Cosmic Ray Propagation Through Turbulent Magnetic Fields

Cosmic Ray Propagation Through Turbulent Magnetic Fields

Ehza Switalska

Submitted in fuffilment of the requirements for the degree of Master of Scíence

School of Chemistry and Physics (Faculty of Sciences)

The University of Adelaide

September 2006 I)ecleration

I certify that this thesis does not incorporate, without acknowledgement, any material previously submitted for any degree, or diploma in any university and that to the best of my knowledge and beliet it does not contain any material previously published, or written by another person, except where due reference is made in the text.

I give consent to this copy of my thesis, when deposited in the University Library, being made available for loan and photocopylng.

Eliza Switalska

07110106 Acknowledgements

I would like to thank my supervisor Prof. Roger Clay for his guidance and never ending patience and support. Special thanks to Melanie Johnston-Hollitt for her advice and use of astronomical data.

Many, many thanks to Mariusz and to my family and friends (especially Juliet) for their continuous support and encouregement.

J CONTENTS

Abstract 8

Preface 9

CHAPTER 1

Cosmic Rays t2

1.1 Introduction 12 1.2 Overview of Cosmic Rays 13

1.2.1 Cosmic Ray Arrival Directions t6 r.2.2 GZK Cut-off t6 1.2.3 High-Energy Cosmic Ray Propagation t7 1.2.4 Cosmic Ray Acceleration t9

CHAPTER 2

Magnetic Fields 2t

2.1 The Galactic and Extragalactic Magnetic Fields 2t 2.2 The origin of the Galactic Magnetic Field 2t 2.3 Methods in Determining Magnetic Field Strengths 23

2.3.1 The Zeeman Effect 23 2.3.2 Optical Dust Absorption and Polarization of Sub-mm and mm Emission 24 2.3.3 Synchrotron Radiation 25 2.3.4 Radio Faraday Rotation 28 2.3.5 Factors in the I)etermination of the Rotation Measure of Galaxies 32 2.3.6 Faraday Depolarization 34 2.3.7 Energy Equipartition between Magnetic Fields and Cosmic Rays 35

2.5 Magnetic Field Turbulence 36

2.5.1 Kolmogorov Scale Turbulence 37

2.6 Magnetic Fields in Galaxy Clusters 39

2.6.1 The Local Group of Galaxies 4l 2.6.2 The Virgo Cluster and Nearby Superclusters 43 2.6.3 Radio and X-Ray Methods for Probing Cluster Magnetic Fields 46

CHAPTER 3

RADIO ASTRONOMY 49

3.1 Introduction 49 3.2 Observational Techniques and Instruments Used 49

4 3.2.1 The Very large Array (VLA) 51 3.2.2 The Australia Telescope Compact Array 53

3.3 Radio Synthesis Imaging 54

CHAPTER 4

Estimation of the Scale of the Turbulent Magnetic Field in Galaxy Clusters 55

4.1 Introduction 55 4.2 Galaxy Clusters and the 43667 Cluster 56

4.2.1 Radio Interferometer - Amplitude vs Baseline Length 57 4.2.2 Amplitude vs Baseline ptot of 43667 58

4.3 Procedure 60

4.3.1 Modified A3ó67 Ptot 60 4.3.2 Analysis of the Modified Spectrum 6l 4.3.3 Results 6r

CHAPTER 5

Modelling of Cosmic Ray Propagation in a Magnetic Field 64

5.1 Previous Studies in Propagation Simulations 64 5.2 Generation of Random Magnetic Fields 66 5.3 Simulation of a Turbulent Magnetic Field 68

CHAPTER 6

Processes Involved in Generating Simulated Turbulent Magnetic Fields and a Study of the Effects of Field Scale Variations 70

6.1 Introduction 70 6.2 Random Magnetic Field Component 7l 6.3 Results of Scale Length Variation 7l

6.3.1 Properties of the Magnetic Field Model 72

6.4 Effects of Clumpiness on the Magnetic Field 83

6.4.1 Small-Scale Clumpiness Results 83 6.4.2 Large-Scale Clumpiness Results 86 6.4.3 Conclusion 88

CHAPTER 7

Cosmic Ray Propagation Through Turbulent Magnetic Fields 90

7.1 Computer Simulation of Propagation 90

5 7.2 Procedure 90 7.3 Results 92

7.3.1 Results for Propagation through Large Scale Turbulence 92 7.3.2 Results for Propagation through Small Scale Turbulence 93 7.3.3 Comments on the Results 94

CHAPTER 8

Conclusion 96

References 99

Bibliography 101

APPENDIX A r02

Diffusion 102

APPENDIX B 105

Programs 105

6 List of Figures and Tables Page

Figure 1(a) Cosmic ray spectrum A 14 Figure 1(b) Cosmic ray spectrum B 15 Figure 2(a) Faraday rotation diagram Figure 2(b) Plot of integratedpolanzation and position angle JJ Figure 2(c) Rotation measure sky map 34 Figure 2(d) Illustration showing the Universe within 500,000 l.y. 42 Figure 2(e) Illustration showing the Universe within 5 million l.y. 43 Figure 2(f) Illustration showing the Universe within 100 million 1.y. 44 Figure 2(g) Illustration showing the Universe within I billion l.y 45 Figure 3(a) Aerial view of the VLA 52 Figure 3(b) The Australia Telescope Compact Array (ATCA) 53 Figure 4(a) Amplitude vs baseline plot (43667) 59 Figure a@) Modified amplitude vs baseline plot (A3667) 60 Figure 4(c) Plots representing the logarithms of amplitude vs spacing 62 Figure 6(a)-(c) B* distribution vs spacing plot for various turbulence sclae 74-76 Figure 6(d)-(h) Histogram plots for the various iteration values 79-8r Figure 6(i)-(l) Original and clumpy field histograms in small scales 84-85 Figure 6(m)-(o) Plots for clumpy field histograms in large scales 87-88 Figure 7(a) Plot of the cosmic ray energy vs the number of cycles in large scales 93

Table 6(a) Integrated field variations through the various iterations 77 Table 6(b) The mean and standard deviation values obtained for iterations 82 Table 7(a) Results for Large Scale Turbulence 92 Table 7(b) Results for Small Scale Turbulence 94

7 Abstract

This study begins with an overview of cosmic ray literature in relation to the origin and propagation of high-energy particles in extragalactic space. It follows with a review of recent methods used in the observation and measurement of extragalactic magnetic fields and describe the radio astronomy techniques used. The later chapters describe computer simulations, which were used for producing turbulent magnetic field models and then follow on to show how these can be used to estimate the probable turbulent scales of such fields. These models are then applied to data from an actual galaxy cluster, where the turbulence spectrum of the Abell galaxy cluster (using data from a recent observational study of A3667) is estimated. The final part of this thesis is concerned with the study of cosmic ray propagation within such turbulent magnetic fields. This is achieved by altering the turbulent scale-lengths and observing the behaviour of cosmic rays within, both large ans small-scale magnetic fields, thereby giving an indication as to the actual paths they may follow in space

and consequently providing some clues as to their origins.

8 Preface

Cosmic rays are high energy charged particles which propagate from their sources to us through the turbulent magnetic fields which permeate the Universe. Over the years, as far back as the 1930s, models have been proposed for cosmic ray propagation in both the galactic and extragalactic environments. The one underlying requirement in all these models is the understanding of the strength and structure of the galactic and extragalactic magnetic fields. This is because cosmic rays are highly energetic charged particles and their propagation is therefore affected while travelling within these fields. Honda (1987) has described a method for simulating a turbulent magnetic field such that it contains a Kolmogorov spectrum of scales. The effect and extent of these interactions is examined in this study of propagation in such fields.

Cosmic rays, being highly energetic charged particles, can be greatly affected by cosmic magnetic fields. Therefore a greater understanding of the properties of such fields is currently needed to provide a more accurate glimpse into the propagation of cosmic rays. There are many ongoing studies concerned with the estimation and measurement of extragalactic magnetic fields, but many still disagree and thus make it more difficult to interpret cosmic ray propagation correctly. The magnetic fields themselves are turbulent with irregularities of various scales. This turbulence is of major concern in this particular study. Propagation simulations, which follow high energy particles through the various turbulence scales, will provide a better understanding of the affects these irregularities may have on the paths of the cosmic rays and are therefore discussed and analyzed in this work. Sophisticated propagation studies date back to the late 50s and 60s, and two of the major studies, which were conducted during these years are relevant here. These two particular studies looked at propagation in two extremes. The first, carried out by Gleeson and Axford (1967), studied the propagation of cosmic rays with a gyroradius which was large compared to the scale of the irregularities. This study involved the "scattering centers" approach. The other study was conducted by Jokipii and Parker (1969), and considered propagation where the gyroradius was small compared to the magnetic irregularity scales. Some years later, Honda (1987), extended propagation studies to the case where the gyroradius and the scale of the irregularity of the magnetic field were comparable. In this more complicated case, a numerical approach had to be adopted instead of the previous analytical ones applied in the earlier studies. Therefore, as in this later study, a numerical

9 computer simulation was used to generate an irregular magnetic field model, before following particle propagation through the treld.

For any of these approaches to provide useable results, as much information as possible is needed on the real astrophysical magnetic fields. There are many methods which can be used to obtain this information, the main one being Faraday rotation measures (RMs). With this technique, linearly polarized radiation from extragalactic sources is used to estimate an integrated field strength for distant objects. However there can be some uncertainty in the determination of RMs, due to noise and low degrees of source polarization, as well as measurement errors which arise from the shape of the object being studied (i.e. galaxies with radio lobes where the different components show up in different ranges of wavelengths).

In general the field strength of the smaller scale irregularities in the magnetic field is believed to follow a Kolmogorov spectrum -k-s/3 lwhere k is the turbulence wave number) within the homogenous regions, but additional "clumpiness" may occur and this needs to be investigated further, which is what this work aims to do. Most studies that have been carried out tend to assume a simple Kolmogorov scale turbulence and fail to consider the effect that clumpiness may have on the measurement of the field and the resulting propagation of cosmic rays. One needs to first be able to interpret the RMs for these clumpy scenarios in order to be able to carry out accurate measurements of the field and to study the cosmic ray propagation in such regions.

In the propagation studies in the following thesis, we generate some simple magnetic fields, with Kolmogorov type turbulence scales and consider their properties. Then we observe the effects these irregularities have on the overall magnetic field structure. The next step is to make these irregularities appear clumpy and see how this affects the properties of propagation in the simulated magnetic field. Understanding of the effects of magnetic field scales will help us to estimate the propagation in real life scenarios. To the extent that real fields are more likely to follow these models, this will facilitate the accurate interpretation of RMs.

We will then look at a particular object, in this case the galaxy cluster A3667, in order to use a realistic scenario where the proposed models can be applied and tested with greater accuracy. We will use radio astronomy data to obtain an estimate for the magnetic field

10 turbulence properties for the fields, which are present in A3667, and use this study to access the realism of our studies of the propagation of cosmic rays through such a structure.

For this study, which is mainly concerned with extragalactic sources, such as galaxy clusters, the techniques and progress made in the field of radio astronomy is crucial in 'We obtaining a more accurate picture of cosmic ray propagation. need to be aware of the obstacles that can arise in this field of research and have an understanding of the way measurements are obtained and interpreted in order to have access to the most up to date measurement data.

11 CHAPTER 1 Cosmic Rays

1.1 Introduction

Cosmic ray research can be thought to have begun in I9l2 when Victor Hess and two assistants flew in a balloon to an altitude of about 5,000 m and discovered evidence of a very penetrating radiation coming from outside our atmosphere. This was the first observation of what we now call cosmic rays, high-energy charged particles that constantly bombard the earth. The primary cosmic radiation consists of protons, but also includes other nuclei, antiprotons, electrons and positrons. It is still not known, exactly where these particles come from, how they are accelerated, or how they propagate through space, but there are many theories. It is presently believed that supernovae ate aî important source of cosmic rays, at least up to energies of 10ls eV. The existence of cosmic ray particles up to ultra-high energies (tl0'o eV) is now beyond any doubt, since the advent of large extensive air shower affays. However, the existence of such particles imposes a challenge for the discovery of their sources and their nature, due to the fact that particles with such high energies are believed to be diffrcult to produce by any astrophysical source with any known mechanisms. As cosmic rays travel through space, they undergo a number of transformations due to their interaction with matter, magnetic fields and radiation. These are reflected in the cosmic ray energy spectrum, anisotropy and composition. In reconstructing the history of cosmic rays, we are actually investigating events and processes in the far reaches of space. The propagation of cosmic rays through astrophysical magnetic fields is the major theme of this thesis and will also be looked at, with regard to some specific cases of turbulent magnetic fields, which are believed to have a considerable effect on their paths through extragalactic space.

l2 1.2 Overview of Cosmic Rays

Let us first discuss their composition. Cosmic rays include essentially all of the elements in the periodic table; about 89% of the nuclei are hydrogen (protons), IÙyo helium, and about lo/o heavier elements. The common heavier elements (such as carbon, oxygen, magnesium, silicon, and iron) are present in about the same relative abundances as in the solar system, but there are important differences in elemental and isotopic composition that provide information on the origin and history of galactic cosmic rays. For example there is a significant overabundance of the rare elements Li, Be, and B produced when heavier cosmic rays such as carbon, nitrogen, and oxygen fragment into lighter nuclei during collisions with the interstellar gas. The isotope Ne22 is also overabundant, showing that the nucleos¡mthesis of cosmic rays and solar system material have differed. Electrons constitute about Io/o of galactic cosmic rays.

The observed cosmic ray spectrum covers about 11 orders of magnitude in energy, from 10e eV to 1020 eV. Close to the lower energy limit, the spectrum turns over due to the effect of the heliosphere on inward propagation to the Earth, which inhibits the arrival of lower energy particles. The actual form of the spectrum outside the heliosphere is not well known below 1010 eV, but could be a downward extension of the power law, which extends upwards from that point up to an energy between 1015 eV and 1016 eV, with a differential index of aboú -2.7. At higher energies, due to a lack of sufficient numbers of events, the statistical confidence in the form of the spectrum diminishes. It steepens somewhat at about

1016 eV, with a new index of about -3.2 and flattens a little at about 1018'5 eV. However, at this time there is no concrete evidence for a further steeping at 6x10le eV. This feature is expected, due to interactions with the microwave background, the GZK cut-off (Clay, 2002).

The spectrum is thus described by a power law oc E-v with two breaks at the "knee" (see Figurel(a)) at = 4 x 10ls eV, and at the "ankle", at 5 x 1018 eV. Above the knee, the spectrum steepens from a differential power law index y = -2.7 to = -3.2. Above the ankle, the spectrum flattens again to a power law index y = -2.8, (fig 1(b)). Cosmic rays with energies above the ankle cannot be confined by the Galactic magnetic field. The lack of likely sources in our Galaxy, plus observations that the beam is close to isotropic, suggests that the ankle marks a cross-over from a Galactic component to one of extragalactic origin. Data from the Fly's Eye experiment, also suggests that the chemical composition may be dominated by heavy nuclei up to the ankle and by protons beyond, (Isola and Sigl, 2002).

13 On the following two pages are the figures representing the cosmic ray spectrum for two slightly different energy ranges.

tfÈ Fi 1ü1 l [.t¡ rl il frör t il. t ô .D x Þ å" þ t¿- ü lt3 c*ol I oot. o h ut {¿ ô I Ê .l c I aû {t rl a f¡ flr I 0 B'ar {t I \ C aì!. 10

l0r1 1011 10r3 10,14 l0r5 l0lt l0l? lOtB lor0 '1020 Enrgy êl Nuchu¡ fufj

Figure 1(a) - Cosmic ray spectrum A This figure represents a spectrum composed of measurements from different instruments to cover the energy range from l0li to 1020eV. It shows the flux (numbers of cosmic rays passing through a surface) at different energies. The graph is logarithmic; a straight line indicates that the number of cosmic rays is proportional to the energy to some power, called the spectrøl index . In the case of cosmic rays, there are fewer particles at higher energies (the numbers go DOWN as the energy goes UP) so the spectral index is negative, (Sokolsky, P. 1989).

The graph has been flattened to emphasise its features through multiplying by E".

14 T 104 t Fluxes of Cosmic Roys fJ} 101 01 6 t x t0-t 4 +- (1 pcrtlcle per mr-Êetondì E= -{ 1 0 t\

-l 1t + .+ 'f]

0 1 0 Knee b (l purticle Fer m'-yeor)

-1S v 10

t0-l.l

-tÐ 10

-22 10

Ankl+ t0-2ã (1 porticle por krnr-yeor)

t0-lE ros roto rott rot" rot" rotn rot" rotu rott rot" r0tt"hîi_,lft'

Figure 1(b) - Cosmic ray spectrum B

This is another representation of the cosmic ray spectrum, which notes the detection (arrival on Earth) frequency ofparticles ofvarious energies, (Bhattacharjee and Sigl, 2000).

15 1.2.1 Cosmic Ray Arrival Directions

Cosmic ray paths are affected by magnetic fields and, at lower cosmic ray energies, the fìelds of the Earth and the heliosphere are important. At energies above 10rr eV, the magnetic fields of the Milky Way galaxy need to be considered. The cosmic rays are deflected by a lesser amount as their energy, or their rigidity increases and they follow looping, spiral paths. The scale of these paths increases in proportion to the cosmic ray energy and inversely with the charge on the particles. At cosmic ray energies over l0ls eV, this scale begins to be comparable to the major dimensions of our galaxy, (Clay and Dawson, 1991).

1.2.2 GZK Cut-off

The Greisen-Zatsepin-Kuzmin limit (GZK cut-off) is a theoretical upper limit on the energy of cosmic rays from distant sources. This limit was computed in 1966 by Kenneth Greisen, Vadim Kuzmin and Georgiy Zatsepin, based on interactions predicted between the cosmic

ray and the photons of the cosmic microwave background radiation (see section 1.2.3 for a detailed description of this mechanism). They predicted that cosmic rays with energies over the threshold energy of 5x10le eV would interact with cosmic microwave background photons to produce pions. This would continue until their energy fell below the pion production threshold. Therefore, it would imply that extragalactic cosmic rays with energies greater than this threshold energy should never be observed on Earth. Now, if Ultra High Energy Cosmic Rays, (UHECR) with energies above 1018 eV, are of extragalactic origin, a cut-off in the cosmic ray spectrum is expected, due to the fact that UHECR are assumed to be protons accelerated in powerful distant astrophysical sources. For sources further away than a few dozen Mpc, this would cause a break, or steepening in the cosmic ray spectrum around the threshold energy of 5x10le eV. Such a feature in the energy spectrum would result from astrophysical proton sources distributed homogenously in the universe. The GZK cut-off is represented by the predicted rather sharp suppression of the flux, at energies above Eczr- 5 x 10le eV., (Watson, 2001). However, even now, it is not yet clear whether this effect is observed, or not, due to the discrepancy between the results of two of the largest experiments, Akeno Giant Air Shower Array (AGASA) and High Resolution Fly's Eye (HiRes). AGASA consists of an array of ground based radiation detectors, whereas

HiRes detects the nitrogen fluorescence emission from the passage of particles through our

16 atmosphere. AGASA reports a higher number of events above 867çthan expected with the cut-off, while HiRes reports a flux consistent with the GZK feature, (De Marco, et al., 2003). De Marco, and colleagues developed a numerical simulation of the propagation of UHECR in the CMB and used it to determine the significance of the GZK featare in the spectrum of UHECR measured by AGASA and HiRes. They found a systematic shift in the flux, which may be interpreted as a systematic error in the relative energy determination of about 30%. After the correction for these systematics, the two experiments are broadly in agreement. After considering the statistical signifìcance of the spectra, De Marco et al., found that, with the low statistical significance of either the excess flux seen by AGASA, or the discrepancies between AGASA and HiRes, it is not justifiable to claim either the detection of the GZK feature, or the extension of the UHECR spectrum beyond 867ç at the present time, (De Marco, et a1.,2003). The resolution to this problem may be found at the completion of the Pierre Auger project, which will combine the two existing complementary detection techniques used by HiRes and AGASA. The simulated spectra for Auger show that if the energy region where statistical fluctuations dominate the spectrum is moved to - 6 1020 eV, by the inclusion of significantly more data from a very large collecting area, it will allow a clear identification of the GZKfeatare, (De Marco, et a1.,2003).

1.2.3 High-Energy Cosmic Ray Propagation

Understanding the influence of the large-scale magnetic fields on cosmic ray propagation is vital for deducing reliable information on their sources. The cosmic ray injection power of the sources believed to be needed to maintain the observed highest energy cosmic ray flux, depends directly on our knowledge of the large-scale magnetic fields in the vicinity of our galaxy. Magnetic fields not only determine the cosmic ray intensity, but also shape the energy spectrum of UHECR. Depending on the typical field strength, protons with energies greater than 1020 eV, are hardly deflected during propagation, whereas, for example, particles of E : l0l8 eV may have a propagation process dominated by diffusion, (Stanev, e/ a1.,2003).

To obtain a detailed understanding of cosmic ray propagation, we need to have an accurate understanding of the effects of galactic and extragalactic magnetic fields on the motion of charged particles. This must be done through measurement of the irregular and regular magnetic fìeld components and a determination of possible charged particle motion through

I7 the observed field. Since the magnetic irregularities are most likely a consequence of random turbulence, a statistical treatment is necessary, (Jokipii, 1966). Early treatments of charged particle motion in an irregular field, were based on the concept of magnetic scattering centres. These treatments rwere an analogy to ordinary diffusion, where particles random walk through the magnetic field, being scattered at the assumed scattering centres. The resulting motion can be described by the use of a diffusion coefficient, or mean-square displacement as a function of time, which remains an undetermined parameter. A quantitative description of this process can be found in Appendix A. In Jokipii's 1966 paper, however, a statistical description of the particle motion is obtained in terms of directly observable properties of the fluctuating field, rather than in terms of an unknown mean free path. The cosmic ray particles are taken to move in a time-independent magnetic field B, which is a random function of position. The Fokker-Planck coefficients, which describe the mean motion of the particle and its scattering in pitch angle, are derived from the particle equations of motion and are expressed in terms of the two-point correlation function of the field, or, alternatively, in terms of the Fourier spectrum of the irregularities. The diffusion tensor is derived and finally, application of the formalism to magnetic field observations is discussed, (Jokipli, 1966). A further discussion of these methods can be found in section 3.2. If we consider the existence of large scale intervening magnetic fields with strength B - 0.1 - 1 pG, a possible explanation for the lack of identification of astronomical sources in the direction of the highest energy events, presents itself. Over a large propagation path his type of field would provide sufficient angular deflection even for high energies and could explain the large scale isotropy of arrival directions observed by the AGASA experiment, as due to diffusion, (Isola and Sigl, 2002).

UHE protons, which undergo interactions with the CMB may readily be transformed into

neutrons as a result of the photo-pion production process. These neutrons will also interact with the CMB, which once again results in pion production. During this process, the neutron may transform back into a proton. Therefore, a UHE cosmic ray may spend a significant amount of time as a neutron and this should be considered when studying the propagation of cosmic rays through extragalactic magnetic fields. For a proton (considering only single pion production) the following two processes are possible:

18 p+y --> p+fto p+y -)n+ft+

The branching ratio for these two processes varies as a function of the comic ray rest frame photon energy and it is found that on average the ratio 3:2 is in favour of a proton not being transformed into a neutron, (Lampard, et al.,1997).

Similarly for a neutron, the two possible processes are: n+y-+n+no n+y -+ p+7r

The branching ratio here, is taken to be 3:2 in favour of the neutron not being transformed into a proton. The free neutron is an unstable particle, which decays into a proton within a short time and this needs to be considered when modelling the propagation of UHE cosmic rays. In its own rest frame, the characteristic decay time of the neutron is about 15 minutes. However, for a neutron of energy 10'o eV, y = 10ll and the decay length is approximately 0.9 Mpc, (Lampard, et al.,1997)., sufficient to cross our local group of galaxies, but not to reach the nearest major group, the Virgo cluster.

1.2.4 Cosmic Ray Acceleration

Cosmic rays with energies up to 1Ola eV are thought to arise, predominantly, through shock acceleration in supernovae remnants (SNR) in our galaxy. Acceleration to somewhat higher energies, may be possible, but these energies would still not be high enough to explain the

smooth extension of the spectrum to 1018 eV (Protheroe,1996).

In 1951 Fermi (1951) suggested that cosmic rays might be accelerated by scattering off magnetised clouds. This statistical acceleration relied on the fact that head-on collisions between a charged particle and a cloud result in a net transfer of energy to the particle. Statistically, such collisions are more likely than the reverse, when the clouds are moving away, and there is a resulting net acceleration. However, the process is slow, the energy gain per collision depending on the speed of the cloud (in units of c) squared. It is thus

referred to as second order acceleration. More recently, it has been recognised that acceleration in a shock front can be linear in the relative velocity and 'fìrst order' diffusive shock acceleration, in which particles are scattered

19 back and forth across a relativistic shock front (accelerating each time) is now the most

popular cosmic ray acceleration mechanism, (Kirk, et. al., 2001). The best studied shocks

for this purpose are those associated with supernova remnants but, at lower energies,

heliospheric shocks have been studied and, at higher energies, shocks in active galaxies are the subject of ongoing study.

These processes are associated with magnetic scattering and so rigidity, momentum per unit charge, is an important parameter. This means that more highly charged particles ('heavier'

nuclei) may be accelerated to the highest energies possible with the mechanism. It is then expected that the cosmic ray composition will be preferentially heavy when an acceleration mechanism is close to its high energy limit. For instance, if the cosmic ray spectral knee is an acceleration artefact, it would be expected that a composition change to heavier nuclei would be found as the knee was crossed.

20 CHAPTER 2 Magnetic Fields

2.1 The Galactic and Extragalactic Magnetic Fields

Intergalactic magnetic field measurements are obtained from several methods; star-light polarization, the Zeeman effect, the rotation measures (RMs) of extragalactic radio sources, the pulsar RMs and radio polarization observations, as well as the newly implemented sub- mm and mm polarization capabilities. The Zeeman effect, as well as the sub-mm and mm emission techniques tend to be used for the study of magnetic fields in Galactic molecular clouds, but the field is only observed at high intensities, (Han and Wielebinski,2002). For more distant objects the usual approach that is available for the observation of magnetic fields, is observations of radio polarization resulting from s¡mchrotron emission by high energy electrons in the magnetic fields. This shows the fields in the jets, or lobes of radio galaxies, or quasars. Magnetic fields in clusters of galaxies are reflected by the observed structure of radio halos, or the RM distribution of background objects.

2.2 The origin of the Galactic Magnetic Field

The Milky Way and other late-type galaxies are permeated with (i) cosmic ray gas, (ii) magnetic fields, which generate polarized s5mchrotron radiation, (iii) ionized (and neutral) interstellar gas, which causes Faraday rotation in the radio and (iv) interstellar dust grains, which align with the interstellar field and induce linear polai'zation in the optical starlight by selective absorption, or scattering. The above make it possible to observationally trace the large- and small-scale magnetic structure in galaxies, (Kronberg,1994). Our own Galactic magnetic field was discovered in 1949, by optical polarization observations. Polarization observations of diffuse Galactic radio background emission in 1962 conftrmed the existence of a Galactic magnetic field. However, the majority of the information on the Galactic magnetic field comes from the analysis of rotation measures of extragalactic radio sources and pulsars. This can be used to construct a 3-D magnetic field structure in the Galactic halo and the Galactic disk. In the Galactic centre, radio synchrotron

2l spurs show the existence of a poloidal field. The polarization mapping of dust emissions and Zeeman observations in the central molecular zone, reveal a toroidal magnetic field parallel to the Galactic plane. Optical polarization and multifrequency radio polarization data, show that the large-scale magnetic field in the nearby galaxies, follows their spiral anns, or dust lanes, (Han and Wielebinski,2002).

It now seems clear that spiral galaxies generally possess large-scale magnetic fields whose evolution and possibly origins, are controlled by induction effects in the partially ionized interstellar gas. Turbulent motions with scales below about 100 pc are present in this gas, and so the observed ubiquity of the large-scale galactic magnetic fields, coherent over scales of at least 1 kpc, requires special explanation, (Beck, et al.,1996).

Faraday rotation measurements indicate that the Galactic Magnetic Field (GMF) in the disk of the Galaxy has a spiral structure with field reversals at the optical Galactic arms. In their recent paper, (Yoshiguchi et a1.,2004) use a bisymmetric spiral field (BSS) model. The

solar system is located at a distance 11 :,RtO : 8.5 kpc from the centre of the Galaxy in the Galactic plane. The local regular magnetic field in the vicinity of the solar system is

assumed to be Bro¡u,.:1.5 ¡rG in the direction l:90o + p, (l represents the latitude within the galactic plane) where the pitch angle is p : -19o. In polar coordinates (rtrq), the strength of the spiral field in the Galactic plane is given by

B(,,,,ç)= r. *,[, - o 2.r [#) ^ ;)

where Bo: 4.4 FG, ro : 10.55 kpc, and þ: ll tan p : -5.67. In this model the field

decreases with Galactocentric distance as llrl¡, and it is zero for 11¡ > 20 kpc. In the region around the Galactic centre (rtt < 4 kpc) the field is highly uncertain, and thus is assumed to be constant and equal to its value atr¡¡:4 kpc, (Yoshiguchi et a1.,2004). Some observations show that the field in the Galactic halo is much weaker than that in the disk, however, these results may be uncertain as there is a viewing problem which makes it diflrcult to find out the magnetic field strength of the central part of our o\ryn galaxy. In his work, Yoshiguchi et al., assume that the regular field corresponds to an A0 dipole field. An A0 mode is defined as an axisymmetric dynamo mode with vertical symmetry and is believed to be operating in the thick disk, or halo, of our Galaxy. Such a dynamo, creates

22 toroidal fields of opposite sign above and below the Galactic plane and poloidal fields of dipole structure perpendicular to the plane in the Galactic Centre region, (Han, et a1.,1997). In spherical coordinates (r,O,

Bx = -3pcsin gcos Scosg I 13

B, = -3ltosin.9cos gsing l13

B, I s)t r' 2.2 = ttc(t - "o.' where p6 - 184.2 pG kpc3 is the magnetic moment of the Galactic dipole. The dipole fìeld is very strong in the central region of the Galaxy, but is only 0.3 ¡rG in the vicinity of the solar system, directed toward the north Galactic pole. There is a significant turbulent component B,rn of the Galactic magnetic field. Its freld strength is difficult to measure and results found in literature are in the range of B'.un : 0.5... ..28,"e, (Yoshiguchi et a1.,2004).

2.3 Methods in I)etermining Magnetic Field Strengths

It is necessary to point out, that from verifying the existence of a magnetic field to actually measuring the field strength, there is a significant and non-trivial step. Below is a description of some of the methods presently employed in the estimation of magnetic field strengths.

2.3.1 The Zeeman Effect

The Galactic magnetic field strength may be estimated from the Zeeman splitting of the 2l- cm neutral hydrogen line. In the presence of magnetic fields, atomic energy levels are split into larger number of levels and the spectral lines are also split, this is known as the Zeeman effect.

23 One direct way of measuring the strength of a uniform magnetic freld is to measure Zeeman splitting of a transition in the interstellar gas. The observed frequency of a transition is given by:

| = r,,, + eB(4nmc)-t H, (2.3) where v'',n is the frequency in the absence of a fìeld, m is the electron mass, and B is in

Gauss, (Kronberg, 1994).

The pattern and amount of splitting are a signature of the magnetic field strength. The observational problem is a significant one, since the splitting amounts to only 28 GHz T-t, l0 for the 2lcm line and the expected magnetic field strengths are only about 10-e - 10 T. Thus, radio spectrometers must be sensitive enough to detect splittings of about 10 Hz in

1420 l|;lHz. Fortunately, when the magnetic field runs parallel to the line of sight, Zeeman splitting results in two circularly polarized components, with opposite serìses of circular polarization, on opposite sides of the line centre. The splitting is always much less than the width of the absorption line and therefore the technique adopted is to observe a radio or millimetre absorption line and to search for an excess of oppositely circularly polarized radiation on either side of the line centre, (Longair, 1994).

2.3.2 Optical Dust Absorption and Polarization of Sub-mm and mm Emission

Optical polarization, observed on Earth, of stars located behind dusty molecular clouds, is due to dichroic extinction by dust grains aligned by a cloud magnetic field, (Vallée,1997). Its study can give us valuable information on the structure of magnetic fields within which the dust is embedded. The dense cores within a cloud are opaque at optical wavelengths, prohibiting optical polat'rzation observations of background stars and restricting optical polarimetry to the tenuous parts of cloud halos. However, polarization measurements in the infrared, far-infrared, sub-mm and mm have made it possible to return to the dust polanzation method of measuring magnetic fields. No scattering is expected at these wavelengths and therefore such observations, of linear polanzation from the thermal emission of magnetically aligned dust grains, provide a relatively easy means of exploring the magnetic field morphology in the emission by dust particles aligned by the magnetic field, (Han and Wielebinski, 2002). This method is used mainly in the study of magnetic

24 fields in galactic molecular clouds. Here, interstellar dust (intermingled with the gas on all scales) can absorb starlight, thereby being heated to significant temperatures, but also shielding molecules and thus avoiding their dissociation. The thermal dust continuum emission traces the temperature-weighted column density of dust grains and wide-ranging network of dusty filaments, which can be observed at mm, sub-mm, far-infrared bands.

Over 90o/o of the dust in galaxies is cold and radiates in the sub-mm and mm range where it can be detected by multi-channel bolometer systems. Sub-mm is a very difficult domain to access for astronomy on the ground. It is only possible in a few narrow bands from 2.5 -

30¡rm. At À > 300pm, observations are only possible from space platforms, the atmosphere being completely opaque.

A project operating from the South Pole, where the observations at these wavelengths are exceptionally good, has been developed through the Centre for Astrophysical Research in Antarctica (CARA). The observations are carried out using the Antarctic Submillimeter Telescope and Remote Observatory (AST/RO), which is a 1.7-meter telescope and it has been in operation since January 1995. It is a general-purpose telescope for astronomy and aeronomy studies at wavelengths between 200 and 2000 microns. The Submillimeter Polarimeter for Antarctic Remote Observing (SPARO), which is a 9-pixel lambda : 450 micron polarimetric imager is also used in the mapping of interstellar magnetic fìelds,

(CARA, 1999).

2.3.3 Synchrotron Radiation

Synchrotron radiation is associated with the acceleration suffered by electrons as they spiral around a magnetic freld. The force felt by a charged particle in a magnetic field is perpendicular to the direction of the field and to the direction of the particle's velocity. The net effect of this is to cause the particle to spiral around the direction of the field. Since circular motion represents acceleration (i.e., a change in velocity), the electrons radiate photons of a characteristic energy, corresponding to the radius of the circle. For non- relativistic motion, the radiation spectrum is simple and is called "cyclotron radiation". The frequency of radiation is simply the gyration frequency, which is given in terms of the

magnetic field as

v:eB/mc

25 where B is the field strength, e is the electric charge, m is the particle (electron) mass, and c is the speed of light. Cyclotron and synchrotron radiation are strongly polarized and the detection of polarization is regarded as strong observational evidence for synchrotron or cyclotron radiation.

This is therefore the relativistic equivalent of cyclotron radiation, the crucial distinction being time dilation and Doppler beaming effects. Its details are outlined below.

Synchrotron Emission

Over large distances, extragalactic magnetic fields are observed from the emission from non-thermal particles at centimetre wavelengths. These particles are typically electrons travelling close to the speed of light. As they travel through a homogenous magnetic field B with veloci ty v = c , the Lorentz force perpendicular to the magnetic field will cause them to perform circular motion arouncl tle field with frequency

,'=nY¡*") (2.4)

where e is the electron charge, y is the Lorentz factor, m is its rest mass and c is the speed of light. This circular motion is superposed on the linear motion of the particle along the field, thus it will travel on a helical trajectory.

In its rest frame the electron will experience an acceleration a : a1 perpendicular to its velocity v. The magnitude of the acceleration is given by u.,- : osVr. Using the Larmor formula for the power emitted by an accelerated particle with charge q (which is covariant if the radiation process is forward-backward synmetric), the rate of energy loss is given by:

dw =_zq'lol' =_2qoy'B' ,,? (2.s) dt-- kt ,s;'t

Averaging over an isotropic distribution of pitch angles we arrive at an average loss rate for the Lorentz factor per particle of,

26 4qo B'yt dy (2.6) dt gct mt

or expressed in terms of the particle momentum:

dp 4qoB'p' Ap' (2.7) dt 9cu mo

A4 where A= ^*n, ,.82 is the syrchrotron loss term, and v particre: c. From this we can easily 9m!co derive the synchrotron loss time of a particle with an initial momentum p6 , which we define as the time it takes the particle to lose half of its energy due to synchrotron losses in a constant magnetic field:

tto,, = (PoA)-t (2.8)

(Rybicki and Lightm an, 197 9).

Emitted Synchrotron Radiation

The radiation emitted by the electron is of variable frequencies, but most is concentrated at:

€l m" vx By' (2.e) L/L"l-

B - magnetic field strength

elm"- electron charge to mass ratio y - relativistic electron's gamma factor

27 The main factor to note about synchrotron radiation is that it is emitted within an angle lly of the instantaneous momentum vector of the electron and is strongly polarized. When one views along the magnetic field lines, the synchrotron emission is at a minimum, as the acceleration is perpendicular to the line of sight. However, the emission reaches a maximum when it is viewed perpendicular to the magnetic field lines The radiation is only seen when the momentum vector of the electron has a component directed along the line of sight towards the observer (Longair, 1994). The spectral density of s¡mchrotron radiation, e(v), depends on N(E), the energy distribution of relativistic electrons. \When N(E) is known, the magnetic field strengths can be estimated.

2.3.4 Radio Faraday Rotation

A particularly important technique for studying magnetic fields is Faraday rotation. This is a magneto-optical phenomenon, or an interaction between light and a magnetic field. The rotation of the plane of polaization is proportional to the intensity of the component of the magnetic field in the direction of the beam of light. The relation between the angle of rotation of the polat'rzation and the magnetic freld in a diamagnetic material is:

B:vBd (2.t0) where

B is the angle of rotation (in radians)

B is the magnetic flux density in the direction of propagation (in teslas) d is the length of the path (in metres) where the light and magnetic field interact v is the Verdet constant for the material. This empirical proportionality constant (in units of radians per tesla per metre) varies with wavelength and temperature and is tabulated for various materials, (Wikimedia Foundation, Inc., 2006).

28 B

/) / d ,.

Diagram 2(a) Illustrated relation between the angle of rotation of the polarization and the magnetic field in a diamagnetic material (Wikipedia, 2006).

In an astrophysical environment the above mentioned phenomena, results from the fact that the modes of propagation of radio waves in a magnetised plasma) ate elliptically polarized in opposite senses. That is, they are right- and left-hand elliptically polarized waves. This arises because under the influence of a perturbing electric field, the electrons are constrained to move in spiral paths about the magnetic field direction. Therefore, when a linearly polarized signal is incident upon a magnetoactive medium, it can be resolved into equal components of oppositely handed elliptically polarized radiation.

The Faraday effect is imposed on light over the course of its propagation from its origin to the Earth, through the interstellar medium. Here, the effect is caused by free electrons and can be charactenzed as a difference in the refractive index seen by the two circularly polarized propagation modes. Hence, in contrast to the Faraday effect in solids or liquids, interstellar Faraday rotation has a simple dependence on the wavelength of light (),), namely:

þ:RMl where the overall strength of the effect is charactenzedby RM, the rotation measure. This in turn depends on B, and the number density of electrons, n", both of which may vary along the propagation path, to give:

29 d 2.Ila !""Bds 0

where, e is the charge of an electron m is the mass of an electron c is the speed of light in a vacuum

The "Rotation Measure" (RM) of the linearly polarized emission of radio sources can be re- written as:

RM = A7l(L.Ê) = g.lxtosJ^r"a ¡¡dlradm-2 (2.11b) where \ is the rotation (radians) of the plane of polarization measured at wavelength À(m), N"(cm-3) is the local thermal electron density of non-relativistic electrons, (in galaxy clusters this is determined from X-ray surface brightness profiles of the hot (T-108K) diffuse gas, which fills the cluster). ,B¡¡ the line-of-sight component of B parallel to the line of sight, measured in tesla, and / the path length þc) along the line of sight, (Kronberg, 1994).

Thus, measurement of the quantity X/¡,'? (RM) (in radians p"r rn'¡ gives information about the integral of N"Br¡ along the line of sight. In addition the sign of the rotation gives information about the weighted mean direction of the magnetic field in the line of sight.

Here, \Me use the standard convention for the extragalactic case:

lf X/)'"2 < 0, B is directed away from the observer, lf xI,"2 > 0, B is directed towards the observer

Many galactic and extragalactic radio sources produce linearly polarized emissions and therefore, by measuring the variation of the position angle of the electric vector with frequency, estimates of JN"nzdl may be obtained for many different lines of sight through the galaxy.

30 The partially ionised interstellar gas is permeated by the galactic magnetic field and hence constitutes a magnetised plasma. Under typical interstellar conditions, both plasma frequency, vo : 8.98 Ne'/2Hz and the gyrofrequency, vs : 2.8 x 1010 B ]Hz, where B is measured in tesla, are much less than typical radio frequencies, 107> v > 10llHz. Under these conditions, the position angle of the electric vector of linearly polarized radio emission is rotated on propagating through a region in which the magnetic field is uniform. This angle depends on the combination of the quantity of magnetised plasma between the obseler and the magnetic field strength and direction. The amount of angular rotation of the wave also depends on the observing wavelength, so observations at many wavelengths reveal the amplitude of the intervening magnetic field and its direction, providing the electron density is known, (Vallée, 1997).

If the source of the radiation is a pulsar, a rough estimate of the strength of the magnetic field between the source and the telescope may be made by combining the measurements of the Faraday rotation of the linearly polarized emission (rotation measure), with measurements of the delay time in the arrival of the radio signals as a function of frequency

(dispersion measure). The dispersion measure gives an estimate of JN"dt.

We can therefore obtain a weighted estimate of the strength of the field in the line of sight:

/- \ rolalion-measure tp \- lu"a,,dt (2.r2) \ tt t dispersion measure N"dl - J Unfortunately the above analysis only applies to the very simplest magnetic field configurations and it becomes complicated when account is taken of the real magnetic field structure within the source region. For example, if there are irregularities or fine structure in the B-field and plasma distribution, we have to add the contribution of each region to the total polari zation. In the pulsar case, estimates of the column density of free electrons, JN"dl may be obtained from the delay time in the arrival of the radio signals as a function of frequency, but suffer from similar problems of non-uniform plasma density. For these tests to be practicable, it is essential to observe sources, which emit sharp pulses over a wide range of frequencies.

31 2.3.5 Factors in the Determination of the Rotation Measure of Galaxies

Because of measurement errors in 1 (the position angle of the linearly polarized radiation at wavelength ì,) and because the observed orientation of the polarization plane is ambiguous by nn radians, it is necessary to take polarization measurements at several wavelengths (at least three) in order to obtain a reliable RM, as for example, in the case of a galaxy, or of a quasar where different components (eg. nucleus, lobes) show up in different ranges of wavelengths, the different wavelengths are necessary to solve for the rotation measure of a unique physical regime. Unlike equipartition estimates, which are insensitive to the presence of field reversals within the volume observed by the telescope beam, the observed value of Faraday rotation will decrease with increasing number of reversals. Although the filled apertures of single-dish telescopes are sensitive to all spatial structures above the resolution limit, synthesis instruments such as the VLA cannot provide interferometric data at short spacings. This shortcoming results in some blindncss to cxtcnded emission. Missing large-scale structures in maps of Stokes parameters Q and lJ, can systematically distort the polarization angles and hence the RM distribution, so that the inclusion of additional data from single-dish telescopes in all Stokes parameters is required, (Beck, et a1.,1996).

Both long and short intervals of 1,2 between data values are necessary. Short intervals test for a possibly high RM and thereby reduce the nn ambiguities, while the longer intervals define the slope more accurately, (Simard-Normandin, 1981). This is illustrated in figure 2(b), which shows data for 3C 353. This is the fourth strongest radio galaxy in the 3C catalogue (flux density 57Jy at 1.4 GHz). The host is a elliptical galaxy at z:0.0304, a dominant member of the Zwicky cluster.

J¿ Br0

2 3

RM=ló-9 Rtl* ¡7.1 RM =17 t þ + t'

sps tto x' X' ,* ¡(" Pqr

+ + 770 ùû ü û r80

90

200 {00 ô00 800 t000 I 100 {00 ó00 800 1000 ¡ 0 ?00 {00 600 800 }000 æ {.*') ñ(c.tl N(.m?)

1 5 ó RM"lt t RH=10.2 é¡0 RHEr2-l i- h þ

+ plr pl' r pFo f tr"

{ lt0 {r ++ {r +

EO rB0

90

r00 100 300 ,t{t0 toD Ð ro0 200 r00 Joo 500 600 0 t00 2{x} 300 ,|fr 50t ìt{cnt} Àt (.m'1) N(crt)

Figure 2(b) - Plot of integrated polarization and position angle (see Simard-Normandin, 1981)

Plots of the integrated polarizationp (open squares) and position angle 1 (filled squares) vs. ¡.2 for 3C 353, showing the best fit rotation measure.

The above figure illustrates that the true y-?"2 variation for a source, sometimes deviates from the simple linear Faraday Rotation Law,

x(X'):1e + @M)À2 (2.13)

Uncertainty in RM determination is unlikely to give a totally false RM provided there are sufficient data. Errors can also occur from noise and if there is low polarization.

Figure 2(c) shows a map of rotation measures of 976 extragalactic radio sources. The filled circles represent magnetic fields pointing towards us and the open circles represent magnetic field directions away from us, (Longair, 1994). These data are plotted in galactic coordinates and illustrate the facts that the galactic field has a strong component along our spiral arm, but also a significant random component.

JJ þE

le Ácuenbâ"r¡ âql 'ernlcrd e¡duus sFil q 'sâseeJcep uolluzlJelod ¡o eer?ep }eu sq1 'uot8er âcrnos eqt qSnorq+ sverperu : e Ãqpelelor sr uo4¿rpur erilJo uollezlrelod¡o eueld âql uãtI^A

'âcJnos

eql se^eel uorlerper âql se se¡8ue luereJJrp o¡ dn ppu uorSer eql qEnorql sqldep luerãJJrp urog Eu4eulSuo sJolco^ uoqezu€lod eq1 'uor8er ãcJnos sql qEnorql uo:4ezlrelod ¡o euuld '(@)Z ârl1Jo uoqelor lerluelsqns sr orâq] esn¿ceq'renemoq'sql8uelenem 3uo1 tv ernErg ees) 'otez o¡ spus¡ qlSuele^e,i\\ eql w orãz ol spuel qcrr.{1v\ 'r\ ol ¡euor¡odord sl uotEe¡ eq} ultlll^{

uoqelor,'(eperu¿ I€rrrelur âsn¿câq sl sF{I 'sercuenbe4 q8noue q31q le pezue¡od,(11tg ureruer uN Ilrlvr uorlerp€r or{1 'ur-ro3:run ere rfirsuep uurseld eql pue gr pleg cr¡eu8eut âql qclql\ ur'1. ezrs 'sâsueJcur 3:o uor8er ¿ tuo{ seluur?uo uorssnue orpeJ erü JI q}8ue¡e.tem âql se 'pezuelodep eruoceq o1 pelcedxe sr s¡eu8rs orper Jo uorlpzuelod ¡o eueld eql 'uotlelor ol uoqlppe uI

uoprzlrulodeqrtuperug 9't'Z

'(gg6¡ '8;equor¡¡

ãr urpu€tr¡roN ur punoJ oslu'¿66¡ ''1o 7a'ue¡1) 's¡41¿ e,tr1e8eu'sloqur,{s uedo pue str [U e^I]ISod luese¡de-t sloqru,(s pe¡¡rg 'ezrs 1se3.re1 eqr roJ lruul;ooE '00€>lnul;0çt"""""'o9>ltruxltot 'o€>lnàlto lnrul 'socrnos ulll.otls ctrletuurÁsl1uv (c)Z ern8tg Jo sel€cs ¿ lueserder s¡oqu,{g orper crlcelu8erlxe '(q ',{ìs ¡4¡U

,06- = ? :dS cl ,09- = 4 .09- =

a , o ct

Õ t o I /r Oo a a I rO I oa o o t q a I a I o b o .oq a a + t Ia o a .c o a a t I ( P .09=? 009=9 o06 = I ið\ which significant depolarization is observed, provides information about the integral of

N the source region. A point to note is that, whereas the rotation of the plane of "B,,lthrotgh source to the Earth, the polarization provides information about the !w"n,,dl , from the depolarization provides information about the source regions themselves. This process is often referred to as, Faraday depolaization, (Longair,1994). Any heterogeneity (in density or in magnetic field) within the magnetic slab increases the observed depolarization. For positrons, the Faraday rotation is of opposite sign, so that there is no Faraday depolarization

in a pure electron-positron plasma. For protons, the Faraday rotation is negligible and does

not compensate for that of the electrons, (Fraix-Burnet, 2000). When taking measurement of galactic magnetic fields, the degree of polarization p and the observed Faraday rotation measure, are very sensitive to the source structure, particularly to whether the source is symmetric along the line of sight. The RM may change sign in a certain wavelength range in an asyrnmetric slab, even when the line-of-sight magnetic field has no reversals. Faraday depolarization in a purely regular magnetic field can be much stronger than suggested by the low rotation measures. A twisted regular magnetic field may result in p inueasing with 1". This behaviour has been detected in several galaxies,

(Sokoloff, et al., I 998).

2.3.7 Energy Equipartition between Magnetic Fields and Cosmic Rays

Synchrotron radiation from highly relativistic particles is an old and well known phenomena. At extreme non-relativistic energies, the primary contribution to the emissivity (or absorption coefficient) is the well known cyclotron radiation at the first and perhaps, first low harmonics. There does not exist a simple formula for the intermediate range, for the evaluation of the emissivity from particles with an arbitrary energy spectrum and pitch angle distribution. The usual practice is to use numerical methods to evaluate these quantities at mildly relativistic energies, (Petrosian, 1981). The low harmonics of the cyclotron radiation, will either be self-absorbed, or absorbed by the surrounding plasma in the case of most astrophysical problems. Consequently, only the radiation emitted at high harmonics, where the optical depth t < 1 is of interest. Petrosian (1981) finds that there exists a simple approximate method for the evaluation of the directivity and spectrum of synchrotron radiation from an ensemble of particles in a given magnetic field, at such high harmonics.

35 His paper presents the results of this method for the total emissivity at these harmonics. Available analyses of the polarization of s¡mchrotron sources assume the synchrotron emissivity scales as e æ82r. However, the scaling of synchrotron emissivity with a magnetic field is different under the widely used assumptions of energy equipartition and pressure equilibrium between magnetic fields and cosmic rays. In this case, the number density of energetic particles, scales as B2 if the energy densities of magnetic fields and cosmic rays are completely correlated and the emissivity scaling becomes, e : cB'B', (2.14) where C is a constant. The stronger dependence of the s¡mchrotron emissivity on magnetic fields, increases the importance of regions with strong B. Therefore the degree of polarization under the scaling (2.14) must be generally larger than that of e = cB'r. This affects both the degree of polarization as ), ---+ 0 and the Faraday effects in any inhomogeneous regular and/or random magnetic field, (Sokoloffl 1998).

2.4 Random Magnetic Fields

The interstellar medium is turbulent and therefore the embedded magnetic fields must contain random small-scale components, (Beck, et al., 1996). However, available observational and theoretical knowledge of random magnetic fields and their maintenance in the interstellar medium (ISM) is rather poor. Instead, crude descriptions in terms of global quantities such as mean magnetic energy are usually applied. Here, the widely used concept of equipartition between the magnetic and kinetic energy in the turbulence is often used and implies that the rms, random magnetic field strength, is given by B :., B"r= (Fv2¡ts)l/2 with v the rms turbulent velocity and P the plasma mass density. The equipartition value is significant in that the Lorentz force is expected to become comparable to the forces driving the turbulent flow as equipartition is approached, (Beck, et a1.,1996).

2.5 Magnetic Field Turbulence

To some extent, astrophysical objects are coupled to the intergalactic gas. The strength of the coupling depends on the conductivity of the gas. Considering a closed loop of material

36 within the gas, with magnetic field (B ) lines passing through the loop. If the loop undergoes a deformation, the magnetic flux passing through the loop will also change. Therefore a current will flow through the loop, (Faraday's law). A magnetic field generated by this current contributes to the magnetic of the loop, thus conserving the flux. Conservation of the magnetic flux depends on the conductivity of the gas. For a highly

conductive gas, flux freezing occurs, where the É lines of force are tied to the underlying matter, which will exhibit turbulent flow, thus increasing turbulence in the magnetic flux. Turbulence is the random mixing of the magnetic flux, i.e. tangling of the field lines. Some

causes of such turbulence are supernova shocks, and galaxy collisions in the case of clusters galaxy mergers. In the gaseous disc of a galaxy, at least |0o/o of the total energy released,

eg. through collisions, goes into the kinetic energy of motion of the interstellar gas.

2.5.1 Kolmogorov Scale Turbulence

The following discussion is based on the introduction by Ahmadi, 2003. Turbulence has a wide range of length (time) scales. Fluctuation energy is produced in large eddies (with low wave numbers), then transferred into smaller and smaller eddies and energy flows down the spectrum to the high wave number region. The energy is mainly dissipated into heat at the smallest eddies (of the order of the Kolmogorov scales). The dissipation rate, e, is roughly

equal to the fluctuation energy production rate.

Suppose the large-scale velocity fluctuation of turbulence is u and the corresponding length scale is A. Then the rate of production (or dissipation) of fluctuation energy is given by:

1 u' (2.rs) ^ Equation (2.15) implies that large eddies lose a significant fraction of their energy in a time period of r¡/4. It should be noted that the direct viscous dissipation rate depends on the

coefficient of viscosity (v) and the mean turbulent velocity gradient (* ) urr¿ ls oy

2 ôu u 2 = l----= (2.16) oy^ N

and the ratio:'

JI 2 u 2 V. 1 (2.r7) v : :'t 1 - u' Ren ^ where uL Re (2.18) v

is the characteristib Reynolds number, which is the ratio of inertial forces, as described by Newton's second law of motion, to viscous forces. If the Reynolds number is high, inertial forces dominate and turbulent flow exists. If it is low, viscous forces prevail, and laminar (non-turbulent streamline flow in parallel layers ) flow results.

Large-scale turbulent motion is roughly independent of viscosity. The small-scale, however, is controlled by viscosity. The small-scale motions are also statistically independent of relative slow large-scale turbulent fluctuations (and/or mean motions). According to Kolmogorov (Universal Equilibrium Theory), the small-scale turbulence is in equilibrium (independent of large-scale) and is controlled solely with e and v. Using dimensional arguments, Kolmogorov defined the length, time, and velocity scales of the smallest eddies of turbulence. These are: 7ul ..tt2 ry=t-lut l"ot .r=f ,r=(rr)''o (2.1g) \t ) \e)

Using equation (2.15), it follows from (2.19) that,

I / o 0 = Re^-3 ,!n = Ren-' '' ,I = R"n-" (2.20)

For eddies much smaller than the energy containing eddies and much larger than dissipative

eddies (of the order of Kolmogorov scales), turbulence is controlled solely by the dissipation

rate e l1l""O the size of the eddy (wavenumber k). In this subrange, \k)

38 "]'

n(t) *[_[;) _ ,2 3¡ s 3 (2.2r) =y:k k In the derivation of (2.21), a dimensional argument is used and the eddy size

appried [;),"- = [;)"'is Equation (2.21) is the famous -5l3 law of Kolmogorov, (Ahmadi, 2003), which we will apply below to astrophysical magnetic fields.

2.6 Magnetic Fields in Galaxy Clusters

Galaxy clusters are the largest vinalized structures in the universe and consist of up to thousands of galaxies. In the early 1970s, observations of clusters revealed atmospheres of hot (107-108 K) highly ionised and low density (=10-3cm-3) gas, containing thermal electrons. These atmospheres are found to extend to Mpc radii and dominate the baryonic mass of the systems 110r3-10r4 M"), (Carilli and Taylor,2002). About l0o/o of galaxies in clusters produce emission at radio wavelengths above a power of 1023 W Hz-l at 20cm. This is observable by telescopes, such as the VLA interferometer. Jets, lobes and tails of radio- emitting plasma are ejected from galaxy cores (on a parsec-scale) and travel outward, often extending hundreds of kilo-parsecs, (Burns, 1998). The gas, in which the galaxies are embedded, emits X-rays by thermal free-free (thermal bremsstrahlung) radiation. The density of the thermal electrons (typically 10-6-10-2 be estimated from the "--3¡ "un luminosity of the X-ray emission, therefore the magnetic field strengths can be estimated using rotation measure techniques. Some clusters of galaxies show the presence of wide diffuse radio emissions (radio halos, or relics) associated with the intra-cluster medium rather than any cluster galaxy. Under minimum energy assumptions, it is possible to calculate an equipartition magnetic field strength averaged over the entire halo volume. The cluster magnetic field structure is made more complex, however, by the fact that the cluster environment is a violent one, (due to galaxy and galaxy group merges and collisions), and therefore, turbulence is produced within the magnetic fìelds of the intracluster medium. For many years not much was known about cluster magnetic fields and only in the last decade,

39 has it become clear that these fields are ubiquitous in cluster atmospheres, and that they may

even play an important role in the dynamics of galaxy clusters, (Carilli and Taylor,2002).

Clusters of galaxies contain hot intracluster gas, which is a remnant of the clouds out of which galaxies form. These clouds were heated by the energy released during their gravitational collapse. Some of the gas then cooled into normal galaxies, but in the case of massive galaxies and clusters most of the heat was preserved in a quasi-hydrostatic gaseous

atmosphere. The temperature of this hot atmosphere is approximately the virial temperature (i.e. K); its mass is gleater than that contained in the visible stars of a cluster. As the gas still loses its energy in the form of radiation, we can observe it in the X-ray waveband. The abundance of the intracluster gas is not primordial, but metal-enriched; in fact, for most elements the abundance is quite identical to the abundance of our solar system. This might be the result of processes like supernova explosions and tidal rippings of galaxy gas, flung

into intracluster space. Because of the interactions between the gas particles, this hot intracluster gas loses its energy (i.e. cools down) by means of bremsstrahlung and line emission. As these are 2-body processes, the radiative cooling time is shortest where the gas is densest: in the centre of the

cluster. In the absence of gravity, the drop in temperature would cause a drop of the gas pressure. In the centre of a cluster, however, gas pressure and gravitational attraction are in equilibrium; therefore, the gas density has to rise to maintain the pressure necessary for supporting the outer layers of gas. To cause its density to rise, the cooled gas has to flow inwards. As the densest gas, which cools quickest, is already concentrated in the centre of the cluster, the inward flow will start at the centre, soon followed by the outer layers. This flow of gas is called the cooling flow, (Heijden, 1998).

For a few galaxy clusters, those with extensive cooling flows, high resolution Faraday rotation measure mapping of extended sources (embedded within the cooling flow zones), have, in combination with X-ray data, produced magnetic field strength estimates of 10-40

¡rG, ordered on scales varying from 100-0.5 kpc. Lower limits of field strengths in the range

0.1 -1 ¡rG have been suggested from recent detections of both excess (over thermal) extreme ultraviolet (EUV) and hard X-ray (HEX) emissions in some clusters, (Clarke, et al. 2001).

40 2.6.1 The Local Group of Galaxies

The Milky Way is a member of a group of galaxies termed the Local Group that contains approximately 20 bright galaxies. Overall it is today considered to have more than 30 members. The number has increased with the continued discovery of new dwarf galaxies in

the cluster, including at least one dwarf in the process of being consumed by the Milky Way, (Milan,2000). The largest galaxies in the local group are the spirals Andromeda (M31) and

the Milky Way. These galaxies are spread in a volume of nearly 3 Mpc in diameter, centred somewhere between the Milky Way and M31. Membership is not certain for all these

galaxies, and there are other possible candidate members.

In the future, interaction between the member galaxies and with the cosmic neighbourhood will continue to change the Local Group. Some astronomers speculate that the two large spirals, our Milky Way and the Andromeda Galaxy, frày perhaps collide and merge in some distant future, to form a giant elliptical. In addition, there is evidence that our nearest big cluster of galaxies, the Virgo Cluster, will probably stop our cosmological recession away from it and accelerate the Local Group toward itself so that it will finally fall and merge into this huge cluster of galaxies, (Frommert and Kronberg,2003).

On the following page there is a graphic representation of the Universe within 500, 000 light

years from the Sun, showing the Satellite Galaxies.

41 'lfll-l Utllt 1,,,

I r lrlllrl Irr.,,1¡/ Llr rL ' L ,',.:tl

#f'l ill.: l \¡',r,3r,

l,:,: tlil ¡rrr I i I r,,,r ¡¡l ',,-rîL,;l]i,\

l,rr,'r1l l,'1 r,lr:ll'r rl, 1. Ì ,r l,l

Figure 2(d) The Universe within 500,000 light years, showing the Satellite galaxies. Number of large galaxies within 500,000 light years : l. Number of dwarf galaxies : 9. Number of stars : 225 billion. The Milky V/ay is surrounded by several dwarf galaxies, typically containing a few tens of millions of stars, which is insignificant compared with the number of stars in the Milky Way itself. This map shows the closest dwarf galaxies, they are all gravitationally bound to the Milky Way requiring billions of years to orbit it, (Powell, 2003).

Next is a representation of the Universe within 5 million light years from the Sun, showing the Local Group of Galaxies.

42 1 nrillli¡rr lt¡

IJtìl iIt- I1,,rrrrh

t l!- ll rL ¡l l.r l- I L 1':'tr 1 4 i' 1,,1 1 1r-l I

I l I r-- I r,l .;' '. 1,1 l,: ,. 't1' " "'" y,l'r,itc,n-re'j'¡ l-i,:i,:i:,,, l ? I Jrl -' T rr,¡rr,rrll¡m l-..¡1.¡l,l P llr llr L lllrl 1 ll lll i r, I I

1 t- a [,i Il.r':' :i

-: ,i rll l:il ll-l " Itu,r,¡¡ l:

,,rÍltll,ì I I f t r,,,,r,-r¡t

I rrr -tr rf,rr

Figure 2(e) The Universe within 5 million light years, showing the local group of galaxies. Number of large galaxies within 5 million light years:3. Number of dwarf galaxies: 37. Number of stars: 700 billion. The Milky Way is one of thre e large galaxies belonging to the group of galaxies called the Local Group which also contains several dozen dwarf galaxies. Most of these galaxies are depicted on the map, although most dwarf galaxies are so faint, that there are probably several more ìvaiting to be discovered, (Powell, 2003).

2.6.2 The Virgo Cluster and Nearby Superclusters

The Virgo Cluster contains some 2000 member galaxies and dominates the local intergalactic neighbourhood. It also represents the physical centre ofour Local Supercluster (also called the Virgo or Coma-Virgo Supercluster). As such, it influences all the galaxies and galaxy groups by the gravitational attraction of its enoÍnous mass. Therefore many galaxies have already fallen, and many will do so in the future, into the cluster. Current data on the mass and velocity of the Virgo cluster indicate that the Local Group is not far enough away from the cluster to escape its gravitational pull and therefore, its recession from Virgo will probably be halted at one time. Then the Local Group will merge into, or be swallowed up by the cluster.

43 Estimates for the magnetic field in the Local Superclustet vary, but Ryu and Bierman, 1998, have proposed that the observational upper limit on the strength 8,.,,', of an extragalactic magnetic field, such as the Local Supercluster, obtained from Faraday rotation observations of distant sources approximates to 8,.,,.', <1 pG,

Below is a representation of the Universe within 100 million light years from the Sun, showing the Virgo Supercluster.

Figure 2(f) The Universe within 100 million light years, showing the Virgo Supercluster. Number of galaxy groups within 100 million light years - 200. Number of large galaxies - 2,500. Number of dwarf galaxies : 25,000. Number of stars : 200 trillion. Our galaxy is just one of thousands that lie within 100 million light years. The above map shows how galaxies tend to cluster into groups, the largest nearby cluster is the Virgo

cluster a concentration of several hundred galaxies which dominates the galaxy groups around it. Collectively, all of these groups of galaxies are known as the Virgo Supercluster. The second richest cluster in this volume of space is the Fomax Cluster, but it is not nearly as rich as the Virgo cluster. Only bright galaxies are depicted on the map, our galaxy is the dot in the very centre, (Powell, 2003).

And finally \Me have the representation of the Universe within 1 billion light years from the

Sun, showing the neighbouring Superclusters of Galaxies.

44 Figure 2(g) The Universe within I billion light years, showing the neighbouring Superclusters. Number of Superclusters within I billion light years: 100. The number of galaxy groups:240,000. Number of large galaxies : 3 million. Number of dwarf galaxies : 30 million. Number of stars : 250,000 trillion. Galaxies and clusters of galaxies are not uniformly distributed in the Universe, instead they collect into vast clusters and sheets and walls of galaxies interspersed with large voids in which very few galaxies seem to exist. The map above shows many of these superclusters including the Virgo supercluster - the minor supercluster of which our galaxy is just a minor member. The entire map is approximately 7 percent of the diameter of the entire visible Universe, (Powell, 2003).

45 2.6,3 Radio and X-Ray Methods for Probing Cluster Magnetic Fields

In the past, X-ray observations have revealed that the space between galaxies within clusters is filled with a very hot thermal (million degree) plasma that can be interpreted through X- ray observations. Relativistic particles and magnetic fields are seen in radio halos in regions around radio galaxies and through Faraday rotation of radiation from background radio

sources, (Böhringer, 1 995). Intracluster magnetic fields are most easily detected through synchrotron radiation. However, synchrotron emissivity does not provide us with the field strength unless there is an independent knowledge of the true number density of relativistic electrons. Apart from being able to detect the synchrotron emitting halo, the intracluster magnetic fields can also be examined using the method of Multi-frequency polarization mapping of the synchrotron halo emission. This emission, due to the thermal gas, will undergo differential Faraday rotation and

depola.'izaÍion as a function of radio wavelength. The following equation:

: L7 8.1x los (2.22) RM /(Ll): !n"B,,dlradm-2 where; 1is the position angle of the linearly polarized radiation at wavelength )', n" is the thermal electron density in cm-3,

Bz is the line-of-sight magnetic field strength in microgauss, and I is the path length in kiloparsecs,

shows that in order to estimate the uniform magnetic field strength from Faraday rotation, an independent measurement of the free electron density, and knowledge of its weighted distribution along the line of sight is required. This rotation measure contains the non- relativistic electron density and an unknown number of field reversals. Once the magnetic

field strengths through Faraday rotation and an independent measurement of the ICM (intra-

cluster medium) thermal electron density are obtained, an estimate of lBl can be determined, (Clarke, et al,200I). The thermal electron density in a cluster can be determined from X-

ray surface brightness profiles of the hot (T - 108 K), diffuse (n" * (43r 1.7)'10 'h)'r'cm-'¡

gas that fills the cluster potential (value here based on the X-ray ROSAT obseruations of the A22I8 cluster of galaxies) (Uyanrker, et al., 1997), where n" is the central electron density, the Hubble parameter Ho:75 hl5kmr-' Mp"-t and q0: 0.5.

46 One must point out that strong magnetic fields (up to a0 pG) have been claimed for the ICM in galaxy clusters, especially those with strong cooling flows and these are based on studies of radio source rotation measures (RMs). This evidence has recently been re-examined and it is suggested that no claims for such strong, cluster-wide fields can be supported against likely alternative explanations for the observations, (Rudnick and Blundell,2004). Rudnick and Blundell show that RM variations in embedded sources have a significant contribution from the medium local to the sources themselves, which would have to be removed before seeking evidence for cluster-pervading fields. They also find that statistical conclusions are based not on background sources, but on sources embedded in the clusters and others with unreliable RMs. They believe that when such inappropriate sources are eliminated from existing samples, only marginal evidence remains for cluster-wide fields from the RM data. They conclude that such high magnetic fields, as derived in recent literature, which are based on RM studies are therefore unreliable. The authors do believe

that some progress is possible with the VLA, but the real break-throughs will probably come with the Square Kilometre Array (SKA), and perhaps the Low Frequency (10 - 240 MHz) radio telescope (LOFAR). As a counter argument to this, Johnston-Hollitt and Ekers (2004) have taken the subsample of the galaxies presented in their paper, as well as those of Clarke (2000), which are background to the clusters, and performed a Kolmogorov-Smirnoff (KS) test to assess if these data are drawn from the sample population as the control galaxies of Clarke (2000). The KS test rejected the null hypothesis at greater than the 99%o conftdence level demonstrating that the two samples are not drawn from the same population. In addition, they also considered the hypothesis that the RMs derived from embedded (within galaxy

clusters) and background sources (themselves not within the clusters) might be drawn from

the same population. In this case the null hypothesis could not be rejected and at least on a statistical level, there is no difference in the value of RMs derived from background or

embedded cluster sources. Thus, despite concerns over the validity of the embedded galaxies as probes, there is no statistical evidence that this class of sources can give rise to significantly different RMs to background galaxies. However, both a combined embedded and cluster background sample gives rises to significantly different RMs than the control sample. And even if using only background sources there is a still statistically signifìcant

excess RM detected along lines of sight through X-ray luminous clusters when compared with other lines of sight, (Johnston-Hollitt and Ekers ,2004).

47 It should be duly noted that in view of such contradictory opinions, one has to be very careful when using the current magnetic field estimations for other studies and be aware of the error margins due to our current technological limitations in accurate observational techniques.

48 CHAPTER 3 RADIO ASTRONOMY

3.1 Introduction

Radio astronomy is now about 70 years old. Karl Jansky made the first detection of cosmic static in 1932, which he identified as emission from our own Milky Way Galaxy. The concentration of emission towards the Galactic Plane was demonstrated, a few years later, by Grote Reber, who made the first rough map of the northern sky at metre wavelengths. Jansky's discovery of radio emission from the Milky Way is now seen as the birth of the new science of radio astronomy. Most astronomers remained unaware of this momentous event for at least the next decade, and its full significance only became apparent with the major discoveries in the 1950s and 1960s of the 2I cm hydrogen line, quasars, the pulsars and the cosmic microwave background. These are now fully assimilated into astronomy, and radio

is now regarded as one among the several tools available to astronomers in their pursuit and understanding of the astrophysics of our Galaxy, neutron stars, black holes or cosmology in general. Radio techniques have proved to be critical to our understanding of astrophysical magnetic field structure.

3.2 Observational Techniques and Instruments Used

The basic relation between the angular resolution 0 and the aperture (or diameter) D of a

telescope is 0*1"/ D radians, where 1" is the wavelength of observation. For the radio domain l, is - 10ó times larger than in the optical, which would imply that one has to build a radio telescope a million times larger than an optical one to obtain the same angular resolution. Technological advances have permitted a significant increase in the size of single radio

dishes. However, the sheer weight of the reflector and its support structure has set a practical limit of about 100 metres for fully steerable parabolic single dishes. Examples are the Effelsberg 100-m dish near Bad Münstereifel in Germany, completed in 1972, and the Green Bank Telescope in West Virginia, USA. The spherical 305-m antenna near Arecibo, Puerto Rico, is the largest single dish available at present. However, it is not steerable and is

built in a natural and close-to-spherical depression in the ground. It has a limiting angular resolution of - 1' at the highest operating frequency (8 GHz). Apart from increasing the dish

49 size, one may also increase the observing frequency to improve the angular resolution. However, the D in the above formula is the aperture within which the antenna surface is accurate to better than - 0.11", and the technical limitations imply that the bigger the antenna, the less accurate the surface. In practice this means that a single dish never achieves a resolution of better than - 10"-20", even at sub-mm wavelengths, (Andernach,1999).

Single dishes do not offer the possibility of close to instantaneous imaging as do interferometers, which use the Fourier transform of the fringe visibilities. Instead, several other methods of observation can be used with single dishes, all with both advantages and disadvantages. If one is interested merely in integrated parameters (flux, polarisation, variability) of a (known) point source, one can use "cross-scans" centred on the source. If one is very sure about the size and location of the source (and its neighbourhood) one can even use "on-off' scans, i.e. point on the source for a while, then point to a neighbouring patch of "empty sky" for comparison. This is usually done using a pair of feeds and measuring their difference signal. However, to take a real image with a single dish it is necessary to raster the field of interest, by moving the telescope e.g. along right ascension (RA), back and forth, each scan shifted in declination (dec) with respect to the other by an amount of no more than - 40o/o of the half-power beam width (HPBV/) if the map is to be fully sampled. The biggest advantage of this raster method is that it allows the map size to be adjusted to the size of the source of interest, which can be several degrees in the case of large radio galaxies or supernova remnants (SNRs). Using this technique a single dish is capable of tracing (in principle) all large-scale features of very extended radio sources. One may say that it "samples" spatial frequencies in a range from the map size down to the beam width. This depends critically on the way in which a baseline is fitted to the individual scans. The simplest way is to assume the absence of sources at the map edges, set the intensity level to zero there, and interpolate linearly between the two opposite edges of the map. A higher- order baseline is able to remove the variable atmospheric effects more efficiently, but it may

also remove real underlying source structure. For example, the radio extent of a galaxy may be significantly underestimated if the map was made too small

In contrast to single dishes, interferometers often have excellent angular resolution (again

0--)"1 D, but now D is the maximum distance between any pair of antennas in the anay). However, the field of view is FOV *Ll d, where d is the size of an individual antenna.

50 Thus, the smaller the individual antennas, the larger the field of view, but also the worse the sensitivity. Very large numbers of antennas increase the design cost for the array and the cost of the on-line correlator, to process the signals from a large number of interferometer pairs. An additional aspect of interferometers is their reduced sensitivity to extended source components, which depends essentially on the smallest distance, Say D',¡n, between two antennas in the interferometer arcay. This is often calledtheminimum spacingor shortest baseline. Roughly speaking, source components larger than - Xl Drnin radians will be attenuated by more than 50% of their flux, and thus practically be lost. A multitude of "cosmetic treatments" of interferometer data have been developed, both for the "uv-" or visibility data and for the maps (i.e. before and after the Fourier transform), mostly resulting from 20 years of experience with the most versatile and sensitive radio interferometers currently available, the Very Large Array (VLA) and its more recent VLBI counterparts the European VLBI Network (EVN), and the Very Large Baseline Array (VLBA), (Andernach, 1999).

3.2.1 The Very large Array (VLA)

The Very Large Array (VLA), is an Earth-rotation synthesis affay consisting of 27 radio

antennas arranged in a Y pattern. Each antenna is 25 metres in diameter and can be moved along the three railway tracks pointing roughly north, south-east and south-west. There are four standard configurations of antennas along the array arrns, arranged in such a way as to

give a good range of baselines and angular resolutions at 1420 }i{Hz down to the order of a

second of arc.

The telescopes are switched between configurations every few months and are occasionally used in intermediate hybrid configurations. Figure 3(a) shows aerial views of the VLA in D-configuration. The N:27 antennas give N(N-I)12:351 baselines and provide a total sensitivity equal to that of a single 130-meter dish. The maximum spatial resolution of the configuration used is determined by the longest baseline, (Werner, 2002).

51 Figure 3(a): Aerial views of the VLA. On top; in its D (most compact) configuration

(Courtesy of the National Radio Astronomy Observatory- http://www.vla.nrao.edu)

52 3.2.2 The Australia Telescope Compact Array

The Australia Telescope Compact Array (ATCA) radio telescope is located at the Paul V/ild Observatory, 25 kilometres west of the town of Narrabri, Australia. The telescope is an anay of six identical dishes, which commonly operate in aperture synthesis mode to produce radio images. The compact array is a part of the Australia Telescope National Facility network of radio telescopes, which consists of the Compact Array and the Parkes and Mopra radio telescopes. The array is frequently operated together with the 64m dish at the Parkes

Observatory and a single dish at Mopra, in order to form a very long baseline interferometry array. The lay-out of the antenna stations is an important design characteristic of a synthesis array. A range of antenna anay configurations are needed to cope with the trade-offs that different observations require between brightness sensitivity, resolution and observing speed. For an array with a modest number of antennas, complementary configurations are needed to fill the u-v plane

Figure 3(b). The Australia Telescope Compact Anay (ATCA), at the Narrabri Observatory, is an array of six

22-m antennas used for radio astronomy. It is located about 25 km west of the town of Narrabri in rural NSW

(about 500 km north-west of Sydney). It is operated by the Australia Telescope National Facility, a division of CSIRO. (Courtesy of the National Radio Astronomy Observatory - htç://www.vla.nrao.edu)

53 3.3 Radio Synthesis Imaging

The key to radio interferometry is to transform the observed visibilities (which consist of measured interference pattern amplitudes and phases for every baseline as a function of time) into a flux-calibrated map of the source with absolute celestial coordinates. This can be illustrated, by defining a coordinate system (u, v, w), in which the various baselines of the interferometer move with the Earth's rotation, as seen from the observed source in the (ct, ô) system. The w component (defined to point in the direction of the source) is a measure of the non-flatness of the aïray. Therefore, in the cases of multiple telescopes such as the VLA or Australia Telescopq w can be mostly ignored and u and v are measured in units of observing wavelength. Conversions betweenthe (u, v) and (ct, ð) systems are carried out by Fourier transforms. Correlated data from each pair of antennas are taken and plotted as intensity points in the uv plane (Fourier domain). These can then be inverse Fourier transformed to the image domain to produce a radio image. Fourier analysis permits us to investigate information in a domain other than that in which we actually observe the data. No information need be lost when changing from the spatial to the frequency domain. Each baseline of an interferometer can be thought of as defining a point in the uv-plane, or aperture plane. As the Earth rotates, these points follow elliptical paths, called uv îacks The pattern formed by all baselines is the uv coverage, which determines the scales of any observable structure. It is thus crucial that the (u, v) plane be evenly sampled, with as few gaps as possible, so that an accurate representation of the source being observed can be recovered, (Wemer, 2002).

In the following chapter, we will discuss Australia Telescope observations of the cluster A3661. The angular frequency data (relating to Fourier amplitude) versus antenna baseline measurements obtained in these observations, (Johnston-Hollitt, 2003 study) will be used to investigate the magnetic turbulence scales in the cluster.

54 CHAPTER 4 Estimation of the Scale of the Turbulent Magnetic Field in Galaxy Clusters

4.1 Introduction

It is not expected that intergalactic magnetic fields would be purely regular and they are usually thought of as having a significant turbulent component, with scales up to the order of tens or hundreds of kiloparsecs. Therefore, it is usual to picture the fields as having turbulence spatial spectra, usually assumed to be Kolmogorov in form, so that the smaller scale length components have lower associated field strengths (Lampard et al., 1997). We will find that the dominant scattering scale for cosmic ray particles will tend to be the largest scale under consideration, although the total magnetic field strength at any point will depend on the distribution of strengths among the various contributing scales. Therefore, when calculating propagation paths for charged cosmic ray particles in galactic and intergalactic space, we will assume that there is a turbulent component to the magnetic field. We will also assume that the turbulence has an associated distribution of scales, which follow a Kolmogorov spectrum. We shall see later, that the detailed distribution of magnetic turbulence scales is not critical, provided that the turbulence is dominated by the larger scales. However, I had an opportunity, through access to radio astronomical data, to check the plausibility that the Kolmogorov spectrum is followed in at least one astrophysical environment, the turbulent magnetic field of the galaxy cluster A3667 . Those data will now be discussed.

55 4.2 Galaxy Clusters and the L3667 Cluster

'We saw that in order to estimate the magnetic field strength along the line of sight of a galaxy cluster, using measured rotation measures (RMs), certain information needs to be available. One of these is the electron density, n".and its distributionalong the line of sight. As mentioned earlier, an independent estimate of the overall ne) can be obtained from X-ray data. A large fraction of the mass of clusters of galaxies is in the form of an X-ray emitting gas in the temperature range 107 - 108 K. The primary X-ray emission from the intracluster medium (ICM) is thermal bremsstrahlung and line radiation, due to this hot diffuse gas. It is assumed that the X-ray emission through the bremsstrahlung process works well to trace the cluster potential and therefore such X-ray image data can be used to obtain an independent estimate of n".

As mentioned earlier, a lot of the radio emissions, from distant objects are generated by the s¡mchrotron process through the interaction of relativistic electrons with magnetic fields. Measurement of this Faraday rotation can give information on parameters of the field and plasma density which are unobtainable by other means. As discussed, observing a background radio source through a galaxy cluster at different wavelengths and measuring the position angle of the polarisation as a function of wavelength allows one to determine the rotation measure. Then, by considering various RMs for distant sources behind a galaxy cluster, which also has measurements of X-ray emission, it is possible to obtain an estimate of the integrated field along the line of sight at various locations in the cluster (Johnston- Hollitt,2003).

Maps of the Faraday rotation measure provide a window through which one can glimpse the turbulent magnetized intracluster medium. However, they give only a projected and partial view of the cluster magnetic field configuration. Therefore statistical methods have to be used to decipher the Faraday signal in terms of magnetic field properties. The magnetic field strength derived from the dispersion of rotational measure values depends, not only on the geometrical factors of the cluster, but critically, on the magnetic autocorrelation length. The magnetic autocorrelation length is not identical to and is usually shorter than, the autocorrelation length of the rotation measure fluctuations. However, both of these can be

measured from such maps with the right analysis method, (EnBlin, 2003).

56 4.2.1 Radio Interferometer - Amplitude vs Baseline Length

When observations are made with a radio interferometer, an interference pattern is observed which has a phase and amplitude which depend on the instantaneous orientation and magnitude of the interferometer baseline, and on the angular components of the source brightness distribution. Thus, the amplitude and phase of the interference pattern, which is produced by a source when observed with a given antenna spacing, is related to the radio brightness distribution across the source by a Fourier transform. A single measurement with a single pair of antennas in an interferometer gives a single datum to be placed in the so called uv-plane, which is the Fourier-transform plane of the angular distribution of the source on the sky. Correlated data from the interferometer antennas, with different orientations and baselines relative to the source, are taken and recorded as intensity and phase points in the uv-plane (Fourier domain). In simplified terms, these can then be inverse Fourier transformed back to the image domain to produce a radio image. Therefore, the source brightness distribution may be obtained by Fourier s5mthesis, provided the amplitudes and phase angles of the interference pattern are known at a sufficient number of spacings. The Fourier synthesis is described by:

i.e. B, =i o,cos(a, +at)do (east-west axis) (4.1) 0 where

Bt - brightness distribution in right ascension (arbitrary units) rrl - pulsatance of the sinusoidal interference pattern corresponding to the antenna spacing A, - amplitude of the interference pattern 0, - phase of pattern

If the distribution is symmetrical, 00, is zero and the equivalent syrnmetrical distribution can be obtained. This is the cosine transform of the amplitude spectrum.

57 4.2.2 Amplitude vs Baseline plot of 43667

The data used for the following section came from a study conducted by Johnston-Hollitt, 2003. The observations were taken at the Australia Telescope Compact Array (ATCA), at the Narrabri Observatory, an array of six22-m diameter antennas. These data were obtained from observations carried out during the period Feb 1999 - Nov 2000, of the A3667 cluster at 1.4 GHz using a 6 km configuration for the array. The A3667 is a unique cluster, with two so-called radio 'relics'. These are regions of diffuse radio emission observed on the edge of galaxy clusters. The emission is due to synchrotron emission from high energy electrons within the cluster magnetic fields. S¡mchrotron emission intensity is proportional to the magnetic field energy density and so the radio image structure reflects the distribution of magnetic field over the cluster. The data in this analysis refer to approximately one third of the cluster area.

The physical picture of this cluster is constantly evolving as more multi-wavelength data are obtained and therefore, a lot of the history surrounding its dynamics remains a mystery. However, it is in arare position of having two diffuse radio emission regions and it is a rich, X-ray luminous, southern galaxy cluster, (Johnston-Hollitt, 2003). Using the data obtained, the total polanzation and position angle of the brightest part in the source is calculated as part of the work done by this study.

Each dish at the ATCA can be combined as a pair with every other dish to maximise the number of spacings between them. This maximises the size-scales in the source, about which the interferometer can obtain information. The rotation of the Earth under the source, changes the distances from the source to each dish. Therefore, data can be obtained over time, along two elliptical arcs in the u-v plane. More antennas give more elliptical arcs, often crossing, but unless the antennas actually touch or overlap, there will always be gaps containing no data in the u-v plane. The corresponding syrthesized antenna beam has side- lobes because of the missing u-v data. There are helpful data-reduction techniques that amount to model fitting or to interpolating or extrapolating across these gaps. The interference pattern generated by each pair, as their signals are combined is fed to a computer device called a correlator, which electronically merges these multiple patterns, (MIT Haystack Observatory,2005). A result of this correlation process, is that a power 'When spectrum in terms of u-v become accessible. using this telescope, D is of the order of

58 the largest antenna spacing, 6 km. The radio image of 43667 will have a resolution of order l"/D and magnetic cloud scale sizes smaller than the beam will be subject to blurring and averaging out. Below is the representation of the power spectrum for different spatial scales. The graph shows the amplitude vs baseline plot of the A3667 galaxy cluster, which relates to the power spectrum for different angular scales. This then relates to the amplitudes of the various turbulence scales present in the image.

I a366?-02.1316 7.4?4O GHz

O

rþ OC; E

Ê <{ c;

N o

o

OA 0 10 1ã 20

1rr2 + .,2¡tl2 ¡k).¡

Figure 4(a) Amplitude vs baseline plot of 43667 _02.1376b at 1.4240 GHz, (Johnston-Hollitt, 2003)

In the plot, u and v represent the antenna spacing in orthogonal directions, measured in units of observing wavelength. The coordinate system, in which the baselines move, is represented by (r, v,w), where w points in the direction of the source.

59 4.3 Procedure

4.3.1 Modified 43667 Plot

Data giving the correlated amplitudes for A3667 were only available to me in the raw form of the plot of figure 4(a). I did not have access to the raw data for the individual measurements. These were needed in order to cary out the necessary calculations, which would provide me with the information on the structure of the magnetic field scales in the source. In order to obtain data points for analysis from the A3667 plot, a digitisation technique was used to isolate approximately 1500 points from the original data and obtain their coordinates. Below is the result of the modified Amplitude vs. Spacing plot, in which the selected data points are displayed.

Modified 43667 Plot

1.2

1 o E O.B ä o6 E 0.4 0.2 0 0 10 20 30 Baseline (kI)

Figure 4(b) Modihed amplitude vs baseline plot, obtained from the data provided in (Figure 4(a)).

60 4.3.2 Analysis of the Modified Spectrum

The spectrum of amplitudes versus antenna baseline relates to the Fourier components making up the image of the source in terms of angular scales of (wavelength/baseline). Figures 4(a,b) clearly show that the largest image scales (inversely related to the antenna baselines) are associated with the greatest amplitudes (as one expects for a Kolmogorov picture) and there is a continuous distribution of amplitudes down to small image scales, which we will see also supports the idea that Kolmogorov turbulence is present. Thus, the next step in the process was to examine the form of the spectrum in more detail through obtaining a logarithmic plot of the spectrum. This is because a spectrum of the Kolmogorov form is a power law distribution relating the physical scales of the turbulence to energy associated with those turbulence scales. In that case, a power law form would also apply to an associated (wavelength/baseline) spectrum.

4.3.3 Results

Below is the logarithmic plot of the spectrum, showing the logarithm of the amplitude versus the logarithm of the spacing, followed by a replot of the data in a form suitable for deriving a Kolmogorov type of spectrum. The negative spectral index (m) of the Kolmogorov spectrum (energy versus wave number) is expected to be 5/3.

6l log Amplitude vs log spacing m =- 1.66

o T' a J a a a =q a a I a a a o E a o) -20 0 o 0 -2.5 aa a ô -

l5 logu=logspacing

Modified 43667 Log Plot

1 1 2 (\ t aa O o) t E ¡" aa .Ë o o o. a O E a

o) a a o a J

a aoa -6 Log(1/Spacing) y= 1.3017x-1.6366

Figure 4(c): Graphs representiug the logarithrns of the amplitude functious vs antetna spacing for Australia Telescope observations of clustel A3667.

After the above data were analysed, the resulting straight line fit through the data points produced the result of m - -513. This resulting spectrun, was thus consistent with Kolmogorov-like spectra, confirming the above predictions for the cluster magnetic field. We will see later that, for the purposes of cosmic ray propagation measuretnents, the

62 important consideration is that there is a domination of the spectrum by large turbulence scales. It is still interesting that turbulence data do support a model with Kolmogorov structure.

63 CHAPTER 5 Modelling of Cosmic Ray Propagation in a Magnetic Field

5.1 Previous Studies in Propagation Simulations

This section will discuss some of the past approaches to the simulation of cosmic ray propagation through astrophysical magnetic fields. Cosmic ray propagation modelling is essentially a simulation of the diffusion (see Appendix A) of charged particles in a turbulent magnetic field. In the past, several methods to approximate the complicated diffusion physics have been employed and will be discussed here. They mainly refer to specific ranges for the ratio of particle gyroradius to magnetic turbulence scale'

Onc such method assumes that cosmic rays travel in a uniform magnetic field and are scattered by scattering centres (hard point scattering) with an isotropic angular distribution, (Gleeson and Axford,1967). In this method, the turbulence of the magnetic field is replaced by the scattering centres and results are expressed as functions of the scattering mean free path.

Another method, used by Jokipii and Parker (1969), stresses the stochastic nature of the magnetic field itself. In this approximation, a low energy cosmic ray cannot travel along the direction perpendicular to the magnetic field beyond the gyroradius. If there are not many scattering centres, the propagation process may be considered as one-dimensional diffusion along the field lines. However, the magnetic field lines themselves, "walk" randomly in the turbulent magnetic field causing the cosmic rays to spread into 3-D space. In this case, the diffusion coefficient in the direction perpendicular to the magnetic field, should be proportional to the one parallel to it. Cosmic ray propagation thus depends very much on the features of the Galactic and intergalactic magnetic field, such as the strength of the average and irregular field and the irregularity scale. Recent estimations of magnetic field strengths of specific galaxy clusters can be found in Clarke, et al, (2000) and more general estimations in Beck, et al, (1997).

When the turbulent field strength becomes large enough that the cosmic ray gyroradii are comparable or smaller than the scale of the field irregularity, the motion of the cosmic rays

64 changes to one dimensional propagation along a'diffusive' field. The effects of the turbulent magnetic field on the cosmic ray motion perpendicular to the average field tend to cancel each other, if they are integrated along the gyro-rotation orbit.

The propagation of cosmic rays, as studied by Jokipii and Parker (1969) and Gleeson and Axford (1967), was taken in two extremes. Gleeson and Axfords' method applies to a large gyroradius where the process can be described by the scattering centres and mean free path, while the treatment of Jokipii and Parker is applicable in the limit of a small gyroradius.

Another study presented by Honda, (1987), considers the case where the galactic gyroradius and the scale of the irregularity of the magnetic field are comparable. For typical field strengths of < 1 ¡rG, this gyroradius is in the range - 3-300 pc for cosmic rays in the 1016- 1018 eV region. In this region, there is no simple way to study the propagation analytically, therefore, a numerical computer simulation is used. The irregularity of the galactic magnetic freld is simulated and the cosmic ray motion is followed, starting from a single source. The diffusion tensor is calculated in the simulation. Using the fact that the diffusion tensor K has three independent components, (K,,,Ka,Kr) as shown by Gleeson and Axford,1967, the analytic form as a function of mean free path is given by,

(s. 1)

(s.2)

(s.3 )

Here Ç¡ is the parallel component of K, Kr is the perpendicular component in the plane and Kr is the component perpendicular to both.

)" is the mS (mean free path), co is the gyrofrequency and c is the speed of light. Because

Honda does not use the idea of point scattering of cosmic rays, the mfo is not used at the

65 beginning of the study. Instead it is derived as the result of the diffusion process. In the simulation the two diagonal elements Kl1and K1 are calculated. Consistency is checked, by comparing these formulae and if it's maintained, the mfp is calculated from the formula.

In Honda's method, the turbulent magnetic field is generated, conserving the flux by maintaining the relation,

div B:0

In cases where the gyroradius is smaller than the dominant scale of the irregularity (L0), scales of irregularity smaller than L6 can be important. The field strength of the smaller %, scale irregularities is assumed to follow the Kolmogorov spectrum - k (Honda, IgSl).

In this simulation, the magnetic field is assumed to be static. This means that both the effects of convection and Fermi type acceleration (1't order) are neglected. Any electric field is also neglected, assuming B.E : 0. This is the same as the assumption that the magnetic field is static.

5.2 Generation of Random Magnetic Fields

There is a difficulty in the simulation with irregular magnetic fields of making the flux lines conserved. This diffrculty can be avoided by generating the 'vector potentials' by random numbers. To generate the random magnetic fields, the following steps are used by Honda.

First, a lattice is embedded in the simulation space, whose constant is chosen as the scale of

irregularity of the É - net¿. Secondly, the vector potential A : (A*, Ay, A,) is generated for

each lattice point, with the probability given by, exp(-A¡lt) dA; (i:x,y,z) (5.4) t - parameter to adjust the irregular É field strength. Thirdly, the vector potential at points other than the lattice points, are given by linear interpolation. And finally, the irregular magnetic fields are given by, ôB : rot(Ã) (5.5)

and the total fields are given as the sum of the irregular magnetic field and the average É - field (8").

66 In order to simulate the turbulent B-fields with some spectrum like the Kolmogorov one, a series of lattices with different lattice constants is prepared. And the irregular fields are summed up with the weight given by the spectrum. The motion of the cosmic rays is determined by the equation of motion, rf n üxB) 4=q([*dt rr or ú :e(E+vxB) (s.6) dt p7 is used for the parallel component of p and p1 for the perpendicular component in the plane determined by B and P and p1 for the component perpendicular to both.

When the magnetic field is uniform, the momentum components ( p'tt, p't, p'r) aftr.r a time At are given by,

p't: ptcos(a;Ar)- prsin(r't\t) 1s.za¡

p'¡ : p tsin(arAr)+ p, cos(aÀt) 1s.zu¡ p'tt= ptt $.7c) r eBc (s.8) Q):-:- ERs

E - energy R, - gyroradius

e - charge

Equations (5.7), are used to follow the direction of the cosmic ray, but for B, the value

averaged along the line of path of the cosmic ray is taken.

The step time (At) for solving (5.6) was chosen to be At : 1/l0ol. This was chosen so that

the error caused by it would be small in the numerical calculations and would also save computer time. In all cases (cAt) is kept smaller than the scale of irregularity, (Honda,

I 987).

67 5.3 Simulation of a Turbulent Magnetic Field

Let us now use the ideas from the previous section to simulate a turbulent magnetic field. We will consider a cubic lattice embedded in space. The lattice has a length scale L0, equal to the largest length scale of turbulence assumed. At each point k within the lattice, values for (A*k, Ayk, Ark; the Cartesian components of the Vector Potential are sampled from a probability distribution of the form,

p(Aik) : rl¡n1"*p (-e,u/¡a¡ ) (s.e)

The turbulent magnetic field can be generated with. V.8,,, : 0 . This is done, by finding a suitable form of the vector potential Ã,o,, , defining

X (5.10) É tail v A ratl

Now lAl is chosen such that, the magnitude of the random magnetic field 8,,,, is as required.

The cosmic ray at any point in time will be located in a particular cube of length scale Ls. To find É,,,, , the curl of the vector potential must be found. In cartesian coordinates this is found using the formula,

ü'Ã,o,,=(uo/ur-uo,Á,)0.(uo/u,-u"'/u,)r.(u"'Á,-u"/*1u (s.11)

Many different length scales are associated with turbulence. A length scale of turbulence can be considered as a wavelength over which fluctuations in field strength occur. To treat the transfer of energy from the largest length scale of turbulence L¡, in the magnetic field, to the

smaller scales, an approximation first stated by Kolmogorov (1961) is used (Lee,7993). As we already discussed, Kolmogorov stated that the turbulence in an incompressible fluid

should exhibit a dissipationless energy cascade, from the large scale of turbulence, Ls, where the energy is injected, to a length scale, /¡, where energy is dissipated due to viscous losses.

68 The result of this is that we find an energy spectrum E(k) of the turbulence which has the form: E(k) æ k-s/3 (5.12)

Thus the enorgy spectrum is a power spectrum with an index of -513, where k is the wavenumber. In the simulation, the vector potentials corresponding to the various turbulence scales are added to ensure that this requirement is satisfied.

69 CHAPTER 6

Processes Involved in Generating Simulated Turbulent Magnetic Fields and a Study of the Effects of Field Scale Variations

6.1 Introduction

Cosmic ray propagation depends very much on the features of the galactic and intergalactic magnetic fields. These include such factors as the strength of the regular and irregular field components, the detailed structure of the regular field, and the scales of any irregularities. To determine the extent to which such features affect the propagation, we need to be able to use models to simulate such fields, with variable turbulence scale lengths, and to estimate the effects of varying their scales and strengths. This chapter will first examine the integrated magnetic field strength along a line of sight, such as would be observed in studies of rotation measures of distant sources, which are observed through a magnetised plasma. Such observations actually measure the integral of the product of the electron plasma density and the line of sight field component. Later, it will consider the addition of clumpiness into the integral, although the integral of just the freld component will be considered in the first instance.

These line of sight observations through turbulent fields will integrate to small values, and this work is intended to examine the extent to which a simple interpretation of rotation measure observations can identify correct field parameters. That is, to what extent various field strength components and scales in the field structure will average out in a predictable way. In chapter 7, the study will continue to examine the effect of turbulence scales on cosmic ray propagation through magnetic fields, in order to determine whichever range of

scale lengths (compared to a simple gyroradius) in a Kolmogorov spectrum are important.

A simple assumption, when examining statistical data, is to expect that an integration process will increase data fluctuations only in proportion to the square root of the number of independent samples accumulated. The total integral of the data set will increase directly with that number, such that the relative fluctuation will then decrease with the square root of the number of samples. Observations of rotation measures provide averages over the path length of the radio waves through a magnetised plasma, such that many small-scale cycles

70 are covered and many fewer large-scale cycles. The final results will then depend in different ways on the averaging of the different scale length components in any random field. The study wished to examine the conditions within which such averaging applies to the integrated magnetic field along a line of sight and, later, how it applies to propagation properties.

6.2 Random Magnetic Field Component

A probability density function seryes to represent an ideal probability distribution. A probability density function can be seen as a "smoothed out" version of a histogram. If one measures values of a random variable repeatedly and produces a histogram depicting relative frequencies of output ranges, then this histogram will resemble the random variable's prob ability density.

In these simulations, parameters of the field and the resulting cosmic ray propagation were recorded multiple times in order to build up a probability distributions for different physical situations. Below is an examination of some probability distributions through the accumulation of histograms for integrations of field strength through magnetic field structures.

6.3 Results of Scale Length Variation

Firstly, the case of integrating the line of sight component of a magnetic field through a region of turbulent magnetic field is examined. The probability distribution of the magnitude of the integrated field along a one dimensional path through a large-scale three dimensional random fìeld was estimated. The expectation was that, for some field models, this integration would result in an integrated line of sight field strength which increased with the square root of the distance. The first step was to generate a random magnetic field with predetermined turbulence properties. Using a simulation program based on the ideas in

chapter 5, three dimensional random magnetic fields were generated, and their components were determined along a straight line path, chosen to be the x-direction. This produced a progression of accumulated random components, which were added and recorded as the

7I integrated magnetic field in the direction of travel. The integral of B* could then be determined as a function of distance, D (where D is in parsecs) travelled. The program was executed through a number of steps, with 1000 steps being equivalent to 1 pc. Results were found for various random fields to show the integrated B*-components as a function of distance (in pc) in each case. This enabled one to examine the statistical distributions for the integrated field strength.

6.3.1 Properties of the Magnetic Field Model

Many different scale lengths are associated with a turbulent magnetic field. These are usually described by a probability distribution, such as a Kolmogorov spectrum. For convenience, a length scale of turbulence can be considered as a wavelength, or cycle, over which fluctuations in field strength occur. In this simulation, there are two extreme ways of changing the distance through which the magnetic field is integrated: either by changing the number of smallest, or largest, cycles. Embedded in the magnetic field modelling program, are five different loops, which represent different scale lengths of the B-field. As will be seen below, these correspond to scales of lpc, 3pc, 9pc, 27pc and 81pc. The largest scale is roughly the size of super- bubbles, typical ofthe larger structures found in our galactic plane. Super-bubbles are large (-100 pc across) shells in the interstellarmedium createdbythe combined action of stellar winds and supernova explosions of massive stars, (Dunrte, et al., 2001). By examining the effects of the various loops (of various scale sizes compared to the particle gyroradius when propagation is considered) one can then (see chapter 7) examine the effect of different turbulence scale lengths on the propagation.

The successive loops in the progtam correspond to different distances through space and are used to make up the spectrum of scales in the Kolmogorov turbulence. These are used to build up the turbulent magnetic field by appropriate weighting of their magnetic field contributions. The loops contain numbers of increments of distance which then define the physical size scales involved. If the number of increments within a certain loop is changed, then the physical distance over which the magnetic field acts is changed. If we label the various program loops (representing the smaller turbulence scales) by ti, the various i represent the variations within the corresponding scale length (which is labelled t for the

smallest and T for the largest) of the magnetic field, (t¡-t5 representing the smallest to largest

72 number of cycles which will be used to study the effect of changes within the smallest turbulence scales). By setting the number of increments of distance, which make up a particular cycle, one can vary how the magnetic field is made up from the dìfferent scale lengths. Changing the number of increments within a program loop has the effect of changing how quickly the magnetic field changes.

Each time the original program completed a full cycle, it would run through 81000 steps of

0.001 pc length, a total of 81 pc for this model. The overall dimensions covered by the field could be extended (such as extending it to fill a complete 'galaxy') by continuing with further full size cycles such as that which was just described above. For instance, the data displayed in figure 6(a) (see below) was made up by completing 10,000 of the elemental cycles just described, a total path of 810 kpc.

As described above, the aim at this stage was to find the integrated field strength along a line of sight through the field (taken as the x direction). This involved taking the components B" at each incremental position (separated by 0.001 pc) and progressively adding them through a long path in the turbulent field. The question to be asked is whether the integrated value increases in a physically understandable way, given the statistical properties of the turbulence. Initially, that is whether it increases as the square root of some characteristic distance. If the magnetic field has an underlying strength of 10 ¡rG, we might expect a resulting integrated strength to increase with the root of the distance, i.e. the resulting integral would be a weighting constant of 10 ¡rG times the square root of some characteristic number of steps or cycles in units of microgauss-steps. This result might then be used for instance, to interpretFaraday rotation data, which results from a weighted (by the plasma density) average of the line of sight freld component.

As described above, the smallest scale size used was determined by the number of the smallest incremental steps. Initially this number was one thousand 0.001pc steps. The next scale up was three times larger. Larger scales were built up by adding cycles after each preceding cycle had completed three iterations. At any given time, a weight (or field strength component) could be given to each scale, and those field strengths added to make up the actual field at any point. The lowest scale changed rapidly with position but, as with most turbulence models, made the smallest contribution to the total field, whilst the higher turbulence scales were updated after step numbers which were larger in size. After this

13 process, the whole program repeated to cover any required total field dimension. Therefore, each time the basic program completed a single iteration, it again looped through 81000 increments to complete a further field iteration covering all scale lengths. The fundamental of distance in the turbulence field was then equivalent to a distance of 81pc for this basic model.

The plot below, figure 6(a) represents the integrated line of sight magnetic field at increasing distances along various paths through a turbulent magnetic field with a Kolmogorov scale size of 81 pc.

'l*106

o E À Ê E- 5x1 0'

Ê q¡^ ocË q)!

l=q :c¡ @= =cË 54) õts os !61ñØ -5x1 cr OÉi, :d

-1x106 0 2xl 4x1 6x1 1x109

Total path through the turbulent field (arbitrary units)

Figure 6(a) The value ofthe integrated B, against the total distance coveredby the integration for a number of paths through a magnetic field. The distance integrated through field is in units of

program steps (1000 steps : lpc), giving a maximum distance of 810 kpc. Many integrations are

shown.

The number of steps in the lowest (and smallest) level of the cycles could be varied in order to increase the number of steps after which the 'B*-total' was obtained, for plotting as in hgure 6(a). We might expect a resultant integrated strength along a path to be of the order of

74 l0pGx 8x108 microgauss-steps. This is then a number which could be used, for instance, to interpret Faraday rotation data, which have results depending on an integral of the magnetic field (multiplied by the plasma electron density) along the line of sight.

This program was executed to build up characteristic statistical distributions of the integrated magnetic field strength. Having based this work on the random walk problem, we rwere looking for a distribution of integrated field strengths, which would have a mean close to zero (the fields were random and so should have no preferred forwards or away direction) and would have a spread, which increased with the square root of the distance. The standard deviation and mean of histograms of that spread, at various total path lengths, were calculated and, thus, the spread of the distributions estimated. In all cases, the mean of the distributions were equal to zero, within statistical limits.

Below are some examples of the developing data for different numbers of steps included in the smallesl scales of the turbulent field. Later it will proceed to observe how changes made to larger scales affect the distribution of B* integrals of the field. In the program, changing the number of steps in the smallest turbulent scales has the effect of changing the physical lengths of any larger cycles but not their total number since each large cycle now contains a changed number of (small) steps. As a result, this process investigates the effect on the integration of changing the number of small-scale cycles. This integration would, in observational terms, relate to the observed rotation measure'

75 3x1 06

é, 2x1 0c o t^

1x106 o6l OE bD ,: éü9 Ëo 0 2o oão -êÉãc€ ãq, o. -111tÊ !G|õts L€ø b¡R :EO -2K1 UË oy

-jx1 tE o 1 000 2000 3000 40Ðo 50uo

Total path through the turbulent field (arbitrary units) Figure 6(b): The value of the integrated B* against the total distance covered by the integration. The distance through the smallest turbulence scale is labelled as t3 where this corresponds to 50 cycles within the smallest scale length loop. In this instance we are still observing the effects of a slight increase (double the previous case) of the number of steps carried out in the smallest scale turbulent cell.

8x1 06 É ê¡ 6r1 û6 õ,ËEa

4x1 0E ocq o€ ã¡ .9 trO 2x106 !(¡

ã() 0 €q !6t 6ø L ú6 -2x] 06 .=áEàn oy -4Kl 06

-6x1 0Ë o toou 4000 6000 EOU0 'l oooo

Total path through the turbulent field (arbitrary units) Figure 6(c ): The value of the integrated B* against the total distance coveredby the integration. The distance through the smallest turbulence scale is labelled as t3 where this corresponds to 100 cycles within the smallest scale length loop.

76 Figures 6(b)-6(c) each show examples of the integrated magnetic field along the line of sight along multiple paths through other randomly generated magnetic fields. In each of these particular cases, path length changes correspond to the changes in the loop representing the smallest scale lengths which were chosen as 50 and 100 cycles. The integrated field may be positive or negative since the field itself has randomly chosen directions and amplitudes, but the spread has a characteristic distribution at a particular distance. The mean integrated field over all the paths is close Io zero as expected. It is the spread which is currently of interest, and this can be described by the standard deviation of the values of the final integrated line of sight field. This is given in column 4 of table 6(a) (below) which presents the results of these various integrated field strength simulations. These are generated by the variation of the t¡ (number of steps in the smallest turbulence scale) value as shown below.

ti Number of Number of Standard Ratio of standard smallest smallest scale Deviation (pG- deviations for a

scale cycles steps) total path length

cycles making up a within the complete smallest turbulence cycle loop cycle

t¡ 13 1053 3.17x105 2.08 t2 25 202s 6.59x10'

t¡ 50 4050 1 .1 5x1 0o t.75 2.r7

t+ 100 8100 2.49x10o 2.13

ts 200 16200 5.3x10o

Table 6 (a) Integrated field variations through the various t¡ iterations (see text)

77 An examination of the above table shows that doubling the number of steps in the program, (by manipulating the small-scale loop, t¡) doubles the spread, and does not produce the anticipated {n spread in variance for the integrated random magnetic field. Instead, the spread increases in proportion to the total path length. This is the result to be expected if the averaging process of the smallest scale components of the magnetic field is not important in contributing to the integrated line of sight field value. If we change the number of smallest cycles, (13, or 25 or 50 or 100 or 200) and keep the same number of larger cycles, the total number of cycles (the smallest increment) is (that number) x81x2x500x100, which corresponds to a distance of (that number)*81kpc. The result is that the integrated value depends proportionally on the total distance integrated over (and this contains a fixed number of large scale cycles). this is despite the fact that the small scale variations of the magnetic fields cancel out pretty well. The total effect of the number of large scales is the same in this model, we are just increasing the lengths of the large scales.

The increase in the number of steps simply extends the lengths of the larger scales in the iterations. The number of those iterations does not change, but their integrated weight simply increases with the total length observed. Hence, there is no obvious effect on the observed rotation measures from the small scales of the magnetic field distribution. Thus, rotation measures do not provide significant information on the smallest turbulence scales. The effect of scale fluctuation is usually incorporated into the total RM values, however, they cannot always be detected accurately and therefore, it is difficult to estimate at what scale- level, the rotation measures become significantly effected by the magnetic field structure.

Next, we again show the distribution of the integrated line of sight field through a turbulent magnetic field for a distance of 810 kpc. However, in the following cases, the data correspond to varying iterations which represent magnetic fìeld modifications within the larsest turbulent scale. The varying field scales are symbolised by the T¡ values shown in the table 6(b). As in the smallest scale-length case, here the various i represent the variations of the corresponding scale length (in this instance, the largest) of the magnetic freld, (Tr-Ts) representing the smallest to largest number of cycles within the largest turbulence scales, which increments by a factor of two. Each consecutive histogram shows the result of doubling the number of iterations from those of the previous one. Therefore we are observing the effects of the systematic increase in the path length of the turbulent magnetic field through the manipulation of the held structure with the largest turbulence scale.

78 In generating the histograms, the program was executed 100 times for each iteration value. Once a statistically appropriate number of the different data plots was produced (for the various iteration values of a particular loop), histograms of the results were generated and the variance in the spread was calculated. The plots below of the histograms show how the spread distribution (standard deviation) changes with the change in the number of cycles within the program representing the scales of the turbulent magnetic freld. Their parameters are summarizedin table 6(b).

?ô

t5

tü

,5

Ð - I .Ox1 -5,oKl 0 5.8H1 1.OK1 1.5t{1 oG Integrated magnetic field component in x-direction (arbitrary units)

Figure 6(d) Histogranr ofthe integrated values ofBx for straight line paths through nragnetic fields with Kohnogorov turbulence. The distance through the largest scale is Tl as discussed in the text.

79 3ô

4E

20

t5

IO

R

o -lx'l -oK1 Þ 5¡cl 1k10Ê

Integrated magnetic held component in x-direction (arbitrary units)

Figure 6(e) Histogram of the integrated values of B^ for straight line paths through magnetic fields with Kolmogorov turbulence. The distance through the next consecutive large scale is T2 as discussed in the text.

,'¡\

15

lo

5

0 - Lflx1 -Þ.oKl 5,Ei{1 'l .5x106

Integrated magnetic field component in x-direction (arbitrary units)

Figure 6(f) Histogram of the integrated values of B* for straight line paths through magnetic fields with Kolmogorov turbulence. The distance through the next consecutive large scale is T3 as discussed in the text.

80 t5

lo

5'

0 -'l Kl Þ lKl zxloË

Integrated magnetic field component in x-diroction (arbitrary units)

Figure 6(9) Histogram of the integrated values of B* for straight line paths through magnetic fÏelds with Kolmogorov turbulence. The distance through the next consecutive large scale is Ta as discussed in the text.

12

10

E

4

-2x :lx D lKl zkì QË

Integrated magnetic field component in x-direction (arbitrary units)

Figure 6(h) Histogram of the integrated values of B^ for straight line paths through magnetic fields with Kolmogorov turbulence. The distance through the next consecutive large scale is T5 as discussed in the text.

81 Below I have the tabulated results of the data represented in figures 6(d)-6(h)) above.

T¡ Iteration Mean Standard Ratio of standard Deviation Value deviations Tr 25 -2983.37 t77413

1.3

Tz 50 -t474r.9 23t240

1.5

T¡ 100 -19629.5 353209

1.5

T+ 200 -47 561.2 545022

1.1 T5 400 -r5916t 639407

Table 6(b). The mean and standard deviation values obtained for T;

From an examination of these results, we can see that the integrated magnetic field increases only with the size and statistics of the largest structures in the field. In this case the anticipated {n variance in the spread is noticeable in the above variations of the program loops which represent the largest-scale lengths. These variations are represented by the various T¡ (Tr-Ts) representing the smallest to largest variations in the scale turbulences. The results in table 6(b) show that as the large-scale length was doubled (i.e. doubling the number in the larger cycle), the value of the spread distribution increased by of steps - ^lZ. The scale is proportional to the total number of iterations (Ti), which make up the large scale structures, and the statistical variations in the integrated magnetic field depend on the total number of large scales present in the integration. If one changes the number of largest cycles, by adding to the total number of times the whole process is repeated, the change to the integrated field is through the square root of the total number of large scale cycles. It

makes no difference what is happening to the small cycles, those effects just cancel out. Any

apparent effects are simply due to the extension to the path length and are independent of any turbulence effects.

82 6.4 Effects of Clumpiness on the Magnetic Field

The field generated in the previous section was randomly generated and of uniform structure. To better simulate another possible scenario, I wanted to look at the same randomly generated magnetic field, but this time with the inclusion of "clumpiness" within the field's structure. Such clumpiness might be associated with calculating a rotation measure from a magnetic field which is embedded within a clumpy, non uniform, plasma. The integrated field could remain the same, but its distribution in space would no longer be smooth. In the practical case of rotation measures, this integrated product of field strength and plasma density would no longer be smooth. This was achieved by summing up the integrated freld at larger intervals, which were then weighted appropriately to ensure that the increased weight would compensate for the decreased rate of making summations in the integrated quantities. Since the fields were turbulent, this might be expected to change the statistical properties of the total integrations. In this particular case, the summation for the B* integration took place at every fifth step in the various cycles, and that fifth step was weighted by a factor of five to ensure that the integrated value was only changed by the type of magnetic field structure, not by limited sampling of the field. This had the effect of making the field appear clumpy. As stated previously, the various cycles within the program represented the different scale lengths present in a magnetic field. Therefore the aim here was to place the modified steps successively within each of the loops. This allowed me to determine to what effect, if any, there was from the additional (clumpy) steps when the field was integrated, and whether the effect on the resultant field was present at the small scale lengths, when compared to lumpiness being present atlarger scale lengths.

6.4.1 Small-Scale Clumpiness Results

This section examines the results found when clumpiness was introduced within the small scale-lengths in the field. Below (figures 60) - 6(l)) are the histogram representations for the magnetic field with the clumpiness introduced. Here t¡1¡-3¡ represent the variations within the small-scale turbulence in the magnetic field, where (1-3) correspond to the three smallest turbulence scales within which clumpiness was added.

83 20

15

1t

5

0 -1-oKr 06 -s,oKl 05 o s-o*ros l.o*roö r.s*ro6

Integrated magnetic field component in x-direction (arbitrary units)

Figure 6(i) Histogram of the integrated values of B* for straight line paths through magnetic fields with Kolmogorov turbulence. For comparison, this is the original histogram without any added clumpiness. 1tr

20

15

10

5

ô -t.sxt oÊ -1.0x1 -5.0x1 5.ox1os 1.ox1o 1 ,5x10S

Figure 6O Histogram of the integrated values of B" for straight line paths through magnetic f,relds with Kolmogorov turbulence. The distance through the smallest turbulence scale is labelled as t¡¡ where this corresponds to clumpiness being added within the smallest loop of the small-scale turbulent field.

84 1n

15

to

t

0 -1.0x1 -5.0x105 0 5,0x1 1 .Dx1o6 1 ,5x 1 o6

Integrated magnetic field component in x-direction (arbitrary units)

Figure 6(k): Histogram of the integrated values of B* for straight line paths through magnetic fields with Kolmogorov turbulence. The distance through the smallest turbulence scale is labelled as t¡2 where this corresponds to clumpiness being added within the second smallest loop of the small-scale turbulent field.

70

15

1t

5

t -1.0x1ü6 -5.0x1 0 5.0x 1 1.0x1 1 .5x'106

Integrated magnetic field component in x-direction (arbitrary units)

Figure 6(l): The value of the integrated B* against the total distance covered by the integration. The distance through the smallest turbulence scale is labelled as t¡3 where this corresponds to clumpiness being added within the third smallest loop of the small-scale turbulent field.

85 In the above examples, the clumpiness was introduced into a number of the smaller scale- lengths turbulences of the field. This did not cause an)¡ significant chanse in the spread distribution of the field, leading to the conclusion that the form of the clumpiness in the small-scale turbulence scales has no noticeable effect on the resulting measurements of the field over that of a uniform field structure. It is also likely to have no noticeable effect on the propagation of particles within such fields, which have gyoradii larger that the scale of the clumpiness. We will now generate histograms, representing the spread after introducing clumpiness within the large scale-lengths of the field.

6.4.2 Large-Scale Clumpiness Results

Below are the histogram representations for the magnetic field with the clumpiness introduced within the large-scale field. Here T;¡-2 represent the variations within the large scale turbulence in the magnetic field, where (l-2) correspond to the two different clumpiness scales being introduced within the largest scale-length loop. Here, 1 represents a three-fold increase in clumpiness and 2, represents a five-fold increase in the clumpiness within the largest-scale turbulent scale. In terms of the program, this means that an extra 3 then 5 cycles were added within the largest-scale program cycle.

86 20

1s

10

5

0 -1.OKl -s oK1 05 o s-ox105 r -oxr oo r -sK1oõ

Integrated magnetic field component in x-direction (arbitrary units)

Figure 6(m) Histogram of the integrated values of B^ for straight line paths through magnetic frelds with Kolmogorov turbulence. As before, for comparison, this is the original histogram without any added clumpiness.

I

al

4

2

0 -6x1 00 -4x106 -zx106 o ?x1o6 4x'l o Bx1 06

Integrated magnetic field component in x-direction (arbitrary units)

Figure 6(n) Histogram of the integrated values of B^ for straight line paths through magnetic fields with Kolmogorov turbulence. The distance through the largest turbulence scale is labelled as T¡1 where this corïesponds to a three-fold increase in clumpiness within the largest loop.

87 1?

10

8

6

4

?

0 6 1 x10 2xl06 -Jxt 06 -2x1 06 -1x106 o

Integrated magnetic held component in x-direction (arbitrary units)

Figure 6(o) The Histogram of the integrated values of B* for straight line paths through magnetic fields with Kolmogorov turbulence. The distance through the largest turbulence scale is labelled as T;2 where this corresponds to a five-fold increase in clumpiness within the largest loop.

From the above histograms we can see that the introduction of clumpiness at the larger turbulence scales, does show an effect on the field distribution and this can be seen through the increase in the standard deviation produced, which \¡/as not visible in the small-scale case. As clumping may also be associated with the magnetic field stratcture, we may expect to observe significant changes on particle paths within such a field.

6.4.3 Conclusion

The main observation here, is the fact that the clumping of the field at the small-scale lengths (figures 60) - 6(1)) has very little or no effect on the spread of the integrated magnetic field through different paths. This is compatible with the earlier results for the integrated field over changes in length of the small-scale turbulence. It makes no difference how that turbulence is distributed, just the total path length, which it represents. It is only at the large- scale length turbulences (i.e. Kolmogorov scales) that one can see the clumping having a

88 signifìcant effect, figures 6(n)-6(o). This, for example, also relates to our ability to use rotation measures to study small plasma cloud structures. It will later be seen that it also has considerable implications for particle propagation and needs to be included in aîy calculations of path estimations of charged particles. Therefore overall, in relation to cosmic ray propagation, this would imply that small-scale turbulence in the magnetic field has very little, or no effect, on the particle and hence its direction. At the larger turbulence scales, however, the effects becomes quite significant and needs to be taken into consideration when tryrng to determine the origin and travel path of cosmic ray particles, which interact with extragalactic magnetic frelds.

89 CHAPTER 7 Cosmic Ray Propagation Through Turbulent Magnetic Fields

7.1 Computer Simulation of Propagation

The final step in this study was to consider cosmic ray trajectories and how they would be affected by a turbulent magnetic field. This was done in such a way as to simulate the behaviour of cosmic rays within a simple computer simulated galaxy model, in which turbulent magnetic fields with different scales were present. Then, by studying the particle's path and confinement time within the galaxy, I could make some predictions regarding the effects of turbulence scales on the propagation of cosmic rays in a galactic environment. The previous section has discussed turbulent magnetic fields in a general astrophysical context, we are now going to examine the effect of such turbulence in the context of cosmic ray

propagation in a galaxy such as our orwn.

7.2 Procedure

For this part of the propagation program a simple galactic model was chosen. It consisted of a disc of 10,000 parsec radius and of 1000 parsecs thickness. This particular model was chosen in order to simulate a simplified galaxy as a simple box containing only turbulent magnetic fields with a characteristic strength of 1 microgauss and a largest scale size of 81 pc

as discussed above. We will also consider the situation with the same field strength but only small scale turbulence. In this case, no account was taken of the spiral arm structure. Regular fields in spiral arrns are currently believed to be no stronger than the corresponding turbulent field. The aim here, was to obtain results for cosmic ray propagation through both large-scale and small-scale magnetic field turbulence scenarios. This was done over a range of cosmic ray energies and a Kolmogorov turbulence spectrum was used. As we have seen, such a Kolmogorov spectrum has its scales weighted such that the magnetic energies associated with the scales increase with scale to the porwer of 5/3. To achieve this, the magnetic field strength was weighted by the scale-length of the power of 516. In the program there are factors of 3 between the scale sizes so the relative weights of the scales were

3st6:2.5

90 The program monitored the number of times it passed through its internal loops corresponding to the number of cycles of magnetic field traversed by the test particle, before the particle left the galaxy. Thus, the loops represented the number of cycles the cosmic ray completed, whilst it remained within the "galaxy". The following table and figures show the results of these calculations. The 'Number"' value is representative of the number of loops executed in the program, which translates into the number of cycles the cosmic ray propagates through in a turbulent magnetic field for two scenarios: large scale turbulence and small scale turbulence within a disc (analogous to a galaxy). These correspond to cases discussed in the previous chapter.

9T 7.3 Results

In the first scenario, I looked at propagation within the large-scale magnetic field turbulence. To do this, the small-scale turbulence was removed from within the computer program for this part of the simulation.

7.3.1 Results for Propagation through Large Scale Turbulence

Directly below (Table 7(a)) are the results obtained in the large-scale simulation showing how the number of cycles of the cosmic ray within the galaxy, relates to a decrease in the particle's energy. Figure 7 (a) is a graphical representation of this result.

Number" E (eV)

280 4.8x 10r8

280 3.2x 10r8

284 2.3x 10rð

292 1.8*10r8

297 1.4x 10r8

341 I .2x l0ru

380 9.8* 10t7

414 8.3* 10r7

445 7.2*l0t't

Table 7(a) - Results for Large Scales Turbulence, where Number" corresponds to the number of cycles the cosmic ray propagates through before exiting the "galaxy".

92 Large Scale Turbulence

500 450 a 400 a 350 E 300 a a o ËE 250 Es 200 õ9 150 3Eo.= 100 dË 50 Zot 0 0 I 2 3 4 5 6

Energy of Particle (x 10^18 eV)

Figure 7(a) - Plot of the Energy of the cosmic ray vs the number of cycles of containment within the galaxy

Within the large-scale propagation, we can see the turbulence start to take effect as the energies are reduced below 3x10l8eV, where the particle's confinement time within the disc is increased by the effect of the freld. This increase in time is observed through the number of turbulence cycles the particle propagates through before exiting the galaxy disc. Above

3x1018 eV, the test particle leaves the galaxy in a relatively straight path and no confinement is evident. This relates to a confinement time model for the structure of the cosmic ray energy spectrum at the "ankle". The value of the containment term above about 3x1018 eV corresponds closely to the time required for a particle to leave the galaxy in an almost direct

path.

7.3.2 Results for Propagation through Small Scale Turbulence

Below, in Table 7(b), we see the results obtained in the small-scale simulation showing once again how the number of cycles of the cosmic ray within the galaxy, relates to a decrease in the particle's energy when propagating within the small magnetic field turbulence.

93 Number" E (eV)

280 4.8x 10ìE

280 3.2* 10r8

280 2.3* 10tE

280 1.8* 10'ö

280 1 .4x 1 0r8

280 1.2x10rU

280 9.8x10r7

280 8.3x 10'7

28r 7.2x70t7

Table 7(b) - Results for Small Scale Turbulence, where Number" corresponds to the number of cycles the cosmic ray propagates through before exiting the "galaxt'' .

In the case of only small scale turbulence, there is little confinement and, over this range, the energy of the particle has very little effect on propagation. One can only see a very slight effect of any galactic confinement at energies below 7*1017eV, where the particle propagates through a single extra program loop before exiting the galaxy disc.

7.3.3 Comments on the Results

Honda's 1987 study, discussed earlier, showed that in the two cases where the gyroradius, Re

is less than the scale of irregularity, Ls, i.e. Ro < Lo and R6 > Lo the results for within the irregular region differ quite considerably. He simulated the motion of cosmic rays in a turbulent magnetic field also in order to calculate the diffusion tensor. This showed that the diffusion tensor obtained showed a different feature in the region where the gyroradius is smaller then the turbulence scale, compared to the region where it is larger.

V/hat we have observed in this particular study, is that in the case of large-scale turbulence it is clearly seen that the cosmic ray particle experiences confinement at the lower energies.

94 Above an energy of 3x10r8eV, the particle has a low propagation time, which is similar to the case in which there is only small scale turbulence. There is then no effective containment. At energies below 3x10l8eV, the cosmic ray is confined within the disc for a longer period of time and goes through many more loops before exiting the disc. In the case of small-scale turbulence, it can be seen that the turbulence has no effect on the confinement time of the cosmic ray, as the number of loops the particle goes through before exiting the disc does not increase. Therefore, small-scale magnetic field turbulence in this type of scenario does not appear to have any significant effect on the cosmic ray propagation

in a galaxy, where the overall field strengths are the same.

95 CHAPTER 8 Conclusion

In this study, with the aid of computer simulations, I examined the properties and effects of some turbulent magnetic field models, which were needed in order to perform numerical calculations of cosmic ray propagation through extragalactic magnetic fields. In order to do this, I firstly used the computer simulations to obtain estimates of the scale and structure of the turbulent magnetic freld in galaxy clusters.

To study the magnetic field structure of galaxy clusters, I used data, which came from a study conducted by Johnston-Hollitt, 2003. The data came from observations of the galaxy cluster A3667 with observations taken at I.4 GHz. The aim of this process was to examine the observable form of the turbulence-scale spectrum of the cluster magnetic field. Since the spectrum of the Kolmogorov form is a power-law distribution, relating the physical scales of the turbulence to the energy associated with these scales, this can be directly related to the A3667 cluster turbulence spectrum, which is observed as an associated wavelength-baseline spectrum. After the results were analysed, it was found that the resulting A3667 turbulence was consistent with Kolmogorov like spectra with a scale index

of -513, therefore confirming and supporting the use of a Kolmogorov model. Since I now had an estimate of the type of turbulence one can expect in extragalactic magnetic fields, I could take the next step and generate some simulated turbulent magnetic fields of this type and study the effects of field scale variations in these models. This was

done in order to obtain a better understanding of the effects such turbulences could have on the propagations of high energy cosmic rays through extragalactic and galactic space. Therefore, this part of the project consisted of comparing the various scenarios in which the magnetic field scale was altered, with the introduction of clumpiness into the turbulent magnetic field at small and large scale-lengths. What was consequently found, was that the presence of clumpiness in the field in the small-scale lengths, had very little, or no effect on

the measured distribution of the integrated field. Hence, it made no difference as to how the turbulence was distributed, just the total path length, which it represented. At the large-scale turbulences (i.e. Kolmogorov scales) however, one could see that the clumpiness was having a significant effect on the measured integrated field distribution. The scales considered for large-scale turbulence were of the order of the particle's

96 gyroradius as it travels through the galaxy. Therefore, even though small scale turbulence in the magnetic fields may have very little effect on cosmic ray propagation, the results of this simulation, imply that the large-scale turbulences definitely need to be taken into account when determining the direction of travel of cosmic ray particles. Finally, to confirm the above observations, I simulated the propagation of a cosmic ray particle through turbulent magnetic fields, representing both, the small-scale and large-scale turbulent field scenarios. For this part of the propagation simulation, a simple galactic model was chosen. It consisted of a disc of parsec radius and 1000 parsec thickness. I then observed the effect on the particle's confinement within the galaxy, when travelling through the two turbulence models. In the large-scale propagation, it was observed that the particle's confinement within the galaxy was clearly decreased at energies above 3x10l8eV, where the particle started to lose confinement and begun to exit the disc. Below this energy, the cosmic ray remained confined within the galaxy. At the small-scales, the propagation of the particle was not seen to be effected in any way that was significant to the overall propagation.

Based on the results obtained in this thesis, it has been shown that cosmic ray directions are quite significantly depended on the Kolmogorov-type, large-scale turbulent magnetic fields present in extragalactic space and these need to be taken into account when studying the propagation and sources of the cosmic rays reaching the Earth.

Further work could be carried out by applyng this simulation model to areas of known cosmic rays sources. At the moment we are still waiting on some significant data from the Auger Project, which will provide more accurate cosmic ray source directions. Also some improvements in rotation measure studies are needed. This may be made possible with the next generation of radio telescopes. Telescopes, such as the VLA have certain limitations. Depending on the field of study, there may be a bias to objects that are unusually luminous, or unusually nearby, sources that are not too extended, or in a particular red shift range.

These are due to constraints on the VLA design that were accepted early on. Many of these constraints can be greatly relaxed today. The scientific capabilities of the VLA, for example, can be transformed by returning it to the state of the art in sensitivity, frequency coverage, angular and spectral resolution. An expanded VLA could be over one hundred times faster at high frequencies, several times more frequency-agile, and one hundred times better at resolving details at all frequencies, for substantially less than the replacement cost of the instrument. With these advances we will be able to provide more accurate and more

97 popularly agreed upon measurements (there is still some disagreement amongst experts in these fields) of magnetic field strengths in extragalactic sources and get a more accurate picture of the turbulent magnetic field strengths and scales and hence in the accurate determination of cosmic ray propagation through extragalactic space and hence their origin.

98 REF'ERENCES

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EnBlin, Torsten, Vogt, Corina and Pfrommer, Christoph, 2003, Proceedings of The Riddle of Cooling Flows in Galaxies and Clusters of Galaxies: EI4, May 31- June 4, 2003.

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Johnston-Hollitt, Melanie,2003, "Detection of magentic fields and diffuse radio emission in Abell 3667 and other rich southern clusters of galaxies", PhD Thesis, University of Adelaide.

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99 Kronberg, Philipp P., 1994, Rep. Prog. Phys., 57 ,325

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100 BIBLIOGRAPHY

Ahmadi, G., 2003, "Turbulence", Clarkson University lhttp: I I www.clarkson. edu/fluidfl ow/courses/me3 26]

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101 APPBNDIX A

DIFFUSION

Method 1- calculating the mean square displacement z'Q) Diffusion regarded as the random walk problem. If a molecule starts at z:0, the z- component of its position vector, after a total of N displacements is then given by: ,=fË, (1) i=1

Because of the random direction of each displacemeú,|, = 0 so thatZ:0.

On the other hand, we get the dispersion relation j -s q +>>46 e) Tir j

(2) to From statistical independen"", 6,6 , = 46 , = 0 so that reduces "' = N6' (3) The mean square displacement É'per step can readily be computed. The z-component of this displacement in time tis ( =v"t .

Hence, 1'=¡?t' Ø) But, by symmetry vl = /rvz. Now by

(s) ¿*1*lnp

where,

2 1 (kr)' D_ (6) 3J; Poo m

is the diffusion coefficient.

Vy'e get,

¡, :i"-/, =*l[e-,,u,du=2r2 (7) ooo !.r

hence,

t02 -) 2-. t : " (8) -v'1'J

Since each displacement between collisions requires a mean time t, the total number N of displacement occurringin atime t is equal to th. Hence, eq. (3) gives, for the mean-square z-component, 4ù:rl7ot (e)

Method 2

We can also calculate the mean-square displacement, by purely macroscopic reasoning based on the diffusion equation,

9!, = Dq+" (ro) ôt ô22 n1:n1(z) (the mean number of molecules which depends on position) Now we imagine that atotal of N, labelled molecules per unit area are introduced at time t: 0 in an infinitesimally thick slab near z: 0. The molecules then proceed to diffuse.

Conservation of the total number of molecules requires that,

N (1 1) [nrQ,t)dz = -æ at all times.

By definition, one also has 1- z'1t¡: (12) * !_f ",(z,t)dz

To find how z2 depends on t, rve multiply the diffusion equatiotr, og (10) by z2 and integrate oveÍ z. This yields,

T,'%0"=pT"'!* (1 3) * ôt _'- ôz' By (12), the left side gives,

*¡ ôn, r, ¿, = = N+(Ò (14) rôt llr'n,a,otJ ot

The right side of (13) can be simplified by successive integrations by part,

103 ï:#-:lr*),- _-,T;þ*

@ =o-2lrn,]: *z lndz -@ =0+2Nr sincenr ^u(u"/ur)-o as lzl-+æ then,eq. (13)becomes, u (,n)=r, (1 s) ôt or z2 :2Dt (16)

Since z2 = 0 for t : 0, the constant of integration has been set to zerc. By comparing (16) with (9), we obtain o=Lv'î (t7) J

or o =!vt 3

See (Reif, 1965)

to4 APPENDIX B

Programs

c Random magnetic field generation and cosmic ray propagation

real pi integer*4 iseed common/seed/iseed c iseed=!23456779 iseed=0 do 990 iem = 2,10,1

c e/m is em. 1 is for 10*x17eV ie em=lx(10**t7/E) em = 0.1x(0.02 + sqrt(sqft(L4t4216))x*iem -1.0) c print*, iem,em EnergY= 1 '9x 10** 17lem c em= 100.0 aaa=0.0 fact= 1.0 pr¡nt*,"E= ",LOE+L7 /em," eV"," ¿¿¿=",aaa," ¡¿ç¡=",fact, $ a," loge(E)= ", log(Energy) c open (f ile='skyma p',status='new', u n it= 20) pi = 3.14159 xc=0. yc=0. zc=O. vzc=0.000001 c do 900 ii=1,10000 c print*,vxctvyc,vzct 180./Pi* c aata n (vzclsqrt(vxcx*2 +vyc**2+0.000 1 )) c visinm/s c x,y] are in 10xx6m

c ...... c starting directions isigi=0 isigii=0

c select 'thet, phi' from Galactic Centre

c This will go in 10 degree intervals of phi do 900 jjj = 1,35,1 ajll = IJJ

c This will be used to step in theta do 900 kkk = 1,8,1 akkk = kkk

c Do the stepping 100 phi= 2.*pi*ajjj/36.O 110 sthet = sin(pi/2./90. * 10.* akkk) cthet = sqrt(1.-sthet**2)

tthet= sthet/cthet thet=ata n (tthet) c if(abs(thetx1B0./pi).1t.60) goto110

vx=3x 10**B*sthet*cos(phi) vy=3x 10**8*sthetxsin(phi) vz=3x 10xxB*cthet c if(ran(iseed).gt.0.5) vz= - l.*vz avz = vz c print*,thet* 180./pi,phi* 1BO. / pi,vz

105 c c Staft at Bkpc c x=8.5* 10.**3*3.>k 10.** 10 c Staft at Okpc x=0. Y=0' z=O. xc = 0.0 yc = 0.0 zc = 0.0 c print*, x,y,z,jjj,kkk

do 500 i=1,100000,1 c Print out every 10000 steps - every 100pc c This will also be the time to recalculate the large scale field c print*,i,xc,yc,zc,aPhi*180./Pi do 500 üj=1,2 do 500 j100=1,3 do 500 j30=1,3 c print*,Breg,bx,zc,vzc do 500 j10=1,3 do 500 j3=1,3 do 500 j1=1,1000,1 c...... c Terminate the propagation? ç *** Fix the D¡sk thicknest x*x (in parsecs)

* rc= Sq rt(XC* x 2+y C'r'r 2+ zC* 2) if(rc.gt. 1000.) goto 600 c Check to see if we have reached the edge of the 'galaxy' c if(rc.ge.1500.) goto 700 if(rc. ge. 15000.) goto 600 yc=yc+0.00001

c ...... c local mag field determination

c find the polar coordinates aphi = atan(xclyc) if(yc.lt.o.) aphi = aphi+3.14159 r=sqrt(xc**2+yc* *2) c Breg is the mag of the underlying regular field c Sun is at B.5kpc compared with 10kpc for Anthony's program c aaa=0.0 c aaa set at the top of the Program c aaa=L4 c shift the sp¡ral arm with aaa (not particularly necessary) Breg=2'15 c if(r.|t.7500.) Breg=9. if(r.|t.100.) Breg=9' c if(r.gt.8000.) Breg=3.6 az=abs(z) c into pc az=az*0.00001*0.0000U3. c Breg=greg*exp(-1.*azl100000.)

c Bran is the random field component c Assume that it is intrinsically 5x the regular

c Calculate the components of the random f¡eld c the various components are calculated when the various c step counters return to one

if(j1.eq.1) goto 001 goto 150

001 a1lx=exp(rand(iseed)) a 12x=exp(rand(iseed)) da lx=a12x-a 1 1x a 1 1Y=s¡P1t. nd( iseed))

106 a12Y=s¡P1.tnd(iseed)) da 1Y=¿ 12t-u t tt a 1 1z=exp(rand(iseed)) a 122=exp(ra nd (iseed ) ) dalz=aI2z-aIIz b1x=da 1z-da ly b1Y=¿¿ 1*-Outt b1z=da 1y-da 1x if(j3.eq.1) goto 003 goto 150 003 a31x=exp(rand(iseed)) a32x=exp(rand (iseed )) da3x=a32x-a31x a3 1 Y=s¡P1.. nd(iseed)) a32Y=sYPltt nd (iseed)) da3Y=¿32U-u3tt a3 1z=exp(rand(iseed)) a32z= exp (r an d ( i seed ) ) da3z=a322-a3Lz b3x=da3z-da3y b3Y=6u3*-Ou" b3z=da3y-da3x if(j10.eq.1) goto 010 goto 150 010 a101x=exp(rand(iseed)) a 102x=exp(ra nd(iseed)) da 1Ox=a 102x-a 101x a 10 1Y=s¡P1¡and ( iseed)) a 102Y=s¡P1¡and (iseed )) da 10Y=¿192t-a101Y a 1012=exp(rand(iseed)) a 1022=exp(ra nd(iseed)) daL0z=alO2z-a 1012 b10x=da1Oz-da 10y b10y=da10x-da 102 b10z=da 10y-da 10x if(j30.eq.1) goto 030 goto 150 030 a30 lx=exp(rand(iseed)) a302x=exp(rand ( iseed)) da 30x=a302x-a30 1x a30 1Y=s¡P1¡a nd(iseed)) a302Y=s¡P1¡a nd(iseed)) da30Y= ¿362Y-a301Y a30 1z= exp(rand (iseed)) a3022=exp(rand (iseed )) da3Oz=a3O2z-a3012 b30x =da30z-da30y b30y=da30x-da30z b30z=da 30y-da30x if(j100.eq.1) goto 101 goto 150 101 a1001x=exp(rand(iseed)) a 1002x=exp(rand (iseed)) da 100x=a 1002x-a 1001x a 1001y=qvp( ra nd(iseed)) a 1002Y= g¡Plrand(iseed)) da 100Y=¿ 1902Y-a 1001Y a 100 1z=exp(ra nd (iseed)) a 10022=exp(ra nd (iseed)) da 1002=a 10022-aIOOLz b100x=da 1 002-da 100y b100y=¿¿190x-da 1002 b1002=da 100y-da 100x goto 150

150 continue c Kolmogorov spectrum has energies *x5/3 so we c weight the mag fields by length **5/6 c factors of 3 between the sizes so weights are 3x*5/6=2.5 c Small Scale

t07 bxran = b1x/38. 3 + b3x/ 1 5. 5 + b1 0x / 6.I9 +b30x/ 2.5 byran = b1y/38. 3 +b3y/ 1 5. 5+ b10y / 6.I9 +b30y / 2.5 bzra n = b 1 z/38. 3 + b3 zl L5.5 +btoz/ 6. t9 + b30z/ 2.5 c Large Scale c bxran= b100x c byran= b100y c bzran= b1002 c b is in tesla = 10**10 microgauss c bx,by,bz originally in microgauss c fact changes the relative strengths of the random and non-rand c fields c fact = 0.1 c fact is set at the top of the program c fact=1.0 is the status quo

bY= bYranxBreg*fact bx= bxran*Breg*fact bz = O. +bzranx Breg*fact c limit reg field to some distance above the plane if (a2.|t.1000.0) goto 432 c if (a2.|t.1000.0) goro 432 c This is a test to remove the reg field by= bYra nx Breg*fact bx = bxra n* Breg*fact

432 continue c print*,"Breg -", Breg,byran,bxran,bzran,fact," bx =",bx, c $ " by =",by," bz -",b2 c if (r.1t.7900) goto 486 c printx,aphi,avz,zc,vz c 486 continue c convert to Tesla bx= bx*1./100000/100000 by= byx 1./100000/100000 bz=bz* t. / L00000/ 1 00000 c...... c calculate new positions and t¡mes c Time units of 10**6 seconds (30 per (light) yr, 100/pc ) c But the distances are in units of 10**6m x=x+vx* 10x*6/10r

vx=vx+ (vyx bz-vz*by)*em'>ß 10* *6 vy=vy+ (vz* bx-vx* bz)*em* 10t< *<6 vz=vz+ (vx*by-vyxbx) *emx 1 0* *6 c ...... c Cont¡nues at the speed of light * * * v = sq rt(vx* 2 *vY* 2*u.* r, y¡= (y¡*3* 10**8)/v vy=(vy*3x 10**8)/v y¿= (yzxfx 10**8)/v c Speeds in terms of c vxc=vx/10*x8l3. vyc=vyl10**B/3. vzc=vz/LO**8/3. c distances in pc xc=x/3./10**8/ IO**2 yc=y/3./IO**8/IO**2 7ç=7/J ./ LO**8/ t0x*2

108 500 continue c 700 if(rc.lt.8499.) goto 900 700 cont¡nue c The vector to the point is x,y,z, ie the radial c vector. The direction of travel is vxc,vyc,vzc c so the scalar produce is r.vc cos(theta) where c theta is the angle between r and vc' c prlnt *, i,phi,cthet,jjj,kkk goto 900 c scalp = xc*vxc+Yc*vYc+zcxvzc c scalp=5calp/sqrt(xc**2+yc**2+zc**2) c scalp=scalp/sqrt(vxc**2+vyc**2+vzc**2) c scalp=180.*acos(scalp)/pi c angplan ids the angle between the galactic plane and the c velocity vector * * + y c'É * 2 + z(F a n g p l a n = xc* vxc+ yc* vyc/sq rt( xc* 2 2) * {< ang pla n = a ngpla n/sq¡t(vxcx 2+vyct( 2+ vzc**2) angplan= 180.*acos(angPlan)/Pi c print*,ii,zc, c print*,ii ,zc,scalP, * c a 180./pi*ata n(vzclsq rt(vxc**2+vycx 2+0.000 1 )) c write(20,*)ii,zc,scalP, * x c a 1 80./pi*atan (vzclsqft(vxc* 2+vyc* 2+0.000 1 ))

600 continue c 600 is for leaving the plane by 5o0pc c print*,i,xctyctzc isigi =isigi +1 isigii =isi9i¡ +i 900 continue c print*,i,xc,yc,zc,thetx 180./pi, ph i* 180./pi print*, t.OE+ L7 / em,"eV ","isigi=",isigi," ii =",isigii print*, "Breg = ", Breg print*, "bxran=", bxran," byran=", byran," bzran-", bzran 990 continue end

109