized using the NFW (Navarro, Frenk, &

White 1997) density profile, where we follow the method of Kazantzidis et al. (2006).

The NFW functional form is used for r ≤ r200:

ρDM(r) =ρs

r/rs(1 + r/rs)2 , r ≤ r200 . (3.1)

Outside of r200, to keep the mass of the halo finite and to avoid an unphysical dis-

tribution function, we implement an exponential cutoff that turns off the profile on a

scale rdecay, a free parameter which we set to 0.1r200:

ρDM(r) =ρs

c200(1 + c200)2

(

r

r200

)κ

exp

(

−r − r200rdecay

)

, r > r200 . (3.2)

Here r200 is the radius within which the mean mass density is 200 times the cosmic

mean, rs is the NFW scale radius, and c200 ≡ r200/rs is the NFW concentration

parameter. We require that at r200 the profile and its logarithmic slope be continuous.

This is achieved via the parameter κ:

κ = −(1 + 3c200)

(1 + c200)+

r200rdecay

(3.3)

Choosing r200 as the truncation radius for the NFW profile is artificial but suffices

for the purposes of this study. Single cluster tests that we have performed (as well

as those performed for other studies, e.g. Ricker and Sarazin 2001) demonstrate the

density profile is stable for the length of the run, with minor steepening of the profile

near the cutoff radius. In addition, radii larger than ∼ r200 are inaccessible to current

X-ray observations due to the low densities at such radii.

To initialize the gas density we choose a Burkert profile:

ρgas(r) =ρc

(1 + (r/rc)2)(1 + r/rc), (3.4)

which originally (Burkert 1995) was given as a fitting function for dark matter profiles

of dwarf galaxies but is also a good fit to the gas density profiles of non-cooling-

core clusters (A. Kravtsov, private communication). The more traditional β-model

19

Table 3.1. Initial Cluster Parameters

Cluster M200 (M⊙) r200 (kpc) c200 nc (cm−3) rc (kpc)

1 6.0 × 1014 1739.79 5.0 0.020 198.83

2 3.0 × 1014 1380.87 7.0 0.062 98.65

(Cavaliere & Fusco-Femiano 1976) is a poorer fit to the gas density profile at large

r in both simulations and observations (Vikhlinin et al. 2006; Hallman et al. 2007;

Nagai et al. 2007b). (However, we follow Ota et al. (2004) and Zhang et al. (2005)

and fit β-model profiles to the X-ray surface brightness profiles in our simulated

observations.)

The gas density is also fitted to an exponential taper at r200, extending to the

radius at which it equals the mean baryonic density of the universe. For the gas

profiles, we choose the core radius rc to be roughly half the scale radius of the DM

profile (Ricker & Sarazin 2001). We fix the gas mass fraction at r200 to 0.12 in line

with observations of real clusters (e.g., Vikhlinin et al. 2006, Sanderson et al. 2003),

and the measurement of Ota et al. (2004).

From these parameters the central gas densities ρc can be determined. We deter-

mine the pressure, temperature, and internal energy profiles by assuming our func-

tional forms for dark matter and gas density and numerically integrating the equation

of hydrostatic equilibrium. The values used for the cluster parameters are given in

Table 3.1.

After the radial profiles are determined it remains to set up the distribution of

positions and velocities for the dark matter particles. Here we follow the procedure

outlined in Kazantzidis et al. (2006). For the particle positions a random deviate

u is uniformly sampled in the range [0,1] and the function u = MDM(r)/MDM(rmax)

is inverted to give the radius of the particle from the center of the halo. For the

particle velocities, the procedure is less trivial. Many previous investigations have

made use of the “local Maxwellian approximation.” In this procedure at a given

radius the particle velocity is drawn from a Maxwellian distribution with dispersion

20

σ2(r), where the latter quantity has been derived from solving the Jeans equation

(Binney & Tremaine 1987). It has been shown that this approach is not sufficient to

accurately represent the velocity distribution functions of dark matter halos with a

central cusp such as the NFW profile (Kazantzidis et al. 2004). To accurately realize

particle velocities, we choose to directly calculate the distribution function via the

Eddington formula (Eddington 1916):

F(E) =1√8π2

[

∫ E

0

d2ρ

dΨ2

dΨ√E − Ψ

+1√E

(

dρ

dΨ

)

Ψ=0

]

(3.5)

where Ψ = −Φ is the relative potential and E = Ψ − 12v2 is the relative energy of

the particle. We tabulate the function F in intervals of E interpolate to solve for

the distribution function at a given energy. Particle speeds are chosen from this

distribution function using the acceptance-rejection method. Once particle radii and

speeds are determined, positions and velocities are determined by choosing random

unit vectors in ℜ3.

Following Czoske et al. (2002) and Jee et al. (2007) we assume a mass ratio of

2:1 for the clusters. In our simulation the clusters are initialized so that their centers

are separated by the sum of their respective r200 values (approximately 3 Mpc), and

they are given an initial relative speed vrel = 3000 km/s, which is the inferred relative

velocity of the two components of Cl 0024+17 measured from the redshift distribution

of the cluster galaxies (Czoske et al. 2002). This value is approximately 2vff for

the clusters, where vff is the free-fall velocity from infinity. Such an initially high

relative speed might be difficult to achieve in a ΛCDM universe (see, e.g. Hayashi &

White 2006). However, our choice is motivated by our desire to match the observed

conditions of the Cl 0024+17 if the merger scenario is correct. For our box size of

14.29 Mpc we achieve a finest resolution of ∆x = 13.96 kpc.

21

3.3 Results

3.3.1 Qualitative Description

Figures 3.1 and 3.2 show slices of density and temperature, respectively, through

the z = 0 coordinate plane at different times in the simulation (the x axis is the

collision axis). From t = 0.0-0.9 Gyr the clusters approach each other and a shock

front forms. At t = 1.0 Gyr the core of the smaller cluster collides with the core of

the larger cluster, driving a shock wave forward into the larger core and displacing

the larger core’s gas from the collision axis into streams of cold gas. There is also a

stream of dense, cold gas that is pulled directly between the two dark matter halos.

Later on, the heads of these streams of gas, along with hotter gas from between

the clusters, begin to fall into the potential of the larger halo and by t = 2.0 Gyr

have fallen in completely. As for the gas core of the smaller cluster, it is penetrated

shortly after the collision by a reverse shock, but this shock weakens as it traverses

the core and so the latter remains cool, in agreement with similar investigations of

high-speed cluster mergers by Takizawa (2005) and Milosavljevic et al. (2007). In

this process some of the gas of the smaller cluster is also ram-pressure stripped, and

a contact discontinuity forms that is similar to the one observed in the 1E0657-56,

and in the simulations of the latter (Springel & Farrar 2007; Mastropietro & Burkert

2008). After penetrating the larger’s gas core the smaller core is forced to lag behind

its dark matter core by ram pressure until it encounters regions of lower density (t >

1.2 Gyr). At this point the gas falls back toward the dark matter core but overshoots

the potential and begins to slosh back and forth inside it. At late times the gas of

the clusters is still settling into their respective dark matter potential wells, and the

temperature structure of the gas is still complicated. Due to the high initial velocity

of the collision, the clusters are in an unbound orbit and by the end of the simulation

the separation between their respective dark matter halo centers is still increasing.

This is in contrast to many previous works involving mergers (e.g. Ricker & Sarazin

2001; Poole et al. 2006), which assumed an initial velocity v ≈ vff .

22

Figure 3.1 Slices of gas density through the z = 0 coordinate plane for times t = 0.0,

0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 Gyr. Each panel is 10 Mpc on a side.

Figure 3.2 Slices of gas temperature through the z = 0 coordinate plane for times t

= 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 Gyr. Each panel is 10 Mpc on a side.

23

Figure 3.3 Mock 40 ksec raw counts image of the simulation viewed along the line of

centers of the clusters in the 0.5-5.0 keV band. The image has been smoothed using

a Gaussian with a fixed kernel radius of 3 pixels. Color scale units are counts/pixel.

The time is t = 3.0 Gyr, ∼2 Gyr after the collision.

3.3.2 Mock X-ray Observations

For our simulated Chandra observations we use the simulated ACIS-S detector and

HRMA mirror system, as in the real Chandra observation of Cl 0024+17. Each obser-

vation is ∼40 ks of simulated exposure time, comparable to the Chandra observation

of Cl 0024+17 (39 ks).

We view the system along the line of centers. Since we do not know the current

state of the actual collision, we set each individual observation as if it were being

observed at a redshift z = 0.395. At this redshift for the assumed cosmology 1” =

6.39 kpc. Figure 3.3 shows an example of a smoothed mock X-ray image. For each

observed time we extract a radial surface brightness profile and a radial temperature

profile.

24

Table 3.2. Fitted Cluster Parameters for a Single β-model

ta rcb S0

c β χ2/d.o.f.

0.0 97.65+0.18−0.96 (3.81+0.02

−0.02) × 10−6 0.798+0.001−0.003 407.61/195

2.0 172.04+7.58−1.73 (1.42+0.03

−0.03) × 10−7 0.714+0.017−0.004 305.14/195

2.5 235.21+2.24−13.91 (1.25+0.03

−0.03) × 10−7 0.964+0.009−0.031 491.04/195

3.0 150.24+1.86−12.50 (1.43+0.03

−0.03) × 10−7 0.733+0.005−0.027 423.58/195

aSimulation time (Gyr)

bCore radius (kpc)

cCore surface brightness (counts s−1cm−2arcsec−2)

Cluster surface brightness profiles are extracted for the energy range 0.5-5.0 keV,

within a circle centered on the surface brightness peak with a radius of 400” (≈ 2500

kpc). Figure 3.4 shows the surface brightness profiles for times t = 2.0, 2.5, and 3.0

Gyr. The profiles are fitted with the aforementioned β-model (Cavaliere & Fusco-

Femiano 1976). We fit with a single β-model and a sum of two β-models. The χ2-

statistic is used to fit the profiles. We follow Ota et al. (2004) and Jee et al. (2007) in

fixing one of the β parameters to unity, as there exists a strong degeneracy between

this parameter and its corresponding core radius rc. We include the background

constant as a free parameter in the single β-model fit, but freeze this value for the

double β-model fit. The fitted profiles for the t = 3.0 Gyr case are shown in Figure

3.5. The values of the fitted parameters for epochs t = 2.0, 2.5, and 3.0 Gyr are given

in Tables 3.2 and 3.3. Errors are quoted at the 90% confidence level.

The projected temperature profiles for t = 2.0, 2.5, and 3.0 Gyr are shown in

Figure 3.6. The profiles were generated by extracting spectra from 4 annular regions

in the 0.5-7.0 keV band centered on the peak of the surface brightness profile. Each

is fitted with a MEKAL model, using the χ2-statistic with grouping to ensure at least

15 photons per bin. The radial extent of the temperature profiles is 125” (≈ 800 kpc),

which is slightly greater than the radial range for which the temperature profile was

extracted in Ota et al. (2004). Single-temperature fits to the spectrum of the whole

cluster in the 0.5-7.0 keV band for t = 2.0, 2.5, and 3.0 Gyr are given in Table 3.4.

25

Figure 3.4 Surface brightness vs. projected radius for times t = 2.0, 2.5, and 3.0 Gyr.

Table 3.3. Fitted Cluster Parameters for a Double β-model

ta rc,1b S0,1

c rc,2b S0,2

c β2 χ2/d.o.f.

0.0 89.24+3.16−0.32 (3.73+0.04

−0.04) × 10−6 204.65+1.32−4.00 (7.03+0.07

−0.07) × 10−7 0.981+0.003−0.014 260.01/193

2.0 39.91+34.41−17.45 (8.18+3.41

−3.39) × 10−8 270.00+6.34−4.78 (1.11+0.02

−0.02) × 10−7 0.970+0.010−0.025 245.83/193

2.5 79.28+9.04−5.36 (2.26+0.17

−0.17) × 10−7 282.00+11.97−5.57 (7.70+0.21

−0.21) × 10−8 0.970+0.021−0.021 226.75/193

3.0 82.12+8.12−3.94 (2.51+0.16

−0.16) × 10−7 316.24+26.16−4.89 (4.94+0.16

−0.16) × 10−8 0.980+0.015−0.049 205.86/193

aSimulation time (Gyr)

bCore radius (kpc)

cCore surface brightness (counts s−1cm−2arcsec−2)

26

Figure 3.5 Single and double β-model fits for t = 3.0 Gyr, ∼2 Gyr after the collision.

1.5

2

2.5

3

3.5

4

4.5

5

5.5

6

0 100 200 300 400 500 600 700 800

T (

keV

)

r (kpc)

t = 2.0 Gyrt = 2.5 Gyrt = 3.0 Gyr

Figure 3.6 Temperature profile vs. projected radius for times t = 2.0, 2.5, and 3.0

Gyr.

27

Table 3.4. Fitted Average Spectral Temperatures

t (Gyr) Tspec (keV) χ2/d.o.f.

2.0 2.67+0.17−0.16 140.62/177

2.5 2.83+0.23−0.18 147.98/174

3.0 2.71+0.20−0.19 128.07/167

3.4 Discussion

3.4.1 The Surface Brightness and Temperature Profiles

The feature in the radial profile of Cl 0024+17 pointed out by Ota et al. (2004) and

Jee et al. (2007) was that a double β-model fit is a better fit to the data, as a single

β-model fit underestimates the central surface brightness. The surface brightness

profile of Cl 0024+17 and the single and double β-model fits to the profile are shown

in Figure 3.7. The difference between the fits is significant; there is a difference in

the χ2-statistic of ∼60 for a difference of two degrees of freedom.

In our mock cluster observations, we find that at later times, it is also possible to

make a distinction between the two surface brightness components of the two clusters

from the difference in the fitted χ2-statistic (see Tables 3.2 and 3.3). At the t = 2.0

Gyr epoch, ∆χ2 ≈45 for a difference of only 2 degrees of freedom, and at later times

is even larger (∆χ2 ≈200-260 for a difference of two degrees of freedom). We also note

that we can use the ∆χ2-test to distinguish between the two clusters in our initial

conditions, which we also show in Tables 3.2 and 3.3.

In Cl 0024+17, the fitted spectral temperature of T = 4.47 keV seems low for a

system undergoing a collision or merger. In our mock observations we note a similar

phenomenon; the fitted temperatures in the mock observations are significantly lower

than what might be expected given the temperatures observed in the simulation.

In our simulated system, the hottest cluster gas is in the larger cluster and is at a

temperature of ∼ 5-6 keV at later times. However, the measured temperatures in

the mock X-ray observations barely reach ∼ 3 keV. A close look at the temperature

28

1e-11

1e-10

1e-09

1e-08

1 10 100 1000 10000

S (

cts/

sec/

cm2 /k

pc2 )

r (kpc)

DataSingle β-model fit

Double β-model fit

Figure 3.7 Cl 0024+17 X-ray surface brightness profile and single and double β-model

fits. Reproduced with permission from Ota et al. (2004).

distribution in the simulation itself (Figure 3.8) reveals that the highest-temperature

gas is confined to the central region of the larger cluster, and the denser gas of the

smaller cluster is colder, around 3 keV. In projection along the line of sight, the lower-

temperature, denser gas “washes out” the higher-temperature, less dense gas, and the

complicated, two-component temperature structure of the clusters is lost because of

the strong density dependence of the X-ray emission (∝ ρ2). Figure 3.9 shows the

profile of the mass-weighted temperature over the same range as Figure 3.6 for t = 3.0

Gyr. The lack of temperatures higher than ∼ 3 keV demonstrates that the denser,

colder gas provides the dominant contribution to the X-ray emission.

The single-temperature fits are in general agreement with our projected tem-

perature profiles. Attempting to fit the cluster temperature using a two-temperature

model resulted in no significant decrease in the χ2-statistic from the single-temperature

fits, so on the basis of this analysis we cannot distinguish the two different average

temperatures of the two clusters.

At late times there is still some moderate variation in the projected temperature

29

Figure 3.8 Slice through the z = 0 coordinate plane of the simulation of temperature

at t = 3.0 Gyr, ∼2 Gyr after the collision. Density contours are logarithmically

spaced by 2.5×, between n = 10−6 − 10−2cm−3. The size of the figure is 14 Mpc by

6 Mpc.

1.8

1.9

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

0 100 200 300 400 500 600 700

T (

keV

)

r (kpc)

Figure 3.9 Radial profile of mass-weighted temperature at t = 3.0 Gyr, ∼2 Gyr after

the collision.

30

3

3.5

4

4.5

5

5.5

6

0 100 200 300 400 500 600

T (

keV

)

r (h50-1 kpc)

Figure 3.10 Cl 0024+17 Temperature profile. Average profile is shown as a dashed

line at T = 4.47 keV. Reproduced with permission from Ota et al. (2004).

profile with time, indicative of the continuing evolution of the system, though the

average temperature of the cluster is roughly constant with time. In Ota et al.

(2004), the observed isothermal temperature profile of Cl 0024+17 (reproduced in

Figure 3.10) was taken as evidence that the merger was not recent (i.e., it was more

than a few Gyr ago). Our results show that if viewed along the line of sight ∼2 Gyr

after the merger, the temperature structure of the system can appear more regular in

projection than it actually is (e.g., the structure seen in Figure 3.2). If the results of

Jee et al. (2007) regarding a ringlike dark matter structure indicate the clusters are

being viewed along the line of sight shortly after the merger, our results show that

the system can still appear relatively relaxed. This agrees with previous results of

mock X-ray observations of simulations of galaxy cluster mergers (e.g., Ricker and

Sarazin 2001, Poole et al. 2006).

Finally, our observed spectral temperature from our simulations of T ∼ 2.6 − 2.8

keV is somewhat lower than the T ∼ 4.47 keV for Cl 0024+17 measured by Ota et

al. (2004) or the T ∼ 4.25 keV measured by Jee et al. (2007). If the collision scenario

31

for Cl 0024+17 is correct, this indicates that the initial conditions for our simulation

are not identical to the pre-merger conditions for the real cluster. A higher impact

velocity or a change in the masses of the clusters could account for the temperature

difference.

3.4.2 The Reliability of Hydrostatic Mass Estimates

Though the temperature and surface brightness profiles are undergoing consider-

able evolution even at late times, it is still relevant to ask how accurate a mass estimate

would be under the assumption of hydrostatic equilibrium. As previously noted, Ota

et al. (2004) and Jee et al. (2007) made estimates of the mass of Cl 0024+17 using sin-

gle and double β-model fits to the surface brightness profile together with an isother-

mal temperature profile. Although our temperature profile is not strictly isothermal,

existing methods for temperature deprojection assume spherical symmetry, so we do

not attempt such a deprojection here. If we assume a cluster temperature Tspec given

by the spectral temperature fit to the entire cluster, we can arrive at a rough estimate

for the mass of the system under the assumption of hydrostatic equilibrium.

For a β-model fit to the cluster gas, the projected cluster mass within a cylindrical

volume for a given temperature, β, and core radius rc can be estimated by (Ota et

al. 1998; Jee et al. 2005)

MX,β(R) = 1.78 × 1014β(

T

keV

)

(

R

Mpc

)

R/rc√

1 + (R/rc)2M⊙ , (3.6)

where R is the projected radius, T is the gas temperature and β is the β-model

index. We compute the estimated mass within the arc radius of the lensed galaxy

in Cl 0024+17, rarc = 35”/223.65 kpc. For the double β-model fits we assume the

same temperature for both cluster components, and we add the mass contributions

from the two fits together. Table 3.5 shows the estimated masses from the single and

double β-model fits for times t = 2.0, 2.5, and 3.0 Gyr. Errors are quoted at the 90%

confidence level. It is clear from the table that assuming a single-cluster model for

the system will underestimate the projected mass by a factor ∼2-3, whereas assuming

a double-cluster model will estimate the projected mass to within ∼10%. This is in

32

Table 3.5. Estimated and Exact Masses at R = 35”

t (Gyr) MX,β(M⊙) MX,2β(M⊙) Mactual(M⊙) MX,β/Mactual MX,2β/Mactual

2.0 (6.03+0.42−0.36) × 1013 (1.71+0.08

−0.08) × 1014 1.74 × 1014 0.35 0.98

2.5 (7.50+0.61−0.58) × 1013 (1.74+0.10

−0.08) × 1014 1.65 × 1014 0.45 1.05

3.0 (6.57+0.49−0.54) × 1013 (1.62+0.10

−0.08) × 1014 1.55 × 1014 0.42 1.05

agreement with the results of Jee et al. (2007).

3.4.3 The Effect of A Slightly Off-Axis Collision

In our simulated head-on collision the gas of the clusters is significantly disrupted

and by the end of the simulation is still settling into the potential wells created by the

dark matter. If instead of colliding head-on the encounter were allowed to be slightly

off-axis, this would result in less disruption of the cluster gas cores. Presumably the

gas would relax more quickly and an even more accurate mass determination could

be made under the assumption of hydrostatic equilibrium. In addition, in a slightly

off-axis collision viewed along the line of sight the two cluster components would be

more clearly distinguished in the X-ray emission. Two-component fits to both the X-

ray surface brightness and the temperature of the different components would enable

a more accurate mass determination.

3.4.4 Implications for X-ray Surveys

These results have important implications for X-ray surveys. Neighboring galaxy

clusters viewed in superposition, whether undergoing mergers or not, may bias tem-

perature measurements based on spectral fitting. This can have an effect particu-

larly on X-ray-based scaling relations. For example, previous simulated observations

of merging clusters have demonstrated that for mergers occurring along the line of

sight, the clusters appear to have higher temperatures for their mass (Ricker & Sarazin

2001; Poole et al. 2007). Our investigation shows that clusters viewed in superposi-

33

tion can make a cluster appear colder than it actually is if the colder cluster’s gas is

significantly denser. In addition, discrepancies between hydrostatic and lensing mass

estimates may in some cases be attributed to the existence of multiple components

viewed along the line of sight. as is apparent in the case of Cl 0024+17. Detailed

analysis of X-ray observations would be required to discern such superpositions.

One possible way to correct for and identify such superposition effects is by using

the optical redshifts of the cluster galaxies to identify separate cluster components, as

Czoske et al. (2001, 2002) did in the case of Cl 0024+17. To reliably identify such sep-

arate cluster components requires accurate redshifts. Because spectroscopic redshifts

are only feasible for the nearest or brightest cluster galaxies, cluster surveys must

rely on photometric redshift estimators. Large optical surveys such as zCOSMOS

(Lilly et al. 2007), Combo-17 (Wolf et al. 2003), and the upcoming Dark Energy Sur-

vey (The Dark Energy Survey Collaboration 2005) have photometric redshift errors

σz ∼ (0.02 − 0.05)(1 + z).

If the only redshift determination for Cl 0024+17 were via photometry, would the

separation between the two components still be discernible? To answer this question

we constructed a mock “galaxy” catalog from our simulation. We chose dark matter

particles from the two clusters as proxies for the cluster galaxies (120 from the smaller

cluster, 240 from the larger) and assumed a value for σz. We constructed redshift

histograms for the resulting galaxy distributions and fitted them to both a Gaussian

distribution and a sum of two Gaussian distributions:

fsingle = A0e−(z−µ0)2/2σ2

0 (3.7)

fdouble = A1e−(z−µ1)2/2σ2

1 + A2e−(z−µ2)2/2σ2 . (3.8)

The fitted parameters for these distributions for varying σz, given a choice of mock

galaxies from our simulation, are shown in Table 3.6. For low values of σz, the mean

values of the two redshift distributions are statistically distinguishable, but for higher

values they are not. The histograms shown in Figures 3.11 and 3.12 demonstrate

that for σz ∼ 0.005 − 0.01 the skewness of the distribution of galaxies is evident,

but when the redshift error increases to σz ∼> 0.02, the resulting galaxy distribution is

indistinguishable from that of a single cluster.

34

Table 3.6. Fitted parameters for Redshift Histograms with Varying σz

σz µ0 σ0 χ2/d.o.f.single µ1 σ1 µ2 σ2 χ2/d.o.f.double

0.005 0.9975 0.0040 12.6/10 1.002 0.0040 0.9959 0.0052 5.88/7

0.01 0.9958 0.0086 31.68/9 0.9998 0.0086 0.9889 0.0089 14.7/6

0.02 0.9936 0.0175 8.64/9 0.9947 0.0175 0.9742 0.0126 9.84/6

0.08 0.9521 0.0634 6.65/7 0.9733 0.0634 0.9260 0.0794 3.56/4

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.98 0.985 0.99 0.995 1 1.005 1.01 1.015

N(z

)/N

tot

z

AllCluster 1Cluster 2

0

0.05

0.1

0.15

0.2

0.25

0.96 0.97 0.98 0.99 1 1.01 1.02 1.03

N(z

)/N

tot

z


Figure 3.11 Number of redshifts per bin vs. redshift, σz = 0.005 and σz = 0.01.

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.94 0.96 0.98 1 1.02 1.04 1.06

N(z

)/N

tot

z


0

0.05

0.1

0.15

0.2

0.25

0.8 0.85 0.9 0.95 1 1.05 1.1 1.15 1.2

N(z

)/N

tot

z


Figure 3.12 Number of redshifts per bin vs. redshift, σz = 0.02 and σz = 0.08.

35

Table 3.7. ∆χ2 for Different Exposure Times and Redshifts

texp (ks) z = 0.395 z = 1.0

10 177.69/96 - 113.21/94 99.20/96 - 86.05/94

40 425.31/195 - 203.44/193 242.51/195 - 173.29/193

240 1497.05/195 - 439.33/193 770.37/195 - 287.46/193

At a redshift of z ≈ 0.4 (Cl 0024+17), the expected photometric redshift errors

from optical surveys would be too large at this time to distinguish between the two

cluster components (σz ∼ 0.03 − 0.07), and at a redshift of z ≈ 1 it would be worse

(σz ∼ 0.04 − 0.1). Identification of line-of-sight mergers and collisions from red-

shift distributions likely will be restricted to clusters for which we have spectroscopic

redshifts available.

Our X-ray fitting results might be taken to suggest that X-ray surface brightness

profiles could be used in place of galaxy redshift distributions to detect clusters seen

in superposition. To test this hypothesis, we placed our simulated clusters at the

epoch t = 3.0 Gyr (∼2 Gyr after the collision) at two redshifts, the current redshift

of z = 0.395 and a redshift of z = 1.0. The same analysis of the surface brightness

profiles as above was performed for exposure times of 10 ks, 40 ks, and 240 ks, the

former being a more typical exposure time for a given cluster in an X-ray survey. We

focus here on the difference in the χ2 between the single and double β-model fits. The

resulting ∆χ2 for the fitted surface brightness models corresponding to the different

exposure times for the different redshifts are shown in Table 3.7. At z = 0.395, it

is possible to use the ∆χ2 between the two models to distinguish between the two

clusters for all exposure times. At z = 1.0, for an exposure time of 10 ks, the ∆χ2

is ≈ 14 for a change of 2 degrees of freedom, and for 40 ks ∆χ2 ≈ 70. Therefore,

we find that at exposure times more typical of X-ray surveys, even at high redshift

we can distinguish between the two cluster components, though with more difficulty.

This is potentially a promising avenue for identifying merging clusters and clusters in

projection.

Chapter 4

Dark Matter Rings In Colliding

Clusters

4.1 Introduction

Recently, a weak lensing analysis presented by Jee et al. (2007) revealed a ringlike

structure in the projected matter distribution. They proposed that dark matter

from the cores of the clusters had been disrupted and ejected from the systems by

the collision, and they demonstrated that such features could be reproduced in a

simulation of a collision of two pure dark matter halos. From this they suggested the

current state of the system is ∼1-2 Gyr past the pericentric passage of the cluster

cores. Ringlike features due to collisions are not without precedent. “Ring galaxies”

provide another astrophysical situation in which “particle” rings are produced due to

collisions (e.g., the Cartwheel Galaxy). We look to the previous N -body simulations

of these phenomena to guide our understanding of what might lead to such a feature

in the dark matter in clusters of galaxies. Two major differences exist between the

dark matter particles in galaxy clusters and star “particles” in disk galaxies. In

the case of disk galaxies, the stars are concentrated in a disk, and the velocities

are essentially circular. In galaxy clusters, dark matter particles form a roughly

spheroidal distribution and have isotropic to radially anisotropic velocity dispersions.

To investigate the effects of the second of these differences we also perform a set of

36

37

pure dark matter (N -body only) simulations with a physically motivated dark matter

profile and variations on the velocity distribution of the dark matter particles.

In Section 4.2 we describe the parameter space of initial conditions that we em-

ploy for these simulations. In Section 4.3 we present slices through the dark matter

density, projected density maps, and projected density profiles of the simulations af-

ter the collision. Finally, in Section 4.4 we discuss the results and the implications

for the purported ringlike feature in the dark matter of Cl 0024+17. For this set of

simulations we assume a flat ΛCDM universe with h = 0.7, Ωm = 0.3.

4.2 Initial Conditions

For ease in setting up the particle distribution functions, we use a Hernquist profile

(Hernquist 1990):

ρDM(r) =M0

2πa3

1

r/a(1 + r/a)3, (4.1)

which has a mass profile that converges to M0 as r → ∞:

M(r) = M0r2

(r + a)2(4.2)

This form of the dark matter density profile is chosen because of the mathematical

simplicity of the corresponding distribution functions (as is shown below) and its

resemblance to the NFW profile in that as r → 0, ρ(r) ∝ r−1.

To determine the effects of a varying velocity anisotropy on the dark matter fea-

tures in the simulation, we allow the 3D velocity dispersion of the particles to vary

from the isotropic form σr = σθ = σφ. We assume σθ = σφ and parametrize the

anisotropy using β ≡ 1 − σ2θ

σ2r, which is taken to be constant over the entire radial

range. Table 4.1 shows the values of β that were used for each simulation, and the

corresponding ratio σθ

σr.

In order to initialize the particle velocities, we sample the particle distribution

function (hereafter DF) directly. The DF F(x,v) is assumed to obey the following

relation:

38

ρ(r) =∫

F(x,v)d3v (4.3)

The DF of any steady-state, spherically symmetric system has a dependence on

the phase space coordinates that comes in only through the “integrals of motion” Eand L, where E = ψ− 1

2v2 is the relative energy and L = rvt is the angular momentum

of a particle (Binney & Tremaine 1987). In these expressions, ψ(r) = −φ(r) is the

relative gravitational potential, and vt is the tangential velocity.

For a constant velocity anisotropy, the DF takes the specific form (Binney &

Tremaine 1987):

F(E , L) = L−2βfE(E) (4.4)

For a Hernquist profile, the energy-dependent part of the DF is (Baes & Dejonghe

2002)

fE(E) =2β

(2π)5/2

Γ(5 − 2β)

Γ(1 − β)Γ(72− β)

E5/2−β2F1

(

5 − 2β, 1 − 2β,7

2− β; E

)

(4.5)

where 2F1 is the hypergeometric function. For half-integer values of β this function

can be expressed in terms of rational functions. To ease our investigation of this

parameter space we therefore choose values of β from this set, specifically the values

β = 1/2, -3/2, -5/2, -17/2. We also include an isotropic setup (β = 0) and a setup

where all velocities are initially circular (β = −∞). Given the mass density function

and the distribution function, we derive initial positions and velocities for the particles

using the acceptance-rejection method.

Finally, following Czoske et al. (2002) and Jee et al. (2007) we assume a mass

ratio of 2:1 for the clusters. In all of our simulations the clusters are initialized so

that their centers are separated by the sum of their respective radii R (3 Mpc), and

they are given an initial relative speed vrel = 3000 km/s, which is the inferred relative

velocity of the two components of Cl0024+17 (Czoske et al. 2002). The values of the

halo parameters are given in Table 4.2.

39

Table 4.1 Velocity Anisotropy

Simulation β σθ/σr

S1 1/2 0.707S2 0 1S3 -3/2 1.58S4 -5/2 1.87S5 -17/2 3.08S6 −∞ ∞

Table 4.2 Initial Cluster Parameters

Cluster M0 (M⊙) R (kpc) a (kpc)1 6.0 × 1014 2000.0 400.02 3.0 × 1014 1000.0 200.0

We refine the adaptive mesh using the dark matter density to resolve the cores

of the clusters and other overdensities. For our box size of 10h−1 Mpc we achieve a

minimum zone spacing of ∆x = 9.77h−1 kpc.

4.3 Results

For our set of simulations we have used the same cluster profiles, masses, initial

separation, and relative velocity, but we have varied the anisotropy of the dark matter

velocity dispersion. In each simulation, from t = 0.0-0.9 Gyr the dark matter cores

accelerate toward each other and pass through each other. The resulting sudden

impulse on the dark matter particles boosts their energies and causes a significant

fraction of the mass in the cores to be ejected from the centers. Figure 4.1 shows

slices through the z = 0 coordinate plane of the dark matter density in the β = −5/2

simulation at different epochs. After the pericentric passage the outer regions of

both halos have expanded, with a trail of dark matter strung between the centers. It

can be seen that the smaller halo is more disrupted, with a larger amount of mass

40

Figure 4.1 Slices through the z = 0 coordinate plane of dark matter density at the

epochs t = 0.0, 0.5, 1.0, 1.5, 2.0, 2.5, and 3.0 Gyr for the β = −5/2 simulation. Each

panel is 10 Mpc on a side.

redistributed to the outer portion of the halo.

Figures 4.2 through 4.7 show slices though the z = 0 coordinate plane of the dark

matter density, the projected dark matter density, and the projected density profile for

each simulation at the epoch t = 1.5 Gyr (≈ 0.6 Gyr after the collision, defined as the

time when the centers of the halos are coincident). At this epoch, our simulations show

no ring feature (defined as a ”bump” in the radial distribution of dark matter particles

projected onto the sky; i.e. an increase followed by a decrease in this distribution) for

initial velocity distributions that are radially anisotropic (β = 0.5) and isotropic orbits

(β = 0). For initial velocity distributions that are increasing tangentially anisotropic

(β = −0.5,−2.5,−8.5), our simulations show a more pronounced shoulder but no

ring. Only for an initially circular velocity distribution (β = −∞) do our simulations

show a ring. Figure 4.8 shows the angle-averaged projected density profile at t = 1.5

Gyr for all of the different initial values of β. The initial profile is also shown for

comparison.

Even in the simulation in which our simulations show a ring (β = −∞), the feature

41

0.0×100

5.0×108

1.0×109

1.5×109

2.0×109

0 0.5 1 1.5 2

κ (M

O• kp

c-2)

r (Mpc)

Figure 4.2 t = 1.5 Gyr snapshot of the β = 1/2 simulation. Left: Slice through the

z = 0 coordinate plane of dark matter density. Center: Dark matter density projected

along the collision axis. Right: Projected density profile.

0.0×100

5.0×108

1.0×109

1.5×109

2.0×109

0 0.5 1 1.5 2

κ (M

O• kp

c-2)

r (Mpc)

Figure 4.3 t = 1.5 Gyr snapshot of the β = 0 simulation. Left: Slice through the

z = 0 coordinate plane of dark matter density. Center: Dark matter density projected

along the collision axis. Right: Projected density profile.

0.0×100

5.0×108

1.0×109

1.5×109

2.0×109

0 0.5 1 1.5 2

κ (M

O• kp

c-2)

r (Mpc)

Figure 4.4 t = 1.5 Gyr snapshot of the β = -3/2 simulation. Left: Slice through

the z = 0 coordinate plane of dark matter density. Center: Dark matter density

projected along the collision axis. Right: Projected density profile.

42

0.0×100

5.0×108

1.0×109

1.5×109

2.0×109

0 0.5 1 1.5 2

κ (M

O• kp

c-2)

r (Mpc)




0.0×100

5.0×108

1.0×109

1.5×109

2.0×109

0 0.5 1 1.5 2

κ (M

O• kp

c-2)

r (Mpc)




0.0×100

5.0×108

1.0×109

1.5×109

2.0×109

0 0.5 1 1.5 2

κ (M

O• kp

c-2)

r (Mpc)

Figure 4.7 t = 1.5 Gyr snapshot of the β = −∞ simulation. Left: Slice through



43

is short-lived. Figure 4.8 shows that the feature is prominent at epochs t = 1.5 Gyr

and t = 2.0 Gyr (≈ 0.6 and ≈ 0.95 Gyr after the collision). However, by t = 2.5 Gyr

and t = 3.0 Gyr (≈ 1.6 and ≈ 2.1 Gyr after the collision), it has become insignificant.

4.4 Discussion

4.4.1 Relevance to Observations of Cl 0024+17

Our simulations show that for physically and observationally motivated dark mat-

ter particle distributions (NFW-like) and initial dark matter velocity distributions

with a wide range of angular anisotropies, no ring feature forms as in the scenario

of Jee et al. (2007). The collision scenario for Cl 0024+17, as mentioned before, is

supported by the results of Czoske et al. (2002), who demonstrated the existence of

a bimodal redshift distribution in the cluster galaxies. They also showed that a sim-

ulation of such a collision can explain the observed flattening of the density profile in

the cluster core even if the initial profiles are cuspy. However, they did not mention

the existence of a ring feature in their paper.

Jee et al. (2007) presented results of a lensing analysis of Cl 0024+17. In addition,

they presented the results of simulations of head-on collisions of galaxy cluster dark

matter halos. The dark matter density distributions of the galaxy clusters in their

study were not NFW-like with central cusps, but softened isothermal spheres with

ρ(r) ∝ [1+(r/rc)2]−1; the initial velocity distribution of the particles was not reported.

In these simulations, a clear ring-like feature is evident. In contrast, our simulations

show no ring-like feature even for initial particle velocity distributions that are highly

tangentially anisotropic; only for an initially circular velocity distribution do our

simulations show a ring. As we discuss below, even modestly tangentially anisotropic

velocity distributions are not expected for the dark matter particles in galaxy clusters.

This, together with the results of our simulations imply that the production of ring-

like features in galaxy cluster collisions is likely to be rare.

44

0.0×100

5.0×108

1.0×109

1.5×109

2.0×109

0 0.5 1 1.5 2

κ (M

O• kp

c-2)

r (Mpc)

Initial Profileβ = 1/2

β = 0β = -3/2β = -5/2

β = -17/2β = -∞

0.0×100

5.0×108

1.0×109

1.5×109

2.0×109

0 0.5 1 1.5 2

κ (M

O• kp

c-2)

r (Mpc)

t = 1.5 Gyrt = 2.0 Gyrt = 2.5 Gyrt = 3.0 Gyr

Figure 4.8 Left: Angle-averaged DM density profiles at t = 1.5 Gyr into the simu-

lation (∼0.6 Gyr post-collision) for varying β. As β is made more negative a more

pronounced shoulder-like feature appears in the density profile but a ring does not

appear except in the β = −∞ case. Right: Angle-averaged DM density profiles at

the epochs t = 1.5, 2.0, 2.5, and 3.0 Gyr for the β = −∞ simulation. Note that the

ringlike feature appears prominently at t = 1.5 Gyr (∼0.6 Gyr after the collision) but

is a transient feature that has completely disappeared at later times.

45

4.4.2 Comparison with “Ring Galaxies”

The phenomenon of “ring galaxies” (e.g., the Cartwheel Galaxy) is explained

by a similar mechanism, that of a smaller, more compact galaxy colliding with a

disk galaxy (Lynds & Toomre 1976). N -body simulations of galaxy-galaxy collisions

have demonstrated this phenomenon, for which the collisionless “particles” are stars

(Theys & Spiegel 1977; Toomre 1978). In such simulations, the velocity distributions

of the stars are set up to be tangentially anisotropic, or β < 0, and most of the

orbits are circular. Also, the stars are concentrated into a disk-shaped structure

rather than a spheroidal structure. Lynds & Toomre (1976) demonstrated that such a

collision causes a crowding of particles on tangential orbits. In our simulations, as β is

made more negative a more pronunced shoulder appears in the projected dark matter

density, but a ring appears only when the initial velocity distribution is circular.

However, ΛCDM simulations of structure formation have demonstrated that in the

inner regions of dark matter halos the particle velocities are nearly isotropic (β ∼ 0),

and they become more radially anisotropic in the outskirts of the halo (Cole & Lacey

1996). Our results show that under such conditions no ring forms. If the formation

of dark-matter ring features in collisions between galaxy clusters is dependent upon

tangentially anisotropic velocities of the particles, then it would require fine-tuned,

high-angular momentum initial conditions in the dark matter distribution, which is

unrealistic under the standard CDM paradigm.

Chapter 5

A Parameter Space Exploration:

Variations in Mass Ratio and

Impact Parameter

5.1 Introduction

In the previous sections we discussed the simulation of a particular merging galaxy

cluster system, Cl 0024+17. In this system, the line of sight is believed to be along the

merger axis. Because of this, projection effects become extremely important. More

specifically, we saw that the two clusters have X-ray emission that is better fit by a

sum of two surface brightness models rather than one, and that the denser, colder

gas of the smaller cluster dominates the hotter, more rarefied gas of the larger, co-

aligned cluster. In addition, the estimated hydrostatic mass of the system was more

accurately determined under the assumption of two cluster components rather than

one. Moving beyond the specific merger scenario for Cl 0024+17, we now proceed

to investigate a larger parameter space of merger initial conditions. Several previous

investigations of merger parameter spaces (e.g. Roettiger et al. 1997; Ricker & Sarazin

2001; Ritchie & Thomas 2002; Poole et al. 2006, 2007, 2008) have emphasized the

effects of merging on cluster morphology and observables. We build on this work

with a new set of simulations. The set of simulations presented here is the highest-

46

47

resolution parameter study of idealized binary cluster mergers performed with an

AMR-based PPM code to date, with a resolution 3-4 times higher than the most

recent investigation (Ricker & Sarazin 2001).

In Section 5.3.2 we extend our analysis beyond these previous works in projecting

the cluster observables along the different axes of the simulation and comparing the

results. In doing so we seek to more definitively constrain the effects of projection

on the features seen in the ICM during cluster mergers as well as the variations of

spectral temperature, luminosity, and hydrostatic mass estimates. In Section 5.4.2

we briefly discuss these results, expanding on the effect of the various parameters in

the surface brightness profile and temperature fitting on the mass estimates.

In the case of Cl 0024+17, identifying multiple galaxy components in redshift

space was key to the development of the line-of-sight merger scenario for this cluster.

Various kinds of substructure tests exist for discerning the existence of multiple com-

ponents, and they are useful for determining the dynamical state of the collisionless

component of the galaxy cluster. A popular test that has been used in the analysis

of recent cluster mergers is the κ-statistic, proposed by Colless & Dunn (1996), and

used recently to test for substructure in the galaxy clusters Abell 1201 (Owers et al.

2008a) and Abell 3667 (Owers et al. 2008b). In Section 5.3.3 we create simulated

galaxy maps and compute this statistic throughout the merger in the different pro-

jections to determine the power that this statistic has to detect the separate merging

components. In Section 5.4.3 we identify the conditions in binary mergers which are

most likely to yield positive identifications of substructure using this test.

Finally, we use these simulations to examine the degree of mixing of the clus-

ters’ ICM that occurs as a result of merging. Previous investigations (e.g., Ritchie

& Thomas 2002; Poole et al. 2008) have suggested that merging does not mix the

cluster ICM efficiently. However, it is known that the various methods for solving

the equations of hydrodynamics result in different degrees of mixing. While previous

investigations of mixing due to merging have employed the Lagrangian “smoothed-

particle hydrodynamics” (SPH) formulation, our simulations have been performed

using an Eulerian PPM formulation. We measure the amount of mixing that occurs

in our simulations in Section 5.3.4 and in Section 5.4.4 we discuss the implications

48

for the impact of the different simulation methods on the result, particularly on the

resulting cluster entropy and cooling time profiles.

Here we assume a flat ΛCDM cosmology with h = 0.7, Ωm = 0.3, and Ωb =

0.02h−2.

5.2 Methods and Analysis

5.2.1 Initial Conditions

For these idealized cluster merger simulations, we choose initial conditions based

on information gleaned from cosmological simulations and cluster observations. Since

we are interested in the effects of merging on relaxed systems, our initial clusters are

configured to be systems consistent with observed relaxed clusters and cluster scaling

relations. For the temperature normalization, we choose clusters that lie along the

M500 − TX relation of Vikhlinin et al. (2008), which is

M500 = M0E(z)−1(TX/5 keV)α (5.1)

with M0 = 3.02 × 1014h−1M⊙ and α = 1.53. The dependence on the redshift z is

encapsulated in the function E(z) ≡ H(z)/H0. For the total gas mass within r500,

we we use the relation

fg(h/0.72)1.5 = 0.125 + 0.037 log10M15 (5.2)

where M15 is the cluster total mass, M500, in units of 1015h−1M⊙.

For our grid of simulations we have chosen a set of masses representative of galaxy

clusters and groups. The chosen masses are given in terms of M200, corresponding to

the mass within which the average density is 200ρcrit(z). The mass M200 is related to

the radius r200 via

M200 =4π

3[200ρcrit(z)]r

3200 (5.3)

For the total mass distribution (gas and dark matter) we choose an NFW (Navarro,

Frenk, & White 1997) profile:

ρtot(r) =ρs

r/rs(1 + r/rs)2 . (5.4)

49

Where the scale density ρs and scale radius rs are determined by the following con-

straints:

rs =r200c200

(5.5)

ρs =200

3c3200ρcrit(z)

[

log(1 + r/rs) −r/rs

1 + r/rs

]−1

(5.6)

The NFW functional form is carried out to r200. For radii r > r200 the mass density

follows an exponential profile:

ρtot(r) =ρs

c200(1 + c200)2

(

r

r200

)κ

exp

(

−r − r200rdecay

)

(5.7)

where κ is set such that the density and its first derivative are continuous at r = r200

and rdecay = 0.1r200.

The total mass distribution uniquely determines the gravitational potential, and

via the constraint of hydrostatic equilibrium of the gas within this potential, the pres-

sure profile is uniquely determined. To determine the physical quantities of density

and temperature, more information is needed. The “entropy” of gas in clusters is

often parametrized by S = kBTne−2/3. Observations and cosmological simulations

have revealed that for much of the radial range of galaxy clusters the entropy of the

cluster gas follows a power-law form S ∝ rα where α = 1.0 − 1.3. For many clusters,

there is also an “entropy floor” in the inner regions. However, we make this small in

order to make our models consistent with relaxed galaxy clusters. To reproduce this

we choose an entropy profile of the form

S(r) = S0 + S1

(

r

0.1r200

)α

(5.8)

where we set α = 1.1. With this information in hand, we write the condition for

hydrostatic equilibrium as follows:

dP

dr= −ρg

dΦ

dr(5.9)

kB

µmp

d(ρgT )

dr= −ρg

dΦ

dr(5.10)

kB

µmp

d

dr

[

T(

T

S

)3/2]

= −(

T

S

)3/2 dΦ

dr(5.11)

50

Following other authors (Poole et al. 2006), we solve this differential equation by

imposing T (r200) = 12T200, where

kBT200 ≡GM200µmp

2r200(5.12)

is the so-called “virial temperature” of the cluster. With the temperature profile in

hand we may use it and the entropy to determine the gas density profile, and so also

determine the dark matter density profile via ρDM = ρtot − ρg.


Cluster M200 r200 c200 fg TX S0 S1

(M⊙) (kpc) (keV) (keV cm2) (keV cm2) Np

C1 6 × 1014 1552.25 4.5 0.1056 4.97 9.62 192.40 5,000,000C2 2 × 1014 1076.27 4.7 0.0879 2.42 5.08 101.60 1,684,119C3 6 × 1013 720.49 5.1 0.0686 1.10 2.73 54.60 513,137

Particle positions and velocities are set up in the same manner as in 3.2. The

simulation parameters for each simulated cluster are given in Table 5.1.

For all of the simulations, we set up the two clusters within a cubical computational

domain of width L = 10h−1 Mpc on a side. The distance between the cluster centers

is given by the sum of their respective r200. Vitvitska et al. (2002) demonstrated

from cosmological simulations that the average infall velocity for merging clusters is

vin(rvir) = 1.1Vc, where Vc =√

GMvir/rvir is the circular velocity at the virial radius

rvir for the primary cluster. For all of our simulations, this is chosen as the initial

relative velocity.

For this investigation we explore a parameter space in mass ratio of the clusters

R and in initial impact parameter b. The set of simulations in this parameter space

is detailed in Table 5.2. Taking the virial mass of our primary system as M200 =

6×1014M⊙, we perform mergers involving virial mass ratios of R = 1:1, 1:3, and 1:10

(however, we present an analysis and discussion of the 1:10 mass-ratio simulations for

the results on ICM mixing and cluster entropy profiles only, leaving the analysis and

discussion of these simulations with respect to the other aspects of this current study

51


Simulation R b (kpc)S1 1:1 0.0S2 1:1 464.43S3 1:1 932.28S4 1:3 0.0S5 1:3 464.43S6 1:3 932.28S7 1:10 0.0S8 1:10 464.43S9 1:10 932.28

for future work). We also vary the impact parameter b for our runs. Vitvitska et al.

(2002) also demonstrated that for merger encounters the average tangential velocity

is v⊥ ≃ 0.4VC . Instead of splitting our initial velocity into radial and tangential

components, we choose to initialize our systems with impact parameters that result

in relative tangential velocities consistent with the results of Vitvitska et al. (2002).

Since the velocity v⊥ represents an average velocity, we select a range of tangential

velocities consistent with this relation. Figure 5.1 shows the initial merger geometry

and how the tangential and radial components of the velocity are related to the initial

choice of impact parameter.

The highest AMR resolution of this set of simulations is ∆x = 5h−1 kpc, which

is ∼3-4 times higher than the resolution in Ricker & Sarazin (2001). The number of

dark matter particles in each simulation is given in Table 5.1.

5.2.2 Mock X-ray Observations

For the mack X-ray observation generation for these simulations we have chosen

an exposure time texp = 60 ks, and situate the clusters at a redshift z = 0.1. At

this redshift, the spatial extent of interest is larger than the corresponding angular

size covered by the Chandra ACIS-S chips that MARX simulates. To cover an area

corresponding to at least R ∼ 3 Mpc, we create three ”observations” of our cluster at

52

Figure 5.1 Schematic representation of initial merger geometry.

three different pointings for each flux map, one at the center of the primary cluster’s

emission and two centered approximately 1 Mpc away from the center, offset in right

ascension. For the spatial analysis these observations are merged using the CIAO

script merge all and for the spectral analysis the spectra from the different observa-

tions are co-added. Figure 5.2 shows some examples of raw counts images generated

from the mock Chandra observations.

To perform the surface brightness and spectral fits, we first adopt the surface

brightness peak of the cluster as the center of our system. For the fitting of the

X-ray spectral temperature and the calculation of the X-ray luminosity, we use “core-

corrected” regions of our simulated observations. It has been shown that quantities

computed with the central core region excised (Vikhlinin et al. 2008; Kravtsov et al.

2006; O’Hara et al. 2007; Poole et al. 2007) (or otherwise accounted for; such as with

two-temperature models, see Ikebe et al. (2002); Shang & Scharf (2009) for examples)

are more successful at preserving the observed mass scaling relations. This is due to

the fact that temperatures and luminosities at these radii are affected by radiative

53

Figure 5.2 Examples of simulated Chandra observations. Images are of raw counts

taken in 60 ks exposures. Left: A Bullet Cluster-like cold front preceded by a shock

front. Center: Two galaxy cluster cores sideswiping each other at the first core

passage. Right: “Sloshing” cold fronts produced by the passage of a smaller cluster,

displacing the gas from the primary cluster’s potential well.

cooling and not directly related to the depth of the cluster potential well, and that the

temperature profiles outside these regions show a greater degree of similarity outside

the cluster core (Vikhlinin et al. 2006, 2008). Therefore, as in Vikhlinin et al. (2008);

Nagai et al. (2007b), we define an annular region of [0.15r500, r500] as our extraction

region. We then extract the spectrum from all of the photons detected in this region.

The photon spectrum is fitted to a MEKAL model in the 0.6-7.0 keV band with the

redshift z = 0.1 and the metallicity Z = 0.3Z⊙ held fixed. From this spectral fit the

temperature and the luminosity of the photons are determined. This procedure is

performed for simulation snapshots equally spaced by ∆t = 100 Myr.

To estimate the mass of the cluster under the assumption of hydrostatic equi-

librium, it is necessary to have both a model for the gas density and temperature

structure of the cluster as a function of radius. As mentioned above, for this investi-

gation we choose the β-model (Cavaliere & Fusco-Femiano 1976) for the gas density

and assume isothermality, with the temperature taken to be equal to our fitted spec-

tral temperature Tspec. In previous investigations, isothermal β-model mass models

have been shown to underestimate the masses of clusters by as much as ∼ 25%−40%

(e.g. Bartelmann & Steinmetz 1996; Kay et al. 2004; Hallman et al. 2006; Rasia et al.

2006; Poole et al. 2007; Piffaretti & Valdarnini 2008, and others). However, the use of

54

this model still enables us to examine the trends in mass due to the effect of mergers

viewed from differing projections. In a future extension of this work we employ other

gas density and temperature profiles for fitting to our simulated clusters that will

more accurately reflect the structure of the ICM.

Initially r500 is assumed to be bracketed within the radial range of the flux map.

For the current value of r500 the surface brightness and temperature fits are performed

and M500 is calculated. This procedure is iterated, varying r500, until the average

estimated density enclosed by r500 is

ρ500 =3M500

4πr5003(5.13)

In many cases, this determination of r500 will be the only one available (in the absence

of other observational constraints), so we choose to define it in this way rather than

computing it directly from simulations.

5.2.3 Caveats

Two caveats must be made regarding the set of simulations we present and the

analysis of the results. The first is the chosen parameter space. Our parameter

space includes configurations that may not be very likely in the real universe, such

as precisely head-on mergers and mergers of equal mass. Such configurations are

nevertheless useful because they provide interesting limiting cases that bracket the

variations of the physical quantities seen over the course of the mergers.

Similarly, the second caveat regards the chosen sky projections of the merging

systems. The mergers are set up such that the orbits of the cluster centers of mass lie

in the x−y plane at the coordinate z = 0. We have chosen to project along the three

axes of the simulation box, and as a result some of the projections chosen are not

very likely, such as those where the line of sight is precisely along the line of centers of

the clusters. Keeping this in mind, these configurations still serve as useful limiting

cases that can illustrate the range of differences in the results expected from viewing

a cluster merger in different projections.

55

5.3 Results

5.3.1 Qualitative Description

For the R = 1:1 and R = 1:3 simulations we have performed we produce a set of

maps of X-ray surface brightness, temperature, and mass density in the x, y, and z

projections of our simulation box (excepting the two cases of the head-on mergers,

where due to the rotational symmetry it is only necessary to project along the x

and z-axes). To produce these maps, we choose a direction of projection, re-grid the

adaptive mesh to the highest resolution in the simulation, and integrate the relevant

quantities along the line of sight. We use these maps to describe the stages of the

merger along the different lines of sight. These figures are found in Appendix B.

For the surface brightness maps we simply integrate our aforementioned flux maps

over the range of photon energy from 1-10 keV. For the projected density maps, we

integrate the sum of the dark matter and gas density over the line of sight. For the

temperature maps, we want to match closely the spectral temperatures that would be

fitted in a real temperature map. Typically this is done by defining some appropriate

weighting for computing a temperature measure averaged over many components

viewed along a line of sight:

〈T 〉 =

∫

VwTdV

∫

VwdV

(5.14)

For this purpose, measures such as the emission measure-weighted temperature (TEM, w =

ρ2) or the emission-weighted temperature (TE, w = ρ2Λ(T )) have commonly been em-

ployed. Spectra of simulated galaxy clusters consistently show that these measures

are higher than the fitted spectral temperature Tspec by about ∼ 20%−30% (Gardini

et al. 2004; Mazzotta et al. 2004). In this work, for the temperature maps we employ

the “spectroscopic-like” temperature weighting (Tsl, w = ρ2T−α), which Mazzotta et

al. (2004) found provides a close approximation to fitted spectral temperatures to

multicomponent spectra for α = 0.75.

For approximately the first ∼ 5 Gyr of the mergers the systems undergo significant

evolution. It is in this period of time that many structures are formed that resemble

56

observed structures in real clusters. In the following discussion of the early evolution

of the mergers we address the equal-mass mergers and the 1:3 mass-ratio mergers

separately.

1:1 Mass Ratio Mergers

For our equal-mass mergers, there is no distinction between the primary and the

secondary system, so either cluster could be chosen without any loss of specificity. In

the head-on case (simulation S1), as seen in the z-projection (Figures B.1 and B.2),

the shocked gas in between the clusters forms a flattened “pancake”-like structure (t ≈1.6 Gyr). As the shocks depart, this gas expands and cools adiabatically. An observer

at this epoch would notice that the X-ray peak is separated from the two dark matter

peaks, with maximum separation of approximately 1 Mpc. Later, a bridge of colder

gas is also formed in between the two dark matter halos as they drag in ICM away

from the central gas concentration and from the surrounding regions of the halos.

During this time (t ≈ 2.0-2.4 Gyr), the X-ray isophotes have a boxy, diamond-shaped

appearance, whereas a temperature map would reveal a “clover-leaf” pattern, with

the high-temperature regions forming the “leaves”, as noted by other authors (Poole

et al. 2006). Eventually, the halos reach their maximum separation and begin to fall

back toward the center of the composite system, driving a weaker, secondary shock

(t ≈ 2.8-3.2 Gyr). The halos then undergo several oscillations around the center (t ≈3.2-3.6 Gyr), driving successively smaller shocks and throughly mixing the central

ICM. As the DM halos undergo these oscillations they drag dense clumps of ICM

with them, creating large “lobes” of gas that persist for ∼2 Gyr.

In the x-projection for the head-on case (Figures B.3 and B.4), the picture is

considerably simpler but nevertheless demonstrates characteristics of a merger. The

surface brightness countours are very regular throughout the entire merger, which is

to be expected due to the high symmetry of the merger geometry and the chosen

projection. No prominent “edges” in the surface brightness are seen, as in this pro-

jection the shocks are completely hidden. The lobes seen in the z-projection are also

not seen along the line of sight. However, as the shocks and cool gas expand outward

57

from the center, they produce ringlike temperature variations centered on the cluster

core. In this extreme case of an equal-mass merger viewed along the line of sight,

this complicated temperature variation with radius may be the only signature in the

X-ray observations of the gas that a merger is occurring. Other signatures, such as

detecting distinct groups in the galaxy redshift distribution (such as in the case of

Cl 0024+17) would possibly provide evidence for the two cluster components.

In both cases where there is an initial impact parameter (simulations S2 and

S3), the initial evolution is similar to the head-on case. In the z-projection (Figures

B.5, B.6, B.11, and B.12), two shocks are created that are seen clearly in both the

temperature maps and the surface brightness contours. However, for the rest of the

encounter, very different structures are formed. The cluster gas cores survive the

initial encounter, though they are ram-pressure stripped (t ≈ 1.2-1.6 Gyr). The

stripped material forms two distinct types of features, also noted by Poole et al.

(2006). The first is a cool bridge of gas that is strung between the two gas cores.

The second are large “plumes” that extend from each gas core. These features are

formed by stripped gas that moves outward to larger radii, adiabatically expanding

and cooling (t ≈ 2.0-2.8 Gyr). As this gas expands out into the surrounding ICM,

the denser, cooler gas comes into pressure equilibrium with the more rarified, hotter

gas, forming contact discontinuities that are seen as cold fronts. The fronts develop

curvature as they expand outward due to the gravitational influence of the cluster

cores, and the plumes appear to extend outward from them. The curvature of the

fronts is more pronounced in the case with the larger impact parameter. In the z-

projection, these plumes extend to radii of r∼> 1-1.5 Mpc. These features last for

approximately ≈ 1.5-2 Gyr before the features come into thermal equilibrium with

the surrounding ICM.

In the y-projection of the off-center mergers (Figures B.7, B.8, B.13, and B.14),

initially the merger appears much like the head-on case, with the two gas cores ap-

proaching each other with shocked gas in between. However, unlike the head-on case,

because of the non-zero impact parameter the two gas cores survive the initial en-

counter, forming small cold fronts that are clearly seen (t ≈ 1.6 Gyr). Though the

shocks show a clear temperature discontinuity in this projection, the jump in surface

58

brightness across the shock is smooth. After the shocks pass, the cold fronts expand

into two large “lobe” features on either side of the center of the system (t ≈ 2.4-

3.2 Gyr), which correspond to the plumes seen in the z-projection, and grow to a

width of ∼ 1 Mpc. Unlike the shock fronts, at the cold fronts there are clear jumps

in surface brightness as well as temperature in this projection.

In the x-projection (Figures B.9, B.10, B.15, and B.16), we see a very similar

sequence in evolution to the y-projection. The gas appears shock-heated but the

surface brightness discontinuities associated with the shock are completely hidden.

However, the cold fronts associated with the plumes are still prominent. In this

projection, the isophotes associated with these features have a “boxy” appearance.

1:3 Mass Ratio Mergers

For the 1:3 mass mergers, the results are very different. In these scenarios the

unequal mass ratio breaks the symmetry seen the in previous examples, leading to a

wider variety of features seen in the intracluster gas.

In the head-on case (simulation S1), in the z-projection (Figures B.17 and B.18),

the gas from the smaller cluster drives a shock into the larger cluster, pushing the

smaller cluster’s gas forward and laterally. A contact discontinuity forms as a cold

front behind the shock front, which is composed of the gas from the smaller core but

is partially draped by gas from the larger core. In this projection, at times t ∼ 1.4-

1.5 Gyr the appearance of the merging clusters is very similar to that of the “Bullet

Cluster”, (1E 0657-56).

It can also be seen in this projection that the gas of the larger cluster which has

been stripped and pushed away from the merger axis forms streams which extend

from the tip of the cold front away from it in the direction of the receding dark

matter halo of the primary (t ≈ 1.6 Gyr). The tails of these streams eventually fall

back into the dark matter potential of the primary cluster, colliding and heating the

gas (t ≈ 2.0-2.4 Gyr). At later times, the gas of the secondary begins to encounter

regions of lower density, and consequently experiences less ram pressure. Without

this restricting force, the gas is pulled forward into the DM potential and overshoots

59

it. As it makes it to the other side, the gas is accelerated forward into the incident

gas and is gravitationally accelerated from behind, resulting in the cool gas becoming

Rayleigh-Taylor unstable and breaking up the cool core (t ≈ 2.0-2.4 Gyr). Shortly

after this the secondary dark matter core falls back towards the primary, driving a

smaller, weaker shock through the primary’s gas, dragging the secondary’s cold gas

behind (t ≈ 2.8-3.6 Gyr).

In the x-projection (Figures B.19 and B.20), the results are very similar to the

head-on case for the equal-mass merger, with the shocks hidden from view in the

surface brightness and the radial temperature variations. In this case the surface

brightness contours appear more disrupted due to the break-up of the primary’s core

by the secondary’s.

In the off-center cases (simulations S5 and S6), the most significant effect of the

initial stages of the merger is to initiate sloshing of the central gas of the primary

cluster in its gravitational potential. As seen in the z-projection (Figures B.21, B.22,

B.27, and B.28), the secondary passes the primary’s core, the latter is pulled toward

the former and the gas and dark matter components begin to separate (t ≈ 1.6-

2.0 Gyr). This is because the gas encounters ram pressure from the shocked wake of

gas trailing the secondary, while the dark matter is collisionless. After the secondary

has passed, the gas which was previously held back is now pulled back toward the

center of the potential. However, this gas overshoots the DM peak, and begins to slosh

around in the potential well (t ≈ 2.0-3.2 Gyr), creating distinctive mushroom-head

shaped formations and cold fronts in the gas that last for ∼ 1-1.5 Gyr. In the case of

the larger impact parameter (S6), these features are larger and more distinctive.

As for the gas core of the secondary, it initially trails its DM peak, held back by

ram pressure, but as it enters regions of lower pressure overshoots the DM potential

well and forms a large plume cold front similar to those seen in the equal-mass mergers

(t ≈ 1.6-2.4 Gyr). As the secondary cluster begins to finally merge with the primary,

this gas forms a cold, low-entropy stream that accretes gas onto the primary for a few

Gyr. Meanwhile, the secondary’s dark matter core drives a smaller shock through

the gas of the primary, heating the gas and disrupting the sloshing cold fronts (t ≈3.2-3.6 Gyr).

60

As seen from the y-projection (Figures B.23, B.24, B.29, and B.30), the sloshing

cold fronts that are prominent in the z-projection do not appear. The cold front of

the secondary’s gas does appear, first appearing as a cool lobe that grows to a size

of ∼ 0.5 Mpc which then expands to a cool wake of gas that trails the secondary’s

dark matter core as it makes its return approach to the primary. The shock created

by the second passage of the secondary appears briefly in this projection.

Finally, from the x-projection (Figures B.25, B.26, B.31, and B.32), though the

initial shocks are hidden from view in the surface brightness contours, the sloshing

cold fronts do appear. In this projection they take on a distinctive mushroom shape

with a front forming the “head” (the gas initially held back by the ram pressure of

the shock) and a secondary front forming the “stem” of the mushroom (the gas that

falls back towards the dark matter core of the primary). As in the z-projection, this

feature persists for ∼ 1-1.5 Gyr before it is destroyed by the second interaction of the

primary with the secondary.

5.3.2 Time Dependence of Global Cluster Observables and

Derived Quantities

In this section we detail the change of global cluster observables and derived quan-

tities with time in different projections. In each case, the quantities plotted are those

determined within the overdensity radius r500 as specified above. In their investiga-

tion of the evolution of the X-ray temperature (TX) and the X-ray luminosity (LX),

Ricker & Sarazin (2001) and Poole et al. (2007) noted that both the X-ray temper-

ature and luminosity of the merging system undergo significant transient increases.

Generally, these jumps in TX and LX occur at two points in the merger evolution:

first, at the moment of initial closest core passage (or moment of core collision in

the case of the head-on mergers), and secondly at the second closest core passage (or

the recollapse of the system in the head-on cases). We observe this general pattern

here as well, but find that the magnitude of the second transient is smaller than in

these previous investigations and the magnitude of the transient increases depend in

general on the direction of projection.

61

It is crucial to point out at the outset that the determination of these quantities

is done in a way that is different from either Ricker & Sarazin (2001) or Poole et

al. (2007). In the former, only the temperature and luminosity evolution was pre-

sented, and these quantities were calculated over the entire computational domain

(and therefore were independent of the chosen direction of projection). In the latter,

the temperature, luminosity, and mass evolution were presented within the varying

r500(t), which was determined from the actual mass distribution from the simula-

tion. In this analysis, we have chosen to derive r500 in each mock observation by a

process of iteration until the estimated average density enclosed by r500 is equal to

〈ρ〉 = 500ρcrit(z). In the absence of other information from which to estimate the

appropriate overdensity radius and mass (e.g. weak lensing), this would be the only

way an observer would be able to estimate these quantities. This way of determining

M500 and r500 therefore serves to make a more direct comparison with X-ray mass

estimates of clusters. It also means that the results of the temperature and surface

brightness fits are coupled to each other, which was not the case in the previous

investigations.

X-ray Spectral Temperature

In the evolution of the X-ray spectral temperature within the radial range [0.15r500, r500],

there are some features common to all projections and simulations. The first is an

initial gradual period of evolution followed by a strong transient increase in the tem-

perature (∆T ∼ 5-10 keV), which lasts approximately 1-1.5 Gyr. This results from

the compression and heating of the gas due to the shock associated with the initial

core passage. A significant drop in temperature follows, at least back to the initial

value and in some cases lower. This drop is due to the brief period of expansion

and cooling that follows the shocks. Following this drop, there is a second transient

increase that is significantly smaller, with an increase typically on the order of 1 keV,

which lasts from 1-2 Gyr, as the cluster cores make their second passage. Following

this there is a gradual rise in the temperature that levels off at very late epochs, due

to the accretion of material back onto the merger remnant that had been ejected by

62

the collision to higher radii.

There are also some general features that are common across simulations in the

same projection. At the beginning of each simulation, the temperature in the x-

projection is lower than that measured in either the y or z-projections, which are

measured to be the same initially in each simulations. This be a small effect, on

the order of a few percent (as in simulation S1), or it can be a large effect, up to

∼1 keV (as in simulation S5). This is a result of the superposition of the two cluster

components along the line of sight. The presence of gas at a cooler temperature than

the primary cluster along the line of sight ensures that the measured temperature will

be cooler than that of the primary cluster, especially if the cooler gas is denser, due

to the ρ2-dependence of the X-ray luminosity. Therefore this effect is most drastic

as the clusters are viewed along the x-direction, and if the masses of the two clusters

are unequal. Towards the end of the simulation, the measured temperature of the

final merger remnant is measured to be different in the different projections with

variations on the order of a few to several percent. The cluster that results from

each merger is not spherically symmetric; it is elongated along the primary axis of

the merger. Projecting these triaxial clusters along the different directions results in

slightly different measured spectral temperatures.

In simulation S1, we have the very highly artificial and unlikely scenario of an

equal-mass merger with zero impact parameter, but this simulation can still serve as

a useful limiting case. In this case the temperature evolution with time in both the

x and z-projections is very similar in shape and magnitude; the largest deviations

are due to the slight time difference in the transient increases (∆T ∼ 1-2 keV, or

∼ 16 − 33%), and shortly after the second transient (∆T ∼ 0.75 keV, or ∼ 13%).

In the equal-mass, off-center cases (simulations S2 and S3), larger deviations appear.

The most obvious of these are the initial difference in the measured temperature (as

discussed above, here on the order of 10%), and the magnitude of the first transient.

As the impact parameter is increased, the magnitude of the transient as measured in

the z-projection is reduced with respect to the x and y-projections, by about 1-3 keV

(∼ 10−30%). The reason for this is the fact that as the impact parameter is increased,

the cool core of the secondary cluster which continues to trail the shock front is not

63

10

0 2 4 6 8 10

TX (

keV

)

t (Gyr)

xz

10

0 2 4 6 8 10

TX (

keV

)

t (Gyr)

xyz

10

0 2 4 6 8 10

TX (

keV

)

t (Gyr)

xyz

10

0 2 4 6 8 10

TX (

keV

)

t (Gyr)

xyz

Figure 5.3 X-ray spectral temperature (TX) vs. time for the 1:1 mass-ratio mergers.

From left to right are simulations S1, S2, and S3. All projections are shown.

cut out of the spectral fit in the z-projection as it is in the other projections. Finally,

as seen in Ricker & Sarazin (2001) and Poole et al. (2007), the peak temperature of

the initial transient decreases as the impact parameter goes up.

In the 1:3 mass-ratio simulations, the general evolution is similar, but the differ-

ences in the fitted spectral temperature in the different directions are more signifi-

cant. The most striking difference is the initial temperature difference between the

x-projection and the others, which is as large as 1-1.5 keV (∼ 20 − 25%), due to

the temperature difference between the two cluster components. In simulation S4,

the head-on case, after this initial evolution, the initial temperature increase in the

two projections is the same, mirroring the equal-mass, head-on case. After the first

transient, the temperature measured in the x-projection is again lower than the tem-

perature measured in the z-projection by a difference of ∼> 0.5-1 keV (∼ 10 − 20%),

as the cooler gas of the secondary cluster begins to fall back to the primary and is co-

aligned with the latter’s hotter gas. The short dip in temperature during the second

transient as seen in the z-projection is due to the bulk of the hottest gas briefly “cut

out” of the temperature fit by the core-correction. The temperature evolution in the

off-center mergers (simulations S5 and S6) is similar to the corresponding equal-mass

cases, with peak transient temperatures lower in the z-projection and a decrease in

peak transient temperature with increasing impact parameter. Between the different

mass-ratio sets, the temperature transients are larger in the equal-mass cases than in

64

10

0 2 4 6 8 10

TX (

keV

)

t (Gyr)

xz

10

0 2 4 6 8 10

TX (

keV

)

t (Gyr)

xyz

10

0 2 4 6 8 10

TX (

keV

)

t (Gyr)

xyz

Figure 5.4 X-ray spectral temperature (TX) vs. time for the 1:3 mass-ratio mergers.


the 1:3 mass-ratio cases.

X-ray Luminosity

In the evolution of the X-ray luminosity with time, there are similar transient

increases to that of the X-ray temperature. A noticeable difference, however, in these

transient increases is a transient decrease in the middle of the luminosity jump. This

effect is also seen in Poole et al. (2007). The origin of this decrease is straightforwardly

understood: the luminosity of the X-ray photons shown here is “core-corrected” (see

Section 5.2.2), and the decreases in luminosity occur when enough high-density, high-

temperature gas dips within r = 0.15r500 of the cluster center, and as the contribution

to the luminosity from gas at these projected radii on the sky is not included, the

measured luminosity decreases for a short period of time.

In the 1:1 mass-ratio simulations, early on one sees a significant difference in

the evolution of the luminosity (Figure 5.5). At the beginning of the simulation,

the luminosity measured in the x-projection is nearly 112-2 times as large as that

measured in the other projections. This is due to the effect of the contribution to

the luminosity from the two cluster components superimposed upon each other. As

the clusters approach, the luminosity increases in both cases, but the peak luminosity

undergoes different evolution with time in the different projections.

In simulation S1, the initial rise of the luminosity peak is very high in the z-

65

1044

1045

0 2 4 6 8 10

L X(0

.5-2

.0 k

eV)

(erg

/s)

t (Gyr)

xz

1044

1045

0 2 4 6 8 10

L X(0

.5-2

.0 k

eV)

(erg

/s)

t (Gyr)

xyz

1044

1045

0 2 4 6 8 10

L X(0

.5-2

.0 k

eV)

(erg

/s)

t (Gyr)

xyz

Figure 5.5 X-ray luminosity (LX) vs. time for the 1:1 mass ratio mergers. From left

to right are simulations S1, S2, and S3. All projections are shown.

projection, as the two gas cores of the clusters approach, and after the brief decline in

luminosity due to the core-correction the following rise due to the reappearance of the

shocked gas outside of the core-corrected region is similar in magnitude. However, in

the x-projection the initial peak is about 1.7 times less luminous. This is due to the

fact that in the x-projection the high-density peaks of the clusters are always in the

core-corrected region, and the smaller increase in luminosity is due to the gas outside

this region on the sky that is shock-heated and compressed. After the core-correction

decline the luminosity rises to match that of the z-projection. A similar situation

exists for the second luminosity peak. The luminosity decline in the x-projection

from the first peak is greater and the rise is weaker than in the z-projection, with the

former’s peak approximately 1.3 times weaker than the latter’s. After briefly coming

to the same luminosity at approximately t = 5 Gyr, the luminosity measured in the

z-projection rises and the luminosity in both projections levels off, with the value

measured in the z-projection approximately 13% higher than that measured in the

x-projection. This effect is related to the asphericity of the merger remnant. In the

different projections, the overdensity radius r500 will be estimated differently due to

the assumption of spherical symmetry being violated. The estimated r500 at these late

epochs is larger in the x-projection in the z-projection, resulting in a larger surface

area on the sky being included in the determination of the luminosity.

In the off-center cases (simulations S2 and S3), the luminosity is higher at the

66

first transient than in the head-on case due to the fact that more of the high-density

X-ray emitting gas is outside of the core-corrected region. As the impact parameter is

increased, the shapes of the luminosity transients become different, as the secondary

cluster’s core crosses the core-corrected area at different times and for different dura-

tions in each projection. After the first transient the luminosity in the x-projection

persists at a higher value than that in the z-projection (≈ 1.6 times) due to the super-

position of the two components. At later times, the luminosity in all three projections

levels off to a relatively constant value approximately twice the initial value as seen

in the z-projection. At these late epochs, the differences in the luminosities in the

different projections are very small, within 3%.

In the 1:3 mass-ratio simulations, there are strong parallels with the 1:1 mass-

ratio simulations if one compares the simulations with identical impact parameter in

terms of the luminosity evolution (Figure 5.6). In simulation S4 the first transient

in the z-projection is larger than the x-projection by a factor of ∼2, and the second

transient is also larger in the z-projection by about 1.6 times in the largest deviation.

At late epochs the luminosity measured in the z-projection is higher than that in the

x-projection by about 6%. In the off-center cases (simulations S5 and S6), as the

impact parameter is increased the shape of the initial luminosity transient as seen in

the initial projection is different, similar to the equal-mass cases. In simulation S6 the

secondary cluster essentially spends all of its time (until accretion onto the primary)

outside the core-corrected region as seen in the z-projection, and as a result there is

no dip in luminosity seen in the transient In these cases the deviations in luminosity

between the different projections after the initial transient are minor, within a few to

several percent.

Estimated Hydrostatic Masses

The accurate estimation of masses under the assumption of hydrostatic equilib-

rium is essential to constructing the M − TX and LX − M galaxy cluster scaling

relations. Since mergers drive the clusters away from the equilibrium state, it is im-

portant to quantify the degree to which the hydrostatic mass estimate deviates from

67

1044

1045

0 2 4 6 8 10

L X(0

.5-2

.0 k

eV)

(erg

/s)

t (Gyr)

xz

1044

1045

0 2 4 6 8 10

L X(0

.5-2

.0 k

eV)

(erg

/s)

t (Gyr)

xyz

1044

1045

0 2 4 6 8 10

L X(0

.5-2

.0 k

eV)

(erg

/s)

t (Gyr)

xyz

Figure 5.6 X-ray luminosity (LX) vs. time for the 1:3 mass ratio mergers. From left

to right are simulations S4, S5, and S6. All projections are shown.

the true value. For this purpose we have fitted β-models to the X-ray surface bright-

ness profiles at each epoch of observation and computed the estimated hydrostatic

mass from the β-model parameters and the fitted spectral temperature.

From previous works (most notably Poole et al. 2007) it would be expected that

large increases in the estimated mass would be expected at the same points that there

are transient increases in the temperature and luminosity. This is generally true in

our simulations but the results are somewhat different, due mainly to the different

determination of the overdensity radius r500. In Poole et al. (2007) the estimated mass

was taken as the mass evaluated from the β-model estimate at the r500 which was

determined from the actual mass within the simulation. Here (as mentioned above)

both M500 and r500 are derived from the hydrostatic mass estimate. Therefore the

evolution of the estimated M500 will depend significantly on the evolution of the fitted

β-model parameters and the fitted spectral temperature. We discuss the effects of

the variations in the β-model parameters on the estimated masses in section 5.4.2.

Figures 5.7 and 5.8 shows the evolution of the estimated M500 with time. The

estimates in the different projections are compared. Initially, in all the simulations,

the estimated mass in the x-projection is less than that in the other projections, in

line with the temperature dependence of the estimated mass. This difference ranges

from around 10% in the head-on cases to around 100% in some of the off-center cases.

In the equal-mass mergers, we see strong transients in mass at the first core

68

1014

1015

1016

0 2 4 6 8 10

MX

, β(r

≤ r

500)

(M

O•)

t (Gyr)

xz

1014

1015

1016

0 2 4 6 8 10

MX

, β(r

≤ r

500)

(M

O•)

t (Gyr)

xyz

1014

1015

1016

0 2 4 6 8 10

MX

, β(r

≤ r

500)

(M

O•)

t (Gyr)

xyz

Figure 5.7 Estimated β-model mass within r500 vs. time for the 1:1 mass ratio mergers.


passage and weaker transients at the second core passage, in line with the behavior of

the temperature jumps. In the head-on simulation there is a difference between the

peak mass of the second transient in the two projections; the mass estimated in the x-

projection is larger by about a factor of 3. In the off-center cases, the mass estimated

in the z-projection is approximately three times higher than in the x-projection in

the first transient increase. These differences are largely due to the variation in the

fitted β-model parameters, particularly the logarithmic slope parameter β, due to the

linear dependence of the estimated mass on this parameter.

In the 1:3 mass-ratio mergers, the two transient increases in mass are more equal

in magnitude than those in the equal-mass mergers. In the head-on case, there are

large differences in the magnitude and shape of the two transients in the different

projections. In the off-center cases, though the second transient increase in mass

is similar in magnitude and shape in the different projections, the first transient is

radically different. In fact, in simulation S6, the simulation with the largest impact

parameter, the first transient is very small in all of the projections and in the z-

projections barely appears at all. The lack of a significant jump in estimated mass

in this case reflects the fact that in due to the high mass ratio and large impact

parameter, the primary cluster is not as significantly disturbed and the resulting

change in estimated mass is weaker than in the other cases. This is discussed more

fully in Section 5.4.2.

69

1014

1015

1016

0 2 4 6 8 10

MX

, β(r

≤ r

500)

(M

O•)

t (Gyr)

xz

1014

1015

1016

0 2 4 6 8 10

MX

, β(r

≤ r

500)

(M

O•)

t (Gyr)

xyz

1014

1015

1016

0 2 4 6 8 10

MX

, β(r

≤ r

500)

(M

O•)

t (Gyr)

xyz

Figure 5.8 Estimated β-model mass within r500 vs. time for the 1:3 mass ratio mergers.


At the end of each simulation, the estimated masses at the end of the simulation

differs in the different projections. The differences in mass range from 44% in the

worst case (simulation S4) to 13% in the best case (simulation S6). This results

primarily from the fitting of spherically symmetric β-models to the triaxial clusters

that result at the end of the simulation. The different fitted parameters in the different

projections contributes to the differences in mass, in addition to the small differences

in temperature noted in Section 5.3.2.

Besides comparing the estimated masses from the different projections to each

other, it is also instructive to compare the estimated mass in each projection to

the actual mass projected within the estimated r500. In the analysis of the density

and temperature structure of Cl 0024+17 that were presented in Ota et al. (2004)

and Jee et al. (2007), it was the projected mass that was used in the estimates

and comparison to the mass derived from lensing. Assuming the gas is isothermal,

spherically distributed, and in hydrostatic equilibrium, the density of matter at a

radius r is estimated from the β-model as (Ota et al. 2004):

ρβ(r) =3kTβ

4πr2Gµmp

[

3r2

r2 + r2c

− 2r4

(r2 + r2c )

2

]

(5.15)

From this profile a projected mass density profile can be derived:

MX,β(R) = 2π∫

∞

−∞

∫ R

0ρβ

(√R′2 + z′2

)

R′dR′dz′ (5.16)

70

=3πkTβ

2GµmP

R2

√

R2 + r2c

(5.17)

Figures 5.9 through 5.14 show the comparison of the estimated projected mass

within r500 to the actual projected mass within r500. Some general features about these

figures can be noted. Not surprisingly, the isothermal β-model significantly underesti-

mates the projected mass at the beginning of all simulations in the x-projection, due

to the implicit assumption of a single cluster when there are two projected along this

direction. Secondly, at the time of the first transient in each simulation the projected

mass within r500 is significantly overestimated (with the exception of the z-projection

in simulation S6, Figure 5.14, right panel), in some cases by an order of magnitude.

However, in general the second transient increase in mass is still an overestimate of

the mass within the projected r500.

The β-model is a poor representation of the gas density profiles of some clusters,

and most temperature profiles of galaxy clusters have been shown to not be isother-

mal. We therefore expect that excepting cases where these two conditions are met

(including the conditions of spherical symmetry and hydrostatic equilibrium) that

masses estimated under these assumptions will be inaccurate. Near the end of our

simulations, when the clusters have merged and the merger remnant has become viri-

alized, we find that the accuracy of the hydrostatic mass estimate from the β-model

varies in the our suite of simulations. In the equal-mass, head-on simulation, the

estimated mass is greater than the actual mass by ∼ 10 − 20%, depending on the

direction of projection. In the equal-mass, off-center cases, the estimated mass is less

than the actual mass by ∼ 30 − 45%. The mass estimates at late times for the 1:3

mass-ratio simulations are more accurate. In the head-on case, in the x-projection

the isothermal β-model overestimates the mass by ∼ 6%, while in the z-projection

the mass is underestimated by ∼ 15%. In the off-center cases, the projected mass is

estimated to within ∼ 10%.

Finally, for the reasons outlined in the previous paragraph the estimated mass

under the assumptions of the isothermal β-model is expected to be inaccurate for

relaxed clusters with cool cores, which exhibit temperature profiles that increase

71

1014

1015

1016

0 2 4 6 8 10

M(r

≤ r

500)

(M

O•)

t (Gyr)

EstimatedActual

1014

1015

1016

0 2 4 6 8 10

M(r

≤ r

500)

(M

O•)

t (Gyr)

EstimatedActual

Figure 5.9 Estimated and actual masses within r500 vs. time for simulation S1. Left:

x-projection. Right: z-projection.

from the cluster center followed by a decline at outer radii. In Poole et al. (2007)

the estimated deprojected mass within r500 was measured to be less than the actual

mass by ∼ 25− 40%. In this case, at the beginning of the simulations the projected

masses within r500 actually overestimate the actual projected mass within the same

radius by ∼ 10%, in projections where only one cluster is seen along the line of

sight. This apparent contradiction is resolved by re. The first is that our simulated

total mass density profiles are of the NFW form at radii r ≤ r200, and at larger radii

the density declines exponentially (Section 5.2.1). Real mass density profiles should

continue a power-law-like form until the density reaches the average mass density

of the universe (?Tavio et al. 2008). If this was allowed to occur, the mass of our

profiles would increase by approximately ∼20. From the mass density profile derived

from the isothermal β-model (Equation 5.15), as r → ∞ the estimated mass density

ρβ(r) ∝ r−2, and this density is projected along the line of sight with no truncation.

Therefore the overestimate in the estimated projected mass at the beginning of the

simulation results from our choice of truncation of the NFW density profile at r200 and

the assumption from the β-model that the mass density profile maintains a power-law

logarithmic slope of -2 as r → ∞.

72

1014

1015

1016

0 2 4 6 8 10

M(r

≤ r

500)

(M

O•)

t (Gyr)

EstimatedActual

1014

1015

1016

0 2 4 6 8 10

M(r

≤ r

500)

(M

O•)

t (Gyr)

EstimatedActual

1014

1015

1016

0 2 4 6 8 10

M(r

≤ r

500)

(M

O•)

t (Gyr)

EstimatedActual


x-projection. Center: y-projection. Right: z-projection.

1014

1015

1016

0 2 4 6 8 10

M(r

≤ r

500)

(M

O•)

t (Gyr)

EstimatedActual

1014

1015

1016

0 2 4 6 8 10

M(r

≤ r

500)

(M

O•)

t (Gyr)

EstimatedActual

1014

1015

1016

0 2 4 6 8 10

M(r

≤ r

500)

(M

O•)

t (Gyr)

EstimatedActual



1014

1015

1016

0 2 4 6 8 10

M(r

≤ r

500)

(M

O•)

t (Gyr)

EstimatedActual

1014

1015

1016

0 2 4 6 8 10

M(r

≤ r

500)

(M

O•)

t (Gyr)

EstimatedActual


x-projection. Right: z-projection.

73

1014

1015

1016

0 2 4 6 8 10

M(r

≤ r

500)

(M

O•)

t (Gyr)

EstimatedActual

1014

1015

1016

0 2 4 6 8 10

M(r

≤ r

500)

(M

O•)

t (Gyr)

EstimatedActual

1014

1015

1016

0 2 4 6 8 10

M(r

≤ r

500)

(M

O•)

t (Gyr)

EstimatedActual



1014

1015

1016

0 2 4 6 8 10

M(r

≤ r

500)

(M

O•)

t (Gyr)

EstimatedActual

1014

1015

1016

0 2 4 6 8 10

M(r

≤ r

500)

(M

O•)

t (Gyr)

EstimatedActual

1014

1015

1016

0 2 4 6 8 10

M(r

≤ r

500)

(M

O•)

t (Gyr)

EstimatedActual



74

5.3.3 A Cluster Substructure Test: The κ-statistic

In the case of Cl 0024+17, the primary evidence for the purported two cluster

components is the bimodal redshift distribution (Czoske et al. 2002). Identifying

independent cluster components using galaxy redshifts is a possible technique for

determination of merger geometries, particularly for cluster mergers that occur at an

orientation nearly parallel to the line of sight.

Several methods for detecting substructure in cluster galaxy distributions exist.

A popular technique is the κ-statistic, formulated by Colless & Dunn (1996). In this

technique, the velocity distribution of each localized group of galaxies (defined as a

galaxy and its n nearest neighbors in sky coordinates) is compared to the velocity

distribution of the whole cluster via a standard K-S two-sample test (Press et al.

1992). This technique has been used to detect substructure in a variety of galaxy

clusters. Additionally, the test utilizes information in three dimensions (two sky

coordinates plus one redshift coordinate), and provides a more robust way to detect

substructure in clusters rather than just relying on the 2-D positions of the galaxies

alone. The κ-statistic has been used recently in the analysis of observed galaxy

cluster mergers (Owers et al. 2008a,b) to determine the existence of substructure in

the galaxy distribution and correlate this information with the structure seen in the

X-ray surface brightness and temperature maps.

What would a hypothetical observer, able to view a galaxy cluster merger for

the billions of years of its duration, observe in terms of the evolving substructure

of the merger? In particular, how long would the separate components be clearly

distinguished? After the system has coalesced into a single merger product, to what

degree would the system still exhibit significant substructure, and for how long? In

order to address these questions, we construct galaxy distributions for our simulated

clusters as we did in the case of Cl 0024+17. A similar study involving idealized

merger scenarios testing a variety of statistical tests for substructure in galaxy clusters

was carried out in Pinkney et al. (1996). The differences between these cluster models

and real merging galaxy clusters are important to note. The most obvious differences

are the assumed initial spherical symmetry of the clusters, the lack of field galaxies and

75

other clusters in projection, and the existence of only two merging cluster components.

We pick 300 particles from the simulation, identifying them as galaxies, and we

project their positions and velocities along the three different axes of the simulation

box, excepting the head-on merger cases where symmetry permits us to ignore the

y-axis. The number of particles are chosen from each halo in proportion to the halo’s

virial mass, and the particle positions lie within r200 of each cluster’s center. Finally,

we convert the line-of-sight velocities to redshifts, using z = 0.1 as the base redshift.

From these galaxy maps, we can construct the κ-statistic, which is defined as

κn =Ngals∑

i=1

−log[PKS(D > Dobs)] (5.18)

where D is the K-S statistic for a local group of galaxies and Dobs is the corresponding

value for all of the galaxies in the cluster. Each local group of galaxies corresponds to

the n =√

Ngals nearest neighbors in the 2D galaxy distribution on the sky. Thus, κn

is just the negative log-likelihood that there is no localized deviation in the velocity

distribution on the scale of galaxy groups of n nearest neighbors. In order to get a

measure of the significance of the κ-statistic (P(κ > κobs)), it is necessary to run a

set of Monte Carlo models to determine the probability that we could get a higher

value of the statistic than that which is observed by chance. This is easily done

by randomly shuffling the velocities between the cluster members. The motivation

behind this choice is that for a three-dimensional substructure test such as the κ-

statistic the null hypothesis is no correlation between position and velocity (Pinkney

et al. 1996), which means that the velocity distribution should be the same locally

as globally (within counting statistics). It is important to recognize that the way the

significance is formulated, low values of P(κ > κobs) correspond to strong evidence of

substructure.

Figures 5.15 and 5.16 show the computed κ-statistic over the length of the simu-

lation for simulations S1-S6 simulations. In these plots the computed statistic in the

different projections is shown, as well as lines marking the points of first DM core

passage (tfirst), the epoch of greatest separation (tapo), the time of the second core

passage (tsecond), and the time beyond which the system appears as a single DM halo

in the z-projection (tsingle).

76

200

400

600

800

1000

1200

1400

0 2 4 6 8 10

κ

t (Gyr)

xz

tfirsttapo

tsecondtsingle

200

400

600

800

1000

1200

1400

0 2 4 6 8 10

κ

t (Gyr)

xyz

tfirsttapo

tsecondtsingle

200

400

600

800

1000

1200

1400

0 2 4 6 8 10

κ

t (Gyr)

xyz

tfirsttapo

tsecondtsingle

Figure 5.15 Computed κ-statistic vs. time for the 1:1 mass ratio mergers. From left

to right are simulations S1, S2, and S3. All projections are shown, as well as the

significant epochs tfirst, tapo, tsecond, and tsingle, which are defined in the text.

The largest variations in the statistic occur in the early stages of the merger. In

the x-projection, the statistic is highest as the clusters approach, and continues at

a significant value after the first core passage. However, as the cores approach their

greatest separation it falls to its lowest values. There is another, smaller peak in

the value of the statistic as the cluster cores make their second approach. In the

y-projection, for the mergers with an initial offset, the statistic begins at a low value

but rises after the initial core passage, as the clusters begin to accelerate along this

axis. The value of the statistic quickly falls until the clusters approach their greatest

separation. In the z-projection, where the line of sight is perpendicular to the merger

plane, the value of the statistic maintains a low value at all times throughout the

merger. At later times in all the projections the statistic maintains a similar low

value.

Figures 5.17 and 5.18 show the significance P(κ > κobs) computed via the Monte

Carlo realizations of the shuffled cluster redshifts for all projections, also marked with

the epochs mentioned above.

In these figures we find that the lowest values of P(κ > κobs) generally correspond

to the highest values of κ from the previous set of figures, with values of P ranging from

zero to a few percent. Though these smallest values appear to correspond uniquely

with instances of real substructure, the value of P can take on widely varying values

throughout the simulation.

At later times in all cases and projections, the κ-statistic fluctuates around a low

77

200

400

600

800

1000

1200

1400

0 2 4 6 8 10

κ

t (Gyr)

xz

tfirsttapo

tsecondtsingle

200

400

600

800

1000

1200

1400

0 2 4 6 8 10

κ

t (Gyr)

xyz

tfirsttapo

tsecondtsingle

200

400

600

800

1000

1200

1400

0 2 4 6 8 10

κ

t (Gyr)

xyz

tfirsttapo

tsecondtsingle

Figure 5.16 Computed κ-statistic vs. time for the 1:3 mass ratio mergers. From left

to right are simulations S4, S5, and S6. All projections are shown, as well as the

significant epochs tfirst, tapo, tsecond, and tsingle, which are defined in the text.

0.2

0.4

0.6

0.8

0 2 4 6 8 10

P(! > !obs)

t (Gyr)

z

0.2

0.4

0.6

0.8

P(! > !obs)

xtfirsttapo

tsecondtsingle

0.2

0.4

0.6

0.8

0 2 4 6 8 10

P(! > !obs)

t (Gyr)

z

0.2

0.4

0.6

0.8

P(! > !obs)

y

0.2

0.4

0.6

0.8

P(! > !obs)

xtfirsttapo

tsecondtsingle

0.2

0.4

0.6

0.8

0 2 4 6 8 10

P(! > !obs)

t (Gyr)

z

0.2

0.4

0.6

0.8

P(! > !obs)

y

0.2

0.4

0.6

0.8

P(! > !obs)

xtfirsttapo

tsecondtsingle

Figure 5.17 Computed significance of the κ-statistic vs. time for the 1:3 mass ratio

mergers. From left to right are simulations S1, S2, and S3. All projections are shown,

as well as the significant epochs tfirst, tapo, tsecond, and tsingle, which are defined in the

text.

78

0.2

0.4

0.6

0.8

0 2 4 6 8 10

P(! > !obs)

t (Gyr)

z

0.2

0.4

0.6

0.8

P(! > !obs)

xtfirsttapo

tsecondtsingle

0.2

0.4

0.6

0.8

0 2 4 6 8 10

P(! > !obs)

t (Gyr)

z

0.2

0.4

0.6

0.8

P(! > !obs)

y

0.2

0.4

0.6

0.8

P(! > !obs)

xtfirsttapo

tsecondtsingle

0.2

0.4

0.6

0.8

0 2 4 6 8 10

P(! > !obs)

t (Gyr)

z

0.2

0.4

0.6

0.8

P(! > !obs)

y

0.2

0.4

0.6

0.8

P(! > !obs)

xtfirsttapo

tsecondtsingle

Figure 5.18 Computed significance of the κ-statistic vs. time for the 1:3 mass ratio

mergers. From left to right are simulations S4, S5, and S6. All projections are shown,

as well as the significant epochs tfirst, tapo, tsecond, and tsingle, which are defined in the

text.

79

value close to ≈ 350-400. However, in the plots of significance even at late times we

see dramatic fluctuations over time. Presumably at later times as the cluster relaxes,

the evidence for substructure should subside, but this is not clearly the case.

Pinkney et al. (1996) noted that three-dimensional substructure tests suffer from

the effects of velocity dispersion gradients. Since the κ-test relies on differences in

the local distribution of galaxy velocities when compared to the global distribution,

gradients in the velocity dispersion can create a significant difference. This can result

in a higher value for the statistic and a higher significance (a low value of P (κ > κobs)),

even for a cluster without substructure.

To quantify the influence of velocity dispersion gradients in the case of the κ-

statistic we run Monte Carlo simulations of single galaxy clusters, with the galaxy

positions drawn from a NFW density profile. The cluster model is chosen to resemble

the configuration of our primary cluster in all simulations, with a mass M = 6.0 ×1014M⊙. We split the simulations into two groups. One set which has galaxy velocities

drawn from a velocity dispersion profile that corresponds to the solution to the Jeans

equation with the anisotropy parameter β = 0 (Binney & Tremaine 1987):

d(ρσ2r )

dr+

2βρσ2r

r= −dΦ

dr(5.19)

where ρ(r) is the NFW density profile, Φ(r) is the corresponding gravitational poten-

tial, and σr(r) is the resulting velocity dispersion profile. In addition, we construct

another set that has galaxy velocities drawn from a velocity dispersion profile that

is assumed to be constant over the radial range. In both sets, we vary the NFW

concentration parameter c = r200/rs, which has the effect of varying the spatial den-

sity of the galaxies as a function of radius as well as the velocity dispersion profile.

As c is increased, the velocity dispersion profile declines more steeply with radius

in the outer regions of the cluster. For each value of the concentration and velocity

dispersion type we simulate 100 clusters.

For these single cluster models, the null hypothesis is that the κ-test should reveal

no significant substructure. If this null hypothesis is produced correctly in our single

clusters, we would expect the κ-test to find one simulated cluster out of 100 to be

significant at the 1% level or better, five clusters out of 100 to be significant at the 5%

80

Table 5.3. κ-statistic Tests on Simulated Single Clusters

NFW Dispersion Constant Dispersion

c P ≤ 1% P ≤ 5% P ≤ 10% µP P ≤ 1% P ≤ 5% P ≤ 10% µP

2.5 0 3 8 0.49±0.08 1 8 14 0.46±0.08

5 3 6 11 0.44±0.08 3 6 12 0.49±0.09

10 6 12 18 0.40±0.08 1 6 8 0.55±0.08

20 6 12 31 0.26±0.06 0 3 5 0.49±0.08

level or better, and ten clusters out of 100 to be significant at the 10% level or better.

Nevertheless, some statistical scatter around these numbers is expected because of

the finite sample sizes involved (100 simulated clusters for each concentration and

dispersion model). This scatter can be quantified with χ2 statistics, where variations

of 1.65σ (90% significance) for the counts in the 1%, 5%, and 10% bins are 1.64, 3.68,

and 5.2 respectively, for N = 100.

In addition, if the null hypothesis is being produced correctly, we should expect

the centroid of the distribution of the significances to reflect this. The centroid

significance is the probability that a sample drawn randomly from the null hypothesis

will give a substructure signal greater than or equal to the actual sample. For this

null hypothesis, a centroid of 0.5 significance is expected, within the errors.

Table 5.3 shows the numbers of simulated clusters with significances at the levels

of 1%, 5%, and 10%, along with the centroid of these significances, as concentration

is varied for clusters with a varying velocity dispersion and a constant velocity disper-

sion. This data reveals that as the concentration is increased, in the cases where the

velocity dispersion is a function of radius the number of simulated clusters with high

significance increases, reflected in the increasing numbers of clusters at the signifi-

cance levels of 1%, 5%, and 10%, and the decrease in the centroid of the significances

from the expected value of 0.5. However, in the cases where the velocity dispersion

does not vary with radius, the centroid of the significances remains close to 0.5 and

81

we find the expected numbers of clusters at the given significance levels. This shows

that variations in velocity dispersion can result in indications of substructure from

the κ-test even if no substructure due to individual cluster components is present.

5.3.4 ICM Mixing

An interesting question from the perspective of merging galaxy clusters is whether

or not such merging efficiently mixes the ICM. In order to investigate the degree of

mixing of the gas of our initial systems, we have made use of “mass scalars”. These

quantities are defined as Yi = ρi/ρ, where ρi is the partial density, and are advected

along with the fluid:∂(ρYi)

∂t+ ∇ · (ρYiv) = 0 (5.20)

For the primary and secondary systems we define a mass scalar for each that is

initially set equal to unity for radii within r200. We then examine the degree of mixing

of these two components at later times. To parameterize this mixing we follow Ritchie

& Thomas (2002) and define the degree of mixing as

M = 1 −∣

∣

∣

∣

∣

ρ1 − ρ2

ρ1 + ρ2

∣

∣

∣

∣

∣

(5.21)

Given the above definition for the mass scalars, this relation simply reduces to

M = 1 −∣

∣

∣

∣

Y1 − Y2

Y1 + Y2

∣

∣

∣

∣

(5.22)

For example, a value of M = 1.0 indicates equal amounts of the two cluster gas

components in a cell (Y1/Y2 = 1), a value of M = 0.5 indicates that the cluster gas

is mixed in a ratio of Y1/Y2 = 3 or Y1/Y2 = 1/3, and a value of M = 0.0 indicates

the gas is not mixed at all, and that only the gas from one cluster is present.

For this investigation we include a set of R = 1:10 simulations with the same set

of impact parameters (see Tables 5.1 and 5.2). Figure 5.19 shows the resulting mixing

M with gas density contours overlaid at t = 10 Gyr. For the equal mass mergers, the

gas is well-mixed in all cases of varying impact parameter out to a radius of nearly ∼1

Mpc. For the 1:3 and 1:10 mass ratio mergers, the gas is also mixed but this mixing

becomes less efficient, particularly in the primary’s core, as the impact parameter

82

Figure 5.19 The degree of gas mixing (as defined in equation 5.22) in each simulation,

at a time t = 10 Gyr. At the top, from left to right, are S1, S2, and S3, in the middle,

from left to right, are S4, S5, and S6, and at the bottom, from left to right, are S7,

S8, and S9. Each panel is 4 Mpc on a side.

83

increases. In the head-on case with a 1:3 mass ratio, there is significant mixing but

in the b = 464 kpc case the central gas is only mixed in a 3-to-1 ratio and in the

b = 932 kpc case the central gas is hardly mixed at all. Similarly, in the head-on

case for the 1:10 mass ratio the central gas is mixed in a 3-to-1 ratio, but in the

off-center cases the central gas is not mixed at all. This result is easily understood;

the central gas in the primary cluster is the lowest-entropy material in the system. In

the off-center mergers, the core is not completely disrupted but only displaced from

the center and as the system as a whole relaxes this material sinks back to the center

of the newly-reconstituted potential well.

5.4 Discussion

5.4.1 Comparisons of Simulated Mergers to Real Clusters

Though we have not attempted to specifically model any particular system, we

can still draw comparisons to real galaxy clusters that show evidence of current or

recent merging. Recent studies have demonstrated the power of combining observa-

tional techniques to constrain merger scenarios (e.g. Bradac et al. 2008; Owers et al.

2008a,b).

MACS J0025.4-1222

In Bradac et al. (2008) described a joint lensing and X-ray analysis for the cluster

MACS J0025.4-1222 similar to that done for 1E 0657-56 in Clowe et al. (2006). The

cluster appears to be comprised of two subclusters of nearly equal mass undergoing

a merger. They demonstrated a clear separation between the dark matter of the two

subclusters, and the X-ray emitting ICM, which is situated almost entirely between

the two subclusters, as seen in Figure 5.20. The surface brightness distribution is

elongated along the merging axis, and there are no X-ray peaks associated with the

dark matter peaks. From the spectroscopic redshift analysis it was determined that

the merger must be occurring in a plane nearly coincident with the plane of the sky.

In addition, the spectroscopic radial temperature profile reveals that the central gas

84

Figure 5.20 MACS J0025.4-1222: Optical, X-ray (pink), and lensing (blue).

is cool (T ∼ 4 keV) and the outer regions are hotter (T ∼ 8 keV).

The merging cluster MACS J0025.4-1222 appears to be similar to our equal-mass,

head-on cluster merger simulation. The clusters drive a shock in between them,

which results in a large, pancake-shaped, compressed mass of gas in the center of the

system. This gas expands and cools, and then begins to collapse toward the merger

axis under the combined gravitational effect of the two dark matter cores. At later

times, as the dark matter cores are oscillating around the center and are relaxing to a

single remnant, as viewed from the z-projection the system looks very similar to the

observed state of MACS J0025.4-1222, as shown in Figure 5.21. The masses of the

different cluster components cited in Bradac et al. (2008) are similar to the masses in

85

Figure 5.21 Projected mass density (blue) and X-ray emission (log-spaced yellow

contours) for the equal-mass, head-on merger. The time is t = 2.2 Gyr into the

merger, which is ∼ 0.8 Gyr after the collision. At this stage the merger appears similar

to MACS J0025.4-1222. (Image credit: Chandra X-ray Observatory, NASA/SAO)

this simulation. Though one might expect the gas in such a merging system to be of

high temperature due to the shocks, at a time t = 2.2 Gyr, which is ∼ 0.8 Gyr after

the collision, the central ICM is cool, with a projected temperature of T ∼ 4 keV,

whereas at larger projected radii (R ≈ 600) the projected temperature is hotter, at

T ∼ 7 keV. A comparison between the temperature profile of MACS J0025.4-1222 and

the projected spectroscopic-like temperature profile at this point in the simulation is

shown in Figure 5.22. A longer X-ray exposure of MACS J0025.4-1222 will enable a

more detailed description of the temperature of the ICM and will help to constrain

the precise nature of the merger scenario.

86

0 20 40 60 80 100

46

810

Temperature / keV

Radius / arcsec

3

4

5

6

7

8

9

10

11

0 10 20 30 40 50 60 70 80 90 100

T (

keV

)

r (arcsec)

Figure 5.22 left: Spectroscopic temperature profile of the X-ray gas. Open diamonds

show the projected values, filled circles deprojected values (assuming spherical sym-

metry). Reproduced from Bradac et al. (2008) with permission. right: Projected

spectroscopic-like temperature profile for the equal-mass, head-on merger, at the

epoch t = 2.2 Gyr, which is ∼ 0.8 Gyr after the collision.

Abell 2163

Another interesting cluster which appears to be in a state of merging is Abell 2163,

which is one of the hottest and most X-ray luminous clusters known (?). Spectral

temperature and X-ray surface brightness maps presented in Govoni et al. (2004)

and Million & Allen (2008) demonstrate a cluster without any prominent edges in

surface brightness but with a complicated temperature structure. There appears to

be a remnant of a cool core surrounded by hot, shocked gas. Govoni et al. (2004)

suggested that the merger is occurring at a large angle to the sky plane.

A possible scenario for such a merger is suggested by our S5 simulation. Figure

5.23 shows the temperature map and surface brightness contours at a time t = 1.3

Gyr into the simulation along the x-projection, shortly after the pericentric passage of

the cluster cores. The remnant of the cool core is clearly seen, surrounded by shocked

gas, while the evidence of the shock itself in the surface brightness contours is hidden

in this projection. A spectroscopic analysis of the cluster galaxy velocities would

help determine if the merger geometry is indeed oriented near the line of sight. In

Figure 5.24 a bubble-plot of the galaxies at the same epoch along the same projection

87

1.0 12.0

kT (keV)

Figure 5.23 Possible merger scenario for Abell 2163. Left: Temperature map and

surface brightness contours from a Chandra observation of A2163, reproduced with

permission form Govoni et al. (2004). Right: Temperature map and surface brightness

contours in the x-projection from simulation S5 at a time t = 1.6 Gyr into the merger.

is shown. In these plots the galaxy positions are plotted, represented by circles or

“bubbles”, and the radius of each bubble is proportional to the individual galaxy’s κ-

value, R ∝ −log[P(κ > κobs)]. The bubbles are colored according to the cluster they

originated from. The value of the κ-statistic for this configuration is high (κ ≈ 530.41)

and it is significant (P(κ > κobs) = 0), so finding a similar result for the real cluster

could help confirm the merger geometry.

Abell 520

Since the galaxies within clusters are expected to behave (to a first approximation)

as a collisionless system of particles, we should expect their distribution to trace that

of the unseen dark matter, though much more sparsely sampled. However, A galaxy

cluster that has proven to be a mystery as of late is Abell 520. Abell 520 is a merging

system at z = 0.2 that has a bow shock in the cluster outskirts that is trailed by a

cold clump (Markevitch et al. 2005), similar to the configuration seen in the “Bullet

88

-2500

-2000

-1500

-1000

-500

0

500

1000

1500

2000

-1500 -1000 -500 0 500 1000 1500 2000

y sky

(kp

c)

xsky (kpc)

PrimarySecondary

Figure 5.24 Simulated bubble-plot for possible merger scenario for Abell 2163. Galax-

ies are projected along the x-direction from simulation S5 at a time t = 1.6 Gyr into

the merger. Red galaxies are from the primary cluster, blue are from the secondary

cluster. The bubble radius R = −10 log[P(D > Dobs)].

Cluster”, 1E 0657-56. However, the mysterious nature of this system consists in the

fact that weak lensing studies (e.g. Mahdavi et al. 2007; Okabe & Umetsu 2008)

revealed the existence of a “dark core”, a local peak in the projected total mass

density with no corresponding galaxy distribution. The estimate of the gas mass at

the corresponding location from the X-ray data is not sufficient to account for the

mass peak (Mahdavi et al. 2007). In addition, there is a region to the east of the

cluster center with a high red galaxy luminosity but no corresponding peak in the

total matter distribution.

The puzzle of this system exists in the fact that because of the similar dynamics

of the galaxies and the dark matter to find them significantly separated, even in a

merging system, is not to be expected (see Figure 5.25 for images of galaxy positions

overlaid on projected mass density contours from our simulations). Additionally, sim-

ple two-body merging simulations such as ours do not indicate that such a dark core

would form in a merging system. We observe that although after the first pericen-

tric passage that some of the dark matter is strung between the two systems due to

89

Figure 5.25 Galaxy positions and projected mass density in the head-on mergers

shortly after initial core passage. Left: equal-mass merger, at the epoch t = 1.5 Gyr.

Right: 1:3 mass-ratio merger, at the epoch t = 1.3 Gyr.

their mutual gravity, it does not appear that there is enough material to form such a

distinct core (see Figure 5.26 for examples), though further work is needed to verify

this quantitatively. Because of these complications, it has been suggested that the

true nature of Abell 520 is a cluster that is forming at the crossing of three filaments

(Girardi et al. 2008), where the dark core could be a component of a third subcluster.

Such a three-body merger is a simple extension of the kinds of simulations presented

in this investigation, but the parameter space is vast and considerable effort would

be required to match the conditions of Abell 520.

5.4.2 Evolution of Luminosity, Temperature, and Mass Esti-

mates

It was seen in Section 5.3.2 that the direction of projection can have a non-

negligible effect on the observed quantities TX and LX and the estimated hydrostatic

mass Mβ . These changes can contribute to the scatter in the scaling relations between

these quantities. The most significant changes before the merger came from the

superposition of multiple components along the same line of sight, which for the

90

Figure 5.26 Projected mass density in the head-on mergers shortly after initial core

passage. Left: equal-mass merger, at the epoch t = 1.5 Gyr. Right: 1:3 mass-ratio

merger, at the epoch t = 1.3 Gyr.

head-on cases is throughout the entire merger in the x-projection and for the off-

center cases is at the points in the simulation where the secondary cluster passes in

front of the primary in the x and y-projections. In these cases the lower-temperature

ICM will bias the fitted temperature Tspec lower than the highest temperature along

that line of sight, sometimes by deviations as high as ∆T ≈ 1 keV. For spectra with a

large number of counts it may be possible to fit a two-temperature model to determine

the temperatures of the separate cluster components if there are indications that such

a configuration exists. During the transient increases, the differences could be on the

order of ∼ 10 − 30%. The existence of subclusters projected in front of or behind

other clusters will also lead to an increase in the measured luminosity, though this

effect should be easier to identify and correct for unless the cluster cores are closely

aligned. Smaller deviations in the measured and estimated quantities resulted from

the asphericity of the merger remnant at late epochs, but these deviations were most

significant if the remnant was viewed along the axis on which the merger occurred,

reducing the likelihood that this would be a large effect in most cases.

It was also seen that throughout the merger and after the merging has completed

that fitting surface-brightness profiles in the different projections can result in different

fitted parameters and different estimated masses from these parameters due to the

91

violation of the assumptions of hydrostatic equilibrium and spherical symmetry. It

is helpful when considering the mass estimations from the β-model fits the effect of

the variations of the different parameters that go into the model on the resulting

estimated mass. Referring back to Equation 2.16, the total mass within r500 is

M500 =3βkTXrc

Gµmp

[

(r500/rc)3

1 + (r500/rc)2

]

(5.23)

From the formula, the mass scales linearly with TX . This is the primary contributor to

the spikes in the estimated mass, which usually correspond to the transient increases

in the temperature curves. However, the β-model parameters are just as important

in determining the estimated mass. The initial, relaxed surface brightness profiles

with cuspy radial profiles in gas density tend to yield β-model fits with small core

radii (rc ∼< 50 kpc) and small logarithmic slope parameter (β∼< 0.5). The effect of

the mergers is a tendency to increase both of these parameters. The estimated mass

has a simple linear scaling on the β-parameter. The scaling with respect to the core

radius depends is slightly more complicated, and depends on the radius r. Figure

5.27 shows the radial profile of the estimated mass for an isothermal β-model for

varying values of the core radius rc, with the other parameters held fixed and scaled

so that 3βkTX/Gµmp = 1. At low radii (r ≪ rc), Mβ ∝ r−2c , so increasing the core

radius decreases the mass at a fixed radius. However, at higher radii (r ≫ rc), the

dependence of the estimated mass on the core radius cancels out.

The seemingly peculiar situation in the mass curve in the z-projection in sim-

ulation S6 (see Figure 5.8, right panel) can be explained by consideration of these

effects. In this case, the large mass ratio and large impact parameter correspond to

a relatively small initial disturbance of the primary’s gas, and the cluster retains a

small core radius and β-parameter, even though the temperature is increased. As

a result, the average density falls at a lower radius to the value where the average

density enclosed by r500 is equal to the overdensity value than in some of the other

cases.

Extensions to the work presented here would be to include comparisons between

the core-corrected and non-core-corrected measurements of TX and LX for the dif-

ferent projections or to investigate other interesting lines of sight in addition to the

92

0.001

0.01

0.1

1

10

100

1000

10 100 1000

m(r

)

r (kpc)

rc = 50 kpcrc = 100 kpcrc = 200 kpcrc = 400 kpc

Figure 5.27 β-model mass profiles for varying values of the core radius rc. All other

parameters are held fixed and scaled so that 3βkTX

Gµmp= 1.

93

ones aligned with the axes of the simulation box. Additionally, a comparison of the

evolution of the merging systems in the LX − TX , LX −Mβ, and Mβ − TX planes

to existing scaling relation studies of real clusters as in Poole et al. (2007), but in

the different projections can further clarify to what degree the choice of projection

has on the scatter in these relations. In line with our study of Cl 0024+17, a logical

next step would be to fit two cluster surface brightness and temperature models to

the X-ray emission at key points and projections in the simulation. In particular, an

interesting case to examine would be one similar to Cl 0024+17, but with a nonzero

impact parameter (see Section 3.4.3). Presumably it should be easier to distinguish

the two cluster components in both surface brightness and temperature in such a

case.

Finally, an extension of this work should include more accurate representations of

the gas density and temperature profiles of clusters than the isothermal β-model. This

would involve more freedom in the modeling of the gas density profile, particularly

at radii near the cluster center and at large radii, where deviations from the β-

model are seen in many clusters, as well as modeling the radial dependence of the

gas temperature. For example, (Vikhlinin et al. 2006) developed profiles for fitting

gas density and temperature of a set of relaxed clusters that have been used to fit

cluster profiles generated from cosmological simulations (Nagai et al. 2007a), fitting

cluster profiles from catalogs for scaling relation studies (Vikhlinin et al. 2008), and

also in the fitting of density profiles derived from S-Z observations of galaxy clusters

(Mroczkowski et al. 2008).

5.4.3 Substructure Tests and Merger Geometry

In the case of Cl 0024+17, the identification of two cluster redshift components

was key to discerning the probable existence of two merging clusters along the line

of sight. In that case, only the redshifts themselves were used in this determination.

Pinkney et al. (1996) noted that one-dimensional tests (those involving only redshifts)

are most sensitive along the line of sight, so such a test is well-suited for Cl 0024+17.

For more general situations, additional tests may be necessary for discerning the

94

presence of substructure. The κ-test is a three-dimensional test, so it is sensitive to

both differences in spatial position and in galaxy redshift.

Our simulated galaxy maps and corresponding measurements of the κ-statistic

show that this is a potentially useful test under the right circumstances. Along lines

of sight not perpendicular to the merger axis, the test can yield significant results

for substructure depending on the geometry. Pinkney et al. (1996) noted that for

three-dimensional substructure tests that they become less sensitive to lines of sight

along the merger axis itself (the x-projection) even though the line-of-sight velocity

difference of the two cluster components is stronger in that direction, since the test

also depends on a spatial separation between subclumps. This fact is also confirmed

in our κ-statistic tests. At the beginning stages of the merger, where the statistic is

the largest, the value of the statistic is larger as the initial impact parameter for the

merger is increased. This is similarly the case in the y-projection when the merger

has an initial impact parameter, though the period during which the statistic carries

a significant value is shorter.

Not surprisingly, the test is more insensitive to lines of sight viewed perpendicular

to the merger axis (the z-projection). In these cases, the line-of-sight velocities of the

two components are entirely due to the velocities of the galaxies within the cluster, so

significant deviations from the mean cluster redshift would not be expected. Over the

course of each merger when viewed in this projection the deviations in the measured

statistic with time do not show the large changes that are seen in the other projections.

There is also a slight difference in sensitivity between the equal-mass merger cases

and the 1:3 mass ratio cases. In the equal-mass cases the κ-test is generally less

sensitive to the presence of substructure. This is due to the fact that in the equal-mass

cases it is more likely that there will be an equal number of nearest neighbors from

both cluster components for a given galaxy at any position in the field, making the

local velocity distribution at any point less distinguishable from the global distribution

from the standpoint of the K-S test. The best example of this is the head-on, equal-

mass merger, where the test is not particularly sensitive in either the x or the z-

projections (see Figures 5.15 and 5.17, left panel).

These differences in sensitivity are also shown by visual inspection of the galaxy

95

Figure 5.28 Galaxy bubble plots for the 1:1 mass ratio merger with zero impact

parameter. The radius of each bubble is R = −10 log[P(D > Dobs)], and the bubbles

are colored according to the originating cluster. The chosen epoch is t = 5.5 Gyr.

Left: x-projection. Right: z-projection.

Figure 5.29 Galaxy bubble plots for the 1:1 mass ratio merger with impact parameter

b = 464 kpc. The radius of each bubble is R = −10 log[P(D > Dobs)], and the

bubbles are colored according to the originating cluster. The chosen epoch is t = 1.2

Gyr. Left: x-projection. Center: y-projection. Right: z-projection.

96

Figure 5.30 Galaxy bubble plots for the 1:3 mass ratio merger with zero impact

parameter. The radius of each bubble is R = −10 log[P(D > Dobs)], and the bubbles

are colored according to the originating cluster. The chosen epoch is t = 1.6 Gyr.

Left: x-projection. Right: z-projection.

“bubble plots” themselves. Figures 5.28 through 5.31 show bubble plots of the simu-

lated cluster galaxies at specific epochs in some specific cases where the differences in

the different projections are significant. The radius of each bubble is proportional to

the individual κ-value for that galaxy, and they are colored according to which cluster

they originated in. At epochs where the substructure is evident in projections closer

to the line of sight, the two subclumps are identifiable as groups of large bubbles.

At the same epoch in the projection perpendicular to the merger plane, the bubbles

are much smaller, indicating that the test is far less sensitive in this direction even

though the substructure is still present.

However, our tests with single clusters demonstrate that in general this test should

be applied with caution, due to the fact that positive indications of substructure can

result from gradients in the galaxy velocity dispersion even if there is no substructure

present. This is likely not a problem in the cases of Owers et al. (2008a) and Owers

et al. (2008b), as they demonstrated other signatures of substructure and merging

components. This falls in line with the suggestion of Pinkney et al. (1996), who

97

Figure 5.31 Galaxy bubble plots for the 1:3 mass ratio merger with impact parameter

b = 932 kpc. The radius of each bubble is R = −10 log[P(D > Dobs)], and the

bubbles are colored according to the originating cluster. The chosen epoch is t = 0.9

Gyr. Left: x-projection. Center: y-projection. Right: z-projection.

emphasized that a battery of tests should be employed to further confirm the existence

of real substructure.

A further interesting comment that may be made about these results is the fact

that they demonstrate that the κ-statistic would likely be very insensitive to the

substructure in the particular case of Cl 0024+17, due to the alignment along the

line of sight of the two cluster components and due to the small mass ratio (1:2) of

these components. Coupled with the results of Section 3.4.4, where it was shown that

distinguishing between two cluster components from the galaxy redshift data alone is

difficult unless the redshift errors are very small, this result reinforces the conclusion

that fitting multiple models to the X-ray surface brightness distribution is a more

powerful way to identify cluster components coaligned on the sky.

5.4.4 ICM Mixing, Cluster Entropy Profiles, and Cooling

Time

In previous investigations of ICM mixing using controlled merger simulations, it

has been reported that the ICM component in these simulations does not efficiently

mix. Ritchie & Thomas (2002), using SPH simulations, defined the degree of mixing

98

as we have above and reported that the ICM only mixed well in the centers of merger

remnants. Poole et al. (2008) investigated mixing from the perspective of metal-

licity gradients, and found no significant flattening of the metallicity profile (which

would imply mixing of gas of different metallicities) following a merger in any of their

simulations. Ricker & Sarazin (2001) did not investigate mixing, and in that case

the reduced resolution of their simulations would have resulted in the suppression of

mixing when compared to our simulations, due to the fact that turbulent eddies and

vortices are not as well-resolved as in our simulations.

Our simulations demonstrate precisely the opposite, that significant mixing can

occur as a result of major merging. One possible reason for the discrepancy is the

choice of method to perform the simulations. The simulations of Ritchie & Thomas

(2002) and Poole et al. (2008) were performed using SPH methods. A recent com-

parison of galaxy cluster merger simulations from a SPH code (GADGET-2) and a

PPM code (FLASH) revealed that in the SPH simulations mixing of the two cluster

components is inhibited, whereas in the PPM simulations the cluster components are

thoroughly mixed (Mitchell et al. 2008).

It has been demonstrated recently by Agertz et al. (2007) that spurious pressure

forces can result at density gradients in SPH simulations, resulting from overestima-

tions of the gas density of particles approaching a high-density region. As a result

the growth of RT and KH instabilities is inhibited. In addition, Dolag et al. (2005)

and Wadsley et al. (2008) showed that the artificial viscosity used in most SPH im-

plementations acts to damp turbulent motions. Therefore, in a galaxy cluster merger

simulation, both of these effects can inhibit the mixing of gas from the two cluster

components. In contrast, PPM simulations such as ours produce the right results for

this physical setup, noting the fact that in real clusters other physical mechanisms

(such as viscosity and magnetic fields) will come into play.

In general, it has been shown that the results of non-radiative cosmological sim-

ulations depend on the numerical scheme adopted for hydrodynamics. Mesh-based

Eulerian codes (such as FLASH) systematically produce higher entropy gas cores than

do particle-based hydrodynamic approaches (see, e.g., Frenk et al. 1999; Dolag et al.

2005; O’Shea et al. 2005). Mitchell et al. (2008) concluded that the extra gas mixing

99

in the PPM simulations was responsible for this difference. In their simulation the

time period of greatest difference between the PPM and SPH codes was the timescale

of merger evolution during which large vortices and turbulent eddies were present in

the PPM simulation, mixing the gas.

In our nine simulations the final central entropy varies as both the impact pa-

rameter and cluster mass ratio is varied. Figure 5.32 shows the final entropy profile

for each simulation, with the initial entropy profile of the primary cluster shown for

comparison. All of the final profiles exhibit a large entropy core where the entropy

flattens out in the center, and the value of the central entropy is significantly larger

than the initial value, with S0 ∼ 200-500 keV cm2. This is in contrast to the entropy

profiles formed by mergers in SPH simulations, as shown by Mitchell et al. (2008).

We also note a correlation between the central entropy in our simulations and the

degree of mixing, which is most evident in the 1:3 and the 1:10 mass-ratio mergers. In

cases of zero impact parameter, the smaller, denser cluster (the secondary) penetrates

the gas of the larger and allows for significant mixing of the central ICM, nearly 100%

for the R = 1:3, b = 0 kpc case (simulation S4) and approximately 60% for the R =

1:10, b = 0 kpc case (simulation S7). It is in these cases where the greatest increase

in core entropy is seen. However, in the off-center cases, most of the gas from the

secondary is stripped as it proceeds along its trajectory, and so though there is mixing

in the cluster outskirts the core of the primary is considerably less mixed, ranging

from about 50% in simulation S5 to 20% in simulation S6 to essentially no core mixing

at all in the R = 1:10 off-center cases (simulations S8 and S9). It is in these cases

where we see the least increase in core entropy.

Mitchell et al. (2008) speculated that heating from recent merger events might

be responsible for the observed clusters that do not possess cool, low entropy cores,

but rather large high-entropy cores such as the final merger products seen in our

simulations. The entropy of the gas in the ICM is closely related to the cooling time,

the total energy of the gas divided by the energy loss rate. It is given by (Zhang &

Wu 2003):

100

r (kpc)

10 100 1000

Sg

as (

ke

V c

m2 )

100

1000

InitialS1S2S3

r (kpc)

10 100 1000

Sg

as (

ke

V c

m2 )

100

1000

InitialS4S5S6

r (kpc)

10 100 1000

Sgas (

ke

V c

m2 )

100

1000

InitialS7S8S9

Figure 5.32 Entropy profiles at t = 10 Gyr, compared to the initial profile. Left: En-

tropy profiles for the 1:1 mass-ratio simulations. Center: Entropy profiles for the 1:3

mass-ratio simulations. Right: Entropy profiles for the 1:10 mass-ratio simulations.

tc = 2.869 Gyr

(

1.2

g

)

(

S

100 keV cm2

)

12(

ne

cm−3

)−23

(5.24)

where ne is the electron number density, S is the gas entropy, and g is the Gaunt

factor. Higher-entropy gas at the same density will have longer cooling times. In

Figure 5.33 we show the cooling time profiles for the nine simulations at t = 10 Gyr,

with the age of the universe marked on the plot for comparison. In all of the simula-

tions, the central cooling times are less than the age of the universe (tage = 13.6 Gyr),

with the highest cooling times ≈ 11 Gyr. This indicates that single mergers are not

sufficient on their own accord to account for the class of clusters with large, high-

entropy cores. However, frequent accretion of small-mass subclusters onto larger

clusters might be able to maintain the entropy floor sufficiently high for longer peri-

ods. This should be a subject for future study, possibly involving a controlled setup

with multiple mergers. One important caveat to note for these results is that there is

no cooling in our simulations, so the central entropies and cooling times shown here

are upper limits.

101

r (kpc)

10 100 1000

t co

ol (

Gyr)

10

100 Initial

S1

S2

S3

tage

= 13.6 Gyr

r (kpc)

10 100 1000

t co

ol (

Gyr)

10

100 Initial

S4

S5

S6

tage

= 13.6 Gyr

r (kpc)

10 100 1000

t co

ol (

Gyr)

10

100 Initial

S7

S8

S9

tage

= 13.6 Gyr

Figure 5.33 Cooling time profiles at t = 10 Gyr, compared to the initial profile. Left:

Cooling time profiles for the 1:1 mass-ratio simulations. Center: Cooling time profiles

for the 1:3 mass-ratio simulations. Right: Cooling time profiles for the 1:10 mass-ratio

simulations.

Chapter 6

Summary and Conclusions

Clusters of galaxies represent an active area of current research in astrophysics,

involving measurements at many wavelengths as well as high-resolution computer

simulations involving diverse physical processes. Since they are the largest coherent

structures to form since the Big Bang, they contain a representation of the different

kinds of matter in the universe and the physical processes that couple this material

together. In addition to this, the dependence of cluster masses as a function of

redshift is a sensitive function of cosmological parameters, and measurements of this

provide powerful constraints on these parameters when used in conjunction with other

methods.

The expectation from the so-called “bottom-up” model of structure formation im-

plies that galaxy clusters formed from the aggregation of smaller objects, and that

this process of merging is still ongoing. Observationally, it is in fact the case that

many clusters are seen in various stages of merging with other clusters. Recent in-

vestigations of this process from a collection of observational techniques has shown

that in these merger events the different kinds of material undergoes significantly

divergent evolution. The cluster gas is shock-heated and compressed, increasing the

temperature and luminosity of the cluster as seen in X-rays. In addition, ram-pressure

stripping and sloshing of gas in the clusters’ gravitational potential results in the for-

mation of cold fronts. The collisionless dark matter component separates from the

gas, eventually undergoing violent relaxation that drives further changes in the ob-

102

103

served state of the gas. Merging provides an opportunity to characterize the physical

processes involved in galaxy clusters. However, merging also drives the gas away from

its equilibrium state, changing the temperature, luminosity, and density structure and

complicating the estimation of cluster masses from cluster scaling relations.

A prominent and interesting example of this process is the galaxy cluster Cl 0024+17,

which is thought to be a system of two clusters undergoing a merger that is viewed

along the collision axis. We have performed a simulation of a high-speed head-on

collision of galaxy clusters and created mock X-ray observations, viewing the clus-

ters from the same direction. Previous investigations of this system had focused on

the collisionless dynamics of the galaxies and the dark matter; we also include gas

in our simulation. The mock X-ray observations of our simulation indicate that the

X-ray emitting gas is still undergoing moderate evolution at times ∼1-2 Gyr after the

collision. However, much of the complicated structure in density and temperature is

obscured due to projection effects. The X-ray surface brightness profile is better fit

by a superposition of β-models than a single β-model at times t = 1-2 Gyr after the

collision, in agreement with the situation observed in Cl 0024+17. The cluster gas

in the center appears colder than it really is due to the projection of denser, colder

gas along the line of sight of the hotter gas. The temperature profile exhibits only

moderate evolution with time, though the temperature distribution of the clusters

in the simulation has significant structure. Though a mass estimate of the cluster

assuming one cluster component in hydrostatic equilibrium underestimates the mass

by a factor of ∼2-3, a mass estimate based on the assumption of two galaxy clusters in

superposition comes close (∼10%) to the actual projected mass of the system within

the arc radius rarc, in agreement with the mass measurements made of the cluster

Cl 0024+17 under the same assumptions. These results may be valuable when look-

ing at clusters discovered in X-ray surveys. With current photometric redshift errors

it will not likely be possible to distinguish such line-of-sight collisions from single

clusters discovered in X-ray surveys at high redshift by identifying separate galaxy

concentrations. However, our results show that it may be possible to distinguish

merging clusters and clusters in projection from single clusters by testing multiple

model fits against single model fits, though this will become more difficult at higher

104

redshifts.

In addition, a recent lensing analysis of Cl 0024+17 revealed possible evidence

of a ring-like dark matter structure. This has been interpreted as the result of a

collision between two clusters of galaxies along the line of sight, a scenario for which

there are other lines of evidence in this system. We have performed simulations of

collisions between galaxy cluster dark matter halos to test this hypothesis, using den-

sity profiles for the dark matter halos motivated by observations and simulations and

investigating a parameter space of initial velocity distribution where we vary the ini-

tial velocity anisotropy. Our simulations show that a more pronounced “shoulder”

feature appears in the projected dark matter density when we make the velocities

of the dark matter particles more tangentially anisotropic, by analogy with the phe-

nomena of ”ring galaxies.” Our simulations show that, although the collision ejects a

large amount of mass from the dark matter cores, a ring-like features does not form,

even for initial velocity distributions that are highly tangentially anisotropic. Only

when the initial velocity distribution is circular does a ring form. Since we do not

expect dark matter particles in clusters of galaxies to have tangentially anisotropic

velocities, our investigation of this parameter space leaves us without an explanation

for the ring-like feature that has been reported in the dark matter distribution of

Cl 0024+17.

Lastly, we have also investigated a more general merger parameter space, where

we have performed simulations of binary mergers with the initial mass ratio and

impact parameter of the clusters varied, and follow the merging process until a single,

virialized cluster results. The initial models for the clusters are representative of

relaxed clusters observed by the Chandra X-ray telescope and the merger parameters

represent a parameter space expected from observations and cosmological simulations.

This set of simulations has the highest resolution achieved for a study of this kind

using an AMR-based PPM code.

For each simulation, we project the observed X-ray flux emitted by the hot cluster

plasma along each direction of the simulation box and generate simulated Chandra

observations for snapshots of the simulation throughout the duration of the merger.

We then perform a standard analysis of these simulated observations, fitting surface

105

brightness profiles to the cluster emission and models to the cluster spectra. From

these models we determine the evolution of the cluster temperature, luminosity, and

estimated mass in each projection. We find in line with previous studies that, at

the first and second pericentric passage of the cluster cores, these quantities undergo

transient increases, but that the amplitude depends in general on the direction of

projection. Cluster mergers projected along different lines of sight on the sky can

result in sizable temperature differences as much as ∼10 − 30% during the merging

period and on the order of a few to several percent after the cluster has relaxed

to a single remnant. Similarly, measured luminosities can differ by sometimes as

much as 1-2 times during the merger and as much as ∼ 13% at the end of the merger.

Estimated cluster masses depend very strongly on the fitted parameters of the surface

brightness and spectral models, and the different fitted parameters that result from

the fits in the different projections can result in differences in estimated mass of as

much as a factor of 2-3 in some cases during the merger, and typically ∼15 − 45%

at the end of the merger. We find in agreement with previous studies that the

isothermal β-model does not typically give an accurate estimate of the cluster mass

in any projection, even for relatively relaxed systems, with a deviations of anywhere

from 6% to 45% depending on the merger scenario and direction of projection. More

accurate fits using more complex models of the density and temperature structure of

the ICM during the cluster mergers could potentially reduce some of these differences,

and will be the subject of future study.

We also create mock galaxy maps from our simulations, and employ the κ-statistic,

a substructure test, to test for the existence of the merging cluster components as

recent authors have done for real clusters. We find that the κ-statistic test is most sen-

sitive to projections with significant line-of-sight redshift differences, with the caveat

that the test becomes less sensitive the closer together the two cluster components

are on the sky. Therefore this test is most sensitive to merging clusters with non-zero

impact parameter which are being viewed close to the direction of their mutual ap-

proach. We also find that the sensitivity of the test increases if the two subclusters

have a large mass ratio. However, the usefulness of this test is complicated by the

fact that clusters with significant velocity dispersion gradients can produce a positive

106

signal for substructure even if no substructure is actually present. Therefore this test

is best used in conjunction with other tests for substructure.

Finally, we also find that in general the hot gas components of the two merging

clusters become very mixed in the final merger remnant, in contrast to some previous

works. This is a consequence of the use of the Eulerian PPM algorithm for solving

the equations of hydrodynamics, which allows for efficient mixing in contrast to the

widely-used Lagrangian SPH algorithm. We also find that this mixing becomes less

efficient with higher mass ratio and larger initial impact parameter. These simulations

produce entropy profiles with high entropy floors and cluster cores with large core

radii. The amount of mixing is related to the entropy floor of the final cluster entropy

profile, with the simulations that involve less efficient mixing for a given mass ratio

resulting in lower entropy profiles in general. We find that the heating produced

in these mergers can raise the central cooling times of galaxy clusters up to tcool ∼10 Gyr, even for mergers with a high mass ratio (R = 1:10). This possibly indicates

that a series of mergers may be able to maintain clusters with high entropy floors

against the effects of cooling, and should be the subject of future study.

A set of simulations such as these provides direction for a pathway forward for

further investigations, involving the inclusion of more physical process in the cluster

gas (and possibly the dark matter), such as explicit viscosity and/or magnetic fields,

or multiple mock observational techniques, such as maps of radio emission or more

accurate modeling of the cluster galaxies. Building on these results, it should be

possible in the near future to place more constraints on the nature of dark matter

and the physics of the ICM, as well as help to further quantify the biases on cluster

observables to identify systematic effects on the cluster scaling relations.

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Appendix A

Mock Observation Generation and

Verification Study

In order to ensure the accuracy of our MARX simulations, we have performed a

few simple verification tests. We have constructed two models: one, a gas sphere with

a constant temperature and a β-model density profile, and the other, two gas spheres,

each with different temperatures and β-model parameters. We then subjected them

to the same analysis as our simulated clusters in FLASH. In the single-cluster case, we

verify that the surface brightness fits we recover the core radius rc, β-parameter, and

central surface brightness S0. Also, we verify in the spectral fit that we recover the

model temperature for the fit to the whole cluster, the normalization of the spectrum,

and the isothermal temperature profile. For the two-cluster case, we verify that the

two cluster components can be distinguished and that the corresponding β-model

parameters are recovered. For these tests we consider the model verified if we can

recover the model parameters within the 1σ errors.

For the spatial fitting, we find that as we increase the exposure time of the ob-

servation, systematic effects become more important. At an exposure time of texp =

240 ks we recover the fitted parameters for both the single and double β-model cases.

This corresponds to the highest exposure time for our simulated clusters. These re-

sults are shown in Tables A.1 and A.2. However, as we go to higher exposure times,

we find that we cannot recover the input models. Table A.3 shows the number of

114

115

Table A.1. Fit to Single Cluster Test, texp = 240 ks

rca β S0

b χ2/d.o.f.

Model 100 1 8.26 × 10−9 ...

Fitted 100.50+3.02−0.50 1.010+0.005

−0.021 (8.18+0.11−0.11) × 10−9 197.92/195

aCore radius (kpc)

bCore surface brightness (counts s−1cm−2arcsec−2)

parameters which are not recovered and the corresponding reduced χ2-statistic for

increasing exposure time. For an exposure time of 1 Ms, we can verify the single

β-model but cannot verify the double β-model, and for an exposure time of 4 Ms we

cannot verify either.

Possible systematic effects may include the modeling of the PSF (which is energy

and spatially dependent), ACIS read-out errors and the precise dependence of the

exposure map on the input spectrum. We note that for the fits that we cannot recover

the parameters the resulting χ2-statistic is large, indicating that the systematic errors

are dominating the statistical errors. We also note that even for these cases we find

the differences in the fitted parameters and the true parameters are at most on the

order of a few percent.

For the spectral fitting we find that up to a high exposure time of 4 Ms that we

recover the input temperature and the normalization of the spectrum, as well as the

isothermal temperature profile. The results of the fit are shown in Table A.4 and the

temperature profile is shown in Figure A.1.

116

Table A.2. Fit to Double Cluster Test, texp = 240 ks

rc,1a S0,1

b rc,2a β2 S0,2

b χ2/d.o.f.

Model 50 3.78 × 10−8 200 0.7 1.38 × 10−8 ...

Fitted 47.14+2.94−4.35 (3.90+0.27

−0.27) × 10−8 198.00+2.43−6.76 0.701+0.029

−0.003 (1.40+0.02−0.02) × 10−8 213.93/193

aCore radius (kpc)

bCore surface brightness (counts s−1cm−2arcsec−2)

Table A.3. Systematic Trends in Radial Fits with Increasing Exposure Time

texp (ks) Nsingle Ndouble χ2/d.o.f.single χ2/d.o.f.double

240 0 / 3 0 / 5 1.01 1.11

1000 0 / 3 1 / 5 1.07 1.20

4000 2 / 3 5 / 5 1.88 2.20

Table A.4. Spectral Fit for a Single Cluster, texp = 4 Ms

Tspec (keV) N

(

10−14

4π[DA(1+z)]2

∫

nenHdV

)

(cm−5) χ2/d.o.f.

Model 4.0 2.547 × 10−4 ...

Fitted 4.01+0.09−0.09 (2.554+0.021

−0.021) × 10−4 91.40/424

117

3.6

3.7

3.8

3.9

4

4.1

4.2

0 100 200 300 400 500 600 700 800

T (

keV

)

r (kpc)

Fitted Temperature ProfileInput Temperature

Figure A.1 Isothermal cluster temperature profile. Dashed line is input temperature.

Appendix B

Projected Temperature and Mass

Density Maps

This appendix contains the projected spectroscopic-like temperature and mass

density maps discussed in 5.3.1. For all of the maps, the X-ray surface brightness

contours in the 1-10 keV band are overlaid, and the spacing is logarithmic. Each

individual panel in the figures is 3 Mpc on a side, and the center of each panel

is determined by the X-ray surface brightness peak in this projection. For each

simulation we begin with the z-projection, as the projected quantities are viewed

perpendicular to the merger plane in this direction and the nature of the features is

most clear.

118

119

1.0 8.0

kT (keV)

Figure B.1 “Spectroscopic-like” temperature maps in the z-projection for simulation

S3 with surface brightness contours in the 1-10 keV band overlaid. The snapshots

taken are 0.8, 1.2, 1.6, 2.0, 2.4, 2.8, 3.2, and 3.6 Gyr after the beginning of the

simulation. Each panel is 3 Mpc on a side.

2.0 × 106 2.0 × 109

log κ (M⊙ kpc−2)

Figure B.2 Projected mass density maps in the z-projection for simulation S1 with

surface brightness contours overlaid. The times shown are the same as in B.1. Each


120

1.0 8.0

kT (keV)

Figure B.3 “Spectroscopic-like” temperature maps in the x-projection for simulation

S1 with surface brightness contours overlaid. The times shown are the same as in

B.1. Each panel is 3 Mpc on a side.

2.0 × 106 2.0 × 109


Figure B.4 Projected mass density maps in the x-projection for simulation S1 with



121

1.0 8.0

kT (keV)





2.0 × 106 2.0 × 109





122

1.0 8.0

kT (keV)

Figure B.7 “Spectroscopic-like” temperature maps in the y-projection for simulation



2.0 × 106 2.0 × 109


Figure B.8 Projected mass density maps in the y-projection for simulation S2 with



123

1.0 8.0

kT (keV)




2.0 × 106 2.0 × 109





124

1.0 8.0

kT (keV)








125

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kT (keV)







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kT (keV)







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