Non-Thermal Emission from Galaxy Clusters Predictions for X-Ray Satellites

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University of Amsterdam

MSc Physics

Gravitation AstroParticle Physics Amsterdam

Master Thesis

Non-Thermal Emission from Galaxy Clusters

Predictions for X-Ray Satellites

by

Richard Tony Bartels 10508333 June 2014 60 ECTS August 2013 - June 2014 Supervisor: Dr Shin’ichiro Ando Daily Supervisor: Dr Fabio Zandanel Examiner: Dr Jacco Vink

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Abstract

Clusters of galaxies are the largest gravitationally bound structures in the Universe and the latest ones to form. Large-scale diffuse synchrotron emission is observed in many clusters proving the presence of relativistic electrons and magnetic fields in the intra-cluster medium. The same population of electrons can inverse-Compton scatter off the photons of the cosmic microwave background. This can generate non-thermal hard X-ray emission, on top of the thermal X-ray bremsstrahlung observed in all clusters. However, so far, non-thermal hard X-ray detections have been claimed in just a few clusters and are not confirmed. A definitive detection of the inverse-Compton emission from galaxy clusters would allow us to disentangle the magnetic fields and relativistic electron distributions. Upper limits on this emission can be used to place lower limits on the magnetic field. In this Master’s thesis, we estimate upper limits for the volume averaged magnetic field that still allow for a detectable non-thermal hard X-ray signal by next-generation X-ray telescopes, in particular ASTRO-H and the already launched NuSTAR, for all known radio halos and relics.

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Acknowledgements

There are various people that I would like to thank for different reasons. First of all, I would like to thank Tera for another year of support and especially for cheering me up when frustration gets the better of me. Also the kittens you offered a temporary home at our place, without me being aware of that, where a nice form of distraction in the final weeks of this project. Finally, there are certain people I have to thank at the university. Thanks to Jacco Vink for being willing to be my second reader and for useful suggestions. Thanks to everyone in Shin’ichiro’s ’thursday’ group for listening to my presentation. Especially to Irene, who has been very helpful. Also thanks to the other Master’s students, with whom I had a lot of fun this year. Thanks to the faculty at GRAPPA, Christoph and Shin’ichiro in particular, for having faith in my future research career. Finally, thanks to my supervisors for all their help. To Shin’ichiro for being willing to guide my project, even though he already had 4 students under his wings. And Fabio, to whom I am most thankful, for being the best supervisor I could have wished for.

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List of Figures

1.1 Abell 1689 . . . 2

1.2 Radio emission in clusters . . . 2

1.3 Non-thermal emisson . . . 4

2.1 Cooling processes . . . 15

2.2 Loss timescales . . . 19

3.1 APEC metallicity dependence . . . 27

3.2 Normalisation for radio halos . . . 29

3.3 APEC normalisation for relics . . . 30

3.4 ASTRO-H sensitivity curves . . . 34

3.5 ASTRO-H sensitivity curve scaling . . . 35

3.6 Spectrum Analysis . . . 36 4.1 1E0657-56 . . . 46 4.2 A0085 . . . 47 4.3 AS753 . . . 48 4.4 A1367 . . . 49 4.5 Coma . . . 51 4.6 A1914 . . . 52 4.7 A2255 . . . 53 4.8 A2319 . . . 54 4.9 A2744 . . . 56 4.10 A3667 . . . 57 4.11 A4038 . . . 58 4.12 MACSJ0717.5+3745 . . . 59 4.13 ZwCl0008.8-5215 . . . 61

4.14 Broken Power Law Spectra . . . 63

4.15 NuSTAR vs. ASTRO-H . . . 66

A.1 Equipartition condition . . . 73

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List of Tables

2.1 γ for maximum loss timescale . . . 18

3.1 Thermal data . . . 33

3.2 ASTRO-H Properties . . . 33

4.1 Cluster Sample . . . 39

4.2 Results for halos . . . 41

4.3 Results for relics . . . 42

4.4 Equipartition magnetic field estimates . . . 44

4.5 Results Broken Power Law . . . 62

4.6 NuSTAR Properties . . . 65

B.1 Radio Data: Halos . . . 77

B.2 Radio Data: Relics . . . 80

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Abbreviations

AGN Active Galactic Nucleus

ATCA Australia Telescope Compact Aarray

cgs centimetre gram second

CMB Cosmic Microwave Background

CR Cosmic Ray

CRe Cosmic Ray electron

CRp Cosmic Ray proton

DM Dark Matter

d.o.f. Degree of Freedom

EoM Equation of Motion

EUV Extreme UltraViolet

FoV Field of View

FWHM Full Width at Half Maximum

HPD Half-Power Diameter

HXI Hard X-ray Imager

HXR Hard X-Rays

IC Inverse Compton

ICM Intra-Cluster Medium

KN Klein-Nishina

LOFAR Low-Frequency Array for Radio Astronomy

NFW Navarro-Frenk-White

NuSTAR Nuclear Spectroscopic Telescope Aarray

NVSS NRAO VLA Sky Survey

RM Rotation Measure

REXCESS Representative XMM-Newton Cluster Structure Survey

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Abbreviations viii

SKA Square Kilometre Array

SXI SoftX-ray Imager

SXR Soft X-Rays

SXS SoftX-ray Spectrometer

VLA Very Large Array

VSSRS Very Steep Spectrum Radio Source

WENSS Westerbork Northern Sky Survey

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Physical Constants

speed of light c = 2.997 924 58_{× 10}10_{cm s}−1

elementary charge e = 4.803 205× 10−10_esu

gravitational constant G = 6.673_{× 10}−8_cm3_g−1_s−1

Planck constant h = 6.626 068 85× 10−27_{erg s}

Planck constant, reduced ~ = 1.054 571 73× 10−27erg s

Boltzmann constant kb = 1.380 649× 10−16erg K

electron mass me = 9.109 382 15× 10−28g

classical electron radius r0 = 2.817 940 29× 10−12cm

Thomson cross-section σT = 0.665 245 856 barn

present day CMB temperature T0 = 2.726 K

Average CMB energy density UCM B = 4.19× 10−13(1 + z)4 erg cm−3

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Symbols

a acceleration cm s−2

B Magnetic field esu cm−2

γ Lorentz factor νc Critical frequency Hz νg Gyration frequency Hz P Power erg s−1 q Charge esu r200 Virial radius kpc rc Core radius kpc

UB Magnetic energy density erg cm−3

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Chapter 1

Introduction

Clusters of galaxies are the largest virialized structures in the universe and as such the latest ones to form according to current paradigm of ΛCDM and hierarchical structure

formation. Their mass is typically of the order of _{∼ 10}15_M

, most of which consists of

dark matter, about 70−80%. The remaining mass is baryonic matter which is distributed

among galaxies (_{∼ few%) and a hot gas called the intra-cluster medium (ICM) which}

makes up 15− 20% of the cluster mass. The thermal plasma component of the ICM

emits in X-rays mainly through bremsstrahlung. This is one of the many ways to identify clusters (Fig.1.1).

Being such enormous systems, clusters are excellent probes in the study of both cosmol-ogy and astrophysics. Cluster mergers are the most energetic phenomena since the Big

Bang, with 1063_{− 10}64_{erg being dissipated into shocks and ICM motions per cluster}

crossing. Therefore, clusters are powerful laboratories for the study of high energy as-trophysics, which is the focus of this thesis. In particular it will deal with non-thermal emission coming from galaxy clusters. It is believed that a fraction of the energy that is dissipated during cluster mergers can go into non-thermal plasma components, such as cosmic rays (CRs) and magnetic fields. Evidence for non-thermal plasma components can indeed be found in clusters that appear to have undergone a recent merger. This

evidence comes in the form of large-scale diffuse radio emission. Large regions (∼ Mpc)

of diffuse radio emission, so-called radio halos, can be found at the center of clusters. Additionally, diffuse emission is observed at the outskirts of clusters. This kind of

emis-sion is called a radio relic (Fig. 1.2). They differ from haloes in their morphology. It

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Chapter 1. Introduction 2

is often elongated and appears to trace a shock front (Feretti et al.,2012;Brunetti and

Jones,2014).

Figure 1.1: An image the cluster Abell 1689. The purple colour scale is X-ray emission from the ICM. Galaxies are observed in the optical and depicted in yellow (http:

//chandra.harvard.edu/photo/2008/a1689/).

Figure 1.2: Left: The X-ray emission from the ICM in Abell 3562 with radio emission in white contours (Giacintucci et al.,2005). Right: The ’Sausage’ radio relic in CIZA J2242.8+5301 (R¨ottgering et al.,2013). In terms of morphology it appears to trace a shock front. The colour coding is radio emission.

If charged particles travel through a magnetic field they are accelerated and therefore

will radiate (Fig. 1.3b). In the case of highly relativistic electrons this radiation is

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Chapter 1. Introduction 3 radio emission is observed in clusters proves the presence of both relativistic electrons and magnetic fields. Moreover, the observed radio spectrum follows to a large extent a power law. As a result, the spectrum of the relativistic electrons producing the emission

must also follow a power law (see section 2.1). Particles that do not follow a

Maxwell-Boltzmann distribution are referred to as non-thermal particles and their emission is dubbed non-thermal emission. The general shape of a Maxwell-Boltzmann distribution

compared to a power law is plotted in Fig. 1.3a.

Diffuse synchrotron radio emission has now been established in a few dozen clusters. Yet, radio observations alone are insufficient to disentangle the electron distribution from the magnetic field. However, these relativistic electrons are also expected to produce X-rays through inverse-Compton (IC) scattering. If they scatter off cosmic microwave background (CMB) photons these gain some energy at the expense of the electron and

thereby become X-ray photons (Fig.1.3c). Given the average energy of the non-thermal

electrons as deduced from radio observations, the emission is expected to be mainly in hard X-rays (HXRs), i.e. > 10 keV. Be that as it may, hard X-rays from clusters have not yet been detected. Non-thermal emission in X-rays has been searched for extensively (e.g.Feretti and Neumann,2006;Rephaeli et al.,2006;Wik et al.,2011) and whereas some detections have been claimed, more recent observations do not confirm them (Ajello et al., 2009, 2010; Wik et al., 2012; Ota et al., 2013; Wik et al., 2014). Intense background emission from the thermal plasma is the most prominent obstruction in detecting IC radiation in the X-ray spectra of clusters.

This project sets out to study the detectability of IC radiation from galaxy clusters by next generation satellites. In particular the detectability by the ASTRO-H telescope

which is currently scheduled for launch in 20151 ₍_{Takahashi et al.}_,₂₀₁₀_{). In order to do}

so the entire spectrum in both radio and X-ray is studied and compared to the back-ground radiation from the thermal plasma and the sensitivity estimates for ASTRO-H. A firm detection of X-rays from non-thermal electrons will not only give us a better understanding of magnetic fields and the particle spectra in clusters, but will also help in studying the injection mechanism of these non-thermal particles. In this work, how-ever, we take a phenomenological approach and are fully agnostic towards the injection mechanism of non-thermal particles. The only assumption that is made is that the same electrons that emit in radio should also be responsible for X-rays through IC radiation.

1

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Chapter 1. Introduction 4

Energy

#

P

articles

log-log Power law

Maxwell-Boltzmann

(a) Maxwell-Boltzmann and power law distributions

(b) Synchrotron radiation (c) Inverse Compton scattering Figure 1.3: Top panel: Different particle distributions. The units on the axes are arbitrary. The red line is a thermal (Maxwell-Boltzmann) distribution which is de-scribed by N (v)_∼ me

2πkT

3

2_v2_e−mev2

2kT . The green line is a power law, often

represen-tative for a non-thermal distribution, and can be described by N (γ) _{∼ K}0γ−p, with

γ being the Lorentz gamma factor. For relativistic particles the Lorentz factor is a dimensionless measure of the energy: γ _≈ P

m. Bottom left: Synchrotron emission

(http://abyss.uoregon.edu/~js/glossary/synchrotron_radiation.html).

Bot-tom right: Inverse Compton scattering (http://www.astro.wisc.edu/~bank/).

Additionally, for the first time, we study the IC radiation for all clusters where a halo or relic is detected, and where radio data is present for at least two different frequencies such that the radio spectral index can be determined.

This work is set up as follows: in chapter 2 the theory behind synchrotron radiation,

inverse Compton scattering and their spectra due to a power law distribution of electrons are studied. In addition, we discuss the effects of electron cooling on the resulting

spectrum. In chapter 3 we discuss how we model the thermal background and analyse

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Chapter 1. Introduction 5 this thesis. We work in centimetre gram second (cgs) units and adopt a cosmological

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Chapter 2

Theoretical Background:

Radiative Processes

In order to understand the radiation coming from the ICM we have to understand its origin. Emission coming from the thermal pool of electrons is mostly due to thermal bremsstrahlung, while the relativistic electron population generates synchrotron and IC emission. We discuss the latter two mechanism below followed by a discussion of electron cooling. Our method for modelling the thermal bremsstrahlung is discussed in section

3.2.

2.1 Synchrotron Radiation

The discussion on synchrotron radiation is mostly based on Blumenthal and Gould

(1970) and Rybicki and Lightman (1979). Another useful, but more concise, discussion

can be found inPinzke (2010).

2.1.1 Motion of a Particle in a Magnetic Field

Synchrotron radiation is due to a charged particle of mass m and charge q, an electron in our case, moving in a magnetic field B. In classical electrodynamics the equations of

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Chapter 2. Theoretical Background 7 motion (EoMs) of such a system are:

d dt(γmv) = q cv× B (2.1a) d dt(γmc 2_{) = qv}_{· E = 0.} _(2.1b)

These equations can be derived by equating Newton’s equation for the four-force with the equation for the Lorentz four-force

meaµ= e cF

µ

νUν. (2.2)

Here Fνµ is the electromagnetic field tensor, Uν the four-velocity, e the electron charge

and methe electron mass. Eq. (2.1b) implies that the Lorentz factor, γ, and thus also the

norm of the velocity,|v|, are constant. From Eq. (2.1a) it follows that the velocity parallel to the magnetic field (vk) is constant and consequently|v⊥|2=|v|2− |vk|2 = constant. Therefore, the particle’s motion is helical, with its acceleration being perpendicular to both the magnetic field and its velocity vector. Its gyration frequency is given by

νg= eB

2πγmec

. (2.3)

2.1.2 Spectrum of a Single Electron

For non-relativistic accelerated charges, the total emitted power is given by Larmor’s formula

P = 2

3 q2_a2

c3 (2.4)

with a the three vector acceleration. However, for our purpose we want to look at relativistic electrons. In order to find the emitted power for relativistic particles we have to go to the instantaneous rest frame of the particle. The idea is that if we find a covariant expression in this frame, it must be valid in any frame. It turns out that

Eq. (2.4) is already in covariant form, but it can be cast in a more convenient form:

P = 2e 2 3c3γ 4_(a2 ⊥+ γ2 0 a2_k) = 2 3cr 2 0γ2B2|v|2sin2θ. (2.5)

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Chapter 2. Theoretical Background 8

Here we have introduced the classical radius of the electron r0= e

2

mec2 and the so-called

pitch angle θ, which is the angle between the velocity vector v and the magnetic field B. Integrating over all pitch angles yields the total power emitted:

Psync(γ) = 4 3σTcβ 2_γ2_U B (2.6) where UB = B 2

8π is the energy density of the magnetic field, B.

We will now look at the spectrum emitted by a single electron. There are two things that should be mentioned. First, there exists a beaming effect for the radiation emitted by relativistic particles. The emitted synchrotron radiation is confined to a cone with opening angle 2/γ or, in other words, the emitted radiation makes at most an angle ∼ 1/γ with the instantaneous velocity vector. This effect is at the origin of the

crit-ical frequency and the appearance of the modified Bessel function in Eq. (2.7) below.

Secondly, there is a difference between the emitted and received power for pitch

an-gles θ _{6= π/2, this difference comes from a Doppler shift due to the particle’s spiralling}

motion. For a detailed discussion see pp. 261-262 in Blumenthal and Gould (1970).

Ultimately, Preceived(ν) = Pemitted(ν)/ sin2θ. The emitted spectrum for a single electron becomes: Pemitted(ν, γ, θ) = √ 3e3B sin θ mec2 Fsyn( ν νc ), Fsyn(x) = x Z +∞ x dξK5/3(ξ) (2.7)

with K5/3(ξ) the modified Bessel function of second kind and νc the critical frequency

given by, νc = 3 2γ 3_ν gsin θ = 3 4π eBγ2_{sin θ} mec ≈ 4.199 B 1 µG γ2sin θ Hz. (2.8)

Above the critical frequency the spectrum rapidly falls.

2.1.3 Spectrum for a Distribution of Electrons

Finally, we have to take into account the electron distribution, which is a function of the energy of the cosmic-ray electrons (CRes) and the pitch angle. The former can be expressed in terms of the Lorentz factor. Our assumption is that the non-thermal

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electrons follow a power law in some energy range, [γmin, γmax], which is based on

observations of halos and relics. We can write for the number of electrons per unit volume,

N (γ, θ) = K0

sin θ

2 γ

−p_. _(2.9)

The factor sin θ₂ comes from assuming the electrons are distributed isotropically with

respect to the pitch angle. Blumenthal and Gould (1970) point out that electrons

emit-ting from some fixed region of space are only observed to emit for a fraction of sin2θ.

Therefore, the observed distribution is related to the actual distribution by Nobs(γ, θ) =

sin2θN (γ, θ). As a result: Pemitted(ν, γ, θ)N (γ, θ) = Preceived(ν, γ, θ)Nobs(γ, θ). The

to-tal emissivity is now given by multiplying the distribution by Eq. (2.7) and integrating

over pitch angle and energy: ˜ Psync(ν) = Z γ2 γ1 Z π 0 Pemitted(ν, γ, θ)N (γ, θ)dγdθ ≈ Z γ2 γ1 Z π 0 2.344_{× 10}−22sin θFsyn( ν νc )N (γ, θ)dγdθ. (2.10)

2.1.4 Magnetic Field Orientation

Before moving on to compute the flux density we observe on earth, we should make a

comment about the orientation of magnetic fields in relation to Eq. (2.10). This is the

standard result for the power coming from a population of electrons as can be found in

the literature (e.g.Blumenthal and Gould,1970;Rybicki and Lightman,1979). However,

it does not account for the orientation of the magnetic field. So far this has been mostly

ignored in the literature, apart from a work byMurgia et al.(2010). In order to account

for this effect we include an extra factor:

δ(θ_{− θ}0)H(θ 0₎

4π dΩ

0_. _(2.11)

Here θ0 _{is the angle between the line of sight and the magnetic field and H(θ}0_{) is the}

distribution of the magnetic field orientations. δ(θ− θ0_{) accounts for the fact that, due}

to the beaming effect, the emitted photons are confined to a narrow solid angle with an opening angle equal to θ, the pitch angle. Therefore, only if the the angle between the magnetic field and the line of sight is approximately the pitch angle will radiation be

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Chapter 2. Theoretical Background 10 that are tangled on sizes much smaller than the volume under consideration, such as in

radio halos and relics, H(θ0_{) = 1, corresponding to an isotropic distribution of magnetic}

fields. Eq. (2.11) then reduces to sin θ₂ after integrating over θ0. On the other hand for

a magnetic field that is completely ordered H(θ0_{) = δ(θ}0 _{− x) and Eq. (}_2.11_{) becomes}

δ(θ− x)sin x

2 . As a result, we modify Eq. (2.10):

Psync(ν) = Z γ2 γ1 Z π 0 Z π 0 Pemitted(ν, γ, θ)N (γ, θ)δ(θ− θ0) H(θ0₎ 4π dγdθdΩ 0 ≈ Z γ2 γ1 Z π 0 Z π 0 2.344_{× 10}−22sin θFsyn( ν νc )N (γ, θ)δ(θ_{− θ}0)H(θ0)sin θ 0 2 dγdθdθ 0_. (2.12) 2.1.5 Photon Spectrum

Ultimately, we want to determine the flux density Sν which is what we observe with

our telescopes. This is given in units of energy per unit time, area and frequency, and

can be obtained by multiplying Eq. (2.12) by the emitting volume and dividing by the

surface of a sphere that is characterised by the luminosity distance DL1:

Ssync(ν)

_erg

cm2_{s Hz}

= Psync(ν)· V/(4πDL2). (2.13)

Photon spectral index Let us briefly look at the spectral index of the photon

spectrum. The synchrotron kernel function, F (x) (Eq.2.7), peaks at x_{∼ 1 (}Blumenthal

and Gould,1970), where x = _νν

c with νc∼ γ

2_{. Therefore, the radio photon spectrum is}

proportional to ν _{∼ γ}2_{. Looking at the spectrum (Eq.}_2.13_{) one finds}

Ssync(ν)∼ νγ−(p+2)dγ

∼ νγ−(p+1) ∼ ν−p−12 .

So the resulting spectrum is Ssyn(ν)∝ ν−α where α = p−1₂ .

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2.2 Inverse Compton Radiation

Another process that should take place in the ICM is IC scattering. Non-thermal elec-trons inverse Compton scatter off CMB photons, thereby transferring some of their momentum to the photons. As a result, the CMB photons can be up-scattered to the X-ray regime. In this section we will first discuss inverse Compton scattering of a single electron followed by a discussion of the spectrum resulting from CMB photons scattering off a population of electrons.

2.2.1 IC scattering in the Thomson limit

In the case of IC scattering we can discern between two limits. The first one is the

Thomson limit, which corresponds to 0_mec2 in the rest frame of the electron, where

0is the energy of the incoming photon. The corresponding cross-section is the Thomson

cross-section, σT ≡ 8π₃ r20, which is independent of the energy of the incoming photon.

The second regime is described by the Klein-Nishina formula, which will be covered in more detail below.

Using energy and momentum conservation the energy transfer from a photon to an

electron in its rest frame (K0) can be shown to be

0₁= 0 1 + 0 mec2(1− cos θ) (2.14) where 0 _and0

1 are the initial and final energy of the photon, respectively. Eq. (2.14)

can be expressed in terms of wavelength using = hν and λ = _νc as

λ0₁_{− λ = λ}c(1− cos θ), (2.15)

where λc= _mch is the Compton wavelength.

The scattering rate dN1

dt is, by considering the scattering rate in the rest frame, K

0_{, and} then boosting to the lab frame K,

dN1 dt = γ −1dN10 dt0 = c Z σ(1_{− β cos θ)dn.} (2.16)

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Chapter 2. Theoretical Background 12 Here n is the distribution of photons. For an isotropic distribution of photons with

energies that satisfy the Thomson limit (σ = σT) this reduces to:

dN1

dt = σTcn (2.17)

Since the background photons in our problem consist mostly of CMB photons we are dealing with an isotropic blackbody distribution at temperature T. The number of pho-tons per unit volume per unit initial photon energy is then given by

n() = 1

π2_(~c)3

2 exp(/kbT )− 1

. (2.18)

Next, we will consider the total energy-loss rate of the electrons, which corresponds

to the energy going into X-ray photons. By definition this equals _h1idN_dt with dN_dt =

σTc R

n()d. Following Eqs (7.9 - 7.16) of Rybicki and Lightman (1979), we find that

the total power going into up-scattered photons equals: PIC(γ) = 4 3σTcγ 2_β2Z _n()d ≡ h1i dN dt . (2.19)

This leads to a mean energy of the Compton up-scattered photons of

h1i = 4 3γ

2_hi _where _{hi =}

R

n()d R

n()d = 2.70kbT. (2.20)

So we can write the total power going into X-ray photons in the Thomson regime as:

PIC(γ) = 4 3σTcβ 2_γ2_U CM B. (2.21) Here UCM B = 2.7kbT0 R n()d(1 + z)4 ≈ 4.19 × 10−13_{(1 + z)}4_{erg cm}−3 _{is the CMB}

energy density with T0 the CMB temperature at present. From Eqs (2.6) and (2.21) we

note that the ratio of synchrotron to IC luminosity at low redshift is Psync(γ) PIC(γ) = UB UCM B ≈ 0.095 B 1 µG 2 . (2.22)

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Chapter 2. Theoretical Background 13 Finally, the total spectrum resulting from inverse Compton scattering of photons off a relativistic electron in the Thomson limit is given by:

dN1(γ, ) dtd1 = 3 4 σTcn() γ2 f 1 4γ2 1 s erg2 = 1.4958× 10−14n() γ2f 1 4γ2 , (2.23)

where f (x) = 2x ln x + x + 1_{− 2x}2. This function is known as the IC kernel function

(see Eq. 2.24).

2.2.1.1 Klein-Nishina limit

The photon spectrum described by Eq. (2.23) is only valid in the Thomson limit, where

γ mec2 in the K frame. For a CMB photon this corresponds to γ ∼ 109. The

complete energy range is described by the Klein-Nishina (KN) formula. The IC kernel function describing the resultant photon spectrum in the KN regime is

FIC(, γ, 1)≡ dN1(γ, ) dtd1 = 3σTc 4γ2 n() × 2q ln q + (1 + 2q)(1_{− q) +}1 2 (Γq)2 1 + Γq(1− q) (2.24) where, Γ = 4γ mec2 , q = 1 Γ(γmec2− 1) . (2.25)

The range of allowed energies follows purely from the kinematics of the problem:

_≤1 ≤

4γ2

1 + Γ or 1

1

4γ2 ≤ q ≤ 1. (2.26)

2.2.2 Inverse Compton Spectrum

In order to find the total Compton spectrum we have to convolve the electron

distribu-tion, N (γ) = K0γ−p, with the IC kernel function and integrate over the electron energy

given by γ and the initial photon spectrum characterised by . Note that we omitted the pitch angle dependence of the electron distribution, since this is only relevant for interactions with magnetic fields and in the case of IC scattering thus integrates out to

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Chapter 2. Theoretical Background 14 unity. The power per unit of energy per unit of volume becomes:

PIC(1) = 1 dNtot dtd1dV = Z ∞ 0 Z γmax γmin N (γ)FIC(, γ, 1)dγd. (2.27)

Similarly to what we did for the synchrotron radiation, we can also find a flux density related to the inverse Compton scattering:

SIC(1) = PIC(1)· V/(4πDL2). (2.28)

Photon spectral index The reasoning behind finding the spectral index related to

inverse Compton scattering is analogous to the argument provided earlier for synchrotron

radiation. Using the average energy of an IC scattered photon _h1i ∼ γ2, it follows that

SIC(1)∼ 1γ−(p+2)dγ ∼ −p−1

2

1 .

Thus again α = p−1₂ . For the full details please see eitherBlumenthal and Gould(1970)

orRybicki and Lightman (1979).

2.3 The Electron Spectrum

Clusters of galaxies are excellent at hosting cosmic rays (CRs). For particles with

en-ergies . 106GeV the diffusion time is longer than the Hubble time, this makes it likely

that CRs are very abundant in the ICM (Berezinsky et al.,1996;Sarazin,1999;Brunetti

and Jones,2014). Moreover, cosmic ray protons (CRps) have cooling times that are of the same order as the diffusion time, cooling mainly through collisions with other pro-tons. As a result, CRps are expected to accumulate in clusters during their assembly history. On the other hand, CRes have cooling times that are much shorter. At energies of γ . 100, they cool through Coulomb cooling, whereas at higher energies γ & 100, IC

and synchrotron cooling dominate (see Fig. 2.1).

In this study we take an agnostic approach for the origin of CRes. Their spectrum is assumed to be a power law, this is supported by radio observations. Moreover, it was shown that the observed power law is related to the electron distribution through

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(a)

(b)

Figure 2.1: Top: The diffusion and cooling times plotted as a function CR energy. CRps can accumulate in the cluster for the Hubble time, while CRes have much shorter cooling times (figure from Brunetti and Jones (2014)). Bottom: Energy loss rate for electrons. At low energies Coulomb cooling dominates whereas at high energies IC losses dominate. The plot is for ne = 10−3cm−3, B = 1 µG and z = 0 (figure from

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Thus, γmin and γmax correspond to a break in the spectrum at low and high energies,

respectively. In particular, cooling processes can alter the shape of the spectrum over

time. In our modelling, we will allow γmin and γmax to be free parameters, except when

there is not enough data to fit for them, in which case they are fixed to theoretically

motivated values (e.g. Sarazin,1999).

The two main models for injecting CRes into the ICM are primary electron models and secondary, or hadronic, models.

Primary electrons can be injected into the ICM as a power law through various

possi-ble accelerating mechanisms, such as in Active Galactic Nuclei (AGN). Sarazin(1999)

discusses various primary electron models. If electrons are injected in a single event, their initial power law spectrum would soon start to deviate due to cooling, producing, amongst others, a cutoff. However, primary electrons from a single injection cannot be

uniquely responsible for radio halos with sizes _{∼ 1 Mpc since their diffusion length is of}

the order_{∼ 10 kpc (}Liang et al.,2002). Turbulent reacceleration models are proposed as

a solution. They argue that seed CRes are reaccelerated through turbulence, produced

on _{∼ Mpc scale during cluster mergers. First, this allows the CRes to maintain their}

energy levels. In addition, since these processes are not very efficient, the time scales on which they occur are short. Thereby it links radio halos to recent cluster mergers.

Finally, the electrons are predicted to have a maximum energy γ _{∼ 10}5_{, producing a}

cutoff in the spectrum (see Feretti et al., 2012, and references therein). Sarazin(1999)

also discusses a continuous injection model, to which reacceleration can be compared during the timescale of the merging event. In such a model the electron population reaches a steady state. It is then shown that, starting from an initial population with spectral index p, the population will over time steepen at high energies to a steady state

solution of p + 1.2 _{Moreover, at low energies, typically γ = b}

Coulomb∆t, the spectrum

will flatten to p− 1. Here bCoulomb is the loss function due to Coulomb cooling and will

be discussed in more detail below.

It has been suggested that a different form of reacceleration can explain the radio emis-sion in radio relics, which typically have a low Mach number and therefore should not

be efficient particle accelerators. Pinzke et al. (2013) showed how reacceleration can

explain the observed spectrum. Reacceleration in relics would typically be a first order Fermi process, accelerating particles through shocks rather than turbulence.

2

This would lead to a difference of 1₂ in the photon spectral index of the injected and observed spectrum. If αinj =

pinj−1

2 and pobs= pinj+ 1 then αobs=

pobs−1 2 = pinj−1+1 2 = αinj+ 1 2.

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Chapter 2. Theoretical Background 17 In secondary models CRes are produced through p-p collisions. These hadronic collisions lead to pions which in return produce electrons and gamma rays through the following

channels (Dennison,1980;Pinzke and Pfrommer,2010):

π0_{−→ 2γ} (2.29a)

π±_{−→ µ}±+ νµ−→ e±+ νe+ νµ+ νµ (2.29b)

Although secondary models are not viable for relics, since the proton density is low at

the cluster outskirts (Feretti et al., 2012), they are popular for explaining radio halos.

Unlike CRes, CRps can diffuse through the cluster volume without cooling. Therefore, CRes can be injected in-situ. Although the hadronic model is attractive for various

reasons (e.g. Pfrommer and Enßlin (2004)), the non-detection of γ-rays (see Eq.2.29)

from clusters to date puts heavy constraints on hadronic models (Zandanel and Ando,

2013).

2.3.1 Loss Functions

In this section we will discuss the energy losses of CRes in clusters following Rephaeli

(1979) and Sarazin(1999). The loss rate function of a single particle is defined as

dγ

dt =−b(γ, t) (2.30)

from which one can then define the lifetime3 _as

tloss≡ γ

b(γ) (2.31)

The total loss rate function consists of parts due to bremsstrahlung, Coulomb cooling, IC scattering and synchrotron radiation, i.e. b = bbremsstrahlung+bCoulomb+bIC+bsynchrotron.

3_{By lifetime we mean the time it takes for the electron to loose a large fraction of its energy and go}

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Chapter 2. Theoretical Background 18 The different loss functions for these processes are:

bbremsstrahlung(γ, ne)≈ 1.51 × 10−16neγ (ln γ + 0.36) s−1 (2.32a) bCoulomb(γ, ne) = 1.2× 10−12ne 1.0 +ln(γ/ne) 75 s−1 (2.32b) bIC(γ) = 4 3 σT mec γ2UCM B ≈ 1.35 × 10−20γ2(1 + z)4s−1 (2.32c) bsynchrotron(γ, B) = 4 3 σT mec γ2UB ≈ 1.3 × 10−21γ2 B 1 µG 2 s−1 (2.32d)

Figure 2.1b shows that for typical cluster values of the electron density and magnetic

field, Coulomb losses dominate at relatively low energies and IC losses dominate at the highest energies.

To see how the total lifetime (Eq. 2.31) depends on the electron density and magnetic

field we calculate the value of the Lorentz factor that maximises the lifetime of a CRe.

Our reference scenario is z = 0, B = 1 µG and ne = 1× 10−3cm−3 for which γ = 301

maximises the cooling time. Next, we vary either ne or B with respect to the reference

scenario and see how γ changes. The results are given in table2.1. In addition, Fig.2.2

provides the cooling time as a function of γ for the parameter values in table 2.1.

B( µG) γ 0.01 315 1 301 2 268 5 171 10 97

(a) Magnetic field dependence

ne( cm−3) γ 10−1 2549 10−2 ₉₁₁ 10−3 301 10−4 97 10−5 ₃₁

(b) Electron density dependence Table 2.1: The Lorentz factor for which the loss timescale (Eq. 2.31) is maximised. The reference scenario is z = 0, B = 1 µG and ne= 1× 10−3cm−3 for which γ = 301.

One parameter is varied to see the effect on the Lorentz factor.

2.3.1.1 γmin

Given the values from table 2.1 and figure 2.2, in case there is not enough data to

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Chapter 2. Theoretical Background 19 0 1 2 3 4 5

log

₁₀

γ

106 107 108 109 1010

Time

(yr)

B = 0.01µG B = 1µG B = 2µG B = 5µG B = 10µG (a) 0 1 2 3 4 5

log

₁₀

γ

105 106 107 108 109 1010 1011

Time

(yr)

ne= 1× 10−1cm−3 ne= 1× 10−2cm−3 ne= 1× 10−3cm−3 ne= 1× 10−4cm−3 ne= 1× 10−5cm−3 (b)

Figure 2.2: Cooling time as a function of γ for various values of the magnetic field and the electron density. The default values are the same as those in table2.1.

radio halos. For radio relics we set γmin= 200; slightly smaller than for halos since the

electron density is expected to be lower at the cluster outskirts, however, this does not have a major impact on our results as we will see. Note that for these values of the

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2.3.1.2 γmax

Deciding upon an upper limit for the power law is somewhat harder. We settle for a

value of γmax = 2× 105 in case there is no spectral steepening in radio observations.

This value corresponds to an average energy of IC scattered photons of _h1i ' 104keV

(Eq. 2.20). We note that, e.g., Pinzke and Pfrommer (2010) consider electrons up to

E _{∼ 10}2TeV, corresponding to γ _{∼ 10}8, which is based on the maximum electron

energy observed in young supernova remnants. For our purpose such high energies are not necessary, because only the position of the cutoff in our spectrum would change. Only if one starts to study the gamma-ray spectrum should we consider changing our allowed energy range. However, for the observed radio spectral index very-high energy

emission is almost unimportant. Finally, fixing the cutoff energy to γ = 2_{× 10}5 _{is in}

accordance with the cutoff in halos predicted by turbulent reacceleration models, as discussed earlier (Feretti et al.,2012).

2.4 Analytical Magnetic Field Estimates

In sections 2.1 and 2.2 we discussed how the spectrum due to synchrotron radiation

and inverse Compton scattering is computed. In our analyses, we assume a power

law spectrum of electrons, N (γ) = K0γ−p, in some range [γmin, γmax]. However, the

magnetic field can also be estimated analytically without modelling the spectrum by

assuming the power law extends from 0 to _{∞. In this case the synchrotron power}

becomes (Blumenthal and Gould,1970, Eq. (4.59)):

Psyn(ν) = 4πK0e3B(p+1)/2 mc2 3e 4πmc (p−1)/2 a(p)ν−(p−1)/2, (2.33)

and for IC radiation (Blumenthal and Gould,1970, Eq. (2.65)):

PIC(1) = 1 dNtot dtd1dV = 3πσT h3_c2 K0(kT ) (p+5)/2_b(p)−(p+1)/2 1 PIC(ν) = 3πσT h2_c2 K0(kT ) (p+5)/2_{b(p) (hν)}−(p−1)/2_. (2.34)

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The functions a(p) and b(p) are given by (seeBlumenthal and Gould (1970) Eq. (4.60)

& (2.66)): a(p) = 2(p−1)/2√3Γ3p−1₁₂ Γ3p+19₁₂ Γp+5₄ 8π1/2_Γp+7 4 (2.35a) b(p) = 2p+3 (p2_{+ 4p + 11)Γ}p+5 2 ζp+5₂ (p + 3)2_{(p + 1)(p + 5)} (2.35b)

Since Psyn and PIC are related to the same CRe population, we can use Eq. (2.33)

and (2.34) to solve for the magnetic field. Let sr be the radio flux density at a given

frequency and fx the X-ray flux4. For a pure power law they can be defined through

fx h erg cm2_s i = kx Z νmax νmin ν_x−αdνx (2.36a) sr h _erg cm2_{s Hz} i = krνr−α (2.36b)

where kx and kr are normalisation factors. Using,

3mekb3σT 4h2_e3 = 2.46× 10 −19 esu cm2_K3 4πmeckb 3eh = 4962 esu cm2_K

we obtain (Harris and Romanishin,1974;Sarazin,1988;Longair,2011):

fxνr−α sr Rνmax νmin ν −α x dνx = 2.46× 10 −19_T3 CM B B b(p) a(p) 4.96× 103_T CM B B α (2.37)

where TCM B = (1 + z)3T0. In case one uses the differential flux in X-ray, rather than

the flux, this expression becomes: sxνr−α srνx−α = 2.46× 10 −19_T3 CM B B b(p) a(p) 4.96_{× 10}3_T CM B B α (2.38) with sx erg cm2_{s Hz} = kxνx−α.

Correction for Isotropically Distributed Magnetic Fields Eq. (2.37) and (2.38)

are frequently used in the literature to provide lower limits on the volume averaged magnetic field in clusters (e.g. Ajello et al. (2009, 2010); Wik et al. (2009); Ota et al.

(2013)). However, these expression do not properly take into account the orientation

4

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of the magnetic fields as discussed in section 2.1.4. Here we calculate the correction to

Eq. (2.35a) for an isotropic distribution of magnetic fields.

The gamma function in Eq. (2.35a) arise from the integral over pitch angles as performed

inBlumenthal and Gould(1970) (see Eqs. 4.58-4.60). In general an integral of this form evaluates to Z π 0 dθ(sin θ)p+n2 =√π Γp+n+2₄ Γp+n+4₄ .

In the literature one finds n = 3. Here one sin θ comes from the dependence of the

power on the pitch angle (Eq. 2.7). Another sin θ comes from the isotropic distribution

of pitch angles (Eq. 2.9). Finally, there is a sinp−12 θ coming from the integral over γ.

To illustrate this last point recall that we have

Z ∞ 0 dγγ−px Z ∞ x dξK_5/3(ξ), where x_≡ ν νc and νc∝ γ 2_{sin θ. Therefore, γ}_{∝ x}−1 2 sin− 1 2 θ and dγ∝ sin− 1 2 θx− 3 2dx. So

overall the above integral is proportional to

sinp−12 θ Z ∞ 0 dxxp−12 Z ∞ x dξK5/3(ξ).

Combining all the terms we have sinp+32 θ. However, as discussed in section2.1.4, there is

an extra sin θ₂ term if one considers an isotropic distribution of magnetic field orientations. Thus, the integral over pitch angles becomes,

1 2 Z π 0 dθ(sin θ)p+52 = √ π 2 Γp+7₄ Γp+9₄ .

Therefore the correct factor a(p) to be used in expression (2.37) and (2.38) when

esti-mating magnetic fields is:

a(p) = 2(p−1)/2√_3Γ3p−1 12 Γ3p+19₁₂ Γp+7₄ 16π1/2_Γp+9 4 . (2.39)

As a result of this modification, lower limits on the magnetic field in cluster halos and relics have been underestimated. Consequently the gap between the magnetic field

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Chapter 2. Theoretical Background 23 strength estimates through equipartition and Faraday rotation measurements on the one side, and IC non-detection on the other, becomes smaller.

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Chapter 3

Methods

This work aims at making predictions for the detectability of non-thermal IC radiation from radio halos and radio relics. We model the radio and X-ray spectrum by fitting

Eq. (2.13) and (2.27) to flux density measurements from the literature. In order to

predict when IC emission becomes detectable, we model the thermal bremsstrahlung and compare this to the ASTRO-H sensitivity curves. We argue that IC emission is detectable if it dominates over the background components. This provides a lower limit

on the detectable value of the electron distribution normalisation, K0. Since Ssyn∝ K0B

this consequently yields a maximum value for the magnetic field B that allows for the detection of IC radiation. All steps are discussed in detail below.

3.1 Cluster Selection and Data

We analyse all known radio haloes and relics based on the September2011-Halo and

September2011-Relic collection from Feretti et al. (2012), with the condition that they

have at least two radio measurements at different wavelengths in order to determine the spectral index of the power law. We searched the literature for radio data that was published after this date. This led to some new radio data for previously studied clusters and, in addition, it led to the inclusion of 1RXS J0603.3+4214, A3411 and ACT-CL J0102-4915 which all host one or more radio relics. In this section, our use of archival radio and non-thermal X-ray measurements is discussed. Our full sample of

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Chapter 3. Methods 25

halos and relics, including X-ray upper limits if available, can be found in table4.1. All

radio data, including the relevant references, can be found in appendix B.

Radio Data Radio data is presented as a flux density in units of Jansky1 ( Jy). In

the literature one often comes across the term integrated flux density, which means that the flux density is integrated over the entire source. This is the value we are interested in. Flux density integrated over the full source must be a fixed quantity, independent of the telescope and its beam size. Moreover, we are interested in the volume averaged magnetic field, therefore we should consider the radio emission from the full volume. Some radio data is presented without errors, in these cases we adopt an error of 10%

unless mentioned otherwise. For more on radio astronomy see for exampleWilson et al.

(2009).

3.1.1 Non-Thermal X-Ray Data

Studying non-thermal inverse Compton emission from galaxy clusters faces the problem of a huge background due to the thermal plasma. However, at high energies & 10 keV, so-called hard X-rays (HXRs), the thermal spectrum quickly falls off and non-thermal emission could eventually dominate over thermal emission. A small number of detections

of non-thermal X-rays have been claimed in the literature (Rephaeli and Gruber,2002;

Nevalainen et al.,2004;Fusco-Femiano et al.,2004;Rephaeli et al.,2006;Rephaeli et al.,

2008;Murgia et al., 2010). However, more recent observations have ruled out most of

these detections and placed upper limits on the non-thermal flux from clusters (Ajello

et al., 2009, 2010; Wik et al., 2011; Ota et al., 2013; Wik et al., 2014). Additionally, some attempts have been made to detect non-thermal emission in soft X-rays (SXRs). A non-thermal excess has in fact been observed, however, it has not been exclusively

attributed to cosmic rays (Sarazin and Lieu, 1997; Bonamente et al., 2002). In our

analysis we use upper limits on non-thermal X-rays if available (e.g. Million and Allen,

2009;Ajello et al.,2009,2010;Wik et al.,2012).

As a result of the limited sensitivity of our current telescopes, upper limits in X-rays are

determined as the excess flux in some large energy range (e.g. 20−80 keV or 50−100 keV).

However, in order to model accurately the current lower limits on the magnetic field, we

1

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Chapter 3. Methods 26 have to determine the flux density at some energy. Therefore, we model the flux back

to a flux density by applying Eq. (2.36a):

fx h erg cm2_s i = kx Z νmax νmin ν_x−αdνx.

The flux, fx, and the energy interval, [νmin, νmax], are given in the literature. For the photon spectral index we use the value corresponding to the best fit in determining the

upper limit (for values and references see table4.1). Using these values to determine kx

we can then find the flux density through:

sx(ν)

erg cm−2s−1Hz−1= ν−αkx. (3.1)

Finally, unless mentioned otherwise, the upper limits we take from the literature are at 99% c.l. or 3σ.

3.2 Background Modelling

The most important background for our purposes is thermal bremsstrahlung. In order to

model this component we use an APEC model as provided in the XSPEC code (Arnaud,

1996).

3.2.1 APEC model

The Astrophysical Plasma Emission Code (APEC)2 _{models emission spectra from}

collisionally-ionized diffuse gas and is used to model the thermal emission in clusters. Four parameters have to be provided. The plasma temperature in keV, which we take

from the literature (Ota, 2001; Chen et al., 2007), the metallicity in solar units, the

redshift, and the normalisation in cm−5_{. We fix the metallicity to 0.30 for all clusters}

in our analysis. Whereas this value can vary a lot from cluster to cluster, 0.30 is a

reasonable average value which is also used in other studies in the literature (e.g. Ota

et al.,2013). Moreover, for our purpose we do not require to have exact knowledge of the background, a good estimate should suffice. Changing the metallicity mostly influences

the intensity of the atomic transitions, as can be seen from Fig.3.1.

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Chapter 3. Methods 27 10−13 10−12 10−11 10−10 10−9

ν

· S

ν

[erg

/cm

2

/s

]

Z1= 1.0Z Z2= 0.3Z Z3= 0.0Z 1016 ₁₀17 ₁₀18 ₁₀19

ν[Hz]

0 10 20 30

Difference

[%]

Z1−Z2 Z1 × 100 Z2−Z3 Z2 × 100

Figure 3.1: Plasma emission in Coma for different metallicities, Z = 0.0Z, Z =

0.3Z and Z = 1.0Z. The other parameters are kept fixed (z = 0.0231, kT = 9.0 keV

and the normalisation is 0.51 cm−5_{). The difference is of the order 5% except at the}

atomic transitions (e.g. iron line) and the high frequency end.

3.2.1.1 Thermal Gas Density

The normalisation for the APEC model contains information about the ICM gas

distri-bution. The exact definition, with DA the angular diameter distance to the source in

cm, and ne(r) and nH(r) the electron and hydrogen number densities in cm−3, is:

10−14 D2

A(1 + z)2 Z

r2ne(r)nH(r)dr. (3.2)

The electron and hydrogen density are not completely independent and are both de-creasing functions of the radial distance to the cluster core. For our purpose the

elec-tron number density is well described by the phenomenological beta model (Cavaliere

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Chapter 3. Methods 28 ne(r) = n0 1 + r rc 22!− 3 2β , (3.3)

where n0 and rcare the core electron number density and the core radius. Some clusters

are described better by a multi-component beta model,

ne(r) =  X i n2_0,i 1 + r rc,i 22!−3βi   1 2 . (3.4)

The parameters of the beta profile, n0, rc and β are taken from the literature (e.g.Ota,

2001;Pinzke et al.,2011).

Clusters without a gas density model. If there exists no gas density model for a

cluster in the literature we model the gas density using the phenomenological model by

Zandanel et al.(2014) which is based on X-ray observations of the representative

XMM-Newton cluster structure survey (REXCESS) sample (Croston et al.,2008;Pratt et al.,

2009). The model is only dependent on the cluster mass (M500), providing an average

gas profile at all masses by using an observational gas fraction-mass relation. Moreover, in case there is no cluster temperature available in the literature we use Eq. (4) and

(5) and table (7) in Mantz et al.(2010) to estimate the cluster temperature. In a few

cases where no mass estimate is present in the literature, we model the mass using the

LX− M500or Lbol− M500relation, also defined in the above-mentioned expressions from

Mantz et al. (2010). Here LX is the ROSAT 0.1− 2.4 keV luminosity, which for many

sources can be found inVoges et al.(1999). Finally, in one case we determine M500 from

M200 assuming a Navarro-Frenk-White (NFW) profile (Navarro et al., 1997) and using

theDuffy et al.(2008) concentration-mass relation.

Thermal emission in the halo region. For halos, we integrate over the entire

cluster, where we set the cluster boundary to be at the virial radius, r200.3 It is

impor-tant to note that for a more accurate approximation of the plasma emission we should integrate only over the volume of diffuse radio emission. The radius of a radio halo,

typ-ically_{∼ 0.5 Mpc, is smaller than the cluster virial radius. However, this has only minor}

3

Strictly speaking, this is not the virial radius, but the radius where the average density of the cluster is 200 times the critical density.

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10

−12

10

−11

10

−10

ν

· S

ν

[erg

/s

/cm

2

]

N1: 0.0528cm−5 N2: 0.0257cm−5

10

17

₁₀

18

₁₀

19

ν[Hz]

0.0

0.2

0.4

0.6

0.8 R

=

N2 N1

Figure 3.2: Top: plasma emission for A0754 computed for two different volumes. The green line is the emission from a sphere with radius r200 = 2360 kpc from the cluster

centre. The red line corresponds to a radius of 300 kpc, which is similar to the region of radio emission. The temperature, redshift and metallicity are set to 9.0 keV, 0.0542 and 0.3 Z, respectively. Bottom: the ratio between the two lines. By integrating out

to the virial radius we overestimate the plasma emission by about a factor of 2. This ratio can be trivially deduced from the values of the normalisation.

influence on the normalisation. We choose to integrate up to the virial radius, slightly overestimating the thermal emission, and thereby obtain more conservative predictions for the detectability of IC emission. We show that the effect is small for the cluster

A0754 in Fig.3.2.

Thermal emission in the relic region. As opposed to radio halos, we do take

into account the weakening of plasma emission for radio relics. Relics are located at the cluster outskirts, so thermal emission is expected to give a much smaller background there. Therefore, it is important to model this more accurately. For this purpose we integrate over part of the volume of a sphere. The following formula is used for the normalisation in this case,

10−14 2D2 A(1 + z)2 (1_{− cos θ)} Z Rcc+0.5Rh Rcc−0.5Rh r2ne(r)nH(r)dr. (3.5)

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Chapter 3. Methods 30 ��

��

Figure 3.3: The geometry that is used in determining the normalisation of the APEC model for radio relics (Eq. 3.5). Rcc corresponds to the distance between the relic

and the cluster center as projected on the sky. Rv is approximately the relics largest

linear size and Rhis the relics ’width’. θ is given by tan−1

0.5Rv

Rcc−0.5Rh

. Eq. (3.5) thus integrates over a solid angle represented by the dashed line. Relic image taken from

R¨ottgering et al.(2013).

Figure 3.3illustrates how we define the region over which we integrate and the

param-eters used in Eq. (3.5). Rv is the relics largest linear size and Rcc its distance from

the cluster centre as projected on the sky. For most relics these values can be found in

Feretti et al.(2012). Rhis the width of the box we use to enclose the relic.

Correspond-ingly we define the surface of the relic to be Rv× Rh. For relics that are classified as

roundish by Feretti et al. (2012) we use Rh = Rv, for elongated relics we estimate Rh

based on radio images. Finally, we define the angle over which the solid angle subtends

as θ = tan−1 0.5Rv

Rcc−0.5Rh

and integrate from Rcc− 0.5Rh to Rcc+ 0.5Rh. Since we

inte-grate over a region that covers a larger region of the sky than our relic region (Rv× Rh)

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3.2.1.2 PyXspec

To obtain the plasma spectra we use PyXspec4, an object oriented python interface to the

XSPEC spectral-fiiting program5_{. In table}_3.1_{, the values for the redshift, normalisation}

and temperature used in this work are shown. We model the background from 0.01₋

82.5 keV in 100 bins.

Cluster z Norm. kT Reference

( cm−5) ( keV) Halos 1E0657-56f 0.296 1.61 × 10−2 14.20 6 A0520 0.199 1.56 × 10−2 7.10 7, 8 A0521a 0.2533 7.80 × 10−3 5.85 11, 13 A0665 0.1819 2.71 × 10−2 6.96 2 A0697 0.282 2.12 × 10−2 8.19 2 A0754 0.0542 5.28 × 10−2 9 1 A1300 0.3072 1.15 × 10−2 8.33 2 A1656 0.0231 5.1 × 10−1 8.38 1 A1758 0.279 1.15 × 10−2 6.88 2 A1914 0.1712 2.52 × 10−2 10.53 1 A2163 0.203 3.6 × 10−2 13.29 3 A2218 0.1756 1.3 × 10−2 7.63 2 A2219 0.2256 3.33 × 10−2 9.22 2 A2255 0.0806 3.57 × 10−2 6.87 1 A2256 0.0581 1.03 × 10−1 7.50 1 A2319 0.0557 1.99 × 10−1 9.20 1 A2744 0.308 1.82 × 10−2 8.95 2 A3562 0.049 4.68 × 10−2 5.16 1 CL0217+70a,b _0.0655 _{9.90 × 10}−3 3.54 14 MACSJ0717.5+3745 0.5458 3.82 × 10−4 11.60 10 PLCK G171.9-40.7a 0.27 1.30 × 10−2 10.65 12 RXCJ1514.9-1523a,b _0.22 _{1.20 × 10}−2 7.83 13 RXCJ2003.5-2323a,b 0.3171 6.90 × 10−3 7.62 13 Relics 1RXS J0603.3+4214a,c 0.225 6.40 × 10−4 7.80 16 A0013 0.0943 2.98 × 10−3 6 5 A0085 0.0551 5.1 × 10−3 6.10 1

Continued on next page

4

https://heasarc.gsfc.nasa.gov/xanadu/xspec/python/html/.

5

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Table 3.1 – continued from previous page

( cm−5) ( keV) A0521a 0.2533 3.20 × 10−4 5.85 11, 13 A0610ab,e _0.0954 _{1.30 × 10}−4 2.44 21 AS753a 0.014 4.20 × 10−4 2.50 20 A0754 0.0542 8.5 × 10−4 9 1 A1240 N 0.159 3.70 × 10−6 6 4 S 0.159 1.50 × 10−6 6 A1300 0.3072 6.80 × 10−4 8.33 2 A1367 0.022 2.70 × 10−4 3.55 1 A1656 0.0231 8.5 × 10−4 8.38 1 A1664 0.1283 6.00 × 10−4 6.80 7 A2048a,b,d _0.0972 _{4.00 × 10}−4 4.21 19 A2061 0.0784 6.60 × 10−4 4.52 9 A2063 0.0349 3.60 × 10−3 3.68 1 A2255 0.0806 3.20 × 10−4 6.87 1 A2256 0.0581 2.3 × 10−2 7.50 1 A2345ab E 0.1765 2.60 × 10−4 6.51 13 W 0.1765 8.50 × 10−4 6.51 A2433a,b,e 0.108 2.70 × 10−5 1.80 18 A2744 0.308 5.4 × 10−5 8.95 2 A3376 E 0.0456 5.40 × 10−3 4.30 1 W 0.0456 3.10 × 10−5 4.30 A3411a 0.1687 3.20 × 10−3 6.40 22 A3667 NW 0.0556 6.60 × 10−4 7 1 A4038 0.03 2.80 × 10−4 3.15 1 ACT-CL J0102-4915a E 0.87 2.70 × 10−2 14.50 15 NW 0.87 2.70 × 10−2 14.50 SE 0.87 2.70 × 10−2 14.50 CIZAJ2242.8+5301a,b,c 0.1912 1.10 × 10−5 5.55 17 PLCK G280+32.0a,c _N _0.39 _{7.40 × 10}−5 12.86 17 S 0.39 5.20 × 10−7 12.86 ZwCl0008.8-5215a,b E 0.1032 1.30 × 10−4 4.98 13 W 0.1032 9.80 × 10−6 4.98 a

Gas density fromZandanel et al.(2014).

b

Temperature from T − M500 relation (Mantz et al.,2010).

c_M

500 from ROSAT LX− M500 relation (Mantz et al.,2010). d

M500from bolometric Luminosity (Mantz et al.,2010). e

M500 scaled from M200.

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Table 3.1 – continued from previous page

( cm−5) ( keV)

f

Normalisation taken straight from the reference.

Table 3.1: Thermal data for our cluster sample. References: [1]Chen et al. (2007);

Pinzke et al. (2011); [2] Ota (2001); [3] Ota et al. (2013); [4] Barrena et al. (2009);

Cavagnolo et al. (2009); [5]Juett et al. (2008) [6] Wik et al.(2014)}; [7]Govoni et al.

(2001); [8] Govoni et al.(2004); Vacca et al.(2014); [9] Marini et al. (2004); [10] Ma

et al. (2008); [11] Ferrari et al. (2006); [12] Collaboration et al. (2011); [13] Planck

Collaboration et al.(2013); [14]Brown et al.(2011b); [15]Menanteau et al.(2012); [16]

Ogrean et al.(2013); [17] Piffaretti et al.(2011); [18]Popesso et al. (2007); [19]Shen

et al.(2008); [20]Subrahmanyan et al.(2003); [21]Yoon et al.(2008); [22]van Weeren

et al.(2013).

3.3 ASTRO-H Sensitivity

ASTRO-H is a next generation X-ray satellite that is scheduled for launch in 2015. The instruments that are of interest for this study are the Hard X-ray Imager (HXI) and the Soft X-ray Imager (SXI). All instruments are co-aligned and will operate simultaneously (Takahashi et al.,2012). The properties of aforementioned instruments, and also of the

Soft X-ray Spectrometer (SXS), are given in table 3.2.

SXS SXI HXI

Energy range (keV) 0.3-12.0 0.4-12.0 5-80 Angular resolution (arcmin) 1.3 1.3 1.7 Field of view (arcmin2₎ _3.05_{× 3.05} ₃₈_{× 38} ₉_{× 9}

Energy resolution (eV) 5 150 < 2000 (@6keV) (@60keV) Effective area (cm2₎ _50/225 _214/360 ₃₀₀

(@0.5/6keV) (@30keV) Instrumental background 2_{× 10}−3_/0.7_{× 10}−3 _0.1/0.1 ₆_{× 10}−3_/2_{× 10}−4†

(s−1_keV−1_FoV) _(@0.5/6keV) _(@0.5/6keV) ₂_{× 10}−3_/4_{× 10}−5‡

(@10/50keV)

†_{for 4 layers, relevant for hard x-rays (20}

− 80keV)

‡_{for 1 layer, relevant for soft x-rays (< 30keV)}

Table 3.2: Properties of the relevant ASTRO-H instruments, the Soft X-ray Spectrom-eter (SXS), Soft X-ray Imager (SXI) and the Hard X-ray Imager (HXI). Values adopted from the ASTRO-H Quick Reference (http://astro-h.isas.jaxa.jp/ahqr.pdf).

To estimate the detectability of the non-thermal X-ray component in galaxy clusters we make use of the sensitivity curves as published by the ASTRO-H collaboration. These

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are shown figure3.4for both point and extended sources (ASTRO-H Quick Reference6,

Takahashi et al. (2010,2012)). In our analysis we use the sensitivity curve for 1 Ms of observation. For different observation times one can assume the sensitivity to improve as√time (see Fig.3.5).

(a)

(b)

Figure 3.4: Sensitivity curves for ASTRO-H. The curves correspond to 1 Ms of obser-vation time. Top plot refers to point sources, whereas the bottom plot is for uniform sources of 1 degree2_{. The figure is taken from}_{Takahashi et al.}₍₂₀₁₀_).

6

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Chapter 3. Methods 35 1018 ₁₀19

ν[Hz]

10−12 10−11 10−10 10−9 10−8

ν

· S

ν

[erg

/sec

/cm

2

]

100 ks 1 Ms 1Ms×√10

Figure 3.5: Comparison of the 100 ks HXI detection limits from the ASTRO-H quick reference (http://astro-h.isas.jaxa.jp/ahqr.pdf) and the 1 Ms curve from

Taka-hashi et al. (2010). Scaling this latter curve by a factor of √10 reproduces to 100 ks

line.

The bottom plot in Fig.3.4 is for a uniformly distributed source of 1 degree2. Since the

field of view (FoV) of HXI is 90 _{× 9}0_{, this means that the number of photons observed}

by HXI is _60×609×9 × Sphotons cm−2s−1keV−1, where S is the HXI sensitivity for a

1 degree2 _{source (Fig.} _3.4b_{). For our analysis, we will assume that the halos and relics}

are uniform sources, whereas in reality they are clearly non-uniform. Emission will be higher in certain regions and ASTRO-H could in fact observe emission from a smaller

region. Let us assume we have a source of a certain size on the sky, x arcmin2. We can

then rescale the sensitivity curve assuming that the source is uniform by x arcmin2

3600 arcmin2 × S

photons cm−2s−1keV−1. (3.6)

As long as the source size is larger than the HXI FoV, we can assume this scaling to hold since it will yield the same number of photons for the HXI instrument. However, several sources in our sample are smaller than the HXI FoV. Nevertheless, we use the same scaling, since the source should in that case be resolvable as an extended object.

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Chapter 3. Methods 36 there our scaling starts to fall short, making HXI too sensitive.

Concluding, for sources _{≥ 10 arcmin}2 _{we use Eq. (}_3.6_{) and the point source sensitivity}

is used otherwise7.

3.4 Analysis of the Spectrum

Having established the theoretical framework for synchrotron and inverse Compton ra-diation, a model for the background and the sensitivity for our instrument of reference, ASTRO-H, we now provide a step-by-step walkthrough of spectral analysis. Abell 754

is used as an example. For illustrative purposes all steps are presented in Fig.3.6.

107 ₁₀9 ₁₀11 ₁₀13 ₁₀15 ₁₀17 ₁₀19 ₁₀21 ν[Hz] 10−16 10−15 10−14 10−13 10−12 10−11 10−10 ν · Sν [erg /sec /cm 2] _{Non-thermal spectrum} BV= 0.06µG 10−10 10−8 10−6 10E[keV]−4 10−2 100 ₁₀2 ₁₀4 (a) 107 ₁₀9 ₁₀11 ₁₀13 ₁₀15 ₁₀17 ₁₀19 ₁₀21 ν[Hz] 10−16 10−15 10−14 10−13 10−12 10−11 10−10 ν · Sν [erg /sec /cm 2] X-ray background BV= 0.06µG Plasma 10−10 10−8 10−6 10E[keV]−4 10−2 100 ₁₀2 ₁₀4 (b) 107 ₁₀9 ₁₀11 ₁₀13 ₁₀15 ₁₀17 ₁₀19 ₁₀21 ν[Hz] 10−16 10−15 10−14 10−13 10−12 10−11 10−10 ν · Sν [erg /sec /cm 2] ASTRO-H sensitivity Takahashi et al. (2010) BV= 0.06µG Plasma ASTRO-H 1Ms 10−10 10−8 10−6 10E[keV]−4 10−2 100 ₁₀2 ₁₀4 (c) 107 ₁₀9 ₁₀11 ₁₀13 ₁₀15 ₁₀17 ₁₀19 ₁₀21 ν[Hz] 10−16 10−15 10−14 10−13 10−12 10−11 10−10 ν · Sν [erg /sec /cm 2] Non-thermal spectrum detectable by ASTRO-H BV= 0.06µG Plasma ASTRO-H 1Ms BV= 0.31µG 10−10 10−8 10−6 10E[keV]−4 10−2 100 ₁₀2 ₁₀4 (d)

Figure 3.6: Analysis of the non-thermal spectrum in a cluster. Data corresponds to Abell 754 (Bacchi et al.,2003;Ajello et al.,2009).

First, Eq. (2.27) is fit to the upper limit in X-ray if available. To do this we fix the high

energy cutoff to γmax = 2× 105, as, unless the energy of the upper limit in X-ray is close

7

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Chapter 3. Methods 37 to the cutoff energy, fixing the cutoff does not impact our result. From this fit we fix the

normalisation of the electron distribution, K0. Next, Eq. (2.13) is fit to the available

radio data. If spectral steepening is observed, the lower and/or higher cutoff are kept free during the fit. The spectral index is taken from the literature, unless it returns a poor fit, in which case we determine a new spectral index. The resultant spectrum

provides the current lower limit on the magnetic field (see Fig.3.6a).

Next, we compare the resulting broadband spectrum to the thermal background (Fig.3.6b)

and ASTRO-H sensitivity (Fig.3.6c). We use the intersection between the thermal

back-ground and ASTRO-H sensitivity to fix a new normalisation of the electron distribution. This point is the lower limit on the normalisation detectable by ASTRO-H, whilst not being obscured by bremsstrahlung emission. If no upper limits on the X-ray flux exist for a cluster, we start from this step.

Finally, we fit Eq. (2.13) to the radio data, again. The resulting value of the magnetic

field is the highest volume averaged value for which non-thermal IC radiation will still be detectable by ASTRO-H. All results are presented in the next chapter.

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Chapter 4

Results and Discussion

This chapter contains the results of our analysis. Our full halo and relic sample is given

in table 4.1, including the approximate area they cover on the plane of the sky, and the

upper limit on the non-thermal X-ray flux where presents. Results for halos are shown

in table4.2and for relics in table4.3. The first column of these tables contains the name

of the cluster. The photon spectral index observed in radio can be found in the second column. Columns 3-5 contain the current lower limit on the magnetic field and the values

for γmin and γmax. Note that there are only lower limits for clusters that have an upper

limit on the IC flux. Moreover, we only fit for γmax and/or γmin if spectral steepening

is observed, otherwise they are fixed to 2_{× 10}5 _{and 300 respectively. Columns 6-8 are}

essentially the same as 3-5, but now the magnetic field (Bu.l.) corresponds to the upper

limit for which ASTRO-H can still detect non-thermal X-rays within 1 Ms of observation

time. Columns 9 and 10 give χ2 and the number of degrees of freedom, respectively.

These refer solely to the fit to the radio data, i.e. the last step described in section3.4.

Finally, columns 11 and 12 contain the flux density and energy at which the ASTRO-H sensitivity curve intersects the thermal background spectrum. Comments on individual

clusters are given in sections 4.1. Section 4.3contains a more general discussion.

Non-Thermal Emission from Galaxy Clusters Predictions for X-Ray Satellites

University of Amsterdam

MSc Physics

Gravitation AstroParticle Physics Amsterdam

Master Thesis

Non-Thermal Emission from Galaxy Clusters

Abstract

Acknowledgements

Contents

List of Figures

List of Tables

Abbreviations

Physical Constants

Symbols

Chapter 1

Introduction

Chapter 2

Theoretical Background:

Radiative Processes

2.1

Synchrotron Radiation

2.2

Inverse Compton Radiation

2.3

The Electron Spectrum

log

γ

Time

(yr)

log

γ

Time

(yr)

2.4

Analytical Magnetic Field Estimates

Chapter 3

Methods

3.1

Cluster Selection and Data

3.2

Background Modelling

ν

·

S

[erg

/cm

/s

]

ν[Hz]

Difference

[%]

10

10

10

ν

·

S

[erg

/s

/cm

]

10

10

10

ν[Hz]

0.0

0.2

0.4

0.6

0.8

R

=

������������

�����

3.3

ASTRO-H Sensitivity

ν[Hz]

ν

·

S

₁₀

₁₀

��

��